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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

Atanassov’s Intuitionistic Fuzzy Programming Method for Heterogeneous Multiattribute Group Decision Making With Atanassov’s Intuitionistic Fuzzy Truth Degrees Shu-Ping Wan and Deng-Feng Li

Abstract—The aim of this paper is to develop a new Atanassov’s intuitionistic fuzzy (A-IF) programming method to solve heterogeneous multiattribute group decision-making problems with A-IF truth degrees in which there are several types of attribute values such as A-IF sets (A-IFSs), trapezoidal fuzzy numbers, intervals, and real numbers. In this method, preference relations in comparisons of alternatives with hesitancy degrees are expressed by A-IFSs. Hereby, A-IF group consistency and inconsistency indices are defined on the basis of preference relations between alternatives. To estimate the fuzzy ideal solution (IS) and weights, a new A-IF programming model is constructed on the concept that the AIF group inconsistency index should be minimized and must be not larger than the A-IF group consistency index by some fixed A-IFS. An effective method is developed to solve the new derived model. The distances of the alternatives to the fuzzy IS are calculated to determine their ranking order. Moreover, some generalizations or specializations of the derived model are discussed. Applicability of the proposed methodology is illustrated with a real supplier selection example. Index Terms—Atanassov’s intuitionistic fuzzy set (A-IF), fuzzy set, mathematical programming, multiattribute group decision making (MAGDM), uncertainty.

I. INTRODUCTION ULTIATTRIBUTE group decision making (MAGDM) is an important research field in decision science, operational research, and management science. The linear program-

M

Manuscript received October 12, 2012; revised January 19, 2013; accepted March 8, 2013. Date of publication March 15, 2013; date of current version March 27, 2014. This work was supported in part by the Key Program of the National Natural Science Foundation of China under Grant 71231003; in part by the National Natural Science Foundation of China under Grant 71061006, Grant 61263018, Grant 71171055, and Grant 71001015; in part by the Program for New Century Excellent Talents in University (the Ministry of Education of China, NCET-10-0020); in part by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant 20113514110009; in part by the “Science and Technology Innovation Team Cultivation Plan of Colleges and Universities in Fujian Province”; in part by the Humanities Social Science Programming Project of Ministry of Education of China under Grant 09YGC630107; in part by the Natural Science Foundation of Jiangxi Province of China under Grant 20114BAB201012; and in part by the Science and Technology Project of Jiangxi Province Educational Department of China under Grant GJJ12265 as well as the “Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics.” S.-P. Wan is with the College of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China (e-mail: shupingwan@ 163.com).. D.-F. Li is with the School of Management, Fuzhou University, Fuzhou, Fujian 350108, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2013.2253107

ming technique for multidimensional analysis of preference (LINMAP) [26] is an effective method for solving MAGDM problems. The LINMAP method is based on pair-wise comparisons of alternatives given by the decision maker (DM) and generates the best alternative, which has the shortest distance to the ideal solution (IS). In the LINMAP [26], 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 [1], [35]. A more realistic approach could be to use fuzzy sets [5], [25], [33], [35], [36], Atanassov’s intuitionistic fuzzy (A-IF) sets (A-IFSs) [1]–[6], [29], [32], [34], and linguistic variables [8], [10]–[12], [15]–[22], [24], [31] to model human judgments. Therefore, extending the LINMAP to suit MAGDM problems under A-IF environments is of a great importance for scientific research work and real applications. Extensions of the LINMAP may be roughly divided into two types. The first type is to consider the assessment information of attribute values, which is expressed with fuzzy sets, linguistic variables, or A-IFSs instead of real numbers [15], [19], [21], [22], [31]. Li and Sun [21] and Li and Yang [22] used linguistic variables to assess alternative attributes. These linguistic variables are transformed into positive triangular fuzzy numbers (TFNs). Xia et al. [31] transformed the linguistic variables into positive trapezoidal fuzzy numbers (TrFNs). Hereby, the fuzzy LINMAP was proposed to solve multiattribute decisionmaking (MADM) problems. Li [15] and Li et al. [19] developed the linear programming methodologies to solve MADM and MAGDM problems in which the attribute values of alternatives are expressed with A-IFSs. The second type of extension is to represent the truth degrees of comparisons of alternatives by fuzzy numbers [25]. In the classical LINMAP [26] and the fuzzy LINMAP [15], [19], [21], [22], [31], the DM gives pairwise comparisons of alternatives with crisp truth degrees 0 or 1. In reality, however, the DM is not sure enough in all comparisons and may express his/her opinion with fuzzy truth degree. Sadi-Nezhad and Akhtari [25] considered the fuzzy truth degree as a TFN and proposed the possibility LINMAP for MAGDM problems. Nevertheless, it is found that there exist some big mistakes in [25, Sec. 6.3]. In real decision situations, DMs have their unique characteristics with regard to knowledge, skills, experience, and personality, which implies that they may express their judgments by means of different preference representation formats. Thus, it is usual and necessary to take into consideration heterogeneous

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WAN AND LI: ATANASSOV’S INTUITIONISTIC FUZZY PROGRAMMING METHOD

information in MAGDM problems. Such a type of MAGDM problem is called a heterogeneous MAGDM, which has drawn much attention from a wide spectrum of disciplines [20]. In general, there are some challenges for the DMs in providing precise and complete weight preference information due to time pressure and lack of knowledge. Namely, weight preference information in MAGDM problems is usually incomplete. For example, real supplier selection problems often involve multiple different types of attributes (or factors) such as development capability, product quality, technology level, delivery time as well as price. They may be ascribed to a kind of MAGDM problem. The research and development capability of a supplier may be assessed by using an A-IFS. Product quality may be represented by a TrFN. Delivery time may be measured by using an interval. The price may be expressed with a real number. Moreover, it is very difficult for DMs to give these attribute weights accurately due to subjective and objective reasons. Thus, the supplier selection problem is a typical kind of heterogeneous MAGDM problem with incomplete preference information. Recently, some methods for solving MAGDM problems with incomplete preference information have been studied in [18], [20], [21], [23]. However, these methods are not applicable to heterogeneous MAGDM problems. On the other hand, due to the influence of subjective and objective factors, the DMs usually give the pair-wise comparisons of alternatives with some hesitancy degrees. This realization has motivated the extension of group LINMAP models to solve heterogeneous MAGDM problems with incomplete weight information in which hesitancy degrees of alternative comparisons are taken into consideration. However, there exist three major difficulties and challenges to solving such heterogeneous MAGDM problems. The first is how to represent rational preference relation sets of alternative comparisons with hesitancy degrees. The second is how to define the group consistency and inconsistency indices, when hesitancy degrees of alternative comparisons are considered. The third is how to construct adequate mathematical programming models and hereby establish corresponding effective solving methods. As far as we know, there has been no investigation into the aforementioned MAGDM problems. The purpose of this paper is to extend the LINMAP to solve heterogeneous MAGDM problems, which involve A-IFSs, TrFNs, intervals, and real numbers. As stated earlier, it seems better to use A-IFSs to capture hesitancy degrees of alternative comparisons. Therefore, the preference relation sets of alternative comparisons with hesitancy degrees are expressed with A-IFSs. Hereby, the A-IF group consistency and inconsistency indices are defined on the basis of the preference relations between alternatives given by the DMs. IS and attribute weights are estimated through a newly constructed A-IF mathematical programming model, which is solved by using the developed A-IF mathematical programming method with A-IFSs. Thus, the alternatives are ranked according to their distances to the IS. It should be emphasized that the IS is unknown a priori. According to the prospect theory [28], in the real decision process, DMs do not always pursue the maximization of the expected utility (or satisfaction and dissatisfaction degrees), whereas they often weight the decision gain and loss risk according to some

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reference points. The decision may be changed with the change of the chosen reference points. Hence, to avoid the subjective randomness of selecting reference points, the IS should not be artificially given a priori. The rest of this paper is organized as follows. Section II briefly reviews some concepts and denotations such as A-IFSs and TrFNs. In Section III, heterogeneous MAGDM problems with incomplete weight information are formulated, and the normalized method is given. In Section IV, a new A-IF group, LINMAP, is developed to solve the aforementioned MAGDM problems. Some generalizations of the constructed A-IF mathematical programming model are discussed in Section V. The developed methodology in this paper is illustrated with a real supplier selection example and comparison analysis is conducted in Section VI. Section VII concludes this paper.

II. BASIC CONCEPTS AND NOTATIONS Definition 1 [1]: Let A = {ui , μA (ui ), νA (ui )|ui ∈ U } and B = {ui , μB (ui ), νB (ui )|ui ∈ U } be two A-IFSs in a finite universe of discourse U = {u1 , u2 , . . . , um }. Then, we stipulate the following: 1) A + B = {ui , μA (ui ) + μB (ui ) − μA (ui )μB (ui ), νA (ui )νB (ui )|ui ∈ U }. 2) AB = {ui , μA (ui )μB (ui ), νA (ui ) + νB (ui )−νA (ui ) νB (ui )|ui ∈ U }. 3) λA = {ui , 1 − (1 − μA (ui ))λ , (νA (ui ))λ |ui ∈ U } (λ > 0). The weighted Minkowski distance between A-IFSs A and B is defined as follows:   m 1 dq (A, B) = wi (|μA (ui ) − μB (ui )|q + |νA (ui ) 2 i=1

1/q (1) −νB (ui )| + |πA (ui ) − πB (ui )| ) q

q

where wi (i = 1, 2, . . . , m) are weights of u i ∈ U , which satisfy the normalized conditions: wi ∈ [0, 1] and m i=1 wi = 1. When q = 1, q = 2, and q → +∞, corresponding d1 (A, B), d2 (A, B), and d+∞ (A, B) are called the weighted Hamming, Euclidean, and Chebyshev distances, respectively [27]. If an A-IFS A contains only one element, i.e., |A| = 1, then A is usually denoted as A = μA , υA  for short. The score and accuracy functions of A = μA , υA  are defined as S(A) = μA + υA and H(A) = μA − υA , respectively [6]. Definition 2 [6]: Assume that A = μA , υA  and B = μB , υB  are two A-IFSs. Let S(A), S(B), H(A), and H(B) be their score and accuracy functions, respectively. Then, 1) if S(A) < S(B), then A is smaller than B, which is denoted by A < B; 2) if S(A) = S(B), then a) if H(A) = H(B), then A is equal to B, which is denoted by A = B; b) if H(A) < H(B), then A is smaller than B, which is denoted by A < B.

