Quality-based supplier selection and evaluation using fuzzy data

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Quality-based supplier selection and evaluation using fuzzy data

Ming-Hung Shu

a

, Hsien-Chung Wu

b,*

a

Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan b

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan

a r t i c l e

i n f o

Article history:

Received 4 December 2008

Received in revised form 20 April 2009 Accepted 21 April 2009

Available online 3 May 2009 Keywords:

Supplier selection Fuzzy number Resolution identity Fuzzy ranking method Fuzzy preference relation Optimization

a b s t r a c t

Since fuzzy quality data are ubiquitous in the real world, under this fuzzy environment, the supplier selection and evaluation on the basis of the quality criterion is proposed in this paper. The Cpkindex has been the most popular one used to evaluate the quality of supplier’s products. Using fuzzy data col-lected from q P 2 possible suppliers’ products, fuzzy estimates of q suppliers’ capability indices Cpkiði ¼ 1; 2; . . . ; qÞ are obtained according to the form of resolution identity that is a well-known theo-rem in fuzzy sets theory. Certain optimization problems are formulated and solved to obtaina-level sets for the purpose of constructing the membership functions of fuzzy estimates of Cpki. These membership functions are sorted by using a fuzzy ranking method to choose the preferable suppliers. Finally, a numer-ical example is illustrated to present the possible application by incorporating fuzzy data into the quality-based supplier selection and evaluation.

1. Introduction

Nowadays rising customers’ expectations as well as increasing the product quality are becoming an important strategic priority in the pretty competitive global business environment. Manufac-turers must produce the correct products at the accurate time and deliver them promptly to customers to sustain their competi-tive advantage in the marketplace (Hensler, 1994;Hutt & Speh, 2006). Manufacturers increasingly purchase components from suppliers or hire contract manufacturers to produce necessary parts, and they assemble these parts to deliver finished products to customers. In the automotive industry, the cost of components and parts purchased from outside vendors have increased up to 50% of their revenues (Weber, Current, & Benton, 1991). The high technology firms spend more than 80% of total product costs on purchasing materials and services (Burton, 1988; Carr & Pearson, 1999). Obviously, the quality of parts obtained from suppliers determines the quality of the finished products produced by man-ufacturers as well as the customers’ satisfaction and loyalty. There-fore, the evaluation of supplier performance and selection of suppliers are becoming major challenges faced by the manufactur-ing and purchasmanufactur-ing managers (ASQC, 1981).

Assessing a group of suppliers and selecting one or more of them are a complex task because various criteria must be consid-ered in the decision-making process such as quality, cost, goodwill, service, delivery time, and environmental impact (Humphreys,

Wong, & Chan, 2003). According to research conducted byDickson

(1966), quality and delivery are two of the most demanded items

by component suppliers. Twenty ﬁve years after Dickson’s research,Weber et al. (1991)still considered quality to be of ‘‘ex-treme importance” and delivery to be of ‘‘considerable impor-tance”. According to Weber’s research on the Just-In-Time (JIT) model, the importance of quality and delivery remains the same.

Pearson and Ellram (1995)surveyed 210 members of the National

Association of Purchasing Management (NAPM), who were ran-domly selected from the listings of electronic firms in the two-digit SIC code 38, and they indicated that quality is the most important criterion in the selection and evaluation of suppliers for both the small and large electronic firms that were surveyed. Moreover, according to the survey of current and potential outsourcing end-users by the OutsourcingInstitute (2003), the top 10 factors in ven-dor selection are commitment to quality, price, reference/reputa-tion, flexible contract terms, scope of resources, additional value-added capability, cultural match, existing relationship, location, and others. Quality is still the most important factor for selecting the preferred suppliers. Furthermore,Olhager and Selldin (2004)

investigated the strategies and practices in the supply chain man-agement using the sample of 128 Swedish manufacturing ﬁrms, and concluded that many aspects are important when companies choose supply chain partners, but quality is the most important criterion. In other words, based on the above works, quality can be seen as a fundamental factor for supplier evaluation among var-ious criteria.

Quality affects the productivity and business performance in both industrial and customers’ organizations. Much evidence

*Corresponding author.

E-mail address:[email protected](H.-C. Wu).

