application of fuzzy goal programming & f-promethee

J. Basic. Appl. Sci. Res., 3(2)1115-1127, 2013

© 2013, TextRoad Publication

ISSN 2090-4304 Journal of Basic and Applied

Scientific Research www.textroad.com

*Corresponding Author: Masoud Tavakoli, MSC, Department of Industrial Engineering, Arak Branch, Islamic Azad University, Arak, Iran. Email: [email protected]

Application of Fuzzy Goal Programming & F-PROMETHEE Approaches in Evaluating and Selecting the Best

Suppliers in Supply Chain

Masoud Tavakoli1*, Akbar Alam Tabriz2, Reza Farahani3, Ehsan Rezapour1

1MSC, Department of Industrial Engineering, Arak Branch, Islamic Azad University, Arak, Iran

2Associate Professor, Department of Industrial Management, Shahid Beheshti University, Tehran, Iran 3Young Researchers Club, Arak Branch, Islamic Azad University, Arak, Iran

ABSTRACT

The goal of this paper is offering a decision making style for selection of suppliers of supply chain system. During the recent years, selection of suitable suppliers of supply chain was a key subject. Since correct selection of suppliers can decrease costs and increase credit and profit of companies, thus those companies deal with suppliers and contractors must notice this subject and become familiar with suppliers selection techniques. To select suitable suppliers, many qualitative and quantitative criteria such as price, quality, flexibility, delivery performance,… must be considered. In this paper, at first we determine criteria indicated by experts for selection of suitable suppliers. Then we obtain weight of each criterion by AHP method, Goal Programming Approach, and related software such as LINDO. Then we rank the suppliers by PROMETHEE method. All calculations are done by fuzzy logic. This research was implemented in HEPCO Co. at Arak industrial city in order to evaluate and select the best suppliers. KEYWORDS: Decision Making, Suppliers Selection, Supply Chain, AHP, Goal Programming, PROMETHEE,

Fuzzy Logic.

1. INTRODUCTION Supply chain management is a process of planning, implementing, and controlling the

operations of the supply-chain network catering to the requirements of customers (purchasers) as efficiently as possible (Bhattacharya, Geraghty, Young, 2010).

Today, SCM has become the most important area for both the factory owners and researchers. Success in SCM is the most important point for achieving growth and progress between other competitors. The main task of SCM is selecting the best supplier to diminish product process cost, diminish the risk while improving supply chain quality, and to maintain a long period of cooperation with suppliers (Khaleie, Fasanghari, Tavassoli, 2012).

The supplier selection process is the most significant variable in the effective management of modern supply-chain networks as it helps in achieving high quality products and customer satisfaction (Gonza ı̌lez, Quesada, Monge, 2004). Therefore, companies have to find the most suitable suppliers to achieve competitive advantages. Researchers have proposed various methods for selecting suppliers. An approach is offered in this paper to evaluate and select suppliers by FGP, PROMETHEE. This approach ranks suppliers by inputs and outputs. This research was implemented in HEPCO Co. in Arak industrial city in order to evaluate and select the best suppliers. The scientific goals of this research are recognizing suitable criteria to evaluate and select the best suppliers in supply chain.

Finding suitable criteria to evaluate and select suppliers and weights of criteria by FGP method. Finding and ranking best suppliers by fuzzy PROMETHEE approach

1.1. Examination of the first goal of research

One of the most important and complex steps in evaluation and selection of suitable suppliers is determination of selection criteria. If selection is not precise, the final result may not be correct and will incur huge costs to the company. To avoid this, at first a complete and integrated list of effective criteria must be provided.

The list of supplier selection criteria in this research was provided by distributing questionnaires between company experts, library studies, databases, and internet sites.

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Tavakoli et al., 2013

Regarding to the excess amount of criteria, we were obliged to select the most important ones to decrease time and cost. To do this, a brainstorming session was held by presence of experts. Finally four criteria were selected by consensus. The expert team comprised of 4 higher managers of HEPCO Co. plus the researcher. These 4 managers were selected so as to be familiar with company needs and evaluation and selection of suppliers.

In this research, different methods were used to determine weights of suppliers evaluation criteria, from which FAHP method was selected. This method is used when decision-making has multiple competitor criteria. These criteria may be quantitative or qualitative, and the base of this method is on pair comparisons.