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Let m ˜ = (l, m1 , m2 , r) be a TrFN, whose membership function is given as follows: ⎧ (x − l)/(m1 − l), (l ≤ x < m1 ) ⎪ ⎨ (m1 ≤ x ≤ m2 ) μm˜ (x) = 1, (2) ⎪ ⎩ (r − x)/(r − m ), (m < x ≤ r). 2 2 The closed interval [m1 , m2 ], and l and r are the mode, and lower and upper limits of m, ˜ respectively. It is easily seen that a TrFN m ˜ is reduced to a real number m if l = m1 = m2 = r. Conversely, a real number m can be written as a TrFN m ˜ = (m, m, m, m). A TrFN m ˜ is reduced ˜ is positive if to a TFN m ˜ = (l, m1 , r) if m1 = m2 . A TrFN m l ≥ 0 and r > 0. Furthermore, a positive TrFN m ˜ is normalized if l ≥ 0 and r ≤ 1. ˜ = (n1 , n2 , n3 , n4 ) be two Let m ˜ = (m1 , m2 , m3 , m4 ) and n TrFNs. Then, the vertex method is defined to calculate the Euclidean distance between them as follows: d(m, ˜ n ˜) =

1 [(m1 − n1 )2 + 2(m2 − n2 )2 + 2(m3 − n3 )2 + (m4 − n4 )2 ]. 6

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

p ≥ 0. Hence, a heterogeneous MAGDM problem can and zij p )m ×n (p = 1, 2, . . . , Q), be concisely expressed as Ap = (yij which are referred to as decision matrices usually used to represent the heterogeneous MAGDM problem. In general, the attribute subsets Ft can be divided into two subsets Ftb and Ftc , where Ftb and Ftc are, respectively, the sets of benefit and cost attributes, Ft = Ftb ∪ Ftc and Ftb ∩ Ftc = ∅(t = 1, 2, 3, 4). The attribute ratings should be normalized to ensure their compatibility. Due to the fact that the ratings of the attributes in F1 are represented by A-IFSs, whose membership and nonmembership degrees belong to the interval [0, 1], these attribute ratings need not to be normalized. p p p p = (a p For the ratings yij ij , bij , cij , dij ) (fi ∈ F2 ), according to benefit attributes and cost attributes, their normalized values can be computed as follows:

ripj = (apij , bpij , cpij , dpij ) ⎧ p m a x p m a x p m a x p m a x (ai j /di , bi j /di , ci j /di , di j /di ), if fi ∈ F2b ⎪ ⎪ ⎨

cipj bipj aipj dipj = (4) ⎪ , 1 − m a x , 1 − m a x , 1 − m a x , if fi ∈ F2c ⎪ ax ⎩ 1 − dm di di di i

ax where dm = max{d p i ij |j = 1, 2, . . . , n; p = 1, 2, . . . , Q}. p p (3) = [e p For the ratings yij ij , gij ] (fi ∈ F3 ), their normalized values can be computed as follows:  p m ax p m ax III. HETEROGENEOUS MULTIATTRIBUTE GROUP DECISION [eij /gi , gij /gi ], if fi ∈ F3b MAKING PROBLEMS WITH INCOMPLETE WEIGHT INFORMATION rp = [ep , g p ] = ij ij ij p m ax [1 − gij /gim ax , 1 − e p ], if fi ∈ F3c ij /gi Suppose that there are Q DMs DP (p = 1, 2, . . . , Q), who (5) have to choose one of or rank n alternatives xj (j = 1, 2, . . . , n) p m ax = max{g |j = 1, 2, . . . , n; p = 1, 2, . . . , Q}. where g i ij based on m attributes fi (i = 1, 2, . . . , m). Let us denote p p = zij (fi ∈ F4 ), their normalized values For the ratings y ij an alternative set by X = {x1 , x2 , . . . , xn } and an atcan be computed as follows: tribute set by F = {f1 , f2 , . . . , fm }. Let us denote F1 =  p m ax {f1 , f2 , . . . , fi 1 }, F2 = {fi 1 +1 , fi 1 +2 , . . . , fi 2 }, F3 = {fi 2 +1 , zij /zi , if fi ∈ F4b p p fi 2 +2 , . . . , fi 3 }, and F4 = {fi 3 +1 , fi 3 +2 , . . . , fm }, respec= zij = (6) rij p 1 − zij /zim ax , if fi ∈ F4c tively, where 1 ≤ i1 ≤ i2 ≤ i3 ≤ m. Namely, F is divided into four subsets Ft (t = 1, 2, 3, 4) in which attribute values are where zim ax = max{z pij |j = 1, 2, . . . , n; p = 1, 2, . . . , Q}. expressed with A-IFSs, TrFNs, intervals, and real numbers, p )m ×n are transformed Thus, the decision matrices Ap = (yij ∩ F = ∅ (t, l = 1, 2, 3 , 4; t =  l) respectively. In addition, F p t l 4 )m ×n (p = into the normalized decision matrices Rp = (rij and t=1 Ft = F , where ∅ is an empty set. Let us denote M1 = p p p p 1, 2, . . . , Q), where rij = yij = μij , υij  for i ∈ M1 . {1, 2, . . . , i1 }, M2 = {i1 + 1, i1 + 2, . . . , i2 }, M3 = {i2 + 1, p p p T , r2j , . . . , rm Denote r pj = (r1j j ) , which is sometime rei2 + 2, . . . , i3 }, M4 = {i3 +1, i3 +2, . . . , m}, and M = {1, 2, as an alternative xj for the DM DP . In other words, . . . , m}. Let ω = (ω1 , ω2 , . . . , ωm )T be the weighting vector, garded p and x r j have the same meaning. j where ωi (i = 1, 2, . . . , m) are the weights mof the attributes fi , which satisfy the normalized conditions: i=1 ωi = 1 and ωi ≥  IV. ATANASSOV’S INTUITIONISTIC FUZZY GROUP LINEAR ω 0. Let Λ0 = {ω| m i=1 i = 1, ωi ≥ ε for i = 1, 2, . . . , m}, PROGRAMMING TECHNIQUE FOR MULTIDIMENSIONAL where ε > 0 is a sufficiently small positive number, which A NALYSIS OF PREFERENCE MODELS AND METHOD WITH ensures that the weights generated are not zero as it may be the A TANASSOV ’S INTUITIONISTIC FUZZY TRUTH DEGREES OF case in the LINMAP [26]. ALTERNATIVE COMPARISONS Let us denote the incomplete preference information structure of attribute weights by Λ, which may consist of some sets of the A. Atanassov’s Intuitionistic Fuzzy Group Consistency and five basic sets Λs (s = 1, 2, 3, 4, 5) in Λ0 [18]. Inconsistency Measurements Let the ratings of alternatives xj on attributes fi given by ∗ p p ) is unknown Suppose that the IS r ∗ = (r1∗ , r2∗ , . . . , rm the DMs DP be expressed by yij . Stipulate if i ∈ M1 , yij is a priori and needs to be determined, where ri∗ is the p p p = μpij , υij ; if i ∈ M2 , yij = (a p expressed as an A-IFS yij ij , ideal rating of the attribute fi (i = 1, 2, . . . , m). Namely, p p p p p b p ij , cij , dij ) is expressed as a TrFN; if i ∈ M3 , yij = [eij , gij ] is if i ∈ M1 , ri∗ = μ∗i , υi∗  is an A-IFS; if i ∈ M2 , ri∗ = p p expressed as an interval; and if i ∈ M4 , yij = zij is expressed as (a∗i , b∗i , c∗i , d∗i ) is a TrFN; if i ∈ M3 , ri∗ = [e∗i , gi∗ ] is p p p p p a real number, where 0 ≤ a p an interval; and if i ∈ M4 , ri∗ = zi∗ is a real number, ij ≤ bij ≤ cij ≤ dij , 0 ≤ eij ≤ gij ,

WAN AND LI: ATANASSOV’S INTUITIONISTIC FUZZY PROGRAMMING METHOD

where μ∗i ≥ 0, υi∗ ≥ 0, μ∗i + υi∗ ≤ 1 (i ∈ M1 ), 0 ≤ a∗i ≤ b∗i ≤ c∗i ≤ d∗i ≤ 1 (i ∈ M2 ), 0 ≤ e∗i ≤ gi∗ ≤ 1 (i ∈ M3 ), and 0 ≤ zi∗ ≤ 1 (i ∈ M4 ) . Using (1) and (3), the square of the weighted Euclidean distance between r pj and r ∗ can be calculated as follows: Sjp =

i1  ωi i=1

+

2

p p [(μpij − μ∗i )2 + (υij − υi∗ )2 + (πij − πi∗ )2 ]

i2  ωi p [(aij − a∗i )2 + 2(bpij − b∗i )2 6 i=i +1 1

+ 2(cpij − c∗i )2 + (dpij − d∗i )2 ] +

i3  ωi p [(eij − e∗i )2 2 i=i +1 2

p + (gij − gi∗ )2 ] +

m 

p [ωi (zij − zi∗ )2 ]

(7)

i=i 3 +1 p πij

μpij

p υij ,

πi∗

μ∗i

υi∗ .

