Fuzzy linear programming approach to multiattribute decision making with multiple types of attribute values and incomplete weight information

Applied Soft Computing 13 (2013) 4333–4348

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Fuzzy linear programming approach to multiattribute decision making with multiple types of attribute values and incomplete weight information Deng-Feng Li a,∗ , Shu-Ping Wan b a b

School of 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 20 September 2012 Received in revised form 14 June 2013 Accepted 23 June 2013 Available online 3 July 2013 Keywords: Multiattribute decision making Fuzzy set Possibility linear programming Fuzzy multi-objective optimization Uncertainty Decision support

a b s t r a c t In the classical Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP), the decision maker (DM) gives the pair-wise comparisons of alternatives with crisp truth degree 0 or 1. However, in the real world, DM is not sure enough in all comparisons and can express his/her opinion with some fuzzy truth degree. Thus, DM’s preferences are given through pair-wise comparisons of alternatives with fuzzy truth degrees, which may be represented as trapezoidal fuzzy numbers (TrFNs). Considered such fuzzy truth degrees, the aim of this paper is to develop a new fuzzy linear programming technique for solving multiattribute decision making (MADM) problems with multiple types of attribute values and incomplete weight information. In this method, TrFNs, real numbers, and intervals are used to represent the multiple types of decision information. The fuzzy consistency and inconsistency indices are defined as TrFNs due to the alternatives’ comparisons with fuzzy truth degrees. Hereby a new fuzzy linear programming model is constructed and solved by the possibility linear programming method with TrFNs developed in this paper. The fuzzy ideal solution (IS) and the attribute weights are then obtained. The distances of alternatives from the fuzzy IS can be calculated to determine their ranking order. The implementation process of the method proposed in this paper is illustrated with a strategy partner selection example. The comparison analyzes show that the method proposed in this paper generalizes the classical LINMAP, fuzzy LINMAP and possibility LINMAP. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The classical Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) developed by Srinivasan and Shocker [1] is an effective and a simple method for solving multiattribute decision making (MADM) problems. The LINMAP is based on pair-wise comparisons of alternatives given by the decision maker (DM) and generates the best compromise alternative as the solution that has the shortest distance to the ideal solution (IS). In the LINMAP, all the decision data are known precisely or given as crisp values. However, under many conditions, crisp data are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information are vague or fuzzy in nature and as such it may not be appropriate to represent them by accurate numerical values. A more realistic approach could be to use fuzzy sets [2], intuitionistic fuzzy (IF) sets (IFSs) [3,4] and linguistic variables [5–7] to model human judgments. Therefore, extending the LINMAP to suit the fuzzy or IF environments is of a great importance for scientific researches and real applications [8–12]. Li and Yang [8] used linguistic variables to assess an alternative on qualitative attributes. These linguistic variables are transformed into positive triangular fuzzy numbers (TFNs) and hereby the LINMAP was developed for multiattribute group decision making (MAGDM) with linguistic variables. Xia et al. [9] also applied linguistic variables to capture fuzziness in decision information and processes by means of a fuzzy decision matrix. The linguistic variables were represented by trapezoidal fuzzy numbers (TrFNs). A fuzzy LINMAP was proposed for solving MADM problems under fuzzy environments. Li and Sun [10] transformed linguistic variables into TFNs and extended the LINMAP

∗ Corresponding author at: School of Management, Fuzhou University, No. 2 Xueyuan Road, Daxue New District, Fuzhou District, Fuzhou, Fujian 350108, China. Tel.: +86 0591 87892973; fax: +86 0591 87892973. E-mail addresses: [email protected], [email protected] (D.-F. Li). 1568-4946/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2013.06.019

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for solving MAGDM problems with linguistic variables and incomplete weight preference information. Using IFSs to express the attribute values, Li [11] and Li et al. [12] respectively extended the LINMAP for MADM and MAGDM under IF environment. The real-life decision making problems often involve multiple different attribute values. In the assessment process, alternatives have to be evaluated on different attributes, which may be qualitative and quantitative. Due to the DM’s knowledge area and the nature of evaluated attributes, the assessments provided by the DM may be measured in different formats such as real numbers, intervals, TrFNs, linguistic variables and IFSs. Consequently, MADM problems with multiple types of attribute values have drawn much attention from a wide spectrum of disciplines [13–16]. With ever increasing complexity in many real situations, there are often some challenges for the DM to provide precise and complete preference information due to time pressure, lack of knowledge (or data) and the DM’s limited expertise about the problem domain. In other words, usually weights are totally unknown or partially known a priori. Namely, weight preference information in MADM problems is usually incomplete. Recently, there are some methods for solving MADM and MAGDM problems with incomplete preference information [10,17–21]. However, the methods [10,17–21] are not applicable to the MADM problems with multiple types of attribute values and the methods [13–16] cannot deal with the MADM problems with incomplete weight information. In the classical LINMAP [1] and the fuzzy LINMAP [8–12], the DM gives the pair-wise comparisons of alternatives in the form of the ordered pairs with crisp truth degree 0 or 1. However, in the real world, DM is not sure enough in all comparisons and can express his/her opinion with a fuzzy degree of truth. Sadi-Nezhad and Akhtari [22] considered the fuzzy truth degree as a TFN and transformed the information of decision matrix into TFNs. They proposed the possibility LINMAP in group decision making. Wan and Li [23] firstly introduced IFSs to depict the fuzzy truth degrees of alternatives’ comparison and developed a fuzzy LINMAP for solving heterogeneous MADM problems. As Wan and Li [23] pointed out that there exist some big mistakes in the definitions, notations, operations, and possibilistic programming model in [22]. To overcome the above disadvantages, the aim of this paper is to extend the possibilistic LINMAP for solving MADM problems with multiple types of attribute values and incomplete weight information. The main works lie in two aspects. On the one hand, a TrFN permits two parameters to represent the most possible values while a TFN uses a single parameter to represent the most possible value. Namely, a TFN is a special case of a TrFN. Therefore, a TrFN is not only valuable for modeling imprecision but also easy to reflect the ambiguous nature of subjective judgments. TrFNs are used to capture fuzzy truth degree information about pair-wise comparisons of alternatives in this study. On the other hand, TrFNs, intervals and real numbers are used to represent the multiple types of attribute values. Considered the comparisons of alternatives with fuzzy truth degrees, the fuzzy consistency and inconsistency indices are defined as TrFNs. The fuzzy IS and attribute weights are then obtained through constructing a new fuzzy linear programming model which is solved by the possibility linear programming method with TrFNs developed in this paper. Compared with the existing LINMAP [1,8–12,22,23], the method proposed in this paper has the following differences and advantages:

(1) The classical LINMAP [1] and the fuzzy LINMAP [8–12] did not consider the DM’s preferences on the pair-wise comparisons of alternatives with fuzzy degrees of truth. In other words, they only considered the crisp truth degrees 0 or 1. In fact, due to the complexity of decision problems and fuzziness of human’s thinking, there exist some uncertainty and fuzziness when DM gives the pair-wise comparisons of alternatives. Consequently, it is very natural and reasonable to introduce fuzzy numbers to represent the information of fuzzy truth degrees. This is a great innovation and main motivation of our paper. (2) This paper utilizes TrFNs to represent the fuzzy truth degrees which can better reflect the ambiguous nature of subjective judgments on the pair-wise comparisons of alternatives given by DM, while [22] used TFNs to express fuzzy truth degrees. Furthermore, the MADM problems studied in this paper involve multiple types of attribute values, whereas that studied in [22] considered only single type of attribute values. (3) To obtain the fuzzy IS and vector of attribute weights, the linear programming model constructed in this paper is fuzzy. We technically develop a new method to solve this kind of fuzzy linear programming models with TrFNs. However, though the constructed linear programming model in [22] is fuzzy linear programming with TFNs, Sadi-Nezhad and Akhtari did not propose any new method to solve it. (4) Under some conditions, the proposed method in this paper can be reduced to the classical LINMAP [1], fuzzy LINMAP [8–12] and the possibility LINMAP [22]. Namely, the LINMAP [1,8–12,22] is a special case of this paper’s method (see Sections 5.2 and 5.3 in detail). (5) Wan and Li [23] considered the hesitance degree of alternatives’ comparison and represented the fuzzy truth degrees as IFSs, while this paper expresses the fuzzy truth degrees as TrFNs. The former constructed fuzzy linear programming with IFSs, whereas the latter constructed fuzzy linear programming with TrFNs. Both fuzzy linear programming models have completely different solving methods. Therefore, the decision principles and motivations for both papers are remarkably different.

The rest of the paper is structured as follows. In Section 2, the distance between TrFNs is defined and the interval objective programming models are introduced. In Section 3, the fuzzy MADM problems with multiple types of attribute values are described, and the normalization method is discussed as well as incomplete weight information structures. In Section 4, in order to solve such MADM problems, a new fuzzy linear programming model is constructed and solved by the developed new possibility linear programming method with TrFNs. The proposed method is illustrated with a real strategy partner selection example and comparison analyzes are conducted in Section 5. Conclusion is given in Section 6.

2. Distances for trapezoidal fuzzy numbers and interval objective programming In this section, some preliminaries about trapezoidal fuzzy numbers and interval objective programming are firstly introduced.

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2.1. Distances for trapezoidal fuzzy numbers ˜ is a special fuzzy subset on the set R of real numbers. Let m ˜ = (l, m1 , m2 , r) be a TrFN, whose membership function A fuzzy number m is given as follows:

m˜ (x) =

⎧ x−l ⎪ ⎪ ⎪ ⎨ m1 − l

(l ≤ x < m1 ) (m1 ≤ x ≤ m2 )

1

⎪ ⎪ ⎪ ⎩ r−x

r − m2

(1)

(m2 < x ≤ r)

˜ l and r are the lower and upper limits of m ˜ [24]. The closed interval [m1 , m2 ] is the mode of m. ˜ = (l, m1 , m2 , r) is reduced to a real number m if l = m1 = m2 = r. Conversely, a real number m can be written It is easily seen that a TrFN m ˜ = (m, m, m, m). A TrFN m ˜ = (l, m1 , m2 , r) is reduced to a TFN m ˜ = (l, m1 , r) if m1 = m2 . as a TrFN m ˜ = (l, m1 , m2 , r) is called a normalized positive TrFN if it is ˜ = (l, m1 , m2 , r) is called a positive TrFN if l ≥ 0 and r > 0. Furthermore, m m positive and l ≥ 0 and r ≤ 1. ˜ = (m1 , m2 , m3 , m4 ) and n˜ = (n1 , n2 , n3 , n4 ) be two TrFNs. Then the vertex method is defined to calculate the distance between Let m them as follows [9]:



˜ n) ˜ = d(m,

1 [(m1 6

− n1 )2 + 2(m2 − n2 )2 + 2(m3 − n3 )2 + (m4 − n4 )2 ].

