advisor:prof. chu, ta-chung student: chen, chun chi
DESCRIPTION
A Maximizing Set and Minimizing Set Based Fuzzy MCDM Approach for the Evaluation and Selection of the Distribution Centers. Advisor:Prof. Chu, Ta-Chung Student: Chen, Chun Chi. Outline. Introduction Fuzzy set theory Model development Numerical example Conclusion. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
1
A Maximizing Set and Minimizing Set Based Fuzzy
MCDM Approach for the Evaluation and Selection of
the Distribution Centers
Advisor:Prof. Chu, Ta-Chung
Student: Chen, Chun Chi
3
Introduction
• Properly selecting a location for establishing a distribution center is
very important for an enterprise to effectively control channels,
upgrade operation performance, service level and sufficiently
allocate resources, and so on. • Selecting an improper location of a distribution center may cause
losses for an enterprise.
• Therefore, an enterprise will always conduct evaluation and
selection study of possible locations before determining
distribution center.
Introduction (cont.)
• Evaluating a DC location, many conflicting criteria must be considered:
1. objective – these criteria can be evaluated quantitatively, e.g. investment cost.
2. subjective – these criteria have qualitative definitions, e.g. expansion possibility, closeness to demand market, etc.
• Perez et al. pointed out “Location problems concern a wide set of fields where it is usually assumed that exact data are known, but in real applications is full of linguistic vagueness.
4
Introduction (cont.)
• Fuzzy set theory, initially proposed by Zadeh, and it can effectively resolve the uncertainties in an ill-defined multiple criteria decision making environment.
• Some recent applications on locations evaluation and selection can be found, but despite the merits, most of the above papers can not present membership functions for the final fuzzy evaluation values and defuzzification formulae from the membership functions.
• To resolve the these limitations, this work suggests a maximizing set and minimizing set based fuzzy MCDM approach.
5
6
Introduction (cont.)
Purposes of this paper:
• Develop a fuzzy MCDM model for the evaluation and selection of
the location of a distribution center.
• Apply maximizing set and minimizing set to the proposed model in
order to develop formulae for ranking procedure.
• Conduct a numerical example to demonstrate the feasibility of the
proposed model.
Fuzzy set theory
• 2.1 Fuzzy sets
• 2.2 Fuzzy numbers
• 2.3 α-cuts
• 2.4 Arithmetic operations on fuzzy numbers
• 2.5 Linguistic values
7
8
2.1 Fuzzy set
• The fuzzy set A can be expressed as:
• (2.1)
where U is the universe of discourse, x is an element in U, A is a fuzzy set in U, is the membership function of A at x. The larger , the stronger the grade of membership for x in A.
} |))( ,{( UxxfxA A
xf A
xf A
9
2.2 Fuzzy numbers
• A real fuzzy number A is described as any fuzzy subset of the real line R with membership function which possesses the following properties (Dubois and Prade, 1978):
– (a) is a continuous mapping from R to [0,1];
– (b)
– (c) is strictly increasing on [a ,b];
– (d)
– (e) is strictly decreasing on [c ,d];
– (f)
where, A can be denoted as .
Af
Af
; ] ,( ,0)( axxf A
Af
; , ,1)( cbxxf A
Af
; ) ,[ ,0)( dxxf A
dcba , , ,
10
2.2 Fuzzy numbers (Cont.)
• The membership function of the fuzzy number A can also be expressed as:
(2.2)
where and are left and right membership functions of A,
respectively.
Af
, ,0
,),(
,1
,),(
)(
otherwise
dxcxf
cxb
bxaxf
xfR
A
LA
A
)(xf LA )(xf R
A
11
2.3 α-cut
• The α-cuts of fuzzy number A can be defined as:
(2.3)
where is a non-empty bounded closed interval contained in R and can be denoted by , where and are its lower and upper bounds, respectively.
1 ,0 , | xfxA A
A
] , [ ul AAA
lA uA
12
2.4 Arithmetic operations on fuzzy numbers
• Given fuzzy numbers A and B, , the α-cuts of A and B are
and respectively. By interval arithmetic, some main operations of A and B can be expressed as follows (Kaufmann and Gupta, 1991):
– (2.4)
– (2.5)
– (2.6)
– (2.7)
– (2.8)
RBA ,
] , [ ul AAA ], , [
ul BBB
] , [ uull BABABA
] , [ luul BABABA
] , [ uull BABABA
] , [ )(
l
u
u
l
B
A
B
ABA
RrrArArA ul , ,
13
2.5 Linguistic variable
• According to Zadeh (1975), the concept of linguistic variable is very useful in dealing with situations which are complex to be reasonably described by conventional quantitative expressions.