=1− − and =1− − where Assume that the DM DP expresses the preference relations of alternatives as the A-IFSs

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An index (Sjp − Skp )− is defined to measure inconsistency between the ranking order of alternatives xk and xj determined by Skp and Sjp and the preference relation given by the DM DP preferring xk to xj as follows: ˜ Cp (k, j)(Skp − Sjp ), if Sjp < Skp (9) (Sjp − Skp )− = 0, if Sjp ≥ Skp . Obviously, the ranking order of xk and xj determined by Sjp and Skp based on (ω, r ∗ ) is consistent with the preference relation, which is given by DP if Sjp ≥ Skp . Hence, (Sjp − Skp )− is defined to be 0. On the other hand, the ranking order of xk and xj determined by Sjp and Skp is inconsistent with the preference relation given by DP if Sjp < Skp . Thus, (Sjp − Skp )− is defined to be C˜p (k, j)(Skp − Sjp ). The inconsistency index is rewritten as (Sjp − Skp )− = C˜p (k, j) max{0, Skp − Sjp } . Hereby, an A-IF group inconsistency index is defined as follows: ˜= B

C p (k ,j )

(Sjp − Skp )−

=

Q 



[C˜p (k, j) max{0, Skp − Sjp }]. (10)

p=1 (k ,j ) ∈ Ω ˜ p< 0 , 1 >

(8)

where (k, j) expresses an ordered pair of the alternatives xk and xj in which the DM DP prefers xk to xj , denoted by xk xj for short; C˜p (k, j) = μC˜ p (k ,j ) , υC˜ p (k ,j )  is an AIFS, whose membership and nonmembership degrees satisfy the condition μC˜ p (k ,j ) + υC˜ p (k ,j ) ≤ 1. An (α, β) cut set of ˜ p is defined as Ω ˜ < α ,β > = {(k, j)| μ ˜ Ω ≥ α, υ ˜ ≤ p



p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

˜ p = {(k, j), C˜p (k, j)|xk xj with C˜p (k, j) Ω ( k, j = 1, 2, . . . , n)}

Q 

C p (k ,j )

β( k, j = 1, 2, . . . , n)}, where α ∈ [0, 1], β ∈ [0, 1], and α + ˜ < 0,1> = {(k, j)| μ ˜ ˜ p is Ω β ≤ 1. Then, the support of Ω p C p (k ,j ) ≥ 0, υC˜ p (k ,j ) ≤ 1( k, j = 1, 2, . . . , n)}. The preference relations given by the DMs are pair-wise comparisons of alternatives on whole rather than on each attribute, which reflect opinions of DMs between alternatives. Cardinal˜ < 0,1> , i.e., the number of alternative pairs in ˜ < 0,1> | of Ω ity |Ω p p < 0,1> ˜ Ω , is at most Cn2 = n(n − 1)/2. Usually, the preference p ˜ p are of partial order. In some situations, relations given by Ω the DM would not be able to specify total preference relations. Namely, the DM only gives some pair-wise comparisons be˜ < 0,1> | < C 2 . tween alternatives, i.e., |Ω p n If the weighting vector ω and IS r ∗ are chosen by DMs already, then using (7) the DM DP can calculate the squares of ˜ < 0,1> the weighted Euclidean distance between each (k, j) ∈ Ω p p p ∗ < 0,1> ˜p and r as Sk and Sj , respectively. For each (k, j) ∈ Ω , xk is closer to the IS than xj if Sjp ≥ Skp . Therefore, the ranking order of xk and xj determined by Sjp and Skp based on (ω, r ∗ ) is consistent with the preference relation given by the DM DP . Conversely, if Sjp < Skp , then (ω, r ∗ ) is not chosen properly since it results in the ranking order of xk and xj determined by Skp and Sjp based on (ω, r ∗ ) being inconsistent with the preference relation given by DP . Thus, (ω, r ∗ ) should be chosen so that the ranking order of xk and xj determined by Skp and Sjp is consistent with the preference relation provided by the DMs.

In a similar way, an index (Sjp − Skp )+ to measure consistency can be defined as follows: ˜ Cp (k, j)(Sjp − Skp ), if Sjp ≥ Skp p p + (11) (Sj − Sk ) = 0, if Sjp < Skp . Obviously, (11) can be rewritten as (Sjp − Skp )+ = C˜p (k, j) max{0, Sjp − Skp }. Hence, an A-IF group consistency index is defined as follows: ˜= G

=

Q 



p=1

˜ p< 0 , 1 > (k ,j )∈Ω

Q 



(Sjp − Skp )+

[C˜p (k, j) max{0, Sjp − Skp }]. (12)

p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

˜ and G ˜ are A-IFSs since the DMs’ Remark 1: It is noted that B preference relations are given through using alternative comparisons with hesitancy degrees, whereas the inconsistency and consistency indices that are defined in the LINMAP [26] and fuzzy LINMAP [15], [19], [21], [22], [31] are real numbers. B. Atanassov’s Intuitionistic Fuzzy Mathematical Programming Models To determine the weighting vector and the IS, i.e., (ω, r ∗ ), the A-IF mathematical programming model may be constructed as follows: ˜ min{B} ˜ ˜ ˜≥h G−B s.t. ω∈Λ

(13)

˜ = μ˜ , υ˜  is an A-IFS given by the DMs a priori, and where h h h Λ is given as in Section III.

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˜ is made in the sense ˜−B ˜≥h Note that the constraint G of Definition 2. The aim of (13) is to minimize the A-IF group ˜ under the condition in which B ˜ is smaller inconsistency index B ˜ by the Athan or equal to the A-IF group consistency index G ˜ It is derived from (9)–(12) that IFS h. ˜−B ˜= G

Q 



[C˜p (k, j)(Sjp

−

Skp )].

−

1

+

(14)

−

p=1 (k ,j )∈Ω ˜ p< 0 , 1 > 0,1> ˜< , let λpk j = max{0, Skp − Sjp }. Then, For each (k, j) ∈ Ω p λpk j ≥ 0 and λpk j ≥ Skp − Sjp . Thus, (13) can be further transformed into the A-IF mathematical programming model as follows: ⎧ ⎫ Q ⎨ ⎬  p C˜p (k, j)λk j min ⎩ ⎭ < 0,1> p=1 (k ,j )∈Ω ˜p

⎧ Q   ⎪ ⎪ ˜ ⎪ [C˜p (k, j)(Sjp − Skp )] ≥ h ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω˜ p< 0 , 1 > ⎪ ⎨ s.t. S p − S p + λp ≥ 0 ((k, j) ∈ Ω ˜ < 0,1> , p = 1, 2, · · · , Q) p k kj ⎪ j ⎪ ⎪ ⎪ p ⎪ λ ≥ 0 ((k, j) ∈ Ω ˜ < 0,1> , p = 1, 2, . . . , Q) ⎪ p ⎪ kj ⎪ ⎩ ω ∈ Λ. (15) Let tpjk = Sjp − Skp and ⎧ μi = ωi μ∗i , υi = ωi υi∗ (i ∈ M1 ) ⎪ ⎪ ⎪ ⎨ ai = ωi a∗ , bi = ωi b∗ , ci = ωi c∗ , di = ωi d∗ (i ∈ M2 ) i i i i (16) ∗ ∗ ⎪ ei = ωi ei , gi = ωi gi (i ∈ M3 ) ⎪ ⎪ ⎩ zi = ωi zi∗ (i ∈ M4 ). Using (7), we have 1 1 p 2 p 2 ωi [(μpij )2 − (μpik )2 + (υij ) − (υik ) 2 i=1

i

tpjk =

p 2 p 2 + (πij ) − (πik ) + 2(μpij − μpik ) p p + 2(υij − υik )] −

i1 

p p μi [2(μpij − μpik ) + (υij − υik )]

i=1

−

i1 

p p υi [2(υij − υik ) + (μpij − μpik )]

i=1

+

i2 1  ωi [(apij )2 − (apik )2 + 2((bpij )2 6 i=i +1

−

(bpik )2 )

1

+ 2((cpij )2 − (cpik )2 )

+ ((dpij )2 − (dpik )2 )] −

i2 1  (ap − apik )ai 3 i=i +1 ij 1

−

i2 2  (bp − bpik )bi 3 i=i +1 ij 1

i2 i2 2  1  (cpij − cpik )ci − (dp − dpik )di 3 i=i +1 3 i=i +1 ij 1

i3 

1 2 i=i

p 2 p 2 ωi [(epij )2 − (epik )2 + (gij ) − (gik ) ]

2 +1

i3 

[(epij − epik )ei ] −

i=i 2 +1

+

m 

i3 

p p [(gij − gik )gi ]

i=i 2 +1 p 2 p 2 [ωi ((zij ) − (zik ) )] − 2

i=i 3 +1

m 

p p [(zij − zik )zi ].

i=i 3 +1

(17) Then, (15) can be written as the following A-IF mathematical programming model:

min

⎧ Q ⎨ ⎩



p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

⎧ Q  ⎪ ⎪ ⎨ s.t.

