Selection of logistics center location using Axiomatic Fuzzy Set and TOPSIS methodology in logistics management

Expert Systems with Applications 38 (2011) 7901–7908

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Short Communication

Selection of logistics center location using Axiomatic Fuzzy Set and TOPSIS methodology in logistics management Ye Li a,b,⇑, Xiaodong Liu a,b, Yan Chen a a b

Transportation Management College, Dalian Maritime University, Dalian 116026, PR China School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, PR China

a r t i c l e

i n f o

Keywords: Logistic center location Axiomatic Fuzzy Set Technique for order preference by similarity to ideal solution Criteria Evaluation system

a b s t r a c t This article presents a comprehensive methodology for the selection of logistic center location. The proposed methodology consists of two parts: (i) AFS (Axiomatic Fuzzy Set) clustering method (Liu, Wang, & Chai, 2005) has been studied further to effectively evaluate logistics center location, and (ii) TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)-based final selection. The criteria, which are relevant in the selection of logistics center site, have been analyzed and identified, and the logistics center site evaluation system is built by using modern principles of town planning and logistics. A case fifteen regional logistics center cities and thirteen criteria are studied and the numerical results show that the proposed evaluation framework is reasonable to identify logistics center location, and it is effective to determine the optimal logistics center location even with the interactive and interdependent criteria/ attributes.  2010 Elsevier Ltd. All rights reserved.

1. Introduction The outsourcing of selection of logistics center location to modern logistics has become a common practice. The commonly known methodology for selection consist of two aspects, the first is determining the nature, which combines Analytic Hierarchy Process (AHP) with fuzzy comprehensive evaluation to criteria evaluation (e.g., Carranza, Hale, & Faassen, 2008; Paul, 1998; Sung, Chang, & Lee, 1999). The second is fixing quantify, which has continuous model and discrete mode (Goldengorin, Ghosh, & Sierksma, 2003; Klose & Drexl, 2005; Yang, Ji, Gao, & Li, 2007). Holmberg (2001) reviewed some of the contributions to the current state of facility location models for distribution system. Aikens (1985) considered selection of non-linearly transport costs, and using branch and bound method based on a dual ascent and adjustment procedure to solute. Tompkins and White (1984) introduced a method which used the preference theory to assign weights to subjective factors by making all possible pair wise comparisons between factors. Zhou, Min, and Gen (2002) investigated balanced allocation of customers to multiple distribution centers with a genetic algorithm approach. Syam (2002) investigated a model and methodologies for the location problem with logistical components. Chen (2001) was proposed a stepwise ranking procedure to determine the ranking

⇑ Corresponding author at: Transportation Management College, Dalian Maritime University, Dalian 116026, PR China. E-mail address: [email protected] (Y. Li). 0957-4174/$ - see front matter  2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.12.161

order of all candidate locations. A fuzzy preference relation matrix is constructed to represent the intensity of the preferences of one plant location over another. Cluster analysis is used for clustering a data set into groups of similar individuals. It is one of the major techniques in pattern recognition. Since Zadeh (1965) proposed fuzzy sets that use the idea of partial membership described by a membership function, many fuzzy clustering methods have been introduced. In popular fuzzy theories, the membership functions are often given subjectively by personal intuitions and the logic operations are implemented by a kind of triangular norms or shortly t-norm which is chosen in advance and independent of the distribution of the original data. However, in real world applications, fuzzy phenomena exist throughout nature and extensively within human society that it is impossible or difficult to define the member ship functions just by personal intuitions. In addition, different logic operator choices and membership function selections may lead to different results for the same data set. In order to cope with the above issues, the authors in Liu, Pedrycz, and Zhang (2003), Liu (1998a, 1998b, 1998c), Liu, Chai, and Wang (2007a, 2007b), Xu, Liu, and Chen (2009) proposed and developed the Axiomatic Fuzzy Set (AFS) theory, in which fuzzy sets (membership functions) and their logic operations are directly determined by a consistent algorithm according to the distributions of original data and the semantics of the fuzzy concepts. Recently, AFS theory has been developed further and applied to fuzzy decision trees (Liu & Pedrycz, 2007), credit rating analysis (Liu & Liu, 2005) and fuzzy clustering analysis (Ding, Liu, & Chen, 2006), etc.

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Most publications in selection of logistics center location focus on single or several important factors and seldom takes into account the more factors, the difficult calculation (Wang, 1992). With further research, appearing fuzzy comprehensive evaluation, and AHP, and Rough Set et al. methodology, which only evaluates from some special issue, and each has self-merit and disadvantage. If using one methodology to evaluate, the reliability of result will weak. In this paper, the comprehensive evaluation methodology of the selection of region logistics center location is proposed. The evaluation system is built, and the score of each city is calculated by AFS, according the factor score to clustering, and the final selection is determined by TOPSIS. 2. AFS theory 2.1. AFS algebras In Liu (1998c) defined was a family of molecular lattices, the AFS algebras, denoted as EI, EII, . . ., EIn, E#I, algebras. For more details the readers can refer to Liu et al. (2003), Liu (1998b, 1998c), Liu et al. (2003). Next these AFS algebras were applied to study the lattice valued representations of fuzzy concepts. The following example serves as an introductory illustration of the AFS algebra.

