Supplier selection using fuzzy quality data and their applications to touch screen

Supplier selection using fuzzy quality data and their applications to touch screen

Bi-Min Hsu

a

, Ching-Yi Chiang

b

, Ming-Hung Shu

b,*

a

Department of Industrial Engineering and Management, Cheng Shiu University, Kaohsiung 833, Taiwan b

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

a r t i c l e

i n f o

Keywords: Fuzzy number Resolution identity Fuzzy preference relation Optimization

a b s t r a c t

The purchasing function directly affects the competitive ability of a firm. Since the determination of suit-able suppliers from a set of suppliers has become a key strategic consideration, managers need to peri-odically evaluate suppliers on the basis of their products quality to select suppliers whose quality characteristics of products meet the standards. The quantification of the process capability is effective to understand the quality of the units shipped from a supplier. While fuzzy data commonly exist in our real world, the quality-based supplier selection with fuzzy quality data is proposed in this paper. We apply the resolution identity result, a well-known method used in fuzzy sets theory, in terms of solv-ing the nonlinear programmsolv-ing problems with bounded variables to construct the membership function of a fuzzy capability-index estimate for each supplier. The preferred suppliers are selected by using a ranking method of fuzzy preference relations of suppliers. Finally, a case study of touch screens is pro-vided to describe the applicability that incorporates the fuzzy data into the problem of quality-based sup-plier selection and evaluation.

Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction

It is well recognized that suppliers play a crucial role in the pro-duction chain and hence in the long term viability of a company. Close working relationships with high performing suppliers are essential in modern production environments. Just-in-time, total quality management, and flexible manufacturing systems have be-come part of the standard vocabulary in management theory.

Sup-plier selection decisions are an important component of

production and logistics management for many firms. Such deci-sions entail the selection of individual suppliers to employ, and the determination of order quantities to be placed with the se-lected suppliers. Selecting right suppliers significantly reduces the material purchasing cost and improves corporate competitive-ness, which is why many experts believe that the supplier selection is the most important activity of a purchasing department (2005). Supplier selection is one of the most critical activities of pur-chasing management in a supply chain, because of the key role of supplier’s performance on cost, quality, delivery and service in achieving the objectives of a supply chain. With increasingly com-petitive global world markets, companies are under intense pres-sure to find ways to cut production and material costs to survive and sustain their competitive position in their respective markets. Therefore, an efficient supplier selection process and evaluation of supplier performance are becoming major challenges faced by the

manufacturing and purchasing, it needs to be in place and of signif-icant importance for successful supply chain management.

Usually, quality is a critical concern for most manufacturers while purchasing materials. The need of high-quality suppliers has always been an important issue for many manufacturing orga-nizations (1991). With reference toDickson (1966), quality and delivery are two of the most demanded items by component sup-pliers. Similarly,Weber, Current, and Benton (1991) considered quality to be of ‘‘extreme importance” and delivery to be of ‘‘con-siderable importance”. In additions, Weber’s research on the Just-In-Time (JIT) model, the importance of quality and delivery remains the same. In another study,Pearson and Ellram (1995) surveyed 210 members of the National Association of purchasing management (NAPM), they were randomly selected from the list-ings of electronic firms, 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. Addi-tionally, there are many researchers studied about the supplier selection topic in the past period.Table 1summarizes the results from various papers. Obviously, quality can be regarded as a funda-mental factor for supplier evaluation among various criteria.

Much evidence suggest that high quality has a positive impact upon significantly increasing profitability, through lowing operat-ing costs and improvoperat-ing market share (Chen & Chen, 2009; Garvin, 1988; Maani, 1989; Phillips, Chang, & Buzzell, 1983; Voehl, Jackson, & Ashton, 1994).Kane (1986)stated that the quantification of the process mean (

l

) and variation ð

r

2_{Þ is essential to understand the}

quality of the units produced from a manufacturing process.

Tagu-0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.02.106

*Corresponding author.

E-mail address:[email protected](M.-H. Shu).

Contents lists available atScienceDirect

Expert Systems with Applications

chi emphasized the loss occurred in a product’s worth when its key quality characteristic deviates from the customers’ target

s

¼ ðUSL þ LSLÞ=2, where USL and LSL stand for the upper and lower specification limits, respectively, and the values of USL and LSL are determined by decision-makers. In order to take into account these basic parameters that have been widely used to measure the man-ufacturing processes performance or supplier potentials, Hsiang and Taguchi (1985)introduced Cpmindex defined as

Cpm¼ USL  LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

2_{þ ð}

_l

_

_s

_Þ2 q ; ð1Þ

sometimes called the Taguchi index or loss-based capability.Table 1 lists the various values of Cpmand its corresponding maximum

pos-sible nonconformities in parts per million (PPM). The value of Cpmis

varied from the lower value of 1.00 to the upper value of 2.00 with increments of 0.05 at each step. For example, if a process has capa-bility with CpmP1:2, then the production yield would be at least

99.968%. In other words, the number of the nonconformities is less than 318.2 PPM. Cpm PPM Cpm PPM Cpm PPM Cpm PPM 0.95 4371.923 1.30 96.193 1.55 3.319 1.80 0.067 1.00 2699.796 1.35 51.218 1.60 1.587 1.85 0.029 1.10 966.848 1.40 26.691 1.65 0.742 1.90 0.012 1.20 318.217 1.45 13.614 1.70 0.340 1.95 0.005 1.25 176.835 1.50 6.795 1.75 0.152 2.00 0.002

It is natural to investigate the problem of supplier selection and evaluation for the cases with q ðq P 2Þ candidate suppliers based on the Cpm index. Let Pi be the population of supplier i with the

mean

l

i and variance

r

2i for i ¼ 1; 2; . . . ; q. The capability index

Cpmiof supplier i can be defined as follows: Cpmi¼ USL  LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

2 i þ ð

l

i

s

iÞ2 q ð2Þ for i ¼ 1; 2; . . . ; q.

