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Research ArticleStock Selection into Portfolio by Fuzzy Quantitative Analysisand Fuzzy Multicriteria Decision Making
Satit Yodmun and Wichai Witayakiattilerd
Department of Mathematics Faculty of Science King Mongkutrsquos Institute of Technology Ladkrabang Bangkok 10520 Thailand
Correspondence should be addressed to Wichai Witayakiattilerd wichaiwikmitlacth
Received 28 November 2015 Revised 26 February 2016 Accepted 7 April 2016
Academic Editor Igor L Averbakh
Copyright copy 2016 S Yodmun and W Witayakiattilerd This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
This paper presents a stock selection approach assisted by fuzzy procedures In this approach stocks are classified into groupsaccording to business types Within each group the stocks are screened and then ranked according to their investment weightobtained from fuzzy quantitative analysis Groups were also ranked according to their group weight obtained from fuzzy analytichierarchy process (FAHP) and technique for order preference by similarity to ideal solution method (TOPSIS) The overall weightfor each stock was then derived from both of these weights and used for selecting a stock into the portfolio As a demonstrationour analysis procedures were applied to a test set of data
1 Introduction
Presently investors are more interested in investing in stocksand bonds than keeping their money in the bank because ityields a higher returnHowever this higher return also comeswith higher risk investors may lose some of their investmentget a lower-than-expected return or get a lower return thanthat from another type of investmentTherefore they have toanalyze a stock carefully before investing in it
In addition to several established approaches to stockanalysismdashsuch as quantitative fundamental analysis tech-nical analysis and stochastic analysismdashnew analytical toolshave been developed and widely used including ones that arebased on Brownian movement fuzzy logic and the analytichierarchy process
The analytic hierarchy process (AHP) is a multicriteriadecision-making approach and is a structured techniquefor organizing and analyzing complex decisions based onmathematics and psychology It was developed by Saaty inthe 1970s to help one make decision when one is facedwith the mixture of qualitative quantitative and sometimesconflicting factors that are taken into consideration AHP hasbeen very effective in making complicated often irreversible
decisions It has been extensively studied and refined sincethen (eg [1ndash11] and references therein)
Fuzzy sets and fuzzy logic especially are of wide interesttoday They are effective tools for modeling in the absenceof complete and precise information complex businessfinance and management systemsThe subjective judgementof experts who have used fuzzy logic techniques producesbetter results than the objective manipulation of inexactdata The concept of a fuzzy set is a reflection of realityreflection which serves as a point of departure for thedevelopment of theories which have the capability to modelthe pervasive imprecision and uncertainty of the real worldAs applied to stock analysis (eg [12ndash15] and referencestherein) fuzzy logic uses integrated experiential knowledgeof human experts to make better quantitative estimates notpossible with classical logic based on robust mathematicalprinciples
By reason of vagueness of boundaries of stock datain future and the attendant imprecision uncertainty andpreference of decision makers therefore fuzzy logic andAHP seem suitable for this problem This paper proposes anapproach to stock analysis based on calculated weights fromfuzzy quantitative analysis and fuzzy multicriteria decision
Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2016 Article ID 9530425 14 pageshttpdxdoiorg10115520169530425
2 Advances in Operations Research
making The idea of using fuzzy quantitative analysis andfuzzymulticriteria decisionmaking to imply final investmentweights for the stock selection into portfolio is different fromthe previous works The practicality of the approach wasdemonstrated by an application to a test set of data
2 Preliminaries
21 Fuzzy Logic Application and Definitions Fuzzy logicwas introduced by Zadeh [16] and has been widely appliedto problems in various fields of study Many researchersused fuzzy logic in stock market analysis (eg [12ndash15]) anddecision making (eg [1ndash4 6 7 9ndash15 17]) In this study weuse fuzzy logic in both stock market analysis and decisionmaking
In this subsection definitions of the fuzzy logic terms andconcepts used in this study are described below
Definition 1 Given a crisp set 119860 of a universe U a fuzzy set on 119860 is defined as
= (119909 119906 (119909)) | 119909 isin 119860 where 119906 (119909) isin [0 1] (1)
and 119906 is a membership function
Definition 2 Given a fuzzy set an 120572-cut set denoted by[]120572 for all 120572 isin [0 1] is defined as
[]120572
=
119909 isin 119860 | 119906 (119909) ge 120572 0 lt 120572 le 1
119909 isin 119860 | 119906 (119909) gt 0 120572 = 0
(2)
Definition 3 Let be a fuzzy set under the membership 119906 Rrarr [0 1] and is a fuzzy number if it satisfies the followingconditions
(1) is a normal fuzzy set that is exist119909 isin R 119906(119909) = 1
(2) is a convex fuzzy set that isforall120582 isin [0 1]forall1199091 1199092isin R
119906(1205821199091+ (1 minus 120582)119909
2) ge min119906(119909
1) 119906(1199092)
(3) For every 120572 isin [0 1] []120572 = [119886 119887] for some closedinterval [119886 119887]
Given an RF fuzzy number space condition (3) ofDefinition 3 ensures that every isin RF can be represented bya closed interval []120572 = [119906(120572) 119906(120572)] where 119906 119906 [0 1] rarr R
are functions that satisfy the following conditions
(1) 119906 is a bounded left continuous and nondecreasingfunction on [0 1]
(2) 119906 is a bounded right continuous and no-increasingfunction on [0 1]
(3) 119906(120572) le 119906(120572) for all 120572 isin [0 1]
Definition 4 = [119906(120572) 119906(120572)] is a positive fuzzy number thatcan be represented by the expression gt 0 if 119906(0) gt 0
Definition 5 Given 119886119871 le 1198861198721 le 1198861198722 le 119886119880 a trapezoidal fuzzynumber is a fuzzy number whose membership function119911(119909) is defined by
119911 (119909) =
119909 minus 119886119871
1198861198721 minus 119886119871 119886119871
le 119909 le 1198861198721
1 1198861198721 le 119909 le 119886
1198722
119909 minus 119886119880
1198861198722 minus 119886119880 1198861198722 le 119909 le 119886
119880
0 otherwise
(3)
and represented by the expression = ⟨119886119871 1198861198721 1198861198722 119886119880⟩
Definition 6 A trapezoidal fuzzy number =
⟨119886119871
119886119872
119886119872
119886119880
⟩ is called a triangular fuzzy numberand expressed as = ⟨119886119871 119886119872 119886119880⟩
Note For any real number 119886 119886 = ⟨119886 119886 119886⟩ = ⟨119886 119886 119886 119886⟩
Definition 7 Given any two positive fuzzy numbers =⟨119886119871
1198861198721 1198861198722 119886119880
⟩ and = ⟨119887119871
1198871198721 1198871198722 119887119880
⟩ and a realpositive number 119901 isin R+ operations oplus ⊖ otimes and ⊘ between and 119887 and an operation ⊙ between and 119901 are defined asfollows
oplus = ⟨119886119871
+ 119887119871
1198861198721 + 1198871198721 1198861198722 + 1198871198722 119886119880
+ 119887119880
⟩
⊖ = ⟨119886119871
minus 119887119880
1198861198721 minus 1198871198722 1198861198722 minus 1198871198721 119886119880
minus 119887119871
⟩
otimes = ⟨119886119871
119887119871
11988611987211198871198721 11988611987221198871198722 119886119880
119887119880
⟩
119901 ⊙ = ⟨119901119886119871
1199011198861198721 1199011198861198722 119901119886119880
⟩
⊘ = ⟨119886119871
1198871198801198861198721
11988711987221198861198722
1198871198721119886119880
119887119871⟩
(4)
Definition 8 Given two trapezoidal fuzzy numbers =
⟨119886119871
1198861198721 1198861198722 119886119880
⟩ and = ⟨119887119871 1198871198721 1198871198722 119887119880⟩ the distancebetween and represented by the symbol 119889( ) is definedas119889 ( )
= radic1
4[(119886119871 minus 119887119871)
2
+ (1198861198721 minus 1198871198721)2
+ (1198861198722 minus 1198871198722)2
+ (119886119880 minus 119887119880)2
]
(5)
For convenience 119868119899= 1 2 119899 is defined for further
use in this paper
Definition 9 = (119894119895)119898times119899
is a fuzzy matrix if 119894119895are fuzzy
numbers for all 119894 isin 119868119898and 119895 isin 119868
119899
Definition 10 = (119894119895)119899times1
is a fuzzy vector when all 119894=
⟨119898119871
119894 1198981198721
119894 1198981198722
119894 119898119880
119894⟩ 119894 isin 119868
119899 are trapezoidal fuzzy numbers
The aggregation of represented by agg is defined as
agg
= ⟨
119899
min119894=1
119898119871
119894 1
119899
119899
sum
119894=1
1198981198721
1198941
119899
119899
sum
119894=1
1198981198722
119894119899max119894=1
119898119880
119894⟩
(6)
Advances in Operations Research 3
22 Consistency Fuzzy Matrix In this subsection we intro-duce the definition of consistency fuzzy matrix and consis-tency index which was developed by Ramik [3 4]
Definition 11 Let 119860 = (119886119894119895)119899times119899
be an 119899 times 119899matrix where 119886119894119895gt
0 for all 119894 119895 isin 119868119899and 119860 is a reciprocal matrix if 119886
119895119894= 1119886119894119895for
all 119894 119895 isin 119868119899
Definition 12 Let119860 = (119886119894119895)119899times119899
be an 119899times119899matrix where 119886119894119895gt 0
for all 119894 119895 isin 119868119899and 119860 is a consistency matrix if there exist
weight vectors 119908 = (119908119894)119899times1
119908119894gt 0 for all 119894 isin 119868
119899 where
sum119899
119894=1119908119894= 1 and 119886
119894119895= 119908119894119908119895for all 119894 119895 isin 119868
119899
Definition 13 Let = (119894119895)119899times119899
be an 119899times119899 fuzzy matrix where119894119895gt 0 are fuzzy numbers for all 119894 119895 isin 119868
119899and is a reciprocal
fuzzy matrix if 119895119894= 1 ⊘
119894119895for all 119894 119895 isin 119868
119899
In particular if every member of = (119894119895)119899times119899
is atriangular fuzzy number
119894119895= ⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ is a reciprocal
fuzzy matrix if 119895119894= ⟨1119886
119880
119894119895 1119886119872
119894119895 1119886119871
119894119895⟩ for all 119894 119895 isin 119868
119899
Definition 14 Let = (119894119895)119899times119899
be an 119899times119899 fuzzymatrix where119894119895= [119886119894119895(120572) 119886119894119895(120572)] gt 0 for all 119894 119895 isin 119868
119899and is a consistency
fuzzy matrix if there exist 119886120572119894119895isin [119886119894119895(120572) 119886119894119895(120572)] for all 119894 119895 isin 119868
119899
and some 120572 isin [0 1] with which 119860 = (119886120572119894119895)119899times119899
is a consistencymatrix that is there exist119908120572 = (119908120572
119894)119899times1
119908120572119894gt 0 for all 119894 isin 119868
119899
where sum119899119894=1119908120572
119894= 1 and 119886120572
119894119895= 119908120572
119894119908120572
119895for all 119894 119895 isin 119868
119899
According to Definition 14 since 119908120572119894gt 0 for all 119894 isin
119868119899 there exist fuzzy vectors = (
119894)119899times1
where 119908120572119894isin
[119908119894(120572) 119908
119894(120572)] gt 0 for all 119894 isin 119868
119899 These vectors are called fuzzy
weight vectorsIt is clear that if is a fuzzy consistency matrix then it is a
fuzzy reciprocal fuzzymatrix and is not a fuzzy consistencymatrix if it is not a fuzzy reciprocal fuzzy matrix Becauseof these reasons construction of a fuzzy consistency matrixusually starts by first constructing a reciprocal fuzzy matrix Ramik and Korviny [4] proposed a method for calculatingfuzzy weight vector = (
119894)119899times1
for a fuzzy reciprocal matrix = (
119894119895)119899times119899
where 119894119895=⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ for all 119894 119895 isin 119868
119899by
using the method of geometric mean 119896= ⟨119908
119871
119896 119908119872
119896 119908119880
119896⟩
are defined for all 119896 isin 119868119899 where
119908119871
119896= 119862119871sdot
(prod119899
119895=1119886119871
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
119908119872
119896=
(prod119899
119895=1119886119872
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
119908119880
119896= 119862119880sdot
(prod119899
119895=1119886119880
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
(7)
119862119871= min119894isin119868119899
(prod119899
119895=1119886119872
119894119895)1119899
(prod119899
119895=1119886119871
119894119895)1119899
119862119880= max119894isin119868119899
(prod119899
119895=1119886119872
119894119895)1119899
(prod119899
119895=1119886119880
119894119895)1119899
(8)
In addition Ramik and Korviny [4] defined a consistencyindex for measuring the nearness of a fuzzy reciprocal matrixto the corresponding fuzzy consistency matrix as follows
Definition 15 Let = (119894119895)119899times119899
be a fuzzy reciprocal matrixof which
119894119895= ⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ are triangular fuzzy numbers
evaluated from a scale 119878 = [1120590 120590] for some real number120590 gt 1 the consistency index of represented by the symbol119868120590
119899() is defined as
119868120590
119899() = 119862
120590
119899sdotmax119894119895
max1003816100381610038161003816100381610038161003816100381610038161003816
119908119871
119894
119908119880
119895
minus 119886119871
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816
119908119872
119894
119908119872
119895
minus 119886119872
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816
119908119880
119894
119908119871
119895
minus 119886119880
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
(9)
where = (119894)119899times1
are fuzzy weight vectors and 119894=
⟨119908119871
119894 119908119872
119894 119908119880
119894⟩ for all 119894 isin 119868
119899as expressed in (7) and
119862120590
119899=
1
max 120590 minus 120590(2minus2119899)119899 1205902 ((2119899)2(119899minus2) minus (2119899)(119899minus2)2) 120590 lt (
119899
2)
119899(119899minus2)
1
max 120590 minus 120590(2minus2119899)119899 120590(2minus2119899)119899 minus 120590 120590 ge (
119899
2)
119899(119899minus2)
(10)
If the consistency index 119868120590119899() = 0 the fuzzy reciprocal
fuzzy matrix is absolutely consistent The closer the valueof 119868120590119899() to 0 is the more consistent the matrix is Generally
an acceptable value is 119868120590119899() lt 01 or 10
Theorem 16 (see [4]) If is an 119899 times 119899 fuzzy reciprocal matrixwith triangular fuzzy elements evaluatedwith the scale [1120590 120590]for some 120590 gt 1 then 0 le 119868120590
119899() le 1
23 Financial Ratios A sustainable investment and missionrequires effective planning and financial management
The quantitative stock analysis is a useful tool that willimprove investmentrsquos understanding of financial results andtrends over time and provide key indicators of organizationalperformance Investormay use the quantitative stock analysisto pinpoint strengths and weaknesses of each company thatimpact to its stock
4 Advances in Operations Research
The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875
119899Ratio which are defined as
follows
Definition 17 Let 1198991 1198992 and 119899
3be the number of common
stock preferred stock and treasury stock respectively 119875119905
current price per share and 119864119903119903th-quarter net profit price to
earnings ratio or 119875119864 is defined as
119875
119864=119875119905(1198991+ 1198992minus 1198993)
119864119903
(11)
119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for
Definition 18 Let 119899 be the number of be the number ofregistered share 119860
119905and 119877
119905the asset and liability of the
company respectively and 119875119905current price per share price to
book value ratio or 119875119861119881 is defined as
119875
BV=119875119905
119861119905
(12)
where 119861119905= (119860119905minus 119877119905)119899
119875BV denotes how many times the current stock price iscompared to its account value
Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th
year-end dividend per share 119896 isin 119868119899 and 119875
0the 119899th-quarter
historical price the current target price 119875119899is defined as
119875119899= 1198750(1 + 119903)
119899
minus
119899
sum
119896=1
119863119896(1 + 119903)
119899minus119896
(13)
Definition 20 Let 119875119899be the current target price and 119875 the
