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Knowledge-Based Systems 80 (2015) 131–142

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A survey of approaches to decision making with intuitionistic fuzzy preference relations Zeshui Xu, Huchang Liao ⇑ Business School, Sichuan University, Chengdu, Sichuan 610065, China

a r t i c l e

i n f o

Article history: Received 16 October 2014 Received in revised form 17 December 2014 Accepted 30 December 2014 Available online 9 January 2015 Keywords: Intuitionistic fuzzy preference relations Consistency Consensus measures Decision making Priority vector derivation methods

a b s t r a c t Intuitionistic fuzzy preference relations (IFPRs) have attracted more and more scholars’ attentions in recent years due to their efﬁciency in representing experts’ imprecise cognitions. With IFPRs, people can express their opinions over different pairs of alternatives from positive, negative and hesitative points of view. This paper presents a comprehensive survey on decision making with IFPRs with the aim of providing a clear perspective on the originality, the consistency, the prioritization, and the consensus of IFPRs. Finally, some directions for future research are pointed out. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Decision making takes place everywhere and every time in our daily life. Before making any decision, the ﬁrst thing we should do is to collect sufﬁcient information related to the considered problem. In many cases, the information is determined according to the decision makers’ opinions or the experts’ assessments. Therefore, how to describe the group members’ opinions is very important and it inﬂuences the ﬁnal decision result directly as they often are only able to express their opinions roughly and subjectively. Generally speaking, there are mainly three ways in which the decision makers or the experts can express their opinions: preference orderings, utility values, and preference relations. Preference orderings are a collection of natural numbers which are a permutation of (1, 2, . . ., n) used by the experts for showing the order positions of a set of alternatives in sequence [64]. Utility values are a series of exact real numbers taken from a closed unit interval [0,1] to indicate the preferences of a decision maker towards different outcomes [8]. Preference relation is constructed via pairwise comparisons over the alternatives, and each value in it indicates the preference degree or intensity of one alternative over another [37]. Comparing these three representation tools, the preference orderings are oversimpliﬁed because they contain little information about the experts’ preferences, which makes it inconvenient or impossible for further investigation especially when a group of ⇑ Corresponding author. Tel.: +86 28 85414255. E-mail addresses: [email protected] (Z. Xu), [email protected] (H. Liao). http://dx.doi.org/10.1016/j.knosys.2014.12.034 0950-7051/Ó 2015 Elsevier B.V. All rights reserved.

experts cannot reach a mutually agreeable result. The utility values of the alternatives are sometimes very difﬁcult to be determined. In addition, as pointed out by Winkler [54], utility theory is lacking as a descriptive theory of how people actually behave if left their own devices. Such a descriptive theory will never be prescriptively appealing [53]. However, the preference relations do not have these limitations. It can express an expert’s judgments subjectively according to his/her cognition. With the preference relations, there is no need for the experts to determine the crisp utility values of alternatives over each criterion. Basically, there are two types of preference relations, which are the multiplicative preference relations and the fuzzy preference relations. Based on the multiplicative preference relations, the famous analytic hierarchy process (AHP) method was proposed [37]. In classical AHP method, the 1–9 scale is used as fundamental scale to represent judgments in forms of pairwise comparisons and thus a multiplicative preference relation should be constructed. As all the judgments in a multiplicative preference relation are crisp values which are hard to be exactly furnished in many complex and uncertain cases, a fuzzy preference relation (FPR) was then introduced [36]. The FPR uses a number from the interval [0,1] to characterize the degree of certainty in the preference between a pair of alternatives. A FPR may arise when each expert is not unambiguously certain as to Ai > Aj , or different experts have different opinions as to Ai > Aj in which case a fraction of the number of experts having voted for Ai > Aj is taken as a degree of Ai > Aj [50]. Tanino [50] ﬁrstly gave the formal deﬁnition of FPR as a fuzzy binary relation matrix satisfying reciprocal condition and max–min transitivity. Although the