Conceptually, in evaluating a group of suppliers, the assessment requires knowledge of

l

iand

r

iof each supplier in Eq.(2).

How-ever,

l

i and

r

i usually unknown for i ¼ 1; 2; . . . ; q. In this case,

the sample data must be collected from each supplier which is in order to estimate the value of index Cpmiand to assess/select the

appropriate suppliers. Let xi1;xi2; . . . ;xini be the independent

ran-dom samples from Pifor i ¼ 1; 2; . . . ; q. Generally, continuous data

obtained from the output responses of supplier’s key quality char-acteristics are always assumed to be real numbers as in the studies byPrasad and Calis (1999), Shiau, Chiang, and Hung (1999), Zim-mer, Hubele, and Zimmer (2001), Pearn and Shu (2003), Xekalaki and Perakis (2004)and Hsu and Shu (2008). In this assumption, the statistical point estimate ^cpmiof Cpmiis given as

^ cpmi¼ USL  LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 i þ ðxi

s

Þ2 q ; ð3Þ

where the process mean

l

i in Eq. (2)is switched by the sample

mean xithat is given by  xi¼ 1 ni Xni j¼1 xij

and the process standard deviation

r

iin Eq.(2)is replaced by the

sample standard deviation sithat is given by si¼ 1 ni Xni j¼1 ðxij xiÞ2 " #1=2 for i ¼ 1; 2; . . . ; q.

In a practical situation, the output continuous quantities col-lected from key quality characteristics of suppliers’ products al-ways appear to be somewhat imprecise manner. For example, the data may be given by color intensity pictures or by the read-ings on an analogue measurement equipment, as in the studies of Filzmoser and Vertl (2004) and Viertl and Hareter (2004). In addition, the imprecise data may come from the insufficient sam-ple data such as the observations made with coarse scales, linguis-tic data, or data collected with vague and incomplete knowledge, as described by Sugano (2006), Gulbay and Kahraman (2007), Zhang and Chu (2009), and Lee (2009). In the other study,Hong (2004) and Lee (2001) proposed an estimation of single yield-based index by considering fuzzy numbers when the measurement refers to the decision-making’s subjective determination. Since supplier selection problems is usually involved with preferences which are often vague and imprecise. In this paper, we propose a method for the selection and evaluation of supplier using fuzzy data.

The paper is organized as follows. In Section2, we introduce the basic properties of fuzzy numbers. In Section3, the fuzzy estimate of Cpmifor each supplier is expressed by using fuzzy data. To obtain

the membership function of fuzzy estimate of each supplier, the resolution identity theorem is applied and the membership degree can be obtained by solving optimization problems. In Section4, we provide a ranking method proposed byYuan (1991)to sort the fuz-zy estimates of Cpmi, which makes decision-makers being capable

of selecting the preferable suppliers. In Section5, we demonstrate the application of the proposed methodology to supplier selection and evaluation using fuzzy data. In Section6, the conclusions are presented.

2. Fuzzy numbers

The key idea of fuzzy set theory is that an element has a degree of membership in a fuzzy set. It is defined by a membership function, all the information about a fuzzy set is described by its

Table 1

Attributes for supplier selection.

No. Researcher Attributes for supplier selection

1 Gregory (1986) Quality, production plan and control system, amount of past business, purchasing item, price 2 Wagner, Ettenson,

and Parrish (1989)

Quality is the most important, the second one is delivery, the last one is cost

3 Pacheco (1989) Customer service, product quality, service, delivery, the quality of clerk

4 Houshyar and David (1992)

Price, quality, delivery, transportation cost 5 Chaudhry, Forst, and

Zydiak (1993)

Quality, delivery, price, capacity 6 Lau and Lau (1994) Quality, lead time, price

7 Anderson (1994) Financial status, product quality, geographical location, inventory, facility layout,

administration management, technical capability, delivery

8 Wilson (1994) Quality, service, delivery, price 9 Benion and Redmond

(1994)

Product characteristic is more important then service, supporting, and quality

10 Pearson and Ellram (1995)

Quality and cost are the most important. Then goes for supplier design and technical capability 11 Swift (1995) To emphasis on price, product quality. Under

single source circumstance, it is needed to evaluate technical supporting from supplier and the reliability of product

12 Patton (1996) Price, quality, delivery, service, equipment & technical, company’s financial status 13 Lambert, Ronald, and

Margaret (1997)

The most important attributes are including quality, delivery, and service

membership function. The membership function maps elements (crisp inputs) in the universe of discourse (interval that contains all the possible input values) to elements (degrees of membership) within a certain interval, which is usually [0, 1]. Then, the degree of membership specifies the extent to which a given element belongs to a set or is related to a concept. The most commonly used range for expressing degree of membership is the unit interval [0, 1]. Definition 2.1. A fuzzy subset ~a in a universe of discourse R is characterized by a membership function

e

~aðxÞ which associates

with each element x in R a real number in the interval [0, 1]. The fuzzy subset ~a of R is defined by a function

e

a~:R! ½0; 1, which is

called membership function. We can also write the fuzzy set ~a as fðx;

e

~aðxÞÞ : x 2 Rg. We denote ~aa¼ fx :

e

~aðxÞ P

a

g as the

a

-level

set of ~a, where ~a is the closure of the set fx :

e

~aðxÞ – 0g.

Proposition 2.1 Zadeh (1965). ~a is a convex fuzzy set if and only if fx :

e

~aðxÞ P

a

g is a convex set for all

a

.

Remark 2.1. Let ~a be a fuzzy number. We regard, ~a0, the 0-level set

of ~a as the closure of the set fx :

e

~aðxÞ – 0g. If ~a is a bounded fuzzy

number then ~a0is a compact set.

