Supplier selection and evaluation using generalized fuzzy multi criteria decision making approach

Supplier Selection and Evaluation Using Generalized Fuzzy Multi-Criteria Decision Making Approach Luu Huu Van Department of Industrial Management National Taiwan University of Science a

Supplier Selection and Evaluation Using Generalized Fuzzy Multi-Criteria Decision Making Approach

Luu Huu Van Department of Industrial Management National Taiwan University of Science and Technology

Taipei 10607, Taiwan Email: vanluuhuu82@gmail.com Shuo-Yan Chou

Vincent F Yu Department of Industrial Management National Taiwan University of Science and Technology

Taipei 10607, Taiwan Email: vincent@mail.ntust.edu.tw

Luu Quoc Dat Department of Industrial Management Department of Development Economics

National Taiwan University of Science and Technology University of Economics and Business, Vietnam National University Taipei 10607, Taiwan Hanoi, Vietnam

Email: sychou@mail.ntust.edu.tw Email: datlq@vnu.edu.vn

Abstract—Supplier selection and evaluation plays an

importance role for companies to gain competitive advantage and

achieve the objectives of the whole supply chain To select the

appropriate suppliers, many qualitative and quantitative criteria

are needed consider in the decision process Therefore, supplier

selection and evaluation can be seem as a multi-criteria decision

making (MCDM) problem in vague environment However, most

existing fuzzy MCDM approaches have been developed using

normal fuzzy numbers or converting generalized fuzzy numbers

into normal fuzzy numbers through normalization process This

leads to a restriction in the application of the fuzzy MCDM

approaches In this study, a generalized fuzzy MCDM approach

is proposed to select and evaluate suppliers In the proposed

approach, the ratings of alternatives and important weights of

criteria are expressed in linguistic terms using generalized fuzzy

numbers Then, the membership functions of the final fuzzy

evaluation value are developed To make procedure easier and

more practical, the weighted ratings are defuzzified into crisp

values by employing the maximizing and minimizing set ranking

approach to determine the ranking order of alternatives Finally,

a numerical example is presented to illustrate the applicability

and efficiency of the proposed method

Keywords—Generalized fuzzy numbers, ranking method,

supplier selection

I INTRODUCTION Supplier selection process is one of the most important

components of production and logistics management for many

companies Selection of the appropriate supplier can

significantly lessen purchasing costs, consequently enhance the

enterprise competitiveness in the market, and increase end user

satisfaction [5] The best supplier selection can also create a

great contributes to the quality of goods for company and to

achieve the objectives to overall the supply chain performance

of organizations [4] However, selection of wrong supplier

could be enough to upset the company’s financial and

operation position

The supplier selection process mainly involves evaluation

of different alternative suppliers based on different qualitative and quantitative criteria This process is essentially considered

as a multiple criteria decision making (MCDM) problem which

is affected by different tangible and intangible criteria including price, quality, technology, flexibility, delivery, etc MCDM methods incorporate with fuzzy set theory has been used popular to solve uncertainty in the supplier selection decision process and to it provide a language appropriate to dispose imprecise criteria [1] Fuzzy MCDM approach allows decision makers to evaluate alternatives suing linguistic terms such as high, high, slightly high, medium, slightly low, low, very low or none rather than precise numerical values, allows them to express their opinions independently, and also provides and algorithm to aggregate the assessments of alternatives And the FMCDM approach offers a flexible practical and effective way of group decision making

Although numerous fuzzy MCDM approaches and applications have been investigated in literature [1, 5, 6, 9, 11], most of these approaches have been developed using normal fuzzy numbers or converting generalized fuzzy numbers into normal fuzzy numbers through normalization process [7] This leads to a restriction in the application of the fuzzy MCDM approaches Additionally, [8] pointed out that the normalization process is a serious disadvantage

In this study, a generalized fuzzy MCDM approach is proposed to select and evaluate suppliers In the proposed approach, the ratings of alternatives and importance weights of criteria are expressed in linguistic terms using generalized fuzzy numbers Then, the membership functions of the final fuzzy evaluation value are developed To make the procedure easier and more practical, the weighted ratings are defuzzified into crisp values by employing the maximizing and minimizing set ranking approach to determine the ranking order of alternatives Finally, a numerical example is presented to illustrate the applicability and efficiency of the proposed method