Contents lists available atScienceDirect

Computers & Industrial Engineering

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suggests that high quality has a positive impact upon signiﬁcantly increasing proﬁtability, through lowering operating costs and improving market share (Garvin, 1988; Maani, 1989; Phillips,

Chang, & Buzzell, 1983; Voehl, Jackson, & Ashton, 1994). Kane

(1986)stated that the quantiﬁcation of the process mean and

var-iation is central to understanding the quality of the components produced from a manufacturing process. This fact brings a issue of quality-based supplier selection and evaluation by process capa-bility indices (PCIs) into the main focus of this research.

The ﬁrst PCI appearing in the literature was the precision index and it was proposed byJuran (1974) and Kane (1986)and deﬁned as

Cp¼

USL LSL

6

r

; ð1Þ

where USL stands for the upper speciﬁcation limit, LSL stands for the lower speciﬁcation limit, and

r

stands for the process standard deviation. The index Cp measures process precision (consistency

of quality). However, it does not consider whether the process is centered. By considering the magnitude of process variance as well as the location of process mean, the Cpkindex is deﬁned as Cpk¼ min USL

l

3

r

;

l

LSL 3

r

; ð2Þ

which has been the most popular one used in the manufacturing industry (Kane, 1986; Kotz & Lovelace, 1998).Montgomery (2005)

recommended some minimum capability requirements for per-forming the manufacturing processes under some certain desig-nated quality conditions. For example, CpkP1:33 is for the

existing processes, and CpkP1:50 is for the new processes. On

the other hand, CpkP1:50 is also for the existing processes on

safety, strength, or critical parameter, and CpkP1:67 is for the

new processes on safety, strength, or critical parameter. Finley

(1992)also found that the required values on all critical supplier

processes are 1.33 or higher, and the Cpk values of 1.67 or higher

are preferred. Many companies have recently adopted criteria for evaluating their processes that include more stringent process capability. Motorola’s ‘‘Six Sigma” program essentially requires the process capability to be at least 2.0 to conform the possible 1:5

r

process shift (Harry, 1988).

The supplier certiﬁcations in the manuals of ISO 9000 and QS-9000 include a detailed procedure in evaluating supplier’s products on the basis of the most well-known Cpkindex. For a purchasing

contract, a minimum value of Cpkis usually speciﬁed. If the

pre-scribed minimum Cpkfails to be met, then the supplier is veriﬁed

to be incapable. Otherwise, the supplier is evaluated to be capable. Naturally, we can investigate the supplier selection and evaluation for the case with q P 2 candidate suppliers’ products by using the Cpkindex.

Let Pibe the products population of ith supplier with the mean

l

i and variance

r

2i where i ¼ 1; 2; . . . ; q. The capability index Cpki

indicated the quality of the ith supplier’s products can be deﬁned as Cpki¼ min USL

l

_i 3

iare usually unknown. In this case, to assess the

appropriate suppliers, sample data must be collected from suppli-ers in order to estimate Cpki. Let xi1;xi2; . . . ;xinibe independent

ran-dom samples from Pi for i ¼ 1; 2; . . . ; q. Generally, the underlying

data obtained from the output responses of each supplier’s prod-ucts are always assumed to be real numbers. Such a situation,

Pearn, Kotz, and Johnson (1992), Pearn and Shu (2003) and Prasad

and Calis (1999)have used the statistical point estimate ^cpkion Cpki

by ^ cpki¼ min USL xi 3si ;xi LSL 3si ; ð4Þ

where the mean

l

iin Eq.(3)is substituted by the sample mean xi xi¼ 1 ni Xni j¼1 xij

and the standard deviation

r

iin Eq.(3)is substituted by the sample

standard deviation si si¼ 1 ni 1 Xni j¼1 ðxij xiÞ2 " #1=2 for i ¼ 1; 2; . . . ; q.

However, in the practical situations, data collected from the key quality characteristic of suppliers’ products are often somewhat imprecise (fuzzy). For example, the data may be given by color intensity of pictures or by the readings on an analogue measure-ment equipmeasure-ment, as in the studies ofFilzmoser and Viertl (2004)

and Viertl and Hareter (2004). In addition, the imprecise data

may be given by scarce sample data, e.g., the observations made with coarse scales, linguistic data, or data collected with vague and incomplete knowledge, as discussed byGulbay and Kahraman

(2007) and Sugano (2006).Lee (2001) and Hong (2004)indicated

that the measurements partly carried out by the decision-makers subjective determination can also be seen to be fuzzy numbers. In their studies, the estimation of Cpkindex was proposed using

fuzzy data.