According to above definitions we realize that a convex and nor-malized fuzzy set defined on real numbers whose membership function is piecewise continuous is called a fuzzy number, it must possess the following four properties:

r ~a is normal, i.e., there exists an x 2 R such that

e

~aðxÞ ¼ 1.

s

_e

_{~ais a quasi-concave, i.e.,}

_e

_{a~ðtx þ ð1  tÞyÞ P minf}

_e

_~aðxÞ;

_e

_~aðyÞg for t 2 ½0; 1.

t

_e

_{~ais upper semiconscious, i.e., fx 2 R :}

_e

_{~aðxÞ P}

_a

_{g is a closed} subset of R for each

a

2 ð0; 1.

u The 0-level set ~a0is a closed and bounded subset of R.

Fig. 1 shows a fuzzy number of the universe of discourse R which is both convex and normal.

Proposition 2.2. If ~a a closed fuzzy number then the

a

-level set of ~a is a closed interval which is denoted by ~aa¼ ~aLa; ~aUa

 

. (That is why we call ~a as closed fuzzy number.)

Proof.

e

~a is an upper semiconscious function, so the

a

-level set

~

aa¼ fx :

e

a~ðxÞ P

a

g is a closed set. Since ~a is a convex fuzzy set,

byProposition 2.1, ~aais a closed interval. From Remark 2.1and Proposition 2.2, we have the following definition. h

Definition 2.2. The

a

-level set of ~a, denoted by ~aa, is defined by

~

aa¼ fx 2 R :

e

~aðxÞ P

a

g where

a

2 ð0; 1. The 0-level set ~a0 is

defined as the closure of the set fx 2 R :

e

~aðxÞ > 0g, i.e.,

~

a0¼ clðfx 2 R :

e

~aðxÞ > 0gÞ ¼ clðUa>0~aaÞ.

From above descriptions, ~aais non-empty, closed, bounded and

convex subset of R, it can be defined as ~aa¼ ~aLa; ~aUa

 

, where ~aL

aand

~ aU

aare the lower and upper bounds of the closed interval,

respec-tively (defined byZimmer et al. (2001)).Fig. 2shows fuzzy number ~

a with

a

-level where ~aa1¼ aLn1;aUn1

 

, and ~aa2¼ aLn2;aUn12

 

. Fig. 3 indicates a fuzzy number can be given by a set of nested intervals, the

a

-levels: ~a; ½a₁ ½a_0:7 ½a_0:5 ½a_0:2 ½a₀.

Remark 2.2. Let ~a be a fuzzy number. It is easy to see that ~aL

a describes increasing with respect to

a

on [0, 1] and ~aU

a describes decreasing with respect to

a

on [0, 1]. On the other hand, ~a is said to be a fuzzy real number, if ~a is a fuzzy number and lð

a

Þ ¼ ~aL

a and uð

a

Þ ¼ ~aU

a are continuous functions on [0, 1].

Remark 2.3. Let ~a be a fuzzy number such that its membership function describes strictly increasing on the interval ~aL

0; ~aL1

 

and strictly decreasing on the interval ~aU

0; ~aU1

 

. From the fact of strict monotonicity, we see that lð

a

Þ ¼ ~aL

a and uð

a

Þ ¼ ~aUa are continuous

function on [0, 1]. Hence, we are convinced ~a is also a fuzzy real number.

Resolution Identity is a renowned formula in fuzzy sets theory. The knowledge of the following result is very useful for us to con-struct the fuzzy estimate of Cpmi.

Let g and h be two functions from [0, 1] into R, and let faa¼ ½gð

a

Þ; hð

a

Þ : 0 6

a

61g be a family of closed intervals. Then we can induce a fuzzy set ~a with the membership function

e

a~ðxÞ ¼ sup

06a61

a

1aaðxÞ

by resolution identity theorem.

We say that faa:0 6

a

6_{1g is decreasing if aa#abfor}

a

_>b. Proposition 2.3. Let faa¼ ½gð

a

Þ; hð

a

Þ : 0 6

a

61g be a family of closed intervals and ~a be induced by faag. If a1–; then ~a is a normal

fuzzy set.

Proof. Let r 2 a1. Then

e

~aðxÞ ¼ sup06a61

a

1AaðxÞ ¼ 1. h

Definition 2.3. A triangular fuzzy number ~a can be defined by a triplet (a1, a2, a3) shown inFig. 4. The membership function

e

~aðxÞ

is defined as:

e

a~ðxÞ ¼ ðx  a1Þ=ða2 a1Þ if a16x 6 a2; ða3 xÞ=ða3 a2Þ if a2<x 6 a3; 0 otherwise: 8 > < > :

Fig. 1. A fuzzy number ~a.

Fig. 2. Fuzzy number ~a witha-level.

The triangular fuzzy number ~a can be expressed as ‘‘around a2”

or ‘‘being approximately equal to a2”, where a2is called the core

value of ~a, and a1and a3are called the left and right spread values

of ~a, respectively. The

a

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

aa¼ ½ð1 

a

Þa1þ

a

a2;ð1 

a

Þa3þ

a

a2.

That is, ~aL

a¼ ð1 

a

Þa1þ

a

a2and ~aUa¼ ð1 

a

Þa3þ

a

a2. We also

see that the triangular fuzzy number is a fuzzy real number. 3. Estimation of Cpmiusing fuzzy quality data

The significance of the methodology proposed is that it is capa-ble of dealing with fuzzy data, which are known to represent the uncertainties in the real world. Now, we will present the fuzzy esti-mation of Cpmifor each supplier i using fuzzy data.