FAHP usually is solved by developmental analysis method. Sometimes this method has unsuitable results, so that sometimes this method concludes granting zero weights to low-important factors. FGP method was used in this research to remove this failure. FGP has none of weaknesses of FAHP method and has precise and reliable results. 1.2. Examination of the second goal of research

The second goal of research is finding existing potential suppliers and ranking them according to the criteria. 4 suppliers with most purchase were selected by consensus of experts of HEPCO Co. Then PROMETHEE method was used to rank them. This is a decision support method that has evolved ranking methods. Fuzzy PROMETHEE creates reliable and accessible results without need to excess data. Success of this method is substantially because of its mathematical properties and easy application.

2. LITERATURE REVIEW This section has three subsections, Fuzzy set theory, The AHP method, The goal programming approach and The PROMETHEE approach. 2.1. Fuzzy set theory

Fuzzy set theory was developed to extract primary possible outcomes from information expressed in vague and imprecise terms (Zadeh,1965). A fuzzy set is defined by a membership function used to map an item onto an interval [0,1] that can be associated with linguistic terms (Lee, Hong, &Wang, 2008). A triangular fuzzy number (TFN), a special case of a trapezoidal fuzzy number, is a very popular tool in fuzzy applications. According to the definition by Laarhoven and Pedrycz (1983), a TFN should possess the following features:

Definition 1: A fuzzy number A on X is a TFN if its membership function A (x) : X [0,1] equals:

( 1) / ( 1), 1( ) ( ) / ( ),

0A

x m x mx u x u m m x u

otherwise

(1)

Where l and u are for the lower and upper bounds of fuzzy number A , respectively, and m is median value.

A FTN is denoted as ( , , )A l m u and the following are the operational laws of two TFNs, 1 1 1 1( , , )A l m u

and 2 2 2 2( , , )A l m u , derived as (Kaufmann & Cupta, 1988, 1991): Fuzzy number addition (+):

1 2 1 1 1 2 2 2

1 2 1 2 1 2

( ) ( , , )( )( , , )( , , )

A A l m u l m ul l m m u u

(2)

Fuzzy number subtraction (–):

1 2 1 1 1 2 2 2

1 2 1 2 1 2

( ) ( , , )( )( , , )( , , )

A A l m u l m ul l m m u u

(3)

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Fuzzy number multiplication ():

1 2 1 1 1 2 2 2 1 2 1 2 1 2( ) ( , , )( )( , , ) ( , , )0, 0, 0

i i i

A A l m u l m u l l m m u ufor l m u

(4)

Fuzzy number division (/):

1 2 1 1 1 2 2 2 1 2 1 2 1 2(/) ( , , )(/)( , , ) ( / , / , / )0, 0, 0 i i i

A A l m u l m u l l m m u ufor l m u

(5)

Notably, the computational results of Eqs. (4) and (5) are not TFNs; however, these computational results can be approximated by TFNs. This study adopts a triangular fuzzy number, which is the most common membership function shape. In this research, if W=(l,m,u) is a positive triangular fuzzy number, then its precise defuzzinated value is obtained from:

[( ) ( )]3

L U L M Li i i i i

iW W W W WW

(6)

Views of experts are shown in the table.1 by linguistic criteria and converting them to the triangular fuzzy numbers.

TABLE1: TRIANGULAR FUZZY NUMBERS FOR LINGUISTIC VARIABLES Row Linguistic scale Symbol Triangular fuzzy

number Reverse triangular fuzzy

number 1 Equal importance VL (2,1,1) (1,1,1.2) 2 Nearly important L (5,3,1) (1.1,1.1,3.5) 3 Very important M (7,5,3) (1.1,3.1,5.7) 4 Very very important H (9,7,5) (1.1,5.1,7.9) 5 Absolutely important VH (8,9,9) (1.1,9.1,9.8)

2.2. The AHP method

The weights of attributes are calculated by means of AHP method developed by Saaty (1990). The procedure of AHP weighting can be summarized as follows:

Firstly, pairs of elements of the n-attribute hierarchical framework are compared within pairwise comparison matrixes A. according to Eq. 7:

11 12 1

21 22 2

1 2

1/ , 1, , 1, 2,...,

n

n

n n nn

ij ji ii

a a aa a a

A

a a a

a a a i j n

where, aij can be interpreted as the degree of preference of i-th attribute over j-th attribute; and vice versa. Secondly, each column of the pairwise comparison matrix is divided by sum of entries of the corresponding column to obtain the normalized comparison matrix. The eigenvalues i of this matrix would give the relative weight of attribute i.