A1=(0,0,0.25)=Unimportant
A2=(0,0.25,0.5)=Less important
A3=(0.25,0.50,0.75)=important
A4=(0.50,0.75,1.00)=More important
A5=(0.75,1.00,1.00)=Very important
Figure 2-1. Linguistic values and triangular
fuzzy numbers
0.50.250 0.75 1
A3 A5A4A2A1
14
Model development
• 3.1 Aggregate ratings of alternatives versus qualitative criteria
• 3.2 Normalize values of alternatives versus quantitative criteria
• 3.3 Average importance weights
• 3.4 Develop membership functions
• 3.5 Rank fuzzy numbers
15
Model development
• decision makers
• candidate locations of distribution centers
• selected criteria,
• In model development process, criteria are categorized into three groups:
– Benefit qualitative criteria:
– Benefit quantitative criteria:
– Cost quantitative criteria:
ltDt ,...,2,1, tD
miAi ,...,2,1, iA
jC njC j ,...,2,1,
gjC j ,...,1,
hgjC j ,...,1,
nhjC j ,...,1,
16
3.1 Aggregate ratings of alternatives versus
qualitative criteria • Assume
• (3.1)
– where
– Ratings assigned by each decision maker for each alternative versus
each qualitative criterion.
– Averaged ratings of each alternative versus each qualitative criterion.
, ),,( ijtijtijtijt cbax ltgjmi ,...,1,,...,1,,...,1
)...(1
21 ijlijijij xxxl
x
, 1
1
l
tijtij a
la ,
1
1
l
tijtij b
lb
l
tijtij c
lc
1
1
:ijtx
:ijx
17
3.2Normalize values of alternatives versus quantitative criteria
• is the value of an alternative versus a benefit quantitative criteria or cost quantitative criteria
.
• denotes the normalized value of
•
(3.2)
• For calculation convenience, assume
),,( ijijijij qpoy ,,...,2,1, miAi
,,...,1, hgjj
nhjj ,...,1,
ijx ijy
, ),,(***ij
ij
ij
ij
ij
ijij q
q
q
p
q
ox * max ,ij ijq q j B
, ),,( ijijijij cbax . ,...,1 ngj
* * *( , , ) ,ij ij ij
ijij ij ij
o o ox
q p o * min ,ij ijq o j C
18
3.3 Average importance weights
• Assume
• (3.3)
• where
• represents the weight assigned by each decision maker for
each criterion. • represents the average importance weight of each criterion.
, ),,( jtjtjtjt fedw , Rw jt , ,...,1 nj , ,...,1 lt
)...(1
21 jljjj wwwl
w
, 1
1
l
tjtj d
ld ,
1
1
l
tjtj e
le .
1
1
l
tjtj f
lf
:jtw
:jw
19
3.4 Develop membership functions
• The membership function of final fuzzy evaluation value,
of each candidate distribution center can be developed as follows:
• The membership functions are developed as:
• (3.4)
• (3.5)
• (3.6)
niTi ,...,1,
1 1 1
,g h n
i j ij j ij j ijj j g j h
G w x w x w x
, ])(,)[( jjjjjj ffeddew
. ])(,)[( ijijijijijijij ccbaabx
20
3.4 Develop membership functions (cont.)
• From Eqs.(3.5) and (3.6),we can develop Eqs.(3.7) and (3.8)as follows:
•
(3.7)
•
(3.8)