m1m4m5 is redundant when forming the left side of the fuzzy concept. Let us take into consideration two expressions of the form a: m1m4 + m2m3m5 and m: m2m4 + m2m5. The semantic content of the fuzzy concepts ‘‘a or m’’ and ‘‘a and m’’ can be expressed as follows ‘‘a or m’’: m1m4 + m2m3m5 + m2m4 + m2m5 equivalent to m1m4 + m2m4 + m2m5, ‘‘a and m’’: m1m2m4 + m1m2m4m5 + m2m3m4m5 + m2m3m5 equivalent to m1m2m4 + m2m3m5 The semantics of the logic expressions such as ‘‘equivalent to’’, ‘‘or’’ and ‘‘and’’ as expressed by P Q i2I( m2Aim), Ai # M, i 2 I, can be formulated in terms of the AFS algebra in the following manner. It is known that a lattice is a partially ordered set L in which any two elements a, b 2 L have a least upper-bound (i.e., a _ b) and a greatest lower bound (i.e., a ^ b). A partially ordered set L is called a complete lattice if every subset A # L has a supand an inf, denoted by _a 2 Aa and ^a2Aa, respectively. A complete lattice is called a completely distributive lattice, if one of the conditions shown below (CD1or CD2) holds

 i2I

Example 1. Let X = {x1, x2, . . . , x5} be a set of five cities and their features (attributes) which are described by traffic, communication, candidate land area, candidate land value and freight transport, see Table 1. Let X = {x1, x2, . . . , x5} be a set of five cities, M = {m1, m2, . . . , m5} be a set of fuzzy attributes on X, where m1: ‘‘Attribute1 is good’’, m2: ‘‘Attribute2 is good’’,. . ., m5: ‘‘Attribute5 is good’’. For each set of Q concepts A # M, m2Am represents conjunction of the concepts in A. For instance, A = {m1, m5} # M, Q m 2 Am = m1m6 representing a new fuzzy concept ‘‘traffic and P Q freight transport are good’’. For i2I( m2Aim), which is a formal Q sum of m 2 Aim, Ai # M, i 2 I, is the disjunction of the conjunctions Q represented by m2Aim’s (i.e., the disjunctive normal form of a formula representing a concept). For example, we may have c = m1m5 + m1m3 + m2 which translates as ‘‘traffic and freight transport are good’’or ‘‘traffic is good and candidate land area is big’’ or ‘‘communication is good’’. (the ‘‘+’’ denotes here a disjunction of concepts). While M may be a set of fuzzy or Boolean (two-valued) P Q concepts, every i 2 I( m 2 Aim), Ai # M, i 2 I, has a well-defined meaning such as the one we have discussed above. By a straightforward comparison of the expressions

m3 m4 þ m1 m4 þ m1 m2 m5 þ m1 m4 m5 and m3 m4 þ m1 m4 þ m1 m2 m5 : We conclude that their left side and right sides are equivalent. Considering the terms on the left side of the expression, for any x, the degree of x belonging to the fuzzy concept represented by m1m4m5 is always less than or equal to the degree of x belonging to the fuzzy concept representing by m1m4. Therefore, the term

i2I

^ aij

 ¼ Q _

j2J I

 ðCD1Þ _

 _ aij

ðCD1Þ ^

f2



j2J I

J i2I i

¼ Q ^ f2

J i2I i

 ^ aif ðiÞ ;

i2I



_ aif ðiÞ



i2I

Q where " i 2 I, " j 2 Ji, aij 2 L, and f 2 i2IJi means that f is a mapping f: I ? [i2IJi such that f(i)2Ji for any i 2 I. Let M be a non-empty set. The set EM⁄ is defined by 

EM ¼

( X i2I

Y

!

)

m jAi # M; i 2 I; I is a non-empty indexing set :

m2Ai

Definition 1 (Liu, 1998c). Let M be a non-empty set. A binary relation R on EM⁄ is defined as follows. For Q P Q P Q P P i 2 J( m2Bjm) 2EM⁄, [ i2I( m2Aim)]R[ i2J i2I ( m2Aim), (

Q

m2Bjm)],(i)

"Ai (i 2 I), $ Bh (h 2 J) such that Ai  Bh; (ii) "Bj

(j 2 J), $ Ak(k 2 I) such that Bj  Ak. It is clear that R is an equivalence relation. The quotient set EM⁄/ P Q P Q R is denoted by EM. The notation ( m)= i2J ( m2Bjm) P Q P Q i2I m2Ai means that i2I( m2Aim) and i2J ( m2Bjm) are equivalent under equivalence relation R. Thus the semantics they represent are equivalent. In Example 1, for n = m3m4 + m1m4 + m1m2m5 + m1m4m5, f= m3m4 + m1m4 + m1m2m5 2 EM, by Definition 1 we have P Q n = f. In what follows, each i2I ( m2Aim) 2 EM is called a fuzzy concept.

Table 1 Descriptions of features.

x1 x2 x3 x4 x5

Traffic

Communication

Candidate land area

Candidate land value

Freight transport

M H M VL L

H B.L&M H M L

H M B.M&H H B.H&VH

B.H&VH B.M&H H B.VL&L H

VH H M H B.H&VH

Where VL = Very Low, B.V L&L = Between Very Low and Low, L = Low, B.L&M = Between Low and Medium, M = Medium, B.M&H = Between Medium and High, H = High, B.H&V H = Between High and Very High, V H = Very High.