Let us consider ~xi1; . . . ; ~xinias fuzzy observations (fuzzy data) and

assume all of them are fuzzy real numbers. Therefore, for any given

a

2 ð0; 1, we can get the corresponding real-data ð~xijÞLaand ð~xijÞUafor

i ¼ 1; . . . ; q and j ¼ 1; . . . ; ni. Using the real-valued data

ð~xi1ÞLa; . . . ;ð~xiniÞLa and substituting in Eq. (2), we can obtain the

estimate ^ cL pmia¼ USL  LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 i  L aþ xLia

s

 2 q ; ð4Þ where  xL ia¼ 1 ni Xni j¼1 ð~xijÞLa and s2i  L a¼ 1 ni 1 Xni j¼1 ð~xijÞLa xLia h i2 :

Likewise, using the real-valued data ð~xi1ÞUa; . . . ;ð~xiniÞUa, we can also

obtain the estimate

^ cU pmia¼ USL  LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 i  U aþ x U ia

s

 2 q ; ð5Þ  xU ia¼ 1 ni Xni j¼1 ð~xijÞUa and s2i  U a¼ 1 ni 1 Xni j¼1 ð~xijÞUa xUia h i2

Now, we define the closed interval by Aiaas

Aia¼ min ^cLpmia; ^cUpmia

n o ;max ^cL pmia; ^cUpmia n o h i :

By applying to this result to ‘‘resolution identity” introduced in Sec-tion2, we can obtain the membership function of fuzzy estimate ~ ^ cpmi of Cpmias

e

~^cpmiðrÞ ¼ sup 06a61

a

 1AiaðrÞ: ð6Þ

It should be noted that the fuzzy estimate shown in Eq.(6)is completely different from the fuzzy estimate obtained byParchami & Mashinchi (2007) & Hsu & Shu (2008); in these studies the result of the fuzzy estimate of Cpmiwas obtained by applying Buckley’s

approach using real-value data and Eq.(3)(i.e., fuzzifying a

real-valued function as a fuzzy-real-valued function). However, in this study we take into account fuzzy data. Let us also write Aia¼ ½lið

a

Þ; uið

a

Þ,

where

lið

a

Þ ¼ min ð^cpmiÞLa;ð^cpmiÞUa

n o

and uið

a

Þ ¼ max ð^cpmiÞLa;ð^cpmiÞUa

n o

: ð7Þ

Since each ~xijis a fuzzy real number, it is clear that ð~xijÞLaand ð~xijÞUa

are continuous with regard to ~a on [0, 1]. This also implies that  xL ia; s2n  L ia; xUia, and s2n  U

iaare continuous with regard to ~a on [0, 1].

From these facts, the

a

-level set ~^cpmi

 

aof fuzzy estimate ~^cpmican

be expressed as ~ ^ cpmi  L a; ~^cpmi  U a ¼ ~^cpmi   a¼ r :

e

~^cpmiðrÞ P

a

n o ¼ min a6b61liðbÞ; maxa6b61uiðbÞ ; ð8Þ

where liðbÞ and uiðbÞ refer to Eq.(7). From Eq.(8), the relationship

between ~^cpmi  L aand ~^c L pmiagiven as ~ ^ cpmi  U

a ¼ mina6b61liðbÞ ¼ mina6b61min ~ ^ cpmi  L b; ~ ^ cpmi  U b   : ð9Þ

Also, the relationship between ~^cpmi

 U a and ~^c U pmiagiven as ~ ^ cpmi  U

a ¼ maxa6b61uiðbÞ ¼ maxa6b61max ~ ^ cpmi  L b ; ~^cpmi  U b   : ð10Þ

Proposition 3.1. The fuzzy estimate ~^cpmidefined in Eq.(6)is a fuzzy

real number.

Proof. FromDefinition 2.1, since the closed interval Ai(i.e.,

a

¼ 1)

is not an empty set, condition r is satisfied. Since the

a

-level set ~

^cpmi

 

ain Eq.(8)is a closed subset of R, i.e., a convex subset of

R, conditions s–u are satisfied. This says that ~^cpmis a fuzzy

num-ber. Since lið

a

Þ and uið

a

Þ are also continuous with respect to

a

on

[0, 1]. This shows that ~^cpmiis indeed a fuzzy real number. h

From this section, we present the computational method to ob-tain the membership degree of any given value r of the fuzzy esti-mate ~^cpmi for i ¼ 1; . . . ; q. By applying the result of ‘‘resolution

identity” shown in Section 2, the membership function of ~^cpmi

can be written by

e

~aðrÞ ¼ sup

06a61

a

 1ð~^cpmiÞ_aðrÞ:

Therefore, given any value r of ~^cpmi, its membership degree can

be obtained by solving the following optimization problem

max

a

subject to ~^_cpmiL a 6_{r 6 ~^cpmi}  U a 0 6

a

61; where ~^cpmi  L aand ~^cpmi  U

a are from Eqs.(9) and (10), respectively.

For the notational convenience, we also write

g

ð

a

Þ ¼ ~^cpmi

 L

a¼ mina6b61liðbÞ ¼ mina6b61min ~ ^ cpmi  L b; ~ ^ cpmi  U b   ð11Þ and fið

a

Þ ¼ ~^cpmi  U

a¼ maxa6b61uiðbÞ ¼ maxa6b61max ~ ^ cpmi  L b; ~ ^ cpmi  L b   : ð12Þ

An optimization subroutine called ‘‘fmincon” which is built-in in the commercial software MATLAB, can be used to construct the h-level set ~^cpm

 

h of the fuzzy estimate ~^cpmby solving the nonlinear

programming problems with bounded variables that are given in Eqs.(11) and (12). 1

( )

x a ~ ε ℜ M embershi p function 1 a a2 a3 0 1

( )

x a ε ℜ M embershi p function 1 a a2 a3 0

4. Supplier selection

4.1. Fuzzy preference relation (FPR)

In this section, we follow the ranking method for the set of all fuzzy numbers proposed byYuan (1991)who claimed that her/ his ranking method is very reasonable based on four criteria that are fuzzy preference presentation, rationality of fuzzy ordering, distinguish ability and robustness. The results obtained by Yuan are based on the normal and convex fuzzy sets; that is, the fuzzy sets satisfy condition r and s onDefinition 2.1introduced in Sec-tion2. Now, we use our notations to rewrite the main results were obtained byYuan (1991). Let ~^cpmiand ~^cpmjbe the fuzzy estimates of

Cpmindices of suppliers i and j, respectively. Then we can define a

fuzzy preference relation (FPR) between ~^cpmiand ~^cpmjas FPR ~^cpmi; ~^cpmj   ¼

D

ij

D

ijþ

D

ji ; ð13Þ where

D

ij¼ Z fa:fiðaÞ>gjðaÞg ðfið

a

Þ 

g

jð

a

Þ d

a

Þ þ Z fa:giðaÞ>fjðaÞg ð

g

ið

a

Þ  fjð

a

ÞÞ d

a

;

D

ji¼ Z fa:fiðaÞ>giðaÞg ðfjð

a

Þ 

g

ið

a

Þ d

a

Þ þ Z fa:gjðaÞ>fiðaÞg ð

g

jð

a

Þ  fið

a

ÞÞ d

a

andDijþDji¼R01ðjfið

a

Þ 

g

jð

a

Þj þ jfjð

a

Þ 

g

ið

a

ÞjÞ d

a

.