2016 Eighth International Conference on Knowledge and Systems Engineering (KSE)

The rest of this paper is organized as follows Section 2

briefly reviews the basic definitions and arithmetic operations

of generalized fuzzy number Section 3 develops the fuzzy

MCDM using generalized fuzzy numbers Section 4 presents a

numerical example to demonstrate the feasibility of the

proposed mode Finally, the conclusion and discussion are

presented in section 5

II PRELIMINARIES

A Trapezoidal Fuzzy Numbers

A generalized fuzzy number A = (a, b, c, d;ϖ ) is described

as any fuzzy subset of the real line R with membership

function fAthat can be generally defined as [2]:

(a) f A is a continuous mapping from R to the closed

interval [0, ],ϖ 0≤ ≤ϖ 1;

(b) f x A( ) = 0,for all x∈ −∞( ,a];

(c) f A is strictly increasing on [ , ];a b

(d) f x A( )= ϖ for all , x∈[ ]b c, ;

(e) f A is strictly decreasing on [ , ];c d

(f) f x A( ) 0,= for all x∈(d, ∞],

Where a, b, c and d are real numbers Unless elsewhere

specified, it is assumed that A is convex and bounded (i.e

a d

−∞ < < ∞

B Arithmetic Operations

[2] presented arithmetical operations between generalized

trapezoidal fuzzy numbers based on the extension principle

Let A and B are two generalized trapezoidal fuzzy

numbers, i.e.,A=( , , , ;a a a a w1 2 3 4 A)andB=( , , , ; ),b b b b w1 2 3 4 B

where a a a a b b b and 1, , , , , ,2 3 4 1 2 3 b4 are real values, 0≤w A≤ 1

and 0≤w B≤ 1

Some arithmetic operators between the generalized fuzzy

numbers A and B are defined as follows:

1) Generalized trapezoidal fuzzy numbers addition:

( ) ( , , , ; )( )( , , , ; )

( , , , ; min( , )),

where a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are real values

2) Generalized trapezoidal fuzzy numbers subtraction:

( ) ( , , , ; )( )( , , , ; )

where a a a a b b b1, , , , , ,2 3 4 1 2 3 and b4 are real values

3) Generalized trapezoidal fuzzy numbers multiplication:

Where a=Min(a b a b a1× 1, 1× 4, 4×b a1, 4×b4),

b= a ×b a ×b a ×b a ×b

d= a b a b a× × ×b a ×b

It is obvious that if a a a a b b b and 1, , , , , ,2 3 4 1 2 3 b are all positive 4

real numbers, then:

4) Generalized trapezoidal fuzzy numbers division:

The inverse of the fuzzy number B is

1/B=(1/ ,1/ ,1/ ,1/ ;b b b b w B) where b b b and 1, ,2 3 b are 4

non-zero positive numbers or all non-zero negative real numbers Let a a a a b b b and 1, , , , , ,2 3 4 1 2 3 b be non-zero positive 4

real numbers Then, the division of A and B is as follows:

(/) ( , , , ; )(/)( , , , ; ) ( / , / , / , / ;min( , )),

A B a a a a w b b b b w

a b a b a b a b w w

=

The α- cuts of fuzzy number A can be defined as (Kaufmann

and Gupta, 1991):

{ ( ) }, [0,1]

Aα= xf A x≥α α∈ , where Aα

is a non-empty bounded closed interval contained in R and can be denoted by

Aα=[Alα,Auα] whereAlα and Auα are its lower and upper bounds of the closed interval respectively

D Arithmetic Operations on Fuzzy Numbers

Given fuzzy numbers A and B, where A B R, ,∈ + the

Į-cuts of A and B are α [ α α]

u

l A A

u

l B B

respectively By the interval arithmetic, some main operations

of A and B can be expressed as follows:

(A B⊕ )α=ª¬A lα+B lα, A uα+B u৬ (5)

(A B )α =ª¬A lα−B A uα, uα−B l৬ (6)

(A B⊗ )α=ª¬A B lα⋅ lα, A B uα⋅ u৬ (7)