⎪ ⎪ ⎩

C˜p (k, j)λpk j

p=1



⎫ ⎬ ⎭

˜ tpjk C˜p (k, j) ≥ h

˜ p< 0 , 1 > (k ,j )∈Ω

(18)

˜ < 0,1> , p = 1, 2, . . . , Q) (λpk j , r, ω) ∈ Θ ((k, j) ∈ Ω p

where Θ = {(λpk j, r, ω)|tpjk + λpk j ≥ 0, λpk j ≥ 0 ((k, j) ∈ ˜ < 0,1> , p = 1, 2, . . . , Q), μi ≥ 0, υi ≥ 0, μi + υi ≤ ωi (i ∈ Ω p M1 ), 0 ≤ ai ≤ bi ≤ ci ≤ di ≤ ωi (i ∈ M2 ), 0 ≤ ei ≤ gi ≤ ωi (i ∈ M3 ), 0 ≤ zi ≤ ωi (i ∈ M4 ), ω ∈ Λ} represents the constraint set of the decision variables, and r = (r1 , r2 , . . . , rm ). Remark 2: Noting that μ∗i ≥ 0, υi∗ ≥ 0, μ∗i + υi∗ ≤ 1 (i ∈ M1 ), 0 ≤ a∗i ≤ b∗i ≤ c∗i ≤ d∗i ≤ 1 (i ∈ M2 ), 0 ≤ e∗i ≤ gi∗ ≤ 1(i ∈ M3 ), 0 ≤ zi∗ ≤ 1 (i ∈ M4 ) , and 0 ≤ ωi ≤ 1 (i = 1, 2, · · · , m), it is derived from (16) that μi ≥ 0, υi ≥ 0, 0 ≤ μi + υi ≤ ωi (i ∈ M1 ), 0 ≤ ai ≤ bi ≤ ci ≤ di ≤ ωi (i ∈ M2 ), 0 ≤ ei ≤ gi ≤ ωi (i ∈ M3 ), and 0 ≤ zi ≤ ωi (i ∈ M4 ), which should be imposed on (18). There are no such constraints in the linear programming model in the LINMAP [26]. Remark 3: In (18), the objective function, some constraints’ coefficients, and right-hand vector contain A-IFSs simultaneously. Thus, (18) is called an A-IF mathematical programming model with A-IFSs, whereas all the counterparts [21], [22], [31] are crisp linear programming models. Obviously, tpjk is the linear function of ω and r. There ex ˜ < 0,1> | + 2m − 2i1 + 2i2 + i3 unknown variables ist Q p=1 |Ωp  ˜ < 0,1> | + m − 3i1 + 2i2 + i3 + 1 inequaland at least Q p=1 |Ωp  ˜ < 0,1> |, the more ities in (18). Generally, the larger the Q p=1 |Ωp precise and reliable the determined weighting vector and the IS. C. Solution Method of Atanassov’s Intuitionistic Fuzzy Mathematical Programming Models  According Q p=1

to

Definition 1, the objective function C˜p (k, j)λpk j of (18) is an A-IFS, i.e.,

˜ p< 0 , 1 > (k ,j )∈Ω

WAN AND LI: ATANASSOV’S INTUITIONISTIC FUZZY PROGRAMMING METHOD

 1−

Q 



Level 1: According to Definition 2, usually the score function has higher priority than the accuracy function. Thus, according to (21), the nonlinear programming model at the first level can be easily constructed as follows:

p

(1 − μC˜ p (k ,j ) )λk j ,

p=1 (k ,j )∈Ω ˜ p< 0 , 1 > Q 





λpk j

(υC˜ p (k ,j ) )



and p=1 (k ,j )∈Ω˜ p< 0 , 1 > (18) is also an A-IFS, i.e.,  Q   1−

tpjk C˜p (k, j)

min z1 = 1 − ⎪ ⎩

Q 



p=1 (k ,j ) ∈Ω ˜ p< 0 , 1 >

+ t pj k

(1 − μC˜ p (k ,j ) )

Q 

,

t pj k

(υC˜ p (k ,j ) )

.

(20)

p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

To the best of our knowledge, there is no method to solve such a kind of A-IF mathematical programming models with A-IFSs. Therefore, we develop a method to solve (18) according to the ranking relation of A-IFSs based on the score and accuracy functions. Combining with (19) and (20) and Definition 2, (18) can be transformed into the bi-objective nonlinear programming model as follows: ⎧ Q ⎨   p (1 − μC˜ p (k ,j ) )λk j min z1 = 1 − ⎩ < 0,1> p=1 (k ,j )∈Ω ˜p

+

Q 



λpk j

(υC˜ p (k ,j ) )

p=1 (k ,j )∈Ω ˜ p< 0 , 1 >



(υC˜ p (k ,j ) )

⎫ ⎬ ⎭

λpk j

⎫ ⎪ ⎬ ⎪ ⎭

⎧ Q   p ⎪ ⎪ 1 − (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ p=1 (k ,j ) ∈Ω ⎪ ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   p ⎪ ⎪ ⎪ + (υC˜ p (k ,j ) )t j k ≥ μh˜ + υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j ) ∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎨ Q s.t. (22)   p ⎪ ⎪ 1 − (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ p=1 (k ,j ) ∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ p ⎪ ⎪ − (υC˜ p (k ,j ) )t j k ≥ μh˜ − υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j ) ∈Ω ⎪ ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎩ p 0,1> ˜< (λk j , r, ω) ∈ Θ ((k, j) ∈ Ω , p = 1, 2, . . . , Q). p

Using the existing software packages, we can solve (22), whose optimal objective value is denoted by z10 . Level 2: Combining with z10 , the nonlinear programming model at the second level can be constructed as follows: ⎧ Q ⎨   p (1 − μC˜ p (k ,j ) )λk j min z2 = 1 − ⎩ < 0,1> p=1 (k ,j )∈Ω ˜p



λpk j

(1 − μC˜ p (k ,j ) )

p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

−

Q 

p=1 (k ,j ) ∈Ω ˜ p< 0 , 1 >





p

(1 − μC˜ p (k ,j ) )λk j

of the first constraint in

p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

⎧ Q ⎨  min z2 = 1 − ⎩

⎧ ⎪ ⎨

(19)

p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

Q

305

Q 



p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

p

(υC˜ p (k ,j ) )λk j

−

⎫ ⎬

where z1 and z2 are the score and accuracy functions of the objective function of (18), respectively. In the sequent, a lexicographic method is developed to solve (21) in the sense of Pareto optimality [14].



p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

⎭

⎧ Q   ⎪ tp ⎪ ⎪ 1 − (1 − μ ˜ p (k ,j ) ) j k ⎪ C ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   p ⎪ ⎪ ⎪ + (υC˜ p (k ,j ) )t j k ≥ μh˜ + υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎨ Q s.t. (21)   p ⎪ ⎪ 1− (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   p ⎪ ⎪ ⎪ − (υC˜ p (k ,j ) )t j k ≥ μh˜ − υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎩ 0,1> ˜< (λpk j , r, ω) ∈ Θ ((k, j) ∈ Ω , p = 1, 2, . . . , Q) p

Q 

s.t.

λpk j

(υC˜ p (k ,j ) )

⎫ ⎬ ⎭

⎧ z1 ≤ z10 ⎪ ⎪ ⎪ ⎪ ⎪ Q ⎪   p ⎪ ⎪ ⎪ 1 − (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   p ⎪ ⎪ ⎪ + (υC˜ p (k ,j ) )t j k ≥ μh˜ + υh˜ ⎪ ⎪ ⎨ p=1 ˜ < 0,1>

(k ,j )∈Ω p (23) Q ⎪   ⎪ p ⎪ ⎪ 1− (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ < 0 , 1 > p=1 (k ,j )∈Ω ˜p ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ p ⎪ ⎪ ⎪ − (υC˜ p (k ,j ) )t j k ≥ μh˜ − υh˜ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ⎪ ˜ p< 0 , 1 > ⎪ ⎪ ⎩ p ˜ < 0,1> , p = 1, 2, · · · , Q). (λk j , r, ω) ∈ Θ ((k, j) ∈ Ω p

In (23), adding the constraint z1 ≤ z10 aims to improve z1 . This is the reason why (23) in the second level is introduced after (22). Using the existing software, we can solve (23). Combining with (16), an optimal solution (ω ∗ , r ∗ ) and the optimal value z2∗ of (23) are obtained. Putting (ω ∗ , r ∗ ) into the function z1 , the maximum value z1∗ of z1 is then obtained. It is not difficult to