. ]))()(())((
,))()(())([(2
2
jijijijjjjijjjijij
ijijijijjjjijijijjjijj
fccbffecfecb
daabddeaabdexw
. ]))()(())((
,))()(())(([2
2
jijijijjjjijjjijij
jijijijjjjijijijjjijj
fccbffecfecb
daabddeaabdexw
21
3.4 Develop membership functions (cont.)
• When applying Eq.(3.8)to Eq.(3.4), three equations are developed:
•
(3.9)
•
(3.10)
•
(3.11)
g
1j 1 1
2
11 1 1
2
. ]))()(())((
, ))()(())(([
g
j
g
jjijijijjjjijjjijij
g
jijij
g
j
g
j
g
jijijjjjijijijjjijj
fccbffecfecb
daabddeaabdexw
h
1j 1 1
2
11 1 1
2
. ]))()(())((
, ))()(())(([
g
h
gj
h
gjjijijijjjjijjjijij
h
gjijij
h
gj
h
gj
h
gjijijjjjijijijjjijj
fccbffecfecb
daabddeaabdexw
n
1j 1 1
2
11 1 1
2
. ]))()(())((
, ))()(())(([
h
n
hj
n
hjjijijijjjjijjjijij
n
hjijij
n
hj
n
hj
n
hjijijjjjijijijjjijj
fccbffecfecb
daabddeaabdexw
22
k
hjjiji
h
gjjiji
g
jjiji
k
hjjiji
h
gjjiji
g
jjiji
k
hjjiji
h
gjjiji
g
jjiji
k
hjijijjjjiji
h
gjijijjjjiji
g
jijijjjjiji
k
hjjjijiji
h
gjjjijiji
g
jjjijiji
k
hjijijjjjiji
h
gjijijjjjiji
g
jijijjjjiji
k
hjijijjji
h
gjijijjji
g
jijijjji
fcQ
fcQfcQ
ebPebP
ebPdaO
daOdaO
cbffecDcbffecD
cbffecDfecbC
fecbCfecbC
abddeaBabddeaB
abddeaBabdeA
abdeAabdeA
13
12
11
13
12
11
13
12
11
13
12
11
13
12
11
13
12
11
13
12
11
)()( )()(
))((
))(( ))((
))((
))(( ))((
:Assume
23
3.4 Develop membership functions (cont.)
• By applying the above equations, Eqs.(3.9)-(3.11) can be arranged as Eqs.(3.12)-(3.14)as follows:
• (3.12)
• (3.13)
• (3.14)
. ],[ 112
1112
11
iiiiiiij
g
jj QDCOBAxw
. ],[ 222
2222
21
iiiiiiij
h
gjj QDCOBAxw
. ],[ 332
3332
31
iiiiiiij
n
hjj QDCOBAxw
24
3.4 Develop membership functions (cont.)
• Applying Eqs.(3.12)-(3.14) to Eq.(3.4) to produce Eq.(3.15):
(3.15)
• The left and right membership function of can be obtained as shown in Eq. (3.16) and Eq. (3.17) as follows:
(3.16)
If
(3.17)
If
iG
21 2 3 1 2 3 1 2 3
21 2 3 1 2 3 1 2 3
[( ) ( ) ( ) ,
( ) ( ) ( )] .
i i i i i i i i i i
i i i i i i i i i
G A A C B B D O O Q
C C A D D B Q Q O
; 321321 iiiiii PPPxQOO
. 321321 iiiiii OQQxPPP
a
122
1 2 3 1 2 3 1 2 3 1 2 3
1 2 3
( ) [( ) 4( )( ( ))]( )
2( )i
L i i i i i i i i i i i iG
i i i
B B D B B D A A C x O O Qf x
A A C
122
1 2 3 1 2 3 1 2 3 1 2 3
1 2 3
( ) [( ) 4( )( ( ))]( )
2( )i
R i i i i i i i i i i i iG
i i i
D D B D D B C C A x Q Q Of x
C C A
25
3.5 Rank fuzzy numbers
• In this research, Chen’s maximizing set and minimizing set (1985) is applied to rank all the final fuzzy evaluation values.
• Definition 1.
The maximizing set M is defined as:
(3.18)
The minimizing set N is defined as:
(3.19)
where usually k is set to 1.
. otherwise,0
, ,)( maxminminmax
min xxxxx
xx
xf i
i
RkR
M
, otherwise,0
, ,)( maxminmaxmin
maxxxx
xx
xx
xf i
i
LkL
N
, infmin Sxx
, supmax Sxx
, 1 ini SS , }0)({ xfxS
iAi
26
3.5 Rank fuzzy numbers (cont.)
Definition 2.
The right utility of is defined as:
(3.20)
The left utility of is defined as:
(3.21)
The total utility of is defined as:
(3.22)
iA
iA
. ~1, ))()((sup)( nixfxfAUiAM
xiM
. ~1, ))()((sup)( nixfxfAUiAM
xiN
iA
. ~1, ))(1)((2
1)( niAUAUAU iNiMiT
28
3.5 Rank fuzzy numbers (cont.)
• Applying Eqs.(3.16)~ (3.22), the total utility of fuzzy number can be obtained as:
(3.23)