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Theorem 1 (Liu, 1998c). Let M be a non-empty set. Then (EM, _, ^) forms a completely distributive lattice under the binary compositions P Q P Q _ and ^ defined as follows. For any i2I( m2Aim), i2J( m2Bjm) 2 ⁄ EM

" X i2I

" X i2I

Y m2Ai

Y

!# m

_

i2J

!# m

" X

^

m2Ai

" X i2J

Y

!# m

m2Bj

Y

¼

!# m

¼

m2Bj

X

Y

k2I[J

m2Ck

! m

X

Y

i2I;j2J

m2Ai[Bj

! m

where for any k 2 I [ J (the disjoint union of I and J, i.e., every element in I and every element in J are always regarded as different elements in I [ J), Ck = Ak, if k 2 I, and Ck = Bk, if k 2 J. In Example 1, for

c ¼ m1 m5 þ m1 m3 þ m2 2 EM 







c0 ¼ ðm1 m5 þ m1 m3 þ m2 Þ0 ¼ m01 þ m05 ^ m01 þ m03 ^ m02   ¼ m01 þ m05 m03 ^ m02 ¼ m01 m02 þ m02 m03 m05

c0 , which is the logical negation of c =m1m5 + m1m3 + m2, reads as ‘‘traffic and communication are good’’ or ‘‘communication and freight transport are not good and candidate land area is not big’’. The authors proved that the operator ‘‘0 ’’ is an order-reversing involution of EI P Q algebra EM, if for any i2I( m2Aim) 2 EM,

X i2I

Y

!!0 m

0

¼ ^i2I ð_m2Ai m Þ ¼ ^i2I

m2Ai

X

meanings of the linguistic values, we have the following ordered relations: ‘‘VH’’ > ‘‘B.H&VH’’ > ‘‘H’’ > ‘‘B.M&H’’ > ‘‘M’’ > ‘‘B.L&M’’ > ‘‘L’’ > ‘‘B.VL&L’’ > ‘‘VL’’. By Table 1 and the semantic meanings of the attributes in M, we have

m1 : x4 < x5 < x1 ¼ x3 < x2

m01 : x4 > x5 > x1 ¼ x3 > x2

m2 : x5 < x2 < x4 < x1 ¼ x3 m3 : x2 < x3 < x4 ¼ x1 < x5

m02 : x5 > x2 > x4 > x1 ¼ x3 m03 : x2 > x3 > x4 ¼ x1 > x5

m4 : x4 < x2 < x5 ¼ x3 < x1

m04 : x4 > x2 > x5 ¼ x3 > x1

m5 : x3 < x2 ¼ x4 < x5 < x1

m05 : x3 > x2 ¼ x4 > x5 > x1

Definition 3 (Liu et al., 2003). Let X and M be sets, (M, s, X) be an AFS structure and (X, r, m) be a measure space, where m is a finite and positive measure, mðXÞ ¼ 0; Asi 2 r; x 2 X; i 2 I. For the fuzzy P Q concept g ¼ i2I ð m2Ai mÞ 2 EM, the membership function of g is defined as follows. For any x 2 X,

lg ðxÞ ¼ sup i2I

In our study, let o = 2X, for W 2 2X, m(W) = jWj(jWj is the cardinal number of the set W). Then the equation can be stated as follows:

! 0

m

m2Ai

  m Asi ðxÞ : mðXÞ

lg ðxÞ ¼ sup i2I

 s  A ðxÞ i : jXj

0

If m stands for the negation of the concept m 2 M, then for any fuzzy concept f 2 EM, f0 means the logical negation of f. In Example 1, The algebra system (EM, ^, _, 0 ) as a lattice not only provides a sound mathematical tool for us to study and determine the upper and lower approximations of a fuzzy set, but also ensures that they are the approximations of the fuzzy set of some underlying semantics. Definition 2 (Liu, 1998c; Liu et al., 2005). Let X, M be sets and 2M be the power set of M. Let s:X  X ? 2M. (M, s, X) is called an AFS structure if s satisfies the following axioms:

In Example 1, let g1 = m1,g2 = m2,g3 = m3m4,g4 = m3 + m4 2 EM. From above equation, we can get: s

jA ðx1 Þj ¼ 4=5 ¼ 0:8 jXj s jA ðx1 Þj s For g2 ;A ¼ fm2 g;A ðx1 Þ ¼ fx5 ;x2 ;x4 ;x1 ;x3 g; lg2 ðx1 Þ ¼ ¼ 5=5 ¼ 1:0 jXj jAs ðx1 Þj For g3 ;A ¼ fm3 ;m4 g;As ðx1 Þ ¼ fx2 ;x3 ;x4 ;x1 g; lg3 ðx1 Þ ¼ ¼ 4=5 ¼ 0:8 jXj  s  jAi ðx1 Þj ¼ supf4=5;5=5g ¼ 1:0 For g4 ;A1 ¼ fm3 g;A2 ¼ fm4 g; lg4 ðx1 Þ ¼ sup jXj i¼1;2 s