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

prove that (ω ∗ , r ∗ , z1∗ , z2∗ ) is a Pareto optimal solution of (21). Thus, the attribute weighting vector ω ∗ is obtained as well as the IS r ∗ . Remark 4: The preference information structure Λ and the ˜ may affect the solution of (18). Generally, Λ and h ˜ A-IFS h may be different for specific decision problems. It is easy to see ˜ should at least from (22) and (23) that the choice of Λ and h assure that the feasible set of (22) is nonempty. D. Decision-Making Process of the Atanassov’s Intuitionistic Fuzzy Group Linear Programming Technique for Multidimensional Analysis of Preference Method Based on the aforementioned A-IF group LINMAP model, the algorithm and process of the heterogeneous MAGDM are summarized as follows. Step 1: Identify the evaluation attributes and incomplete weight information structure. Step 2: Express preference relations between alternatives with ˜ p (p = 1, 2, . . . , Q) . the A-IFSs Ω Step 3: Construct the decision matrices Ap and normalize them, i.e., obtain the normalized decision matrices Rp (p = 1, 2, . . . , Q) by using (4)–(6). Step 4: Construct the A-IF mathematical programming model according to (18). Step 5: Obtain the weighting vector ω and the IS r ∗ through solving the constructed model by using the A-IF mathematical programming method developed in Section IV-C. Step 6: Calculate the distances Sjp (j = 1, 2, . . . , n) of alternatives xj from the IS by using (7). Step 7: Determine the ranking orders of the alternatives for the DMs Dp (p = 1, 2, . . . , Q) according to the increasing orders of the distances Sjp (j = 1, 2, . . . , n) . Step 8: Determine the group ranking order of the alternatives and the best alternative by using social choice functions such as Borda’s function [14]. V. GENERALIZATIONS OF THE ATANASSOV’S INTUITIONISTIC FUZZY MATHEMATICAL PROGRAMMING MODEL Equation (18) has several possible generalizations or specializations as follows. 1) If all the weights ωi (i = 1, 2, . . . , m) are known and the purpose is to estimate only the IS, then (18) can still be used provided we dispose of the constraints ωi (i = 1, 2, . . . , m) and regard all weights ωi as known constants. 2) If the IS r ∗ is already known, then merely the weights ωi (i = 1, 2, . . . , m) have to be determined. In this case, using (1), the aforementioned analysis can be extended to the q-power of the weighted Minkowski distance between r pj and r ∗ as follows: Sjpq =

i1  ωi i=1

2

p p [(μpij − μ∗i )q + (υij − υi∗ )q + (πij − πi∗ )q ]

 i2  ωi + (apij − a∗i )q + 2(bpij − b∗i )q 6 i=i +1 1

+

2(cpij

−

c∗i )q

+

(dpij

−

d∗i )q



+

i3  ωi p p [(eij − e∗i )q + (gij − gi∗ )q ] 2 i=i +1 2

+

m 

p ωi (zij − zi∗ )q .

(24)

i=i 3 +1

¯ 1 = (ω1 , ω2 , . . . , ωi 1 )T , ω ¯ 2 = (ωi 1 +1 , Let us denote ω T ¯ ¯ 4 = (ωi 3 +1 , ωi 1 +2 , . . . , ωi 2 ) , ω 3 = (ωi 2 +1 , ωi 2 +2, . . . , ωi 3 )T, ω p p p = (μpij − μ∗i )q + (υij − υi∗ )q + (πij − ωi 3 +2 , . . . , ωm )T , δij p p p p p ∗ q ∗ q ∗ q ∗ q πi ) , σij = (aij − ai ) + 2(bij − bi ) +2(cij − ci ) +(dij − p p p = (epij − e∗i )q + (gij − gi∗ )q , ςijp = (zij − zi∗ )q , T pjk = d∗i )q , ξij q q q q q q p (δ1j − δ1k , δ2j − δ2k , . . . , δi 1 j − δi 1 k )T , H j k = (σip1 +1,j − σip1 +1,k , σip1 +2,j −σip1 +2,k , . . . , σip2 ,j −σip2 ,k )T , Δpjk = (ξip2 +1,j − ξip2 +1,k , ξip2 +2,j − ξip2 +2,k , . . . , ξip3 ,j − ξip3 ,k )T , and Φ pjk = p p T (ςip3 +1,j − ςip3 +1,k , ςip3 +2,j − ςip3 +2,k , . . . , ςm j − ςm k ) . pq pq p T 1 ¯1 + Then, using (24), it follows that Sj − Sk = 2 (T j k ) ω p T p T p T 1 1 ¯ ¯ ¯ 6 (H j k ) ω 2 + 2 (Δj k ) ω 3 + (Φ j k ) ω 4 . Equation (18) can be rewritten as follows:

min

⎧ Q ⎨ ⎩

p=1

 ˜ p< 0 , 1 > (k ,j )∈Ω

C˜p (k, j)λpk j

⎫ ⎬ ⎭

⎧ Q    ⎪ 1 p T 1 ⎪ ⎪ (T j k ) ω ¯ 1 + (H pjk )T ω ¯2 ⎪ ⎪ ⎪ 2 6 ⎪ p=1 (k ,j )∈Ω˜ < 0 , 1 > ⎪ p ⎪   ⎪ ⎪ ⎪ 1 ⎪ p T p T ˜ ˜ ⎪ ¯ ¯ C ω ω (Δ + ) + (Φ ) (k, j) ≥h ⎪ 3 4 p jk ⎪ 2 jk ⎪ ⎪ ⎪ ⎪ ⎨1 p T 1 ¯ 1 + (H pjk )T ω ¯2 s.t. 2 (T j k ) ω (25) 6 ⎪ ⎪ 1 ⎪ p p p T T ⎪ ¯ 3 + (Φ j k ) ω ¯ 4 + λk j ≥ 0 + (Δj k ) ω ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ˜ < 0,1> , p = 1, 2, . . . , Q) ((k, j) ∈ Ω ⎪ p ⎪ ⎪ ⎪ ⎪ p 0,1> ⎪ ˜< λk j ≥ 0 ((k, j) ∈ Ω , p = 1, 2, . . . , Q) ⎪ p ⎪ ⎪ ⎩ ω∈Λ which is an A-IF linear programming model that can still be solved through using the method developed in Section IV-C. It should be noted that simultaneous determination of the weights and the IS is not as simple for the weighted Minkowski distance. However, as mentioned previously, we are able currently to extend the method for joint determination of the weights and IS to the Hamming distance. 3) The analysis in Section IV-A assumes that every DM has equal confidence in each pair-wise comparison judgment. On the other hand, the DM may state his/her confidence on a scale that ranges from “little” confidence to “much” confidence [26]. The objective function of (18) and its constraints can be readily modified to incorporate such measures. 4) It is easy to see that Section IV-A assumes that all DMs have equal importance, i.e., weights. In some situations, however, we may believe that the DMs may have different weights. Suppose that the weighting vector of the DMs is w = (w1 , w2 , . . . , wQ )T ; then (18) can be appropriately

WAN AND LI: ATANASSOV’S INTUITIONISTIC FUZZY PROGRAMMING METHOD

modified to incorporate the weights of the DMs as follows: ⎫ ⎧ Q ⎬ ⎨  C˜p (k, j)λpk j ] [wp min ⎭ ⎩ < 0,1> p=1

˜p (k ,j )∈Ω

⎧ Q  ⎪ ⎪ ⎨ s.t.

⎪ ⎪ ⎩

p=1



˜ wp tpjk C˜p (k, j) ≥ h

˜ p< 0 , 1 > (k ,j )∈Ω

0,1> ˜< (λpk j , r, ω) ∈ Θ ((k, j) ∈ Ω , p = 1, 2, . . . , Q). p

5) If the objective function of (13) is modified to maximize ˜ as it may be the case in the fuzzy LINMAP [15], [31], then G (13) is changed as follows: ˜ max{G} ˜ ˜ ˜≥h G−B s.t. ω ∈ Λ.

Then, set zkm in = min{zk (x∗1 ), zk (x∗2 )} (k = 1, 2) . The linear membership function of zk can be defined as follows:

p=1 (k ,j )∈Ω ˜p

s.t.

⎪ ⎩



p=1 (k ,j )∈Ω ˜ p< 0 , 1 >

˜ [C˜p (k, j)(Sjp − Skp )] ≥ h

max {zk = zk (x)} ⎧ Q   ⎪ p ⎪ ⎪ 1 − (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q ⎪ p   ⎪ ⎪ + (υC˜ p (k ,j ) )t j k ≥ μh˜ + υh˜ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎨ s.t. (29) Q   ⎪ ⎪ t pj k ⎪ ⎪ 1− (1 − μC˜ p (k ,j ) ) ⎪ ⎪ ⎪ < 0,1> ⎪ p=1 ˜ (k ,j )∈Ω p ⎪ ⎪ ⎪ ⎪ Q p   ⎪ ⎪ ⎪ − (υC˜ p (k ,j ) )t j k ≥ μh˜ − υh˜ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎩ p ˜ < 0,1> , p = 1, 2, . . . , Q). (λk j , r, ω) ∈ Θ ((k, j) ∈ Ω p

(26)

Accordingly, (14) is changed as follows: ⎧ ⎫ Q ⎨ ⎬  max C˜p (k, j) max{0, Sjp − Skp } ⎩ ⎭ < 0,1> ⎧ Q  ⎪ ⎨

307

(27)

ω ∈ Λ.

