A fuzzy inhomogenous multiattribute group decision making approach ...

Knowledge-Based Systems 67 (2014) 71–89

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

A fuzzy inhomogenous multiattribute group decision making approach to solve outsourcing provider selection problems Deng-Feng Li a,⇑, Shu-Ping Wan b a b

School of Economics and Management, Fuzhou University, Fuzhou, Fujian 350108, China College of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China

a r t i c l e

i n f o

Article history: Received 10 July 2013 Received in revised form 10 May 2014 Accepted 7 June 2014 Available online 16 June 2014 Keywords: Outsourcing provider Multiattribute group decision making Production operation Fuzzy linear programming Supply chain management

a b s t r a c t Considering various situations and characteristics of supply chain management, we regard the outsourcing provider selection as a type of fuzzy inhomogenous multiattribute group decision making (MAGDM) problems with fuzzy alternatives’ comparisons and incomplete weight information. Hereby we focus on developing a new fuzzy linear programming method for solving such MAGDM problems. In this method, the decision makers’ preferences are given through pair-wise alternatives’ comparisons with fuzzy truth degrees represented as trapezoidal fuzzy numbers (TrFNs). Intuitionistic fuzzy sets, TrFNs, intervals and real numbers are used to express the inhomogenous decision information. Under the condition that the fuzzy positive ideal solution (PIS) and fuzzy negative ideal solution (NIS) are known, the fuzzy consistency and inconsistency indices are defined on the basis of the relative closeness degrees and expressed with TrFNs. The attribute weights are estimated through constructing a new fuzzy linear programming model, which is solved by the developed method of fuzzy linear programming with TrFNs. Through solving the constructed linear goal programming model, we obtain the collective comprehensive relative closeness degrees of alternatives to the fuzzy PIS, which are used to rank the alternatives. The effectiveness of the proposed method is verified with an example of IT outsourcing provider selection. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction In global business environment, many companies are looking forward to outsourcing. They entrust some jobs to external providers. Through proper outsourcing, they can deliver goods to consumers in time and hereby reduce operating costs, increase their focus on internal resources and core activities, and sustain competitive advantages [1]. Information technology (IT, or information system (IS)) outsourcing is a common outsourcing activity for many organizations [2]. Generally, the process of IT outsourcing may be divided into seven phases, involving: (1) IT demand, application status and performance evaluation of department; (2) development and programming of IT; (3) outsourcing strategies; (4) design contract object; (5) select outsourcing providers; (6) contract negotiation, implement and supervise; (7) project approval. The process of IT outsourcing is illustrated with Fig. 1. It is very important that companies scientifically select appropriate outsourcing providers to increase the success rate of ⇑ Corresponding author. Address: School of Economics and Management, Fuzhou University, No. 2, Xueyuan Road, Daxue New District, Fuzhou District, Fuzhou, Fujian 350108, China. Tel./fax: +86 0591 83768427. E-mail address: [email protected] (D.-F. Li). http://dx.doi.org/10.1016/j.knosys.2014.06.006 0950-7051/Ó 2014 Elsevier B.V. All rights reserved.

outsourcing. Selecting outsourcing providers in the fifth phase is a classical problem of decision making in supply chain management. Usually there are many attributes to be considered between different alternatives. Due to the complexity, fuzziness and uncertainty inherent in the evaluated attributes, selecting appropriate outsourcing providers is a difficult task. There are a number of frameworks in the literature offering guidelines and prescriptions on the outsourcing decision. Most of early researchers commonly utilized the transaction cost theory to illustrate outsourcing decisions. However, in recent years, strategy aspects related to core competency, risk analysis and organizational flexibility have becoming important. As a result, this trend has led researchers and industries to become more interesting in multi-criteria decision making (MCDM) or multi-attribute decision making (MADM) methods for outsourcing.

1.1. Review for decision methods of selecting outsourcing providers Lin et al. [3] proposed a hybrid MCDM method for outsourcing vendor selection through combining a case study of a semiconductor company in Taiwan. Combining a decision making trial and evaluation laboratory (DEMATEL) with the analytical network

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Project approval

Implement and supervise

Contract negotiation

Design contract object

Select outsourcing providers

Outsourcing strategies

Development and programming of IT

IT demand

Application status of IT

Performance evaluation of department

Fig. 1. The process of IT outsourcing.

process (ANP) method, Hsu et al. [4] proposed a novel hybrid model for the selection of an outsourcing provider. Ho et al. [5] integrated the quality function deployment, fuzzy set theory and analytic hierarchy process (AHP) to evaluate and select the optimal third-party logistics service providers. Chen et al. [6] presented the fuzzy preference ranking organization method for enrichment evaluation to evaluate four potential suppliers on seven criteria and four decision makers through using a realistic case study. Chen and Wang [7] developed the fuzzy Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method for the strategic decision of optimizing partners’ choice in IT outsourcing projects. Fan et al. [8] utilized an extended DEMATEL method to identify risk factors of IT outsourcing with interdependent information. Combining the DEMATEL, ANP with zero-one goal programming, Tsai et al. [9] developed a MCDM method for sourcing strategy mix decision in IT projects. From a policy-maker’s perspective, Tjader et al. [10] researched the offshore outsourcing decision making. Buyukozkan and Cifci [11] proposed a novel hybrid MCDM method based on the fuzzy DEMATEL, fuzzy ANP and fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to evaluate green suppliers. Chou et al. [12] developed a fuzzy MCDM method to evaluating IT investments. The aforementioned methods seem to be effective and applicable for selecting outsourcing providers. Nevertheless, they have three disadvantages. The first is that most of these methods assume that the attribute weights are completely given a priori. In many real decision situations, there are some difficulties or challenges for the decision maker (DM) to provide precise and complete preference information due to time pressure, lack of knowledge (or data), and limited expertise about the problem domain. In other words, usually weights are totally unknown or partially known a priori [13–15]. The second is that these methods only considered single DM for the process of outsourcing provider selection and ignored the function of group decision making. With ever increasing complexity and business competitiveness, companies often engage more and more DMs (or experts) in participating in making decision to increase the success of outsourcing. Therefore, outsourcing provider selection may be regarded as a type of multiattribute group decision making (MAGDM) problems. The

third is that these methods seldom considered the inhomogenous evaluation information. The real-life decision making problems often involve multiple different types/formats of attribute values such as real numbers, intervals, trapezoidal fuzzy numbers (TrFNs), linguistic terms [16] and intuitionistic fuzzy sets (IFSs) [17] due to DMs’ knowledge areas, work backgrounds and manners/habits. Consequently, MAGDM problems may include multiple different formats of decision information. Such a type of MAGDM problems is called the inhomogenous MAGDM problems, which have drawn some attention from a spectrum of disciplines [16,18–22]. For instance, ratings of research and development capability of an outsourcing provider can be expressed with IFSs, the product quality and technological level can be assessed by TrFNs, the flexibility and delivery time can be represented by intervals, the assessment of price can be expressed with real numbers. Moreover, it is very difficult for the DMs to accurately give these attribute weights due to various subjective and objective reasons. Thus, the outsourcing provider selection problems belong to a type of inhomogenous MAGDM problems with incomplete weight information. 1.2. The motivation of this paper To reflect the vague characteristic of human thinking and evaluation, the fuzzy set theory [23] should be incorporated into decision making models [24–29]. TOPSIS [30] and Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) [31] are two commonly-used decision making methods. TOPSIS is based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). However, TOPSIS can only be used to solve the decision making problems with attribute weights completely known a priori. If attribute weights are completely unknown or partially/incompletely known, TOPSIS cannot be used. LINMAP is based on pair-wise alternatives’ comparisons given by the DM and generates the compromise alternative as the solution which has the shortest distance to the PIS. Nevertheless, LINMAP neglects the importance of the NIS in the decision making process. Integrating the advantages of TOPSIS

D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89

and LINMAP to develop a new method for solving outsourcing provider selection problems is a valuable and an important research. In LINMAP [31] and its extensions [32,33], DM’s preferences are usually given through pair-wise alternatives’ comparisons with crisp truth degrees 0 or 1. However, in real situations, DM is not absolutely sure about all pair-wise alternatives’ comparisons and hereby may express his/her opinion with a fuzzy degree of truth. These realizations motivate the development of methods for dealing with incomplete weight information, multiple formats of attribute values and fuzzy truth degrees of alternatives’ comparisons. The purpose of this paper is to develop a new fuzzy linear programming method for solving such MAGDM problems through combining TOPSIS with LINMAP. Since a TrFN permits two parameters to represent the most possible values, it seems to be suitable for modeling imprecision and reflecting the ambiguous nature of subjective judgments. In this paper, TrFNs are used to capture imprecise or uncertain truth degree information about alternatives’ comparisons. IFSs, TrFNs, intervals and real numbers are used to express different types (or formats) of decision information. Under the condition that the fuzzy PIS and the fuzzy NIS are simultaneously given a priori, the individual comprehensive relative closeness degree is used to assess each alternative. Thus, the fuzzy group consistency and inconsistency indices are defined as TrFNs since the fuzzy truth degrees of alternatives’ comparisons are represented by TrFNs. The attribute weights are estimated through constructing a new fuzzy linear programming model, which is solved by the developed fuzzy linear programming method with TrFNs. Finally, by solving a goal programming model, we obtain the collective comprehensive relative closeness degrees of alternatives to the fuzzy PIS, which are used to generate the ranking order of the alternatives. Ma et al. [16] established a fuzzy MAGDM process model and thereby developed a fuzzy MAGDM decision support system (called Decider), which can handle information expressed in linguistic terms, boolean values, as well as numerical values to assess and rank a set of alternatives within a group of DMs. Similarly, the models and method for heterogeneous MAGDM proposed in this paper also can be used to design and develop decision support system (DSS) to verify the actual application (see Section 6). 1.3. The main features and innovations of this paper Compared with TOPSIS [30], LINMAP [31–33] and the MCDM methods of selecting outsourcing providers [3–12], the method proposed in this paper has the following remarkable features: (1) TOPSIS [30], LINMAP [31–33] and the MCDM methods [3–12] did not consider the DM’s preferences on alternatives’ comparisons with fuzzy truth degrees. In other words, they only considered the crisp truth degrees 0 or 1. In fact, due to the complexity of decision making problems themselves and fuzziness of human thinking, there exist some uncertainty and fuzziness when DM gives pair-wise alternatives’ comparisons. Consequently, it is very natural and reasonable to introduce fuzzy numbers to represent fuzzy truth degrees. This paper utilizes TrFNs to represent the fuzzy truth degrees which can better reflect the ambiguous nature of subjective judgments on alternatives’ comparisons given by DMs. (2) This paper studies the MAGDM problems involving multiple different types of attribute values, whereas TOPSIS [30], LINMAP [31–33] and the MCDM methods [3–12] studied the decision making problems with only single type of attribute values.