current stock price 119875119875119899is called price per target price ratio
represented by the symbol 119875119875119899
119875119875119899denotes how many times the current stock price is
compared to the current target price
3 Stock Selection Procedure
This section presents the proposed stock selection procedurewhich is done in the following 3 main steps
Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock
Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group
Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market
31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for
each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions
(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group
(2) More recent data reflect current trend better thanearlier ones
(3) Fuzzy rules are flexible and depend on expert infor-mation
The specific steps of the fuzzy analysis are as follows
Step 11 This step involves screening in only 119898 individualstocks (119878
1 1198782 119878
119898) in the same industrial group of which
sufficient financial data are provided for calculating 119875119864119875BV and 119875119875
119899of 119899 earlier years up to the present
Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894
119896)
and (119875119875119899)(119878119894
119896) for all 119894 isin 119868
119899and 119896 isin 119868
119898 where 119878119894
119896denotes the
119896th stock in the 119894th year
Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878
119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896)
119896 isin 119868119898 from the following equations
(119864
119875)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119864
119875) (119878119894
119896)
(119875
BV)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
BV) (119878119894
119896)
(119875
119875119899
)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
119875119899
) (119878119894
119896)
where 119908119894=
2119894
119899 (119899 + 1) 119894 isin 119868
119898
(14)
Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875
119899and a fuzzy set119882 of the investment weights from
(119864119875)119908
(119878119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896) 119896 isin 119868
119898
Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows
Advances in Operations Research 5
Rule-1 if 1199091is 11and 119909
2is 12and 119909
3is 13then 119910 is
1
Rule-2 if 1199091is 21and 119909
2is 22and 119909
3is 23then 119910 is
2
Rule-119902 if 1199091is 1199021and 119909
2is 1199022and 119909
3is 1199023then 119910 is
119902
1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875
119899
and 1198821 respectively and
1198961 1198962 and
1198963 119896 isin 119868
119902 are
linguistic terms of 119864119875 119875BV 119875119875119899 and 119882
1 respectively
that is 119864119875 = 11 21
1199021 119875BV =
12 22
1199022
119875119875119899= 13 23
1199023 and119882 =
1 2
119902
Step 16This step involves importing 119864119875 119875BV and 119875119875119899of
the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861
Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908
119911= 119906B(119911) int
119861
119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int
119861
119911119908119911119889119911 =
int119861
119911119906B(119911) 119889119911 int119861
119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group
32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)
AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω
119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus
1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω
119899is replaced by
a set of fuzzy preference values Ω120575119899= 1
120575 1(119899 minus 1)
13120575 12120575 1 2120575 3120575 (119899 minus 1)
120575 120575 where
120575= ⟨119896 minus 120575 119896 119896 +
120575⟩ and 1120575= 1⊘
120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868
119899
and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]
and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers
Applying this technique to 1198991decision makers 119899
2deci-
sion criteria and 1198993industrial groups as alternate choices the
analysis steps are as follows
Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899
1 is compared to another
decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as
120593 (119894 119895) =
119894119895 exist119894119895isin Ω119899 119895 gt 119894
1 119895 = 119894
1 ⊘ 120593 (119895 119894) 119895 lt 119894
(15)
The decision makerrsquos preference matrix = (119894119895)1198991times1198991
is areciprocal matrix where
119894119895=
120593 (119894 119895) 119894 lt 119895
1 119894 = 119895
1 ⊘ 120593 (119895 119894) 119894 gt 119895
(16)
Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =
(119894119895)1198991times1198991
) 119889119896= ⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ is a fuzzyweight vector for
all 119896 isin 1198681198991
where
119908119871
119889119896= 119862119871sdot
(prod1198991
119895=1119886119871
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119872
119889119896=
(prod1198991
119895=1119886119872
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119880
119889119896= 119862119880sdot
(prod1198991
119895=1119886119880
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
(17)
with
119862119871= min119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119871
119894119895)11198991
119862119880= max119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119880
119894119895)11198991
(18)
If its consistency index 1198681205901198991
() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21
Step 23 This step involves decision makers 1198891 1198892
1198891198991
constructing decision criteria 1198881 1198882 119888
1198992
for evaluatingindustrial groups 119866
1 1198662 119866
1198993
where 119888119894 119894 = 1 119899
2 is
constructed from investment weight of 1198993individual groups
6 Advances in Operations Research
given by decision makers in the term of linguistic terms (seeTable 1)
Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887
119895119894119896= ⟨119887119871
119895119894119896 1198871198721
119895119894119896 1198871198722
119895119894119896 119887119880
119895119894119896⟩ 119895 isin 119868
1198993
119894 isin 1198681198992
and 119896 isin 1198681198991
which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888
1 1198882 119888
1198992
shown in(19)
Decision Criteria for Evaluating Industrial Groups 1198661
1198662 119866
1198993
Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111
211
119899311
112
212
119887119899312
119887111198991
119887211198991
119887119899311198991
1198882
1198661
1198662
1198661198993
121
221
119899321
122
222
119887119899322
119887121198991
119887221198991
119887119899321198991
1198881198992
1198661
1198662
1198661198993
111989921
211989921
119899311989921
119887111989922
119887211989922
119887119899311989922
119887111989921198991
119887211989921198991
119899311989921198991
= (19)
Step 24 This step involves decision makers 1198891 1198892 119889
1198991
evaluating decision criteria 1198881 1198882 119888
1198992
constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (
119894119895)1198992times1198991
for evaluation is then obtainedwhere
119894119895isin VL LMLMMHHVH for all 119894 isin 119868
1198992
and119895 isin 1198681198991
as shown in (20)
Evaluation of Decision Criteria 1198881 1198882 119888
1198992
Consider
11988911198892sdot sdot sdot 119889
1198991
11988811112sdot sdot sdot 11198991
11988822122sdot sdot sdot 21198991
=
1198881198992
1198992111989922sdot sdot sdot 11989921198991
(20)
Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector
119889= (
119889119896)119899times1
where 119889119896= ⟨119908
119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ =
⟨119908119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from Step 22 Equation (21)
shows these multiplication results
Decision Criteria Based on Weights of Decision MakersConsider
1198891
1198892
sdot sdot sdot 1198891198991
119888111otimes 1198891
12otimes 1198892sdot sdot sdot 11198991
otimes 1198891198991
119888221otimes 1198891
22otimes 1198892sdot sdot sdot 21198991
otimes 1198891198991
= 119908
1198881198992
11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991
otimes 1198891198991
(21)
Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (
119889119896)119899times1
where 119889119896=
⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ = ⟨119908
119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from
Step 22 The multiplication results are in (22)
Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111otimes 1198891
211otimes 1198891
119899311otimes 1198891
112otimes 1198892
212otimes 1198892
119887119899312otimes 1198892
111198991
otimes 1198891198991
211198991
otimes 1198891198991
119887119899311198991
otimes 1198891198991
1198882
1198661
1198662
1198661198993
121otimes 1198891
221otimes 1198891
119899321otimes 1198891
122otimes 1198892
222otimes 1198892
119887119899322otimes 1198892
121198991
otimes 1198891198991
221198991
otimes 1198891198991
119887119899321198991
otimes 1198891198991
1198881198992
1198661
1198662
1198661198993
111989921otimes 1198891
211989921otimes 1198891
119899311989921otimes 1198891
119887111989922otimes 1198892
119887211989922otimes 1198892
119887119899311989922otimes 1198892
119887111989921198991
otimes 1198891198991
119887211989921198991
otimes 1198891198991
119887119899311989921198991
otimes 1198891198991
= 119908
(22)
Advances in Operations Research 7
Table 1
Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩
Low (L) ⟨01 02 03⟩
Medium low (ML) ⟨02 03 04⟩
Medium (M) ⟨03 04 06 07⟩
Medium high (MH) ⟨06 07 08⟩
High (H) ⟨07 08 09⟩
Very high (VH) ⟨08 09 1 1⟩
Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows
119888119894= ⟨119908119871
119888119894 1199081198721
119888119894 1199081198722
119888119894 119908119880
119888119894⟩ (23)
where 119908119871119888119894= min1198991
119896=1119888119871
119908119894119896 1199081198721119888119894= (1119899
1) sum1198991
119896=11198881198721
119908119894119896 1199081198722119888119894=
(11198991) sum1198991
119896=11198881198722
119908119894119896 119908119880119888119894= max1198991
119896=1119888119880
119908119894119896 for all 119894 isin 119868
1198992
119908= (119908119895119896)1198992times1198991
and 1198991is the number of decision makers
Equation (24) shows these aggregation results
Weights of Decision Criteria 1198881 1198882 119888
1198992
Consider
1198881
1198882sdot sdot sdot 119888
1198992
119882211988811198882sdot sdot sdot
1198881198992
(24)
Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations
119895119894= ⟨119909119871
119895119894 1199091198721
119895119894 1199091198722
119895119894 119909119880
119895119894⟩ (25)
where 119909119871119895119894= min1198991
119896=1119887119871
119908119895119894119896 1199091198721119895119894= (1119899
1) sum1198991
119896=11198871198721
119908119895119894119896 1199091198722119895119894=
(11198991) sum1198991
119896=11198871198722
119908119895119894119896 119909119880119895119894= max1198992
119896=1119887119880
119908119895119894119896 for all 119895 isin 119899
3 119894 isin 119899
2
119908= (119908119895119894119896)11989931198992times1198991
and 1198991is the number of decision makers
These results are shown in (26)
Evaluation Matrix of Industrial Groups 1198661 1198662 119866
1198993
Con-sider
1198881
1198882sdot sdot sdot 119888
1198992
11986611112sdot sdot sdot
11198992
11986622122sdot sdot sdot
21198992
=
1198661198993
1198993111989932sdot sdot sdot 11989931198992
(26)
Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows
= (119895119894)1198993times1198992
119895119894= ⟨
119909119871
119895119894
119909lowast
119894
1199091198721
119895119894
119909lowast
119894
1199091198722
119895119894
119909lowast
119894
119909119880
119895119894
119909lowast
119894
⟩ where 119909lowasti =1198993max119895
119909119880
119895119894
(27)
Then multiplying the normalized matrix with the decisionweights from Step 26 = (V
119895119894)1198993times1198992
where V119895119894=
⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894when 119895 isin 119868
1198993
119894 isin 1198681198992
Industrial Groupsrsquo Evaluation Matrix Consider
11988811198882sdot sdot sdot 119888
1198992
1198661
V11
V12sdot sdot sdot V11198992
1198662
V21
V22sdot sdot sdot V21198992
=
1198661198993
V11989931V11989932sdot sdot sdot V11989931198992
(28)
Step 28 This step involves defining positive ideal solution(119866lowast
) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast
1198992
) and119866minus = (Vminus1 Vminus2 Vminus
1198992
) respectively whereVlowast119894= max1198993
119895V119880119895119894 and Vminus
119894= min1198993
119895V119871119895119894 119895 isin 119868
1198993
119894 isin 1198681198992
= (V
119895119894)1198993times1198992
Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following
119889lowast
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowast
119894) 119895 isin 119868
1198993
119889minus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vminus
119894) 119895 isin 119868
1198993
(29)
where 119889V(V119895119894 Vlowastminus
119894) are calculated in the same way as fuzzy
numbers are calculated according to Definition 8 (depictedin (30))
Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider
1198881
1198882
sdot sdot sdot 1198881198992
119889lowastminus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowastminus
119895)
1198661119889V (V11 V
lowastminus
1) 119889V (V12 V
lowastminus
2) sdot sdot sdot 119889V (V1119899
2
Vlowastminus1198992
) 119889lowastminus
1
1198662119889V (V21 V
lowastminus
1) 119889V (V22 V
lowastminus
2) sdot sdot sdot 119889V (V2119899
2
Vlowastminus21198992
) 119889lowastminus
2
1198661198993
119889V (V11989931 Vlowastminus1) 119889V (V119899
32 Vlowastminus2) sdot sdot sdot 119889V (V119899
31198992
Vlowastminus1198992
) 119889lowastminus
1198993
(30)
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
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MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Advances in Operations Research
making The idea of using fuzzy quantitative analysis andfuzzymulticriteria decisionmaking to imply final investmentweights for the stock selection into portfolio is different fromthe previous works The practicality of the approach wasdemonstrated by an application to a test set of data
2 Preliminaries
21 Fuzzy Logic Application and Definitions Fuzzy logicwas introduced by Zadeh [16] and has been widely appliedto problems in various fields of study Many researchersused fuzzy logic in stock market analysis (eg [12ndash15]) anddecision making (eg [1ndash4 6 7 9ndash15 17]) In this study weuse fuzzy logic in both stock market analysis and decisionmaking
In this subsection definitions of the fuzzy logic terms andconcepts used in this study are described below
Definition 1 Given a crisp set 119860 of a universe U a fuzzy set on 119860 is defined as
= (119909 119906 (119909)) | 119909 isin 119860 where 119906 (119909) isin [0 1] (1)
and 119906 is a membership function
Definition 2 Given a fuzzy set an 120572-cut set denoted by[]120572 for all 120572 isin [0 1] is defined as
[]120572
=
119909 isin 119860 | 119906 (119909) ge 120572 0 lt 120572 le 1
119909 isin 119860 | 119906 (119909) gt 0 120572 = 0
(2)
Definition 3 Let be a fuzzy set under the membership 119906 Rrarr [0 1] and is a fuzzy number if it satisfies the followingconditions
(1) is a normal fuzzy set that is exist119909 isin R 119906(119909) = 1
(2) is a convex fuzzy set that isforall120582 isin [0 1]forall1199091 1199092isin R
119906(1205821199091+ (1 minus 120582)119909
2) ge min119906(119909
1) 119906(1199092)
(3) For every 120572 isin [0 1] []120572 = [119886 119887] for some closedinterval [119886 119887]
Given an RF fuzzy number space condition (3) ofDefinition 3 ensures that every isin RF can be represented bya closed interval []120572 = [119906(120572) 119906(120572)] where 119906 119906 [0 1] rarr R
are functions that satisfy the following conditions
(1) 119906 is a bounded left continuous and nondecreasingfunction on [0 1]
(2) 119906 is a bounded right continuous and no-increasingfunction on [0 1]
(3) 119906(120572) le 119906(120572) for all 120572 isin [0 1]
Definition 4 = [119906(120572) 119906(120572)] is a positive fuzzy number thatcan be represented by the expression gt 0 if 119906(0) gt 0
Definition 5 Given 119886119871 le 1198861198721 le 1198861198722 le 119886119880 a trapezoidal fuzzynumber is a fuzzy number whose membership function119911(119909) is defined by
119911 (119909) =
119909 minus 119886119871
1198861198721 minus 119886119871 119886119871
le 119909 le 1198861198721
1 1198861198721 le 119909 le 119886
1198722