In this paper, we study the quality-based supplier selection and evaluation using fuzzy data, which was not seriously treated by the researchers. The paper is organized as follows. In Section2, we introduce the basic properties of fuzzy numbers. In Section3, we discuss the fuzzy estimate of Cpkiby considering fuzzy quality data

collected from suppliers. Using the form of resolution identity the-orem, the membership function of the fuzzy estimate of Cpki for

each supplier is obtained. In order to compute the membership de-grees, some optimization problems are formulated. In Section 4, we provide computational methods to solve the optimization problems. In Section5, to select the preferable suppliers, a ranking method proposed byYuan (1991)is extended to sort the member-ship functions of fuzzy estimates of suppliers’ Cpkiindices. Finally,

we summarize our proposed method in a step-by-step procedure. An application of light emitting diodes (LEDs) is illustrated as an example.

2. Fuzzy numbers

The fuzzy subset ~a of R is deﬁned by a function n~a:R! ½0; 1,

which is called the membership function. The

a

-level set of ~a, de-noted by ~aa, is deﬁned as ~aa¼ fx 2 R : n~aðxÞ P

a

g for all

a

2(0,1].

The 0-level set ~a0 is deﬁned as the closure of the set

fx 2 R : n~aðxÞ > 0g, i.e., ~a0¼ clðfx 2 R : na~ðxÞ > 0gÞ ¼ clðSa>0~aaÞ.

Deﬁnition 2.1. The fuzzy subset ~a of R is said to be a fuzzy number, if the following conditions are satisﬁed:

(i) ~a is normal, i.e., there exists an x 2 R such that n~aðxÞ ¼ 1.

(ii) n~a is quasi-concave, i.e., n~aðtx þ ð1 tÞyÞ P minfn~aðxÞ; n~aðyÞg

for t 2 ½0; 1.

(iii) n~a is upper semicontinuous, i.e., fx 2 R : n~aðxÞ P

a

g is a

closed subset of R for each

a

2(0,1].

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Since ~aa ~a0 for each

a

2(0,1], condition (iv) shows that the

a

-level sets ~aaare bounded subsets of R for all

Þ ¼ ~aUa are continuous functions on ½0; 1. It implies that ~a is also a fuzzy real number.

In order to construct the fuzzy estimate of Cpki, the following

re-sult is very useful.

Proposition 2.1 (Resolution Identity (Zadeh, 1975)). Let ~a be a fuzzy subset of R with membership function n~a. Then,

n~aðxÞ ¼ sup

a2½0;1

a

1~aaðxÞ;

where 1Ais a characteristic function of set A, i.e., 1AðxÞ ¼ 1 if x 2 A, and

1AðxÞ ¼ 0 if x R A (note that the

a

-level set ~aaof ~a is a usual set).

3. Estimation of Cpkibased on fuzzy data

The signiﬁcance of this study is that the proposed methodology can effectively handle fuzzy data to evaluate the quality of sup-plier’s products. Now, we will present the estimation of Cpki for

each supplier using fuzzy data.

Let ~xi1; . . . ; ~xini be fuzzy observations (fuzzy data). It is

under-stood that the fuzzy data are assumed to be fuzzy real numbers. Therefore, for any given

a

2 ½0; 1, the corresponding real-valued data ð~xijÞLaand ð~xijÞUafor i ¼ 1; . . . ; q and j ¼ 1; . . . ; nican be obtained.

According to Eq.(4), using the real-valued data ð~xi1ÞLa; . . . ;ð~xiniÞ

L

a, we

can obtain the estimate

^ cL pkia¼ min USL xL ia 3sL ia ;x L ia LSL 3sL ia ; ð5Þ where xL ia¼ 1 ni Xni j¼1 ð~xijÞLa and sLia¼ 1 ni 1 Xni j¼1 ð~xijÞLa xLia 2 " #1=2 :

Similarly, using the real-valued data ð~xi1ÞUa; . . . ;ð~xiniÞ

U

ain Eq.(4),

we can also obtain the estimate

^ cU pkia¼ min USL xU ia 3sU ia ;x U ia LSL 3sU ia ; ð6Þ where xU ia¼ 1 ni Xni j¼1 ð~xijÞUa and sUia¼ 1 ni 1 Xni j¼1 ð~xijÞUa xUia 2 " #1=2 :

Now, we deﬁne the closed interval by Aiaas

Aia¼ min ^cLpkia; ^cUpkia

n o ;max ^cL pkia; ^cUpkia n o h i :