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multiplicative preference relations and the fuzzy preference relations have attracted signiﬁcant attention of many scholars due to their efﬁciency, there are still some weaknesses on the fuzzy preference relations and the multiplicative preference relations. Both of them consider only the preference degrees or intensities of one alternative against another. In many cases, if some of the experts are not very familiar with the decision making problem or there exists some incomplete information about some of the alternatives, it might be very difﬁcult for the experts to determine such preference degrees. In 1983, Atanassov [1] extended the traditional fuzzy set into the intuitionistic fuzzy set by considering the degrees of hesitancy. With this type of representation technique, when comparing pairs of alternatives, the experts can express their imprecise cognitions from the positive, negative and hesitative points of view and thus construct an intuitionistic fuzzy preference relation (IFPR) [48,66]. ~ as a preference structure, whose eleXu [66] deﬁned an IFPR R ments are intuitionistic fuzzy numbers (IFNs), denoted as ~rij ¼ ðlij ; v ij ; pij Þ with lij ; v ij 2 ½0; 1; lij þ v ij 6 1; lij ¼ v ji ; lii ¼ v ii ¼ 0:5, for all i; j ¼ 1; 2; . . . ; n; l ¼ 1; 2; . . . ; s. lij means the preference degree of the alternative Ai to Aj ; v ij indicates the nonpreference degree of the alternative Ai to Aj , and pij ¼ 1 lij v ij is interpreted as an indeterminacy degree or a hesitancy degree. As an IFPR can express the opinions of an expert in terms of ‘‘preferred’’, ‘‘not preferred’’, and ‘‘indeterminate’’ aspects, it is more comprehensive and ﬂexible than the fuzzy preference relation and the multiplicative preference relation in expressing an expert’s preferences. Montero and Gomez [34] viewed the preference modeling as a classiﬁcation problem, which is deeply related to intuitionistic fuzzy sets. The IFPR has been implemented into many different aspects. After giving the formal deﬁnition of IFPR, Xu [66] applied it into the process of assessing a set of agroecological regions in Hubei Province, China. In this example, Hubei province was divided into seven agroecological regions and three experts were asked to prioritize them with respect to their comprehensive functions. After establishing the integral processes of intuitionistic fuzzy AHP (IFAHP) method, Xu and Liao [67] then applied this method into the global supplier development problem in which the experts provided their assessments in terms of IFPRs. Xu [62] also used the IFPR to aid the customer to buy a refrigerator. In this example, the evaluator compared each pair of refrigerators and constructed an IFPR to represent the opinions of the evaluator over these refrigerators. The linear programming method were utilized to produce the priorities of these refrigerators. Based on the additive consistency as well as the least squares optimization method and the goal programming method, Gong et al. [23] used the IFPR to analyze and assess the industry meteorological service for China Meteorological Administration. On the other hand, according to the multiplicative consistency of IFPR, Gong et al. [24] took the IFPR as a tool to represent the evaluation information of house buyer and then used the goal programming method to help the buyer to rank the candidate houses. Wang [52] used the IFPR to help a customer to select a new vehicle to buy. He also showed how to use the IFPR to select the international exchange doctoral students. Liao and Xu [26] proposed an automatic procedure to repair the inconsistent IFPR and developed an algorithm to aid the decision makers to analyze the performance of three types of motorcycles. After introducing a new deﬁnition of multiplicative consistency for the IFPR, Liao and Xu [29] implemented the IFPR into the process of selecting a ﬂexible manufacturing system (FMS). Afterward, Liao and Xu [30] developed some fractional models for group decision making with IFPRs and then implemented these methodologies into a group decision making problem concerning the evaluation and ranking of the main factors of electronic learning. To show the efﬁciency of the error-analysis-based method, Xu

[61] applied the IFPR into the supply chain management problem to determine the importance of the factors which can inﬂuence the cooperation among enterprises. Recently, Liao et al. [33] proposed the framework of group decision making with IFPRs and applied the proposed decision making process to select outstanding PhD students for China Scholarship Council. In the process of group decision making with IFPRs, assuming that each IFPR is irreﬂexive, asymmetric and transitive, Dimitrov [18] proved that the aggregation rule [17] maps also each proﬁle of such preferences into an irreﬂexive, asymmetric and transitive intuitionistic fuzzy collective preference relation. In order to better understand the state of the art of IFPRs, this paper provides an extensive and intensive overview on IFPRs, including its originality of concept, transitivity and consistency, priority methods and consensus measures. Based on these objectives, the remainder of this paper is set out as follows: Section 2 reviews the originality of IFPR. After recalling the transitivity of IFPR, Section 3 discusses the state of the art of the consistency of an IFPR, including the different forms of additive consistency and the different forms of multiplicative consistency. In Section 4, a survey concerning the priority methods is given. Section 5 mainly addresses different types of consensus measures for group decision making with IFPRs. The paper ends with some concluding remarks in Section 6. 2. Preference relations and intuitionistic fuzzy preference relation In analytic hierarchy process (AHP), Saaty [37] decomposed a complex multi-criteria decision making (MCDM) problem into a multi-level hierarchic structure of objectives, criteria, sub-criteria and alternatives, and then provided a fundamental scale of relative magnitudes expressed in dominance units to represent judgments in the form of pairwise comparisons. The fundamental scale expresses relative importance of the elements in a level with respect to the elements in the level immediately above it. Deﬁnition 1 [37]. Let A ¼ fA1 ; A2 ; . . . ; An g be a ﬁnite set of alternatives and C ¼ fC 1 ; C 2 ; . . . ; C m g be a set of criteria to compare the alternatives. A fundamental scale for the criteria C j 2 C (j ¼ 1; 2; . . . ; mÞ is a mapping P C j , which assigns to every pair ðAi ; Ak Þ 2 A A a positive real number P C j ðAi ; Ak Þ ¼ pik that denotes the relative intensity with which an individual perceives the criterion C j 2 C in an element Ai 2 A in relation to the other Ak 2 A. In addition, Saaty further developed the 1–9 scale to describe the preferences between alternatives as being either equally, moderately, strongly, very strongly or extremely preferred. These preferences are translated into pairwise weights of one, three, ﬁve, seven or nine, respectively, with two, four, six, eight as the intermediate values (see Table 1 for more details). The 1–9 scale satisﬁes the reciprocal condition, i.e., the intensity of preference of Ai over Ak is inversely related to the intensity of preference of Ak over Ai , that is,

pik ¼ 1=pki ;