(2)

Eq. (1) is an effective and a simple method to calculate the distance between two TrFNs. ˜ = (m1 , m2 , m3 , m4 ) and n˜ = (n1 , n2 , n3 , n4 ) are identical if and only if the distance meaMoreover, it is easily seen that two TrFNs m ˜ n) ˜ = 0. surement d(m, 2.2. Interval objective programming Ishibuchi and Tanaka [25] gave the definitions of the maximization and minimization problems with the interval objective functions, which are introduced in Definitions 1 and 2 as follows. Definition 1.

The maximization problem with the interval objective function is described as follows:

max {˜a}

(3)

s.t. a˜ ∈ ˝ which is equivalent to the following bi-objective mathematical programming problem: ¯ max {a- , 12 (a- + a)}

(4)

s.t. a˜ ∈ ˝ ¯ is an interval and ˝ is a set of constraints in which the variable a˜ should satisfy according to requirements in real situations. where a˜ = [a- , a] Definition 2.

The minimization problem with the interval objective function is described as follows:

min {˜a}

(5)

s.t. a˜ ∈ ˝ which is equivalent to the following bi-objective mathematical programming problem: ¯ 12 (a- + a)} ¯ min {a,

(6)

s.t. a˜ ∈ ˝ 3. Fuzzy MADM problems with multiple types of attribute values and incomplete weight information In this section, the fuzzy MADM problems with multiple types of attribute values are described and the normalization method is present. The incomplete weight information structures are also summarized in detail. 3.1. The description of fuzzy MADM problems with multiple types of attribute values A MADM problem is to choose one of or rank finite alternatives based on the assessment information of multiple different attributes. Let {a1 , a2 , . . ., an } be an alternative set and F = {f1 , f2 , . . ., fm } be an attribute set. Since there are multiple types of attribute values, we divide F into three subsets F1 = {f1 , f2 , . . ., fi1 }, F2 = {fi1 +1 , fi1 +2 , . . ., fi2 } and F3 = {fi2 +1 , fi2 +2 , . . ., fm }, where 1 ≤ i1 ≤ i2 ≤ m, Ft (t = 1, 2, 3) are attribute subsets in which attribute values are expressed with TrFNs, interval values and real numbers, respectively. They satisfy that Ft ∩ Fl = ∅ (t, l = 1, 2, 3 ; t = / l) and

3  t=1

Ft = F, where ∅ is an empty set. Denote M1 = {1, 2, . . ., i1 } , M2 = {i1 + 1, i1 + 2, . . ., i2 } , M3 = {i2 + 1, i2 + 2, . . ., m} ,

M = {1, 2, . . ., m} and N = {1, 2, . . ., n}, where Mt are the corresponding subscript sets of subsets Ft (t = 1, 2, 3). Let the rating of an alternative aj on attribute fi given by the DM be denoted by yij (i = 1, 2, . . ., m ; j = 1, 2, . . ., n). If i ∈ M1 , yij = (bij1 , bij2 , bij3 , bij4 ) is a TrFN; if i ∈ M2 , yij = [eij , gij ] is an interval; if i ∈ M3 , yij = zij is a real number, where 0 ≤ bij1 ≤ bij2 ≤ bij3 ≤ bij4 , 0 ≤

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eij ≤ gij and zij ≥0. Hence, we can elicit a fuzzy decision matrix, denoted by Y = (yij )m×n , which is used to concisely represent the fuzzy MADM problem considered in this paper. 3.2. The normalization method Due to different dimensions and measurements of attributes, the attribute values should be normalized to eliminate effect of physical dimensions and measurements on the final decision. For simplicity, we partition the attribute subsets Ft into two mutually exclusive and complete subsets: Ftb and Ftc , where Ftb and Ftc are respectively the sets of benefit attributes and cost attributes, Ft = Ftb ∪ Ftc and Ftb ∩ Ftc = ∅ (t = 1, 2, 3). In the sequent, we give the following normalization method. For ratings yij = (bij1 , bij2 , bij3 , bij4 ) (fi ∈ F1 ) according to benefit attributes and cost attributes, the normalized values rij = (bij1 , bij2 , bij3 , bij4 ) can be computed as follows:

rij = (bij1 , bij2 , bij3 , bij4 ) =

⎧  bij1 bij2 bij3 bij4 ⎪ ⎪ if fi ∈ F1b , , , ⎪ ⎨ 

bi4 max bi4 max bi4 max bi4 max

bij4 bij3 bij2 bij1 ⎪ ⎪ ⎪ ,1 − ,1 − ,1 − ⎩ 1− b b b b i4 max

where bi4 max =

max{bij4 |j

i4 max

i4 max

i4 max



(7) if fi ∈ F1c

= 1, 2, · · ·, n}.

Similarly, for ratings yij = [eij , gij ] (fi ∈ F2 ), the normalized values rij = [eij , gij ] can be computed as follows:



rij = [eij , gij ] =

[eij /gi max , gij /gi max ] if fi ∈ F2b [1 − gij /gi max , 1 − eij /gi max ] if

(8)

fi ∈ F2c

where gi max = max{gij |j = 1, 2, . . ., n}.

For ratings yij = zij (fi ∈ F3 ), the normalized values rij = zij can be computed as follows:



rij = zij =

zij /zi max

if fi ∈ F3b

1 − zij /zi max

if fi ∈ F3c

(9)

where zi max = max{zij |j = 1, 2, . . ., n}. By Eqs. (7)–(9), the fuzzy decision matrix Y = (yij )m×n can be normalized into matrix R = (rij )m×n . Thus, the alternative aj may be fully expressed with the elements of the jth column in matrix R. Denote r j = (r1j , r2j , . . ., rmj ). Then, rj and aj have the same meaning and may be interchangeably used. 3.3. Incomplete weight information structures In decision making process, the importance of different m attributes should be taken into account. Suppose ωi is the relative weight of attribute fi , satisfying the normalization condition, i.e., ω = 1 and ωi ≥ 0 (i = 1, 2, . . ., m). Denote a weight vector by ω = (ω1 , ω2 , . . ., i=1 i

m

ωm )T . Let 0 be a set of all weight vectors with ωi ≥ ε for all i (i = 1, 2, . . ., m), i.e., 0 = {ω| i=1 ωi = 1, ωi ≥ε for i = 1, 2, . . ., m}, where ε > 0 is a sufficiently small positive number which ensures that the weights generated are not zeros as it may be the case in the LINMAP [1]. Thus each weight of 0 is not smaller than a given sufficiently small positive number ε. In some real decision situations, the DM may specify some preference relations on weights of attributes according to his/her knowledge, experience and judgment. Such information of attribute weights is incomplete. Usually incomplete information of attribute weights can be obtained according to partial preference relations on weights given by the DM and has several different structure forms. Summarized earlier research [17–20], these weight information structures may be expressed in the following five basic relations among attribute weights, which are denoted by subsets s (s = 1, 2, 3, 4, 5) of weight vectors in 0 , respectively [21]. (1) A weak ranking: 1 = {ω ∈ 0 |ωt ≥ ωj for all t ∈ T1 and j ∈ J1 } , where T1 and J1 are two disjoint subsets of the subscript index set M = {1, 2, . . ., m} of all attributes. Thus, 1 is a set of all weight vectors in 0 with the property that the weight of an attribute in the set T1 is greater than or equal to that of an attribute in the set J1 . (2) A strict ranking: 2 = {ω ∈ 0 |ˇtj ≥ ωt − ωj ≥ ˛tj for all t ∈ T2 and j ∈ J2 } , where ˛tj > 0 and ˇtj > 0 are constants, satisfying ˇtj > ˛tj ; T2 and J2 are two disjoint subsets of M. Thus, 2 is a set of all weight vectors in 0 with the property that the weight of an attribute in the set T2 is greater than or equal to that of an attribute in the set J2 but their difference does not exceed some range, i.e., a closed interval [˛tj , ˇtj ]. (3) A ranking with multiples: 3 = {ω ∈ 0 |ωt ≥  tj ωj for all t ∈ T3 and j ∈ J3 } , where  tj > 0 is a constant; T3 and J3 are two disjoint subsets of M. Thus, 3 is a set of all weight vectors in 0 with the property that the weight of an attribute in the set T3 is greater than or equal to  ij times of that of an attribute in the set J3 . (4) An interval form: 4 = {ω ∈ 0 | j ≥ ωj ≥ j for all j ∈ J4 } , where  j > 0 and j > 0 are constants, satisfying  j > j ; J4 is a subset of M. Thus, 4 is a set of all weight vectors in 0 with the property that the weight of an attribute in the set J4 does not exceed some range, i.e., a closed interval [ j ,  j ]. (5) A ranking of differences: 5 = {ω ∈ 0 |ωt − ωj ≥ ωk − ωl for all t ∈ T5 , j ∈ J5 , k ∈ K5 and l ∈ L5 } , where T5 , J5 , K5 and L5 are four disjoint subsets of M. Thus, 5 is a set of all weight vectors in 0 with the property that the difference between weights of attributes in the sets T5 and J5 , is greater than or equal to that of attributes in the sets K5 and L5 . In reality, usually the preference information structure  of attribute importance may consist of several sets of the above basic sets s (s = 1, 2, 3, 4, 5). For example, the DM may provide a preference information structure expressed as follows [21]:

 = {ω ∈ 0 |0.15 ≤ ω1 ≤ 0.55, 0.2 ≤ ω2 ≤ 0.65, 0.1 ≤ ω3 ≤ 0.35, ω2 ≥1.2ω1 , 0.02 ≤ ω2 − ω3 ≤ 0.45},

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which may be decomposed into the following three basic subsets: 2 = {ω ∈ 0 |0.02 ≤ ω2 − ω3 ≤ 0.45}, 3 = {ω ∈ 0 |ω2 ≥1.2ω1 } and



4 = {ω ∈ 0 |0.15 ≤ ω1 ≤ 0.55, 0.2 ≤ ω2 ≤ 0.65, 0.1 ≤ ω3 ≤ 0.35}, where 0 = {ω = (ω1 , ω2 , ω3 )T |ω1 + ω2 + ω3 = 1, ωi ≥ ε (j = 1, 2, 3)} . In





other words, the information structure  consists of the above three sets 2 , 3 , and 4 . 4. The possibility LINMAP with fuzzy truth degrees on comparisons of alternatives To solve the above fuzzy MADM problems with multiple types of attribute values and incomplete weight information, we develop a possibility LINMAP model with fuzzy truth degrees about the comparisons of alternatives in this section. 4.1. Fuzzy consistency and inconsistency measurements ∗ ) is unknown a priori and needs to be determined, where r ∗ is the best rating on attribute Suppose that the fuzzy IS r ∗ = (r1∗ , r2∗ , . . ., rm i ∗ fi (i = 1, 2, . . ., m). Namely, if i ∈ M1 , ri , is a TrFN, i.e., ri∗ = (b∗i1 , b∗i2 , b∗i3 , b∗i4 ); if i ∈ M2 , ri∗ is an interval, i.e., ri∗ = [ei∗ , gi∗ ]; if i ∈ M3 , ri∗ is a real number, i.e., ri∗ = zi∗ . Where, 0 ≤ b∗i1 ≤ b∗i2 ≤ b∗i3 ≤ b∗i4 ≤ 1 (i ∈ M1 ), 0 ≤ ei∗ ≤ gi∗ ≤ 1(i ∈ M2 ) and 0 ≤ h∗i ≤ 1 (i ∈ M3 ). According to Eq. (2), the square of the weighted Euclidean distance between the alternative r j = (r1j , r2j , . . ., rmj ) and the fuzzy IS ∗ ) can be calculated as follows: r ∗ = (r1∗ , r2∗ , . . ., rm