For g1 ;A ¼ fm1 g;A ðx1 Þ ¼ fx4 ;x5 ;x1 ;x3 g; lg1 ðx1 Þ ¼

AX1:"(x1, x2)2 X  X, s(x1, x2) # s(x1, x1); AX2:"(x1, x2), (x2, x3)2 X  X, s(x1, x2) \ s(x2, x3) # s(x1, x3). 2.2. The new AFS clustering method X is called universe of discourse; M is called a concept set and s is called a structure. Let X be a set of objects and M be a set of simple concepts on X. If s: X  X ? 2M is defined as follows: for any (x, y) 2 X  X

sðx; yÞ ¼ fmjm 2 M; ðx; yÞ 2 Rm g 2 2M where Rm is the binary relation of simple concept m 2 M(refer to Definition 2). Then (M, s, X) is an AFS structure. Now we prove this. For any (x1, x2) 2 X  X, if m 2 s(x1, x2), we know that (x1, x2) 2 Rm. Because each m 2 M is a simple concept, we have (x1, x1) 2 Rm. This implies that s(x1, x2) # s(x1, x1) and AX1. For (x1, x2), (x2, x3) 2 X  X, if m 2 s(x1, x2) \ s(x2, x3), then (x1, x2), (x2, x3) 2 Rm. Since m is a simple concept, we have (x1, x3) 2 Rm, i.e., m 2 s(x1, x3). This implies s(x1, x2) \ s(x2, x3) # s(x1, x3). Therefore (M, s, X) is an AFS structure. By the above discussion, an AFS structure based on a data set can be established, as long as each concept in M is a simple concept on X. Let us continue with Example  1, in which X = {x1,x2, . . ., x5} is the set of five suppliers and M ¼ m1 ; m01 ; . . . ; m5 ; m05 , where m1: ‘‘Attribute1 is good’’, m01 : ‘‘Attribute1 is not good’’;. . ., m5: ‘‘Attribute5 is good’’, m05 : ‘‘Attribute1 is not good’’. For the semantic

Cluster analysis is a very useful classification tool. It has been used frequently in product position, strategy formulation, market segmentation studies and business system planning. Further, we could discriminate one or more strategies from airfreight industry and to comprehend the competitive situation deeply. AFS clustering method imitates the clustering procedure of human being, shown as Fig. 1. By analyzing the algorithm, we find that the following two issues (i.e. the algorithm of steps 1 and 4 shown in Fig. 1) result in the lower performance of the algorithm: (a) The description of each object can not character it well. In Liu et al. (2005), the author choose all the feasible fuzzy description of each object, but they may be so ‘‘rough’’ that the ‘‘bad’’ descriptions are included. And the bad description always makes the clustering accuracy lower. In this paper, we just choose one best description from all the feasible descriptions through the selecting method. (b) The fuzzy cluster validity index which is used to select the best clustering result in Liu et al. (2005) only considers the clarity of the boundary among the clusters. However, the number of the clusters is also an important factor to influence the clustering results. Thus, in the new validity index of this paper, both the clarity of the boundary and the number of the clusters are considered.

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Fig. 1. Clustering procedure realized by human beings.

2.2.1. The elementary fuzzy clustering method realized via AFS fuzzy logic STEP 1: Find fuzzy set # = _b2^ b,x 2 X,l_b2^b(x) is the highest degree of x belonging to any cluster, due to# being the maximum element in (^)EI. In order to produce a welldefined clustering result, eachx should belong to # to the highest extent. Proposition 1 outlines the properties of the fuzzy set #. STEP 2: Find the fuzzy description of each object: for each x 2 X; and the fuzzy description nx of x, which is dx for the Boolean case. For fuzzy set nx 2 (^)EI; where (^)EI is the sub EI algebra generated by ^; not only is lnx(x) approachingl_b2^b(x), but alsolnx(y) is as small as possible for y 2 X, y – x. In other words, x can be distinguished bynx from other objects in X at the highest extent. STEP 3: Evaluating the similarity between objects based on the fuzzy descriptions: apply nx the fuzzy description of each x 2 X to establish the fuzzy matrix M = (mij) on X = (x1;x2; :::;xn); where mij the similarity degree between xi and xj which is defined as follows: for any xi; xj 2 X, mij ¼ minflnxi ^nxj ðxi Þ; lnxi ^nxj ðxj Þg. Theorem 1 demonstrates that there exists an integer rsuch that (Mr)2 = Mr, i.e., fuzzy matrix Q = Mr can yield a partition tree with equivalence classes. STEP 4: Cluster according to the determined similarity degrees:

let Q = Mr = (qij) and the Boolean matrix Q a ¼ qaij ; where qaij ¼ 1 () qij P a; the threshold a 2 [0, 1]. Fora 2 [0, 1]; xi; xj 2 X; xi; xj are in the same cluster for given threshold a if and only if qaij ¼ 1. For some xi 2 X, if qaij ¼ 0, then the clustering label of xi cannot be determined for fuzzy attributes in ^ under the threshold a. STEP 5: Select the well-delineated clustering results: for each clusterC # X under the threshold a, the fuzzy description of C, nC is defined as follows.