Likewise, (18) is converted into the A-IF mathematical programming model as follows: ⎧ ⎫ Q ⎨ ⎬  C˜p (k, j)λpjk max ⎩ ⎭ < 0,1> p=1 (k ,j )∈Ω ˜p

⎧ Q  ⎪ ⎪ ˜ ⎪ tpjk C˜p (k, j) ≥ h ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ ⎪ p p 0,1> ⎪ ˜< −tj k + λj k ≥ 0 ((k, j) ∈ Ω , p = 1, 2, . . . , Q) ⎪ p ⎪ ⎪ ⎪ ⎪ p < 0,1> ⎨ ˜p λj k ≥ 0 ((k, j) ∈ Ω , p = 1, 2, . . . , Q) s.t. (28) ⎪ ⎪ μi ≥ 0, υi ≥ 0, μi + υi ≤ ωi (i ∈ M1 ) ⎪ ⎪ ⎪ ⎪ 0 ≤ ai ≤ bi ≤ ci ≤ di ≤ ωi (i ∈ M2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ ei ≤ gi ≤ ωi (i ∈ M3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ≤ zi ≤ ωi (i ∈ M4 ) ω∈Λ which is similarly solved by the method in Section IV-C. 6) In Section IV-C, the lexicographic method is proposed to solve (21). In fact, there are possible other methods to solve (21) such as the following three methods. The first is the max–max method. Let us denote the vector by x = (λpk j , r, ω T ); then the two objective functions zk (k = 1, 2) in (21) are the functions of the decision variable vector x, simply denoted by zk = zk (x). Let (zkm ax , x∗k ) be the optimal solution of the linear programming model as follows:

⎧ 0, if zk < zkm in ⎪ ⎨ μz k (x) = (zk − zkm in )/(zkm ax − zkm in ), if zkm in ≤ zk ≤ zkm ax ⎪ ⎩ 1, if zk > zkm ax . (30) According to the fuzzy multiobjective decision-making method [14], setting μ = 12 [max{μz 1 (x), μz 2 (x)} + 12 (μz 1 (x) + μz 2 (x))], then (21) can be transformed into the linear programming model as follows:

max {μ} ⎧ 2μz k + (μz 1 + μz 2 ) ≤ 4μ (k = 1, 2) ⎪ ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ p ⎪ ⎪ (1 − μC˜ p (k ,j ) )t j k 1 − ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ p ⎪ ⎪ + (υC˜ p (k ,j ) )t j k ≥ μh˜ + υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎨ (31) s.t. Q ⎪   ⎪ ⎪ t pj k ⎪ ⎪ 1− (1 − μC˜ p (k ,j ) ) ⎪ ⎪ ⎪ < 0,1> ⎪ p=1 ˜ (k ,j )∈Ω p ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ tp ⎪ ⎪ − (υ ˜ p (k ,j ) ) j k ≥ μh ˜ − υh ˜ ⎪ C ⎪ ⎪ < 0,1> ⎪ p=1 ˜ (k ,j )∈ Ω ⎪ p ⎪ ⎪ ⎪ ⎩ p ˜ < 0,1> , p = 1, 2, . . . , Q). (λk j , r, ω) ∈ Θ ((k, j) ∈ Ω p

The second is the max–min method. Setting μ = 1 1 2 [min{μz 1 (x), μz 2 (x)} + 2 (μz 1 (x) + μz 2 (x))], then (21) can be transformed into the linear programming model as follows:

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max {μ} ⎧ 2μz k + (μz 1 + μz 2 ) ≥ 4μ (k = 1, 2) ⎪ ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ p ⎪ ⎪ (1 − μC˜ p (k ,j ) )t j k 1 − ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ p ⎪ ⎪ + (υC˜ p (k ,j ) )t j k ≥ μh˜ + υh˜ ⎪ ⎪ ⎪ ⎨ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > s.t. (32) ⎪ Q ⎪   ⎪ p ⎪ ⎪ ⎪ 1− (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   p ⎪ ⎪ ⎪ − (υC˜ p (k ,j ) )t j k ≥ μh˜ − υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎩ p 0,1> ˜< (λk j , r, ω) ∈ Θ ((k, j) ∈ Ω , p = 1, 2, . . . , Q). p The third is the linear weighted average method. Equation (21) can be transformed into the linear programming model as follows: max {λ1 μz 1 (x) + λ2 μz 2 (x)} ⎧ Q   ⎪ p ⎪ ⎪ 1 − (1 − μC˜ p (k ,j ) )t j k ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   ⎪ p ⎪ ⎪ + (υC˜ p (k ,j ) )t j k ≥ μh˜ + υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎨ Q s.t. (33)   ⎪ t pj k ⎪ ⎪ 1 − (1 − μ ) ⎪ C˜ p (k ,j ) ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎪ ⎪ Q ⎪   p ⎪ ⎪ ⎪ − (υC˜ p (k ,j ) )t j k ≥ μh˜ − υh˜ ⎪ ⎪ ⎪ ⎪ p=1 (k ,j )∈Ω ˜ p< 0 , 1 > ⎪ ⎪ ⎩ p ˜ < 0,1> , p = 1, 2, . . . , Q) (λk j , r, ω) ∈ Θ ((k, j) ∈ Ω p where λ = (λ1 , λ2 )T is the weight vector of zk (k = 1, 2), which satisfies the conditions λk ≥ 0 (k = 1, 2) and λ1 + λ2 = 1.

VI. SUPPLIER SELECTION EXAMPLE AND COMPARISON ANALYSIS OF THE OBTAINED RESULTS In this section, a supplier selection example is used to illustrate the proposed method, and then the comparison analysis of the obtained results is conducted. A. Supplier Selection Problem and Analysis There is a company who manufactures automobile parts. The company desires to select a suitable supplier to purchase the key components of its new product. The company invites three experts (i.e., DMs) D1 , D2 , and D3 to form the component purchase group. After preliminary screening, five candidate suppliers x1 , x2 , x3 , x4 , and x5 remain for further evaluation. All

the DMs of the purchase group agree to evaluate these suppliers with the following six attributes, including research and development capability f1 , product quality f2 , technological level f3 , flexibility f4 , delivery time f5 , and price f6 , where the former five attributes are qualitative. The assessments (or ratings) of the suppliers on f1 are divided into two parts: satisfaction degrees and dissatisfaction degrees, which are just the membership degrees and nonmembership degrees of A-IFSs, respectively, i.e., the ratings of the suppliers on f1 can be expressed with A-IFSs. The assessments (or ratings) of the suppliers on f2 and f3 are represented by TrFNs. Due to the uncertainty of the product process, it is better to use the intervals to represent the flexibility f4 and the delivery time f5 . The assessments (or ratings) of the suppliers on f6 can be represented by real numbers. The ratings of all the suppliers on the attributes are given by the three experts as the matrices (A1 ), (A2 ), and (A3 ) as shown at the bottom of the next page. According to the experts’ comprehensions and judgments, the experts provide their preference relations between alternatives ˜ 1 = {(1, 2), C˜1 (1, 2), (3, 1), C˜1 (3, 1), (4, 5), as follows: Ω C˜1 (4, 5), (5, 2), C˜1 (5, 2), (2, 3), C˜1 (2, 3), (4, 3), C˜1 (4, 3)}, ˜ 2 = {(2, 1), C˜2 (2, 1), (3, 2), C˜2 (3, 2), (4, 3), C˜2 (4, 3), Ω ˜ 3 = {(1, 3), C˜3 (1, 3), (2, 5), C˜3 (2, (5, 4), C˜2 (5, 4)},and Ω 5), (4, 5), C˜3 (4, 5), (3, 2), C˜3 (3, 2), (4, 3), C˜3 (4, 3)}, where C˜1 (1, 2) = 0.7, 0.2, C˜1 (3, 1) = 0.6, 0.3, C˜1 (4, 5) = 0.4, 0.5, C˜1 (5, 2) = 0.3, 0.4,C˜1 (2, 3) = 0.5, 0.2,C˜1 (4, 3) = 0.6, 0.2, C˜2 (2, 1) = 0.55, 0.34, C˜2 (3, 2) = 0.35, 0.48, C˜2 (4, 3) = 0.65, 0.2 , C˜2 (5, 4) = 0.42, 0.36, C˜3 (1, 3) = 0.60, 0.28, C˜3 (2, 5) = 0.46, 0.34, C˜3 (4, 5) = 0.72, 0.15, C˜3 (3, 2) = 0.41, 0.45, and C˜3 (4, 3) = 0.28, 0.62. Thus, ˜ 2 , and Ω ˜ 3 are Ω ˜ < 0,1> = {(1, 2), (3, 1), ˜1, Ω the supports of Ω 1 < 0,1> ˜ = {(2, 1), (3, 2), (4, 3), (4, 5), (5, 2), (2, 3), (4, 3)}, Ω 2 ˜ < 0,1> = {(1, 3), (2, 5), (4, 5), (3, 2), (4, 3)}, (5, 4)}, and Ω 3 respectively. The preference information structure Λ of attributes’ importance given by the experts is given as Λ = {ω ∈ Λ0 |3ω1 ≥ ω2 , 0.01 ≤ ω2 − ω3 ≤ 0.2, 0.25 ≤ ω4 ≤ 0.45, ω4 − ω5 ≤ ω1 − ω2 }. Using (4)–(6), A1 , A2 , and A3 can be normalized into (R1 ), (R2 ), and (R3 ) as shown at the top of page 11. ˜ = 0.0001, 0.9, we can construct According to (18), taking h the A-IF programming model with A-IFSs, whose optimal solution can be obtained through using the method that is developed in Section IV-C, where some of its components are obtained as follows: λ112 = λ145 = λ131 = λ143 = 0, λ123 = 0.0000072, λ152 = 1.6528,λ221 = λ243 = λ232 = 0, λ254 = 0.0346,λ313 = λ325 = λ345 = λ332 = λ343 = 0, μ1 = 0.04125, v1 = 0.03407, a2 = b2 = 0, c2 = d2 = 0.06, a3 = b3 = c3 = d3 = 0.30996, e4 = g4 = 0.24184,e5 = 0.00808, g5 = 0.10265, z6 = 0, ω1 = 0.18,ω2 = 0.06,ω3 = 0.05,ω4 = 0.3167, and ω5 = ω6 = 0.1967. Using (16), the IS r ∗ = (r1∗ , r2∗ , . . . , r5∗ ) can be calculated, where r1∗ = 0.2292, 0.1893, r2∗ = (0, 0, 1, 1), r3∗ = (0.6199, 0.6199,0.6199, 0.6199), r4∗ = [0.7637, 0.7637], r5∗ = [0.0411, 0.5219], and r6∗ = z6∗ = 0. Then, using (7), the squares of the distances of the suppliers from the IS r ∗ for the experts (or DMs) D1 , D2 , and D3 can be computed as S11 = 0.0509,S21 = 0.0527, S31 = 0.0450, S41 = 0.0367, S51 = 0.1132, S12 = 0.0538,