Sj =

i1 ω

i

6

2 [(bij1 − b∗i1 )

2 + 2(bij2 − b∗i2 )

2 + 2(bij3 − b∗i3 )

2 + (bij4 − b∗i4 ) ] +

i2 ωi i=i1 +1

i=1

2

[(eij − ei∗ )2 + (gij − gi∗ )2 ] +

m

ωi (zij − zi∗ )2 ]

(10)

i=i2 +1

Due to the influence of various subjective and objective factors, the DM makes judgment on the priority of two alternatives with fuzzy truth degree. Integrating all the judgments, we can formulate the preference relations between alternatives as a fuzzy set of ordered pairs ˜ = {C(k, ˜ ˜ ˜ j)/(k, j)|ak aj with the truth degree C(k, j) ( k, j = 1, 2, . . ., n)}, where C(k, j)/(k, j) expresses an ordered pair of alternatives ak and ˝ ˜ aj that the DM prefers ak to aj (denoted by ak aj ) with the truth degree C(k, j), which is a TrFN defined on the unit interval [0,1], denoted by ˜ 0 = support ˝ ˜ = {(k, j)|C(k, ˜ ˜ C(k, j) = (Ckjl , Ckjm , Ckjm , Ckjr ), satisfying 0 ≤ Ckjl ≤ Ckjm ≤ Ckjm ≤ Ckjr ≤ 1. Let ˝ j) > 0 ( k, j = 1, 2, . . ., n)}. 1

2

1

2

It should be noted that the DM gives the comparison between two alternatives on a whole rather than on each attribute fi (i = 1, 2, . . ., ˜ 0 | of ˝ ˜ 0 , i.e., number of alternative pairs in ˝ ˜ 0 , is at most m). Since the alternative set contains n efficient alternatives, the cardinality |˝ 2 ˜ ˜ Cn = n(n − 1)/2. Usually the relations given by ˝0 are partial order. Even, there exist some intransitivity in ˝0 . In some situations, the DM ˜ 0| < C2. would not be able to specify all the relations, i.e., only give some pair-wise comparisons between alternatives, |˝ n ∗ ) are chosen by the DM already, then using Eq. (10) the DM If the weight vector ω = (ω1 , ω2 , . . ., ωm )T and the fuzzy IS r ∗ = (r1∗ , r2∗ , . . ., rm ˜ 0 and the fuzzy IS r ∗ = (r ∗ , r ∗ , . . ., r ∗ ) can calculate the square of the weighted Euclidean distance between each pair of alternative (k, j) ∈ ˝ 1

2

m

as Sk and Sj . The Sk and Sj are used to rank the alternatives ak and aj , which can be viewed as a kind of objective ranking order, while the ˜ 0 given by DM is a subjective ranking order. Generally, there exists deviation between the subjective and objective ordered pair (k, j) ∈ ˝ ranking orders. To measure such a deviation, we introduce the concepts of fuzzy consistency and inconsistency. For each pair of alternatives, if Sj ≥ Sk , the alternative ak is closer to the fuzzy IS than the alternative aj , and thus ak aj , which is consistent ˜ 0 (means ak aj ) given by the DM. Conversely, if Sj < Sk , then the obtained objective ranking order with the subjective preference (k, j) ∈ ˝ ˜ 0 . Therefore, an index (Sj − Sk )− is defined to measure inconsistency between aj ak is inconsistent with the subjective preference (k, j) ∈ ˝ the subjective and objective ranking orders as follows:



−

(Sj − Sk ) =

˜ j)(Sk − Sj ) C(k,

(Sj < Sk )

0

(Sj ≥Sk )

(11)

Obviously, the ranking order of alternatives ak and aj determined by Sj and Sk based on (ω, r*) is consistent with the preferences given by the DM if Sj ≥ Sk . Hence, (Sj − Sk )− is defined to be 0. On the other hand, the ranking order of alternatives ak and aj determined by Sj ˜ and Sk based on (ω, r*) is inconsistent with the preferences given by the DM if Sj < Sk . Hence, (Sj − Sk )− is defined to be C(k, j)(Sk − Sj ). The − ˜ inconsistency index can be rewritten as (Sj − Sk ) = C(k, j) max{0, Sk − Sj }. Then, a total fuzzy inconsistency index of the DM is defined as follows: B˜ =



(Sj − Sk )− =

˜ (k,j) ∈ ˝ 0



˜ [C(k, j) max{0, Sk − Sj }].

(12)

˜ (k,j) ∈ ˝ 0

In a similar way, an index (Sj − Sk )+ to measure consistency between the subjective and objective ranking orders can be defined as follows:



+

(Sj − Sk ) =

˜ C(k, j)(Sj − Sk )

(Sj ≥Sk )

0

(Sj < Sk )

(13)

˜ j) max{0, Sj − Sk }. Hence, a total fuzzy consistency index of the DM is defined as Obviously, Eq. (13) can be rewritten as (Sj − Sk )+ = C(k, ˜ = G



˜ (k,j) ∈ ˝ 0

(Sj − Sk )+ =



˜ [C(k, j) max{0, Sj − Sk }]

(14)

˜ (k,j) ∈ ˝ 0

˜ are all TrFNs since the DM’s Remark 1. It is noted that the total fuzzy inconsistency index B˜ and the total fuzzy consistency index G ˜ preferences are given through pair-wise comparisons of alternatives with fuzzy truth degrees, which are represented as TrFNs C(k, j). Whereas, the total inconsistency and consistency indices defined in the classical LINMAP [1] and the fuzzy LINMAP [8–12] are real numbers

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since the DM’s preferences in [1,8–12] are given through pair-wise comparisons of alternatives with crisp truth degrees 0 or 1. This is the most difference among [1], [8–12], and this paper. Additionally, Sadi-Nezhad and Akhtari [22] did not explicitly give the inconsistency and consistency indices. 4.2. Fuzzy linear programming models In the classical LINMAP [1], the linear programming model was constructed through minimizing the total inconsistency index under the condition in which the total inconsistency index is smaller than or equals to the total consistency index by a positive constant h given by DM in advance. To suit the fuzzy MADM problems considered in this paper, we construct the fuzzy mathematical programming model as follows: ¯ min{B}

˜ − B≥ ˜ h˜ G s.t. ω∈

(15)

where h˜ is a positive TrFN given by the DM a priori, denoted by h˜ = (h1 , h2 , h3 , h4 ).  is the preference information structure of attribute importance given in Section 3.2. Sadi-Nezhad and Akhtari [22] defined the fuzzy distance between an alternative Ai and the FIS a˜ ∗ = (˜a∗1 , a˜ ∗2 , · · ·, a˜ ∗n ) as Si =

Remark 2.

n

[ωj (˜xij − a˜ ∗j )2 ], i.e., Eq. (8) in [22], where ωj = (ωjL , ωjM , ωjR ) is the weight of the jth attribute, x˜ ij = (aijL , aijM , aijR ) is the fuzzy score of an alternative Ai on the jth attribute, a˜ ∗j = (a∗jL , a∗jM , a∗jR ) is the ideal value of the jth attribute. Obviously, according to the operations of TFNs [26], Si is a TFN since all ωj , x˜ ij and a˜ ∗j (j = 1, 2, · · ·, n) are TFNs. Thus, Sl − Sk is a TFN. However, h in the constraints of Eqs. (9) and (11) in [22] j=1

is a positive constant (i.e., a real number) instead of a positive TFN. Such a hypothesis results in the right-hand side of the corresponding P  equality in the constraint of Eq. (9) must be a positive real number and equal to h, i.e., ˜ (Sl − Sk ) = h, which is contradictory (k,l) ∈ ˝ p=1 p

to the fact that Sl − Sk is a TFN. Thus, all h appeared in [22] should be revised as TFNs, and Eqs. (9) and (11) in [22] are wrong. ˜ − B˜ = It yields from Eqs. (11)–(14) that G as follows: min

⎧ ⎨



˜ (k,j) ∈ ˝ 0

[(Sj − Sk )+ − (Sj − Sk )− ] =



˜ (k,j) ∈ ˝ 0

˜ [C(k, j)(Sj − Sk )]. Then, Eq. (15) can be rewritten

⎫ ⎬ ˜ j) max 0, Sk − Sj C(k, ⎭ 

⎩ ˜ ∈˝ 0 ⎧ (k,j) ˜ ⎨ [C(k, j)(Sj − Sk )]≥h˜

s.t.

(16)

˜

⎩ (k,j) ∈ ˝0 ω∈





˜ 0 , let kj = max 0, Sk − Sj , then for each (k, j) ∈ ˝ ˜ 0 , kj ≥0 and x = (x1 , x2 , . . ., xq )T . Thus, Eq. (16) can be For each pair of (k, j) ∈ ˝ transformed into the following fuzzy linear programming model:



min{

˜ [kj C(k, j)]}

⎧ (k,j) ∈ ˝˜ 0 ˜ ⎪ [(Sj − Sk )C(k, j)]≥h˜ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (k,j) ∈ ˝˜ 0 s.t.

˜ 0) Sj − Sk + kj ≥0 ((k, j) ∈ ˝

(17)

⎪ ⎪ ˜ 0) ⎪ kj ≥0 ((k, j) ∈ ˝ ⎪ ⎪ ⎪ ⎩ ω∈

˜ 0 , let uikj = (b2 − b2 ) + 2(b2 − b2 ) + 2(b2 − b2 ) + (b2 − b2 ), xj ≥ 0(j = 1, 2, . . ., q) and ikj = 2(bij4 − bik4 ). For i ∈ M1 and (k, j) ∈ ˝ ij1 ik1 ij2 ik2 ij3 ik3 ij4 ik4 ˜ 0 , let ikj = (e2 − e2 ) + (g 2 − g 2 ) ıikj = 2(eij − eik ) and  ikj = 2(gij − gik ). For i ∈ M3 and (k, j) ∈ ˝ ˜ 0 , let Analogously, for i ∈ M2 and (k, j) ∈ ˝ ij ik ij ik 2 ),  ikj = (zij2 − zik ikj = 2(zij − zik ). Thus, using Eq. (10), Eq. (17) can be written as the following fuzzy linear programming model:

min{

s.t.