The value of FPR ~^cpmi; ~^cpmj

 

shown in Eq. (13)will be within [0, 1] indicating the degree of preference.Yuan (1991)suggested that if FPR ~^cpmi; ~^cpmj

 

>0:5, then ~^cpmiis more preferred to ~^cpmj. In

a more general setting, the decision-maker can set up the rules to determine the preference between ~^cpmiand ~^cpmj. Let

c

be a

pre-determined value in [0, 1] with

c

>0:5. The rules are suggested and summarized inTable 2.

Likewise, we consider the fuzzy preference relation as FPR ~^cpmj; ~^cpmi

 

. Then the suggestive rules are provided inTable 3. On the basis of the rules ofTables 2 and 3, we are convinced that the rules are reasonable, no any conflict between them due to

FPR ~^cpmj; ~^cpmi

 

¼ 1  FPR ~^cpmi; ~^cpmj

 

: ð14Þ

4.2. Verification and interpretation for FPR

Firstly, from Eq.(14), we see that the rule (iii) is equivalent to the rule (iv). In other words, the rule (iii) can be rewritten as: ~

^

cpmj is more preferred to ~^cpmi with the preference degree

c

. If

FPR ~^cpmi; ~^cpmj

 

<1 

c

. Secondly, it obviously can realize the rule (vi) is the counterpart of the rule (i) according to Eq.(14). In other words, the rule (vi) can be rewritten as: ~^cpmiis more preferred to

~

^cpmjwith the preference degree

c

. If FPR ~^cpmj; ~^cpmi

 

<1 

c

. Finally, from Eq.(14)again, we are convinced that the rule (ii) is equal to the rule (v). In other words, if ~^cpmjis indifferent from ~^cpmiwith the

indifference degree ð

c

;1 

c

Þ, then ~^cpmiis indifferent from ~^cpmjwith

the indifference degree ð

c

;1 

c

Þ, and vice versa. Based on the above observations, it suffices to consider the fuzzy preference relation FPR ~^cpmi; ~^cpmj

 

to select the most preferred supplier. Calculating the value of FPR ~^cpmi; ~^cpmj

 

by resort to the numer-ical integration to obtain the value of Dij and Dji, we can sort

fS1; . . . ;Sqg into fSh1; . . . ;Shqg such that, for i < j, ~^cpmhiis more

pre-ferred to ~^cpmhjor ~^cpmhi is indifferent from ~^cpmhj on the basis of the

above rules (i)–(iii).

According to above descriptions, there are some interesting observations made. We know two suppliers Siand Sjare indifferent

from each other if and if only 1 

c

6 ~^c_{pmi; ~}^c_pmj

 

6

c

. Therefore, we can tell that if

c

is close to 0.5, then we have less chance to con-clude that these two suppliers Si and Sj are indifferent from each

other. On the contrary, if

c

is large, then we have more chance to conclude that the suppliers Si and Sj are indifferent from each

other. For the extreme case by taking the preference degree

c

¼ 0:5, we see that the two suppliers Siand Sjare indifferent from

each other if and only if FPR ~^cpmi; ~^cpmj

 

¼ 0:5 by considering it, we have a great chance to obtain a total order sequence fShi; . . . ;Shqg;

that is, we have a great chance to conclude that Shiis more

pre-ferred to Shjfor any i < j. However, this sequence has a slight

draw-back when the value of FPR ~^cpmhi; ~^cpmhiþ1

 

is close to 0.5. For example, if FPR ~^cpmhi; ~^cpmhiþ1

 

¼ 0:5012, under

c

¼ 0:5, the-oretically, we need to conclude that Shiis more preferred to Shiþ1.

However, from the point of human’s view/intuition that makes us think the suppliers Shiand Shiþ1 should be regarded as

indiffer-ence. FormFig. 5, we can definitely see the curve Shiand curve

Shiþ1 are almost overlapped. The good thing is that we can

over-come this drawback by taking a larger value of

c

and solve this problem.

Moreover, if the value of

c

is taken to be too large, then there will be too many suppliers that are classified as indifference, which will not be helpful for making decision. Therefore, the decision-maker should take a suitable value of

c

. In terms of experts and engineer’s advice/experience, the value of

c

is suggested to assign between 0.55 and 0.65, i.e.,

c

2 ½0:55; 0:65.

Now, let us consider two groups of suppliers A ¼ fShi; . . . ;Shiþng

and B ¼ fShj; . . . ;Shjþmg that are classified as indifference, where

i + n < j; that is, any two suppliers in A (resp. B) are indifferent from each other. As we mentioned before, if the value of

c

is large, then the numbers of A and B will be large. In this case, it is not helpful for the decision-maker to select the preferred suppliers. Yet, one thing can be sure is that FPR ~^cpmhiþr; ~^cpmhjþs

 

>

c

for any Shiþr2 A

and Shiþr2 B. Since

c

is assumed to be large in this case, it can be

Table 2

Suggestive rule to determine the preference from FPR ~^cpmi; ~^cpmj

 

. No. Scenario Better fuzzy

process Preference degree (i) _{FPR ~}^cpmi; ~^cpmj   >c ~^cpmiis more preferred to ~^cpmj