nC ¼ _ nx x2C

Ia ¼

P

1

P 2

Ci2[16k6l C k

lbou ðciÞ

Ci2[16k6l C k ltotal ðciÞ

a

þ

jCj jXj

where nTotal ¼ _16k6l nC k , l > 2. jCj is the number of the clusters, jXj is the number of the objects. The less Ia the better the clustering. Proposition 1 (Liu and Liu, 2005). Let X be an universe of discourse and M be a finite set of simple concepts. Let {ln(x)—n 2 EM} be a set of coherence membership functions of the AFS fuzzy logic system (EM, _,^0 ) and the AFS structure (M, s, X). Let ^ # EM. Then for any b 2 (^)EI, for any x 2 X; lb ðxÞ 6 l_b2^ bðxÞ. Proposition 1 implies that for each x 2 X, the degree of x belonging to fuzzy set _b2^b is the largest of other fuzzy sets in (^)EI. But _b2^b is not the fuzzy description of x, because _b2^b is the maximum fuzzy set in lattice (^)EI and for each y 2 X,y – x the degree of y belonging to fuzzy set _b2^b is also the largest of other fuzzy sets in (^)EI. Therefore for a given x 2 X, we should find the fuzzy set nx in (^)EI such that not only is lnx ðxÞ approaches l_b2^ bðxÞ; but also lnx ðyÞ is as small as possible for each y 2 X and y – x, In what follows, we find the fuzzy set nx in (^)EI for each given x. For e P 0 (in general, e is very small), we define

( e

Ak jlAk ðxÞ P l_b2^ bðxÞ  e; k 2 I; a ¼

Bx ¼

X

) Ai 2 ^

i2I

^ bjH # Bex ; l^b2H bðxÞ P l_b2^ bðxÞ  e b2H n o e ^x ¼ cjc is minimal elemen tBex Bex ¼



Theorem 2 (Liu, 1998c). Let X be an universe of discourse and M be a finite set of simple concepts, (M, s, X) be an AFS structure. Let {ln(x)—n 2 EM} be a set of coherence membership functions of the AFS fuzzy logic system (EM, _, ^0 ) and the AFS structure (M, s, X). Let ^ # EM. For a given x 2 X and a given e > 0; a 2 ^ex , Let #xa ¼ fb 2 ð^ÞEI jb P ag. Then the following observations hold: (1) (2)

txa is a sub-EI algebra of (^)EI l^b2tx bðxÞ P la ðxÞ P l_b2^ bðxÞ  e a

The fuzzy description nC of class Cwhose membership degree lnC ðxÞ is not only the most approachable l_b2^b(x); for each x 2 C; but also lnC ðyÞ is as small as possible for y 2 X, y R C. In other words, the objects in cluster C can be distinguished from other objects in X to the highest possible extent. The fuzzy description of the boundary among the clusters C1;C2; :::;Cl is a fuzzy set nbou 2 EM;

nbou ¼ _ðnC i ^ nC j Þ 16i;j6l;i–j

where nC i ; i = 1;2; :::; l is the fuzzy description for the ith cluster Ci. The clarity of the fuzzy clustering for some the threshold can a be evaluated by Ia a fuzzy cluster validity index defined as follows. For any threshold a 2 [0, 1],

(3) For g 2 (^)EI, if y – X,y – x

lg ðxÞ > l_b2^ bðxÞ  e, then 9a 2 ^ex , for any

lg ðyÞ P l^b2tx bðyÞ P la ðyÞ a

(4) nx ¼ _a2^ex ð^b2txa bÞ P nx ¼ _a2^ax a  In Example 1, Let X ¼ fx1 ; x2 ; . . . ; x5 g; e ¼ 0; M ¼ m1 ; m01 . . . ; 0 0 0 m5 ; m5 g; t ¼ m1 þ m1 þ . . . þ m5 þ m5 , we can get lt(x1) = lt(x2) = . . . = lt (x5) = 1.0 In STEP 1: Forx1 : lm2 ðx1 Þ ¼ lm4 ðx1 Þ ¼ lm5 ðx1 Þ ¼ 1 ¼ l# ðx1 Þ. We have B0x1 ¼ fm2 ; m4 ; m5 g. lm2 m4 m5 ðx1 Þ ¼ 1 ¼ l# ðx1 Þ, and we know that m2m4m5 is the

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minimal element. In K0x1 ¼ fm2 ; m4 ; m5 ; m2 m4 ; m2 m5 ; m4 m5 ; m2 m4 m5 g. So, nx1 = m2m4m5. We 0 also can get others in the   same  way: Bx2 ¼ 0 0 0 0 m1 ; m3 ; nx2 ¼m1 m3 ;Bx3 ¼ m2 ;m5 ;nx3 ¼m2 m05 ; B0x4 ¼m01 m04 ;nx4 ¼m01 m004 ;B0x5 ¼m3 m02 ;nx4 ¼ m3 m02 In STEP 2: The fuzzy relation matrix F is

2

1:0

6 6 6 F¼6 6 6 4

0 1:0

0:6

0

0

0:6

v ij ¼ wj rij ; 

v þ1 ; . . . :; v þn

nC 1 ¼ nx1 ¼ m2 m4 m5 ; nC 2 ¼ nx2 ¼ m1 m03 ; nC 3 ¼ nx3 ¼ m2 m05 ; nC 4 ¼ nx4 ¼ m01 m004 ; nC 5 ¼ nx5 ¼ m3 m02

þ di

¼

( n X

v ij  v

di ¼

I1.0 is the smallest. The best cluster is

2

)1=2 ;

i ¼ 1; . . . ; m;

j ¼ 1; . . . ; n

( n X

v ij  v j

2

)1=2 ;

i ¼ 1; . . . ; m;

j ¼ 1; . . . ; n

j¼1

C 4 ¼ fx4 g;