Finally, the obtained relative weight vector is multiplied by the weight coefficients of the elements at the higher levels, until the top of the hierarchy is reached. The result is global weight vector W of the attributes and can be shown as below:

(7)

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1

2

n

ww

W

w

Since the comparison is based on the subjective evaluation, a consistency ratio is required to ensure the selection accuracy. The consistency index (CI) of the comparison matrix is computed as follows:

max( ) / ( 1)CI n n (9) Where max is the highest eigenvalue of the pairwise comparison matrix. The closer the inconsistency index is to zero, the greater the consistency so the relevant index should be lower than 0.10 to accept the AHP results as consistent (Saaty1990). 2.3. The Goal programming approach

A goal programming (GP) model is useful in dealing with multi-criteria decision problems where the goals cannot simultaneously be optimized. GP allows decision makers to consider several objectives together in finding a set of acceptable solutions and to obtain an optimal compromise. It was first introduced by Charnes and Cooper (1961), and further developed by Lee (1972), Ignizio (1985), and many others (Tamiz, Jones, & Romero, 1998; Chang, 2007). The purpose of GP is to minimize the deviations between the achievement of goals and their aspiration levels (Chang, 2007). Sharma, Benton, and Srivastava (1989) proposed a GP formulation for vendor selection to attain goals pertaining to price, quality and lead-time under demand and budget constraints. Buffa and Jackson (1983) also proposed the use of GP for price, quality and delivery objectives to evaluate vendors. An integrated AHP and preemptive goal programming based multi-criteria decision making (MCDM) methodology is developed by Wang, Huang, and Dismukes (2004) to select the best set of multiple suppliers to satisfy capacity constraint.

Determining precisely the goal value of each objective is difficult for decision makers since possibly only partial information can be obtained (Chen & Tsai, 2001). Some approaches, such as probability distribution, penalty function fuzzy numbers and various types of thresholds, are used to reformulate the GP models in order to incorporate uncertainty and imprecision into the formulation (Chen & Tsai, 2001). Narasimhan (1980) was the first to propose fuzzy goal programming (FGP) by using the fuzzy set theory with preference based membership function to GP. Since then, many achievements have been made in areas of preemptive FGP, weight additive model and stochastic model (Chang, 2007). Some researchers have investigated FGP regarding the problem formulation, the relative importance and the fuzzy priority of the fuzzy goals, and associated solution algorithms (Chen & Tsai, 2001). A review of the past researches on FGP is done by Chen and Tsai (2001) and Chang (2007).

Kim and Whang (1998) investigated the application of tolerance concepts to goal programming in a fuzzy environment by formulating a FGP problem with unequal weights as a single linear programming problem with the concept of tolerance. The model could reflect the decision maker’s view on subjective fuzzy business goals based on his/her experience or intuition. Chen and Tsai (2001) formulated FGP by “incorporating different importance and preemptive priorities by using an additive model to maximize the sum of achievement degrees of all fuzzy goals.” The approach allowed the decision maker to determine a desirable achievement degree for each fuzzy goal and to reflect explicitly the relative importance of these goals. Kumar et al. (2004) presented a fuzzy goal programming approach that considered multiple objectives and dealt with some of the parameters that were fuzzy in nature. A fuzzy mixed integer goal programming was formulated. Three primary goals are minimizing the net cost, minimizing the net rejections, and minimizing the net late deliveries, while the constraints are regarding buyer’s demand, vendors’ capacity, vendors’ quota flexibility, purchase value of items, budget allocation to individual vendor, etc.

Chang (2007) proposed an MCGP approach to solve a multi choice aspiration level (MCAL) problem, in which decision makers can set more aspiration levels to each goal of the multiple

(8)

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objective decision- making problem to find more appropriate resources so as to reach a higher aspiration level in the initial stage of the solution process. The approach is applicable when there is a goal that can be achieved from some specific aspiration levels (i.e., one goal mapping many aspiration levels) (Chang, 2007). The achievement function of MCGP is (Chang, 2007).