119909 minus 119886119880
1198861198722 minus 119886119880 1198861198722 le 119909 le 119886
119880
0 otherwise
(3)
and represented by the expression = ⟨119886119871 1198861198721 1198861198722 119886119880⟩
Definition 6 A trapezoidal fuzzy number =
⟨119886119871
119886119872
119886119872
119886119880
⟩ is called a triangular fuzzy numberand expressed as = ⟨119886119871 119886119872 119886119880⟩
Note For any real number 119886 119886 = ⟨119886 119886 119886⟩ = ⟨119886 119886 119886 119886⟩
Definition 7 Given any two positive fuzzy numbers =⟨119886119871
1198861198721 1198861198722 119886119880
⟩ and = ⟨119887119871
1198871198721 1198871198722 119887119880
⟩ and a realpositive number 119901 isin R+ operations oplus ⊖ otimes and ⊘ between and 119887 and an operation ⊙ between and 119901 are defined asfollows
oplus = ⟨119886119871
+ 119887119871
1198861198721 + 1198871198721 1198861198722 + 1198871198722 119886119880
+ 119887119880
⟩
⊖ = ⟨119886119871
minus 119887119880
1198861198721 minus 1198871198722 1198861198722 minus 1198871198721 119886119880
minus 119887119871
⟩
otimes = ⟨119886119871
119887119871
11988611987211198871198721 11988611987221198871198722 119886119880
119887119880
⟩
119901 ⊙ = ⟨119901119886119871
1199011198861198721 1199011198861198722 119901119886119880
⟩
⊘ = ⟨119886119871
1198871198801198861198721
11988711987221198861198722
1198871198721119886119880
119887119871⟩
(4)
Definition 8 Given two trapezoidal fuzzy numbers =
⟨119886119871
1198861198721 1198861198722 119886119880
⟩ and = ⟨119887119871 1198871198721 1198871198722 119887119880⟩ the distancebetween and represented by the symbol 119889( ) is definedas119889 ( )
= radic1
4[(119886119871 minus 119887119871)
2
+ (1198861198721 minus 1198871198721)2
+ (1198861198722 minus 1198871198722)2
+ (119886119880 minus 119887119880)2
]
(5)
For convenience 119868119899= 1 2 119899 is defined for further
use in this paper
Definition 9 = (119894119895)119898times119899
is a fuzzy matrix if 119894119895are fuzzy
numbers for all 119894 isin 119868119898and 119895 isin 119868
119899
Definition 10 = (119894119895)119899times1
is a fuzzy vector when all 119894=
⟨119898119871
119894 1198981198721
119894 1198981198722
119894 119898119880
119894⟩ 119894 isin 119868
119899 are trapezoidal fuzzy numbers
The aggregation of represented by agg is defined as
agg
= ⟨
119899
min119894=1
119898119871
119894 1
119899
119899
sum
119894=1
1198981198721
1198941
119899
119899
sum
119894=1
1198981198722
119894119899max119894=1
119898119880
119894⟩
(6)
Advances in Operations Research 3
22 Consistency Fuzzy Matrix In this subsection we intro-duce the definition of consistency fuzzy matrix and consis-tency index which was developed by Ramik [3 4]
Definition 11 Let 119860 = (119886119894119895)119899times119899
be an 119899 times 119899matrix where 119886119894119895gt
0 for all 119894 119895 isin 119868119899and 119860 is a reciprocal matrix if 119886
119895119894= 1119886119894119895for
all 119894 119895 isin 119868119899
Definition 12 Let119860 = (119886119894119895)119899times119899
be an 119899times119899matrix where 119886119894119895gt 0
for all 119894 119895 isin 119868119899and 119860 is a consistency matrix if there exist
weight vectors 119908 = (119908119894)119899times1
119908119894gt 0 for all 119894 isin 119868
119899 where
sum119899
119894=1119908119894= 1 and 119886
119894119895= 119908119894119908119895for all 119894 119895 isin 119868
119899
Definition 13 Let = (119894119895)119899times119899
be an 119899times119899 fuzzy matrix where119894119895gt 0 are fuzzy numbers for all 119894 119895 isin 119868
119899and is a reciprocal
fuzzy matrix if 119895119894= 1 ⊘
119894119895for all 119894 119895 isin 119868
119899
In particular if every member of = (119894119895)119899times119899
is atriangular fuzzy number
119894119895= ⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ is a reciprocal
fuzzy matrix if 119895119894= ⟨1119886
119880
119894119895 1119886119872
119894119895 1119886119871
119894119895⟩ for all 119894 119895 isin 119868
119899
Definition 14 Let = (119894119895)119899times119899
be an 119899times119899 fuzzymatrix where119894119895= [119886119894119895(120572) 119886119894119895(120572)] gt 0 for all 119894 119895 isin 119868
119899and is a consistency
fuzzy matrix if there exist 119886120572119894119895isin [119886119894119895(120572) 119886119894119895(120572)] for all 119894 119895 isin 119868
119899
and some 120572 isin [0 1] with which 119860 = (119886120572119894119895)119899times119899
is a consistencymatrix that is there exist119908120572 = (119908120572
119894)119899times1
119908120572119894gt 0 for all 119894 isin 119868
119899
where sum119899119894=1119908120572
119894= 1 and 119886120572
119894119895= 119908120572
119894119908120572
119895for all 119894 119895 isin 119868
119899
According to Definition 14 since 119908120572119894gt 0 for all 119894 isin
119868119899 there exist fuzzy vectors = (
119894)119899times1
where 119908120572119894isin
[119908119894(120572) 119908
119894(120572)] gt 0 for all 119894 isin 119868
119899 These vectors are called fuzzy
weight vectorsIt is clear that if is a fuzzy consistency matrix then it is a
fuzzy reciprocal fuzzymatrix and is not a fuzzy consistencymatrix if it is not a fuzzy reciprocal fuzzy matrix Becauseof these reasons construction of a fuzzy consistency matrixusually starts by first constructing a reciprocal fuzzy matrix Ramik and Korviny [4] proposed a method for calculatingfuzzy weight vector = (
119894)119899times1
for a fuzzy reciprocal matrix = (
119894119895)119899times119899
where 119894119895=⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ for all 119894 119895 isin 119868
119899by
using the method of geometric mean 119896= ⟨119908
119871
119896 119908119872
119896 119908119880
119896⟩
are defined for all 119896 isin 119868119899 where
119908119871
119896= 119862119871sdot
(prod119899
119895=1119886119871
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
119908119872
119896=
(prod119899
119895=1119886119872
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
119908119880
119896= 119862119880sdot
(prod119899
119895=1119886119880
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
(7)
119862119871= min119894isin119868119899
(prod119899
119895=1119886119872
119894119895)1119899
(prod119899
119895=1119886119871
119894119895)1119899
119862119880= max119894isin119868119899
(prod119899
119895=1119886119872
119894119895)1119899
(prod119899
119895=1119886119880
119894119895)1119899
(8)
In addition Ramik and Korviny [4] defined a consistencyindex for measuring the nearness of a fuzzy reciprocal matrixto the corresponding fuzzy consistency matrix as follows
Definition 15 Let = (119894119895)119899times119899
be a fuzzy reciprocal matrixof which
119894119895= ⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ are triangular fuzzy numbers
evaluated from a scale 119878 = [1120590 120590] for some real number120590 gt 1 the consistency index of represented by the symbol119868120590
119899() is defined as
119868120590
119899() = 119862
120590
119899sdotmax119894119895
max1003816100381610038161003816100381610038161003816100381610038161003816
119908119871
119894
119908119880
119895
minus 119886119871
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816
119908119872
119894
119908119872
119895
minus 119886119872
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816
119908119880
119894
119908119871
119895
minus 119886119880
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
(9)
where = (119894)119899times1
are fuzzy weight vectors and 119894=
⟨119908119871
119894 119908119872
119894 119908119880
119894⟩ for all 119894 isin 119868
119899as expressed in (7) and
119862120590
119899=
1
max 120590 minus 120590(2minus2119899)119899 1205902 ((2119899)2(119899minus2) minus (2119899)(119899minus2)2) 120590 lt (
119899
2)
119899(119899minus2)
1
max 120590 minus 120590(2minus2119899)119899 120590(2minus2119899)119899 minus 120590 120590 ge (
119899
2)
119899(119899minus2)
(10)
If the consistency index 119868120590119899() = 0 the fuzzy reciprocal
fuzzy matrix is absolutely consistent The closer the valueof 119868120590119899() to 0 is the more consistent the matrix is Generally
an acceptable value is 119868120590119899() lt 01 or 10
Theorem 16 (see [4]) If is an 119899 times 119899 fuzzy reciprocal matrixwith triangular fuzzy elements evaluatedwith the scale [1120590 120590]for some 120590 gt 1 then 0 le 119868120590
119899() le 1
23 Financial Ratios A sustainable investment and missionrequires effective planning and financial management
The quantitative stock analysis is a useful tool that willimprove investmentrsquos understanding of financial results andtrends over time and provide key indicators of organizationalperformance Investormay use the quantitative stock analysisto pinpoint strengths and weaknesses of each company thatimpact to its stock
4 Advances in Operations Research
The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875
119899Ratio which are defined as
follows
Definition 17 Let 1198991 1198992 and 119899
3be the number of common
stock preferred stock and treasury stock respectively 119875119905
current price per share and 119864119903119903th-quarter net profit price to
earnings ratio or 119875119864 is defined as
119875
119864=119875119905(1198991+ 1198992minus 1198993)
119864119903
(11)
119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for
Definition 18 Let 119899 be the number of be the number ofregistered share 119860
119905and 119877
119905the asset and liability of the
company respectively and 119875119905current price per share price to
book value ratio or 119875119861119881 is defined as
119875
BV=119875119905
119861119905
(12)
where 119861119905= (119860119905minus 119877119905)119899
119875BV denotes how many times the current stock price iscompared to its account value
Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th
year-end dividend per share 119896 isin 119868119899 and 119875
0the 119899th-quarter
historical price the current target price 119875119899is defined as
119875119899= 1198750(1 + 119903)
119899
minus
119899
sum
119896=1
119863119896(1 + 119903)
119899minus119896
(13)
Definition 20 Let 119875119899be the current target price and 119875 the
current stock price 119875119875119899is called price per target price ratio
represented by the symbol 119875119875119899
119875119875119899denotes how many times the current stock price is
compared to the current target price
3 Stock Selection Procedure
This section presents the proposed stock selection procedurewhich is done in the following 3 main steps
Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock
Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group
Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market
31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for
each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions
(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group
(2) More recent data reflect current trend better thanearlier ones
(3) Fuzzy rules are flexible and depend on expert infor-mation
The specific steps of the fuzzy analysis are as follows
Step 11 This step involves screening in only 119898 individualstocks (119878
1 1198782 119878
119898) in the same industrial group of which
sufficient financial data are provided for calculating 119875119864119875BV and 119875119875
119899of 119899 earlier years up to the present
Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894
119896)
and (119875119875119899)(119878119894
119896) for all 119894 isin 119868
119899and 119896 isin 119868
119898 where 119878119894
119896denotes the
119896th stock in the 119894th year
Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878
119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896)
119896 isin 119868119898 from the following equations
(119864
119875)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119864
119875) (119878119894
119896)
(119875
BV)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
BV) (119878119894
119896)
(119875
119875119899
)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
119875119899
) (119878119894
119896)
where 119908119894=
2119894
119899 (119899 + 1) 119894 isin 119868
119898
(14)
Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875
119899and a fuzzy set119882 of the investment weights from
(119864119875)119908
(119878119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896) 119896 isin 119868
119898
Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows
Advances in Operations Research 5
Rule-1 if 1199091is 11and 119909
2is 12and 119909
3is 13then 119910 is
1
Rule-2 if 1199091is 21and 119909
2is 22and 119909
3is 23then 119910 is
2
Rule-119902 if 1199091is 1199021and 119909
2is 1199022and 119909
3is 1199023then 119910 is
119902
1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875
119899
and 1198821 respectively and
1198961 1198962 and
1198963 119896 isin 119868
119902 are
linguistic terms of 119864119875 119875BV 119875119875119899 and 119882
1 respectively
that is 119864119875 = 11 21
1199021 119875BV =
12 22
1199022
119875119875119899= 13 23
1199023 and119882 =
1 2
119902
Step 16This step involves importing 119864119875 119875BV and 119875119875119899of
the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861
Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908
119911= 119906B(119911) int
119861
119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int
119861
119911119908119911119889119911 =
int119861
119911119906B(119911) 119889119911 int119861
119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group
32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)
AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω
119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus
1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω
119899is replaced by
a set of fuzzy preference values Ω120575119899= 1
120575 1(119899 minus 1)
13120575 12120575 1 2120575 3120575 (119899 minus 1)
120575 120575 where
120575= ⟨119896 minus 120575 119896 119896 +
120575⟩ and 1120575= 1⊘
120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868
119899
and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]
and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers
Applying this technique to 1198991decision makers 119899
2deci-
sion criteria and 1198993industrial groups as alternate choices the
analysis steps are as follows
Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899
1 is compared to another
decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as
120593 (119894 119895) =
119894119895 exist119894119895isin Ω119899 119895 gt 119894
1 119895 = 119894
1 ⊘ 120593 (119895 119894) 119895 lt 119894
(15)
The decision makerrsquos preference matrix = (119894119895)1198991times1198991
is areciprocal matrix where
119894119895=
120593 (119894 119895) 119894 lt 119895
1 119894 = 119895
1 ⊘ 120593 (119895 119894) 119894 gt 119895
(16)
Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =
(119894119895)1198991times1198991
) 119889119896= ⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ is a fuzzyweight vector for
all 119896 isin 1198681198991
where
119908119871
119889119896= 119862119871sdot
(prod1198991
119895=1119886119871
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119872
119889119896=
(prod1198991
119895=1119886119872
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119880
119889119896= 119862119880sdot
(prod1198991
119895=1119886119880
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
(17)
with
119862119871= min119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119871
119894119895)11198991
119862119880= max119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119880
119894119895)11198991
(18)
If its consistency index 1198681205901198991
() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21
Step 23 This step involves decision makers 1198891 1198892
1198891198991
constructing decision criteria 1198881 1198882 119888
1198992
for evaluatingindustrial groups 119866
1 1198662 119866
1198993
where 119888119894 119894 = 1 119899
2 is
constructed from investment weight of 1198993individual groups
6 Advances in Operations Research
given by decision makers in the term of linguistic terms (seeTable 1)
Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887
119895119894119896= ⟨119887119871
119895119894119896 1198871198721
119895119894119896 1198871198722
119895119894119896 119887119880
119895119894119896⟩ 119895 isin 119868
1198993
119894 isin 1198681198992
and 119896 isin 1198681198991
which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888
1 1198882 119888
1198992
shown in(19)
Decision Criteria for Evaluating Industrial Groups 1198661
1198662 119866
1198993
Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111
211
119899311
112
212
119887119899312
119887111198991
119887211198991
119887119899311198991
1198882
1198661
1198662
1198661198993
121
221
119899321
122
222
119887119899322
119887121198991
119887221198991
119887119899321198991
1198881198992
1198661
1198662
1198661198993
111989921
211989921
119899311989921
119887111989922
119887211989922
119887119899311989922
119887111989921198991
119887211989921198991
119899311989921198991
= (19)
Step 24 This step involves decision makers 1198891 1198892 119889
1198991
evaluating decision criteria 1198881 1198882 119888
1198992
constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (
119894119895)1198992times1198991
for evaluation is then obtainedwhere
119894119895isin VL LMLMMHHVH for all 119894 isin 119868
1198992
and119895 isin 1198681198991
as shown in (20)
Evaluation of Decision Criteria 1198881 1198882 119888
1198992
Consider
11988911198892sdot sdot sdot 119889
1198991
11988811112sdot sdot sdot 11198991
11988822122sdot sdot sdot 21198991
=
1198881198992
1198992111989922sdot sdot sdot 11989921198991
(20)
Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector
119889= (
119889119896)119899times1
where 119889119896= ⟨119908
119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ =
⟨119908119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from Step 22 Equation (21)
shows these multiplication results
Decision Criteria Based on Weights of Decision MakersConsider
1198891
1198892
sdot sdot sdot 1198891198991
119888111otimes 1198891
12otimes 1198892sdot sdot sdot 11198991
otimes 1198891198991
119888221otimes 1198891
22otimes 1198892sdot sdot sdot 21198991
otimes 1198891198991
= 119908
1198881198992
11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991
otimes 1198891198991
(21)
Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (
119889119896)119899times1
where 119889119896=
⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ = ⟨119908
119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from
Step 22 The multiplication results are in (22)
Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111otimes 1198891
211otimes 1198891
119899311otimes 1198891
112otimes 1198892
212otimes 1198892
119887119899312otimes 1198892
111198991
otimes 1198891198991
211198991
otimes 1198891198991
119887119899311198991
otimes 1198891198991
1198882
1198661
1198662
1198661198993
121otimes 1198891
221otimes 1198891
119899321otimes 1198891
122otimes 1198892
222otimes 1198892
119887119899322otimes 1198892
121198991
otimes 1198891198991
221198991
otimes 1198891198991
119887119899321198991
otimes 1198891198991
1198881198992
1198661
1198662
1198661198993
111989921otimes 1198891
211989921otimes 1198891
119899311989921otimes 1198891
119887111989922otimes 1198892
119887211989922otimes 1198892
119887119899311989922otimes 1198892
119887111989921198991
otimes 1198891198991
119887211989921198991
otimes 1198891198991
119887119899311989921198991
otimes 1198891198991
= 119908
(22)
Advances in Operations Research 7
Table 1
Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩
Low (L) ⟨01 02 03⟩
Medium low (ML) ⟨02 03 04⟩
Medium (M) ⟨03 04 06 07⟩
Medium high (MH) ⟨06 07 08⟩
High (H) ⟨07 08 09⟩
Very high (VH) ⟨08 09 1 1⟩
Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows
119888119894= ⟨119908119871
119888119894 1199081198721
119888119894 1199081198722
119888119894 119908119880
119888119894⟩ (23)
where 119908119871119888119894= min1198991
119896=1119888119871
119908119894119896 1199081198721119888119894= (1119899
1) sum1198991
119896=11198881198721
119908119894119896 1199081198722119888119894=
(11198991) sum1198991
119896=11198881198722
119908119894119896 119908119880119888119894= max1198991
119896=1119888119880
119908119894119896 for all 119894 isin 119868
1198992
119908= (119908119895119896)1198992times1198991
and 1198991is the number of decision makers
Equation (24) shows these aggregation results
Weights of Decision Criteria 1198881 1198882 119888
1198992
Consider
1198881
1198882sdot sdot sdot 119888
1198992
119882211988811198882sdot sdot sdot
1198881198992
(24)
Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations
119895119894= ⟨119909119871
119895119894 1199091198721
119895119894 1199091198722
119895119894 119909119880
119895119894⟩ (25)
where 119909119871119895119894= min1198991
119896=1119887119871
119908119895119894119896 1199091198721119895119894= (1119899
1) sum1198991
119896=11198871198721
119908119895119894119896 1199091198722119895119894=
(11198991) sum1198991
119896=11198871198722
119908119895119894119896 119909119880119895119894= max1198992
119896=1119887119880
119908119895119894119896 for all 119895 isin 119899
3 119894 isin 119899
2
119908= (119908119895119894119896)11989931198992times1198991
and 1198991is the number of decision makers
These results are shown in (26)
Evaluation Matrix of Industrial Groups 1198661 1198662 119866
1198993
Con-sider
1198881
1198882sdot sdot sdot 119888
1198992
11986611112sdot sdot sdot
11198992
11986622122sdot sdot sdot
21198992
=
1198661198993
1198993111989932sdot sdot sdot 11989931198992
(26)
Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows
= (119895119894)1198993times1198992
119895119894= ⟨
119909119871
119895119894
119909lowast
119894
1199091198721
119895119894
119909lowast
119894
1199091198722
119895119894
119909lowast
119894
119909119880
119895119894
119909lowast
119894
⟩ where 119909lowasti =1198993max119895
119909119880
119895119894
(27)
Then multiplying the normalized matrix with the decisionweights from Step 26 = (V
119895119894)1198993times1198992
where V119895119894=
⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894when 119895 isin 119868
1198993
119894 isin 1198681198992
Industrial Groupsrsquo Evaluation Matrix Consider
11988811198882sdot sdot sdot 119888
1198992
1198661
V11
V12sdot sdot sdot V11198992
1198662
V21
V22sdot sdot sdot V21198992
=
1198661198993
V11989931V11989932sdot sdot sdot V11989931198992
(28)
Step 28 This step involves defining positive ideal solution(119866lowast
) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast
1198992
) and119866minus = (Vminus1 Vminus2 Vminus
1198992
) respectively whereVlowast119894= max1198993
119895V119880119895119894 and Vminus
119894= min1198993
119895V119871119895119894 119895 isin 119868
1198993
119894 isin 1198681198992
= (V
119895119894)1198993times1198992
Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following
119889lowast
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowast
119894) 119895 isin 119868
1198993
119889minus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vminus
119894) 119895 isin 119868
1198993
(29)
where 119889V(V119895119894 Vlowastminus
119894) are calculated in the same way as fuzzy
numbers are calculated according to Definition 8 (depictedin (30))
Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider
1198881
1198882
sdot sdot sdot 1198881198992
119889lowastminus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowastminus
119895)
1198661119889V (V11 V
lowastminus
1) 119889V (V12 V
lowastminus
2) sdot sdot sdot 119889V (V1119899
2
Vlowastminus1198992
) 119889lowastminus
1
1198662119889V (V21 V
lowastminus
1) 119889V (V22 V
lowastminus
2) sdot sdot sdot 119889V (V2119899
2
Vlowastminus21198992
) 119889lowastminus
2
1198661198993
119889V (V11989931 Vlowastminus1) 119889V (V119899
32 Vlowastminus2) sdot sdot sdot 119889V (V119899
31198992
Vlowastminus1198992
) 119889lowastminus
1198993
(30)
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Operations Research 3
22 Consistency Fuzzy Matrix In this subsection we intro-duce the definition of consistency fuzzy matrix and consis-tency index which was developed by Ramik [3 4]
Definition 11 Let 119860 = (119886119894119895)119899times119899
be an 119899 times 119899matrix where 119886119894119895gt
0 for all 119894 119895 isin 119868119899and 119860 is a reciprocal matrix if 119886
119895119894= 1119886119894119895for
all 119894 119895 isin 119868119899
Definition 12 Let119860 = (119886119894119895)119899times119899
be an 119899times119899matrix where 119886119894119895gt 0
for all 119894 119895 isin 119868119899and 119860 is a consistency matrix if there exist
weight vectors 119908 = (119908119894)119899times1
119908119894gt 0 for all 119894 isin 119868
119899 where
sum119899
119894=1119908119894= 1 and 119886
119894119895= 119908119894119908119895for all 119894 119895 isin 119868
119899
Definition 13 Let = (119894119895)119899times119899
be an 119899times119899 fuzzy matrix where119894119895gt 0 are fuzzy numbers for all 119894 119895 isin 119868
119899and is a reciprocal
fuzzy matrix if 119895119894= 1 ⊘
119894119895for all 119894 119895 isin 119868
119899
In particular if every member of = (119894119895)119899times119899
is atriangular fuzzy number
119894119895= ⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ is a reciprocal
fuzzy matrix if 119895119894= ⟨1119886
119880
119894119895 1119886119872
119894119895 1119886119871
119894119895⟩ for all 119894 119895 isin 119868
119899
Definition 14 Let = (119894119895)119899times119899
be an 119899times119899 fuzzymatrix where119894119895= [119886119894119895(120572) 119886119894119895(120572)] gt 0 for all 119894 119895 isin 119868
119899and is a consistency
fuzzy matrix if there exist 119886120572119894119895isin [119886119894119895(120572) 119886119894119895(120572)] for all 119894 119895 isin 119868
119899
and some 120572 isin [0 1] with which 119860 = (119886120572119894119895)119899times119899
is a consistencymatrix that is there exist119908120572 = (119908120572
119894)119899times1
119908120572119894gt 0 for all 119894 isin 119868
119899
where sum119899119894=1119908120572
119894= 1 and 119886120572
119894119895= 119908120572
119894119908120572
119895for all 119894 119895 isin 119868
119899
According to Definition 14 since 119908120572119894gt 0 for all 119894 isin
119868119899 there exist fuzzy vectors = (
119894)119899times1
where 119908120572119894isin
[119908119894(120572) 119908
119894(120572)] gt 0 for all 119894 isin 119868
119899 These vectors are called fuzzy
weight vectorsIt is clear that if is a fuzzy consistency matrix then it is a
fuzzy reciprocal fuzzymatrix and is not a fuzzy consistencymatrix if it is not a fuzzy reciprocal fuzzy matrix Becauseof these reasons construction of a fuzzy consistency matrixusually starts by first constructing a reciprocal fuzzy matrix Ramik and Korviny [4] proposed a method for calculatingfuzzy weight vector = (
119894)119899times1
for a fuzzy reciprocal matrix = (
119894119895)119899times119899
where 119894119895=⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ for all 119894 119895 isin 119868
119899by
using the method of geometric mean 119896= ⟨119908
119871
119896 119908119872
119896 119908119880
119896⟩
are defined for all 119896 isin 119868119899 where
119908119871
119896= 119862119871sdot
(prod119899
119895=1119886119871
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
119908119872
119896=
(prod119899
119895=1119886119872
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
119908119880
119896= 119862119880sdot
(prod119899
119895=1119886119880
119896119895)1119899
sum119899
119894=1(prod119899
119895=1119886119872
119894119895)1119899
(7)
119862119871= min119894isin119868119899
(prod119899
119895=1119886119872
119894119895)1119899
(prod119899
119895=1119886119871
119894119895)1119899
119862119880= max119894isin119868119899
(prod119899
119895=1119886119872
119894119895)1119899
(prod119899
119895=1119886119880
119894119895)1119899
(8)
In addition Ramik and Korviny [4] defined a consistencyindex for measuring the nearness of a fuzzy reciprocal matrixto the corresponding fuzzy consistency matrix as follows
Definition 15 Let = (119894119895)119899times119899
be a fuzzy reciprocal matrixof which
119894119895= ⟨119886119871
119894119895 119886119872
119894119895 119886119880
119894119895⟩ are triangular fuzzy numbers
evaluated from a scale 119878 = [1120590 120590] for some real number120590 gt 1 the consistency index of represented by the symbol119868120590
119899() is defined as
119868120590
119899() = 119862
120590
119899sdotmax119894119895
max1003816100381610038161003816100381610038161003816100381610038161003816
119908119871
119894
119908119880
119895
minus 119886119871
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816
119908119872
119894
119908119872
119895
minus 119886119872
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816
119908119880
119894
119908119871
119895
minus 119886119880
119894119895
1003816100381610038161003816100381610038161003816100381610038161003816
(9)
where = (119894)119899times1
are fuzzy weight vectors and 119894=
⟨119908119871
119894 119908119872
119894 119908119880
119894⟩ for all 119894 isin 119868
119899as expressed in (7) and
119862120590
119899=
1
max 120590 minus 120590(2minus2119899)119899 1205902 ((2119899)2(119899minus2) minus (2119899)(119899minus2)2) 120590 lt (
119899
2)
119899(119899minus2)
1
max 120590 minus 120590(2minus2119899)119899 120590(2minus2119899)119899 minus 120590 120590 ge (
119899
2)
119899(119899minus2)
(10)
If the consistency index 119868120590119899() = 0 the fuzzy reciprocal
fuzzy matrix is absolutely consistent The closer the valueof 119868120590119899() to 0 is the more consistent the matrix is Generally
an acceptable value is 119868120590119899() lt 01 or 10
Theorem 16 (see [4]) If is an 119899 times 119899 fuzzy reciprocal matrixwith triangular fuzzy elements evaluatedwith the scale [1120590 120590]for some 120590 gt 1 then 0 le 119868120590
119899() le 1
23 Financial Ratios A sustainable investment and missionrequires effective planning and financial management
The quantitative stock analysis is a useful tool that willimprove investmentrsquos understanding of financial results andtrends over time and provide key indicators of organizationalperformance Investormay use the quantitative stock analysisto pinpoint strengths and weaknesses of each company thatimpact to its stock
4 Advances in Operations Research
The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875
119899Ratio which are defined as
follows
Definition 17 Let 1198991 1198992 and 119899
3be the number of common
stock preferred stock and treasury stock respectively 119875119905
current price per share and 119864119903119903th-quarter net profit price to
earnings ratio or 119875119864 is defined as
119875
119864=119875119905(1198991+ 1198992minus 1198993)
119864119903
(11)
119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for
Definition 18 Let 119899 be the number of be the number ofregistered share 119860
119905and 119877
119905the asset and liability of the
company respectively and 119875119905current price per share price to
book value ratio or 119875119861119881 is defined as
119875
BV=119875119905
119861119905
(12)
where 119861119905= (119860119905minus 119877119905)119899
119875BV denotes how many times the current stock price iscompared to its account value
Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th
year-end dividend per share 119896 isin 119868119899 and 119875
0the 119899th-quarter
historical price the current target price 119875119899is defined as
119875119899= 1198750(1 + 119903)
119899
minus
119899
sum
119896=1
119863119896(1 + 119903)
119899minus119896
(13)
Definition 20 Let 119875119899be the current target price and 119875 the
current stock price 119875119875119899is called price per target price ratio
represented by the symbol 119875119875119899
119875119875119899denotes how many times the current stock price is
compared to the current target price
3 Stock Selection Procedure
This section presents the proposed stock selection procedurewhich is done in the following 3 main steps
Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock
Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group
Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market
31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for
each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions
(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group
(2) More recent data reflect current trend better thanearlier ones
(3) Fuzzy rules are flexible and depend on expert infor-mation
The specific steps of the fuzzy analysis are as follows
Step 11 This step involves screening in only 119898 individualstocks (119878
1 1198782 119878
119898) in the same industrial group of which
sufficient financial data are provided for calculating 119875119864119875BV and 119875119875
119899of 119899 earlier years up to the present
Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894
119896)
and (119875119875119899)(119878119894
119896) for all 119894 isin 119868
119899and 119896 isin 119868
119898 where 119878119894
119896denotes the
119896th stock in the 119894th year
Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878
119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896)
119896 isin 119868119898 from the following equations
(119864
119875)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119864
119875) (119878119894
119896)
(119875
BV)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
BV) (119878119894
119896)
(119875
119875119899
)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
119875119899
) (119878119894
119896)
where 119908119894=
2119894
119899 (119899 + 1) 119894 isin 119868
119898
(14)
Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875
119899and a fuzzy set119882 of the investment weights from
(119864119875)119908
(119878119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896) 119896 isin 119868
119898
Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows
Advances in Operations Research 5
Rule-1 if 1199091is 11and 119909
2is 12and 119909
3is 13then 119910 is
1
Rule-2 if 1199091is 21and 119909
2is 22and 119909
3is 23then 119910 is
2
Rule-119902 if 1199091is 1199021and 119909
2is 1199022and 119909
3is 1199023then 119910 is
119902
1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875
119899
and 1198821 respectively and
1198961 1198962 and
1198963 119896 isin 119868
119902 are
linguistic terms of 119864119875 119875BV 119875119875119899 and 119882
1 respectively
that is 119864119875 = 11 21
1199021 119875BV =
12 22
1199022
119875119875119899= 13 23
1199023 and119882 =
1 2
119902
Step 16This step involves importing 119864119875 119875BV and 119875119875119899of
the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861
Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908
119911= 119906B(119911) int
119861
119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int
119861
119911119908119911119889119911 =
int119861
119911119906B(119911) 119889119911 int119861
119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group
32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)
AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω
119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus
1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω
119899is replaced by
a set of fuzzy preference values Ω120575119899= 1
120575 1(119899 minus 1)
13120575 12120575 1 2120575 3120575 (119899 minus 1)
120575 120575 where
120575= ⟨119896 minus 120575 119896 119896 +
120575⟩ and 1120575= 1⊘
120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868
119899
and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]
and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers
Applying this technique to 1198991decision makers 119899
2deci-
sion criteria and 1198993industrial groups as alternate choices the
analysis steps are as follows
Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899
1 is compared to another
decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as
120593 (119894 119895) =
119894119895 exist119894119895isin Ω119899 119895 gt 119894
1 119895 = 119894
1 ⊘ 120593 (119895 119894) 119895 lt 119894
(15)
The decision makerrsquos preference matrix = (119894119895)1198991times1198991
is areciprocal matrix where
119894119895=
120593 (119894 119895) 119894 lt 119895
1 119894 = 119895
1 ⊘ 120593 (119895 119894) 119894 gt 119895
(16)
Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =
(119894119895)1198991times1198991
) 119889119896= ⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ is a fuzzyweight vector for
all 119896 isin 1198681198991
where
119908119871
119889119896= 119862119871sdot
(prod1198991
119895=1119886119871
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119872
119889119896=
(prod1198991
119895=1119886119872
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119880
119889119896= 119862119880sdot
(prod1198991
119895=1119886119880
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
(17)
with
119862119871= min119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119871
119894119895)11198991
119862119880= max119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119880
119894119895)11198991
(18)
If its consistency index 1198681205901198991
() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21
Step 23 This step involves decision makers 1198891 1198892
1198891198991
constructing decision criteria 1198881 1198882 119888
1198992
for evaluatingindustrial groups 119866
1 1198662 119866
1198993
where 119888119894 119894 = 1 119899
2 is
constructed from investment weight of 1198993individual groups
6 Advances in Operations Research
given by decision makers in the term of linguistic terms (seeTable 1)
Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887
119895119894119896= ⟨119887119871
119895119894119896 1198871198721
119895119894119896 1198871198722
119895119894119896 119887119880
119895119894119896⟩ 119895 isin 119868
1198993
119894 isin 1198681198992
and 119896 isin 1198681198991
which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888
1 1198882 119888
1198992
shown in(19)
Decision Criteria for Evaluating Industrial Groups 1198661
1198662 119866
1198993
Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111
211
119899311
112
212
119887119899312
119887111198991
119887211198991
119887119899311198991
1198882
1198661
1198662
1198661198993
121
221
119899321
122
222
119887119899322
119887121198991
119887221198991
119887119899321198991
1198881198992
1198661
1198662
1198661198993
111989921
211989921
119899311989921
119887111989922
119887211989922
119887119899311989922
119887111989921198991
119887211989921198991
119899311989921198991
= (19)
Step 24 This step involves decision makers 1198891 1198892 119889
1198991
evaluating decision criteria 1198881 1198882 119888
1198992
constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (
119894119895)1198992times1198991
for evaluation is then obtainedwhere
119894119895isin VL LMLMMHHVH for all 119894 isin 119868
1198992
and119895 isin 1198681198991
as shown in (20)
Evaluation of Decision Criteria 1198881 1198882 119888
1198992
Consider
11988911198892sdot sdot sdot 119889
1198991
11988811112sdot sdot sdot 11198991
11988822122sdot sdot sdot 21198991
=
1198881198992
1198992111989922sdot sdot sdot 11989921198991
(20)
Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector
119889= (
119889119896)119899times1
where 119889119896= ⟨119908
119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ =
⟨119908119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from Step 22 Equation (21)
shows these multiplication results
Decision Criteria Based on Weights of Decision MakersConsider
1198891
1198892
sdot sdot sdot 1198891198991
119888111otimes 1198891
12otimes 1198892sdot sdot sdot 11198991
otimes 1198891198991
119888221otimes 1198891
22otimes 1198892sdot sdot sdot 21198991
otimes 1198891198991
= 119908
1198881198992
11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991
otimes 1198891198991
(21)
Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (
119889119896)119899times1
where 119889119896=
⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ = ⟨119908
119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from
Step 22 The multiplication results are in (22)
Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111otimes 1198891
211otimes 1198891
119899311otimes 1198891
112otimes 1198892
212otimes 1198892
119887119899312otimes 1198892
111198991
otimes 1198891198991
211198991
otimes 1198891198991
119887119899311198991
otimes 1198891198991
1198882
1198661
1198662
1198661198993
121otimes 1198891
221otimes 1198891
119899321otimes 1198891
122otimes 1198892
222otimes 1198892
119887119899322otimes 1198892
121198991
otimes 1198891198991
221198991
otimes 1198891198991
119887119899321198991
otimes 1198891198991
1198881198992
1198661
1198662
1198661198993
111989921otimes 1198891
211989921otimes 1198891
119899311989921otimes 1198891
119887111989922otimes 1198892
119887211989922otimes 1198892
119887119899311989922otimes 1198892
119887111989921198991
otimes 1198891198991
119887211989921198991
otimes 1198891198991
119887119899311989921198991
otimes 1198891198991
= 119908
(22)
Advances in Operations Research 7
Table 1
Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩
Low (L) ⟨01 02 03⟩
Medium low (ML) ⟨02 03 04⟩
Medium (M) ⟨03 04 06 07⟩
Medium high (MH) ⟨06 07 08⟩
High (H) ⟨07 08 09⟩
Very high (VH) ⟨08 09 1 1⟩
Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows
119888119894= ⟨119908119871
119888119894 1199081198721
119888119894 1199081198722
119888119894 119908119880
119888119894⟩ (23)
where 119908119871119888119894= min1198991
119896=1119888119871
119908119894119896 1199081198721119888119894= (1119899
1) sum1198991
119896=11198881198721
119908119894119896 1199081198722119888119894=
(11198991) sum1198991
119896=11198881198722
119908119894119896 119908119880119888119894= max1198991
119896=1119888119880
119908119894119896 for all 119894 isin 119868
1198992
119908= (119908119895119896)1198992times1198991
and 1198991is the number of decision makers
Equation (24) shows these aggregation results
Weights of Decision Criteria 1198881 1198882 119888
1198992
Consider
1198881
1198882sdot sdot sdot 119888
1198992
119882211988811198882sdot sdot sdot
1198881198992
(24)
Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations
119895119894= ⟨119909119871
119895119894 1199091198721
119895119894 1199091198722
119895119894 119909119880
119895119894⟩ (25)
where 119909119871119895119894= min1198991
119896=1119887119871
119908119895119894119896 1199091198721119895119894= (1119899
1) sum1198991
119896=11198871198721
119908119895119894119896 1199091198722119895119894=
(11198991) sum1198991
119896=11198871198722
119908119895119894119896 119909119880119895119894= max1198992
119896=1119887119880
119908119895119894119896 for all 119895 isin 119899
3 119894 isin 119899
2
119908= (119908119895119894119896)11989931198992times1198991
and 1198991is the number of decision makers
These results are shown in (26)
Evaluation Matrix of Industrial Groups 1198661 1198662 119866
1198993
Con-sider
1198881
1198882sdot sdot sdot 119888
1198992
11986611112sdot sdot sdot
11198992
11986622122sdot sdot sdot
21198992
=
1198661198993
1198993111989932sdot sdot sdot 11989931198992
(26)
Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows
= (119895119894)1198993times1198992
119895119894= ⟨
119909119871
119895119894
119909lowast
119894
1199091198721
119895119894
119909lowast
119894
1199091198722
119895119894
119909lowast
119894
119909119880
119895119894
119909lowast
119894
⟩ where 119909lowasti =1198993max119895
119909119880
119895119894
(27)
Then multiplying the normalized matrix with the decisionweights from Step 26 = (V
119895119894)1198993times1198992
where V119895119894=
⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894when 119895 isin 119868
1198993
119894 isin 1198681198992
Industrial Groupsrsquo Evaluation Matrix Consider
11988811198882sdot sdot sdot 119888
1198992
1198661
V11
V12sdot sdot sdot V11198992
1198662
V21
V22sdot sdot sdot V21198992
=
1198661198993
V11989931V11989932sdot sdot sdot V11989931198992
(28)
Step 28 This step involves defining positive ideal solution(119866lowast
) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast
1198992
) and119866minus = (Vminus1 Vminus2 Vminus
1198992
) respectively whereVlowast119894= max1198993
119895V119880119895119894 and Vminus
119894= min1198993
119895V119871119895119894 119895 isin 119868
1198993
119894 isin 1198681198992
= (V
119895119894)1198993times1198992
Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following
119889lowast
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowast
119894) 119895 isin 119868
1198993
119889minus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vminus
119894) 119895 isin 119868
1198993
(29)
where 119889V(V119895119894 Vlowastminus
119894) are calculated in the same way as fuzzy
numbers are calculated according to Definition 8 (depictedin (30))
Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider
1198881
1198882
sdot sdot sdot 1198881198992
119889lowastminus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowastminus
119895)
1198661119889V (V11 V
lowastminus
1) 119889V (V12 V
lowastminus
2) sdot sdot sdot 119889V (V1119899
2
Vlowastminus1198992
) 119889lowastminus
1
1198662119889V (V21 V
lowastminus
1) 119889V (V22 V
lowastminus
2) sdot sdot sdot 119889V (V2119899
2
Vlowastminus21198992
) 119889lowastminus
2
1198661198993
119889V (V11989931 Vlowastminus1) 119889V (V119899
32 Vlowastminus2) sdot sdot sdot 119889V (V119899
31198992
Vlowastminus1198992
) 119889lowastminus
1198993
(30)
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Advances in Operations Research
The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875
119899Ratio which are defined as
follows
Definition 17 Let 1198991 1198992 and 119899
3be the number of common
stock preferred stock and treasury stock respectively 119875119905
current price per share and 119864119903119903th-quarter net profit price to
earnings ratio or 119875119864 is defined as
119875
119864=119875119905(1198991+ 1198992minus 1198993)
119864119903
(11)
119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for
Definition 18 Let 119899 be the number of be the number ofregistered share 119860
119905and 119877
119905the asset and liability of the
company respectively and 119875119905current price per share price to
book value ratio or 119875119861119881 is defined as
119875
BV=119875119905
119861119905
(12)
where 119861119905= (119860119905minus 119877119905)119899
119875BV denotes how many times the current stock price iscompared to its account value
Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th
year-end dividend per share 119896 isin 119868119899 and 119875
0the 119899th-quarter
historical price the current target price 119875119899is defined as
119875119899= 1198750(1 + 119903)
119899
minus
119899
sum
119896=1
119863119896(1 + 119903)
119899minus119896
(13)
Definition 20 Let 119875119899be the current target price and 119875 the
current stock price 119875119875119899is called price per target price ratio
represented by the symbol 119875119875119899
119875119875119899denotes how many times the current stock price is
compared to the current target price
3 Stock Selection Procedure
This section presents the proposed stock selection procedurewhich is done in the following 3 main steps
Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock
Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group
Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market
31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for
each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions
(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group
(2) More recent data reflect current trend better thanearlier ones
(3) Fuzzy rules are flexible and depend on expert infor-mation
The specific steps of the fuzzy analysis are as follows
Step 11 This step involves screening in only 119898 individualstocks (119878
1 1198782 119878
119898) in the same industrial group of which
sufficient financial data are provided for calculating 119875119864119875BV and 119875119875
119899of 119899 earlier years up to the present
Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894
119896)
and (119875119875119899)(119878119894
119896) for all 119894 isin 119868
119899and 119896 isin 119868
119898 where 119878119894
119896denotes the
119896th stock in the 119894th year
Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878
119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896)
119896 isin 119868119898 from the following equations
(119864
119875)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119864
119875) (119878119894
119896)
(119875
BV)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
BV) (119878119894
119896)
(119875
119875119899
)
119908
(119878119896) =
119899
sum
119894=1
119908119894(119875
119875119899
) (119878119894
119896)
where 119908119894=
2119894
119899 (119899 + 1) 119894 isin 119868
119898
(14)
Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875
119899and a fuzzy set119882 of the investment weights from
(119864119875)119908
(119878119896) (119875BV)119908(119878
119896) and (119875119875
119899)119908
(119878119896) 119896 isin 119868
119898
Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows
Advances in Operations Research 5
Rule-1 if 1199091is 11and 119909
2is 12and 119909
3is 13then 119910 is
1
Rule-2 if 1199091is 21and 119909
2is 22and 119909
3is 23then 119910 is
2
Rule-119902 if 1199091is 1199021and 119909
2is 1199022and 119909
3is 1199023then 119910 is
119902
1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875
119899
and 1198821 respectively and
1198961 1198962 and
1198963 119896 isin 119868
119902 are
linguistic terms of 119864119875 119875BV 119875119875119899 and 119882
1 respectively
that is 119864119875 = 11 21
1199021 119875BV =