By applying this result to the form of Resolution Identity shown in Proposition2.1, we obtain the membership function of fuzzy estimate ~^cpkiof Cpkias

n~^cpkiðrÞ ¼ sup

06a61

a

1AiaðrÞ: ð7Þ

It should be noted that the fuzzy estimate given in Eq.(7)is en-tirely distinct from the fuzzy estimate obtained byParchami and

Mashinchi (2007) and Hsu and Shu (2008); in their studies, the

result of fuzzy estimate was obtained from Eq.(4)using Buckley’s approach on real-valued data, i.e., fuzzifying a real-valued function to be a fuzzy-valued function. However, in this study, fuzzy data are taken into account.

on ½0; 1. This also implies

that xL ia; ðs2nÞ

L

ia; xUia, and ðs2nÞ U

iaare continuous with respect to

a

on

½0; 1. Therefore, we conclude that ^cL

pkia and ^cUpkia are continuous

with respect to

a

on ½0; 1. From these facts, the

a

-level set ð~^cpkiÞa

of fuzzy estimate ~^cpkican be expressed as ð~^cpkiÞ L a;ð~^cpkiÞ U a h i ¼ ð~^cpkiÞa¼ r : n~^cpkiðrÞ P

a

n o ¼ min a6b61liðbÞ; maxa6b61uiðbÞ ; ð9Þ

where liðbÞ and uiðbÞ are those deﬁned in Eq.(8). From Eq.(9), the

relationships between ð~^cpkiÞLaand ^cLpkiaand ð~^cpkiÞUa and ^cUpkiaare

ð~^cpkiÞLa¼ min_a₆ b61liðbÞ ¼ mina6b61min ^c L pkib^c U pkib n o ð10Þ and

ð~^cpkiÞUa¼ max_a₆_b61uiðbÞ ¼ max

a6b61max ^c L pkib; ^c U pkib n o ; ð11Þ respectively.

Proposition 3.1. The fuzzy estimate ~^cpkideﬁned in(7)is a fuzzy real number.

Proof. From Deﬁnition2.1, since the closed interval A1(i.e.,

a

¼ 1)

is not an empty set, condition (i) is satisﬁed. Since the

a

-level set ð~^cpkiÞain Eq.(9)is a closed subset of R, i.e., a convex subset of R,

conditions (ii)–(iv) are satisﬁed. This implies that ~^cpki is a fuzzy

number. Since lið

a

Þ and uið

a

Þ are continuous on ½0; 1. From Eq. (9), it is found that ð~^cpkiÞLa and ð~^cpkiÞUa are also continuous with

respect to

a

on ½0; 1. This shows that ~^cpkiis indeed a fuzzy real

number. h

4. Computational methods

Now, we will present computational methods for obtaining a membership degree of any given value r of the fuzzy estimate ~

^cpkifor i ¼ 1; 2; . . . ; q. From Proposition2.1, the membership

func-tion of ~^cpkican be expressed as n~^cpkiðrÞ ¼ sup

06a61

a

1ð~^cpkiÞaðrÞ:

Þ ¼ ð~^cpkiÞUa¼ max_a₆ b61uiðbÞ ¼ maxa6b61max ^c L pkib; ^c U pkib n o :

min a6b61min ^c L pkib; ^c U pkib n o

¼ min ^cLpkib; ^cU_pkib

n o

for some b_{2 ½}

_a

_;_{1. Then, we have} min

a6b61

^cL pkib6^c

L

pkib and min a6b61 ^ cU pkib6^c U pkib;

which implies that

min min a6b61 ^ cL pkib;_amin₆ b61 ^ cU pkib

6min ^cL_pkib; ^cU_pkib

n o ¼ min a6b61min ^c L pkib; ^c U pkib n o :

On the other hand, since minf^cL

pkib; ^cUpkibg 6 ^cLpkib and

minf^cL

pkib; ^cUpkibg 6 ^cUpkib, we have min a6b61min ^c L pkib; ^c U pkib n o 6 _min a6b61 ^ cL pkib and min a6b61min ^c L pkib; ^c U pkib n o 6 _min a6b61 ^ cU pkib: It implies that min a6b61min ^c L pkib; ^c U pkib n o 6_min _min a6b61 ^ cL pkib;_amin₆ b61 ^ cU pkib :

This concludes that

min a6b61min ^c L pkib; ^c U pkib n o ¼ min min a6b61 ^_cL pkib;_amin₆_b61^c U pkib :

Similarly, we can also show that

max a6b61max ^c L pkib; ^cUpkib n o ¼ max max a6b61 ^ cL pkib;_amax 6b61 ^ cU pkib :

This completes the proof. h

The optimization problem (MP1) can now be simpliﬁed as

ðMP2Þ max

a

subject to

g

ið

a

Þ 6 r

fið

a

Þ P r

0 6

a

61:

Since

g

ið

a

Þ 6 r or fið

a

Þ P r constraint can be eliminated in the

ways described as follows.