With the preference degreec (ii) _{1 c}6FPR ~^cpmi; ~^cpmj   <c ~^cpmiis indifferent from ~ ^ cpmj

With the indifference degree ðc;1 cÞ (iii) _{FPR ~^c} pmi; ~^cpmj   <1 c ~^cpmiis less preferred to ~^cpmj

With the non-preference degreec

Table 3

Suggestive rule to determine the preference from FPR ~^cpmj; ~^cpmi

 

. No. Scenario Better fuzzy

process Preference degree (i) _{FPR ~}^cpmj; ~^cpmi   >c ~^cpmjis more preferred to ~^cpmi

With the preference degreec (ii) _{1 c}6FPR ~^cpmj; ~^cpmi   <c ~^cpmjis indifferent from ~ ^ cpmi

With the indifference degree ðc;1 cÞ (iii) _{FPR ~^c} pmj; ~^cpmi   <1 c ~^cpmjis less preferred to ~^cpmi

With the non-preference degreec

inferred that Shiþr is preferred to Shjþsvery much. In other words,

the group A is preferred to the group B very much. Therefore, the best case is that the value of

c

is large and the number of the most preferred group of indifferent suppliers is small.

4.3. The procedure for selecting the most preferred supplier using FPR We provide the following step-by-step procedure for supplier selection using the fuzzy preference relation to each ordered pair with fuzzy data and rank the alternatives fS1; . . . ;Sqg:

Step 1: Set the number q of suppliers and the value of

c

such that

c

>0:5.

Step 2: Obtain the fuzzy estimate ~^cpmi for each supplier

i ¼ 1; 2; . . . ; q according to the above computational method.

Step 3: Calculate the value of FPR ~^cpmi; ~^cpmj

 

by using commercial software MATLAB with to get the values ofDijandDji. We

sort fSi; . . . ;Sqg into fShi; . . . ;Shqg such that, for any

i < j; ~^cpmhiis more preferred to ~^cpmhjor ~^cpmhiis indifferent

from ~^cpmhjaccording to the above rules (i)–(iii).

Step 4: The most preferred group of suppliers is fShi; . . . ;Shiþtg,

where any two suppliers are indifferent from each other. Step 5: If

c

is small, then the decision-maker may randomly select one of the supplier from fShi; . . . ;Shiþtg as the most

pre-ferred supplier. If

c

is large, then the decision-maker may provide some other criteria to rank the set of

suppli-ers fShi; . . . ;Shiþtg again until the most preferred supplier

is located. In fact, if the company is allowed to select more than one supplier to supply the required materials, then the decision-maker can use the same procedure to simi-larly select the most preferred suppliers from the most preferred group suppliers fShi; . . . ;Shiþtg.

5. Application of the model to the case study

The objective of this section is to illustrate a case study to dem-onstrate the application of fuzzy sets theory to supplier selection and evaluation. In this example we take into account the fuzzy data.

5.1. Touch screen introduction

Touch screens are display overlays which have the ability to dis-play and receive information on the same screen. The effect of such overlays allows a display to be used as an input device, removing the keyboard and/or the mouse as the primary input device for interacting with the display’s content. The systems are designed to help individuals who have difficulty manipulating a mouse or keyboard. Such displays can be attached to computers or, as termi-nals, to networks. Touch screens also have assisted in recent changes in the design of personal digital assistant (PDA), satellite navigation, and mobile phone devices, making these devices more usable. It is an easy to use input device that allows users to control PC software and DVD video by touching the display screen.

The touch screen uses a glass panel with a uniform conductive ITO (indium tin oxide) coating on the one-side surface. A PET film is tightly suspended over the ITO coating surfaces of a glass panel. The glass substrate and the PET film are separated by tiny, trans-parent insulating dot spacers. The PET film has a hard coating on the outer side and a conductive ITO coating on the inner side. We can clearly see the structure schema of touch screen from Fig. 6.

With the growing number of outdoor touch applications and with the advent of HR (highly reflective) LCDs, the production of touch screens using anti-reflective coatings is increasing. The amount of light which passes through a solid is impacted both by its color absorption and the bending of light at the air/surface interface. For example, a glass window passes about 95% of the light; while the color absorption is minimal, the bending of the light going in and coming out of the glass ‘uses” up about 5% of the light. One can see the impact of this phenomenon as a reflec-tion from the first surface. Oddly, by adding purple tinged anti-reflective materials to glass, the light transmission rate can exceed the ordinary 95%.

Almost all touch screens put a silica coating on the first surface to diffuse the reflective light and reduce the mirror effect. This is called an anti-glare coating, not to be confused with an

anti-reflec-0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Suppliers Shi and Shi+1

Cpm Shi Shi+1

α

Shi Shi+1

Fig. 5. Curve Shiand curve Shiþ1.

tive coating. Anti-glare coatings are relatively inexpensive and en-hance the scratch resistance of the touch panel. However, anti-glare coatings diffuse the image and consequently reduce the sharpness of the display image, reducing visual clarity. The amount of diffusion is measured as the haze factor.

Anti-reflective (AR) coatings are very thin. Measured in ratio of the size of a light wave, these coatings literally trap light within the AR coating. When the touch screen does not reflect light, the image behind the touch panel will be brighter and certainly more easily read. However, creating anti-reflective coatings, which is an extre-mely precise process, is very expensive.

Among these quality characteristics, the light transmission rate is the most critical one.Fig. 7shows the integration schema of LCD and the touch screen.

5.2. An overview company

Company chosen for this research study is a famous open frame manufacturing industry located in the United States California. The company planned to improve the quality of the product. They planned to purchase the quality raw material at low cost and at a short duration of time. Instead of purchasing the material from the single supplier they noted that five alternative suppliers, namely supplier 1 (S1), supplier 2 (S2), supplier 3 (S3), supplier 4

(S4), and supplier 5 (S5) were taken into consideration. Top

man-ager of the company needs to decide to choose the most preferred supplier among these suppliers based on the quality and the fuzzy

sample data. Let us consider the fuzzy sample data of the light transmission rate with size 20 have been collected from each sup-plier listed inTable 4.