˜ [kj C(k, j)]}

˜ ∈˝ 0  ⎧(k,j)   1 T 1 T T T T T T T T T ⎪ ˜ ¯ ¯ C(k, ω ω − o v −  v −  v −  v ) + − ı v −  v ) +  ω ¯ − v ] j) ≥h˜ (u ( ⎪ 5 7 1 1 2 3 4 2 6 3 kj kj kj kj kj kj kj kj kj kj ⎪ 6 2 ⎪ ⎪ ˜ ⎪ (k,j) ∈ ˝ 0 ⎪ ⎪ T T T ⎪ 1 Tω Tω ˜ 0) ⎪ ¯ ¯ 2 − ıkj v5 −  Tkj v6 ) + kj ω (u − oTkj v1 − Tkj v2 − kj v3 − Tkj v4 ) + 12 ( kj ¯ 3 − Tkj v7 + kj ≥0 ((k, j) ∈ ˝ ⎪ 6 kj 1 ⎪ ⎪ ⎪ ⎪ ⎨ kj ≥0((k, j) ∈ ˝ ˜ 0)

⎪ 0 ≤ vi1 ≤ vi2 ≤ vi3 ≤ vi4 ≤ ωi (i = 1, 2, · · ·, i1 ) ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ vi5 ≤ vi6 ≤ ωi (i = i1 + 1, i1 + 2, · · ·, i2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ vi7 ≤ ωi (i = i2 + 1, i2 + 2, · · ·, m) ⎪ ⎪ ⎪ ⎩ ω∈

(18)

D.-F. Li, S.-P. Wan / Applied Soft Computing 13 (2013) 4333–4348

4339

where

⎧ v = ωi b∗i1 , vi2 = ωi b∗i2 , vi3 = ωi b∗i3 , vi4 = ωi b∗i4 (i ∈ M1 ) ⎪ ⎨ i1 ⎪ ⎩

vi5 = ωi ei∗ , vi6 = ωi gi∗ (i ∈ M2 ) vi7 =

(19)

ωi zi∗ (i ∈ M3 ) T

¯ 1 = (ω1 , ω2 , . . ., ωi1 )T , ω ¯ 2 = (ω1i +1 , ωii +2 , . . ., ωi2 )T , ω ¯ 3 = (ωi2 +1 , ωi2 +2 , . . ., ωm )T , ω = (ω ω ¯ 1T , ω ¯ 2T , ω ¯ 3T ) , T

and

T

T

T

v1 = (v11 , v21 , . . .vi1 1 ) , v2 = (v12 , v22 , . . .vi1 2 ) , v3 = (v13 , v23 , . . .vi1 3 ) , v4 = (v14 , v24 , . . .vi1 4 ) , v5 = (vii +1.5 , vi1 +2,5 , . . ., vi2 5 )T , v6 = (vii +1.6 , vi1 +2,6 , . . ., vi2 6 )T , v7 = (vi2 +1.7 , vi2 +2,7 , . . ., vm7 )T , ukj = (u1kg , u2kj , . . ., ui1 kj )T , Okj = (O1kj , O2kj , . . ., Oi1 kj )T , kj = (1kj , 2kj , . . ., i1 kj )T , ϕkj = (ϕ1kj , ϕ2kj , . . ., ϕi1 kj ), kj = ( 1kj , 2kj , . . ., i1 kj )T , T

T

 kj = ( i1 +1,kj , i1 +2,kj , . . ., i2 kj ) , ıkj = (ıi1 +1,kj , ıi1 +2,kj , . . ., ıi2 kj ) ,  kj = (i1 +1,kj , i1 +2,kj , . . ., i2 kj )T , T kj = (i2 +1,kj , i2 +2,kj , . . ., mkj ) , kj = (i2 +1,kj , i2 +2,kj , . . ., mkj )T .

Remark 3. Due to 0 ≤ b∗i1 ≤ b∗i2 ≤ b∗i3 ≤ b∗i4 ≤ 1(i ∈ M1 ), 0 ≤ ei∗ ≤ gi∗ ≤ 1 (i ∈ M2 ), 0 ≤ zi∗ ≤ 1 (i ∈ M3 ) and 0 ≤ ωi ≤ 0 (i = 1, 2, . . ., m), according to Eq. (19), we can obtain the inequalities as follows: 0 ≤ vi1 ≤ vi2 ≤ vi3 ≤ vi4 ≤ ωi (i ∈ M1 ), 0 ≤ vi5 ≤ vi6 ≤ ωi (i ∈ M2 ) and 0 ≤ vi7 ≤ ωi (i ∈ M3 ). which are added into the constraints of Eq. (18). ˜ Remark 4. In Eq. (18), the objective function, some constraints coefficients and right-hand vector contain TrFNs such as C(k, j) and h˜ simultaneously. Thus, Eq. (18) is a fuzzy linear programming model with TrFNs, whereas Eq. (9) in [8], Eq. (16) in [10] and Eq. (11) in [11] are crisp linear programming models due to the fact that fuzziness of the pair-wise comparisons of alternatives was not taken into consideration. ˜ 0 | + 2m + 2i1 + i2 variables need to be determined, i.e., m weights ωi (i = 1, 2, . . ., m), |˝ ˜ 0 | variables Obviously, in Eq. (18) there exist |˝ ˜ 0 , 4i1 variables of (vi1 , vi2 , vi3 , vi4 ) (i = 1, 2, . . ., i1 ), 2(i2 − i1 ) variables of (vi5 , vi6 ) (i = i1 + 1, i1 + 2, . . ., i2 ) and (m − i2 ) varikj ≥0 (k, j) ∈ ˝ ˜ 0 | + 11 inequalities. To determine these variables, the number 2|˝ ˜ 0 | + 11 of the ables of vi7 (i = i2 + 1, i2 + 2, . . ., m), and at least 2|˝ ˜ 0 | (i.e., pair-wise comparison between alternatives) the more precise and inequalities should not be much small. In general, the larger |˝ reliable determining the weight vector and the fuzzy IS. ˜ 0 | + 2m + 2i1 + i2 ,  = (kj ) ˜ ˜ Denote q = |˝ and C˜ = (C(k, j))|˝ ˜ |˝ |×1

0 |×1

0

˜ , where kj and C(k, j) are listed by using the lexicographic method T

T ˜ 0 . Let x = (ω ¯ T1 , ω ¯ T2 , ω ¯ T3 , v1 T , v2 T , v3 T , v4 T , v5 T , v6 T , v7 T ,  ) be a q × 1 vector of unknown variables and according to the subscripts (k, j) ∈ ˝ T T c˜ = (0, 0, . . ., 0, C˜ ) be a q × 1 vector of TrFNs. Analogously, denote



E˜ =

1 6



˜ j)uTkj , C(k,

1 2

˜ (k,j) ∈ ˝



1 6

T T ˜ j) kj , kj , C(k,

˜ (k,j) ∈ ˝



˜ j)oTkj , C(k,



1 6

˜ (k,j) ∈ ˝

˜ j)Tkj , − C(k,



1 6

˜ (k,j) ∈ ˝

1 2

˜ j)Tkj , − C(k,

˜ (k,j) ∈ ˝



T ˜ j)ıkj , − C(k,

1 2

˜ (k,j) ∈ ˝



˜ j) Tkj , − C(k,

˜ (k,j) ∈ ˝

˜ (k,j) ∈ ˝

and

⎛ 1 1 T T − uTkj −  kj −kj 2 ⎜ 6 ⎜ 1 ⎜ − uT − 1  T −T kj ⎜ 6 kj 2 kj ⎜ ⎜ . .. .. ⎜ . ⎜ . . . ⎜ ⎜ − 1 uT − 1  T −T ⎜ 6 kj kj 2 kj ⎜ ⎜ T T T ⎜ −I 0 0 ⎜ ⎜ T T ⎜ −I 0 0T ⎜ ⎜ 0T 0T ⎜ −I T D=⎜ ⎜ −I T 0T 0T ⎜ ⎜ ⎜ 0T −I T 0T ⎜ ⎜ ⎜ 0T −I T 0T ⎜ ⎜ T ⎜ 0 0T −I T ⎜ ⎜ T ⎜ 0 0T 0T ⎜ ⎜ T ⎜ 0 0T 0T ⎜ ⎜ T 0T 0T ⎝ 0 0

T

where 0 = (0, 0,

0 . . . 0)T

T

0

T

1 T o 6 kj

1 T  6 kj

1 T  6 kj

1 T  6 kj

1 T ı 2 kj

1 T  2 kj

Tkj

−1

0

···

1 T o 6 kj

1 T  6 kj

1 T  6 kj

1 T  6 kj

1 T ı 2 kj

1 T  2 kj

Tkj

0

−1

···

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

1 T o 6 kj

1 T  6 kj

1 T  6 kj

1 T  6 kj

1 T ı 2 kj

1 T  2 kj

Tkj

0

0

···

IT

0T

0T

0T

0T

0T

0T

0

0

···

T

T

0

T

T

T

T

T

0

0

···

0T

0T

IT

0T

0T

0T

0T

0

0

···

0

T

0

T

0

T

T

T

0

T

0

0

···

0

T

0

T

0

T

0

T

0

0

···

0

I

0

I

0

T

0

0

T

T

0

T

0 I

0

0

0T

0T

0T

0T

0T

IT

0T

0

0

···

0

0T

0T

0T

0T

0T

IT

0

0

···

T

0

T

0

T

0

T

0

T

0

0

···

0

T

0

T

0

T

0

T

0

0

···

0T

0

0

···

T

0

0

···

T

−I

T

T

−I

0T

0T

IT

T

T

T

I

0

0

and I = (1, 1, . . .,

I

0 1)T

T

0

0

T

−I T 0

T

0T I

T

0T −I

T

0

0



⎞

⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎟ ⎟ −1 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎠ 0

are the column vectors of constants 0 and 1, respectively.

p×q



˜ j) Tkj , 0T , C(k, 1×q

4340

D.-F. Li, S.-P. Wan / Applied Soft Computing 13 (2013) 4333–4348

T

Fig. 1. The strategy to solve min{˜z = c˜ x}.

Then, Eq. (18) can be transformed into the possibility linear programming model with TrFNs in matrix form as follows: min{˜z = c˜ T x}

⎧ E x ≤ h˜ ⎨

s.t.

⎩

(20)

Dx ≤ l x≥0

As  is the linear preference structure of ω, the constraint ω ∈  can also be easily incorporated into the constraints Dx ≤ l, where l is the vector of constants. ˜ simultaneously. ˜ j) and h) In Eq. (20), the objective function, some constraints coefficients and right-hand vector contain TrFNs (e.g. C(k, In what follows, we technically develop a new possibility linear programming method with TrFNs for solving Eq. (20). 4.3. A possibility linear programming method with trapezoidal fuzzy numbers Without loss of generality, Eq. (20) may be generalized as the following possibility linear programming model with TrFNs: min{˜z = c˜ T x}

⎧ Ax ≤ b˜ ⎨

s.t.