8Ai ; Ak 2 A; C j 2 C; j ¼ 1; 2; . . . ; m

ð1Þ

With the 1–9 scale, in general, Saaty pointed out that: Ai C j Ak if and only if pik > 1 where the binary relation ‘‘C j ’’ represents ‘‘be preferred to’’ according to the criterion C j ; Ai C j Ak if and only if pik ¼ 1 where the binary relation ‘‘C j ’’ represents ‘‘be indifferent to’’ according to the criterion C j . If all the pairwise judgments determined by the experts with 1– 9 scale are stored in an n n matrix A ¼ ðaik Þnn , then a multiplicative preference relation is constructed:

Z. Xu, H. Liao / Knowledge-Based Systems 80 (2015) 131–142 Table 1 Comparison between the 1–9 scale and the 0.1–0.9 scale. 1–9 scale

0.1–0.9 scale

Meaning

1/9 1/7 1/5 1/3 1 3 5 7 9 Other values between 1/9 and 9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Other values between 0 and 1

Extremely not preferred Very strongly not preferred Strongly not preferred Moderately not preferred Indifferent Moderately preferred Strongly preferred Very strongly preferred Extremely preferred Intermediate values used to present compromise

Deﬁnition 2 [37]. A multiplicative preference relation R on the set A ¼ fA1 ; A2 ; . . . ; An g is represented by a matrix R ¼ ðr ik Þnn , where r ik is the intensity of preference of Ai over Ak measured using the 1– 9 scale, and satisﬁes the reciprocal condition, i.e.,

r ik ¼ 1=r ki ;

8Ai ; Ak 2 A

ð2Þ

It is noted that all judgments in the multiplicative preference relation corresponding to Saaty’s 1–9 scale are crisp values, which are hard to be exactly determined due to the complexity and uncertainty involved in the real-world decision making problems and incomplete information or knowledge. Later, Orlovsky [36] used the fuzzy membership function lPC : A A ! ½0; 1 as the funj damental scale to express relative importance of the alternatives. The membership degree lPC ðAi ; Ak Þ ¼ bik denotes the relative j intensity with which the alternative Ai is preferred to Ak over the criterion C j 2 C. If all the pairwise comparison values are represented by fuzzy numbers within the interval ½0; 1 and stored in a matrix B ¼ ðbik Þnn , a fuzzy preference relation is determined. However, the fuzzy preference relation proposed by Orlovsky does not satisfy the reciprocal condition. Tanino [50] further proposed the concept of fuzzy preference relation (in his paper, he also named it as fuzzy preference orderings) which satisﬁes the reciprocal condition and can be expressed as follows: Deﬁnition 3 [50]. A fuzzy preference relation B on the set A ¼ fA1 ; A2 ; . . . ; An g is represented by a matrix B ¼ ðbik Þnn , where bik is the intensity of preference of Ai over Ak , and satisﬁes:

bik þ bki ¼ 1;

bij 2 ½0; 1;

8Ai ; Ak 2 A

ð3Þ

In the fuzzy preference relation B ¼ ðbik Þnn , each element is measured by the 0.1–0.9 scale with bik denoting the degree that the alternative Ai is preferred to Ak . Concretely speaking, the case bik ¼ 0:5 indicates that there is indifference between the alternatives Ai and Ak ; bik > 0:5 indicates that the alternative Ai is preferred to Ak , especially, bik ¼ 1 means that the alternative Ai is absolutely preferred to Ak ; bik < 0:5 indicates that the alternative Ak is preferred to Ai , especially, bik ¼ 0 means that the alternative Ak is absolutely preferred to Ai . Fordor and Roubens [20] developed a general axiomatic approach to the deﬁnition of strict preference, indifference and incomparability relations associated with a valued preference relation. The difference between the 1–9 scale (used in the multiplicative preference relation) and the 0.1–0.9 scale (used in the fuzzy preference relation) is shown in Table 1: In either the multiplicative preference relation or the fuzzy preference relation, the preference degree to which an alternative is preferred to another one is represented by a single value in terms of the 1–9 scale or the 0.1–0.9 scale. However, in some situations, if the expert is not very familiar with the decision making problem or

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there contains some incomplete information of the alternatives, it would be very hard for the expert to determine the exact values of the preference degrees. In such cases, the expert may prefer to express his/her imprecise cognition from the positive, negative and hesitative points of view. For example, when getting together to evaluate an alternative, the decision makers may not only present ‘‘agreement’’ or ‘‘disagreement’’ but also ‘‘abstention’’ which indicates the hesitation and indeterminacy over the alternative [67]. The same situation can be found in a voting process as well. Given that the elements in both the multiplicative preference relation and the fuzzy preference relation are single-valued elements, they cannot be used to express the support and objection evidences simultaneously in many practical situations. In order to depict such sorts of realistic situations and to model human’s perception and cognition more comprehensively, Atanassov [1,2] introduced a new extension of fuzzy set, named intuitionistic fuzzy set (IFS), which is characterized by a membership function, a nonmembership function and a hesitancy function. Deﬁnition 4 [1,2]. Let a crisp set X be ﬁxed and let A X be a ﬁxed ~ in X is an object of the set. An intuitionistic fuzzy set (IFS) A following form:

~ ¼ x; l ðxÞ; v A ðxÞjx 2 X A A

ð4Þ

where the function lA : A ! ½0; 1 and v A : A ! ½0; 1 deﬁne the degree of membership and the degree of non-membership of the element x 2 X to the set A, respectively, and for every ~ on X, then x 2 X; 0 6 lA þ v A 6 1 holds. For each IFS A

pA ðxÞ ¼ 1 lA ðxÞ v A ðxÞ

ð5Þ

is called the degree of non-determinacy (uncertainty) of the membership of the element x 2 X to the set A. In the case of ordinary fuzzy sets, pA ðxÞ ¼ 0 for every x 2 X. For convenience, Xu [65] called a ¼ ðla ; v a Þ an intuitionistic fuzzy number (IFN). Szmidt and Kacprzyk [47] justiﬁed that pA ðxÞ cannot be omitted when calculating the distance between two IFSs. With the IFNs, the expert can express his/her opinions over the alternatives from three aspects, which are ‘‘preferred’’, ‘‘not preferred’’, and ‘‘indeterminate’’. The preliminary idea of intuitionistic fuzzy preference relation (IFPR) which consists of a fuzzy preference relation and a hesitance matrix, was given by Szmidt and Kacprzyk [48]. Later, Xu [66] gave a standard deﬁnition of IFPR, which is a judgment matrix whose elements are IFNs, which is shown as follows: Deﬁnition 5 [66]. An intuitionistic fuzzy preference relation (IFPR) on the set X ¼ fx1 ; x2 ; . . . ; xn g is represented by a matrix ~ ¼ ~rij , where ~r ij ¼< ðxi ; xj Þ; lðxi ; xj Þ; v ðxi ; xj Þ; pðxi ; xj Þ > for all R nn i; j ¼ 1; 2; . . . ; n. For convenience, let ~r ij ¼ lij ; v ij ; pij where lij denotes the degree to which the object xi is preferred to the object xj ; v ij indicates the degree to which the object xi is not preferred to the object xj , and pij ¼ 1 lij v ij is interpreted as an indeterminacy degree or a hesitancy degree, with the conditions:

lij ; mij 2 ½0; 1; lij þ v ij 6 1; lij ¼ v ji ; lii ¼ v ii ¼ 0:5; pij ¼ 1 lij v ij ; for all i; j ¼ 1; 2; . . . ; n

ð6Þ

Bustince and Burillo [7] studied the structures of the intuitionistic fuzzy preference relation and analyzed the existent relations between the structures of a relation and the structures of its complementary one. Afterward, Bustince [5] presented several theorems to build reﬂexive, symmetric, antisymmetric, perfect antisymmetric and transitive intuitionistic fuzzy relations. Xu [66] proposed the concept of incomplete IFPR in which some of the

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preference values are unknown. There are also some algorithms to estimate the missing values for the incomplete IFPR [49,57,58,66]. Motivated by the interval-valued fuzzy preference relation proposed by Bustince and Burillo [6], some scholars have also done some researches on interval-valued IFPR [31,55,59]. Xu [63] made a survey on the research results (before 2007) of different kinds of preference relations.

3. Transitivity and consistency of intuitionistic fuzzy preference relations 3.1. Transitivity of IFPR Consistency is one of the most important research topics related to preference relations due to the fact that lacking consistency may lead to the inconsistent conclusions in the process of decision making [32]. Cutello and Montero [12] took the consistency of fuzzy preference relations as fuzzy rationality, and gave an axiomatic basis for deﬁning the concept of fuzzy rationality. Transitivity is a basic characteristic to identify the consistency of a pairwise comparison matrix [15,16,51,67]. Different kinds of transitivity properties have been proposed for multiplicative preference relations and fuzzy preference relations, such as the weak transitivity, the max–min transitivity, the max-max transitivity, the restricted max–min transitivity, the restricted max-max transitivity, the additive transitivity, and the multiplicative transitivity [9–11,25,29,37,49,51,66]. De Baets et al. [15] established the important relationships between the basic properties of the components of a fuzzy preference structure without incomparability. Later, De Baets et al. [14] presented a general framework for studying the transitivity of reciprocal relations, which is named as cycle-transitivity. Considering that Switalski [41] also introduced the FG-transitivity as an immediate generalization of stochastic transitivity, De Baets et al. [13] made a comparison between the framework of cycle-transitivity and the framework of FG-transitivity. Díaz et al. [16] investigated the transitivity bounds in the additive preference structures. Xu [63] reviewed the transitivity properties of a fuzzy preference relation. Genç et al. [22] also gave a detailed analysis over these transitivity properties. ~ ¼ ð~rik Þ ~ Let R nn be an IFPR, where r ik ¼ ðlik ; v ik Þ for all i; k ¼ 1; 2; . . . ; n. By using the comparison law of IFNs [29,43,75], Xu and Liao [67] introduced the following transitivity properties for IFPRs: ~ satis(1) If ~r ij ~r jk P ~r ik , for all i; j; k ¼ 1; 2; . . . ; n, then we say R ﬁes the triangle condition. (2) If ~r ij P ð0:5; 0:5Þ; ~rjk P ð0:5; 0:5Þ ) ~rik P ð0:5; 0:5Þ, for all ~ satisﬁes the weak transitivity i; j; k ¼ 1; 2; . . . ; n, then we say R property. ~ (3) If ~r ik P minf~rij ; ~rjk g, for all i; j; k ¼ 1; 2; . . . ; n, then we say R satisﬁes the max–min transitivity property. ~ (4) If ~rik P maxf~rij ; ~rjk g, for all i; j; k ¼ 1; 2; . . . ; n, then we say R satisﬁes the max-max transitivity property. (5) If ~rij P ð0:5; 0:5Þ; ~r jk P ð0:5; 0:5Þ ) ~r ik minf~r ij ; ~r jk g, for all ~ satisﬁes the restricted max– i; j; k ¼ 1; 2; . . . ; n, then we say R min transitivity property. (6) If ~r ij P ð0:5; 0:5Þ; ~rjk P ð0:5; 0:5Þ ) ~rik P maxf~rij ; ~rjk g, for all ~ satisﬁes the restricted maxi; j; k ¼ 1; 2; . . . ; n, then we say R max transitivity property. The weak transitivity means that if the alternative Ai is preferred to Ak , and Ak is preferred to Aj , then Ai should be preferred to Aj . The max–min transitivity implies that the intuitionistic fuzzy preference value between the alternatives Ai and Aj should be