C 5 ¼ fx5 g:

Then the description of each cluster is obtained as followed: The fuzzy description nC 1 of Cluster C1 is m2 m4m5 with the interpretation: the communication is good, the candidate land value and the freight transport is high; The fuzzy description nC 2 of Cluster C2 is m1 m03 with the interpretation: the traffic is good, but the candidate land area is small; The fuzzy description nC 3 of Cluster C3 is m2 m05 with the interpretation: the communication is good, but the freight transport is not high; The fuzzy description nC 4 of Cluster C4 is m01 m04 with the interpretation: the communication is not good, and the candidate land value is not high; The fuzzy description nC 5 of Cluster C5 is m02 with the interpretation: the communication is not good. 3. TOPSIS methodology TOPSIS method was first developed by Hwang and Yoon (1981). TOPSIS is a multiple criteria method to identify solutions from a finite set of alternatives. The basic principle is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution (Jahanshahloo, Lotfi, & Izadikhah, 2006; Shih, Shyur, & Stanley, 2007). The positive-ideal solution is a solution that maximizes the benefit criteria and minimizes the cost criteria; where as the negative ideal solution maximizes the cost criteria and minimizes the benefit criteria. The procedure of TOPSIS can be expressed in a series of steps (Alev, 2009; Yang & Hung, 2007). STEP 1: Calculate the normalized decision matrix. The normalized value rij is calculated as

i ¼ 1; . . . ; m;

þ j

Similarly, the separation from the negative ideal solution is given as 

i¼1

j

j¼1

C 1 ¼ fx1 g; C 2 ¼ fx2 g; C 3 ¼ fx3 g; C 4 ¼ fx4 g; C 5 ¼ fx5 g: I0:6 ¼ 1:64

ffiffiffiffiffiffiffiffiffiffiffiffiffi ,v u m uX r ij ¼ xij t x2ij ;

¼

where I is associated with benefit criteria, and J is associated with cost criteria. STEP 4: Calculate the separation measures, using the n-dimensional Euclidean distance. The separation of each alternative from the positive ideal solution is given as

When the threshold a is 1.0, there is five clusters:

C 3 ¼ fx3 g;



j

C 1 ¼ fx2 g; C 2 ¼ fx3 g; C 3 ¼ fx4 g; C 4 ¼ fx1 ; x5 g: I0:6 ¼ 1:8

C 2 ¼ fx2 g;

j ¼ 1; . . . ; n

    max v ij ji 2 I ; min v ij ji 2 J ; j j       min v ij ji 2 I ; max v ij ji 2 J ; A ¼ v 1 ; . . . ; v n ¼

Aþ ¼

0

F2 = F. So,Q = F can yield a partition tree with equivalence classes. In STEPS 3 and 4: The threshold can be as a = 0.6, 1.0 When the threshold a is 0.6, there is four clusters:

C 1 ¼ fx1 g;

i ¼ 1; . . . ; m;

STEP 3: Determine the positive ideal and negative ideal solution

3

7 0 7 7 1:0 0 0 7 7 7 1:0 0 5 1:0 0

STEP 2: Calculate the weighted normalized decision matrix by multiplying by the normalized decision matrix by its associated weights. The weighted normalized vij is calculated as

j ¼ 1; . . . ; n

STEP 5: Calculate the relative closeness to the ideal solution. The relative closeness of the alternative Ai with respect to A+ is defined as   þ  C i ¼ di = di þ di ; 

i ¼ 1; . . . ; m þ

Since di P 0 and di P 0, then, clearly, Ci 2 [0, 1]. STEP 6: Rank the preference order. For ranking alternatives using this index, we can rank alternatives in decreasing order. The basic principle of the TOPSIS method is that the chosen alternative should have the ‘‘shortest distance’’ from the positive ideal solution and the ‘‘farthest distance’’ from the negative ideal 4. Numerical example This section will give an example to show the applications of the algorithm. Suppose that a decision maker needs to select at most 15 potential cities and 13 attributes. The relevant data in the problem are listed in Table 2. The fuzzy numbers and the distance function can lead to the deferent clustering results, we selected the deferent fuzzy numbers and the distance function as follows: 4.1. AFS algorithm  STEP 1: Let X ¼ fcA ; cB ; . . . ; cO g; e ¼ 0; M ¼ m1 ; m01 ; . . . ; m13 ; m013 g; t ¼ m1 þ m01 þ . . . þ m13 þ m013 , we can get lt(cA) = lt(cB) = . . . = lt (cO) = 1.0. We just show cD as example:

  B0cD ¼ m02 ; m04 ; m5 ; m06 ; m7 ; m8 ; m9 ; m012 ; m13 ; nC D ¼ m02 m04 m5 m06 m7 m8 m9 m012 m13 STEP 2: F2 = (F2)2. So, Q = F3 can yield a partition tree with equivalence classes. STEP 3 and 4: When threshold a = 0.4000. There are two clusters:

‘‘VL’’: (0, 0, 0, 0), ‘‘B.VL&L’’: (0, 0, 0.1, 0.2), ‘‘L’’: (0, 0.2, 0.2, 0.2), ‘‘B.L&M’’: (0, 0.2, 0.4, 0.5), ‘‘M’’: (0, 0.3, 0.6, 0.7), ‘‘B.M&H’’: (0.3, 0.5, 0.8, 1), ‘‘H’’: (0.6, 0.8, 0.8, 1), ‘‘B.H&VH’’: (0.6, 0.8, 0.9, 1), ‘‘VH’’: (1, 1, 1, 1), and p = 3 for the distance function dp (, ).