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TABLE I BORDA’S SCORES OF THE SUPPLIERS FOR EXPERTS

S22 = 0.0404, S32 = 0.0649, S42 = 0.0225, S52 = 0.0613, S13 = 0.0271, S23 = 0.0412, S33 = 0.0345, S43 = 0.0366, and S53 = 0.0757, respectively. The ranking orders of the five suppliers for the three experts D1 , D2 , and D3 are generated as follows: x4  x3  x1  x2  x5 , x4  x2  x1  x5  x3 , and x1  x3  x4  x2  x5 , respectively. Borda’s scores of the five suppliers can be obtained as in Table I. The ranking order of the five suppliers is generated as x4  x1  x2 ∼ x3  x5 . Therefore, the best selection is supplier x4 . As far as this supplier selection example is concerned, we do ˜ = μ˜ , υ˜ . the sensitivity analysis about the A-IFS h h h 1) When υh˜ = 0.9 is fixed and μh˜ takes any value from the interval [0.0001, 0.001], the obtained weighting vector and IS remain unchanged, i.e., ω = (0.18,0.06, 0.05,0.3167, 0.1967, 0.1967)T and r ∗ = (0.2292, 0.1893, (0, 0, 1, 1), (0.6199, 0.6199, 0.6199, 0.6199), [0.7637, 0.7637], [0.0411, 0.5219], 0). However, when μh˜ takes any value from the interval (0.001, 0.1], the obtained weighting vector and IS are changed to ω = (0.18, 0.06, 0.05, 0.2564, 0.2268, 0.2268)T and r ∗ = (0.0544, 0.3639, (0, 0, 0.6884, 1), (1, 1, 1, 1), [0.8145, 0.8145], [0.2178, 0.2178], 0). ⎛

1 A1 = (yij )6×5

2 A2 = (yij )6×5

and

⎜ ⎜ ⎜ ⎜ (70, 90, 91, 92) =⎜ ⎜ [4, 10] ⎜ ⎝ [65, 88] 119 ⎛ 0.4, 0.5 ⎜ (5, 6, 7, 8) ⎜ ⎜ ⎜ (80, 85, 90, 95) =⎜ ⎜ [4, 7] ⎜ ⎝ [75, 88] 120 ⎛

3 A3 = (yij )6×5

0.5, 0.3 (3, 4, 5, 6)

0.2, 0.3 ⎜ (5, 6, 7, 8) ⎜ ⎜ ⎜ (72, 80, 90, 95) =⎜ ⎜ [6, 8] ⎜ ⎝ [75, 89] 111

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2) When μh˜ = 0.0001 is fixed and υh˜ takes any value from the interval (0.80, 0.9], the obtained weighting vector and IS also remain unchanged. However, when the value of υh˜ is less than or equal to 0.80, the obtained ω and r ∗ change. For example, taking υh˜ = 0.8, we obtain ω = (0.18, 0.06, 0.05, 0.25, 0.23, 0.23)T and r ∗ = (0, 0.2445, (0, 0, 0.7751, 1), (0.5889, 0.5889, 0.5889, 0.5889), [0.8583, 0.9916], [0.2317, 0.2317], 0). The analysis stated previously shows that ω and r ∗ are not ˜ = μ˜ , υ˜  is sensitive to μh˜ . Due to the fact that the A-IFS h h h given a priori, in general, μh˜ should not be too great and take the value less than 0.1. Meanwhile, υh˜ should not be too small and take the value more than 0.80. B. Comparison Analysis With Other Similar Methods The results that are obtained by the method proposed in this paper and other similar methods with crisp truth degrees [21], [22], [31] are compared in this section. 1) As stated earlier, real numbers and intervals can be rewrit˜p, p = ten as TrFNs. Thus, if all C˜p (k, j) = 1 ((k, j) ∈ Ω 1, 2, · · · , Q), F1 = ∅, and Q = 1, then (18) is reduced to the linear programming model constructed in [31]. Namely, the fuzzy LINMAP [31] is a special case of the method that is proposed in this paper. 2) If all the TFNs [21], [22] are rewritten as TrFNs, then the linear programming models that are constructed in [21], [22] are also special cases of (18) with F1 = F3 = F4 = ∅ and C˜p (k, j) ˜ p , p = 1, 2, . . . , Q). = 1 ((k, j) ∈ Ω In the aforementioned supplier selection example, if 0,1> ˜< we assume that C˜p (k, j) = 1 for all (k, j) ∈ Ω p (p = 1, 2, 3) and h = 0.0001, then according to (18), we can

0.6, 0.2 (6, 7, 8, 9)

0.4, 0.4 (5, 6, 7, 8)

0.7, 0.1 (1, 2, 3, 4)

(30, 80, 85, 90) [7, 9]

(50, 60, 75, 85) [4, 9]

(75, 80, 85, 95) [6, 10]

[87, 90] 110

[45, 58] 120

[70, 90] 118

0.3, 0.6 (2, 3, 4, 5)

0.3, 0.3 (3, 4, 5, 6)

0.6, 0.3 (1, 2, 3, 4)

(50, 60, 75, 85) [5, 8]

(30, 80, 85, 90) [3, 6]

(75, 80, 85, 95) [7, 9]

[87, 90] 118

[65, 58] 115

[66, 87] 108

0.5, 0.4 (3, 5, 6, 7) (50, 60, 75, 85)

0.6, 0.2 (4, 5, 6, 7) (74, 80, 82, 85)

0.7, 0.1 (4, 5, 8, 9) (65, 70, 78, 81)

[6, 8] [82, 90] 116

[7, 10] [78, 86] 110

[5, 7] [66, 78] 120

0.3, 0.6 (2, 3, 4, 5)

⎞

⎟ ⎟ ⎟ (80, 85, 90, 95) ⎟ ⎟ ⎟ [2, 8] ⎟ ⎠ [92, 95] 100 ⎞ 0.2, 0.7 (6, 7, 8, 9) ⎟ ⎟ ⎟ (70, 90, 91, 92) ⎟ ⎟ ⎟ [8, 10] ⎟ ⎠ [89, 95] 119 ⎞ 0.5, 0.3 (1, 4, 6, 7) ⎟ ⎟ ⎟ (82, 84, 89, 92) ⎟ ⎟ ⎟ [3, 6] ⎟ ⎠ [65, 90] 105

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⎛

0.5, 0.3 ⎜ [0.333, 0.444, 0.556, 0.667] ⎜ ⎜ ⎜ [0.737, 0.945, 0.958, 0.968] 1 R =⎜ ⎜ [0.4, 1.0] ⎜ ⎜ ⎝ [0.0737, 0.3158] 0.0083

0.6, 0.2

0.4, 0.4

[0.667, 0.778, 0.889, 1] [0.316, 0.842, 0.895, 0.947] [0.7, 0.9]

[0.556, 0.667, 0.778, 0.889] [0.526, 0.632, 0.79, 0.895] [0.4, 0.9]

[0.0526, 0.0842] 0.0083

[0.3895, 0.5263] 0 0.7, 0.1

0.3, 0.6

[0.222, 0.333, 0.444, 0.556] ⎟ ⎟ ⎟ [0.842, 0.895, 0.947, 1] ⎟ ⎟ ⎟ [0.2, 0.8] ⎟ ⎟ ⎠ [0, 0.0316]

[0.111, 0.222, 0.333, 0.444] [0.790, 0.842, 0.895, 0.947] [0.6, 1.0] [0.0526, 0.2632] 0.0167

⎛

0.4, 0.5 ⎜ [0.556, 0.667, 0.778, 0.889] ⎜ ⎜ ⎜ [0.842, 0.895, 0.947, 1] 2 R =⎜ ⎜ [0.4, 0.7] ⎜ ⎜ ⎝ [0.0737, 0.2105] 0

0.1667

0.3, 0.6

0.3, 0.3

[0.222, 0.333, 0.444, 0.556] [0.556, 0.667, 0.778, 0.889] [0.5, 0.8]

[0.333, 0.444, 0.556, 0.667] [0.316, 0.842, 0.895, 0.947] [0.3, 0.6]

[0.0526, 0.0842] 0.0167

[0.0737, 0.3158] 0.0417 0.6, 0.3

0.2, 0.7

[0.0842, 0.3053] 0.1 ⎛

0.2, 0.3 ⎜ [0.556, 0.667, 0.778, 0.889] ⎜ ⎜ ⎜ [0.758, 0.842, 0.947, 1] 3 R =⎜ ⎜ [0.4, 1.0] ⎜ ⎜ ⎝ [0.0737, 0.3158] 0.0083

0.0083

0.5, 0.4 [0.333, 0.556, 0.667, 0.778]