The upper and lower specification limits of light transmission rate are set as USL = 95% and LSL = 85%, respectively. And the target

value is set at

s

¼ 90%. Then the estimates of statistics

xL

ia; sLia; ^cLpmia; xUia; sUia, and ^cUpmiain the

a

-level sense of the five

sup-pliers are shown inTable 5.

5.3. Result analysis

Using the commercial software MATLAB, can calculate the

a

-le-vel sets, the graphs of membership functions of fuzzy estimates ~

^cpmifor i ¼ 1; . . . ; 5 can be constructed and shown asFig. 8.

After pair-wise comparisons were obtained and entered into data matrices. The value ofDij and Dji are tabulated in the 2nd

and 3rd column inTable 6, and the values of the fuzzy preference relation FPR ~^cpmi; ~^cpmj   and FPR ~^cpmj; ~^cpmi   ¼ 1  FPR ~^cpmi; ~^cpmj   for i; j ¼ 1; 2; 3; 4; 5 and i < j are tabulated in the 4th and 5th column in Table 6:

(1) After finding the values of the fuzzy preference relation FPR ~^cpmi; ~^cpmj   and FPR ~^cpmj; ~^cpmi   ¼ 1  FPR ~^cpmi; ~^cpmj   for i; j ¼ 1; 2; 3; 4; 5 and i < j. We draw an analogy of the value of

c

is set to be 0.50. In the light of Step 2 in the above pro-cedure, the sequence of suppliers arranged by considering the preference of selection is given by fS5;S1;S4;S2;S3g. On

the basis of the result of the sorting we are convinced that supplier 5 is the most preferred choice and supplier 3 is the worst choice among all five suppliers with the prefer-ence degree 0.50.

(2) Furthermore, if the value of

c

is set to be 0.65, then the sequence goes for fðS2;S3Þ; ðS2;S4Þ; ðS3;S4Þg, in which S2 is

indifferent from S3, S2is indifferent from S4, and S3is

indif-ferent from S4 with indifference degrees (0.35, 0.65), that

can be written as below.

0:35 6 FPR ~^cpm2; ~^cpm3   60:65 and 0:35 6 FPR ~^cpm3; ~^cpm2   6_0:65 and 0:35 6 FPR ~^cpm2; ~^cpm4   6_{0:65 and} 0:35 6 FPR ~^cpm4; ~^cpm2   60:65 and

Fig. 7. Integration schema of LCD and touch screen.

Table 4

Triangular fuzzy data collected from the suppliers (unit: %).

S1 S2 S3 S4 S5 (87, 90, 93) (90, 93, 94) (88, 91, 92) (89, 90, 91) (88, 91, 92) (88, 92, 93) (91, 92, 95) (89, 90, 93) (91, 92, 94) (89, 91, 93) (89, 92, 95) (88, 91, 95) (86, 89, 92) (87, 89, 91) (86, 89, 92) (90, 91, 92) (90, 91, 92) (90, 91, 92) (85, 89, 97) (89, 90, 91) (86, 90, 93) (91, 92, 95) (89, 90, 93) (90, 92, 93) (88, 91, 93) (88, 91, 92) (92, 93, 94) (90, 91, 92) (85, 89, 97) (90, 91, 92) (90, 93, 94) (92, 94, 95) (80, 82, 83) (91, 92, 93) (91, 92, 93) (90, 91, 92) (91, 93, 94) (89, 91, 92) (88, 89, 96) (89, 91, 92) (88, 91, 93) (91, 93, 94) (89, 91, 92) (90, 91, 95) (90, 91, 92) (88, 92, 93) (90, 91, 92) (88, 89, 90) (77, 79, 83) (88, 89, 90) (89, 90, 93) (91, 92, 93) (89, 90, 91) (92, 93, 94) (82, 86, 88) (93, 94, 95) (93, 94, 96) (91, 92, 94) (91, 92, 95) (90, 91, 92) (88, 90, 92) (89, 91, 93) (87, 89, 91) (90, 91, 95) (88, 90, 91) (86, 89, 90) (87, 91, 92) (85, 89, 90) (90, 91, 92) (85, 89, 90) (91, 93, 94) (92, 94, 95) (90, 92, 93) (91, 92, 95) (91, 92, 94) (88, 89, 92) (88, 91, 94) (86, 89, 92) (92, 93, 94) (88, 89, 92) (90, 92, 93) (93, 94, 95) (91, 92, 93) (92, 94, 95) (90, 92, 93) (89, 91, 92) (90, 91, 92) (88, 89, 90) (91, 93, 94) (89, 90, 91) (89, 91, 93) (90, 93, 94) (88, 91, 92) (91, 93, 94) (89, 90, 92) (80, 85, 89) (79, 81, 85) (77, 79, 83) (90, 91, 92) (87, 89, 90)

0:35 6 FPR ~^cpm3; ~^cpm4   6_{0:65 and} 0:35 6 FPR ~^cpm4; ~^cpm3   60:65:

Clearly, with the preference degree 0.65, supplier S5 is the

most preferred supplier to supply this particular touch screen for manufacturing.

6. Conclusions

Fuzzy logic is a powerful problem-solving methodology with a myriad of applications in embedded control and information pro-cessing. Fuzzy provides a remarkably simple way to draw definite conclusions from vague, ambiguous or imprecise information. In a sense, fuzzy logic resembles human decision making with its abil-ity to work from approximate data and find precise solutions. This paper proposed an approach for the selection of suppliers, which model is capable of handling fuzzy data and was not seriously trea-ted by the researchers.

The quality-based supplier selection and evaluation using the imprecise sample data has been applied for supplier selection. A general method is proposed to obtain the fuzzy estimate of the capability index Cpmiof supplier i using ‘‘resolution identity” in

fuz-zy sets theory.