1

1

min ( )

. . ( ) ( ), 1,2,...,

0, 1,2,...,( ) ( ), 1,2,...,

n

i i ii

m

i i i ij ijj

i i

ij i

w d d

s t f x d d g S B i n

d d i nS B U x i n

XF (F is a feasible set) Where di is the deviation from the target value gi; wi represents the weight attached to the deviation;

max(0, ( ) )i i id f X g and max(0, ( ))i i id g f X are respectively over- and under-achievements of the i-th goal; Sij(B) represents a function of binary serial number; and Ui(x) is the function of resources limitations. For something that is more/higher the better in the aspiration levels, the highest possible value of membership function is 1, based on the fuzzy theory (Charnes & Cooper, 1961). To achieve the maximization of gijSij(B), the flexible membership function goal with aspiration level 1 (i.e. the highest possible value of membership function) is as follows (Chang, 2007):

min

max min

( )1ij ij

i i

g S B gd d

g g

where gmax and gmin are, respectively, the upper and lower bounds of the right-hand side (i.e. aspiration levels) of Eq. (10). For easy calculation, the fractional form of Eq. (11) is:

min1 1( ) 1ij ij i i

i i

g S B g d dL L

where Li = gmax – gmin .

For something that is more/higher the better in the aspiration levels, the highest possible value of membership function is 1, based on the fuzzy theory (Charnes & Cooper, 1961). To achieve the maximization of gijSij(B), the flexible membership function goal with aspiration level 1 (i.e. the highest possible value of membership function) is as follows (Chang, 2007):

min

max min

( )1ij ij

i i

g S B gd d

g g

where gmax and gmin are, respectively, the upper and lower bounds of the right-hand side (i.e. aspiration levels) of Eq. (10). The fractional form of Eq. (13) can also be converted into a polynomial form as Eq.(14).

max1 1 ( ) 1ij ij i i

i i

g g S B d dL L

2.4. The PROMETHEE approach 2.4.1. Theoretical basis: In PROMETHEE the information within each criterion, the preference structure, is based on pairwise comparisons (Brans & Mareschal, 2005). The deviation between the evaluations of two

(10)

(11)

(12)

(13)

(14)

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alternatives on a particular criterion is considered. For small deviations, the decision maker allocates a small preference to the best alternative or possibly no preference if the deviation is negligible. The larger deviation is the larger preference. These preferences may vary between 0 and 1. Thus, for each criterion, there is a function:

( , ) ( ) ( ) , , j j j jP a b F g a g b a b A (15)

for which: 0 ( , ) 1jP a b

Expression (1) gives the preference of a over b for observed deviations between their evaluations on criterion gj(.), in case of a criterion to be maximized. For criteria to be minimized, the preference function should be reversed:

( , ) ( , ), , j j jP a b F d a b a b A (16)

A general criterion gj(.) Pj (a,b) has to be defined for each criterion of type of … preference functions are usually proposed (Brans & Mareschal, 2005), as shown in Table 1. However, other types of generalized criteria may also be considered.

Parameter q is the indifference threshold and represents the largest deviation which is considered as negligible by the decision maker. Parameter p is the preference threshold representing the smallest deviation which is considered as sufficient to generate a full preference. In the Gaussian criterion the preference function is monotonically increasing for all deviations and has no discontinuities. In this case, parameter s defines the inflection point of the preference function.

A number of variations of the PROMETHEE method have been developed. PROMETHEE I provides a partial ranking, including possible incomparabilities, while PROMETHEE II shows a complete ranking of alternatives. PROMETHEE III and PROMETHEE IV provide a ranking based on intervals and in a continuous case, respectively. PROMETHEE V extends the application of PROM- ETHEE II to the problem of selection of several options, given a set of constraints, and PROMETHEE VI aims to represent the human brain. The geometrical analysis for interactive aid (GAIA) module provides a geometrical representation of the results obtained by the PROMETHEE methodology. 2.4.2. PROMETHEE 1 methodology

For the application of PROMETHEE two major actions are needed: the calculation of aggregated preference indices and outranking flows. The aggregate preference indices (a,b) expresses with which degree a is preferred to be over all the criterian, while (b,a) indicates how b is preferred to a. Assuming a,bA:

1

1

( , ) ( , )

( , ) ( , )

k

j jj

k

j jj

a b P a b w

b a P b a w

(17)

(a,b)0 implies a weak global preference of a over b, while (a,b)1 implies a strong global preference of a over b (Brans & Mareschal, 2005). Each alternative faces (n-1) other alternatives in A. Two outranking flows can be defined: (i) The positive outranking flow: 1( ) ( , )

1 x A

a a xn

(18)

representing how an alternative a is outranking all the others. The higher +(a), the better the alternative. (ii) The negative outranking flow: 1( ) ( , )