H B.VL&L H B.VL&L H VL H VL B.H&VH VL VH B.VL&M M H B.VL&L H B.VL&L B.M&H B.VL&L B.H&VH L B.H&VH VL H B.VL&L M

N M

M B.VL&M H B.VL&L M L H VL B.M&H L B.H&VH VL B.H&VH H B.VL&L VH L H VL B.M&H B.VL&L B.H&VH VL H B.VL&L B.M&H B.M&H L H B.VL&L H VL H VL B.H&VH VL B.M&H L M

L K J I

B.M&H L H B.VL&L M L B.M&H B.VL&L B.M&H L H B.VL&L B.L&M H B.VL&L H B.VL&L M L B.H&VH VL B.M&H L B.M&H L M

H G

H B.VL&L H B.VL&L B.M&H B.VL&L M L B.M&H L B.M&H L B.M&H H B.VL&L B.H&VH VL H VL B.H&VH B.L&M B.H&VH VL B.H&VH VL B.M&H

F E

H B.VL&L B.H&VH VL H VL B.H&VH B.VL&L H B.VL&L B.H&VH VL B.M&H H B.VL&L H B.VL&L B.M&H B.VL&L H VL H B.VL&L B.M&H L B.M&H H B.VL&L H B.VL&L B.M&H B.VL&L B.M&H B.VL&L H B.VL&L B.M&H L B.M&H

D C B A

H B.VL&L H B.VL&L H VL B.H&VH B.L&M VH L B.M&H B.VL&M M Weather condition Landform condition Water supply Power supply Solid castoff disposal Communication Traffic Candidate land area Candidate land shape Candidate land circumjacent main line Candidate land land-value Freight transport Fundamental construction investment

Table 2 The fuzzy synthetical judgment of some region.

B.M&H L H B.VL&L B.M&H B.VL&L H VL B.H&VH VL B.M&H L B.M&H

O

Y. Li et al. / Expert Systems with Applications 38 (2011) 7901–7908

VH L VH L H VL B.H&VH B.L&M VH L H B.VL&L M

7906

C 1 ¼ fcA ; cL g; C 2 ¼ fcB ;cC ;cD ;cE ;cF ;cG ;cH ;cI ;cJ ;cK ;cM ;cN ;cO g; I0:4000 ¼ 1:5410 nC A ¼ m02 m03 m4 m5 m06 m7 m8 m9 m10 m011 m12 þ m1 m3 m4 m5 m06 m7 m8 m9 m10 nC 2 ¼ m02 m03 m05 m011 þ m02 m03 m05 m08 m011 þ m02 m04 m5 m06 m7 m8 m9 m012 þ m02 m04 m5 m06 m7 m8 m012 þ m02 m03 m05 m07 m10 m011 þ m02 m03 m05 m6 m7 m08 m10 m011 þ m03 m05 m6 m07 m10 m013 þ m03 m5 m06 m08 m010 m011 þ m01 m2 m03 m05 m6 m08 m10 m012 m13 þ m02 m3 m4 m5 m06 m010 þ m03 m05 m08 m010 m011 þ m02 m03 m05 m08 m010 þ m02 m03 m5 m6 m08 m010 m11 m12 When threshold a = 0.5333. There are five clusters:

C 1 ¼ fcA g; C 2 ¼ fcG g; C 3 ¼ fcL g; C 4 ¼ fcB ; cC ; cD ; cE ; cF ; cH ; cI ; cJ ; cK ; cM ; cN g;

I0:5333 ¼ 2:2181

C 5 ¼ fcO g When threshold a = 0.6000. There are six clusters:

C 1 ¼ fcA g; C 4 ¼ fcL g;

C 2 ¼ fcG g; C 3 ¼ fcD ; cE ; cI ; cJ g; C 5 ¼ fcB ; cC ; cF ; cH ; cK ; cM ; cN g; I0:6000 ¼ 2:4646

C 6 ¼ fcO g When threshold a = 0.6667. There are ten clusters:

C 1 ¼ fcA g; C 2 ¼ fcG g; C 3 ¼ fcD ; cE g; C 4 ¼ fcF ; cH g; C 5 ¼ fcI ; cJ g; C 6 ¼ fcB ; cC ; cM g; C 7 ¼ fcK g; I0:6667 ¼ 2:5516 C 8 ¼ fcL g; C 9 ¼ fcN g; C 10 ¼ fcO g When threshold a = 0.7333. There are twelve clusters: C 1 ¼ fcA g; C 2 ¼ fcB ;cC g; C 3 ¼ fcD g; C 4 ¼ fcE g; C 5 ¼ fcG g; C 6 ¼ fcF ;cH g; C 7 ¼ fcI ;cJ g; C 8 ¼ fcK g; I0:7333 ¼ 2:4503 C9 ¼ fcL g; c10 ¼ fcM g; C 11 ¼ fcN g; C 12 ¼ fcO g

When threshold a = 1.0000. There are two clusters:

C 1 ¼ fcA g; C 2 ¼ fcH g; C 3 ¼ fcD g; C 4 ¼ fcE g; C 5 ¼ fcG g; C 6 ¼ fcF g; C 7 ¼ fcI g; C 8 ¼ fcJ g; I1:0000 ¼ 1:7222 C 9 ¼ fcL g; C 10 ¼ fcK g; C 11 ¼ fcN g; C 12 ¼ fcO g I0.4 is the smallest. The best cluster is C1 = {cA, cL}, C2 = {cB, cC, cD, cE, cF, cG, cH, cI, cJ, cK, cM, cN, cO}. Then the description of each cluster is obtained as followed: The fuzzy description nC1 of Cluster C1 is m02 m03 m4 m5 m06 m7 m8 m9 m10 m011 m12 þ m1 m3 m4 m5 m06 m7 m8 m9 m10 with the interpretation: the landform condition, water supply, communication, candidate land land-value are not good, and power supply, solid castoff disposal, traffic, candidate land area, candidate land shape, candidate land circumjacent, freight transport are good. Or weather condition, water supply, power supply, solid castoff disposal, traffic, candidate land area, candidate land shape, t candidate land circumjacent are good, and communication is not good. The fuzzy description nC2 of Cluster C2 is like above.

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Y. Li et al. / Expert Systems with Applications 38 (2011) 7901–7908

4.2. TOPSIS algorithm STEP 1: For the region is A and L, normalization of original data was dealt. The result as the following: rij= Weather Landform Water Power Solid Communi- Traffic Candidate Candidate Candidate condition condition supply supply castoff cation land area land land disposal shape circumjacent main line A 0.6247 L 0.7809

0.5547 0.8321

0.6247 0.5547 0.7071 0.7809 0.8321 0.7071

0.7071 0.7071

0.7071 0.7809 0.7071 0.6247

0.7071 0.7071

0.7071 0.7071

Candidate Freight Fundamental transport construction land investment landvalue 0.6585 0.7526

0.8944 0.4472

0.7071 0.7071

STEP 2: The weight is wj = [0.38; 0.25; 0.34; 0.23; 0.36; 0.45; 0.42; 0.54; 0.46; 0.38; 0.29; 0.41; 0.50]. And the weighted normalized matrix is as following: vij= Weather Landform Water Power Solid Communi- Traffic Candidate Candidate Candidate condition condition supply supply castoff cation land area land land disposal shape circumjacent main line A 0.2374 L 0.2967

0.1387 0.2080

0.2124 0.1276 0.2546 0.3182 0.2655 0.1914 0.2546 0.3182

0.2970 0.4217 0.2970 0.3373

0.3253 0.3253

0.2687 0.2687

Candidate Freight Fundamental transport construction land investment landvalue 0.1910 0.2182

0.3667 0.1834

0.3536 0.3536

Then the positive ideal and negative ideal solution we can get. STEP 3, 4: The separation of each alternative from the positive ideal solution is as following: 

þ

A L

di

di

Ci

0.2278 0.0694

0.0694 0.2278

0.2335 0.7665

STEP 5: Because dL > dA, we can select the L as the region center location. The selection of B, C, D, E, F, G, H, I, J, K, M, N, O region is as above: Then the result is as following:

þ di  di

Ci

B

C

D

E

F

G

H

I

J

K

M

N

O

0.1619

0.1495

0.1244

0.2843

0.2173

0.2035

0.1891

0.1168

0.1156

0.1918

0.1350

0.1798

0.1632

0.2315 0.5884

0.2904 0.6601

0.2839 0.6953

0.2190 0.4352

0.1739 0.4446

0.2845 0.5830

0.2429 0.5622

0.3255 0.7359

0.2829 0.7098

0.3079 0.6162

0.3039 0.6924

0.2135 0.5429

0.3207 0.6627

Because dI > dJ > dD > dM > dO > dC > dK > dB > dG > dH > dN > dF > dE. We can select the I and Jas the region center location. In conclusion, the L, I and J are selected as logistics center.

5. Conclusion Many practitioners and researchers have presented the advantages of logistics management. In order to increase the competitive advantage, many companies consider that a well-designed and implemented logistics system is important tool. Under this condition, building the closeness and long-term relationships between regions is critical success factor to establish the logistics system. Therefore, selection of logistics center location problem becomes the most important issue to implement a successful logistics system. In general, selection of logistics center location problems adhere to uncertain and imprecise data and fuzzy-set theory is adequate to deal with them. In a decision-making process, the use of linguistic variables in decision problems is highly beneficial when performance values cannot be expressed by means of numerical values. In other words, very often, in assessing of possible regions with respect to criteria and importance weights, it is appropriate to use linguistic variables instead of numerical values.

This article outlined a hybrid method, which incorporates AFS and TOPSIS techniques into an evaluation process, in order to select competitive regions in logistics. This hybrid method emphasizes the following characteristics:  It accounts for both qualitative and quantitative factors that have an impact on logistics performance.  It adopts AFS and TOPSIS techniques to find the performance frontiers from a set of potential regions. Due to the decision-makers’ experience, feel and subjective estimates often appear in the process of selection of logistics center location problem, AFS clustering method has been studied further to effectively evaluate logistics center location. The TOPSIS method is very flexible. According to the closeness coefficient, we can determine not only the ranking order but also the assessment status of all possible regions, and select the suitable logistics center location finally.

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