12 22
1199022
119875119875119899= 13 23
1199023 and119882 =
1 2
119902
Step 16This step involves importing 119864119875 119875BV and 119875119875119899of
the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861
Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908
119911= 119906B(119911) int
119861
119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int
119861
119911119908119911119889119911 =
int119861
119911119906B(119911) 119889119911 int119861
119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group
32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)
AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω
119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus
1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω
119899is replaced by
a set of fuzzy preference values Ω120575119899= 1
120575 1(119899 minus 1)
13120575 12120575 1 2120575 3120575 (119899 minus 1)
120575 120575 where
120575= ⟨119896 minus 120575 119896 119896 +
120575⟩ and 1120575= 1⊘
120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868
119899
and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]
and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers
Applying this technique to 1198991decision makers 119899
2deci-
sion criteria and 1198993industrial groups as alternate choices the
analysis steps are as follows
Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899
1 is compared to another
decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as
120593 (119894 119895) =
119894119895 exist119894119895isin Ω119899 119895 gt 119894
1 119895 = 119894
1 ⊘ 120593 (119895 119894) 119895 lt 119894
(15)
The decision makerrsquos preference matrix = (119894119895)1198991times1198991
is areciprocal matrix where
119894119895=
120593 (119894 119895) 119894 lt 119895
1 119894 = 119895
1 ⊘ 120593 (119895 119894) 119894 gt 119895
(16)
Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =
(119894119895)1198991times1198991
) 119889119896= ⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ is a fuzzyweight vector for
all 119896 isin 1198681198991
where
119908119871
119889119896= 119862119871sdot
(prod1198991
119895=1119886119871
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119872
119889119896=
(prod1198991
119895=1119886119872
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119880
119889119896= 119862119880sdot
(prod1198991
119895=1119886119880
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
(17)
with
119862119871= min119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119871
119894119895)11198991
119862119880= max119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119880
119894119895)11198991
(18)
If its consistency index 1198681205901198991
() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21
Step 23 This step involves decision makers 1198891 1198892
1198891198991
constructing decision criteria 1198881 1198882 119888
1198992
for evaluatingindustrial groups 119866
1 1198662 119866
1198993
where 119888119894 119894 = 1 119899
2 is
constructed from investment weight of 1198993individual groups
6 Advances in Operations Research
given by decision makers in the term of linguistic terms (seeTable 1)
Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887
119895119894119896= ⟨119887119871
119895119894119896 1198871198721
119895119894119896 1198871198722
119895119894119896 119887119880
119895119894119896⟩ 119895 isin 119868
1198993
119894 isin 1198681198992
and 119896 isin 1198681198991
which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888
1 1198882 119888
1198992
shown in(19)
Decision Criteria for Evaluating Industrial Groups 1198661
1198662 119866
1198993
Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111
211
119899311
112
212
119887119899312
119887111198991
119887211198991
119887119899311198991
1198882
1198661
1198662
1198661198993
121
221
119899321
122
222
119887119899322
119887121198991
119887221198991
119887119899321198991
1198881198992
1198661
1198662
1198661198993
111989921
211989921
119899311989921
119887111989922
119887211989922
119887119899311989922
119887111989921198991
119887211989921198991
119899311989921198991
= (19)
Step 24 This step involves decision makers 1198891 1198892 119889
1198991
evaluating decision criteria 1198881 1198882 119888
1198992
constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (
119894119895)1198992times1198991
for evaluation is then obtainedwhere
119894119895isin VL LMLMMHHVH for all 119894 isin 119868
1198992
and119895 isin 1198681198991
as shown in (20)
Evaluation of Decision Criteria 1198881 1198882 119888
1198992
Consider
11988911198892sdot sdot sdot 119889
1198991
11988811112sdot sdot sdot 11198991
11988822122sdot sdot sdot 21198991
=
1198881198992
1198992111989922sdot sdot sdot 11989921198991
(20)
Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector
119889= (
119889119896)119899times1
where 119889119896= ⟨119908
119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ =
⟨119908119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from Step 22 Equation (21)
shows these multiplication results
Decision Criteria Based on Weights of Decision MakersConsider
1198891
1198892
sdot sdot sdot 1198891198991
119888111otimes 1198891
12otimes 1198892sdot sdot sdot 11198991
otimes 1198891198991
119888221otimes 1198891
22otimes 1198892sdot sdot sdot 21198991
otimes 1198891198991
= 119908
1198881198992
11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991
otimes 1198891198991
(21)
Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (
119889119896)119899times1
where 119889119896=
⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ = ⟨119908
119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from
Step 22 The multiplication results are in (22)
Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111otimes 1198891
211otimes 1198891
119899311otimes 1198891
112otimes 1198892
212otimes 1198892
119887119899312otimes 1198892
111198991
otimes 1198891198991
211198991
otimes 1198891198991
119887119899311198991
otimes 1198891198991
1198882
1198661
1198662
1198661198993
121otimes 1198891
221otimes 1198891
119899321otimes 1198891
122otimes 1198892
222otimes 1198892
119887119899322otimes 1198892
121198991
otimes 1198891198991
221198991
otimes 1198891198991
119887119899321198991
otimes 1198891198991
1198881198992
1198661
1198662
1198661198993
111989921otimes 1198891
211989921otimes 1198891
119899311989921otimes 1198891
119887111989922otimes 1198892
119887211989922otimes 1198892
119887119899311989922otimes 1198892
119887111989921198991
otimes 1198891198991
119887211989921198991
otimes 1198891198991
119887119899311989921198991
otimes 1198891198991
= 119908
(22)
Advances in Operations Research 7
Table 1
Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩
Low (L) ⟨01 02 03⟩
Medium low (ML) ⟨02 03 04⟩
Medium (M) ⟨03 04 06 07⟩
Medium high (MH) ⟨06 07 08⟩
High (H) ⟨07 08 09⟩
Very high (VH) ⟨08 09 1 1⟩
Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows
119888119894= ⟨119908119871
119888119894 1199081198721
119888119894 1199081198722
119888119894 119908119880
119888119894⟩ (23)
where 119908119871119888119894= min1198991
119896=1119888119871
119908119894119896 1199081198721119888119894= (1119899
1) sum1198991
119896=11198881198721
119908119894119896 1199081198722119888119894=
(11198991) sum1198991
119896=11198881198722
119908119894119896 119908119880119888119894= max1198991
119896=1119888119880
119908119894119896 for all 119894 isin 119868
1198992
119908= (119908119895119896)1198992times1198991
and 1198991is the number of decision makers
Equation (24) shows these aggregation results
Weights of Decision Criteria 1198881 1198882 119888
1198992
Consider
1198881
1198882sdot sdot sdot 119888
1198992
119882211988811198882sdot sdot sdot
1198881198992
(24)
Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations
119895119894= ⟨119909119871
119895119894 1199091198721
119895119894 1199091198722
119895119894 119909119880
119895119894⟩ (25)
where 119909119871119895119894= min1198991
119896=1119887119871
119908119895119894119896 1199091198721119895119894= (1119899
1) sum1198991
119896=11198871198721
119908119895119894119896 1199091198722119895119894=
(11198991) sum1198991
119896=11198871198722
119908119895119894119896 119909119880119895119894= max1198992
119896=1119887119880
119908119895119894119896 for all 119895 isin 119899
3 119894 isin 119899
2
119908= (119908119895119894119896)11989931198992times1198991
and 1198991is the number of decision makers
These results are shown in (26)
Evaluation Matrix of Industrial Groups 1198661 1198662 119866
1198993
Con-sider
1198881
1198882sdot sdot sdot 119888
1198992
11986611112sdot sdot sdot
11198992
11986622122sdot sdot sdot
21198992
=
1198661198993
1198993111989932sdot sdot sdot 11989931198992
(26)
Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows
= (119895119894)1198993times1198992
119895119894= ⟨
119909119871
119895119894
119909lowast
119894
1199091198721
119895119894
119909lowast
119894
1199091198722
119895119894
119909lowast
119894
119909119880
119895119894
119909lowast
119894
⟩ where 119909lowasti =1198993max119895
119909119880
119895119894
(27)
Then multiplying the normalized matrix with the decisionweights from Step 26 = (V
119895119894)1198993times1198992
where V119895119894=
⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894when 119895 isin 119868
1198993
119894 isin 1198681198992
Industrial Groupsrsquo Evaluation Matrix Consider
11988811198882sdot sdot sdot 119888
1198992
1198661
V11
V12sdot sdot sdot V11198992
1198662
V21
V22sdot sdot sdot V21198992
=
1198661198993
V11989931V11989932sdot sdot sdot V11989931198992
(28)
Step 28 This step involves defining positive ideal solution(119866lowast
) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast
1198992
) and119866minus = (Vminus1 Vminus2 Vminus
1198992
) respectively whereVlowast119894= max1198993
119895V119880119895119894 and Vminus
119894= min1198993
119895V119871119895119894 119895 isin 119868
1198993
119894 isin 1198681198992
= (V
119895119894)1198993times1198992
Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following
119889lowast
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowast
119894) 119895 isin 119868
1198993
119889minus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vminus
119894) 119895 isin 119868
1198993
(29)
where 119889V(V119895119894 Vlowastminus
119894) are calculated in the same way as fuzzy
numbers are calculated according to Definition 8 (depictedin (30))
Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider
1198881
1198882
sdot sdot sdot 1198881198992
119889lowastminus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowastminus
119895)
1198661119889V (V11 V
lowastminus
1) 119889V (V12 V
lowastminus
2) sdot sdot sdot 119889V (V1119899
2
Vlowastminus1198992
) 119889lowastminus
1
1198662119889V (V21 V
lowastminus
1) 119889V (V22 V
lowastminus
2) sdot sdot sdot 119889V (V2119899
2
Vlowastminus21198992
) 119889lowastminus
2
1198661198993
119889V (V11989931 Vlowastminus1) 119889V (V119899
32 Vlowastminus2) sdot sdot sdot 119889V (V119899
31198992
Vlowastminus1198992
) 119889lowastminus
1198993
(30)
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Operations Research 5
Rule-1 if 1199091is 11and 119909
2is 12and 119909
3is 13then 119910 is
1
Rule-2 if 1199091is 21and 119909
2is 22and 119909
3is 23then 119910 is
2
Rule-119902 if 1199091is 1199021and 119909
2is 1199022and 119909
3is 1199023then 119910 is
119902
1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875
119899
and 1198821 respectively and
1198961 1198962 and
1198963 119896 isin 119868
119902 are
linguistic terms of 119864119875 119875BV 119875119875119899 and 119882
1 respectively
that is 119864119875 = 11 21
1199021 119875BV =
12 22
1199022
119875119875119899= 13 23
1199023 and119882 =
1 2
119902
Step 16This step involves importing 119864119875 119875BV and 119875119875119899of
the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861
Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908
119911= 119906B(119911) int
119861
119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int
119861
119911119908119911119889119911 =
int119861
119911119906B(119911) 119889119911 int119861
119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group
32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)
AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω
119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus
1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω
119899is replaced by
a set of fuzzy preference values Ω120575119899= 1
120575 1(119899 minus 1)
13120575 12120575 1 2120575 3120575 (119899 minus 1)
120575 120575 where
120575= ⟨119896 minus 120575 119896 119896 +
120575⟩ and 1120575= 1⊘
120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868
119899
and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]
and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers
Applying this technique to 1198991decision makers 119899
2deci-
sion criteria and 1198993industrial groups as alternate choices the
analysis steps are as follows
Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899
1 is compared to another
decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as
120593 (119894 119895) =
119894119895 exist119894119895isin Ω119899 119895 gt 119894
1 119895 = 119894
1 ⊘ 120593 (119895 119894) 119895 lt 119894
(15)
The decision makerrsquos preference matrix = (119894119895)1198991times1198991
is areciprocal matrix where
119894119895=
120593 (119894 119895) 119894 lt 119895
1 119894 = 119895
1 ⊘ 120593 (119895 119894) 119894 gt 119895
(16)
Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =
(119894119895)1198991times1198991
) 119889119896= ⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ is a fuzzyweight vector for
all 119896 isin 1198681198991
where
119908119871
119889119896= 119862119871sdot
(prod1198991
119895=1119886119871
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119872
119889119896=
(prod1198991
119895=1119886119872
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
119908119880
119889119896= 119862119880sdot
(prod1198991
119895=1119886119880
119896119895)11198991
sum1198991
119894=1(prod1198991
119895=1119886119872
119894119895)11198991
(17)
with
119862119871= min119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119871
119894119895)11198991
119862119880= max119894isin1198681198991
(prod1198991
119895=1119886119872
119894119895)11198991
(prod1198991
119895=1119886119880
119894119895)11198991
(18)
If its consistency index 1198681205901198991
() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21
Step 23 This step involves decision makers 1198891 1198892
1198891198991
constructing decision criteria 1198881 1198882 119888
1198992
for evaluatingindustrial groups 119866
1 1198662 119866
1198993
where 119888119894 119894 = 1 119899
2 is
constructed from investment weight of 1198993individual groups
6 Advances in Operations Research
given by decision makers in the term of linguistic terms (seeTable 1)
Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887
119895119894119896= ⟨119887119871
119895119894119896 1198871198721
119895119894119896 1198871198722
119895119894119896 119887119880
119895119894119896⟩ 119895 isin 119868
1198993
119894 isin 1198681198992
and 119896 isin 1198681198991
which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888
1 1198882 119888
1198992
shown in(19)
Decision Criteria for Evaluating Industrial Groups 1198661
1198662 119866
1198993
Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111
211
119899311
112
212
119887119899312
119887111198991
119887211198991
119887119899311198991
1198882
1198661
1198662
1198661198993
121
221
119899321
122
222
119887119899322
119887121198991
119887221198991
119887119899321198991
1198881198992
1198661
1198662
1198661198993
111989921
211989921
119899311989921
119887111989922
119887211989922
119887119899311989922
119887111989921198991
119887211989921198991
119899311989921198991
= (19)
Step 24 This step involves decision makers 1198891 1198892 119889
1198991
evaluating decision criteria 1198881 1198882 119888
1198992
constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (
119894119895)1198992times1198991
for evaluation is then obtainedwhere
119894119895isin VL LMLMMHHVH for all 119894 isin 119868
1198992
and119895 isin 1198681198991
as shown in (20)
Evaluation of Decision Criteria 1198881 1198882 119888
1198992
Consider
11988911198892sdot sdot sdot 119889
1198991
11988811112sdot sdot sdot 11198991
11988822122sdot sdot sdot 21198991
=
1198881198992
1198992111989922sdot sdot sdot 11989921198991
(20)
Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector
119889= (
119889119896)119899times1
where 119889119896= ⟨119908
119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ =
⟨119908119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from Step 22 Equation (21)
shows these multiplication results
Decision Criteria Based on Weights of Decision MakersConsider
1198891
1198892
sdot sdot sdot 1198891198991
119888111otimes 1198891
12otimes 1198892sdot sdot sdot 11198991
otimes 1198891198991
119888221otimes 1198891
22otimes 1198892sdot sdot sdot 21198991
otimes 1198891198991
= 119908
1198881198992
11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991
otimes 1198891198991
(21)
Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (
119889119896)119899times1
where 119889119896=
⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ = ⟨119908
119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from
Step 22 The multiplication results are in (22)
Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111otimes 1198891
211otimes 1198891
119899311otimes 1198891
112otimes 1198892
212otimes 1198892
119887119899312otimes 1198892
111198991
otimes 1198891198991
211198991
otimes 1198891198991
119887119899311198991
otimes 1198891198991
1198882
1198661
1198662
1198661198993
121otimes 1198891
221otimes 1198891
119899321otimes 1198891
122otimes 1198892
222otimes 1198892
119887119899322otimes 1198892
121198991
otimes 1198891198991
221198991
otimes 1198891198991
119887119899321198991
otimes 1198891198991
1198881198992
1198661
1198662
1198661198993
111989921otimes 1198891
211989921otimes 1198891
119899311989921otimes 1198891
119887111989922otimes 1198892
119887211989922otimes 1198892
119887119899311989922otimes 1198892
119887111989921198991
otimes 1198891198991
119887211989921198991
otimes 1198891198991
119887119899311989921198991
otimes 1198891198991
= 119908
(22)
Advances in Operations Research 7
Table 1
Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩
Low (L) ⟨01 02 03⟩
Medium low (ML) ⟨02 03 04⟩
Medium (M) ⟨03 04 06 07⟩
Medium high (MH) ⟨06 07 08⟩
High (H) ⟨07 08 09⟩
Very high (VH) ⟨08 09 1 1⟩
Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows
119888119894= ⟨119908119871
119888119894 1199081198721
119888119894 1199081198722
119888119894 119908119880
119888119894⟩ (23)
where 119908119871119888119894= min1198991
119896=1119888119871
119908119894119896 1199081198721119888119894= (1119899
1) sum1198991
119896=11198881198721
119908119894119896 1199081198722119888119894=
(11198991) sum1198991
119896=11198881198722
119908119894119896 119908119880119888119894= max1198991
119896=1119888119880
119908119894119896 for all 119894 isin 119868
1198992
119908= (119908119895119896)1198992times1198991
and 1198991is the number of decision makers
Equation (24) shows these aggregation results
Weights of Decision Criteria 1198881 1198882 119888
1198992
Consider
1198881
1198882sdot sdot sdot 119888
1198992
119882211988811198882sdot sdot sdot
1198881198992
(24)
Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations
119895119894= ⟨119909119871
119895119894 1199091198721
119895119894 1199091198722
119895119894 119909119880
119895119894⟩ (25)
where 119909119871119895119894= min1198991
119896=1119887119871
119908119895119894119896 1199091198721119895119894= (1119899
1) sum1198991
119896=11198871198721
119908119895119894119896 1199091198722119895119894=
(11198991) sum1198991
119896=11198871198722
119908119895119894119896 119909119880119895119894= max1198992
119896=1119887119880
119908119895119894119896 for all 119895 isin 119899
3 119894 isin 119899
2
119908= (119908119895119894119896)11989931198992times1198991
and 1198991is the number of decision makers
These results are shown in (26)
Evaluation Matrix of Industrial Groups 1198661 1198662 119866
1198993
Con-sider
1198881
1198882sdot sdot sdot 119888
1198992
11986611112sdot sdot sdot
11198992
11986622122sdot sdot sdot
21198992
=
1198661198993
1198993111989932sdot sdot sdot 11989931198992
(26)
Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows
= (119895119894)1198993times1198992
119895119894= ⟨
119909119871
119895119894
119909lowast
119894
1199091198721
119895119894
119909lowast
119894
1199091198722
119895119894
119909lowast
119894
119909119880
119895119894
119909lowast
119894
⟩ where 119909lowasti =1198993max119895
119909119880
119895119894
(27)
Then multiplying the normalized matrix with the decisionweights from Step 26 = (V
119895119894)1198993times1198992
where V119895119894=
⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894when 119895 isin 119868
1198993
119894 isin 1198681198992
Industrial Groupsrsquo Evaluation Matrix Consider
11988811198882sdot sdot sdot 119888
1198992
1198661
V11
V12sdot sdot sdot V11198992
1198662
V21
V22sdot sdot sdot V21198992
=
1198661198993
V11989931V11989932sdot sdot sdot V11989931198992
(28)
Step 28 This step involves defining positive ideal solution(119866lowast
) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast
1198992
) and119866minus = (Vminus1 Vminus2 Vminus
1198992
) respectively whereVlowast119894= max1198993
119895V119880119895119894 and Vminus
119894= min1198993
119895V119871119895119894 119895 isin 119868
1198993
119894 isin 1198681198992
= (V
119895119894)1198993times1198992
Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following
119889lowast
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowast
119894) 119895 isin 119868
1198993
119889minus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vminus
119894) 119895 isin 119868
1198993
(29)
where 119889V(V119895119894 Vlowastminus
119894) are calculated in the same way as fuzzy
numbers are calculated according to Definition 8 (depictedin (30))
Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider
1198881
1198882
sdot sdot sdot 1198881198992
119889lowastminus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowastminus
119895)
1198661119889V (V11 V
lowastminus
1) 119889V (V12 V
lowastminus
2) sdot sdot sdot 119889V (V1119899
2
Vlowastminus1198992
) 119889lowastminus
1
1198662119889V (V21 V
lowastminus
1) 119889V (V22 V
lowastminus
2) sdot sdot sdot 119889V (V2119899
2
Vlowastminus21198992
) 119889lowastminus
2
1198661198993
119889V (V11989931 Vlowastminus1) 119889V (V119899
32 Vlowastminus2) sdot sdot sdot 119889V (V119899
31198992
Vlowastminus1198992
) 119889lowastminus
1198993
(30)
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Advances in Operations Research
given by decision makers in the term of linguistic terms (seeTable 1)
Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887
119895119894119896= ⟨119887119871
119895119894119896 1198871198721
119895119894119896 1198871198722
119895119894119896 119887119880
119895119894119896⟩ 119895 isin 119868
1198993
119894 isin 1198681198992
and 119896 isin 1198681198991
which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888
1 1198882 119888
1198992
shown in(19)
Decision Criteria for Evaluating Industrial Groups 1198661
1198662 119866
1198993
Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111
211
119899311
112
212
119887119899312
119887111198991
119887211198991
119887119899311198991
1198882
1198661
1198662
1198661198993
121
221
119899321
122
222
119887119899322
119887121198991
119887221198991
119887119899321198991
1198881198992
1198661
1198662
1198661198993
111989921
211989921
119899311989921
119887111989922
119887211989922
119887119899311989922
119887111989921198991
119887211989921198991
119899311989921198991
= (19)
Step 24 This step involves decision makers 1198891 1198892 119889
1198991
evaluating decision criteria 1198881 1198882 119888
1198992
constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (
119894119895)1198992times1198991
for evaluation is then obtainedwhere
119894119895isin VL LMLMMHHVH for all 119894 isin 119868
1198992
and119895 isin 1198681198991
as shown in (20)
Evaluation of Decision Criteria 1198881 1198882 119888
1198992
Consider
11988911198892sdot sdot sdot 119889
1198991
11988811112sdot sdot sdot 11198991
11988822122sdot sdot sdot 21198991
=
1198881198992
1198992111989922sdot sdot sdot 11989921198991
(20)
Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector
119889= (
119889119896)119899times1
where 119889119896= ⟨119908
119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ =
⟨119908119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from Step 22 Equation (21)
shows these multiplication results
Decision Criteria Based on Weights of Decision MakersConsider
1198891
1198892
sdot sdot sdot 1198891198991
119888111otimes 1198891
12otimes 1198892sdot sdot sdot 11198991
otimes 1198891198991
119888221otimes 1198891
22otimes 1198892sdot sdot sdot 21198991
otimes 1198891198991
= 119908
1198881198992
11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991
otimes 1198891198991
(21)
Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (
119889119896)119899times1
where 119889119896=
⟨119908119871
119889119896 119908119872
119889119896 119908119880
119889119896⟩ = ⟨119908
119871
119889119896 119908119872
119889119896 119908119872
119889119896 119908119880
119889119896⟩ calculated from
Step 22 The multiplication results are in (22)
Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider
1198891
1198892
1198891198991
1198881
1198661
1198662
1198661198993
111otimes 1198891
211otimes 1198891
119899311otimes 1198891
112otimes 1198892
212otimes 1198892
119887119899312otimes 1198892
111198991
otimes 1198891198991
211198991
otimes 1198891198991
119887119899311198991
otimes 1198891198991
1198882
1198661
1198662
1198661198993
121otimes 1198891
221otimes 1198891
119899321otimes 1198891
122otimes 1198892
222otimes 1198892
119887119899322otimes 1198892
121198991
otimes 1198891198991
221198991
otimes 1198891198991
119887119899321198991
otimes 1198891198991
1198881198992
1198661
1198662
1198661198993
111989921otimes 1198891
211989921otimes 1198891
119899311989921otimes 1198891
119887111989922otimes 1198892
119887211989922otimes 1198892
119887119899311989922otimes 1198892
119887111989921198991
otimes 1198891198991
119887211989921198991
otimes 1198891198991
119887119899311989921198991
otimes 1198891198991
= 119908
(22)
Advances in Operations Research 7
Table 1
Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩
Low (L) ⟨01 02 03⟩
Medium low (ML) ⟨02 03 04⟩
Medium (M) ⟨03 04 06 07⟩
Medium high (MH) ⟨06 07 08⟩
High (H) ⟨07 08 09⟩
Very high (VH) ⟨08 09 1 1⟩
Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows
119888119894= ⟨119908119871
119888119894 1199081198721
119888119894 1199081198722
119888119894 119908119880
119888119894⟩ (23)
where 119908119871119888119894= min1198991
119896=1119888119871
119908119894119896 1199081198721119888119894= (1119899
1) sum1198991
119896=11198881198721
119908119894119896 1199081198722119888119894=
(11198991) sum1198991
119896=11198881198722
119908119894119896 119908119880119888119894= max1198991
119896=1119888119880
119908119894119896 for all 119894 isin 119868
1198992
119908= (119908119895119896)1198992times1198991
and 1198991is the number of decision makers
Equation (24) shows these aggregation results
Weights of Decision Criteria 1198881 1198882 119888
1198992
Consider
1198881
1198882sdot sdot sdot 119888
1198992
119882211988811198882sdot sdot sdot
1198881198992
(24)
Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations
119895119894= ⟨119909119871
119895119894 1199091198721
119895119894 1199091198722
119895119894 119909119880
119895119894⟩ (25)
where 119909119871119895119894= min1198991
119896=1119887119871
119908119895119894119896 1199091198721119895119894= (1119899
1) sum1198991
119896=11198871198721
119908119895119894119896 1199091198722119895119894=
(11198991) sum1198991
119896=11198871198722
119908119895119894119896 119909119880119895119894= max1198992
119896=1119887119880
119908119895119894119896 for all 119895 isin 119899
3 119894 isin 119899
2
119908= (119908119895119894119896)11989931198992times1198991
and 1198991is the number of decision makers
These results are shown in (26)
Evaluation Matrix of Industrial Groups 1198661 1198662 119866
1198993
Con-sider
1198881
1198882sdot sdot sdot 119888
1198992
11986611112sdot sdot sdot
11198992
11986622122sdot sdot sdot
21198992
=
1198661198993
1198993111989932sdot sdot sdot 11989931198992
(26)
Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows
= (119895119894)1198993times1198992
119895119894= ⟨
119909119871
119895119894
119909lowast
119894
1199091198721
119895119894
119909lowast
119894
1199091198722
119895119894
119909lowast
119894
119909119880
119895119894
119909lowast
119894
⟩ where 119909lowasti =1198993max119895
119909119880
119895119894
(27)
Then multiplying the normalized matrix with the decisionweights from Step 26 = (V
119895119894)1198993times1198992
where V119895119894=
⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894when 119895 isin 119868
1198993
119894 isin 1198681198992
Industrial Groupsrsquo Evaluation Matrix Consider
11988811198882sdot sdot sdot 119888
1198992
1198661
V11
V12sdot sdot sdot V11198992
1198662
V21
V22sdot sdot sdot V21198992
=
1198661198993
V11989931V11989932sdot sdot sdot V11989931198992
(28)
Step 28 This step involves defining positive ideal solution(119866lowast
) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast
1198992
) and119866minus = (Vminus1 Vminus2 Vminus
1198992
) respectively whereVlowast119894= max1198993
119895V119880119895119894 and Vminus
119894= min1198993
119895V119871119895119894 119895 isin 119868
1198993
119894 isin 1198681198992
= (V
119895119894)1198993times1198992
Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following
119889lowast
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowast
119894) 119895 isin 119868
1198993
119889minus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vminus
119894) 119895 isin 119868
1198993
(29)
where 119889V(V119895119894 Vlowastminus
119894) are calculated in the same way as fuzzy
numbers are calculated according to Definition 8 (depictedin (30))
Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider
1198881
1198882
sdot sdot sdot 1198881198992
119889lowastminus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowastminus
119895)
1198661119889V (V11 V
lowastminus
1) 119889V (V12 V
lowastminus
2) sdot sdot sdot 119889V (V1119899
2
Vlowastminus1198992
) 119889lowastminus
1
1198662119889V (V21 V
lowastminus
1) 119889V (V22 V
lowastminus
2) sdot sdot sdot 119889V (V2119899
2
Vlowastminus21198992
) 119889lowastminus
2
1198661198993
119889V (V11989931 Vlowastminus1) 119889V (V119899
32 Vlowastminus2) sdot sdot sdot 119889V (V119899
31198992
Vlowastminus1198992
) 119889lowastminus
1198993
(30)
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
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MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Operations Research 7
Table 1
Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩
Low (L) ⟨01 02 03⟩
Medium low (ML) ⟨02 03 04⟩
Medium (M) ⟨03 04 06 07⟩
Medium high (MH) ⟨06 07 08⟩
High (H) ⟨07 08 09⟩
Very high (VH) ⟨08 09 1 1⟩
Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows
119888119894= ⟨119908119871
119888119894 1199081198721
119888119894 1199081198722
119888119894 119908119880
119888119894⟩ (23)
where 119908119871119888119894= min1198991
119896=1119888119871
119908119894119896 1199081198721119888119894= (1119899
1) sum1198991
119896=11198881198721
119908119894119896 1199081198722119888119894=
(11198991) sum1198991
119896=11198881198722
119908119894119896 119908119880119888119894= max1198991
119896=1119888119880
119908119894119896 for all 119894 isin 119868
1198992
119908= (119908119895119896)1198992times1198991
and 1198991is the number of decision makers
Equation (24) shows these aggregation results
Weights of Decision Criteria 1198881 1198882 119888
1198992
Consider
1198881
1198882sdot sdot sdot 119888
1198992
119882211988811198882sdot sdot sdot
1198881198992
(24)
Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations
119895119894= ⟨119909119871
119895119894 1199091198721
119895119894 1199091198722
119895119894 119909119880
119895119894⟩ (25)
where 119909119871119895119894= min1198991
119896=1119887119871
119908119895119894119896 1199091198721119895119894= (1119899
1) sum1198991
119896=11198871198721
119908119895119894119896 1199091198722119895119894=
(11198991) sum1198991
119896=11198871198722
119908119895119894119896 119909119880119895119894= max1198992
119896=1119887119880
119908119895119894119896 for all 119895 isin 119899
3 119894 isin 119899
2
119908= (119908119895119894119896)11989931198992times1198991
and 1198991is the number of decision makers
These results are shown in (26)
Evaluation Matrix of Industrial Groups 1198661 1198662 119866
1198993
Con-sider
1198881
1198882sdot sdot sdot 119888
1198992
11986611112sdot sdot sdot
11198992
11986622122sdot sdot sdot
21198992
=
1198661198993
1198993111989932sdot sdot sdot 11989931198992
(26)
Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows
= (119895119894)1198993times1198992
119895119894= ⟨
119909119871
119895119894
119909lowast
119894