(i) If

g

_ið1Þ 6 r 6 fið1Þ, then n~^cpkiðrÞ ¼ 1.

(ii) Since fið

a

Þ P fið1Þ P

g

ið1Þ > r for all

a

2 ½0; 1, if r <

g

ið1Þ,

then the constraint fið

a

Þ P r is redundant. Therefore,

prob-lem (MP2) can be reduced to be an easier optimization problem

ðMP3Þ max

a

subject to

g

ið

a

Þ 6 r

0 6

a

61:

(iii) Since

g

ið

Þ P r

0 6

a

6_1:

a

Þ <

lowþup

2 and go to Step 2.

For problem (MP4), we consider the equivalent constraint

fið

a

Þ 6 r

Since fið

a

Þ is decreasing, fið

a

Þ becomes increasing. Thus, the

above algorithm is still applicable for solving problem (MP4). 5. Quality-based supplier selection

In this section, a problem of selecting the better suppliers from q available suppliers fS1;S2; . . . ;Sqg using fuzzy quality data is

con-sidered.Yuan (1991)proposed a ranking method which is satisﬁed with four reasonable criteria on sorting fuzzy numbers such as fuz-zy preference presentation, rationality of fuzfuz-zy ordering, distin-guish ability, and robustness. These criteria obtained are on the basis of the normal and convex fuzzy sets satisfying conditions in Deﬁnition2.1. This implies that the ranking method can also be ap-plied for the set of all fuzzy numbers.

Now, we generalize Yuan’s fuzzy ranking method to carry out the quality-based supplier selection. Let ~^cpki and ~^cpkjbe the fuzzy

ÞjÞd

a

:

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Obviously, the value of FPRð~^cpki; ~^cpkjÞ in Eq.(12)is within ½0; 1,

which can indicate a degree of preference between ~^cpkiand ~^cpkj. It

_{, then it can be said that ~}^_c_pkj _is indifferent to ~^cpkiwith indifference degrees of ð

c

(v) If 1

c

6FPRð~^c_pkj_{; ~}^c_pki_{Þ 6}

c

, then it can be said that ~^c_pki is indifferent to ~^cpkjwith indifference degrees of ð

c

;1

c

Þ.

(vi) If FPRð~^cpkj; ~^cpkiÞ < 1

c

, then it can be said that ~^cpkj is less

preferable than ~^cpkiwith a non-preference degree of

c

.

By observing the above rules, certain interesting results are summarized as follows:

(1) It is clear that rule (iii) is equivalent to rule (iv). In other words, rule (iii) can also be stated to be if FPRð~^cpki; ~^cpkjÞ <

1

c

, then it can be said that ~^cpkj is more preferable than

~ ^

cpkiwith a preference degree of

c

.

(2) It is clear that rule (vi) is equivalent to rule (i). In other words, rule (vi) can also be described as if FPRð~^cpkj; ~^cpkiÞ <

1

c

, then it can be said that ~^cpki is more preferable than

~ ^

cpkjwith a preference degree of

c

.

(3) Obviously, rule (ii) is equivalent to rule (v). In other words, if ~

^

Using rules (i)–(iii), fS1; . . . ;Sqg can be sorted into fSh1; . . . ;Shqg

for any i < j, then it implies that either ~^cpkhi is more preferable to

~ ^

cpkhj or ~^cpkhi is indifferent to ~^cpkhj. Therefore, we can state some

interesting observations.

(a) The two suppliers Si and Sj are indifferent to each other if

Step 1: Select q possible suppliers and collect quality data from them.

Step 2: Obtain the membership function ~^cpki for each supplier

i 2 f1; 2; . . . ; qg using the fuzzy estimation of Cpkiand the

computational methods described in Sections 3 and 4, respectively.