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(3) This paper defines the fuzzy group consistency and inconsistency indices based on the concept of relative closeness degrees. The attribute weights are objectively determined through constructing a new fuzzy linear programming model, which is solved by the fuzzy linear programming method with TrFNs proposed in this paper. On the contrary, TOPSIS [31–33] and the MCDM methods [3–12] assumed that the attribute weights are given artificially or known a priori. (4) Once attribute weights are determined, we can calculate the individual comprehensive relative closeness degrees of alternatives to the fuzzy PIS and hereby obtain the collective comprehensive relative closeness degrees of the alternatives to the fuzzy PIS through solving the constructed goal programming model. This is the most difference between LINMAP [31–33] and the method proposed in this paper. The main innovations of this paper are summarized as follows: (1) Introducing TrFNs to depict the fuzzy alternatives’ comparisons. (2) Utilizing the relative closeness to replace the distance to fuzzy PIS in defining fuzzy consistency and inconsistency indices and sufficiently considering the fuzzy NIS in the decision process. (3) Constructing goal programming model to objectively derive the collective comprehensive relative closeness degrees. The rest of this paper is organized as follows. Section 2 briefly introduces some basic concepts such as IFSs, interval order relations, interval-valued objective programming and TrFN order relations. Section 3 formulates fuzzy inhomogenous MAGDM problems with fuzzy truth degrees and presents the normalization methods and incomplete weight information structures. In Section 4, a new fuzzy linear programming model is constructed and solved by using the developed fuzzy linear programming method with TrFNs. The proposed method is illustrated with a real IT outsourcing provider selection example and comparison analyses are conducted in Section 5. Section 6 develops a decision support system based on the proposed method. Conclusions are given in Section 7. 2. Some basic concepts and notations In this section, IFSs, interval order relations, interval-valued objective programming and TrFN order relations are introduced to facilitate the discussions. 2.1. Distances between intuitionistic fuzzy sets Let V = {v1, v2, . . ., vm} be a finite universe of discourse. An IFS A in V is an object having the following form [17]:

A ¼ f< v j ; lA ðv j Þ; tA ðv j Þ > jv j 2 Vg; where lA(vj) 2 [0, 1] and tA(vj) 2 [0, 1] are respectively the membership and non-membership degrees of the element vj 2 V to the set A # V so that they satisfy the condition: 0 6 lA(vj) + tA(vj) 6 1. For any IFS A in V, let pA(vj) = 1  lA(vj)  tA (vj), which is called the intuitionistic index of an element vj in the set A. Obviously, 0 6 pA(vj) 6 1 for every vj 2 V. If pA(vj) = 0, then the IFS A is reduced to a fuzzy set, i.e., A = { < vj, lA(vj), 1  lA(vj) > jvj 2 V}. In many decision situations, weights of elements vj 2 V should be taken into account. Assume that each element vj 2 V is assigned Pm ^ ¼ ^ j , where w ^ j 2 ½0; 1 and ^ Let w a weight w j¼1 wj ¼ 1. T

^ 1; w ^ 2; . . . ; w ^ mÞ ðw

be the weight vector of the elements

vj

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(j = 1, 2, . . ., m), where the symbol ‘‘T’’ represents the transpose of a vector or matrix. Let A = { < vj, lA(vj), tA(vj) > jvj 2 V} and B = { < vj, lB(vj), tB(vj) > jvj 2 V} be two IFSs in V. Szmidt and Kacprzyk [34] proposed the weighted Minkowski distance between IFSs A and B as follows: Sq ðA;BÞ ¼

" #1=q m 1X ^ j ðjlA ðv j Þ  lB ðv j Þjq þ jtA ðv j Þ  tB ðv j Þjq þ jpA ðv j Þ  pB ðv j Þjq Þ : w 2 j¼1

~g min fa ~ 2 X; s:t: a which is equivalent to the following bi-objective mathematical programming problem:

; ða þ a Þ=2g min fa ~ 2 X; s:t: a ð1Þ

When q = 1, q = 2, q ? +1, the weighted Minkowski distance Sq(A, B) is accordingly reduced to the weighted Hamming distance, Euclidean distance, and Chebyshev distance.

~ should satwhere X is the set of constraints in which the variable a isfy according to requirements in real situations. 2.4. Order relations over trapezoidal fuzzy numbers

2.2. The order relations over intervals

~ ¼ ðm1 ; m2 ; m3 ; m4 Þ is a special fuzzy subset on the real A TrFN m number set, whose membership function is given as follows:

~ ¼ ½a; a  be an interval, where a and a  are the left and right Let a ~, respectively, and a 6 a . An interval a ~ endpoints of the interval a ~ ¼< mða ~Þ; rða ~Þ >, where mða ~Þ ¼ can also be represented by a Þ=2 is the middle point of the interval a ~, rða ~ Þ ¼ ða   aÞ=2 is ða þ a ~. If rða ~Þ ¼ 0, i.e., a ¼ a , then a ~ ¼ ½a; a  is the radius of the interval a reduced to a real number. ~ is In terms of the fuzzy set [23], the statement ‘‘the interval a ~ may be regarded as a fuzzy relation not greater than the interval b’’ ~ denoted by a ~ Hu et al. [35] defined a fuzzy ~ and b, ~ 6I b. between a partial order relation over intervals, which is given as follows.

8 > < ðx  m1 Þ=ðm2  m1 Þ if m1 6 x < m2 lm~ ðxÞ ¼ 1 if m2 6 x 6 m3 > : ðm4  xÞ=ðm4  m3 Þ if m3 < x 6 m4 ;

~ ¼ ½b; b  be two intervals. The ~ ¼ ½a; a  and b Definition 1. [35]. Let a ~ is regarded as a fuzzy set, whose membership ~ 6I b premise a degree is defined as follows:

8 1 > > >  > ðb > > > : 0:5

where the closed interval [m2, m3], m1 and m4 are the mode, lower ~ respectively. and upper limits of m, ~ ¼ ðm1 ; m2 ; m3 ; m4 Þ is called the positive TrFN if m1 P 0 A TrFN m ~ is normalized if m1 P 0 and m4 6 1. and m4 > 0. A positive TrFN m ~ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n ~ ¼ ðn1 ; n2 ; n3 ; n4 Þ be two TrFNs. Let m Then, the vertex method is defined to calculate the Euclidean distance between them as follows [32]:

~ n ~Þ ¼ dðm;

6b if a  and rða 6b ~Þ > 0 if a 6 b 6 a  ~ > rða  6 b and rðbÞ ~Þ if b 6 a 6 a ~ and a ¼ b; ~Þ ¼ rðbÞ if rða

where 1 is a fuzzy number being less than one, which indicates the ~ ~ is weakly not greater than the interval b. fact that the interval a ~ 6 1. Thus, uða ~ may be explained ~ 6I bÞ ~ 6I bÞ Obviously, 0 6 uða as the acceptability degree of the premise (or order relation) ~ If uða ~ ¼ 0, then a ~ is not accepted. If 0 < ~ 6I b. ~ 6I bÞ ~ 6I b a ~ < 1, the DM accepts the premise a ~ with different ~ 6I b uða~ 6I bÞ ~ ¼ 1, then the ~ satisfactory degrees between 0 and 1. If uða 6I bÞ ~ Namely, the ~ 6I b. DM absolutely satisfies with the premise a ~ is absolutely true. The sym~ 6I b DM believes that the premise a bol ‘‘6I’’ is an interval version of the order relation ‘‘6’’ in the real number set and has the linguistic interpretation ‘‘essentially not greater than’’. In the following, the fuzzy ranking index u is used to define satisfactory equivalent forms of interval-valued inequalities. Definition 2. [36]. A satisfactory equivalent form of an interval~ is defined as follows: ax P b and ~xPI b valued inequality a ~ 6 a, where a 2 [0, 1] is the acceptance degree of the uða~x 6 bÞ I

interval-valued inequality which may be violated. 2.3. Interval-valued objective programming Ishibuchi and Tanaka [37] gave the definition of the minimization problem with the interval-valued objective functions, which are introduced in Definition 3. ~ ¼ ½a; a  be an interval. The minimization Definition 3. Let a problem with the interval-valued objective function is described as follows:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ðm1  n1 Þ2 þ 2ðm2  n2 Þ2 þ 2ðm3  n3 Þ2 þ ðm4  n4 Þ2 : 6 ð2Þ

~ of a TrFN m ~ ¼ ðm1 ; m2 ; m3 ; m4 Þ is The interval expectation EðmÞ defined as follows [38]:

~ ¼ EðmÞ

  1 1 ðm1 þ m2 Þ; ðm3 þ m4 Þ : 2 2

ð3Þ

Inspired by Definitions 1 and 2, a ranking method of TrFNs is defined as follows. ~ and n ~ be two TrFNs. Then, the order relations Definition 4. Let m ~ Pn ~ iff between them are stipulated as follows: (1) m ~ PI Eðn ~ Þ; (2) m ~ 6n ~ iff EðmÞ ~ 6I Eðn ~ Þ; (3) m ~ ¼n ~ iff EðmÞ ~ ¼I Eðn ~ Þ. EðmÞ It can be easily seen from Eq. (3) that the above ranking function (i.e., interval expectation) of TrFNs is linear, i.e., ~ þ bn ~ Þ ¼ aEðmÞ ~ þ bEðn ~ Þ, where a and b are any real numbers. Eðam

3. Fuzzy inhomogenous MAGDM problems with fuzzy truth degrees of alternatives’ comparisons In this section, fuzzy inhomogenous MAGDM problems with fuzzy truth degrees of alternatives’ comparisons are described and the incomplete weight information structures are summarized. 3.1. Representation of fuzzy inhomogenous MAGDM problems with fuzzy truth degrees A MAGDM problem is to choose one of or rank finite alternatives based on the assessment information of multiple attributes. Let {a1, a2, . . ., an} be an alternative set and F = {f1, f2, . . ., fm} be an attribute set. There is a set {D1, D2, . . ., DP} of DMs participating in decision making. Denote the weight vector of DMs by w = (w1, w2, . . ., wP)T, satisfying 0 6 wp 6 1 (p = 1, 2, . . ., P) and PP p¼1 wp ¼ 1. Since there are multiple types of attribute values,

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we divide F into four subsets F 1 ¼ ff1 ; f2 ; . . . ; fi1 g; F 2 ¼ ffi1 þ1 ; fi1 þ2 ; . . . ; fi2 g; F 3 ¼ ffi2 þ1 ; fi2 þ2 ; . . . ; fi3 g and F 4 ¼ ffi3 þ1 ; fi3 þ2 ; . . . ; fm g, where 1 6 i1 6 i2 6 i3 6 m, Ft (t = 1, 2, 3, 4) are attribute subsets in which attribute values are expressed with IFSs, TrFNs, interval values and real numbers, respectively. Ft (t = 1, 2, 3, 4) satisfy the S conditions: Ft \ Fl = ; (t, l = 1, 2, 3, 4; t – l) and 4t¼1 F t ¼ F, where ; is an empty set. Denote M1 = {1, 2, . . ., i1}, M2 = {i1 + 1, i1 + 2, . . ., i2}, M3 = {i2 + 1, i2 + 2, . . ., i3}, M4 = {i3 + 1, i3 + 2, . . ., m}, M = {1, 2, . . ., m} and N = {1, 2, . . ., n}, where Mt are the corresponding subscript sets of the subset Ft (t = 1, 2, 3, 4). Due to the influence of various subjective and objective factors, the DM Dp makes judgment and comparison on the priority between two alternatives with fuzzy truth degree. The DM Dp gives preference relations between alternatives as a fuzzy set e p ¼ fððk; jÞ; ~cp ðk; jÞÞjak p aj with ~cp ðk; jÞ ðk; j ¼ 1; 2; . . . ; nÞg, where X ððk; jÞ; ~cp ðk; jÞÞ expresses an ordered pair of the alternatives ak and

in K0, respectively. In reality, usually the preference information structure K of attribute importance may consist of several sets of the above basic sets Ks (s = 1, 2, 3, 4, 5).

aj that the DM Dp prefers ak to aj (denoted by ak p aj) with the fuzzy truth degree ~cp ðk; jÞ, which is a TrFN defined on the unit   interval [0,1], denoted by ~cp ðk; jÞ ¼ cpkj1 ; cpkj2 ; cpkj3 ; cpkj4 , here

be computed as follows:

cpkj1

cpkj2

cpkj3

cpkj4

~ p is denoted 6 1. The support set of X

06 6 6 6 e 0 ¼ fðk; jÞjl ðk; jÞ > 0 ðk; j ¼ 1; 2; . . . ; nÞg. by X ~cp p

It should be noted that DMs give the comparisons between two alternatives on whole, not on each attribute fi (i = 1, 2, . . ., m). Since the alternative set contains n non-inferior (or effective) alterna   e 0 e 0 (i.e., the number of the alternative tives, the cardinality  X  of X p

p

e 0 ) is at most C 2 ¼ nðn  1Þ=2. Usually, the relations given pairs in X n p e 0 are a partial order. There exist some intransitivity in X e 0 . In by X p

p

many real situations, DMs may not be able to specify all the relations (or comparisons) of the alternatives. Namely, DM Dp only give    e 0 2 some pair-wise alternatives’ comparisons, i.e.,  X  < C ðp ¼ p

n

1; 2; . . . ; PÞ. Let the ratings of alternatives aj (j = 1, 2, . . ., n) on attributes fi (i = 1, 2, . . ., m) given by DM Dp be denoted by ypij , respectively. If D E 0p are IFSs; if i 2 M2, then ypij ¼ i 2 M1, then ypij ¼ l0p ij ; tij   h i 0p 0p 0p 0p 0p bij1 ; bij2 ; bij3 ; bij4 are TrFNs; if i 2 M3, then ypij ¼ e0p are interij ; g ij vals;

if

i 2 M4,

then

ypij

¼

z0p ij

are

real 0p

numbers,

where

0p

0p

0p

3.3. The normalization methods Generally, the attribute subsets Ft can be divided into two subsets F bt and F ct , where F bt and F ct are respectively the sets of benefit attributes and cost attributes, which satisfy the conditions: F t ¼ F bt [ F ct and F bt \ F ct ¼ ; ðt ¼ 1; 2; 3; 4Þ. Since the m attributes may be measured in different ways, the decision matrixes Yp (p = 1, 2, . . ., P) need to be normalized. Usually, the linear transformation is used to convert the different scales of attribute ratings/ values into a comparable D scaleE(e.g., the unit interval [0,1]). 0p For the ratings ypij ¼ l0p attriij ; tij ðfi 2 F 1 Þ, according to benefit D E butes and cost attributes, the normalized values rpij ¼ lpij ; tpij can

r pij

8D E 0p > < l0p if f i 2 F b1 ij ; tij E ¼ D 0p > : t0p if f i 2 F c1 : ij ; lij

  0p 0p 0p 0p Similarly, for the ratings ypij ¼ bij1 ; bij2 ; bij3 ; bij4 ðfi 2 F 2 Þ, the nor  p p p p malized ratings r pij ¼ bij1 ; bij2 ; bij3 ; bij4 are computed as follows:

8 0p

0p 0p 0p bij1 bij2 bij3 bij4 > > > if f i 2 F b2 ; ; ; max max max max < bi4 bi4 bi4 bi4 p

r ij ¼ 0p 0p 0p 0p > bij4 bij3 bij2 bij1 > > 1  max if f i 2 F c2 ; ; 1  max ; 1  max ; 1  max : b b b b i4

mn

represent the fuzzy MAGDM problem with inhomogenous information. 3.2. Incomplete weight information structures Let the weight vector of attributes be denoted by x = (x1, x2, . . ., xm)T, where xi is the relative weight of an attribute fi, which Pm satisfies the normalization conditions: i¼1 xi ¼ 1 and xi P 0 (i = 1, 2, . . ., m). Let K0 be a set of all weight vectors xi P e  with P for all attributes fi (i = 1, 2, . . ., m), i.e., K0 ¼ xj m i¼1 xi ¼ 1; xi P e ði ¼ 1; 2; . . . ; mÞg, where e > 0 is a sufficiently small number which ensures that the generated weights are not zeros as it may be the case in LINMAP [31]. In some real decision situations, DMs can give partial preference relations on attribute weights according to their knowledge, experience and judgments. So information of the attribute weights is incomplete and has several different structure forms. Summarizing earlier researches, Li [15] expressed these weight information structures in the five basic relations among attribute weights, which are denoted by subsets Ks (s = 1, 2, 3, 4, 5) of weight vectors

i4

i4

ð5Þ

i4

n o max 0p where bi4 ¼ max bij4 jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P . For the rath i h i 0p ings ypij ¼ e0p (fi 2 F3), the normalized ratings r pij ¼ epij ; g pij are ij ; g ij computed as follows:

8h i > < e0p =g max ; g 0p =g max if f i 2 F b3 i i ij ij p i r ij ¼ h > max max : 1  g 0p ; 1  e0p if f i 2 F c3 ; ij =g i ij =g i

ð6Þ

n o where g max ¼ max g 0p i ij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P . For the ratp p ings ypij ¼ z0p ij ðfi 2 F 4 Þ, the normalized ratings r ij ¼ zij are computed

as follows:

0p 0p 0p 0 6 l0p ij 6 1; 0 6 tij 6 1; lij þ tij 6 1, 0 6 bij1 6 bij2 6 bij3 6 bij4 ; 0 6 0p 0p e0p ij 6 g ij and zij P 0. Hence, we can elicit the fuzzy decision   ðp ¼ 1; 2; . . . ; PÞ, which are usually used to matrixes Y p ¼ ypij

ð4Þ

( r pij ¼

if f i 2 F b4

max z0p ij =zi

1

max z0p ij =zi

ð7Þ

if f i 2 F c4 ;

n o where zmax ¼ max z0p i ij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P .   Thus, all the fuzzy decision matrixes Y p ¼ ypij

mn

are trans-

formed into the normalized fuzzy decision matrixes as follows:

  Rp ¼ r pij

mn

ðp ¼ 1; 2; . . . ; PÞ: 

Denote the vector r pj ¼ rp1j ; r p2j ; . . . ; rpmj regarded as the alternative aj for DM Dp.

ð8Þ T

, which may usually be

4. Fuzzy linear programming models and method for solving fuzzy inhomogenous MAGDM problems In the following, to solve the above fuzzy inhomogenous MAGDM problems, we focus on developing a new fuzzy linear programming method through combining LINMAP with TOPSIS. 4.1. The comprehensive relative closeness degrees of alternatives  þ þ þ þ T Denote  the fuzzy PIS T by r ¼ þr 1 ; r2 ; . . . ; r m and the fuzzy NIS  by r ¼ r 1 ; r2 ; . . . ; r m , where r i and r i are the best and worst

76

D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89

ratings on the attributes

þ þ fi (i = 1, 2, . . .,m), respectively. Namely, if i 2 M1, then r þ and ri ¼ li ; ti are IFSs, where i ¼ l i ; ti

n

o lþi ¼ max lpij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P ; n o tþi ¼ min tpij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P o

n

o

ð9Þ

li ¼ min lpij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P ; ti ¼ max tpij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P : If i 2 M2, then r þ i ¼ TrFNs, where



þ þ þ þ bi1 ; bi2 ; bi3 ; bi4

and r i ¼

dij pþ dij

p

þ dij

ð19Þ

: p

and

n

p

dpij ¼

ð10Þ 

    bi1 ; bi2 ; bi3 ; bi4

are

n o þ p bik ¼ max bijk jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P ðk ¼ 1; 2; 3; 4Þ

ð11Þ

and

n o  p bik ¼ min bijk jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P ðk ¼ 1; 2; 3; 4Þ:

ð12Þ

 þ þ    If i 2 M3, then r þ and r are intervals, where i ¼ ei ; g i i ¼ ei ; g i

n o eþi ¼ max epij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P ; n o g þi ¼ max g pij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P

ð13Þ

and

n o ei ¼ min epij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P ; n o g i ¼ min g pij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P :

ð14Þ

þ   If i 2 M4, then r þ i ¼ zi and r i ¼ zi are real numbers, where

n o zþi ¼ max zpij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P

ð15Þ

and

n o zi ¼ min zpij jj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; P :

ð16Þ

Using Eqs. (1) and (2), we can calculated the squares of the  Euclidean distances between rpij and rþ i as well as r i as follows: 8  2  2  2  p þ 1 > > þ tpij  tþi þ ppij  pþi > 2 lij  li > > > > 2  2  2  2  >  > p þ p þ p þ p þ > > < 16 bij1  bi1 þ 2 bij2  bi2 þ 2 bij3  bi3 þ bij4  bi4 pþ dij ¼  2  2  > > > > 12 epij  eþi þ g pij  g þi > > > > >  > > : zp  zþ 2 i ij

if i 2 M 2 if i 2 M 3 if i 2 M 4 ð17Þ

and 8  2  2  2  p p p 1    > > l  l þ t  t þ p  p > i i i ij ij ij >2 > >  > 2  2  2  2  > > p  p  p  p  > > < 16 bij1  bi1 þ 2 bij2  bi2 þ 2 bij3  bi3 þ bij4  bi4 p dij ¼  2  2  > > p 1  > > þ g pij  g i > 2 eij  ei > > > >  > > : zp  z 2 i ij

Dpj ¼

if i 2 M 1 if i 2 M 2

p

m X

xi dpij :

ð20Þ

i¼1

It is clear that the bigger Dpj the better the alternative aj (or r pj ) for the DM Dp. 4.2. Fuzzy group consistency and inconsistency indexes If the weight vector x is already chosen by the decision group, then using Eq. (20), the DM Dp can calculate the individual comprehensive relative closeness degrees Dpk and Dpj of each pair of altere 0 with respect to the fuzzy PIS r+. Dp and Dp are natives ðk; jÞ 2 X p j k used to generate the ranking order of the alternatives ak and aj, which may be viewed as an objective ranking order, while the e 0 given by the DM Dp may be regarded as a ordered pair ðk; jÞ 2 X p subjective ranking order. Generally, there exists some deviation between the subjective and objective ranking orders. To measure such a deviation, we introduce the concepts of fuzzy group consistency and inconsistency indexes. e 0 , the alternative ak is closer to the fuzzy PIS For each ðk; jÞ 2 X p than the alternative aj if Dpk P Dpj . Then, the ranking order of the alternatives ak and aj determined by Dpk and Dpj based on x is cone 0 given by the DM sistent with the subjective preference ðk; jÞ 2 X p p p Dp. Conversely, if Dk < Dj , then the obtained objective ranking order aj p ak is inconsistent with the subjective preference e 0 . Therefore, x should be chosen so that the objective ðk; jÞ 2 X p ranking order is as consistent as possible with the subjective preference provided by the DM Dp. e 0 , an index Definition 5. For each pair of alternatives ðk; jÞ 2 X p  ðDpj  Dpk Þ measuring fuzzy inconsistency between the subjective and objective ranking orders is defined as follows:

( 

if i 2 M 1

pþ

Obviously, 0 6 dij 6 dij þ dij , which directly infers that pþ 0 6 dpij 6 1. If dij ¼ 0, then dpij ¼ 1. Thus, the bigger dpij the better the alternative aj (or r pij ). The individual comprehensive relative closeness degree of the alternatives r pj with respect to the fuzzy PIS r+ for the DM Dp is defined as follows:

ðDpj  Dpk Þ ¼

~cp ðk; jÞðDpj  Dpk Þ if Dpk < Dpj 0

if Dpk P Dpj :

ð21Þ

Definition 5 may be explained as follows. If Dpk P Dpj , the objective ranking order determined by Dk and Dj is consistent with the e 0 . Hence, the inconsistency subjective preference ðk; jÞ 2 X (Dj  Dk) is defined as 0. On the other hand, if Dpk < Dpj , the objective ranking order determined by Dk and Dj is inconsistent with the e 0 . Hence, the inconsistency subjective preference ðk; jÞ 2 X    p  Dpj  Dpk is defined as ~cp ðk; jÞ Dpj  Dpk . The inconsistency index can be rewritten as   n o p p p p Dj  Dk ¼ ~cp ðk; jÞ max 0; Dj  Dk . Then, the fuzzy group inconsistency index is defined as follows:

if i 2 M 3 if i 2 M 4 ; ð18Þ

p p p þ þ    where pþ i ¼ 1  li  ti ; pi ¼ 1  li  ti and pij ¼ 1  lij  tij . The relative closeness degrees of rpij with respect to rþ i are defined as follows:

2 3 P   X X 6 7 e¼ B Dpj  Dpk 5 4wp p¼1 ðk;jÞ2 e X 0p 2 3 P  n o X X 6 ~cp ðk; jÞ max 0; Dpj  Dpk 7 ¼ 4wp 5: p¼1 ðk;jÞ2 e X 0p

ð22Þ

D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89

Analogously, we introduce the definition of the fuzzy consistency as follows. e 0 , an index Definition 6. For each pair of alternatives ðk; jÞ 2 X p  þ p p Dj  Dk measuring fuzzy consistency between the subjective and objective ranking orders is defined as follows:

8    þ < ~cp ðk; jÞ Dp  Dp if Dpk P Dpj j k p p Dj  Dk ¼ :0 if Dpk < Dpj : 

ð23Þ

It is easy to see that Eq. (23) can be rewritten as þ n o ¼ ~cp ðk; jÞ max 0; Dpk  Dpj . Hence, the fuzzy group con-

Dpj  Dpk

sistency index is defined as follows:

2

ð24Þ

4.3. Fuzzy linear programming models with TrFNs Solving the fuzzy inhomogenous MAGDM problem becomes determining the weight vector x. Thus, we construct the fuzzy mathematical programming model with TrFNs as follows:

ð25Þ

~ is a positive TrFN given by the group a priori, denoted by where h ~ h ¼ ðh1 ; h2 ; h3 ; h4 Þ; K is the preference information structure of attribute importance stated as in Subsection 3.2. The aim of Eq. (25) is to minimize the fuzzy group inconsistency e under the condition in which the fuzzy group inconsisindex B e is smaller than or equal to the fuzzy group consistency index B ~ e by a TrFN h. tency index G It is derived from Eqs. (21)–(24) that

8 9 P > < = þ   > X X  p eB e¼ G wp Dj  Dpk  Dpj  Dpk > > ; p¼1 : ðk;jÞ2 e X 0p 8 9 < = P >  i> X X h p p ~cp ðk; jÞ Dk  Dj wp : ¼ > > ; p¼1 : 0 e ðk;jÞ2 X p Then, Eq. (25) can be rewritten as follows:

n o e 0 , let kp ¼ max 0; Dp  Dp , then kp P 0 and For each ðk; jÞ 2 X p j kj k kj kpkj P Dpj  Dpk , which infers that Dpk  Dpj þ kpkj P 0. It directly fol  P p p lows from Eq. (20) that Dpk  Dpj ¼ m i¼1 xi dik  dij . Thus, Eq. (26) can be transformed into the fuzzy mathematical programming model as follows:

2 39 P =   > X X 6 7 e¼ min B kpkj ~cp ðk; jÞ 5 4wp > > : ; p¼1 ðk;jÞ2 e X 0p

Remark 1. The above fuzzy group inconsistency and consistency indexes are TrFNs since the fuzzy truth degrees of pair-wise alternatives’ comparisons are represented as TrFNs. Whereas, in LINMAP and its extensions [31–33], the inconsistency and consistency indexes were real numbers since they did not consider the fuzzy truth degrees of alternatives’ comparisons. Additionally, LINMAP and its extensions [31–33] utilized the distances between the alternative and the PIS to define the inconsistency and consistency indexes, while our paper uses the comprehensive relative closeness degrees of the alternative with respect to the fuzzy PIS to define the fuzzy group inconsistency and consistency indexes. These are the most differences of our paper from LINMAP and its extensions [31–33].

e min f Bg ( ~ eB ePh G s:t: x 2 K;

ð26Þ

8 >
X X 6 e¼ ~cp ðk; jÞ max 0; Dpj  Dpk 7 min B 4wp 5 > > : ; p¼1 ðk;jÞ2 e X 0p 8 9 8 > P > < =  i> >X X h > < ~ ~cp ðk; jÞ Dpk  Dpj wp Ph > s:t: p¼1 > : ; > ðk;jÞ2 e X 0p > > : x 2 K:

77

8 >
< Dj  Dj  dj þ dj ¼ 0 ðj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; PÞ s:t: 0 6 Dj 6 1 ðj ¼ 1; 2; . . . ; nÞ > : pþ p dj dj ¼ 0 ðj ¼ 1; 2; . . . ; n; p ¼ 1; 2; . . . ; PÞ: ð34Þ Using the existing software such as Matlab and Lingo to solve Eq. (34), the collective comprehensive relative closeness degrees Dj (j = 1, 2, . . ., n) of the alternatives aj can be obtained. Step 9: The ranking order of all alternatives is generated according to the non-increasing order of Dj (j = 1, 2, . . ., n) and the best alternative from the alternative set is determined. 5. A real example study of IT outsourcing provider selection and comparison analyses To illustrate the method proposed in this paper, an IT outsourcing provider selection problem and analysis are given. Meanwhile, the comparison analyses of the obtained results are conducted in this section.

5.1. An IT outsourcing provider selection example and analysis San’an Optoelectronics Company Limited (San’an for short) is China’s earliest-established and largest production base for full-color ultra-high bright LED epitaxial products and chips with optimum quality. It earned the ‘‘National Model Project of HighTech Industrialization’’ from the National Development and Reform Commission and the ‘‘Leading Company in Semi-Conductor Illumination Projects’’ from the Ministry of Science and Technology of People’s Republic of China. San’an is mainly engaged in the research & development, production and marketing of full-color, ultra-high brightness LED epitaxial products, chips, compound solar cells, and high-power concentrating solar products with internationally leading performance. With its production capacity ranked first in China, the company possesses modern, clean workshops of grade 100 to 10,000, and thousands of pieces of the most advanced equipments in the world for producing epitaxial products and chips, and an automated production line for high-power concentrating solar products. Products of San’an have been widely used in indoor and outdoor lighting, backlighting, displays, traffic signal lights, electronics and aerospace, solar power, and other fields. The products have been used at home and abroad, and have earned an excellent reputation from customers all over the world. To save the operation cost and increase their focus on the core competencies, San’an is engaged in the decision about an IT outsourcing project and intends to select its service providers. San’an invites three experts (i.e., DMs) D1, D2 and D3 to form the decision group. After preliminary screening, there are five potential service providers for further evaluating. The five outsourcing providers are Shanghai Ingens IT Company Limited a1, Beijing Teamsun Technology Company Limited (Teamsun for short) a2, Shenzhen Sinoway IT Outsourcing Company Limited (Sinoway for short) a3, Lenovo a4, and Taiji Company Limited (Taiji for short) a5. Considering the real need of San’an company, the decision group chooses six independent attributes to evaluate the above five outsourcing providers. These attributes are research and development capability f1, product quality f2, technological level f3, flexibility f4, delivery time f5 and price f6, where the former four attributes are qualitative and benefit attributes, whereas the latter two attributes are cost attributes. The assessments (or ratings) of the outsourcing providers on f1 are divided into two parts: satisfaction degrees and dissatisfaction degrees, which are essentially the membership degrees and non-membership degrees of IFSs, respectively. Namely, the ratings of the outsourcing providers on f1 can be expressed with IFSs. The ratings of the outsourcing providers on f2 and f3 are represented by TrFNs. Due to the uncertainty of product process, intervals are suitably used to represent the ratings of the outsourcing providers on f4 and f5. The ratings of the outsourcing providers on f6 can be represented by real numbers. After data eliciting and statistical treatment, the ratings of the five outsourcing providers on the six attributes given by the three experts are obtained as follows:

1 h0:5; 0:3i h0:6; 0:2i h0:4; 0:4i h0:7; 0:1i h0:3; 0:6i B ð3; 4; 5; 6Þ ð6; 7; 8; 9Þ ð5; 6; 7; 8Þ ð1; 2; 3; 4Þ ð2; 3; 4; 5Þ C C B C B B ð70; 90; 91; 92Þ ð30; 80; 85; 90Þ ð50; 60; 75; 85Þ ð75; 80; 85; 90Þ ð80; 85; 90; 95Þ C C; ¼B C B ½4; 10 ½7; 9 ½4; 9 ½6; 10 ½2; 8 C B C B A @ ½65; 88 ½87; 90 ½45; 58 ½70; 90 ½92; 95 0

  Y 1 ¼ y1ij

65

119

110

120

79

118

100

80

D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89

0

Y 2 ¼ ðy2ij Þ65

h0:4; 0:5i

h0:3; 0:6i

h0:3; 0:3i

h0:6; 0:3i

h0:2; 0:7i

1

B ð5; 6; 7; 8Þ ð2; 3; 4; 5Þ ð3; 4; 5; 6Þ ð1; 2; 3; 4Þ ð6; 7; 8; 9Þ C C B C B B ð80; 85; 90; 95Þ ð50; 60; 75; 85Þ ð30; 80; 85; 90Þ ð75; 80; 85; 95Þ ð70; 90; 91; 92Þ C C ¼B C B ½4; 7 ½5; 8 ½3; 6 ½7; 9 ½8; 10 C B C B A @ ½75; 88 ½87; 90 ½65; 88 ½66; 87 ½89; 95 120

118

115

108

119

h0:2; 0:3i

h0:5; 0:4i

h0:6; 0:2i

h0:7; 0:1i

h0:5; 0:3i

and

0

Y 3 ¼ ðy3ij Þ65

1

B ð5; 6; 7; 8Þ ð3; 5; 6; 7Þ ð4; 5; 6; 7Þ ð4; 5; 8; 9Þ ð1; 4; 6; 7Þ C C B C B B ð72; 80; 90; 95Þ ð50; 60; 75; 85Þ ð74; 80; 82; 85Þ ð65; 70; 78; 81Þ ð82; 84; 89; 92Þ C C; ¼B C B ½6; 8 ½6; 8 ½7; 10 ½5; 7 ½3; 6 C B C B A @ ½75; 89 ½82; 90 ½78; 86 ½66; 78 ½65; 90 111

116

110

120

105

respectively.

According to the comprehensions and judgments, the experts provide their fuzzy preference relations between alternatives as e 1 ¼ fðð1; 2Þ; ~c1 ð1; 2ÞÞ; ðð3; 1Þ; ~c1 ð3; 1ÞÞ; ðð4; 5Þ; ~c1 ð4; 5ÞÞ; follows: X e 2 ¼ fðð2; 1Þ; ~c1 ð5; 2ÞÞ; ðð2; 3Þ; ~c1 ð2; 3ÞÞ; ðð4; 3Þ; ~c1 ð4; 3ÞÞg; X ðð5; 2Þ; ~c2 ð2; 1ÞÞ; ðð3; 2Þ ; ~c2 ð3; 2ÞÞ; ðð4; 3Þ; ~c2 ð4; 3ÞÞ; ðð5; 4Þ; ~c2 ð5; 4ÞÞg, and e 3 ¼ fðð1; 3Þ; ~c3 ð1; 3ÞÞ; ðð2; 5Þ; ~c3 ð2; 5ÞÞ; ðð4; 5Þ; ~c3 ð4; 5ÞÞ; ðð3; 2Þ; X ~c3 ð3; 2ÞÞ; ðð4; 3Þ; ~c3 ð4; 3ÞÞg, ~c1 ð1; 2Þ ¼ ð0:1; 0:2; 0:3; 0:4Þ; where ~c1 ð3; 1Þ ¼ ð0:3; 0:4; 0:5; 0:6Þ; ~c1 ð4; 5Þ ¼ ð0:5; 0:6; 0:7; 0:8Þ, ~c1 ð5; 2Þ ¼ ð0:4; 0:5; 0:6; 0:7Þ; ~c1 ð2; 3Þ ¼ ð0:6; 0:7; 0:8; 0:9Þ; ~c1 ð4; 3Þ ¼ ð0:4; 0:5; ~c2 ð2; 1Þ ¼ ð0:55; 0:64; 0:71; 0:80Þ; ~c2 ð3; 2Þ ¼ ð0:20; 0:24; 0:9; 1:0Þ, 0:25; 0:26Þ; ~c2 ð4; 3Þ ¼ ð0:35; 0:36; 0:40; 0:45Þ; ~c2 ð5; 4Þ ¼ ð0:52; 0:55; 0:64; 0:68Þ; ~c3 ð1; 3Þ ¼ ð0:32; 0:45; 0:48; 0:51Þ; ~c3 ð2; 5Þ ¼ ð0:63; 0:64;

0:72; 0:75Þ; ~c3 ð4; 5Þ ¼ ð0:65; 0:68; 0:72; 0:78Þ; ~c3 ð3; 2Þ ¼ ð0:48; 0:52; 0:64; 0:73Þ, and ~c3 ð4; 3Þ ¼ ð0:45; 0:58; 0:62; 0:69Þ. Thus, it is easy to e 0 ¼ fð1; 2Þ; ð3; 1Þ; ð4; 5Þ; ð5; 2Þ; ð2; 3Þ; ð4; 3Þg, X e0 ¼ obtain that X 1 2 e 0 ¼ fð1;3Þ;ð2;5Þ;ð4;5Þ;ð3;2Þ; fð2; 1Þ; ð3; 2Þ; ð4; 3Þ; ð5; 4Þg and X 3 ð4; 3Þg. The decision group gives the preference information structure K of the attribute importance as K = {x 2 K0j3x1 P x2, 0.01 6 x2  x3 6 0.2, 0.25 6 x4 6 0.45, x4  x5 6 x6  x2, 0.1 6 x6 6 0.25}. According to Eqs. (4)–(7), the fuzzy decision matrices Y1, Y2 and Y3 are normalized into R1, R2 and R3 as follows:

0

1 h0:5; 0:3i h0:6;0:2i h0:4; 0:4i h0:7; 0:1i h0:3; 0:6i B ð0:333; 0:444; 0:556; 0:667Þ ð0:667; 0:778; 0:889; 1Þ ð0:556; 0:667; 0:778;0:889Þ ð0:111;0:222; 0:333; 0:444Þ ð0:222; 0:333; 0:444; 0:556Þ C B C B C B ð0:737; 0:945; 0:958; 0:968Þ ð0:316; 0:842;0:895; 0:947Þ ð0:526; 0:632; 0:79; 0:895Þ ð0:790; 0:842; 0:895; 0:947Þ ð0:842; 0:895; 0:947; 1Þ C 1 C; R ¼B B C ½0:7;0:9 ½0:4; 0:9 ½0:6; 1:0 ½0:2; 0:8 ½0:4; 1:0 B C B C @ A ½0:0737; 0:3158 ½0:0526;0:0842 ½0:3895; 0:5263 ½0:0526; 0:2632 ½0; 0:0316 0:0083

0:0083

0

0:0167

0:1667

0

1 h0:4; 0:5i h0:3;0:6i h0:3; 0:3i h0:6; 0:3i h0:2; 0:7i B ð0:556; 0:667; 0:778; 0:889Þ ð0:222; 0:333;0:444; 0:556Þ ð0:333; 0:444; 0:556;0:667Þ ð0:111;0:222; 0:333; 0:444Þ ð0:667; 0:778; 0:889; 1Þ C B C B C B ð0:842; 0:895; 0:947; 1Þ ð0:556; 0:667;0:778; 0:889Þ ð0:316; 0:842; 0:895;0:947Þ ð0:79; 0:842; 0:895; 1Þ ð0:737; 0:947; 0:958; 0:968Þ C 2 B C R ¼B C ½0:5;0:8 ½0:3; 0:6 ½0:7; 0:9 ½0:8; 1:0 ½0:4; 0:7 B C B C @ A ½0:0737; 0:2105 ½0:0526;0:0842 ½0:0737;0:3158 ½0:0842; 0:3053 ½0; 0:0632 0

0:0167

0:0417

0:1

0:0083

and 0

1 h0:2; 0:3i h0:5;0:4i h0:6; 0:2i h0:7; 0:1i h0:5; 0:3i B ð0:556; 0:667; 0:778; 0:889Þ ð0:333; 0:556;0:667; 0:778Þ ð0:444; 0:556; 0:667;0:778Þ ð0:444;0:556; 0:889; 1Þ ð0:111; 0:444; 0:667; 0:778Þ C B C B C B ð0:758; 0:842; 0:947; 1Þ ð0:526;0:632; 0:79; 0:842Þ ð0:779; 0:842; 0:863;0:895Þ ð0:684;0:737; 0:821; 0:856Þ ð0:863; 0:884; 0:936; 0:968Þ C C; R3 ¼ B B C ½0:6;0:8 ½0:7; 1:0 ½0:5; 0:7 ½0:3; 0:6 ½0:6; 0:8 B C B C @ A ½0:0632; 0:2105 ½0:0526; 0:1368 ½0:0947;0:1789 ½0:1789;0:3053 ½0:0526; 0:3158 0:0750

respectively.

0:0333

0:0833

0

0:1250

D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89

By using Eqs. (9)–(16), we obtain the fuzzy PIS and the fuzzy NIS as follows:

rþ ¼ ðh0:7; 0:1i; ð0:6667; 0:7778; 0:8889; 1Þ; ð0:8421; 0:9474; 0:9579; 1Þ; ½0:8; 1; ½0:3895; 0:5263; 0:1667Þ and

81

r ¼ ðh0:2; 0:7i; ð0:1111; 0:2222; 0:3333; 0:4444Þ;  0:3158; 0:6316; 0:7895; 0:8947Þ; ½0:2; 0:6; ½0; 0:0632; 0ÞT : ~ ¼ ð0:0001; 0:0002; 0:0003; 0:0004Þ and the weight vector Take h of DMs w ¼ ð0:2; 0:2; 0:6ÞT . According to Eq. (27), the fuzzy linear programming model with TrFNs is constructed as follows:

  min ~z ¼ 0:2 ð0:1;0:2;0:3; 0:4Þk112 þ ð0:3;0:4;0:5; 0:6Þk131 þ ð0:5;0:6;0:7; 0:8Þk145 þ ð0:4;0:5; 0:6; 0:7Þk152 þ ð0:6;0:7; 0:8; 0:9Þk123 þ ð0:4;0:5; 0:9;    1Þk143 þ 0:2 ð0:55;0:64;0:71;0:80Þk221 þ ð0:20;0:24;0:25;0:26Þk232 þ ð0:35;0:36;0:40; 0:45Þk243 þ ð0:52; 0:55; 0:64; 0:68Þk254 þ 0:6½ð0:32;  0:45; 0:48; 0:51Þk313 þ ð0:63; 0:64; 0:72; 0:75Þk325 þ ð0:65; 0:68; 0:72; 0:78Þk345 þ ð0:48; 0:52; 0:64; 0:73Þk332 þ ð0:45; 0:58; 0:62; 0:69Þk343 8 0:2½ð0:1898x1  0:6923x2 þ 0:7051x3  0:3889x4 þ 0:3143x5  0:4970x6 Þð0:1; 0:2; 0:3;0:4Þ þ ð0:3272x1 þ 0:6334x2  > > > > > > 0:8655x3  0:1222x4 þ 0:6754x5  0:0028x6 Þð0:3; 0:4; 0:5;0:6Þ þ ð0:9545x  0:0588x2  0:0836x3 þ 0:7980x4 þ 0:1871x5  > > > > > > > > 0:9878x6 Þð0:5;0:6; 0:7; 0:8Þ þ ð0:9091x1  0:9412x2 þ 0:7199x3  0:8535x4  0:0078x5 þ 0:5002x6 Þð0:4; 0:5; 0:6;0:7Þþ > > > > > ð0:5170x1 þ 0:0588x2 þ 0:1604x3 þ 0:5111x4  0:9897x5 þ 0:4998x6 Þð0:6;0:7;0:8; 0:9Þ þ ð0:5625x1  0:9412x2 þ 0:7967x3 þ > > > > > > > 0:4556x4  0:8104x5 þ 0:0122x6 Þð0:4; 0:5; 0:9;1:0Þ þ 0:2½ð0:1898x1  0:8824x2  0:8803x3 þ 0:3333x4  0:1095x5 þ > > > > > > 0:0122x6 Þð0:55; 0:64; 0:71; 0:80Þ þ ð0:4745x1 þ 0:2489x2 þ 0:1604x3  0:4762x4 þ 0:3143x5 þ 0:0878x6 Þð0:20; 0:24; 0:25; > > > > > > 0:26Þ þ ð0:3221x1  0:3077x2 þ 0:6451x3 þ 0:9206x4  0:0084x5 þ 0:5921x6 Þð0:35; 0:36; 0:40;0:45Þ þ ð0:8421x1 þ > > > > > > > 1:0000x2 þ 0:0600x3 þ 0:0556x4  0:3162x5  0:6893x6 Þð0:52;0:55;0:64;0:68Þ þ 0:6½ð0:4974x1 þ 0:2489x2 þ 0:0708x3  > > > > > > 0:2619x4 þ 0:0135x5  0:0990x6 Þð0:32; 0:45; 0:48; 0:51Þ þ ð0:2022x1 þ 0:2393x2  0:8688x3 þ 0:6905x  0:2667x5  > > > > > > 0:8411x6 Þð0:63;0:64;0:72;0:75Þ þ ð0:2353x1 þ 0:5058x2  0:4954x3 þ 0:3333x4 þ 0:1963x5  0:8999x6 Þð0:65;0:68;0:72; > > > > > > > > > 0:78Þ þ ð0:3920x1 þ 0:0641x2 þ 0:7405x3 þ 0:2619x4 þ 0:0673x5 þ 0:4410x6 Þð0:48;0:52;0:64;0:73Þ þ ð0:0455x1 þ 0:2024x2  > > > > > 0:3671x3  0:6191x4 þ 0:3957x5  0:4998x6 Þð0:45;0:58;0:62;0:69Þ P ð0:0001; 0:0002; 0:0003;0:0004Þ > > > > > 0:1898x1  0:6923x2 þ 0:7051x3  0:3889x4 þ 0:3143x5  0:4970x6 þ k1 P 0 > > 12 > > > > > 0:3272x1 þ 0:6334x2  0:8655x3  0:1222x4 þ 0:6754x5  0:0028x6 þ k131 P 0 > > > > > > 0:9545x  0:0588x  0:0836x þ 0:7980x þ 0:1871x  0:9878x þ k1 P 0 > > 2 3 4 5 6 45 < s:t: 0:9091x  0:9412x þ 0:7199x  0:8535x  0:0078x þ 0:5002x þ k1 P 0 1 2 3 4 5 6 > 52 > > > > 1 > > > 0:5170x1 þ 0:0588x2 þ 0:1604x3 þ 0:5111x4  0:9897x5 þ 0:4998x6 þ k23 P 0 > > > > > 0:5625x1  0:9412x2 þ 0:7967x3 þ 0:4556x4  0:8104x5 þ 0:0122x6 þ k143 P 0 > > > > > > > 0:1898x1  0:8824x2  0:8803x3 þ 0:3333x4  0:1095x5 þ 0:0122x6 þ k221 P 0 > > > > > > > 0:4745x1 þ 0:2489x2 þ 0:1604x3  0:4762x4 þ 0:3143x5 þ 0:0878x6 þ k232 P 0 > > > > > > > 0:3221x1  0:3077x2 þ 0:6451x3 þ 0:9206x4  0:0084x5 þ 0:5921x6 þ k243 P 0 > > > > > > 0:8421x1 þ 1:0000x2 þ 0:0600x3 þ 0:0556x4  0:3162x5  0:6893x6 þ k254 P 0 > > > > > 3 > > > > 0:4974x1 þ 0:2489x2 þ 0:0708x3  0:2619x4 þ 0:0135x5  0:0990x6 þ k13 P 0 > > > > > 0:2022x1 þ 0:2393x2  0:8688x3 þ 0:6905x  0:2667x5  0:8411x6 þ k325 P 0 > > > > > 3 > > > 0:2353x1 þ 0:5058x2  0:4954x3 þ 0:3333x4 þ 0:1963x5  0:8999x6 þ k45 P 0 > > > > > 0:3920x1 þ 0:0641x2 þ 0:7405x3 þ 0:2619x4 þ 0:0673x5 þ 0:4410x6 þ k3 P 0 > 32 > > > > > 0:0455x þ 0:2024x  0:3671x  0:6191x þ 0:3957x  0:4998x þ k3 P 0 > > 1 2 3 4 5 6 43 > > > > 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 > > k12 ; k31 ; k45 ; k52 ; k23 ; k43 ; k21 ; k32 ; k43 ; k54 ; k13 ; k25 ; k45 ; k32 ; k43 P 0 > > > > > > 3x1 P x2 ; 0:01 6 x2  x3 6 0:2; 0:15 6 x4 6 0:45; x4  x5 P x6  x3 ;0:1 6 x6 6 0:25 > > > > : x1 þ x2 þ x3 þ x4 þ x5 þ x6 ¼ 1; xi P 0:05 ði ¼ 1;2; 3; 4; 5;6Þ ð35Þ

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According to Eq. (32), Eq. (35) can be transformed into the linear programming model as follows:

min

s:t:



Z ¼ 0:3600k112 þ 0:6000k131 þ 0:8400k145 þ 0:7200k152 þ 0:9450k123 þ 0:9900k143 þ 0:2860k221 þ 0:0985k232 þ 0:1630k243 þ  0:2515k254 þ 0:1870k313 þ 0:2840k325 þ 0:2915k345 þ 0:2555k332 þ 0:2480k343