1199091198721
119895119894
119909lowast
119894
1199091198722
119895119894
119909lowast
119894
119909119880
119895119894
119909lowast
119894
⟩ where 119909lowasti =1198993max119895
119909119880
119895119894
(27)
Then multiplying the normalized matrix with the decisionweights from Step 26 = (V
119895119894)1198993times1198992
where V119895119894=
⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894when 119895 isin 119868
1198993
119894 isin 1198681198992
Industrial Groupsrsquo Evaluation Matrix Consider
11988811198882sdot sdot sdot 119888
1198992
1198661
V11
V12sdot sdot sdot V11198992
1198662
V21
V22sdot sdot sdot V21198992
=
1198661198993
V11989931V11989932sdot sdot sdot V11989931198992
(28)
Step 28 This step involves defining positive ideal solution(119866lowast
) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast
1198992
) and119866minus = (Vminus1 Vminus2 Vminus
1198992
) respectively whereVlowast119894= max1198993
119895V119880119895119894 and Vminus
119894= min1198993
119895V119871119895119894 119895 isin 119868
1198993
119894 isin 1198681198992
= (V
119895119894)1198993times1198992
Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following
119889lowast
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowast
119894) 119895 isin 119868
1198993
119889minus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vminus
119894) 119895 isin 119868
1198993
(29)
where 119889V(V119895119894 Vlowastminus
119894) are calculated in the same way as fuzzy
numbers are calculated according to Definition 8 (depictedin (30))
Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider
1198881
1198882
sdot sdot sdot 1198881198992
119889lowastminus
119895=
1198992
sum
119894=1
119889V (V119895119894 Vlowastminus
119895)
1198661119889V (V11 V
lowastminus
1) 119889V (V12 V
lowastminus
2) sdot sdot sdot 119889V (V1119899
2
Vlowastminus1198992
) 119889lowastminus
1
1198662119889V (V21 V
lowastminus
1) 119889V (V22 V
lowastminus
2) sdot sdot sdot 119889V (V2119899
2
Vlowastminus21198992
) 119889lowastminus
2
1198661198993
119889V (V11989931 Vlowastminus1) 119889V (V119899
32 Vlowastminus2) sdot sdot sdot 119889V (V119899
31198992
Vlowastminus1198992
) 119889lowastminus
1198993
(30)
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
8 Advances in Operations Research
Table 2 119864119875 of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383
Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862
119895 and ranking the
industrial groups according to them 119862119862119895are defined as
follows
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
119895 isin 1198681198993
(31)
From the calculation a set of investment weightsfor industrial groups 119882
3= (119908
1 1199082 119908
1198993
) where1199081 1199082 119908
1198993
are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group
33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882
119874119860(119904119894119895) where
119882119874119860(119904119894119895) = 119882
1(119904119894119895) sdot 1198822(119866119895) and119882
1(119904119894119895) are the weight of the
119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the
weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies
4 Application of the Analysis Procedures toa Demonstration Case
As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866
1) consumer product and
service industry (1198662) financial industry (119866
3) industrial
product and technology industry (1198664) property and con-
struction industry (1198665) and resource industry (119866
6) Stocks
from each individual industry were analyzed as follows
Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below
In this group1198665 we use the past 5-year financial fact data
of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom
Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ
Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock
Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875
119899 Tables 2 3 and
4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866
5
Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers
Values of119864119875119875BV and119875119875119899were grouped into 3 levels
low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were
119871 = ⟨119897119871
1198971198721 1198971198722 119897119880
⟩
119872 = ⟨119898119871
1198981198721 1198981198722 119898119880
⟩
119867 = ⟨ℎ119871
ℎ1198721 ℎ1198722 ℎ119880
⟩
(32)
The fuzzy sets of linguistic terms were as follows
119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩
119867119885 = ⟨19 23 100 100⟩
Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Operations Research 9
Table 3 119875BV of STPI
STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27
Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263
Table 4 119875119875119899of STPI
STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27
Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641
Target price in 2014 293056Closing price to target priceratio 07098
Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866
5
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899
243 112 094 238 294 097 167 071 186 083 287 24
Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882
Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882
Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882
Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875
119899 which in this study
were the values of the 22nd of January 2015 shown in Table 6
Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation
Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7
For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8
Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866
1 1198662 119866
6 were analyzedThree decisionmakers
1198891 1198892 1198893constructed 4 decision criteria 119888
1 1198882 1198883 1198884
calculated in the following steps
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Advances in Operations Research
Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom
Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899
255 091 134 234 429 179 155 066 237 108 283 312
Table 7 Investment weights from the analysis procedures
Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084
Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up
1198661
1198662
1198663
1198664
1198665
1198666
11990411
00418 11990412
026 11990413
01276 11990414
00518 11990415(CK) 0084 119904
1600261
11990421
0024 11990422
0169 11990423
01528 11990424
01077 11990425(ITD) 0105 119904
2601258
11990431
01148 11990432
01359 11990433
00282 11990434
01745 11990435(PREB) 0084 119904
3600667
11990441
01704 11990442
01006 11990443
00843 11990444
00528 11990445(SEAFCO) 01091 119904
4602034
11990451
01003 11990452
0004 11990453
00822 11990454
01108 11990455(STEC) 01091 119904
5600315
11990461
0097 11990462
01376 11990463
00841 11990464
01399 11990465( STPI) 01435 119904
6601576
11990471
00764 11990472
01825 11990473
00335 11990474
00916 11990475(SYNTEC) 0084 119904
7602068
11990481
00705 11990482
00104 11990483
00421 11990484
01099 11990485(TRC) 0113 119904
8600638
11990491
01484 11990493
0211 11990494
00825 11990495(TTCL) 0084 119904
9601215
119904101
01565 119904103
02517 119904104
00986 119904105
(UNIQ) 0084
Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining
= (
(1 1 1) (1 2 3) (2 3 4)
(1
31
2 1) (1 1 1) (1 2 3)
(1
41
31
2) (
1
31
2 1) (1 1 1)
) (33)
Step 22 This step involves calculating the fuzzy weightvectors
119889= (
119889119896)3times1
for = (119894119895)3times3
and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩
1198892
= ⟨025869 029712 029712 034012⟩ and 1198893
=
⟨016296 016296 016296 018717⟩ and a consistency index119868120590
3() = 009403
Step 23 This step involves the 3 decision makers 1198891 1198892
1198893evaluating 6 industrial groups 119866
1 1198662 119866
6 according
to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms
VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9
Step 24 This step involves decision makers 1198891 1198892 1198893
evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the
linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10
Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively
Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14
Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V
119895119894)6times4
where V
119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V
119895119894= 119895119894otimes 119888119894 when
119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15
Step 28This step involves stipulating a positive ideal solution(119878lowast
) and a negative ideal solution (119878minus) to be
119878lowast
= [(054 054 054 054)
(0486 0486 0486 0486) (054 054 054 054)
(0432 0432 0432 0432)]
119878minus
= [(0021 0021 0021 0021)
(0021 0021 0021 0021)
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Operations Research 11
Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
06 07 07 08 06 07 07 08 08 09 1 11198662
08 09 1 1 07 08 08 09 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
06 07 07 08 06 07 07 08 06 07 07 081198666
06 07 07 08 07 08 08 09 06 07 07 08
1198882
1198661
06 07 07 08 07 08 08 09 07 08 08 091198662
07 08 08 09 06 07 07 08 06 07 07 081198663
08 09 1 1 08 09 1 1 08 09 1 11198664
06 07 07 08 07 08 08 09 07 08 08 091198665
06 07 07 08 06 07 07 08 07 08 08 091198666
07 08 08 09 07 08 08 09 07 08 08 09
1198883
1198661
07 08 08 09 07 08 08 09 07 08 08 091198662
08 09 1 1 07 08 08 09 07 08 08 091198663
08 09 1 1 08 09 1 1 07 08 08 091198664
07 08 08 09 06 07 07 08 06 07 07 081198665
07 08 08 09 06 07 07 08 07 08 08 091198666
06 07 07 08 06 07 07 08 07 08 08 09
1198884
1198661
06 07 07 08 06 07 07 08 06 07 07 081198662
06 07 07 08 08 09 1 1 07 08 08 091198663
07 08 08 09 07 08 08 09 07 08 08 091198664
08 09 1 1 08 09 1 1 08 09 1 11198665
07 08 08 09 07 08 08 09 07 08 08 091198666
07 08 08 09 06 07 07 08 07 08 08 09
Table 10 Evaluation of fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
08 09 1 1 08 09 1 1 08 09 1 11198882
07 08 08 09 07 08 08 09 07 08 08 091198883
08 09 1 1 08 09 1 1 08 09 1 11198884
05 06 07 08 08 09 1 1 07 08 08 09
(0024 0024 0024 0024)
(0021 0021 0021 0021)]
(34)
Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast
) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively
Step 210 This step involves obtaining the nearness coeffi-cients 119862119862
119895 119895 = 1 6 to the positive ideal solution and the
investment weights shown in Table 18
Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882
119874119860(119904119894119895) where 119894 represents
the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19
From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights
5 Conclusions
The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Advances in Operations Research
Table 11 Fuzzy decision criteria
Criteria Decision maker1198891
1198892
1198893
1198881
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882
03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883
03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884
02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685
Table 12 Fuzzy evaluation of industrial groups
Criteria Industrial group Decision maker1198891
1198892
1198893
1198881
1198661
0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666
0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015
1198882
1198661
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664
0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168
1198883
1198661
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662
0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663
0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664
033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666
0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168
1198884
1198661
0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662
0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664
0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665
033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666
033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168
Table 13 Aggregation of decision criteria
Criteria1198881
1198882
1198883
1198884
Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432
Table 14 Aggregation of evaluation of industrial groups
Group Criteria1198881
1198882
1198883
1198884
1198661
013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662
0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663
0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664
013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665
0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666
0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Advances in Operations Research 13
Table 15 Decision matrix
Group Criteria1198881
1198882
1198883
1198884
1198661
0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662
0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663
0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664
0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665
0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666
0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389
Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889lowast
1= 119889V (1198661 119878
lowast
) 0381572 0348712 0378268 0309819 1418371119889lowast
2= 119889V (1198662 119878
lowast
) 0364995 0346628 0369604 0301308 1382533119889lowast
3= 119889V (1198663 119878
lowast
) 0374072 0326879 0365443 0298251 1364645119889lowast
4= 119889V (1198664 119878
lowast
) 0356869 0348712 0384022 0284309 1373911119889lowast
5= 119889V (1198665 119878
lowast
) 0388341 0351266 0381127 0298251 1418985119889lowast
6= 119889V (1198666 119878
lowast
) 038534 0341527 0389187 0300654 1416708
Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion
Distance Criteria Sum1198881
1198882
1198883
1198884
119889minus
1= 119889V (1198661 119878
minus
) 0223775 0197715 0247374 0174345 0843208119889minus
2= 119889V (1198662 119878
minus
) 0281158 0220808 0276399 017835 0956716119889minus
3= 119889V (1198663 119878
minus
) 0251745 0251745 0278567 0198531 0980589119889minus
4= 119889V (1198664 119878
minus
) 0285192 0197715 0245285 0225285 0953478119889minus
5= 119889V (1198665 119878
minus
) 0221504 0196478 0246015 0198531 0862528119889minus
6= 119889V (1198666 119878
minus
) 0223065 0222647 0218246 0197315 0861274
Table 18 Nearness coefficients to the positive ideal solution
Industrial group 1198661
1198662
1198663
1198664
1198665
1198666
119862119862119895=
119889minus
119895
119889minus
119895+ 119889lowast
119895
0304297 038056 0392965 0380328 0318558 0315015
Weights 0157599 0172877 0176738 0173169 0159816 0159816
Table 19 The final investment weights of all of the stocks in the market
119904119894119895
11990412
119904102
11990493
11990476
11990446
11990472
11990434
11990422
11990423
11990441
11990466
119904101
11990464
119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247
119904119894119895
11990462
11990432
11990491
11990465
11990413
11990426
11990496
11990454
11990484
11990424
11990431
11990485
11990455
119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373
119904119894119895
11990445
11990442
119904104
11990425
11990474
11990461
11990451
11990443
11990463
11990453
11990494
11990415
11990435
119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279
119904119894119895
11990475
11990495
119904105
11990471
11990481
11990436
11990486
11990444
11990414
11990483
11990411
11990473
11990456
119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529
119904119894119895
11990433
11990416
11990421
11990482
11990452
119882119874119860(119904119894119895) 00047 00039 00034 00018 00007
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Advances in Operations Research
used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision
For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand
References
[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011
[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001
[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009
[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010
[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995
[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010
[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013
[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015
[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012
[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009
[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980
[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013
[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007
[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002
[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015
[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of