1 x A

a x an

(19)

expressing how an alternative a is outranking all the others. The lower –(a), the better the alternative. The PROMETHEE I partial ranking is obtained from the positive and negative outranking flows:

1120


( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) (

l

l

l

a b and a b

aP b if a b and a b

a b and a b

al b if a b and a b

a b and a baR b if

a b and a

) ( )b

(20)

where Pl, ll, and Rl represent the preference, inference and incomparability, respectively. In the case of aRlb, a higher power of one alternative is associated to a lower weakness of the other. In this case, the information provided by both flows is not consistent and both alternatives are considered incomparable. 2.4.3. PROMETHEE 2 methodology

In case a complete ranking is necessary, then PROMETHEE II should be chosen. In this case, the net outranking flow is defined as the balance between the positive and the negative outranking flows:

( ) ( ) ( )a a a (21) In PROMETHEE II, all the alternatives are comparable, no incomparabilities remain (Brans & Mareschal, 2005). When (a) > (b), “a” outranks “b” on all the criteria.

TABLE 2: TYPES OF GENERALIZED CRITERIA Type Generalized criterion Definition Parameter

Type 1: Usual criterion

0 0( )

1 0

dP d

d

–

Type 2: Quad criterion

0( )

1

d qP d

d q

q

Type 3: Criterion with linear preference

0 0( ) / 0

1 0

dP d d p d p

d

p

Type 4: Level criterion

0( ) 1/ 2

1

d qP d q d p

d p

p,q

Type 5: Criterion with linear preference and indifference area

0

( )

1

d qd p

P d q d pd p

d p

p,q

Type 6: Gaussian criterion

2

2

0 0( )

1 exp( 02

)

dP d zd d

d

s

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3. PROPOSED METHOLOGY In this research, the proposed methodology is:

Methodology of this research includes 2 phases. Each phase includes 2 steps, which is described below. Phase 1: Finding suitable criteria to select the best suppliers and determination of weights of criteria Stage 1: Finding suitable criteria to select the best suppliers To find suitable criteria to select the best suppliers, views of experts in part supply subject were collected by questionnaires. Thus, 30 questionnaires were distributed among staff of HEPCO Co. After analyzing the questionnaires and obtaining 47 criteria, 4 criteria for selection and evaluation of suppliers were indicated in a brainstorming session. These criteria are:

1. Total production cost (C1) 2. Product quality (C2) 3. Delivery performance (C3) 4. Premium feature (C4)

Stage 2: Finding weights of criteria by FAHP and FGP approach This stage includes 7 steps:

1. Step 1: Formation of pair comparison matrix of criteria based on linguistic variables 2. Step 2: Formation of pair comparison matrix of criteria based on triangular fuzzy numbers 3. Step 3: Separation of pair comparison matrix into three non-fuzzy matrices AL, AM, and AU 4. Step 4: Defining abbreviated signs to simplify FGP model 5. Step 5: Writing final FGP model regarding to the 3 types of limitations 6. Step 6: Solving FGP model by Lindo software and finding fuzzy weights of criteria 7. Step 7: Defuzination of calculated fuzzy weights and obtaining absolute weights

Phase 2: Finding and ranking best suppliers Stage 1: Finding best existing potential suppliers In this research, 4 suppliers with most purchases were selected by consensus of experts:

1. Volvo (A1) 2. Newholland (A2) 3. Kumatsu (A3) 4. Libher (A4)

Stage 2: Ranking suppliers by fuzzy PROMETHEE approach This stage includes 5 steps, which will be described in case study:

1. Step 1: Formation of decision-making matrix 2. Step 2: Defining priority function 3. Step 3: Calculating general ranking (a,b) 4. Step 4: Calculating + and – and partial ranking

Finding suitable criteria to select best suppliers

Finding weights of criteria by FAHP and FGP approach

Recognizing existing potential suppliers

Ranking suppliers by fuzzy PROMETHEE approach

FIG 1. PROPOSED MODEL

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5. Step 5: Calculating and general ranking of suppliers

FIG.2. HIERARCHICAL PROCESS OF SELECTION OF SUPPLIERS

4. CASE STUDY Here the steps of the proposed methodology are described by a case study in HEPCO Co. 4.1. Finding weights of criteria by FAHP and FGP approach 4.1.1 Formation of pair comparison matrix of criteria based on linguistic variables

TABLE3. PAIR COMPARISON MATRIX OF CRITERIA (LANGUAGE VARIABLE) Total production

cost (C1)