( )Į Į Į Į Į

l u u l

A B =ª¬A B A B, º¼ (8)

E Linguistic Values and Fuzzy Numbers

There are decision situations in which the information can not be assessed precisely in a quantitative form but may be in a qualitative one, and thus, the use of a linguistic approach is necessary The concept of linguistic variable is to provide a means of approximating characterization of ill-defined phenomena in a system or model Linguistic values are those values represented in words or sentences in natural or artificial languages, where each linguistic value can be modeled by a fuzzy set

In fuzzy set theory, conversion scales are employed to convert linguistic values into fuzzy numbers Determining the number of conversion scales is generally intuitive and subjective, in this study a five-point scale has been used to convert linguistic values into triangular fuzzy numbers (TFNs),

as introduced in TABLE 1 and TABLE 2

TABLE 1 THE LINGUISTIC TERMS AND RELATED FUZZY NUMBERS

OF EVALUATION RATINGS

Ratings

Linguistic variables TFNs

Very Low (VL) (0.0, 0.1, 0.2)

Low (L) ( 0.1, 0.3, 0.5)

Fair (F) (0.3, 0.5, 0.7)

Good (G) (0.6, 0.7, 1.0)

Very Good (VG) (0.8, 0.9, 1.0)

TABLE 2 THE LINGUISTIC TERMS AND RELATED FUZZY NUMBERS

OF CRITERIA WEIGHTS

Ratings

Linguistic variables TFNs

Unimportant (UI) (0.0, 0.1, 0.3)

Less Important (LI) ( 0.2, 0.3, 0.4)

Important (IM) (0.3, 0.5, 0.7)

More Important (MI) (0.7, 0.8, 0.9)

Very Imprortant (VI) (0.8, 0.9, 1.0)

III MODEL ESTABLISHMENT

A Aggregate the Importance Weights

Let W jt =( ,t u v jt jt, jt,ϖjt), W jt∈R+ be the importance

weights assigned by decision maker D t to criterion C j The

averaged weight w j=( , , ,t u v j j jϖjt) of criterion C assessed j

by committee of k decision makers can be evaluated as:

w = k ⊗ w ⊕w ⊕ ⊕w (10)

1 k

t

k =

= ¦ ,

1

t

u u

k =

1

t

v v

k =

ϖj=min{ ,ϖ ϖ1 2, ,ϖj}

B Agrregate Rating of Alteratives Versus Criteria

Let x ijt =(m n p ijt, ijt, ijt,ϖijt),i=1, 2, , ,m j=1, 2, , ,h

1, 2, , ,

t= k be the suitability rating assigned to alternative Ai

by decision maker D t under criterion C j. The averaged rating

x = m n p ϖ of alternativeAi versus criteria C j

assessed by the committee of k decision makers can be

expressed as follow

1

k

(11) where m ij 1m ijt

k

= , n ij 1n ijt,

k

= p ij 1 p ijt

k

=

ϖ =ij min{ϖ ,ϖ , ,ϖij1 ij2 ijk}

C Aggregate the Weighted Ratings

The membership function Ti i = 1,…, m, j = 1,…, n is the

final fuzzy evaluation value of each alternative:

1

n

j

T

=

=¦ ⊗

(12)

α- Cuts is used to develop the membership function:

1

n

j

=

(13) Thus, the membership function is developed as follows:

ij ij ij x ij ij p ij x p ij

2

2

j ij

j ij

j j ij w ij ij j x ij j

t

α α

α

­°

®

°¯

½°

¾

°¿

2 1

1

2 1

;

n

j j ij ij w x n

j

j

j j ij w ij ij j x ij j

n

j j ij ij w x j

j j ij w ij ij j x ij j

p

u v n

α α

α

=

=

=

⊗ =

­

°°

®

°

°¯

½

°°

¾

°

¦

¦

¦

Suppose that:

1 1

,

n

j

A

=

1 1

,

n

ij j j ij w ij ij j x j

B

=

1 1

,

n

ij j j ij ij w x j

p

=

1 1

,

n

ij j j ij w ij ij j x j

D

=

=¦ª¬ − ⁄ϖ + − ⁄ϖ º¼ 1

1

,

n

j

O m t

=

1 1

,

n j

j

Q p v

=

1

,

n

ij ij j j

n u P

=

=¦ Then we have:

1

,

n

j

Q

=

We have two simplified equation as follows:

i i O i x

Aα +Bα+ − =

Cα +Dα+ − =

From the above two equations, we have:

2

i

B

A

, 2

i

D

C

Since α∈[0,1], then the left and right membership functions

)

(x

f L

T i of T i can be produced as:

( ) [ 2 4 ( )]1/2

2

T i

i

B

x

f

A

α = =− + + − , O i≤ ≤x P i (14)

2

Ti

i

D

x

f

C

α = =− − + − P ≤ i x≤Q i (15)

For convenience, Ti can be expressed as:

( , , , , , , )

T= O P Q A B C D , i= 1, ,m

D Defuzzification

The conversion from a fuzzy set to a crisp number is called

defuzzification Numerous ranking methods have been

investigated to rank the fuzzy numbers in literature This study

employs the ranking method proposed by [3] to defuzzify all

the final fuzzy evaluation values This ranking method in one

of the most commonly used approaches of ranking fuzzy

numbers in fuzzy decision making

E Ranking Obtain

Using [3] ranking method, the total utility value of each

i

A is applied to defuzzify all the final fuzzy evaluation values

i

T as follows:

1

1

2

i

M

i

D C x Q

D

U

C

=

(16) 2

2

2

i

G

i

D

U

C

=

(17) 1

1

[ 4 ( )]

2

i

i i i Li i

G

i

B

U

A

=

(18) 2

2

2

i

M

i

B A x O

B

U

A

=

(19)

where:

1

min max min

4 2

i

Q

−

1

1

1/2 2

max min

4

i

i

i

Q

1

2 2

2

max min

2 4

i

i

D

Q

1

2 2

2

2

2 max min

4 4

i

i

i i

D Q

2

2 2

2

2 2

i

Q

{

}

max min max min min

R

Q

⇔

Similarly, we have:

{

1

2 2 max min max min

max

/ 2 4

R

i

i

D

C x

½

− «¬+ − »¼ °¿¾ (21)

{

2

1/2 max

L

i

i

A

½

{

2 max min

min

4 ( )]

L

i

i

B

A

½

−

Then, the total utility value of T with index of optimism , α= 0.5 is defined as:

2

2

1

0.5

2

2

2

2 / 4 2

i

i

i

i

T

i

i

i

i

D i u

C

D

C

B

A

B

A

=

−

+

­°

®

°¯

½°

¾

°¿ (24) The greater the 0.5( )

u , the bigger fuzzy number A i and the higher its ranking order

IV NUMERICAL EXAMPLE This section applies the proposed approach to solve the supplier selection and evaluation problem to demonstrate the feasibility and applicability of the proposed approach

Assume that a company desires to select the suitable material supplier for the company’s producing strategy After

preliminary screening, three suppliers A1, A2 and A3 are chosen for further evaluation A committee of three decision makers,

D1, D2 and D3, and, has been formed to conduct the assessment

and to select the most suitable supplier using nine criteria:

product/service quality (C1), customer satisfaction (C2),

organization control (C3), technological capability (C4),

relationship closeness (C5), complaints (C6), product/service

warranty period (C7), punctuality of delivery (C8) , unit price

(C9)

The computational procedure is summarized as the

following:

A Step 1: Agrregate the Ratings of Alternatives Versus

Criteria

TABLE 3 presents the suitability ratings of suppliers versus

the nine criteria Using Equation (11) the aggregated suitability

ratings are obtained in the last column of TABLE 3

TABLE 3 SUITABILITY RATINGS OF ALTERNATIVES VERUS

CRITERIA

C1

A1 G G VG (0,667, 0,833, 0,900; 0,9)

A2 F G G (0,500, 0,700, 0,800; 0,8)

A3 G F G (0,500, 0,700, 0,800; 0,8)

C2

A1 G G F (0,500, 0,700, 0,767; 0,8)

A2 F F G (0,400, 0,600, 0,733; 0,8)

A3 G VG G (0,667, 0,833, 0,900; 0,9)