⎩

(21)

Dx ≤ l x≥0

T T ˜ = (˜aij ) , D = (dij ) A , b˜ = (b˜ 1 , b˜ 2 , . . ., b˜ s ) , l = (l1 , l2 , . . ., lp ) , c˜ = (˜c1 , c˜2 , . . ., c˜q )T and x = (x1 , x2 , . . ., xq )T . a˜ ij = s×q p×q (aijl , aijm1 , aijm2 , aijr ), c˜j = (cjl , cjm1 , cjm1 , cjr ) and b˜ i = (bil , bim1 , bim2 , bir )(i = 1, 2, . . ., s; j = 1, 2, . . ., q) are known TrFNs. dij and li (i = 1, 2, · · · , p ; j = 1, 2, · · · , q) are known real numbers. xj ≥ 0(j = 1, 2, · · · , q) are crisp and unknown decision variables, which need to be solved.

where

simplicity,

For

T

denote

Al = (aijl )m×n , Am1 = (aijm1 )m×n , Am2 = (aijm2 )m×n , Ar = (aijr )m×n , c l = (c1l , c2l , . . ., cql )T , c m1 = T

T

(c1m1 , c2m1 , . . ., cqm1 ) , c m2 = (c1m2 , c2m2 , . . ., cqm2 )T , c r = (c1r , c2r , . . ., cqr )T , bl = (b1l , b2l , . . ., bsl ) , bm1 = (b1m1 , b2m1 , . . ., bsm1 ) , bm2 = T

(b1m2 , b2m2 , . . ., bsm2 ) and br = (b1r , b2r , . . ., bsr )T .

In Eq. (21), the fuzzy objective is fully defined by four corner points ((c l )T x, 0), ((c m1 )T x, 1), ((c m2 )T x, 1) and (( c r )T x, 0), geometrically. Thus, minimization the fuzzy objective can be obtained by pushing these four critical points in the direction of the left-hand side. Fortunately, the vertical coordinates of the critical points are fixed at either 1 or 0. The only considerations are the four horizontal coordinates. Therefore, our problem is to solve the mathematical programming model as follows:



min z˜ = (c l )T x, (c m1 )T x, (c m2 )T x, (cr )T x

!

⎧ Ax ≤ b˜ ⎨

s.t.

⎩

(22)

Dx ≤ l x≥0

where z˜ = ((c l )T x, (c m1 )T x, (c m2 )T x, (c r )T x) is a TrFN and may be viewed as the vector of four objective functions (c l )T x, (c m1 )T x, (c m2 )T x and ( c r )T x. In order to keep the TrFN shape (normal and convex) of the possibility distribution, it is necessary to make a little change. Instead of minimizing these four objectives simultaneously, we are going to minimize (c m2 )T x, minimize 12 [(c m1 )T x + (c m2 )T x], minimize [(c r )T x −

(c m2 )T x] and maximize [(c m1 )T x − (c l )T x],, where the last two objective functions are actually relative measures from (c m2 )T x and (c m1 )T x, respectively. The first two objective functions aim at the closed interval [(c m1 )T x, (c m2 )T x], which is the mode of a TrFN z˜ . Since the objective function of Eq. (21) is to minimize c˜ T x, it is natural to minimize the interval [(c m1 )T x, (c m2 )T x] for this objective function.

According to Definition 2, in order to minimize the interval [(c m1 )T x, (c m2 )T x], we need minimize the right endpoint (c m2 )T x and minimize

the middle point

1 [(c m1 )T x 2

+ (c m2 )T x] of this interval simultaneously, depicted as in Fig. 1.

D.-F. Li, S.-P. Wan / Applied Soft Computing 13 (2013) 4333–4348

4341

It can be seen from Fig. 1 that Eq. (22) is converted into the following multi-objective programming model: max{z1 = (c m1 )T x − (c l )T x} min{z2 = (c m2 )T x} min{z3 =

1 [(c m1 )T x 2

+ (c m2 )T x)]}

min{z4 = (c r )T x − (c m2 )T x}

⎧  ⎪ Ax ≤ b˜ ⎪ ⎨

s.t.

(23)

Dx ≤ l ⎪ ⎪ ⎩ x≥0

Remark 5.

Let

x* be the optimal solution of Eq. (23). Since the objective of Eq. (21) z˜ = ((c l )T x, (c m1 )T x, (c m2 )T x, (c r )T x) is a

TrFN, we have (c l )T x ≤ (c m1 )T x ≤ (c m2 )T x ≤ (c r )T x. The first objective of Eq. (23) shows that (c m1 )T x∗ − (c l )T x∗ ≥(c m1 )T x − (c l )T x≥0.

Thus, we obtain (c l )T x∗ ≤ (c m1 )T x∗ . As aforementioned, the second and third objectives of Eq. (23) are equivalent to minimizing the

interval [(c m1 )T x, (c m2 )T x], which yields that (c m1 )T x∗ ≤ (c m2 )T x∗ . Combining (c m2 )T x ≤ (c r )T x with the fourth objective of Eq. (23), we get z 4 = (c r )T x − (c m2 )T x≥0 and thus (c m2 )T x∗ ≤ (c r )T x∗ . Consequently, we have (c l )T x∗ ≤ (c m1 )T x∗ ≤ (c m2 )T x∗ ≤ (c r )T x∗ , i.e., z˜ ∗ = ((c l )T x∗ , (c m1 )T x∗ , (c m2 )T x∗ , (c r )T x∗ ) is also a TrFN, which assures the convex and normal output through solving Eq. (23).

Due to that the constraints of Eq. (23) still contain TrFNs, we combine the fuzzy ranking concepts with the strategy for the fuzzy objective function to deal with the constraints. Thus, the auxiliary multi-objective mathematical programming model is obtained as follows: max{z1 = (c m1 )T x − (c l )T x} max{z2 = −(c m2 )T x} max{z3 = − 12 [(c m1 )T x + (c m2 )T x)]} max{z4 = (c m2 )T x − (c r )T x}

⎧A x ≤ b lˇ lˇ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Am1 ˇ x ≤ bm1 ˇ ⎪ ⎪ ⎨

s.t.

(24)

Am2 ˇ x ≤ bm2 ˇ

⎪ ⎪ ⎪ Arˇ x ≤ brˇ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Dx ≤ l x≥0

where ˇ ∈ [0, 1] is the minimal acceptable possibility, Alˇ = Al + ˇ(Am1 − Al ), Am1 ˇ = Am1 + ˇ(Am1 − Am1 ) = Am1 , Am2 ˇ = Am2 + ˇ(Am2 − Am2 ) = Am2 , Arˇ = Ar − ˇ(Ar − Am2 ), blˇ = bl + ˇ(bm1 − bl ), bm1 ˇ = bm1 + ˇ(bm1 − bm1 ) = bm1 , bm2 ˇ = bm2 + ˇ(bm2 − bm2 ) = bm2 , brˇ = br − ˇ(br − bm2 ). If ˇ is given, Eq. (24) is a multi-objective linear programming model. There are known and classic ways to define a solution for a multiobjective linear programming problem. These ways are based on how and when the articulation of preferences of multiple-objectives is gathered from DM. Next, we utilize the method of fuzzy multi-objective decision [24] to solve this multi-objective linear programming model. Since the objective function zi is the function of the decision variable vector x, simply denoted by zi = zi ( x) (i = 1, 2, 3, 4). Let zimax and x∗i respectively be the maximum objective value and the optimal solution for the following linear programming model: max{zi = zi (x)}

⎧A x ≤ b lˇ lˇ ⎪ ⎪ ⎪ ⎪ ⎪ Am1 ˇ x ≤ bm1 ˇ ⎪ ⎪ ⎪ ⎨

s.t.

Am2 ˇ x ≤ bm2 ˇ

⎪ ⎪ Arˇ x ≤ brˇ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Dx ≤ l x≥0

(25)

4342

D.-F. Li, S.-P. Wan / Applied Soft Computing 13 (2013) 4333–4348

Then, let zimin = min{zi (x∗1 ), zi (x∗2 ), zi (x∗3 ), zi (x∗4 )} (i = 1, 2, 3, 4). The linear membership function of the objective function zi (i = 1, 2, 3, 4) can be computed as follows:

zi (x) =

⎧ 1 ⎪ ⎨ ⎪ ⎩

if zi > zimax

(zi − zimin )/(zimax − zimin ) 0

if zimin ≤ zi ≤ zimax if zi <

(26)

zimin

Thus, according to the fuzzy multi-objective decision theory [24] (pp. 76), we can take the aggregation function F˜ (x) = h( (x)) = 1 1 ˜ is the fuzzy multi-objective optimal point set [min{ z1 (x), z2 (x), z3 (x), z4 (x)} + 4 (z1 (x) + z2 (x) + z3 (x) + z4 (x))], where F 2 and (x) = (z1 (x), z2 (x), z3 (x), z4 (x))T . Let  = 12 [min{z1 (x), z2 (x), z3 (x), z4 (x)} + 14 (z1 (x) + z2 (x) + z3 (x) + z4 (x))], then

the fuzzy optimal solution of the linear programming model max{}

s.t.

⎧ 4 ⎪ ⎪ ⎪ 4zi + zi ≥8 (i = 1, 2, 3, 4) ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪ A x ≤ blˇ ⎪ ⎪ lˇ ⎪ ⎨A x ≤ b m1 ˇ

(27)

m1 ˇ

⎪ ⎪ Am2 ˇ x ≤ bm2 ˇ ⎪ ⎪ ⎪ ⎪ ⎪ Arˇ x ≤ brˇ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Dx ≤ l ⎪ ⎩ x≥0

is the efficient (or Pareto optimal) solution of the following fuzzy multi-objective decision problem [24]: max{ (x)}

s.t.

⎧A x ≤ b lˇ lˇ ⎪ ⎪ ⎪ ⎪ ⎪ Am1 ˇ x ≤ bm1 ˇ ⎪ ⎪ ⎪ ⎨A x ≤ b m2 ˇ

(28)

m2 ˇ

⎪ ⎪ Arˇ x ≤ brˇ ⎪ ⎪ ⎪ ⎪ ⎪ Dx ≤ l ⎪ ⎩ x≥0

Theorem 1.

The Pareto optimal solution of Eq. (28) is a Pareto optimal solution of Eq. (24).

Proof. Assume that x is the Pareto optimal solution of Eq. (28) but not the Pareto optimal solution of Eq. (24). By the definition of the / x ) such that z i ( x ) ≤ z i ( x) (i = 1, 2, 3, 4). Since zi (x) (i = 1, 2, 3, 4) are strictly Pareto optimal solution [24] (pp. 61), there exists x ( x = monotonic increasing functions, we have zi (x ) ≤ zi (x) (i = 1, 2, 3, 4). There is a contradictory with the fact that x is the Pareto optimal solution of Eq. (28). Thus, Theorem 1 holds. Theorem 1 shows that the Pareto optimal solution of Eq. (24) can be obtained by solving Eq. (27). Hence, Eq. (21) is solved by the auxiliary model (i.e., Eq. (24)). Obviously, if all TrFNs in the possibility linear programming model with TrFNs (i.e., Eq. (21)) are TFNs, that is, all aijm1 = aijm2 , cjm1 = cjm2 and bim1 = bim2 (i = 1, 2, . . ., m; j = 1, 2, . . ., n), hereby Am1 = Am2 , c m1 = c m2 , Am1 ˇ = Am2 ˇ and bm1 ˇ = bm2 ˇ , then Eq. (24) is then reduced to the following auxiliary model max{z 1 = (c m1 )T x − (c l )T x} min{z 2 = (c m2 )T x} min{z 4 = (c r )T x − (c m2 )T x}

s.t.