0.6, 0.2 [0.444, 0.556, 0.667, 0.778]

[0.526, 0.632, 0.79, 0.842] [0.7, 0.9]

[0.779, 0.842, 0.863, 0.895] [0.4, 0.9]

[0.0526, 0.0842] 0.0833

[0.3895, 0.5263] 0 0.7, 0.1 [0.444, 0.556, 0.889, 1] [0.684, 0.737, 0.821, 0.856] [0.6, 1.0] [0.0526, 0.2632] 0.0167

construct the corresponding linear programming model in which the optimal solution can be obtained through using the simplex method of linear programming, where some of its components are ω = (0.18, 0.06, 0.05, 0.25, 0.23, 0.23)T and r ∗ = (0.2647, 0.3316, (0, 0, 1, 1), (1, 1, 1, 1), [ 0.5740, 0.9013 ], [0.2005, 0.2005], 0.0111). Analogously, Borda’s scores of the five suppliers can also be obtained as in Table II. Hereby, the ranking order of the five

⎞

[0.667, 0.778, 0.889, 1] ⎟ ⎟ ⎟ [0.737, 0.947, 0.958, 0.968] ⎟ ⎟ ⎟ [0.8, 1.0] ⎟ ⎟ ⎠ [0, 0.0632]

[0.111, 0.222, 0.333, 0.444] [0.79, 0.842, 0.895, 1] [0.7, 0.9]

and

⎞

⎞ 0.5, 0.3 [0.111, 0.444, 0.667, 0.778] ⎟ ⎟ ⎟ [0.863, 0.884, 0.936, 0.968] ⎟ ⎟ ⎟ [0.2, 0.8] ⎟ ⎟ ⎠ [0, 0.0316] 0.1667

suppliers is x4 ∼ x1  x2  x3  x5 and the best selection is suppliers x4 and x1 . These results are little different from those obtained previously. Putting (ω , r ∗ ) into (10) and (12) with C˜p (k, j) = 1 for 0,1> ˜< (p = 1, 2, 3), we can get the group inall (k, j) ∈ Ω p consistency index B = 0.0570 and consistency index G = 0.2143. Namely, the purchase group accepts the ranking order x4 ∼ x1  x2  x3  x5 with the inconsistency B =

WAN AND LI: ATANASSOV’S INTUITIONISTIC FUZZY PROGRAMMING METHOD

TABLE II BORDA’S SCORES OF THE SUPPLIERS FOR EXPERTS

0.0570 and the consistency G = 0.2143. Putting (ω, r ∗ ) into (10) and (12), we can get the A-IF group inconsistency in˜ = 0.4558, 0.2123 and the consistency index G ˜= dex B 0.0730, 0.8974. This means that the purchase group accepts the ranking order x4  x1  x2 ∼ x3  x5 with the in˜ = 0.4558, 0.2123 and the consistency G ˜= consistency B 0.0730, 0.8974. That is to say, there exists some hesitancy degree. Namely, the satisfaction degree of the inconsistency index is 0.4558 and the dissatisfaction degree is 0.2123, whereas the hesitancy degree is 0.3319. Likewise, the satisfaction degree of the consistency index is 0.073 and the dissatisfaction degree is 0.8974, whereas the hesitancy degree is 0.0296. These pieces of information may be useful for the DMs to make rational and right decisions.

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Compared with the possibilistic LINMAP [25], the method that is proposed in this paper has the following features and advantages. 1) In this paper, A-IFSs are utilized to represent the truth degrees, which may reflect well the hesitancy nature of subjective judgments on comparisons of alternatives, whereas, in [25], TFNs are used to express the truth degrees. Furthermore, the MAGDM problems that are studied in this paper involve multiple types of attribute ratings, whereas only single type of attribute ratings (i.e., TFNs) was considered in [13] and [25]. 2) To estimate the IS and weighting vector, the constructed model [i.e., (18)] is an A-IF mathematical programming model with A-IFSs, which is solved through developing a new effective method which is based on the score and accuracy functions of A-IFSs. Moreover, the developed method is applicable to more general A-IF mathematical programming models with A-IFSs. However, there is no new method proposed in [13] and [25]. 3) If the aforementioned mistakes in [25] are revised, all the TFNs are rewritten as TrFNs, and the fuzzy truth degrees are expressed as A-IF truth degrees, then [25, eq. (9)] is still a special case of (18) constructed in this paper with F1 = F3 = F4 = ∅. VII. CONCLUSION

C. Errors and Issues of the Possibilistic Linear Programming Technique for Multidimensional Analysis of Preference and Comparison Analysis It is found that there are some big errors and issues in the definitions, notations, and operations and models in the possibilistic LINMAP [25]. 1) The decision variables λpk l in [25, eqs. (9) and (11)] should be revised as λk l . In fact, Sadi-Nezhad and Akhtari (e.g., [25, eq. (7)]) defined the fuzzy distance betweenan alternative Ai and a∗1 , a ˜∗2 , . . . , a ˜∗n ) as Si = nj=1 [ωj (˜ xij − a ˜∗j )2 ], the IS a ˜∗ = (˜ ∗ ∗ ∗ ∗ ˜j = (aj L , aj M , aj R ) are where ωj = (ωj L , ωj M , ωj R ) and a TFNs, and x ˜ij = (aij L , aij M , aij R ) is the TFN of an alternative Ai on the jth attribute. Obviously, Si is not related to the DM Dp . Hence, λk l = max {0, Sk − Sl } is independent of Dp . 2) According to the operations of TFNs [9], Si is a TFN since ˜ij , and a ˜∗j (j = 1, 2, . . . , n) are TFNs. Thus, Sl − Sk all ωj , x is a TFN. Hence, all λk l in [25, eqs. (9) and (11)] should be nonnegative TFNs instead of nonnegative real numbers. In[25, eq. (9)], h is a positive constant, which results in 3) Q ˜ p (Sl − Sk ) being a real number and equal to h. p=1 (k ,l)∈Ω   Clearly, in fact, Q ˜ p (Sl − Sk ) is a TFN. Thus, h p=1 (k ,l)∈Ω should be a TFN instead of a real number. There is a similar error in the constraint in [25, eq. (11)]. 4) All the inequalities Sk − Sl + λpk l ≥ 0 in [25, eq. (9)] are wrong. These inequalities should be Sl − Sk + λpk l ≥ 0 since λk l = max {0, Sk − Sl } is the inconsistency index. Otherwise, we obtain λk l = max{0, Sl − Sk }, which is the consistency in  p ˜ dex. Thus, min { Q ˜ p Cp (k, l)λk l } in [25, eq. (9)] p=1 (k ,l)∈Ω means minimizing the total consistency index of the group, which is not rational.

In the real decision process, DMs often have some hesitancy degrees of alternative comparisons. Hence, the heterogeneous MAGDM problems considering hesitancy degrees of alternative comparisons are very interesting and common in real situations. A new A-IF group LINMAP has been developed to solve the aforementioned heterogeneous MAGDM problems with incomplete information. In this method, A-IFSs are used to express hesitancy degrees of alternative comparisons. Under some conditions, the proposed method may be reduced to the classical LINMAP [26], fuzzy LINMAP [15], [19], [21], [22], [31], and the possibility LINMAP [25]. Namely, the aforementioned methods are a special case of the method proposed in this paper. Therefore, the method proposed in this paper is of the flexibility and universality. In addition, this paper also gives a particular answer to the open problem of how to study the notion of A-IF linear programming models, which was proposed by Atanassov [2], [3]. Although the method proposed in this paper has been illustrated with the supplier selection example, it is also applicable to group decision-making problems in many areas such as water resource management [7], [23], [30], performance evaluation and environment as well as flood control. The parameters in the constructed models may affect the final decision results. Therefore, combining with updated optimization methods [37], effective parameterized methods (especially analytical methods) will be studied in the future. REFERENCES [1] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 50, pp. 87–96, 1986.

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Shu-Ping Wan was born in 1974. He received the Ph.D. degree in control theory and control engineering from Nankai University, Nankai, Tianjin, China, in 2005. He is currently a Professor with the College of Information Technology, Jiangxi University of Finance and Economics, Nanchang, Jiangxi, China. He has contributed more than 80 papers to professional journals. His current research interests include decision analysis, fuzzy game theory, information fusion, and financial engineering.

Deng-Feng Li was born in 1965. He received the B.Sc. and M.Sc. degrees in applied mathematics from the National University of Defense Technology, Changsha, China, in 1987 and 1990, respectively, and the Ph.D. degree in system science and optimization from the Dalian University of Technology, Dalian, China, in 1995. From 2003 to 2004, he was a Visiting Scholar with the School of Management, University of Manchester Institute of Science and Technology, Manchester, U.K. He is currently a “Minjiang Distinguished” Professor with the School of Management, Fuzhou University, Fuzhou, China. He has authored or co-authored more than 300 journal papers and five monographs. He has co-edited one proceeding of the international conference. His current research interests include fuzzy decision analysis, group decision making, fuzzy game theory, supply chain, fuzzy sets and system analysis, fuzzy optimization, and differential game. Dr. Li has won more than 20 academic achievements and awards, such as the Chinese State Natural Science Award and the 2013 IEEE Computational Intelligence Society IEEE Transactions on Fuzzy Systems Outstanding Paper Award.