)(2
))]()((4)[()([
2
1
321
2
1
3213212
321321
iii
iiiRiiiiiiiii
ACC
OQQxACCBDDBDDI
. ])(2
))]()((4)[()(1
321
2
1
3213212
321321
iii
iiiLiiiiiiiii
CAA
QOOxCAADBBDBBi
, ~1, ))(1)((2
1)( niGUGUGU iNiMiT
iG
29
3.5 Rank fuzzy numbers (cont.)
• is developed as follows:
Assume:
(3.24)
(3.25)
iRx
)(2
)(
321
321
minmax
min
iii
iiiR
ACC
BDD
xx
xxi
.)(2
))]()((4)[(
321
2
1
3213212
321
iii
iiiRiiiiii
ACC
OQQxACCBDDi
)))(()(2( maxmin321maxminmin321 xxBDDxxxACCx iiiiiiRi
2maxmin321minmax ))[(( xxBDDxx iii
.)(2/)])((4 3212
1
321min321 iiiiiiiii ACCOQQxACC
30
3.5 Rank fuzzy numbers (cont.)
• is developed as follows:
Assume:
(3.26)
(3.27)
iLx
)(2
)(
321
321
maxmin
max
iii
iiiL
CAA
DBB
xx
xxi
.)(2
))]()((4)[(
321
2
1
3213212
321
iii
iiiLiiiiii
CAA
QOOxCAADBBi
)))(()(2( minmax321minmaxmax321 xxDBBxxxCAAx iiiiiiLi
2minmax321maxmin ))[(( xxDBBxx iii
. )(2/)])((4 3212
1
321max321 iiiiiiiii CAAQOOxCAA
31
NUMERICAL EXAMPLE
• 4.1 Ratings of alternatives versus qualitative
criteria
• 4.2 Normalization of quantitative criteria
• 4.3 Averaged weights of criteria
• 4.4 Development of membership function
• 4.5 Defuzzification
32
NUMERICAL EXAMPLE
• Assume that a logistics company is looking for a suitable city to set
up a new distribution center.
• Suppose three decision makers, D1, D2 and D3 of this company is
responsible for the evaluation of three distribution center
candidates, A1, A2 and A3.
33
Criteria
Qualitative criteria Quantitative criteria
Benefit Benefit Cost
Expansion possibility
Availability of acquirement
material
Closeness to demand market
Human resource
Square measure of area
Investment cost
Figure 4-1. Selected criteria
34
Unimportant (0.00,0.00,0.25)
Less important (0.00,0.25,0.50)
Important (0.25,0.50,0.75)
More important (0.50,0.75,1.00)
Very important (0.75,1.00,1.00)
Very low (VL) /Very difficult (VD) /Very far (VF) (0.00,015,0.30)
Low (L)/Difficult (D)/Far (F) (0.15,0.30,0.50)
Medium (M) (0.30,0.50,0.70)
High (H)/Easy (E)/Close (C) (0.50,0.70,0.85)
Very high (VH)/Very easy (VE)/Very close (VC) (0.70,0.85,1.00)
Table 4-1 Linguistic values and fuzzy numbers for importance weights
Table 4-2 Linguistic values and fuzzy numbers for ratings
35
4.1 Ratings of alternatives versus qualitative criteria
• Ratings of distribution center candidates versus qualitative criteria are given by decision makers as shown in Table 4-3. Through Eq. (3.1), averaged ratings of distribution center candidates versus qualitative criteria can be obtained as also displayed in Table 4-3.
Candidates Criteria D1 D2 D3Averaged
Ratings
A1
C1 VH H VH (0.63,0.80,0.95)C2 VE E M (0.50,0.68,0.85)C3 C VC VC (0.63,0.80,0.95)C4 M H H (0.43,0.63,0.80)
A2
C1 VH VH H (0.63,0.80,0.95)C2 M M E (0.37,0.57,0.75)
C3 C C VC (0.57,0.75,0.90)C4 VH VH VH (0.70,0.85,1.00)
A3
C1 L L H (0.27,0.43,0.62)C2 VE E VE (0.63,0.80,0.95)C3 M M C (0.37,0.57,0.75)C4 L M H (0.32,0.50,0.68)
36
4.2 Normalization of quantitative criteria
• Evaluation values under quantitative criteria are objective. Suppose values of distribution center candidates versus quantitative criteria are present as in Table 4-4.
Table 4-4 Values of distribution center candidates versus quantitative
criteria
CriteriaDistribution Center Candidates
Units
A1 A2 A3
C5 100 80 90 hectare
C6 2 5 10 million
37
4.2 Normalization of quantitative criteria (Cont.)
• According to Eq. (3.2), values of alternatives under benefit and cost quantitative criteria can be normalized as shown in Table 4-5.
Table 4-5 Normalization of quantitative criteria
CriteriaDistribution Center Candidates
A1 A2 A3
C5 1 0.8 0.9
C6 1 0.4 0.2
38
4.3 Averaged weights of criteria
• The linguistic values and its corresponding fuzzy numbers, shown in Table 4-1, are used by decision makers to evaluate the importance of each criterion as displayed in Table 4-6. The average weight of each criterion can be obtained using Eq. (3.3) and can also be shown in Table 4-6.