⎧ Alˇ x ≤ blˇ ⎪ ⎪ ⎪ ⎪ ⎨ Am ˇ x ≤ bm 1

1ˇ

⎪ Arˇ x ≤ brˇ ⎪ ⎪ ⎪ ⎩ x≥0

provided the constraint Dx ≤ l is removed from Eq. (21). Obviously, the proposed possibility linear programming model with TrFNs in this paper is reduced to the possibility linear programming model with TFNs in [27]. Therefore, the former is the extension of the latter indeed. T

T

T

˜ 0, . . ., 0) and A ˜ of Eq. (21) can be viewed as b˜ = (h, ˜ = (E˜ , 0, . . ., 0) , It is easy to see that Eq. (20) is a special case of Eq. (21) since b˜ and A respectively. Hence, Eq. (20) can be solved through using Eq. (21).

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4.4. Decision making process of the extended possibility LINMAP On the basis of the above analysis, the extended possibility LINMAP is proposed for solving fuzzy MADM problems with multiple types of attribute values and incomplete weight information. The main steps are summarized as follows:

Step 1: Identify all alternatives to be evaluated and evaluation attributes. ˜ of ordered pairs between alternatives. Step 2: Integrate the DM’s preferences to formulate the fuzzy set ˝ Step 3: Elicit the fuzzy decision matrix Y and obtain the normalized decision matrix R by Eqs. (7)–(9). Step 4: Construct the fuzzy linear programming model (i.e., Eq. (18)). ∗ ) through solving Eq. (18) by the possibility linear Step 5: Derive the weight vector ω = (ω1 , ω2 , . . ., ωm )T and the fuzzy IS r ∗ = (r1∗ , r2∗ , . . ., rm programming method with TrFNs developed in Section 4.3. Step 6: Compute the distances Sj of alternatives aj (j = 1, 2, . . ., n) from the fuzzy IS using Eq. (10). Step 7: The ranking order of alternatives is generated according to the increasing order of the distances Sj (j = 1, 2, . . ., n). The best alternative from the alternative set {a1 , a2 , . . ., an } is selected.

5. A real strategy partner selection example and comparison analyses To illustrate the proposed method, a strategy partner selection problem and the analysis process are given. Meanwhile the comparison analyses of the obtained results are also conducted in this section.

5.1. A strategy partner selection problem and the analysis process Strategy management is the management of company strategy. In strategy management, strategy partner selection is a very important part. For a company, selecting the suitable strategy partner is a complex problem which involves multiple different evaluation indexes. Thus, strategy partner selection may be ascribed to a type of MADM problems. The proposed extended possibility method is illustrated with a strategy partner selection process of a vehicle manufacturing company. Jiangling Motors Co., Ltd. (JMC for short) is one of the biggest companies in China commercial vehicle industry and one of China Top 100 Listed Companies for consecutive eight years. In November 1993, JMC successfully issued A share in Shenzhen Stock Exchange and became the first listed company in Jiangxi Province. JMC whole vehicle sales volume reached 200,000 units in 2012. To increase their focus on the core competencies, JMC desires to select a suitable strategy partner for the research and development of its new product. After preliminary screening, five candidate partners, including Volkswagen a1 , Ford a2 , BMW a3 , Mercedes-Benz a4 and FIAT a5 , remain for further evaluation. According the real need and characteristic of the company, the DM (i.e., JMC) considers five attributes to evaluate these partners, including product quality f1 , technological level f2 , flexibility f3 , delivery time f4 and price f5 . f1 , f2 and f3 are qualitative attributes. The assessments for f1 and f2 are represented by TrFNs. Due to the uncertainty of product process, it is better to use the intervals to represent the flexibility f3 and the delivery time f4 . The assessment for price f5 can be represented by crisp numbers. The data and ratings of all partners on every attribute are given by the DM as follows:

⎛

(3, 4, 5, 6)

(6, 7, 8, 9)

(5, 6, 7, 8)

(1, 2, 3, 4)

(2, 3, 4, 5)

119

115

120

118

109

⎞

⎜ (70, 90, 91, 92) (30, 80, 85, 90) (50, 60, 75, 85) (75, 80, 85, 95) (80, 85, 90, 95) ⎟ ⎜ ⎟ ⎟. Y =⎜ [8, 10] [7, 9] [6, 8] [9, 10] [5, 7] ⎜ ⎟ ⎝ ⎠ [87, 90] [75, 88] [76, 89] [92, 95] [65, 88]

˜  According to the DM’s comprehensions  and judgments, the DM provides his/her preference relations between alternatives as ˝ = ˜ ˜ ˜ ˜ ˜ ˜ C(1,2) , C(3,1) , C(4,5) , C(5,2) , C(2,3) , C(4,3) (1,2) (3,1) (4,5) (5,2) (2,3) (4,3)

˜ ˜ , and the corresponding fuzzy truth degrees are as follows: C(1, 2) = (0.1, 0.2, 0.3, 0.4), C(3, 1) =

˜ ˜ ˜ ˜ (0.3, 0.4, 0.5, 0.6), C(4, 5) = (0.5, 0.6, 0.7, 0.8), C(5, 2) = (0.4, 0.5, 0.6, 0.7), C(2, 3) = (0.6, 0.7, 0.8, 0.9), C(4, 3) = (0.4, 0.5, 0.9, 1.0). ˜ is ˝ ˜ 0 = {(1, 2), (3, 1), (4, 5), (5, 2), (2, 3), (4, 3)}. The preference information structure  of attribute importance Thus, the support of ˝ given by the DM is given as  = {ω ∈ 0 |ω1 ≥ 2ω2 , 0.01 ≤ ω2 − ω3 ≤ 0.2, 0.05 ≤ ω4 ≤ 0.3, ω4 − ω5 ≥ ω1 − ω2 }. By using Eqs. (7)–(9), the decision matrix Y can be normalized into the following decision matrix:

⎛ (0.3333, 0.4444, 0.5556, 0.6667) R=

⎜ ⎝

(0.6667, 0.7778, 0.8889, 1.0000)

(0.5556, 0.6667, 0.7778, 0.8889) (0.1111, 0.2222, 0.3333, 0.4444) (0.2222, 0.3333, 0.4444, 0.5556)

(0.7368, 0.9474, 0.9579, 0.9684) (0.3158, 0.8421, 0.8947, 0.9474) (0.5263, 0.6316, 0.7895, 0.8947) (0.7895, 0.8421, 0.8947, 0.9474)

(0.8421, 0.8947, 0.9474, 1.0000)

[0.8000, 1.0000]

[0.7000, 0.9000]

[0.6000, 0.8000]

[0.9000, 1.0000]

[0.2000, 0.7000]

[0.0737, 0.3158]

[0.0526, 0.0842]

[0.0737, 0.2105]

[0.0632, 0.2000]

[0.0000, 0.0316]

0.0083

0.0417

0

0.0167

0.0917

⎞ ⎟. ⎠

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Combining with the above normalized decision matrix, the related vectors in Eq. (18) are computed as follows: T

T

T

T

T

T

T

T

T

T

u12 = (2.6667, −1.0943) , o12 = (0.6667, −0.8421) , 12 = (1.3333, −0.4211) , ϕ12 = (1.3333, −0.2526) , 12 = (0.6667, −0.0421) , u31 = (−1.6296, 1.9890) , o31 = (−0.4444, 0.4211) , 31 = (−0.8889, 1.2632) , T

T

ϕ31 = (−0.8889, 0.6737) , 31 = (−0.4444, 0.1474) , u45 = (0.4444, 0.5651) , o45 = (0.2222, 0.1053) , T

T

T

T

T

45 = (0.4444, 0.2105) , ϕ45 = (0.4444, 0.2105) , 45 = (0.2222, 0.1053) , u52 = (3.2593, −1.0886) , o52 = (0.8889, −1.0526) , T

T

T

T

T

52 = (1.7778, −0.2105) , ϕ52 = (1.7778, −0.2105) , 52 = (0.8889, −0.1053) , u23 = (−1.0370, −0.8947) , o23 = (−0.2222, 0.4211) , T

T

T

T

T

23 = (−0.4444, −0.8421) , ϕ23 = (−0.4444, −0.4211) , 23 = (−0.2222, −0.1053) , u43 = (2.6667, −1.4183) , o43 = (0.8889, −0.5263) , T

T

T

T

T

t43 = (1.7778, −0.8421) , ϕ43 = (1.7778, −0.4211) , 43 = (0.8889, −0.1053) , 12 = (−0.3400, −0.0953) , ı12 = (−0.2000, −0.0421) , T

T

T

T

12 = (−0.2000, −0.4632) , 12 = 0.0017, 12 = 0.0667, 31 = (0.6400, 0.0554) , ı31 = (0.4000, 0.0000) , 31 = (0.4000, 0.2105) , 31 = 0.000069, T

T

T

31 = 0.0167, 45 = (−1.2800, −0.0430) , ı45 = (−1.4000, −0.1263) , 45 = (−0.6000, −0.3368) , 45 = 0.0081, 45 = 0.1500, 52 = (0.7700, 0.0089) T

T

T

T

T

ı52 = (1.0000, 0.1053) , 52 = (0.4000, 0.1053) , 52 = −0.0067, 52 = −0.1000, 23 = (−0.3000, 0.0399) , ı23 = (−0.2000, 0.0421) , 23 = (−0.2000, 0.2526) T

T

T

T

23 = −0.0017, 23 = −0.0833, 43 = (−0.8100, 0.0058) , ı43 = (−0.6000, 0.0211) , 43 = (−0.4000, 0.0211) , 43 = −0.00028, T

T

T

T

T

T

T

T

¯ 1 = (ω1 , ω2 ) , ω ¯ 2 = (ω3 , ω4 ) , ω ¯ 3 = ω5 , v1 = (11 , 21 ) , v2 = (12 , 22 ) , v3 = (13 , 23 ) , v4 = (14 , 24 ) , v5 = (34 , 45 ) , v6 = (36 , 46 ) , v7 =57 . 23 = −0.0333, ω

Take h˜ = (0.0001, 0.0002, 0.0003, 0.0004). According to Eq. (18), the possibility linear programming model with TrFNs is obtained (see Eq. (A.1) of Appendix). Solving Eq. (A.1) by Eqs. (25)–(28) developed in Section 4.3, the components of the optimal solution with ˇ = 0.5 can be obtained as follows: 12 = 31 = 45 = 52 = 23 = 43 = 0,

v11 = 0.0853, v12 = 0.1093, v13 = 0.1621, v14 = 0.2438, v21 = 0.008986, v22 = 0.0816, v23 = 0.1047, v24 = 0.1273, v35 = 0.10705, v36 = 0.1118, v45 = 0.001583, v46 = 0.0016568, ω1 = 0.2894, ω2 = 0.1422, ω3 = 0.1231, ω4 = 0.2999, ω5 =0.1452. By using Eq. (19), the fuzzy IS r ∗ = (r1∗ , r2∗ , . . ., r5∗ ) can be calculated, where r1∗ = (0.2948, 0.3778, 0.5599, 0.8422), r2∗ = (0.0632, 0.5737, 0.7365, 0.8953), r3∗ = [0.8693, 0.9080], r4∗ = [0.0053, 0.0055] and r5∗ = 0.8761. Then, the square of the distance of partner a1 from the fuzzy IS r* can be computed using Eq. (10) as follows: S1 =

1 ω [(b11 6 1

2

2

2

2

2

2

− b∗11 ) + 2(b12 − b∗12 ) + 2(b13 − b∗13 ) + (b14 − b∗14 ) ] + 16 ω2 [(b21 − b∗21 ) + 2(b22 − b∗22 ) + 2(b23 − b∗23 )

2

+ (b24 − b∗24 ) ] + 12 ω3 [(e31 − e3∗ )2 + (g31 − g3∗ )2 ] + 12 ω4 [(e41 − e4∗ )2 + (g41 − g4∗ )2 ] + ω5 (z51 − z5∗ )2 = 0.1472. 2

In a similar way, the squares of the distances of the other partners from the fuzzy IS r* are obtained as follows: S2 = 0.1441,

S3 = 0.1451,

S4 = 0.1478,

S5 = 0.1471.