equal to or greater than the minimum partial values between the alternatives Ai ; Ak and Ak ; Aj . By contrast, the max-max transitivity denotes that the intuitionistic fuzzy preference value between the alternatives Ai and Aj should be equal to or greater than the maximum partial values between the alternatives Ai ; Ak and Ak ; Aj . The restricted max–min transitivity is a special max–min transitivity under the condition that Ai is preferred to Ak and Ak is preferred to Aj . In analogous, the restricted max-max transitivity can be mathematically presented. Comparing the above mentioned transitivity properties, the weak transitivity is the minimum requirement condition to ﬁnd out whether an IFPR is consistent or not. The max-max transitivity is better than the max–min transitivity [22]; however, the max-max transitivity cannot be veriﬁed under reciprocity [9]. Neither the restricted max–min transitivity nor the restricted max-max transitivity implies reciprocity. To reﬂect the reciprocity of an IFPR, the additive transitivity and the multiplicative transitivity properties have been introduced and further taken as the concept to verify the consistency of an IFPR. These led to the concepts of additive consistency and multiplicative consistency for the IFPR. 3.2. Additive consistency of IFPR The concept of additive consistency of an IFPR was motivated by the additive transitivity property of a fuzzy preference relation [50]. If we understand ðbij 0:5Þ to be an intensity of preference of Ai over Aj , then an additive transitivity relation of intensities of preference is deﬁned as:

ðbij 0:5Þ þ ðbjk 0:5Þ ¼ ðbik 0:5Þ;

for all i; j; k ¼ 1; 2; . . . ; n ð7Þ

Let xi (i ¼ 1; 2; . . . ; nÞ be the underlying weights of the alternaP tives and satisﬁes ni¼1 xi ¼ 1; xi 2 ½0; 1, then the elements of an additive consistent fuzzy preference relation can be given as:

bij ¼ 0:5ðxi xj þ 1Þ;

for all i; j; k ¼ 1; 2; . . . ; n

ð8Þ

Based on the additive transitivity of a preference relation, different forms of deﬁnitions for the additive consistency of an IFPR have been proposed. In what follows, let us review these different deﬁnitions of additive consistent IFPR: For each IFS ~r ij ¼ ðlij ; v ij Þ, the condition lij 6 1 v ij ~ ¼ ð~r ij Þ (i; j ¼ 1; 2; . . . ; nÞ always holds. Thus, the IFPR R can be nn transformed into an interval-valued complementary judgment ^ ¼ ð^r ij Þ ^þ where ^r ij ¼ ð^r matrix R ij ; r ij Þ ¼ ½lij ; 1 v ij ði; j ¼ 1; 2; nn . . . ; nÞ,

and

^r ^þ ^þ ^ ^þ ^ ^þ ^ ij þ r ji ¼ r ij þ r ji ¼ 1; r ij P r ji P 0; r ii P r ii P 0:5;

i; j ¼ 1; 2; . . . ; n. Based on the above transformation, Xu [62] introduced the deﬁnition of additive consistent IFPR: ~ ¼ ð~rij Þ Deﬁnition 6 [62]. Let R be an IFPR with ~r ij ¼ ðlij ; v ij Þ nn (i; j ¼ 1; 2; . . . ; nÞ, if there exists a vector x ¼ ðx1 ; x2 ; . . . ; xn ÞT , such that

lij 6 0:5ðxi xj þ 1Þ 6 1 v ij ; for all i; j ¼ 1; 2; . . . ; n where xi 2 ½0; 1 (i ¼ 1; 2; . . . ; nÞ, and an additive consistent IFPR.