In order to derive the membership degree of any given

c

of fuz-zy estimate ~^cpmi, the original problem is transformed into the

opti-mization problems. After obtaining the fuzzy estimates

~ ^

cpm1; . . . ; ~^cpmq

n o

of respective supplier fS1; . . . ;Sqg, we follow the

ranking method proposed byYuan (1991)to choose the most pre-ferred supplier. The application model/case study taken from the touch screen application is provided illustrate the applicability of the proposed methodology.

The results show that the model has the capability to be flexible and deal with fuzzy data to choose their supplier. The final priority of each alternative will lead to a recommended best option. It can be concluded that the model could facilitate decision making, and the approach could help in reduced time consuming efforts in the supplier selection process.

Table 5

Thea-level estimates of Cpmi.

a-level xL ia sLia ^cLpmia xUia sUia ^cUpmia S1 0.0 88.35 2.56 0.44 92.65 1.42 0.55 0.1 88.60 2.48 0.48 92.47 1.45 0.58 0.2 88.85 2.40 0.54 92.29 1.48 0.61 0.3 89.10 2.32 0.59 92.11 1.51 0.64 0.4 89.35 2.24 0.65 91.93 1.56 0.66 0.5 89.60 2.17 0.71 91.75 1.60 0.68 0.6 89.85 2.11 0.77 91.57 1.65 0.69 0.7 90.10 2.05 0.80 91.39 1.71 0.70 0.8 90.35 1.99 0.78 91.21 1.77 0.71 0.9 90.60 1.94 0.75 91.03 1.83 0.72 1.0 90.85 1.90 0.73 90.85 1.90 0.73 S2 0.0 89.90 3.02 0.54 93.45 2.33 0.22 0.1 90.09 2.99 0.55 93.28 2.36 0.24 0.2 90.27 2.96 0.53 93.11 2.39 0.26 0.3 90.46 2.92 0.52 92.94 2.43 0.28 0.4 90.64 2.90 0.50 92.77 2.47 0.30 0.5 90.83 2.87 0.48 92.60 2.51 0.32 0.6 91.01 2.85 0.47 92.43 2.56 0.33 0.7 91.20 2.83 0.45 92.26 2.61 0.35 0.8 91.38 2.81 0.43 92.09 2.67 0.36 0.9 91.57 2.80 0.41 91.92 2.73 0.38 1.0 91.75 2.79 0.39 91.75 2.79 0.39 S3 0.0 87.50 3.50 0.24 91.00 2.94 0.45 0.1 87.69 3.47 0.26 90.84 2.96 0.47 0.2 87.87 3.43 0.28 90.67 2.98 0.48 0.3 88.06 3.40 0.30 90.51 3.00 0.50 0.4 88.24 3.37 0.32 90.34 3.03 0.51 0.5 88.43 3.35 0.34 90.18 3.06 0.53 0.6 88.61 3.32 0.36 90.01 3.09 0.54 0.7 88.80 3.30 0.38 89.85 3.13 0.52 0.8 88.98 3.28 0.40 89.68 3.16 0.49 0.9 89.17 3.26 0.43 89.52 3.21 0.47 1.0 89.35 3.25 0.45 89.35 3.25 0.45 S4 0.0 89.15 3.53 0.39 93.50 3.00 0.17 0.1 89.31 3.48 0.41 93.23 2.93 0.20 0.2 89.47 3.44 0.43 92.95 2.88 0.24 0.3 89.63 3.39 0.45 92.68 2.84 0.27 0.4 89.79 3.35 0.48 92.40 2.83 0.31 0.5 89.95 3.32 0.50 92.13 2.84 0.34 0.6 90.11 3.28 0.50 91.85 2.86 0.37 0.7 90.27 3.25 0.49 91.58 2.91 0.39 0.8 90.43 3.22 0.47 91.30 2.98 0.41 0.9 90.59 3.19 0.46 91.03 3.06 0.43 1.0 90.75 3.16 0.45 90.75 3.16 0.45 S5 0.0 88.35 2.13 0.52 91.65 1.39 0.81 0.1 88.54 2.05 0.57 91.51 1.38 0.85 0.2 88.72 1.97 0.63 91.36 1.37 0.89 0.3 88.91 1.89 0.69 91.22 1.37 0.92 0.4 89.09 1.82 0.75 91.07 1.37 0.96 0.5 89.28 1.74 0.82 90.93 1.37 0.99 0.6 89.46 1.67 0.89 90.78 1.38 1.02 0.7 89.65 1.61 0.96 90.64 1.39 1.05 0.8 89.83 1.55 1.04 90.49 1.40 1.07 0.9 90.02 1.49 1.12 90.35 1.42 1.10 1.0 90.20 1.44 1.11 90.20 1.44 1.11 0.4 0.6 0.8 1 1.2 1.4 1.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cpm S5 S1 S2 S3 S4

α

S1 S2 S4 S5 S3

Fig. 8. The membership functions of fuzzy estimates of five suppliers. Table 6

Value of the fuzzy preference relation.

Pair-wise comparison Dij Dji FPR ~^cpmi; ~^cpmj

  FPR ~^cpmj; ~^cpmi   S1and S2(i = 1 and j = 2) 0.7635 0.0373 0.9534 0.0466 S1and S3(i = 1 and j = 3) 0.7623 0.0343 0.9569 0.0431 S1and S4(i = 1 and j = 4) 0.7509 0.0287 0.9632 0.0368 S1and S5(i = 1 and j = 5) 0.1047 1.0211 0.0930 0.9070 S2and S3(i = 2 and j = 3) 0.1664 0.1646 0.5028 0.4972 S2and S4(i = 2 and j = 4) 0.1589 0.1629 0.4938 0.5062 S2and S5(i = 2 and j = 5) 0.0105 1.3105 0.0079 0.9921 S3and S4(i = 3 and j = 4) 0.1492 0.1550 0.4904 0.5096 S3and S5(i = 3 and j = 5) 0.0098 1.3117 0.0074 0.9926 S4and S5(i = 4 and j = 5) 0.0060 1.3021 0.0046 0.9954

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