The ranking order of partners is generated as a2 a3 a5 a1 a4 . The best selection is the partner Ford a2 . As one of the first companies with foreign investment in Jiangxi Province, with the support of its strategic partner Ford, JMC develops rapidly and gains ground. In 1997, JMC worked with Ford to launch Transit light bus which was the first co-developed vehicle by China and foreign companies in China. Thus, the partner Ford a2 is still the best selection for JMC. It is easily seen that there are some intransitive ˜ For instance, the DM prefers a1 to a2 and a2 to a3 , whereas he/she prefers a3 to a1 . Nevertheless, we can still obtain relations in the set ˝. the ranking order of five partners, which indicates the effectiveness of the method proposed in this paper. 5.2. Comparison analysis with the existing LINMAP In this subsection, we compare the methods [8–12] and the proposed method in this paper. In the methods [8–12], the pair-wise comparisons of alternatives are given in the form of the ordered pairs with crisp truth degree 0 or 1 rather than fuzzy numbers, while this paper sufficiently considers the fuzzy truth degrees on alternatives’ comparison. (1) Xia et al. [9] transformed the linguistic variables into TrFNs and proposed fuzzy LINMAP method for solving MADM problems under fuzzy ˜ 0 ) and the minimization ˜ environments. Since real numbers and intervals can be written as TrFN, if all the TrFNs C(k, j) = 1 ((k, j) ∈ ˝ ˜ then the fuzzy linear programming model (i.e., Eq. (18)) constructed in B˜ of objective function of Eq. (18) is revised as maximization G, this paper is reduced to the linear programming model (i.e., Eq. (16)) in [9]. That is to say, the fuzzy LINMAP proposed in [9] is just a special case of that proposed in this paper. (2) As the stated earlier, a TFN is a special case of a TrFN. If all the TFNs [8,10] are written as TrFNs and single DM is considered, then the linear programming model (i.e., Eq. (9)) constructed by [8] and Eq. (11) constructed by [10] are also special cases of the fuzzy linear programming model (i.e., Eq. (18)) constructed in this paper with attribute subsets F2 = F3 =∅ (i.e., i1 = m). (3) In [11,12], the attribute values are in the single form of IFSs, whereas this paper studies the multiple types of attribute values (i.e., TrFNs, intervals and real numbers). The programming models in [11,12] are only crisp linear programming, whereas that constructed in this paper is fuzzy linear programming with TrFNs.

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(4) Compared with the fuzzy LINMAP [8,9,11,12], Eq. (18) considers some incomplete weight preference information structures. The DM may easily add his/her preference information of attribute weights into Eq. (18), which greatly enhances the facility for the DM. ˜ 0 ) and h = 0.0001, then the corresponding fuzzy linear ˜ j) = 1 for all ((k, j) ∈ ˝ In the above partner selection example, suppose C(k, programming model (i.e., Eq. (18)) is reduced to the linear programming model (see Eq. (A.2) of Appendix). Using the simplex method of the linear programming, we find that Eq. (A.2) has no feasible solution. This shows that introducing the fuzzy truth degrees to represent the pair-wise comparisons of alternatives is of great importance. The fuzzy truth degrees play an important role in the decision results. The crisp truth degrees 0 or 1 for ordered pairs may result in no feasible solution of the constructed linear programming model. Thus, the ranking order of alternatives certainly can not be obtained. 5.3. Comparison analysis with the possibilistic LINMAP Although Sadi-Nezhad and Akhtari [22] considered the fuzzy truth degrees as TFNs and proposed the possibility LINMAP in group decision making, it is found that there are some big mistakes in the definitions, notations and operations and possibilistic programming model (please see [23] for more details). Compared with [22], the proposed method in this paper has some notable features as follows: (1) Though [22] and this paper take the fuzzy truth degrees into consideration, the former represented the fuzzy truth degrees as TFNs while the latter represents the fuzzy truth degrees as TrFNs. TrFNs can better reflect the ambiguous nature of subjective judgments on the pair-wise comparisons of alternatives given by DM. (2) The decision problems studied in this paper involve multiple types of attribute values, whereas that studied in [22] considered only single type of attribute values. Namely, the possibility LINMAP [22] is not appropriate for the heterogeneous decision problems. (3) Likewise, if all the above mistakes in [22] are revised, all the TFNs [22] are written as TrFNs and single DM is considered, then the fuzzy mathematical programming model (i.e., Eq. (9)) constructed by [22] is still a special case of the fuzzy linear programming model (i.e., Eq. (18)) constructed in this paper with attribute subsets F2 = F3 =∅ (i.e., i1 = m). 6. Conclusions The existing LINMAP did not consider the fuzzy truth degrees on the comparisons of alternatives. This paper firstly introduced TrFNs to represent the fuzzy truth degrees and developed a new possibilistic LINMAP for solving the MADM problem with multiple types of attribute values and incomplete weight information. In this method, TrFNs, intervals and real numbers were used to represent the multiple types of assessment information of attributes. In order to determine the fuzzy IS and weights of attributes, the fuzzy inconsistency and consistency indices were defined, which were still the TrFNs rather than real numbers. Thus, the constructed programming model was fuzzy linear programming involving the TrFNs, which was solved by the developed possibility linear programming method with TrFNs. The validity and applicability of the proposed method was illustrated with a real strategy partner selection example. The comparison analyzes showed that the proposed method has some advantages over the classical LINMAP, fuzzy LINMAP and simultaneously generalized the possibility LINMAP. Furthermore, it is expected to be applicable to decision making problems in many areas, especially in situations where the pair-wise comparisons of alternatives involved the fuzzy truth degree and the partial preference weights information of attributes are provided a priori. How to effectively elicit the fuzzy truth degrees on alternatives’ comparisons is a critical issue before applying the proposed method, which will be researched in the near future. 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) and the Science and Technology Project of Jiangxi Province Educational Department of China (No. GJJ12265) and the Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics.

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Appendix. min{˜z = (0.1, 0.2, 0.3, 0.4)12 + (0.3, 0.4, 0.5, 0.6)31 + (0.5, 0.6, 0.7, 0.8)45 + (0.4, 0.5, 0.6, 0.7)52 + (0.6, 0.7, 0.8, 0.9)23 + (0.4, 0.5, 0.9, 1.0)43 }

⎧ (0.4444ω1 − 0.1824ω2 − 0.111111 + 0.140421 − 0.222212 + 0.070222 − 0.222213 + 0.042123 − 0.111114 + 0.007024 ⎪ ⎪ ⎪ ⎪ −0.1700ω3 − 0.0476ω4 + 0.100035 + 0.021145 + 0.100036 + 0.231646 + 0.0017ω5 − 0.066757 )(0.1, 0.2, 0.3, 0.4) ⎪ ⎪ ⎪ ⎪ ⎪ +(−0.2716ω1 + 0.3315ω2 + 0.074111 − 0.070221 + 0.148112 − 0.210522 + 0.148113 − 0.112323 + 0.074114 − 0.024624 ⎪ ⎪ ⎪ ⎪ +0.3200ω3 + 0.0277ω4 − 0.200035 − 0.200036 − 0.105346 + 0.000069444ω5 − 0.016757 )(0.3, 0.4, 0.5, 0.6) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +(0.0741ω1 + 0.0942ω2 − 0.037011 − 0.017521 − 0.074112 − 0.035122 − 0.074113 − 0.035123 − 0.037014 − 0.017524 ⎪ ⎪ ⎪ ⎪ ⎪ −0.6400ω3 − 0.0215ω4 + 0.700035 + 0.063245 + 0.300036 + 0.168446 + 0.0081ω5 − 0.150057 )(0.5, 0.6, 0.7, 0.8)+ ⎪ ⎪ ⎪ ⎪ +(0.5432ω ⎪ 1 − 0.1814ω2 − 0.148111 + 0.175421 − 0.296312 + 0.035122 − 0.296313 + 0.035123 − 0.148114 ⎪ ⎪ ⎪ ⎪ +0.017524 + 0.3850ω3 + 0.0044ω4 − 0.500035 − 0.052645 − 0.200036 − 0.052646 − 0.0067ω5 + 0.100057 )(0.4, 0.5, 0.6, 0.7) ⎪ ⎪ ⎪ ⎪ ⎪ +(−0.1728ω1 − 0.1491ω2 + 0.037011 − 0.070221 + 0.074112 + 0.140422 + 0.074113 + 0.070223 + 0.037014 + 0.017524 ⎪ ⎪ ⎪ ⎪ ⎪ −0.1500ω3 + 0.0199ω4 + 0.10035 − 0.021145 + 0.100036 − 0.126346 − 0.0017ω5 + 0.083357 )(0.6, 0.7, 0.8, 0.9) ⎪ ⎪ ⎪ ⎪ ⎪ +(0.4444ω1 − 0.2364ω2 − 0.148111 + 0.087721 − 0.296312 + 0.140422 − 0.296313 + 0.070223 − 0.148114 + 0.017524 ⎪ ⎪ ⎪ ⎪ ⎪ −0.4050ω3 + 0.0029ω4 + 0.300035 − 0.010545 + 0.200036 − 0.010546 − 0.00027778ω5 + 0.033357 )(0.4, 0.5, 0.9, 1.0) ⎪ ⎪ ⎪ ⎪ ⎪ 0.0002, 0.0003, 0.0004) ≥(0.0001, ⎪ ⎪ 0.4444ω1 − 0.1824ω2 − 0.111111 + 0.140421 − 0.222212 + 0.070222 − 0.222213 + 0.042123 − 0.111114 + 0.007024 ⎪ ⎪ ⎪ ⎪ ⎪ −0.1700ω3 − 0.0476ω4 + 0.100035 + 0.021145 + 0.100036 + 0.231646 + 0.0017ω5 − 0.066757 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −0.2716ω1 + 0.3315ω2 + 0.074111 − 0.070221 + 0.148112 − 0.210522 + 0.148113 − 0.112323 + 0.074114 − 0.024624

s.t.