C3

A1 VG VG G (0,733, 0,867, 0,933; 0,9)

A2 G F F (0,400, 0,600, 0,700; 0,8)

A3 F G G (0,500, 0,700, 0,800; 0,8)

C4

A1 F F G (0,400, 0,600, 0,733; 0,8)

A2 G G F (0,500, 0,700, 0,767; 0,8)

A3 VG G VG (0,733, 0,867, 0,933; 0,9)

C5

A1 VG VG G (0,733, 0,867, 0,933; 0,9)

A2 G VG G (0,667, 0,833, 0,900; 0,9)

A3 G F F (0,400, 0,600, 0,700; 0,8)

C6

A1 F F G (0,400, 0,600, 0,733; 0,8)

A2 F F F (0,300, 0,500, 0,633; 0,8)

A3 G G VG (0,667, 0,833, 0,900; 0,9)

C7

A1 G G VG (0,667, 0,833, 0,900; 0,9)

A2 VG G G (0,667, 0,833, 0,900; 0,9)

A3 G G G (0,600, 0,800, 0,867; 0,9)

C8

A1 G F G (0,500, 0,700, 0,800; 0,8)

A2 G G VG (0,667, 0,833, 0,900; 0,9)

A3 VG G G (0,667, 0,833, 0,900; 0,9)

C9

A1 G G VG (0,667, 0,833, 0,900; 0,9)

A2 VG G G (0,667, 0,833, 0,900; 0,9)

A3 F F G (0,400, 0,600, 0,733; 0,8)

B Step 2: Aggregate the Importance weights

TABLE 4 displays the importance weights of nine criteria

from the three decision makers Using Equation (10) the

aggregated weights of criteria from the decision makers as

shown in the last column of TABLE 4

TABLE 4 THE IMPORTANCE WEIGHTS OF THE CRITERIA AND

THE AGGREGATED WEIGHTS

Criteria

Decision Makers

w ij

D 1 D 2 D 3

C1 AI AI VI (0.733, 0.867, 0.967; 0.9)

C2 AI VI VI (0.667, 0.833, 0.933; 0.9)

C3 I VI I (0.400, 0.600, 0.767; 0.8)

C4 I VI I (0.400, 0.600, 0.767; 0.8)

C5 VI VI AI (0.667, 0.833, 0.933; 0.9)

C6 AI AI VI (0.733, 0.867, 0.967; 0.9)

C7 I I VI (0.400, 0.600, 0.767; 0.8)

C8 I VI VI (0.500, 0.700, 0.833; 0.8)

C9 VI I I (0.400, 0.600, 0.767; 0.8)

C Step 3: Aggregate the weighted ratings and defuzzification

Using Equation (12) to (24), the left, right and total utilities

with Į = 1/2 can be obtained as shown in TABLE 5 It can be

seen that from TABLE 5 the ranking order of the three supplier

is A1 > A3 > A2 Thus, the most suitable suppliers is A1, which has the largest total utility

TABLE 5 THE LEFT, RIGHT AND TOTAL UTILITIES OF EACH

SUPPLIER

A1 0,635 0,343 0,569 0,288 0,530 1

A2 0,593 0,269 0,605 0,351 0,476 3

A3 0,630 0,335 0,574 0,296 0,524 2

V CONCLUSION Supplier selection and evaluation problem is a fuzzy MCDM problem that is affected by several qualitative and quantitative criteria In order to solve the supplier selection problem, this paper has proposed and extension of fuzzy MCDM In the proposed approach, the ratings of alternatives and relative importance weights of criteria for suppliers are expressed in linguistic values, which are represented by generalized fuzzy numbers The membership function of each weighted rating of each supplier for each criterion I then developed To avoid complicated calculations of fuzzy numbers, these weighted ratings are defuzzified into crisp values by using the new maximizing set and minimizing set ranking approach to determine the ranking order of alternatives A numerical example was given to illustrate the applicability of the proposed approach The results indicate that the proposed fuzzy MCDM approach is practical and useful The proposed approach can also be applied to other management problems under similar settings such as lecturer’s performance evaluation, project selection, hospital service quality evaluation, logistics center location selection, etc

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