8 0:6247x1  0:5925x2 þ 0:3032x3 þ 0:8494x4  0:6965x5  0:5931x6 P 0:0003 > > > > > > > 0:7516x1  0:9764x2 þ 0:5895x3 þ 1:041x4  0:9352x5  0:6972x6 P 0:0007  0:0004a > > > > > > > 0:1898x1  0:6923x2 þ 0:7051x3  0:3889x4 þ 0:3143x5  0:4970x6 þ k112 P 0 > > > > > 1 > > > 0:3272x1 þ 0:6334x2  0:8655x3  0:1222x4 þ 0:6754x5  0:0028x6 þ k31 P 0 > > > > > > 0:9545x  0:0588x2  0:0836x3 þ 0:7980x4 þ 0:1871x5  0:9878x6 þ k145 P 0 > > > > > > 1 > > > 0:9091x1  0:9412x2 þ 0:7199x3  0:8535x4  0:0078x5 þ 0:5002x6 þ k52 P 0 > > > > > 0:5170x1 þ 0:0588x2 þ 0:1604x3 þ 0:5111x4  0:9897x5 þ 0:4998x6 þ k1 P 0 > 23 > > > > > > 0:5625x1  0:9412x2 þ 0:7967x3 þ 0:4556x4  0:8104x5 þ 0:0122x6 þ k143 P 0 > > > > > > 2 > > > 0:1898x1  0:8824x2  0:8803x3 þ 0:3333x4  0:1095x5 þ 0:0122x6 þ k21 P 0 > > > > > 0:4745x þ 0:2489x þ 0:1604x  0:4762x þ 0:3143x þ 0:0878x þ k2 P 0 < 1 2 3 4 5 6 32

ð36Þ

> > 0:3221x1  0:3077x2 þ 0:6451x3 þ 0:9206x4  0:0084x5 þ 0:5921x6 þ k243 P 0 > > > > > > > 0:8421x1 þ 1:0000x2 þ 0:0600x3 þ 0:0556x4  0:3162x5  0:6893x6 þ k254 P 0 > > > > > > > 0:4974x1 þ 0:2489x2 þ 0:0708x3  0:2619x4 þ 0:0135x5  0:0990x6 þ k313 P 0 > > > > > > > 0:2022x1 þ 0:2393x2  0:8688x3 þ 0:6905x  0:2667x5  0:8411x6 þ k325 P 0 > > > > > > > > 0:2353x1 þ 0:5058x2  0:4954x3 þ 0:3333x4 þ 0:1963x5  0:8999x6 þ k345 P 0 > > > > > > > 0:3920x1 þ 0:0641x2 þ 0:7405x3 þ 0:2619x4 þ 0:0673x5 þ 0:4410x6 þ k332 P 0 > > > > > 3 > > > 0:0455x1 þ 0:2024x2  0:3671x3  0:6191x4 þ 0:3957x5  0:4998x6 þ k43 P 0 > > > > > > k112 ; k131 ; k145 ; k152 ; k123 ; k143 ; k221 ; k232 ; k243 ; k254 ; k313 ; k325 ; k345 ; k332 ; k343 P 0 > > > > > > > 3x1 P x2 ; 0:01 6 x2  x3 6 0:2; 0:15 6 x4 6 0:45; x4  x5 P x6  x3 ; 0:1 6 x6 6 0:25 > > > > : x1 þ x2 þ x3 þ x4 þ x5 þ x6 ¼ 1; xi P 0:05 ði ¼ 1; 2; 3; 4; 5; 6Þ

Taking a = 0.5 and solving Eq. (36) by using the simplex method of linear programming, we obtain its optimal solution, whose components are as follows: k145 ¼ k131 ¼ k143 ¼ k123 ¼ k345 ¼ k332 ¼ k243 ¼ k232 ¼ k254 ¼ 0; k112 ¼ 0:1508; k152 ¼ 0:2603; k221 ¼ 0:2470; k313 ¼ 3 3 0:0578; k25 ¼ 0:1008; k43 ¼ 0:2090; x1 ¼ 0:0606; x2 ¼ 0:1818; x3 ¼ 0:1718; x4 ¼ 0:2743; x5 ¼ 0:1564; x6 ¼ 0:1551. Then, using Eq. (20), we can compute the individual comprehensive relative closeness degrees of the alternatives (i.e., the outsourcing providers) with respect to the fuzzy PIS as follows:

D11 ¼ 0:4731; D12 ¼ 0:6239; D13 ¼ 0:4914; D14 ¼ 0:4914; D15 ¼ 0:3636; D21 ¼ 0:4196; D22 ¼ 0:1726; D23 ¼ 0:2063; D24 ¼ 0:6238; D25 ¼ 0:6238 and

D31 ¼ 0:6326; D32 ¼ 0:3770;

D33 ¼ 0:6904;

D34 ¼ 0:4814;

D35

¼ 0:4777: According to Eq. (34), the linear goal programming model can be constructed as follows:

( " #) 3 5   X X pþ p min wp dj þ dj p¼1

j¼1

8 p pþ p > < Dj  Dj  dj þ dj ¼ 0 ðj ¼ 1; 2; . . . ; 5; p ¼ 1; 2; 3Þ s:t: 0 6 Dj 6 1ðj ¼ 1; 2; . . . ; 5Þ > : pþ p dj dj ¼ 0 ðj ¼ 1; 2; . . . ; 5; p ¼ 1; 2; 3Þ:

ð36Þ

Solving Eq. (36) by using the simplex method of linear programming, we can obtain the collective comprehensive relative closeness degrees of the alternatives with respect to the fuzzy PIS as follows: D1 = 0.4731, D2 = 0.6239, D3 = 0.4914, D4 = 0.4914, and D5 = 0.3636. Hence, the ranking order of the five outsourcing providers (i.e., alternatives) is generated as a2  a3  a4  a1  a5. The best selection for San’an is Teamsun a2. As a China’s leading integrated IT service provider, Teamsun is the first domestic localized IT service provider with service network across China and parts of the southeast Asia. It is the first company to put forward IT service products in China with its business activities comprising IT product service, development of application software, system integration, value-added distribution

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D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89 Table 1 Computation results for specific values of the parameter.

a

xT(D1, D2, D3, D4, D5)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

(0.0590, (0.0593, (0.0596, (0.0599, (0.0603, (0.0606, (0.0610, (0.0611, (0.0611, (0.0611, (0.0611,

0.1771, 0.1780, 0.1788, 0.1798, 0.1808, 0.1818, 0.1830, 0.1833, 0.1833, 0.1833, 0.1833,

0.1671, 0.1680, 0.1688, 0.1698, 0.1708, 0.1718, 0.1730, 0.1733, 0.1733, 0.1733, 0.1733,

Ranking orders 0.2862, 0.1627, 0.1479) 0.2839, 0.1615, 0.1493) 0.2818, 0.1604, 0.1505) 0.2795, 0.1591, 0.1519) 0.2768, 0.1577, 0.1536) 0.2743, 0.1564, 0.1551) 0.2714, 0.1548, 0.1569) 0.2705, 0.1543, 0.1574) 0.2705, 0.1543, 0.1574) 0.2705, 0.1543, 0.1574) 0.2705, 0.1543, 0.1574)

(0.4745, (0.4742, (0.4745, (0.4737, (0.4734, (0.4731, (0.4728, (0.4218, (0.4218, (0.4218, (0.4218,

and other IT services across cloud computing, mobile internet and information security. It has achieved successful cases in the fields such as telecom, finance, education, manufacture, energy, communication and government. Thus, the wise selection for San’an is Teamsun, which is accordance with our intuition. This analysis shows the rationality and effectiveness of the method proposed in this paper. Analogously, for some specific values of the parameter a 2 [0, 1], we can obtain the weight vectors of attributes, the collective comprehensive relative closeness degrees of the alternatives (i.e., outsourcing providers) and their ranking orders, depicted as in Table 1. From Table 1, when the acceptance degree a 2 [0, 0.6], the best outsourcing provider is Teamsun a2; when the acceptance degree a 2 [0.7, 1], the best outsourcing provider is Lenovo a4 or Taiji a5. Especially, if a 2 [0.7, 1], the weight vectors of the six attributes are identical, i.e., x = (0.0611, 0.1833, 0.1733, 0.2705, 0.1543, 0.1574)T, hereby the collective comprehensive relative closeness degrees of the alternatives (i.e., outsourcing providers) and their ranking orders are completely identical. To recognize the impact of the representation of the attributes (e.g., IFSs, TrFNs, intervals, real numbers) on the final decision results, we further make sensitivity analyses from two aspects as follows: (1) If all IFSs ypij ¼

D

E





l0pij ; t0pij and TrFNs ypij ¼ b0pij1 ; b0pij2 ; b0pij3 ; b0pij4 in

the fuzzy decision matrices Y1, Y2 and Y3 are reduced to interh i h i 0p 0p 0p and ypij ¼ bij2 ; bij3 , respectively, i.e., vals ypij ¼ l0p ij ; 1  tij 0p

0p

0p

0p

bij1 ¼ bij2 and bij3 ¼ bij4 , then the above outsourcing provider selection example only contains two kinds of attribute information: intervals and real numbers. In this case, we use the method proposed in this paper to determine the best outsourcing provider for San’an. For different acceptance degrees a 2 [0, 1], we obtain the weight vectors of the six attributes, which are the same as x = (0.05, 0.06, 0.05, 0.3997, 0.2423, 0.1980)T. Hereby, the collective comprehensive relative closeness degrees of the alternatives (i.e., outsourcing providers) are completely identical, i.e., D1 = 0.4066, D2 = 0.6000, D3 = 0.4993, D4 = 0.4989, D5 = 0.2901, and the ranking orders of the alternatives are completely the same as a2  a3  a4 Da1  a5E.   (2) If all IFSs ypij ¼ l0p t0pij , TrFNs ypij ¼ b0pij1 ; b0pij2 ; b0pij3 ; b0pij4 and ij ; i h 0p 1 2 3 intervals ypij ¼ e0p ij ; g ij in Y , Y and Y are reduced to the real   0p 0p p p 0p numbers ypij ¼ l0p ij þ 1  tij =2; yij ¼ bij and yij ¼ eij , respec0p

0p

0p

0p

0p

0p tively, i.e., bij1 ¼ bij2 ¼ bij3 ¼ bij4 ¼ bij and e0p ij ¼ g ij , then the

above outsourcing provider selection example only contains the real number type of attribute information. In this case, we use the method proposed in this paper to determine the best outsourcing provider for San’an. For different

0.6241, 0.6241, 0.6241, 0.6240, 0.6240, 0.6239, 0.6239, 0.1710, 0.1710, 0.1710, 0.1710,

0.4972, 0.4961, 0.4951, 0.4939, 0.4926, 0.4914, 0.4899, 0.2069, 0.2069, 0.2069, 0.2069,

0.4972, 0.4961, 0.4951, 0.4939, 0.4926, 0.4914, 0.4900, 0.6229, 0.6229, 0.6229, 0.6229,

0.3525) 0.3547) 0.3566) 0.3588) 0.3613) 0.3636) 0.3663) 0.6229) 0.6229) 0.6229) 0.6229)

a2  a3  a4  a1  a5 a2  a3  a4  a1  a5 a2  a3  a4  a1  a5 a2  a3  a4  a1  a5 a2  a3  a4  a1  a5 a2  a3  a4  a1  a5 a2  a4  a3  a1  a5 a4  a5  a1  a3  a2 a4  a5  a1  a3  a2 a4  a5  a1  a3  a2 a4  a5  a1  a3  a2

Table 2 Borda’s scores of the providers for experts. Providers

D1

D2

D3

Borda’s scores

a1 a2 a3 a4 a5

1 4 3 2 0

2 0 1 3 3

2 0 3 1 1

5 4 7 6 4

acceptance degrees a 2 [0,1], we can obtain the weight vectors of the six attributes, which are the same as x = (0.05, 0.06, 0.05, 0.3950, 0.2756, 0.694)T. Then, the collective comprehensive relative closeness degrees of the alternatives (i.e., outsourcing providers) are completely the same as D1 = 0.4147, D2 = 0.5817, D3 = 0.5304, D4 = 0.5007 and D5 = 0.2612, and their ranking orders are completely the same as a2  a3  a4  a1  a5. It is apparent that the generated ranking orders of the outsourcing providers considering two kinds or single type of attribute information are the same as a2  a3  a4  a1  a5, which is greatly different from that obtained by considering multiple different kinds of attribute information (see Table 1). The above sensitivity analyses imply that the final decision results are extremely sensitive to the representation of types of attribute information. Since real-life outsourcing provider selection problems are very complex and full of uncertainty and fuzziness, using multiple kinds of attribute information to evaluate outsourcing providers is more reasonable than using single type or two kinds of attribute information. Therefore, it is very important and necessary to study outsourcing provider selection problems with multiple kinds of attribute information. In the mean time, the sensitivity analyses show that the proposed method is not only suitable for inhomogenous MAGDM problems but also can be directly used to solve MAGDM problems with single type of attribute information.