+0.3200ω3 + 0.0277ω4 − 0.200035 − 0.200036 − 0.105346 + 0.000069444ω5 − 0.016757 ≥0

⎪ ⎪ ⎪ 0.0741ω1 + 0.0942ω2 − 0.037011 − 0.017521 − 0.074112 − 0.035122 − 0.074113 − 0.035123 − 0.037014 − 0.017524 ⎪ ⎪ ⎪ ⎪ ⎪ −0.6400ω3 − 0.0215ω4 + 0.700035 + 0.063245 + 0.300036 + 0.168446 + 0.0081ω5 − 0.150057 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ 0.5432ω 1 − 0.1814ω2 − 0.148111 + 0.175421 − 0.296312 + 0.035122 − 0.296313 + 0.035123 − 0.148114 ⎪ ⎪ ⎪ ⎪ ⎪ +0.017524 + 0.3850ω3 + 0.0044ω4 − 0.500035 − 0.052645 − 0.200036 − 0.052646 − 0.0067ω5 + 0.100057 ≥0 ⎪ ⎪ ⎪ ⎪ −0.1728ω1 − 0.1491ω2 + 0.037011 − 0.070221 + 0.074112 + 0.140422 + 0.074113 + 0.070223 + 0.037014 + 0.017524 ⎪ ⎪ ⎪ ⎪ ⎪ −0.1500ω3 + 0.0199ω4 + 0.10035 − 0.021145 + 0.100036 − 0.126346 − 0.0017ω5 + 0.083357 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ 0.4444ω1 − 0.2364ω2 − 0.148111 + 0.087721 − 0.296312 + 0.140422 − 0.296313 + 0.070223 − 0.148114 + 0.017524 ⎪ ⎪ ⎪ ⎪ ⎪ −0.4050ω3 + 0.0029ω4 + 0.300035 − 0.010545 + 0.200036 − 0.010546 − 0.00027778ω5 + 0.033357 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ 12 ≥0, 31 ≥0, 45 ≥0, 52 ≥0, 23 ≥0, 43 ≥0, ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ v11 ≤ v12 ≤ v13 ≤ v14 ≤ ω1 ⎪ ⎪ ⎪ 0 ≤ v21 ≤ v22 ≤ v23 ≤ v24 ≤ ω2 ⎪ ⎪ ⎪ ⎪ 0 ≤ v35 ≤ v36 ≤ ω3 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ v45 ≤ v46 ≤ ω4 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ v57 ≤ ω5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ω1 ≥2ω2 , 0.01 ≤ ω2 − ω3 ≤ 0.2, 0.05 ≤ ω4 ≤ 0.3, ω4 − ω5 ≥ω1 − ω2 ⎪ ⎩ ω1 + ω2 + ω3 + ω4 + ω5 = 1

(A.1)

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min{12 + 31 + 45 + 52 + 23 + 43 }

s.t.

⎧ 0.0012ω − 0.0167 + 0.3 + 1.0617ω − 0.3236ω − 0.6600ω − 0.0142ω − 0.3333 + 0.2456 − 0.6667 5 57 36 1 2 3 4 11 21 12 ⎪ ⎪ ⎪ ⎪ +0.1404 − 0.6667 + 0.0702 − 0.3333 + 0.0175 + 0.5 − 0.0 + 0.1053 ≥0.0001 22 13 23 14 24 35 45 46 ⎪ ⎪ ⎪ ⎪ ⎪ 0.4444ω − 0.1824ω − 0.1111 + 0.1404 − 0.2222 + 0.0702 − 0.2222 + 0.0421 1 2 11 21 12 22 13 23 − 0.111114 + 0.007024 ⎪ ⎪ ⎪ ⎪ ⎪ −0.1700ω3 − 0.0476ω4 + 0.100035 + 0.021145 + 0.100036 + 0.231646 + 0.0017ω5 − 0.066757 ≥0 ⎪ ⎪ ⎪ ⎪ −0.2716ω ⎪ 1 + 0.3315ω2 + 0.074111 − 0.070221 + 0.148112 − 0.210522 + 0.148113 − 0.112323 + 0.074114 − 0.024624 ⎪ ⎪ ⎪ ⎪ +0.3200ω3 + 0.0277ω4 − 0.200035 − 0.200036 − 0.105346 + 0.000069444ω5 − 0.016757 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ 0.0741ω ⎪ 1 + 0.0942ω2 − 0.037011 − 0.017521 − 0.074112 − 0.035122 − 0.074113 − 0.035123 − 0.037014 − 0.017524 ⎪ ⎪ ⎪ ⎪ −0.6400ω3 − 0.0215ω4 + 0.700035 + 0.063245 + 0.300036 + 0.168446 + 0.0081ω5 − 0.150057 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ 0.5432ω1 − 0.1814ω2 − 0.148111 + 0.175421 − 0.296312 + 0.035122 − 0.296313 + 0.035123 − 0.148114 ⎪ ⎪ ⎪ ⎪ ⎪ +0.017524 + 0.3850ω3 + 0.0044ω4 − 0.500035 − 0.052645 − 0.200036 − 0.052646 − 0.0067ω5 + 0.100057 ≥0 ⎪ ⎪ ⎪ ⎨ −0.1728ω1 − 0.1491ω2 + 0.037011 − 0.070221 + 0.074112 + 0.140422 + 0.074113 + 0.070223 + 0.037014 + 0.017524

(A.2)

⎪ −0.1500ω3 + 0.0199ω4 + 0.10035 − 0.021145 + 0.100036 − 0.126346 − 0.0017ω5 + 0.083357 ≥0 ⎪ ⎪ ⎪ ⎪ 0.4444ω1 − 0.2364ω2 − 0.148111 + 0.087721 − 0.296312 + 0.140422 − 0.296313 + 0.070223 − 0.148114 + 0.017524 ⎪ ⎪ ⎪ ⎪ ⎪ −0.4050ω3 + 0.0029ω4 + 0.300035 − 0.010545 + 0.200036 − 0.010546 − 0.00027778ω5 + 0.033357 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ 12 ≥0, 31 ≥0, 45 ≥0, 52 ≥0, 23 ≥0, 43 ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ v11 ≤ v12 ≤ v13 ≤ v14 ≤ ω1 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ v21 ≤ v22 ≤ v23 ≤ v24 ≤ ω2 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ v35 ≤ v36 ≤ ω3 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ v45 ≤ v46 ≤ ω4 ⎪ ⎪ ⎪ ⎪ 0 ≤ v57 ≤ ω5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ω1 ≥2ω2 , 0.01 ≤ ω2 − ω3 ≤ 0.2, 0.05 ≤ ω4 ≤ 0.3, ω4 − ω5 ≥ω1 − ω2 ⎪ ⎩ ω1 + ω2 + ω3 + ω4 + ω5 = 1

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference, Psychometrica 38 (1973) 337–342. L.A. Zadeh, Fuzzy sets, Information and Control 18 (1965) 338–353. K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 50 (1986) 87–96. T.Y. Chen, A comparative study of optimistic and pessimistic multicriteria decision analysis based on Atanassov fuzzy sets, Applied Soft Computing 12 (8) (2012) 2289–2311. S. Alonso, I.J. Pérez, F.J. Cabrerizo, E. Herrera-Viedma, A linguistic consensus model for Web 2.0 communities, Applied Soft Computing 13 (1) (2013) 149–157. P.D. Liu, F. Jin, Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making, Information Sciences 205 (2012) 58–71. P.D. Liu, F. Jin, X. Zhang, Y. Su, M.H. Wang, Research on the multi-attribute decision-making under risk with interval probability based on prospect theory and the uncertain linguistic variables, Knowledge-Based Systems 24 (4) (2011) 554–561. D.-F. Li, J.B. Yang, Fuzzy linear programming technique for multiattribute group decision making in fuzzy environments, Information Sciences 158 (2004) 263–275. H.C. Xia, D.-F. Li, J.Y. Zhou, J.M. Wang, Fuzzy LINMAP method for multiattribute decision making under fuzzy environments, Journal of Computer and System Sciences 72 (2006) 741–759. D.-F. Li, T. Sun, Fuzzy LINMAP method for multiple attribute group decision making with linguistic variables and incomplete information, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15 (2) (2007) 153–173. D.-F. Li, Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment, Fuzzy Optimization and Decision Making 7 (2008) 17–34. D.-F. Li, G.H. Chen, Z.G. Huang, Linear programming method for multiattribute group decision making using IF sets, Information Sciences 180 (9) (2010) 1591–1609. K. Khalili-Damghani, S. Sadi-Nezhad, A hybrid fuzzy multiple criteria group decision making approach for sustainable project selection, Applied Soft Computing 13 (1) (2013) 339–352. O. Taylan, A hybrid methodology of fuzzy grey relation for determining multi attribute customer preferences of edible oil, Applied Soft Computing 13 (5) (2013) 2981–2989. D.-F. Li, Z.G. Huang, G.H. Chen, A systematic approach to heterogeneous multiattribute group decision making, Computers & Industrial Engineering 59 (2010) 561–572. K.H. Guo, W.L. Li, An attitudinal-based method for constructing intuitionistic fuzzy information in hybrid MADM under uncertainty, Information Sciences 208 (15) (2012) 28–38. 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, Knowledge-Based Systems 23 (2) (2010) 169–181. E. Herrera-Viedma, F. Chiclana, F. Herrera, S. Alonso, Group decision-making model with incomplete fuzzy preference relations based on additive consistency, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 1 (2007) 176–189. V.V. Podinovski, Set choice problems with incomplete information about the preferences of the decision makers, European Journal of Operation Research 207 (2010) 371–379. Z.J. Wang, K.W. Li, W.Z. Wang, An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights, Information Sciences 179 (2009) 3026–3040. D.-F. Li, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information, Applied Soft Computing 11 (2011) 3402–3418. S. Sadi-Nezhad, P. Akhtari, Possibilistic programming approach for fuzzy multidimensional analysis of preference in group decision making, Applied Soft Computing 8 (2008) 1703–1711.

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D.-F. Li, S.-P. Wan / Applied Soft Computing 13 (2013) 4333–4348

[23] S.P. Wan, D.-F. Li, Fuzzy LINMAP approach to heterogeneous MADM considering the comparisons of alternatives with hesitation degrees, Omega: The International Journal of Management Science 41 (6) (2013) 925–940. [24] D.-F. Li, Fuzzy Multiobjective Many-Person Decision Makings and Games, National Defense Industry Press, Beijing, 2003. [25] H. Ishibuchi, H. Tanaka, Multiobjective programming in optimization of the interval objective function, European Journal of Operation Research 48 (1990) 219–225. [26] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [27] Y.J. Lai, C.L. Hwang, A new approach to some possibilistic linear programming problems, Fuzzy Sets and Systems 49 (1992) 121–133.