Quality of product (C2)

Delivery performance

(C3)

Premium feature (C4)

Total production cost (C1)

– Very important Very very important

Nearly important

Quality of product (C2)

Very important – Absolutely important

Absolutely important

Delivery performance (C3)

Very very important Absolutely important – Very important


Nearly important Absolutely important Very important –

4.1.2. Formation of pair comparison matrix of criteria based on fuzzy triangular numbers

12 12 12 1 1 1

21 21 21 2 2 2

1 1 1 2 2 2

1 ( , , ) ( , , )...( , , ) 1 ... ( , , )

( )

( , , ) ( , , ) ... 1

n n n

n n nz n n

n n n n n n

l m u l m ul m u l m u

A a

l m u l m u

TABLE4. PAIR COMPARISON MATRIX FOR WEIGHTS OF INDICES (TRINGULAR NO)

C1 C2 C3 C4 C1 (1,1,1) (1.7,1.5,1.3) (5,7,9) (1,3,5) C2 (3,5,7) (1,1,1) (8,9,9) (8,9,9) C3 (1.9,1.7,1.5) (1.9,1.9,1.8) (1,1,1) (1.5,1.3,1) C4 (1.5,1.3,1) (1.9,1.9,1.8) (1,3,5) (1,1,1)

Selection of best supplier

C1 Production

cost

C2 Product quality

C4 Premium feature

C3 Delivery performance

A1 Volvo

A2 Newholland

A3 Kumatsu

A4 Libher

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4.1.3. Separation of pair matrix into three non-fuzzy matrixes AL, AM, and AU

12 1

21 2

1 2

1 ...1 ...

...

... 1

n

nL

n n

l ll l

A

l l

12 1

21 2

1 2

1 ...1 ...

...

... 1

n

nM

n n

m mm m

A

m m

12 1

21 2

1 2

1 ...1 ...

...

... 1

n

nU

n n

U UU U

A

U U

1.00 0.14 5.00 1.003.00 1.00 8.00 8.000.11 0.11 1.00 0.200.20 0.11 1.00 1.00

LA

1.00 0.20 7.00 3.005.00 1.00 9.00 9.000.14 0.11 1.00 0.330.33 0.11 3.00 1.00

MA

1.00 0.33 9.00 5.007.00 1.00 9.00 9.000.20 0.13 1.00 1.001.00 0.13 5.00 1.00

UA

4.1.4. Defining abbreviation signs to simplify FGP model (table 5).

TABLE5. ABBREVIATION SIGNS Main symbol Substituted symbol

εi+ Ei

εi- Fi

γi+ Gi

γi- Hi

δi Di Wi

L Li Wi

M Mi Wi

U Ui

4.1.5. Writing final FGP model regarding 3 types of limitations

1min ( ) ( )

nT

i i i i ii

J e E E

s.t.

( ) ( 1) 0

( ) ( 1) 0( ) 0

L U L

U L U

M M

A I W n W E E

A I W n WA nl W

1,

1,

1

1, 1, ...,

1, 1, ...,

1

nL Ui j

j j i

nU Li j

j j i

nMi

i

w w i n

w w i n

w

00

, , , , , 0

U M

M L

L

W WW W

W E E

4.1.6. Solving FGP model by Lindo software and finding fuzzy weight of criteria.

L1 0.198451 M1 0.198451 U1 0.0218026 L2 0.62817 M2 0.68385 U2 0.693751 L3 0.015047 M3 0.026971 U3 0.026371 L4 0.061853 M4 0.091329 U4 0.127433

Limitation type 1: for deviation vectors from matrix compatibility condition

Limitation type 2: for normality of fuzzy weight vectors

Limitation type 3: for relationship between lower and higher bounds of fuzzy weight vector

(22)

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TABLE6. FUZZY WEIGHT OF CRITERIA

Criterion for evaluation and selection of suppliers

Fuzzy weight (WL,WM,WU)

Production cost (C1) W1 (0.198451,0.198451,0.218026) Product quality (C2) W2 (0.628170,0.683850,0.693751)

Delivery performance (C3) W3 (0.015047,0.026371,0.026371) Premium feature (C4) W4 (0.061853,0.091329,0.127433)