Step 3: Provide a value of

c

such that

c

2 ½0:55; 0:65. Calculate the values of FPRð~^cpki; ~^cpkjÞ for i; j 2 f1; 2; . . . ; qg and i < j.

Step 4: According to the rules (i)–(iii) stated in Section5. The preferable group of suppliers fSh1; . . . ;Shtg is determined,

where t is the number of preferable suppliers and any two suppliers in the group are indifferent.

Step 5: If t P 2, then the decision-makers may randomly select one of the suppliers as the most preferable supplier. Of course, if the decision-makers decide to select more than one supplier to supply the required products, then the same procedure can be used to select the preferable suppliers. Before providing a numerical example, we introduce a special kind of fuzzy number known as a triangular fuzzy number whose membership function is triangular. However, it should be noted that the proposed methodology is still applicable to the case of any bell-shaped fuzzy number whose membership function is bell-shaped. The membership function of the triangular fuzzy number ~a is deﬁned as n~aðrÞ ¼ ðr a1Þ=ða2 a1Þ if a16r 6 a2 ða3 rÞ=ða3 a2Þ if a2<r 6 a3 0 otherwise; 8 > < > :

The triangular fuzzy number ~a, which is denoted as ~

a ¼ ða1;a2;a3Þ can be said to be ‘‘around a2” or ‘‘being

approxi-mately equal to a2,” where a2is called the core value of ~a, and a1

and a3are called the left and right spread values of ~a, respectively.

The

a

-level set (a closed interval) of ~a is then given as ~

Example 6.1. Since light emitting diodes (LEDs) have a long life span and high intensity of solid-state illumination exhibiting a wide range of colors, the uses of LEDs are growing rapidly in a wide variety of applications such as automotive lighting, computer displays, LCD televisions, signaling and general lighting products. Here, an LED-based lighting ﬁxture (LED-LF) is investigated as an example; the LED-LF is manufactured in Tainan Industrial Park, Taiwan. Due to high demands on the LED-LFs, the company does not have enough production capacity to supply one type of LED components used in the LED-LFs. Therefore, the decision-makers decide to purchase the LED components from some possible suppliers. The luminous intensity of LED sources is a critical characteristic for this type of LEDs. Thus far, all light measurements and rating systems depend on the perception of the human eye or imprecise terminology and calibration standards (Ryer, 1997). This implies that the randomness is not the only aspect of uncertainty

for data collected on the luminous intensity of LED sources; that is, the occurrence of fuzziness introduces another uncertainty that should be taken into account while solving the problem.

Four suppliers are capable of producing this type of LEDs. The decision-makers need to choose preferable suppliers based on the fuzzy sample data of the luminous intensity which have been collected from each supplier with size 20, as listed in Table 1, where the data ~xin¼ ðxin1;xin2;xin3Þ with i ¼ 1; 2; 3; 4 and n ¼ 1; 2; . . . ; 20 are assumed as triangular fuzzy numbers.

The upper and lower speciﬁcation limits of luminous intensity are set at USL ¼ 90 mcd=m2 _{and LSL ¼ 40 mcd=m}2_{, respectively.} Then the statistics of the four suppliers in the

a

-level sense are listed inTable 2.

Following the computational methods described in Section4, the optimization problems can be solved by implementing a method provided by a MATLAB function fmincon and the

mem-Table 1

Triangular fuzzy data collected from the suppliers (unit: mcd=m2_).