Pn

i¼1

ð9Þ

xi ¼ 1. Then, R~ is called

Deﬁnition 6 is easy to be understood if we combine (8) and the above transformation rule. On the other hand, as for an interval^ ¼ ð^r ij Þ , Gong et al. valued complementary judgment matrix R nn

[23] claimed that it is additive consistent if there exists a priority ^ ¼ ðx ^ 1; x ^ 2; . . . ; x ^ n ÞT ¼ ð½xl1 ; xu1 ; ½xl2 ; xu2 ; . . . ; ½xln ; xun ÞT , x vector such that

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^r ij ¼ 0:5 þ 0:2 log 3xi =xj ¼ ½0:5 þ 0:2 log 3

xil =xju

; 0:5 þ 0:2 log 3

xiu =xjl

;

for all i; j ¼ 1; 2; . . . ; n

ð10Þ

^ i can be interpreted as the membership where the priority weight x degree range of the importance of alternative Ai . Based on the ~ ¼ ð~rij Þ abovementioned transformation between the IFPR R nn and its corresponding interval-valued complementary judgment matrix ^ ¼ ð^r ij Þ , it is natural to introduce the following deﬁnition of R nn additive consistent IFPR: ~ ¼ ð~rij Þ Deﬁnition 7 [23]. Let R be an IFPR with ~r ij ¼ ðlij ; v ij Þ nn ^ ¼ ðx ^ 1; x ^ 2; . . . ; x ^ n ÞT ¼ (i; j ¼ 1; 2; . . . ; nÞ, if there exists a vector x T

ð½xl1 ; xu1 ; ½xl2 ; xu2 ; . . . ; ½xln ; xun Þ , such that l

u

l j

u i

lij ¼ 0:5 þ 0:2 log 3xi =xj ;

v ij ¼ 0:5 þ 0:2 log 3x =x

for all i; j ¼ 1; 2; . . . ; n

;

ð11Þ

~ is called an additive consistent IFPR. Then, R

~ ¼ ð~rij Þ Deﬁnition 8 [52]. An IFPR R with ~rij ¼ ðlij ; v ij Þ (i; j ¼ nn 1; 2; . . . ; nÞ is called additive consistent if it satisﬁes the following additive transitivity:

lik þ ljk þ lki ¼ lkj þ lji þ lik ; for all i; j; k ¼ 1; 2; . . . ; n

ð12Þ

~ ¼ ðx ~ 1; x ~ 2; . . . ; x ~ n ÞT ¼ ððxl1 ; xv1 Þ; ðxl2 ; xv2 Þ; . . . ; ðxln ; xvn ÞÞT Let x be an underlying intuitionistic fuzzy priority vector of an IFPR ~ ¼ ð~rij Þ , where x ~ i ¼ ðx ~ li ; x ~ vi Þ ði ¼ 1; 2; . . . ; nÞ are the IFNs, R nn l v ~i ;x ~ i 2 ½0; 1 and x ~ li þ x ~ vi 6 1ði ¼ 1; 2; . . . ; nÞ. x ~ li which satisfy x v ~ i indicate the membership and non-membership degrees and x of the alternative xi as per a fuzzy concept of ‘‘importance’’, respec~ is said to be normalized if it satisﬁes [40,52]: tively. x

xlj 6 xvi ; xli þ n 2 P

j¼1;j–i

n X

xvj ; for all i ¼ 1;2;. .. ; n

ð13Þ

j¼1;j–i

¼

(

lij ; v ij ¼

ð0:5; 0:5Þ;

if i ¼ j

l

l

ð0:5xi þ 0:5xvj ; 0:5xvi þ 0:5xj Þ; if i – j ð14Þ

l

l

where xi ; xvi 2 ½0; 1; xi þ xvi 6 1; P P nj¼1;j–i xvj , for all i ¼ 1; 2; . . . ; n.

Pn

j¼1;j–i

bij =bji ¼ ðbik =bki Þ bkj =bjk ;

Motivated by the multiplicative transitivity and the relationship between the IFPR and its corresponding fuzzy or multiplicative preference relations, several distinct deﬁnitions of multiplicative consistency have been proposed for IFPRs. Based on the transformation relationship between the IFPR ~ ¼ ð~rij Þ R nn and its corresponding interval complementary judg^ ¼ ð^rij Þ , Xu [62] gave the deﬁnition of multiplicament matrix R nn tive consistent IFPR:

nÞ be an IFPR, if there exists a vector x ¼ ðx1 ; x2 ; . . . ; xn ÞT , such that

lij 6

xi 6 1 v ij ; for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n xi þ xj ð17Þ

where xi P 0; ði ¼ 1; 2; . . . ; nÞ; plicative consistent IFPR.

Pn

~ i¼1 xi ¼ 1. Then, we call R a multi-

Moreover, Gong et al. [24] introduced another deﬁnition of multiplicative consistent IFPR, which is also based on the transformation between an IFPR and its corresponding interval-valued fuzzy preference relation: ~ ¼ ð~rij Þ Deﬁnition 10 [24]. Let R be an IFPR with ~r ij ¼ ðlij ; v ij Þ nn ^ ¼ ðx ^ 1; x ^ 2; . . . ; x ^ n ÞT ¼ (i; j ¼ 1; 2; . . . ; nÞ, if there exists a vector x T

ð½xl1 ; xu1 ; ½xl2 ; xu2 ; . . . ; ½xln ; xun Þ , such that

lij ¼

xli

; u

xli þ xj

v ij ¼

xlj xlj þ xui

;

for all i; j ¼ 1; 2; . . . ; n

ð18Þ

After making some transformations over the Eq. (15), Xu et al. [68] proposed another deﬁnition of multiplicative consistent IFPR, which was given based on the membership and non-membership degrees of IFSs: ~ ¼ ðrij Þ Deﬁnition 11 [68]. An IFPR R with nn (i; j ¼ 1; 2; . . . ; nÞ is multiplicative consistent if