Table 4-6 Averaged weight of each criterion
D1 D2 D3 Averaged weights
C1 MI VI IM (0.50,0.75,0.92)
C2 IM MI LI (0.25,0.50,0.75)
C3 LI LI VI (0.25,0.53,0.67)
C4 UI IM VI (0.33,0.50,0.67)
C5 MI VI IM (0.50,0.75,0.92)
C6 VI VI VI (0.75,1.00,1.00)
• Apply Eqs. (3.4)-(3.15) and g=4, h=5, n=6 to the numerical example to produce the following values
For each candidate as displayed in Table 4-7
39
1, 2, 3,i i iA A A 1, 2, 3,i i iB B B 1, 2, 3,i i iC C C
1, 2, 3,i i iD D D 1, 2, 3,i i iO O O 1, 2, 3,i i iP P P 1, 2, 3,i i iQ Q Q
Table 4-7 Values for 1, 2, 3,i i iA A A 1, 2, 3,i i iB B B 1, 2, 3,i i iC C C 1, 2, 3,i i iD D D 1, 2, 3,i i iO O O 1, 2, 3,i i iP P P 1, 2, 3,i i iQ Q Q
A1 A2 A3
Ai1 0.17 0.17 0.17
Ai2 0.00 0.00 0.00
Ai3 0.00 0.00 0.00
Bi1 0.77 0.76 0.62
Bi2 0.25 0.16 0.23
Bi3 0.25 0.10 0.05
Ci1 0.12 0.12 0.12
Ci2 0.00 0.00 0.00
Ci3 0.00 0.00 0.00
A1 A2 A3
Di1 -1.11 -1.11 -1.08
Di2 -0.17 -0.13 -0.15
Di3 0.00 0.00 0.00
Oi1 0.74 0.78 0.49
Oi2 0.50 0.40 0.45
Oi3 0.75 0.30 0.15
Pi1 1.68 1.71 1.28
Pi2 0.75 0.60 0.68
Pi3 1.00 0.40 0.20
A1 A2 A3
Qi1 2.68 2.70 2.23
Qi2 0.92 0.73 0.83
Qi3 1.00 0.40 0.20
• the calculation values for are shown in table 4-8.
40
1 2 3,i i iA A C 1 2 3,i i iB B D 1 2 3,i i iO O O
1 2 3,i i iC C A 1 2 3,i i iD D B 1 2 3,i i iP P P 1 2 3,i i iQ Q O
Table 4-8 Values for 1 2 3,i i iA A C 1 2 3,i i iB B D 1 2 3,i i iO O O 1 2 3,i i iC C A 1 2 3,i i iD D B
1 2 3,i i iP P P 1 2 3,i i iQ Q O
A1 A2 A3
Ai1+Ai2-Ci3 0.17 0.17 0.17
Bi1+Bi2-Di3 1.02 0.92 0.84
Oi1+Oi2-Qi3 0.24 0.78 0.74
Ci1+Ci2-Ai3 0.12 0.12 0.12
Di1+Di2-Bi3 -1.53 -1.34 -1.28
Pi1+Pi2-Pi3 1.43 1.91 1.75
Qi1+Qi2-Oi3 2.84 3.13 2.91
41
4.4 Development of membership function
• Through Eqs. (3.17) - (3.18) , the left, , and right, , membership functions of the final fuzzy evaluation value, , of each distribution center candidate can be obtained and displayed in Table 4-9.
)( xf LGi
)( xf RGi
niGi ,...,1,
Table 4-9 Left and right membership functions of Gi
42
4.5 Defuzzification
• By Eqs. (3.23)-(3.28) in Model development for defuzzification, the total utilities, and can be obtained and shown in Table 4-10.
Table 4-10 Total utilities and
Then according to values in Table 4-10, candidate A2 has the largest total utility, 0.551. Therefore becomes the most suitable distribution center candidate.
),( iT GU iRxiLx
),( iT GUiRx
iLx
2A
Alternatives T1 T2 T3
1.97 2.26 2.12
1.39 1.40 1.33
0.315 0.551 0.517
iRx
iLx
)( iT GU
43
CONCLUSIONS
• A fuzzy MCDM model is proposed for the evaluation and selection of the locations of distribution centers
• Chen’s maximizing set and minimizing set is applied to the model
in order to develop ranking formulae.
• Ranking formulae are clearly developed for better executing the decision making
• A numerical example is conducted to demonstrate the computational procedure and the feasibility of the proposed model.