4.1.7. Defuzination of calculated fuzzy weights and obtaining abstract weights (table7). With using of Eq(6).

TABLE7. THE IMPORTANCE WEIGHT OF EACH CRITERIA. Criterion for evaluation and selection of

suppliers Fuzzy weight Priority

Production cost (C1) 0.18762 Second Product quality (C2) 0.64492 First

Delivery performance (C3) 0.03244 Fourth Premium feature (C4) 0.13503 Third

4.2. Ranking suppliers by fuzzy PROMETHEE approach 4.2.1. Formation of decision-making matrix

TABLE8. LINGUISTIC VARIABLES AND THEIR CORRESPONDING FUZZY NUMBERS. Row Supplier situation Triangular fuzzy number Reverse triangular fuzzy

number 1 Worst (W) (2,1,1) (1,1,1.2) 2 Poor (P) (5,3,1) 1,1.1,3.5) 3 Fair (F) (7,5,3) (1.1,3.1,5.7) 4 Good (G) (9,7,5) (1.1,5.1,7.9) 5 Best (B) (8,9,9) (1.1,9.1,9.8)

TABLE9. THE RATING OF EACH ALTERNATIVE UNDER EACH CRITERION

Criterion Supplier Decision-makers average crisp DM1 DM2 DM3 DM4

Production cost (C1)

A1 G G F F 4 6 8 3.33 A2 G F G F 4 6 8 3.33 A3 F F F G 3.5 5.5 7.5 3.17 A4 F F F F 3 5 7 3

Product quality (C2)

A1 G G G G 5 7 9 3.67 A2 G G G G 5 7 9 3.67 A3 F G F F 3.5 5.5 7.5 3.17 A4 F G F F 3.5 5.5 7.5 3.17

Delivery performance (C3)

A1 G G G G 5 7 9 3.67 A2 G B G G 5.75 7.5 9 3.58 A3 F F F F 3 5 7 3 A4 F F F F 3 5 7 3


A1 G G G G 5 7 9 3.67 A2 F G F G 4 6 8 3.33 A3 p F F F 2.5 4.5 6.5 2.83 A4 p F F F 2.5 4.5 6.5 2.83

4.2.2. Defining priority function: Pj (a,b) calculated with Type 3: Criterion with linear preference at table2 with using Eq.13.

TABLE10. PRIORITY FUNCTION FOR SUPPLIER A1 REGARDING TO THE OTHER SUPPLIERS Criterion Supplier

C1 C2 C3 C4

A1 0 0 0 0 A2 0 0 0.025 0.102 A3 0.05 0.152 0.203 0.254 A4 0.10 0.152 0.20 0.254

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C1 C2 C3 C4

A1 0 0 -0.03 -0.10 A2 0 0 0 0 A3 0.05 0.15 0.18 0.15 A4 0.10 0.15 0.18 0.15


C1 C2 C3 C4

A1 -0.05 -0.15 -0.20 -0.25 A2 -0.05 -0.15 -0.18 -0.15 A3 0 0 0 0 A4 0.05 0 0 0


C1 C2 C3 C4

A1 -0.33 -0.50 -0.67 -0.83 A2 -0.33 -0.50 -0.58 -0.50 A3 -0.17 0 0 0 A4 0 0 0 0

4.2.3. Calculation of total priority (a,b) with using Eq.17.

TABLE14. THE MULTICRITERIA PREFERENCE INDEX. Criterion Supplier

SUM A1 SUM A2 SUM A3 SUM A4 C1 C2 C3 C4

Weight 0.18762 0.64492 0.03244 0.13503 A1 0.00 -0.01 -0.15 -0.52 A2 0.01 0.00 -0.13 -0.47 A3 0.15 0.13 0.00 -0.03 A4 0.16 0.14 0.01 0.00

4.2.4. Calculation of + and – and partial ranking with using Eq.18, 19, 21

TABLE15. THE OUTGOING/LEAVING FLOWS. Supplier A1 A2 A3 A4

Q+ 0.11 0.09 -0.09 -0.34

TABLE16. THE INCOMING/ENTERING FLOWS. Supplier A1 A2 A3 A4

Q– -0.23 -0.20 0.08 0.10

TABLE17. RANKING OF SUPPLIERS Supplier A1 A2 A3 A4

φ 0.33 0.28 -0.18 -0.44 Priority 1 2 3 4

FIG 3. RANKING OF SUPPLIERS

5. CONCLUSION AND OFFERS

Regarding to the analysis, about the results of this research we can say that since correct selection of suppliers can affect cost decrement and credit and profit increment for companies, thus those companies dealing with suppliers and contractors must have special notice to this problem and become familiar with evaluation and selection techniques of suppliers.