(

lij ¼

Besides the additive consistency, there is another kind of consistency, named multiplicative consistency. The multiplicative consistency is based on the multiplicative transitivity of a fuzzy preference relation B ¼ ðbij Þnn , which is characterized as [50]:

ð16Þ

r ij ¼ ðlij ; v ij Þ

xlj 6 xvi , and xli þ n 2

3.3. Multiplicative consistency of IFPR

xi ; for all i; j; k ¼ 1; 2; . . . ; n xi þ xj

~ is called a multiplicative consistent IFPR. Then, R

With the underlying normalized intuitionistic fuzzy priority ~ ¼ ðx ~ 1; x ~ 2; . . . ; x ~ n ÞT , an additive consistent IFPR vector x ~ R ¼ ð~r ij Þnn can be established as:

~r ij

bij ¼

~ ¼ ð~rij Þ Deﬁnition 9 [62]. Let R with ~rij ¼ ðlij ; v ij Þ (i; j ¼ 1; 2; . . . ; nn

Both Deﬁnition 6 and Deﬁnition 7 are based on the transformation rule between the IFPR and its corresponding interval-valued ^ ¼ ð^r ij Þ . Furthermore, Wang complementary judgment matrix R nn [52] introduced a new deﬁnition of additive consistent IFPR by directly employing the membership and nonmembership degrees of IFNs:

n X

then the multiplicative transitivity can be interpreted as the ratio of the preference intensity for the alternative Ai to that of Aj should be equal to the multiplication of the ratios of preferences when using an intermediate alternative Ak . Let xi ði ¼ 1; 2; . . . ; nÞ be the underlying weights of the alternaP tives and satisﬁes ni¼1 xi ¼ 1; xi 2 ½0; 1, then the elements of an multiplicative consistent fuzzy preference relation can be given as:

for all i; j; k ¼ 1; 2; . . . ; n

ð15Þ

The multiplicative transitivity is easy to be understood: if we use bij =bji to represent the ratio of the preference intensity for the alternative Ai to that of Aj , i.e., Ai is bij =bji times as good as Aj ,

ðlik ; lkj Þ 2 fð0; 1Þ; ð1; 0Þg

lik lkj lik lkj þð1lik Þð1lkj Þ ;

(

v ij ¼

0;

0;

otherwise

ðv ik ; v kj Þ 2 fð0; 1Þ; ð1; 0Þg

v ik v kj v ik v kj þð1v ik Þð1v kj Þ ;

otherwise

;

for all i 6 k 6 j

ð19Þ

;

for alli 6 k 6 j

ð20Þ

Later, Liao and Xu [29] pointed out that the deﬁnition in Xu et al. [68] is not reasonable in some cases because with the above consistency conditions, the relationship lij ljk lki ¼ lik lkj lji (for all i; j; k ¼ 1; 2; . . . ; nÞ cannot be derived any more. To circumvent this weakness, they introduced a general deﬁnition of multiplicative consistent IFPR:

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~ ¼ ð~rij Þ Deﬁnition 12 [29]. An IFPR R with ~r ij ¼ ðlij ; v ij Þ nn (i; j ¼ 1; 2; . . . ; nÞ is called multiplicative consistent if the following multiplicative transitivity is satisﬁed:

4.1. Methods to derive the crisp priorities from an IFPR

lij ljk lki ¼ v ij v jk v ki ; for all i; j; k ¼ 1; 2; . . . ; n

This method is based on the additive consistency deﬁnition shown as Deﬁnition 6 as well as the multiplicative consistency definition shown as Deﬁnition 9. Firstly, an IFPR is transformed into its corresponding score matrix S ¼ sð~r ij Þ nn by using the formula:

ð21Þ

Liao and Xu [29] further clariﬁed that the conditions in Deﬁnition 11 satisfy Eq. (21), which implies the consistency measured by the conditions given in Deﬁnition 11 is a special case of multiplicative consistency deﬁned as Deﬁnition 12 for an IFPR. Hence, in general, Deﬁnition 11 is not sufﬁcient and suitable to measure the multiplicative consistency of an IFPR. With the underlying normalized intuitionistic fuzzy priority ~ ¼ ðx ~ 1; x ~ 2; . . . ; x ~ n ÞT , a multiplicative consistent weight vector x ~ ¼ ð~r Þ IFPR R can be established as [29]: ij nn

~r ij ¼

lij ; v ij ¼ l

8 < ð0:5; 0:5Þ;

where xi ; xvi 2 ½0; 1; xi þ xvi 6 1; P P nj¼1;j–i xvj , for all i ¼ 1; 2; . . . ; n.

sð~r ij Þ ¼ lij v ij ;

Pn

j¼1;j–i

l

if i – j

i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n

sð~rij Þ minfsð~rij Þg sð~r ij Þ ¼

i

maxfsð~rij Þg minfsð~r ij Þg i

;

i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; n

i

ð24Þ

ð22Þ

l

xj 6 xvi , and xi þ n 2

The additive consistency is sometimes in conﬂict with the ½0; 1 scale used for providing the preference values, whereas the multiplicative consistency does not have this limitation [10,69].