5.2. Comparison analysis with the method without considering fuzzy truth degrees In this subsection, the method proposed in this paper is compared with the linear programming method without considering fuzzy truth degrees. e 0 ; p ¼ 1; 2; . . . ; PÞ, then the If all the TrFNs ~cp ðk; jÞ ¼ 1ððk; jÞ 2 X p fuzzy linear programming model with TrFNs (i.e., Eq. (27)) is reduced to the crisp linear programming model. In this case, for the above outsourcing provider selection example, taking ~ ¼ 0:0001, the fuzzy linear programming model with TrFNs (i.e., h Eq. (35)) is reduced to the crisp linear programming model as follows:

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  min Z ¼ k112 þ k131 þ k145 þ k152 þ k123 þ k143 þ k221 þ k232 þ k243 þ k254 þ k313 þ k325 þ k345 þ k332 þ k343 8 0:3123x1 þ 0:9009x2 þ 0:6728x3 þ 0:4877x4  0:3214x5  0:6644x6 P 0:0001 > > > > > 0:1898x1  0:6923x2 þ 0:7051x3  0:3889x4 þ 0:3143x5  0:4970x6 þ k112 P 0 > > > > 1 > > > > 0:3272x1 þ 0:6334x2  0:8655x3  0:1222x4 þ 0:6754x5  0:0028x6 þ k31 P 0 > > 1 > > > 0:9545x  0:0588x2  0:0836x3 þ 0:7980x4 þ 0:1871x5  0:9878x6 þ k45 P 0 > > 1 > > > 0:9091x1  0:9412x2 þ 0:7199x3  0:8535x4  0:0078x5 þ 0:5002x6 þ k52 P 0 > > > > > 0:5170x1 þ 0:0588x2 þ 0:1604x3 þ 0:5111x4  0:9897x5 þ 0:4998x6 þ k123 P 0 > > > > > 0:5625x1  0:9412x2 þ 0:7967x3 þ 0:4556x4  0:8104x5 þ 0:0122x6 þ k143 P 0 > > > > > > 0:1898x1  0:8824x2  0:8803x3 þ 0:3333x4  0:1095x5 þ 0:0122x6 þ k221 P 0 > > > > > 0:4745x1 þ 0:2489x2 þ 0:1604x3  0:4762x4 þ 0:3143x5 þ 0:0878x6 þ k232 P 0 > < s:t: 0:3221x1  0:3077x2 þ 0:6451x3 þ 0:9206x4  0:0084x5 þ 0:5921x6 þ k243 P 0 > > > > 0:8421x1 þ 1:0000x2 þ 0:0600x3 þ 0:0556x4  0:3162x5  0:6893x6 þ k254 P 0 > > > > > 0:4974x1 þ 0:2489x2 þ 0:0708x3  0:2619x4 þ 0:0135x5  0:0990x6 þ k3 P 0 > > 13 > > > 0:2022x þ 0:2393x  0:8688x þ 0:6905x  0:2667x  0:8411x þ k3 P 0 > > 1 2 3 5 6 25 > > > > > 0:2353x1 þ 0:5058x2  0:4954x3 þ 0:3333x4 þ 0:1963x5  0:8999x6 þ k345 P 0 > > > > > 0:3920x1 þ 0:0641x2 þ 0:7405x3 þ 0:2619x4 þ 0:0673x5 þ 0:4410x6 þ k332 P 0 > > > > > > 0:0455x1 þ 0:2024x2  0:3671x3  0:6191x4 þ 0:3957x5  0:4998x6 þ k343 P 0 > > > > > k112 ; k131 ; k145 ; k152 ; k123 ; k143 ; k221 ; k232 ; k243 ; k254 ; k313 ; k325 ; k345 ; k332 ; k343 P 0 > > > > > 3x1 P x2 ; 0:01 6 x2  x3 6 0:2; 0:15 6 x4 6 0:45; x4  x5 P x6  x3 ; 0:1 6 x6 6 0:25 > > > : x1 þ x2 þ x3 þ x4 þ x5 þ x6 ¼ 1; xi P 0:05 ði ¼ 1; 2; 3; 4; 5; 6Þ:

Solving Eq. (38) by using the simplex method of linear programming, we get the optimal solution whose components are given as follows: k145 ¼ k131 ¼ k123 ¼ k345 ¼ k332 ¼ k243 ¼ k232 ¼ k254 ¼ 0, k143 ¼ 0:0250; k112 ¼ 0:1255; k152 ¼ 0:3238; k221 ¼ 0:1839; k313 ¼ 0:0712, k325 ¼ 0:1087; k343 ¼ 0:1892; x1 ¼ 0:0513; x2 ¼ 0:1539; x3 ¼ 0:1439; x4 ¼ 0:3330, x5 = 0.2179, x6 = 0.10. Then, the individual comprehensive relative closeness degrees of the alternatives (i.e., outsourcing providers) with respect to the fuzzy PIS can be computed using Eq. (20) as follows:

D11 ¼ 0:4826; D12 ¼ 0:6081; D13 ¼ 0:5450; D14 ¼ 0:5200; D15 ¼ 0:2844; D21 ¼ 0:3807; D22 ¼ 0:1968; D23 ¼ 0:2013; D24 ¼ 0:6272; D25 ¼ 0:6272 and

ð38Þ

D31 ¼ 0:6029; D32 ¼ 0:3916; D33 ¼ 0:6741; D34 ¼ 0:4849; D35 ¼ 0:4025: The ranking orders of the five outsourcing providers for the three experts D1, D2 and D3 are generated as follows: a2  a3  a4  a1  a5, a4  a5  a1  a3  a2 and a3  a1  a4  a5  a2, respectively. The Borda’s scores of the five outsourcing providers can be obtained as in Table 2. Therefore, the ranking order of the five outsourcing providers without considering fuzzy truth degrees is a3  a4  a1  a2  a5 and the best selection is Sinoway a3, which are remarkably different from those obtained by the fuzzy linear programming method with considering fuzzy truth degrees proposed in this paper. The comparisons of ranking orders between both methods are depicted in Fig. 2. This analysis shows that introducing the fuzzy

Order positions

This paper’s method with α = 0.7

1 2

Method without considering fuzzy NIS

3

This paper’s method with α = 0.5

4

Method without considering fuzzy truth degrees

5

This paper’s method with α = 0.6

a1

a2

a3

a4

a5

Alternatives

Fig. 2. Comparisons of ranking orders among the method without considering fuzzy truth degrees, the method without considering fuzzy NIS and this paper’s method.

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D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89 Table 3 Computation results for different weight vectors obtained by TOPSIS.

xT

(C1, C2, C3, C4, C5)

(1/6, 1/6, 1/6, 1/6, 1/6, 1/6) (0, 1/6, 1/3, 1/3, 1/6, 0) (1/8, 1/8, 1/4, 1/4, 1/8, 1/8) (1/4, 1/8, 1/8, 1/8, 1/8, 1/4)

(0.2645, (0.7855, (0.6419, (0.2826,

0.2476, 0.5236, 0.6407, 0.2038,

truth degrees to represent pair-wise alternatives’ comparisons is of great importance. The fuzzy truth degrees can flexibly reflect the preference information of alternatives and thus play an important role in decision making. Naturally, the crisp truth degrees 0 or 1 cannot capture the uncertainties inherent in alternatives’ comparisons. 5.3. Comparison analysis with the method without considering the fuzzy NIS As stated in Introduction, the fuzzy NIS also plays an importance role in real decision process. In this section, if the fuzzy

Ranking orders

0.2643, 0.2896, 0.5951, 0.4837,

0.8205, 0.1872) 0.5941, 0.7011) 0.7300, 0.5225) 0.9577, 0.0465)

a4  a1  a3  a2  a5 a1  a5  a4  a2  a3 a4  a1  a2  a3  a5 a4  a3  a2  a1  a5

P p NIS were neglected, we replace Dpj ¼ m i¼1 xi dij of Eq. (20) with P pþ m p Dj ¼ i¼1 xi dij and adopt the same steps as Subsection 4.5 to solve the above outsourcing provider selection problem. The computation results indicate that for any acceptance degree a 2 [0, 1], the weight vectors of the six attributes obtained by the method without considering the fuzzy NIS are the same as x = (0.0720, 0.2160, 0.2060, 0.1500, 0.1656, 0.1904)T, the collective comprehensive relative closeness degrees of the five outsourcing providers are the same as D1 = 0.5600, D2 = 0.3980, D3 = 0.3980, D4 = 0.9030, and D5 = 0.1208, and their ranking orders are the same as a4  a1  a2  a3  a5, which are remarkably different from those obtained by the method proposed in this paper. The

Decision information input

Decision information process

Set of DMs, Set of alternatives, Set of attributes Fuzzy sets of DMs’ preferences on alternatives comparisons DMs give pair-wise comparisons of alternatives Normalize decision matrixes Decision matrixes Incomplete attribute weight information structures Incomplete attribute weight information

Decision making process

Individual comprehensive relative closeness degrees of alternatives

Fuzzy group consistency and inconsistency indexes

Fuzzy linear programming models with TrFNs

Linear goal programming model

Comparison analysis with the method Attribute weight vector without considering the fuzzy NIS Collective comprehensive relative closeness degrees of alternatives

Comparison analysis with the method without considering fuzzy truth degrees

Ranking order of alternatives Comparison analysis with TOPSIS

Decision result display

Result comparison and analysis

Fig. 3. The modules of the decision support system.

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comparisons of ranking orders between both methods are also depicted in Fig. 2. The main reason is that only the fuzzy PIS is considered whereas the fuzzy NIS is neglected. Therefore, the ranking order obtained by the method without considering the fuzzy NIS cannot well reflect some tradeoff/compromise of relative closeness degrees of the alternatives to the fuzzy PIS as well as the fuzzy NIS. 5.4. Comparison analysis with TOPSIS In this subsection, we also do not consider the fuzzy truth degrees of the alternative’s comparisons and use TOPSIS to solve the above outsourcing provider selection example. The steps of TOPSIS are summarized as follows:

Step 1: Calculate the group decision matrix Y = (yij)mn, where P yij ¼ Pp¼1 ypij =P ði ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; nÞ;

Step 2: Normalize the group decision matrix Y = (yij)mn into the decision matrix R = (rij)mn by using the similar methods to Eqs. (4)–(7); Step 3: Compute the weighted normalized decision matrix R0 = (xirij)mn; Step 4: Determine the fuzzy PIS r+ and the fuzzy NIS r by using the similar methods to Eqs. (9)–(16); þ  Step 4: Calculate the Euclidean distances dj and dj ðj ¼ 1; 2; . . . ; nÞ of the alternatives aj from the fuzzy PIS r+ as well as the fuzzy NIS r; Step 5: Calculate the relative closeness coefficients of the    þ  alternatives to the fuzzy PIS, i.e., C j ¼ dj = dj þ dj ðj ¼ 1; 2; . . . ; nÞ; Step 6: Rank the alternatives according to the non-increasing order of Cj (j = 1, 2, . . ., n). Due to the fact that TOPSIS needs to determine the attribute weight vector a priori, we consider some special cases of the

Fig. 4. Necessary settings of the experts, attributes and alternatives.

Fig. 5. Decision information input for IT outsourcing provider selection.