The result in HEPCO Co. shows that this company must use the 7 obtained criteria to evaluate its suppliers, or at least it must prioritize these 7 criteria than the others.

The second result is that HEPCO Co. must use this priority to rank its suppliers. For example, if these 7 suppliers supply a common part, HEPCO Co. must select the suppliers with higher rank.

The used models in this research are capable for decision-making of managers. However, regarding to the existing limitations in each model, we propose use these model in combination, because by this method, limitation of a model will be covered by another.

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The proposals are:

Fulfillment of this research in other organizations and comparing the results Study of risk and its calculation in decision-making problems Repetition of this research by other new methods Briefing the goal of this research and concentration on the limited goals

REFERENCES

Bhattacharya, A., Geraghty, J., & Young, P. (2010). " Supplier selection paradigm: An integrated

hierarchical QFD methodology under multiple-criteria environment". Applied Soft Computing, 10, 1013-1027.

Brans, J., & Mareschal, B. (2005). PROMETHEE methods. In J. Figueira, S. Greco, & M. Ehrgott (Eds.), Multiple criteria analysis – state of the art surveys. International series in operations research and management sciences, 78, 163–195. New York, USA: Springer.

Buffa, F.P., & Jackson, W.M. (1983). " A goal programming model for purchase planning". Journal of Purchasing and Materials Management, 27-34. Fall.

Chang, C.-T. (2007). "Multi-choice goal programming". Omega: The International Journal of Management Science, 35(4), 389-396.

Charnes, A., & Cooper, W.W. (1961). "Management Model and Industrial Application of Linear Programming". (Vol. 1). New York: Wiley.

Chen, L.-H., & Tsai, F.-C. (2001). "Fuzzy goal programming with different importance and priorities." European Journal of Operational Research, 133, 548-556.

Gonza ı̌lez, M.E., Quesada, G., & Quesada, C.A.M. (2004). "Determining the importance of the supplier selection process in manufacturing: a case study". International Journal of Physical Distribution & Logistics Management, 34 (6), 492-504.

Ignizio, J. P. (1985). " Introduction to linear goal programming". Beverly Hills: Sage University.

Kaufmann, A., & Gupta, M. M. (1991). "Introduction to fuzzy arithmetic: Theory and applications". New York: Van Nostrand Reinhold Company Inc.

Kaufmann, A., & Gupta, M.M. (1988). " Fuzzy mathematical models in engineering and management science". Amsterdam: North-Holland.

Khaleie, S., Fasanghari, M., & Tavassoli, E. (2012). " Supplier selection using a novel intuitionist fuzzy clustering approach". Applied Soft Computing, in Press.

Kim, J.S., & Whang, K.-S. (1998). "A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function. European Journal of Operational Research, 107, 614–624.

Kumar, M., Vrat, P., & Shankar, R. (2004). "A fuzzy goal programming approach for vendor selection problem in a supply chain". Computers and Industrial Engineering, 46, 69–85.

Laarhoven, P.J.M., & Pedrycz, W. (1983). " A fuzzy extension of Saaty’s priority theory". Fuzzy Sets and Systems, 11, 199-227.

Lee, Y.-C., Hong, T.-P., & Wang, T.-C. (2008). "Multi-level fuzzy mining with multiple minimum supports". Expert Systems with Applications, 34, 459-468.

Lee, S.M. (1972). "Goal programming for decision analysis". Philadelphia: Auerbach. Narasimhan, R. (1980). "Goal programming in a fuzzy environment". Decision Sciences, 11,

243-252. Saaty, T.L. (1990). " How to make a decision: the analytic hierarchy process". Eur J Oper Res, 48, 9-26. Sharma, D., Benton, W.C., & Srivastava, R. (1989). " Competitive strategy and purchasing

decisions". In Proceedings of the 1989 annual conference of the decision sciences institute, 1088-1090.

Tamiz, M., Jones, D., & Romero, C. (1998). "Goal programming for decision making: an overview of the current state-of-art". European Journal of Operational Research, 111, 567-581.

Wang, G., Huang, S.H., & Dismukes, J.P. (2004). " Product-driven supply chain selection using integrated multi-criteria decision-making methodology". International Journal of Production Economics, 91, 1-15.

Zadeh, L. A. (1965). " Fuzzy sets". Information and Control, 8, 338-353.

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application of fuzzy goal programming & f-promethee

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