S1 S2 S3 S4 (68.26, 70.46, 72.18) (72.69, 73.96, 75.20) (54.24, 64.27, 70.69) (58.53, 60.58, 66.15) (70.22, 72.88, 73.10) (57.90, 58.69, 60.26) (66.67, 72.00, 73.61) (54.79, 61.16, 69.86) (62.26, 63.52, 66.82) (76.09, 77.08, 79.03) (66.31, 70.38, 74.15) (58.37, 58.92, 72.53) (66.64, 68.10, 70.09) (64.40, 66.80, 68.54) (59.89, 66.92, 71.78) (55.20, 56.13, 66.92) (67.80, 69.17, 70.22) (67.28, 68.07, 69.70) (69.28, 70.23, 73.45) (59.92, 65.08, 65.57) (65.33, 66.79, 68.20) (64.46, 65.61, 67.98) (65.90, 66.92, 67.90) (65.16, 70.91, 73.60) (59.90, 61.71, 62.90) (65.90, 67.48, 68.67) (62.73, 63.03, 64.02) (54.03, 58.64, 61.83) (68.54, 69.38, 70.32) (67.67, 68.12, 69.29) (59.88, 60.12, 62.54) (66.50, 66.63, 72.08) (72.10, 73.28, 74.90) (67.25, 68.62, 68.98) (69.25, 70.03, 71.32) (62.17, 65.77, 66.84) (72.19, 74.26, 75.32) (60.80, 61.01, 62.20) (70.00, 71.39, 72.12) (65.24, 73.35, 79.83) (72.69, 73.96, 75.20) (65.90, 66.92, 67.90) (63.89, 64.74, 66.23) (65.65, 70.76, 72.17) (57.90, 58.69, 60.26) (62.73, 63.03, 64.02) (65.80, 66.85, 67.92) (68.26, 72.26, 75.81) (76.09, 77.08, 79.03) (59.88, 60.12, 62.54) (54.42, 55.75, 57.34) (63.88, 64.98, 71.10) (64.40, 66.80, 68.54) (69.25, 70.03, 71.32) (67.56, 69.47, 70.22) (61.98, 62.58, 74.81) (67.28, 68.07, 69.70) (70.00, 71.39, 72.12) (59.42, 60.12, 61.78) (58.42, 59.82, 63.24) (64.46, 65.61, 67.98) (63.89, 64.74, 66.23) (64.90, 68.48, 69.67) (58.59, 68.87, 70.05) (65.90, 67.48, 68.67) (65.80, 66.85, 67.92) (62.69, 63.22, 65.79) (55.25, 66.51, 68.72) (67.67, 68.12, 69.29) (54.42, 55.75, 57.34) (68.35, 69.72, 70.98) (54.72, 66.88, 74.94) (67.25, 68.62, 68.98) (67.56, 69.47, 70.22) (62.94, 63.51, 64.20) (63.69, 65.03, 75.51) (60.80, 61.01, 62.20) (59.42, 60.12, 61.78) (65.30, 66.96, 67.65) (57.63, 61.54, 66.43) Table 2

Thea-level estimates of Cpki.

a-level S1 S2

xL

ia sLia ^cLpkia xUia sUia ^cUpkia xLia sLia ^cLpkia xUia sUia ^cUpkia

0.0 66.88 4.56 1.69 69.70 4.60 1.47 65.16 5.11 1.62 67.56 5.15 1.45 0.1 67.02 4.57 1.68 69.55 4.61 1.48 65.27 5.12 1.61 67.43 5.16 1.46 0.2 67.16 4.58 1.66 69.41 4.62 1.49 65.37 5.13 1.60 67.29 5.17 1.47 0.3 67.29 4.59 1.65 69.26 4.63 1.49 65.47 5.15 1.59 67.15 5.18 1.47 0.4 67.43 4.61 1.63 69.12 4.63 1.50 65.58 5.16 1.58 67.01 5.19 1.48 0.5 67.57 4.62 1.61 68.97 4.64 1.51 65.68 5.18 1.57 66.88 5.20 1.48 0.6 67.70 4.64 1.60 68.83 4.66 1.52 65.78 5.19 1.55 66.74 5.21 1.49 0.7 67.84 4.66 1.59 68.68 4.67 1.52 65.88 5.21 1.54 66.60 5.22 1.49 0.8 67.98 4.67 1.57 68.54 4.68 1.53 65.99 5.23 1.53 66.47 5.23 1.50 0.9 68.11 4.69 1.56 68.39 4.69 1.53 66.09 5.24 1.52 66.33 5.25 1.50 1.0 68.25 4.71 1.54 68.25 4.71 1.54 66.19 5.26 1.51 66.19 5.26 1.51 S3 S4 0.0 63.97 4.54 1.76 68.17 4.50 1.62 60.40 4.48 1.52 70.40 4.68 1.39 0.1 64.19 4.45 1.81 67.97 4.46 1.65 60.84 4.38 1.58 69.84 4.54 1.48 0.2 64.42 4.38 1.86 67.78 4.42 1.67 61.28 4.32 1.64 69.28 4.44 1.55 0.3 64.64 4.32 1.90 67.58 4.39 1.70 61.73 4.28 1.69 68.73 4.38 1.62 0.4 64.86 4.27 1.94 67.38 4.37 1.73 62.17 4.27 1.73 68.17 4.34 1.67 0.5 65.09 4.24 1.96 67.19 4.34 1.75 62.61 4.30 1.75 67.61 4.34 1.72 0.6 65.31 4.22 1.95 66.99 4.33 1.77 63.05 4.35 1.76 67.05 4.38 1.75 0.7 65.54 4.22 1.93 66.79 4.32 1.79 63.49 4.44 1.76 66.49 4.45 1.76 0.8 65.76 4.24 1.91 66.60 4.31 1.81 63.94 4.56 1.75 65.94 4.56 1.76 0.9 65.98 4.27 1.87 66.40 4.31 1.82 64.38 4.70 1.73 65.38 4.70 1.75 1.0 66.21 4.32 1.83 66.21 4.32 1.84 64.82 4.86 1.70 64.82 4.86 1.70