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Fig. 6. Decision result display for IT outsourcing provider selection.

attribute weight vector. Using TOPSIS, we obtain the relative closeness coefficients of the alternatives and their ranking orders, depicted as in Table 3. From Table 3, it is not difficult to find that the obtained ranking orders of the five alternatives are completely different for different weight vectors. Furthermore, these ranking orders obtained by TOPSIS are remarkably different from those obtained by the method proposed in this paper. The above analysis indicates that weight vectors may result in different ranking orders of alternatives. TOPSIS greatly depends on the selection of attribute weights. However, the method proposed in this paper can objectively determine attribute weights and avoid subjective randomness of weight selection. In addition, TOPSIS cannot deal with the MAGDM problems with fuzzy truth degrees of alternatives’ comparisons. From these aspects, the method proposed in this paper is more reasonable and objective than TOPSIS. 6. A decision support system developed on the proposed method This section is devoted to developing a DSS based on the proposed method in this paper. The DSS is developed by using the Microsoft C#.net language for running on platforms such as Windows. It consists of five main modules (shown in Fig. 3), i.e., Decision information input, Decision information process, Decision making process, Decision result display, and Result comparison and analysis. 6.1. Decision information input This module provides four interfaces: (1) Set of DMs, set of alternatives and set of attributes; (2) DMs give pair-wise comparisons of alternatives; (3) Decision matrixes; (4) Incomplete attribute weight information. The user can conveniently input the decision information through these designed interfaces. In the first interface, the user can set the set of DMs, identify the alternative set and the attribute

set; In the second interface, each DM can give pair-wise comparisons of alternatives with fuzzy truth degrees. In the third interface, each DM can assess the alternatives with respect to attributes in different formats of information, such as IFS, TrFN, interval and real number. In the last interface, the group of DMs provides the attribute weight information according to their experiences, knowledge backgrounds, and specialties. 6.2. Decision information process This module is designed to process the decision information from the first module. We make three interfaces including: (1) Fuzzy sets of DMs’ preferences on alternatives comparisons; (2) Normalize decision matrixes; (3) Incomplete attribute weight information structures. The first interface is used to form the fuzzy sets e p ðp ¼ 1; 2; . . . ; PÞ of preferences on alternatives comparisons. X The second interface provides the normalized decision matrixes Rp (p = 1, 2, . . ., P) according to Eqs. (4)–(7). The third interface manifests the preference information structure K of attribute importance. 6.3. Decision making process Employing the algorithm and decision process developed in Subsection 4.5, the user can solve the fuzzy inhomogenous MAGDM. Therefore, the decision making process is the key module of the DSS. This module is composed of the four interfaces: (1) Individual comprehensive relative closeness degrees of alternatives; (2) Fuzzy group consistency and inconsistency indexes; (3) Fuzzy linear programming models with TrFNs; (4) Linear goal programming model. 6.4. Decision result display This module outputs the decision results involving: (1) Attribute weight vector; (2) Collective comprehensive relative closeness degrees of alternatives; (3) Ranking order of alternatives.

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6.5. Result comparison and analysis To demonstrate the superiorities of the proposed method in this paper, we devise the module of result comparison and analysis. This module can conduct the comparison analyses with three methods: (1) the method without considering the fuzzy NIS; (2) the method without considering fuzzy truth degrees; (3) TOPSIS. Applying the developed DSS to the IT outsourcing provider selection example in Subsection 5.1, the main interfaces are displayed in Figs. 4–6. The company of San’an is satisfied with the decision result. 7. Conclusions The IT outsourcing provider selection is a complex problem, which may be formulated as a type of inhomogeneous MAGDM problems with fuzzy alternatives’ comparisons and incomplete weight information. In this paper, inspired by TOPSIS and LINMAP, we develop a new fuzzy linear programming method for solving such problems. There are some conclusions of the proposed method which are described as follows: (1) Our method is more reasonable and consistent with actual problems since it takes into consideration the fuzzy truth degrees of alternatives’ comparisons, whereas the existing MCDM methods [3–12] such as TOPSIS [30], LINMAP [31] as well as their extensions [32,39] did not consider the fuzzy truth degrees, which are not accordance with real decision situations. (2) Our method uses the individual comprehensive relative closeness degrees to replace the distances of alternatives from the fuzzy PIS to define the fuzzy group consistency and inconsistency indexes. These indexes consider the fuzzy PIS and the fuzzy NIS simultaneously, which can effectively overcome disadvantages of LINMAP and its extensions only considering the fuzzy PIS. (3) Our method can objectively determine the attribute weights, which can make up the disadvantage of TOPSIS. (4) The linear goal programming model is constructed to derive the collective comprehensive relative closeness degrees of alternatives, which are used to rank the alternatives. The example and comparison analysis demonstrate the applicability and effectiveness of the method proposed in this paper. (5) Although a real IT outsourcing provider selection example is analyzed to illustrate the proposed method, it is expected to be applicable to decision making problems in many areas such as risk investment, performance evaluation of enterprises and engineering management. However, the proposed method of the fuzzy linear programming model with TrFNs greatly depends on the parameter a. Therefore, how to reasonably determine the acceptance degree a of the interval-valued inequalities is a valuable and an interesting topic, which will be investigated in near future. In addition, we will continue to improve the developed DSS for solving more heterogeneous MAGDM problems as our future works. Acknowledgments This research was supported by the Key Program of National Natural Science Foundation of China (No. 71231003), the National Natural Science Foundation of China (Nos. 71061006, 61263018, 71171055 and 71001015), the Program for New Century Excellent Talents in University (the Ministry of Education of China,

NCET-10-0020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20113514110009), the Humanities Social Science Programming Project of Ministry of Education of China (No. 09YGC630107), the Natural Science Foundation of Jiangxi Province of China (No. 20114BAB201012), the Science and Technology Project of Jiangxi Province Educational Department of China (Nos. GJJ12265 and GJJ12740), ‘‘Twelve five’’ Programming Project of Jiangxi Province Social Science (No. 13GL17) and the Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics.

References [1] D.Y. Chung, B.C. Choi, Outsourcing and scheduling for two-machine ordered flow shop scheduling problems, Eur. J. Oper. Res. 226 (2013) 46–52. [2] C.Y. Yang, J.B. Huang, A decision model for IS outsourcing, Int. J. Inform. Manage. 20 (2000) 225–239. [3] Y.T. Lin, C.L. Lin, H.C. Yu, G.H. Tzeng, A novel hybrid MCDM approach for outsourcing vendor selection: a case study for a semiconductor company in Taiwan, Expert Syst. Appl. 37 (2010) 4796–4804. [4] B. Feng, Z.P. Fan, Y.Z. Li, A decision method for supplier selection in multiservice outsourcing, Int. J. Product. Econ. 132 (2) (2011) 240–250. [5] W. Ho, T. He, C.K.M. Lee, A. Emrouznejad, Strategic logistics outsourcing: an integrated QFD and fuzzy AHP approach, Expert Syst. Appl. 39 (2012) 10841– 10850. [6] Y.H. Chen, T.C. Wang, C.Y. Wu, Strategic decisions using the fuzzy PROMETHEE for IS outsourcing, Expert Syst. Appl. 37 (2011) 13216–13222. [7] L.Y. Chen, T.C. Wang, Optimizing partners’ choice in IS/IT outsourcing projects: the strategic decision of fuzzy VIKOR, Int. J. Product. Econ. 120 (2009) 233–242. [8] Z.P. Fan, W.L. Suo, F. Bo, Identifying risk factors of IT outsourcing using interdependent information: an extended DEMATEL method, Expert Syst. Appl. 39 (2012) 3832–3840. [9] W.H. Tsai, J.D. Leu, J.Y. Liu, S.J. Lin, M.J. Shaw, A MCDM approach for sourcing strategy mix decision in IT projects, Expert Syst. Appl. 37 (2010) 3870–3886. [10] Y.C. Tjader, J.S. Shang, L.G. Vargas, Offshore outsourcing decision making: a policy-maker’s perspective, Eur. J. Oper. Res. 207 (2010) 434–444. [11] G. Buyukozkan, G. Cifcim, A novel hybrid MCDM approach based on fuzzy DEMATEL, fuzzy ANP and fuzzy TOPSIS to evaluate green suppliers, Expert Syst. Appl. 39 (2012) 3000–3011. [12] T.Y. Chou, S.T. Chou, G.H. Tzeng, Evaluating IT/IS investments: a fuzzy multicriteria decision model approach, Eur. J. Oper. Res. 173 (2006) 1026–1046. [13] F.J. Cabrerizo, I.J. Pérez, E. Herrera-Viedma, Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information, Knowl.-Based-Syst. 23 (2) (2010) 169–181. [14] Z. Zhang, C.H. Guo, A method for multi-granularity uncertain linguistic group decision making with incomplete weight information, Knowl.-Based-Syst. 26 (2012) 111–119. [15] D.-F. Li, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information, Appl. Soft Comput. 11 (2011) 3402–3418. [16] J. Ma, J. Lu, G. Zhang, Decider: a fuzzy multi-criteria group decision support system, Knowl.-Based Syst. 23 (1) (2010) 23–31. [17] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 50 (1986) 87–96. [18] B.K. Wong, V.S. Lai, A survey of the application of fuzzy set theory in production and operations management: 1998–2009, Int. J. Product. Econ. 129 (1) (2011) 157–168. [19] K. Khalili-Damghani, S. Sadi-Nezhad, A hybrid fuzzy multiple criteria group decision making approach for sustainable project selection, Appl. Soft Comput. 13 (1) (2013) 339–352. [20] D.-F. Li, Z.G. Huang, G.H. Chen, A systematic approach to heterogeneous multiattribute group decision making, Comput. Indus. Eng. 59 (2010) 561– 572. [21] K.H. Guo, W.L. Li, An attitudinal-based method for constructing intuitionistic fuzzy information in hybrid MADM under uncertainty, Inform. Sci. 208 (15) (2012) 28–38. [22] H. Zhao, Z.S. Xu, Hybrid fuzzy multiple attribute decision making, Information 12 (5) (2009) 1033–1044. [23] L. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338–356. [24] P. Luukka, Fuzzy similarity in multicriteria decision-making problem applied to supplier evaluation and selection in supply chain, Adv. Artif. Intell. 2011 (2011) 1–9. [25] C.Y. Shen, K.T. Yu, Enhancing the efficacy of supplier selection decision-making on the initial stage of new product development: a hybrid fuzzy approach considering the strategic and operational factors simultaneously, Expert Syst. Appl. 36 (8) (2009) 11271–11281. [26] H. Zhang, L. Yu, MADM method based on cross-entropy and extended TOPSIS with interval-valued intuitionistic fuzzy sets, Knowl.-Based Syst. 30 (2012) 115–120. [27] P. Li, H. Qian, J. Wu, J. Chen, Sensitivity analysis of TOPSIS method in water quality assessment: I. Sensitivity to the parameter weights, Environ. Monit. Assess. 185 (3) (2013) 2453–2461.

D.-F. Li, S.-P. Wan / Knowledge-Based Systems 67 (2014) 71–89 [28] G. Zhang, J. Lu, A linguistic intelligent user guide for method selection in multiobjective decision support systems, Inform. Sci. 179 (14) (2009) 2299–2308. [29] G. Zhang, J. Ma, J. Lu, Emergency management evaluation by a fuzzy multicriteria group decision support system, Stochast. Environ. Res. Risk Assess. 23 (4) (2009) 517–527. [30] C.L. Hwang, K. Yoon, Multiple Attributes Decision Making Methods and Applications, Springer, Berlin, Heidelberg, 1981. [31] V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference, Psychometrica 38 (1973) 337–342. [32] H.C. Xia, D.-F. Li, J.Y. Zhou, J.M. Wang, Fuzzy LINMAP method for multiattribute decision making under fuzzy environments, J. Comput. Syst. Sci. 72 (2006) 741–759. [33] Z.J. Wang, K.W. Li, An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives, Expert Syst. Appl. 39 (2012) 13509–13516.

89

[34] E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets Syst. 114 (3) (2000) 505–518. [35] C.Y. Hu, R.B. Kearfott, A.D. Korvin, V. Kreinovich, Knowledge Processing with Interval and Soft Computing, Springer-Verlag, London, 2008. [36] D.-F. Li, J.X. Nan, M.J. Zhang, Interval programming models for matrix games with interval payoffs, Optim. Methods Softw. 27 (1) (2012) 1–16. [37] H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, Eur. J. Oper. Res. 48 (1990) 219–225. [38] D. Dubois, H. Prade, The mean value of a fuzzy number, Fuzzy Sets Syst. 24 (1987) 279–300. [39] D.-F. Li, Decision and Game Theory in Management with Intuitionistic Fuzzy Sets, Springer-Verlag, Heidelberg, Germany, 2014.