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bership functions of fuzzy estimates of ~^cpki for i ¼ 1; . . . ; 4 are constructed asFig. 1.

Now, the value of

c

is set to 0:50. The values of FPRð~^cpki; ~^cpkjÞ and FPRð~^cpkj; ~^cpkiÞ ¼ 1 FPRð~^cpki; ~^cpkjÞ for i; j ¼ 1; 2; 3; 4 and i < j are computed and listed inTable 3.

According to Step 4 described in the above procedure for the supplier selection, the order of four suppliers is ranked as fS3;S4;S1;S2g. Thus, S3is the most preferable supplier with 0:50 preference degree. If the value of

c

is set to 0:61, then the order is fðS3;S4Þ; ðS2;S1Þg, where (S3and S4) and (S2and S1) are indifferent with indifference degrees of ð0:61; 0:39Þ; that is,

0:39 6 FPRð~^cpk3; ~^cpk4Þ 6 0:61 and 0:39 6 FPRð~^cpk4; ~^cpk3Þ 6 0:61 and

0:39 6 FPRð~^cpk2; ~^cpk1Þ 6 0:61 and 0:39 6 FPRð~^cpk1; ~^cpk2Þ 6 0:61: Thus, we can conclude that S3and S4are the preferable

suppli-ers with a preference degree of 0:61. In this case, the decision-mak-ers may randomly select either S3 or S4 or provide some other

criteria to select the most preferable supplier. Of course, the sup-pliers S3and S4are the best choice if the decision-makers decide

to select more than one supplier to supply this type of LEDs.

7. Conclusions

The fuzzy set theory is a useful method for modeling the prob-lems with fuzzy (imprecise) information that has been recognized as one of uncertainties in the real world. The main contribution is that the methodology proposed in this paper is capable of selecting

the preferable suppliers using fuzzy quality data, which was not seriously treated by the researchers. With the help of resolution identity theorem which is widely used in fuzzy sets theory, compu-tational methods solving certain optimization problems are pro-posed to construct membership functions of fuzzy estimates of suppliers’ capability indices Cpki with i ¼ 1; 2; . . . ; q and q P 2.

Moreover, using membership functions f~^cpk1; . . . ; ~^cpkqg of q

suppli-ers, a fuzzy ranking method proposed byYuan (1991)is extended to select the preferable suppliers. A step-by-step procedure is developed and a real example taken from the application of light emitting diodes (LEDs) is illustrated the applicability of our pro-posed methods.

It is well-known that there are numerous process capability indices that are available for supplier selection. Although we choose the index Cpkiin this paper, we have to remark that the

pro-posed methodology is still applicable to other indices by solving the similar optimization problems.

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1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ˆ

pk

C

α 1 S 2 S S3 S4

Fig. 1. The membership functions of fuzzy estimates ~^cpkiof suppliers.

Table 3

Values of fuzzy preference relation (FPR).

Pairwise comparison M_ij M_ji FPRð~^cpki; ~c^pkjÞ FPRð~^cpkj; ~^cpkiÞ S1and S2(i ¼ 1 and j ¼ 2) 0.2629 0.1716 0.6050 0.3950 S1and S3(i ¼ 1 and j ¼ 3) 0.1013 0.7632 0.1171 0.8829 S1and S4(i ¼ 1 and j ¼ 4) 0.3122 0.8386 0.2713 0.7287 S2and S3(i ¼ 2 and j ¼ 3) 0.0552 0.8084 0.0639 0.9361 S2and S4(i ¼ 2 and j ¼ 4) 0.2493 0.8670 0.2233 0.7767 S3and S4(i ¼ 3 and j ¼ 4) 0.7534 0.6182 0.5494 0.4506

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