Based on the additive consistency of IFPR shown as Deﬁnition 6, the following model is constructed [62]: Model 1

u 1 ¼ min

Crisp priorities, i.e., the traditional precise weighting vector x ¼ ðx1 ; x2 ; . . . ; xn Þ with xi (i ¼ 1; 2; . . . ; nÞ being the underlying weights of the alternatives and satisfying Pn i¼1 xi ¼ 1; xi 2 ½0; 1; Interval-valued priorities, which is in terms of

s:t:

k¼1

þ > > dij 6 1 v ij ; > > > > n > X > > > xi ¼ 1; > xi P 0; > > > i¼1 > > > þ > > > > dij P 0; dij P 0; :

u 2 ¼ min

j ¼ i þ 1; . . . ; n i ¼ 1; 2; . . . ; n 1 i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n

n1 X n X

þ

dij þ dij

i¼1 j¼iþ1

T

of the alternative Ai ; ~ ¼ ðx ~ 1; x ~ 2; . . . ; x ~ n ÞT ¼ Intuitionistic fuzzy priorities, shown as x

The crisp priorities are easy to be understood. As it has been proven that the intuitionistic fuzzy set is mathematically equivalent to the interval-valued fuzzy set, the interval-valued priorities and the intuitionistic fuzzy priorities are mathematically equivalent. In the process of decision making with an IFPR, how to yield the priority vector from the IFPR is a crucial question and has attracted many scholars’ attentions. Different priority vector derivation methods have been proposed. Most of these methods are based on the consistency condition of an IFPR. In the following, let us review the state of the art of these methods. They are classiﬁed by the distinct forms of priorities.

! 8 m X > > > ~ ~ i ¼ 1; 2; . . . ; n 1; 0:5 x ð Þ ÞÞ þ 1 s ð r s ð r > k ik jk > > > k¼1 > > > > þdij P lij ; j ¼ i þ 1; . . . ; n > > > ! > > m > X > > > xk ðsð~rik Þ sð~rjk ÞÞ þ 1 i ¼ 1; 2; . . . ; n 1; > < 0:5

Based on the multiplicative consistency of IFPR shown as Deﬁnition 9, the following model is constructed [62]: Model 2

^ 1; x ^ 2; . . . ; x ^ n ÞT ¼ ð½xl1 ; xu1 ; ½xl2 ; xu2 ; . . . ; ½xln ; xun Þ , where ^ ¼ ðx x ^ i represents the membership degree range of the importance x

T l l l ~ i ¼ ðx ~ li ; x ~ vi Þ ði ¼ 1; ððx1 ; xv1 Þ; ðx2 ; xv2 Þ; . . . ; ðxn ; xvn ÞÞ where x l l v ~i ;x ~ i 2 ½0; 1; x ~i þx ~ vi 6 1. x ~ li 2; . . . ; nÞ is an IFN, and satisﬁes x v ~ and xi indicate the membership and non-membership degrees of the alternative Ai as per a fuzzy concept of ‘‘importance’’, respectively.

n1 X n X þ dij þ dij i¼1 j¼iþ1

4. Priority vector derivation methods based on the consistency of intuitionistic fuzzy preferences After establishing an IFPR, to ﬁnd the ﬁnal result for a decision making problem, the following thing we should do is to derive the underlying priorities from the IFPR. In Saaty [37] ’s traditional AHP method, a scale of priorities is an n-dimensional vector x ¼ ðx1 ; x2 ; . . . ; xn Þ obtained from the multiplicative preference relation and xi is a weight which accurately represents the relative dominance of the alternative Ai among the alternatives in A. As to IFPR, there are mainly three different types of priorities, shown as below:

ð23Þ

Then, the score matrix S ¼ sð~r ij Þ nn is normalized into S ¼ sð~rij Þ nn , where

if i ¼ j

l l 2x 2x : ðxl xv þxil xv þ2 ; xl xv þxjl xv þ2Þ; i i j j i i j j

l

The linear programming method

s:t:

8 m X > > > xk ð1 lij Þðsð~rik Þ sð~rjk ÞÞ > > > k¼1 > > > > þdij P 0; i ¼ 1;2;. .. ;n 1; j ¼ i þ 1;. .. ;n > > > > m > X > > < xk ðv ijsð~rik Þ ð1 v ij Þsð~rjk ÞÞ k¼1 > > þ > dij 6 0; > > > > n > X > > > xi ¼ 1; > xi P 0; > > > i¼1 > > : þ dij P 0; dij P 0;

i ¼ 1;2;. .. ;n 1; j ¼ i þ 1;. .. ;n i ¼ 1;2;. .. ;n 1 i ¼ 1;2;. .. ;n 1; j ¼ i þ 1;. .. ;n

Solving either Model 1 or Model 2, the underlying crisp priorities can be yielded for an IFPR. 4.2. Methods to derive the interval-valued priorities from an IFPR The least squares optimization method and the goal programming method If the priorities are supposed to be in terms of interval-values, based on the additive consistency deﬁnition of an IFPR shown as Deﬁnition 7, it is easy to establish the following least squares optimization model to derive the interval-valued priorities [23]:

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Model 3

Min J 1 ¼

n X n X

xli 35ðlij 0:5Þ xuj

2

þ xui 35ð0:5v ij Þ xlj

2

i¼1 j¼1

8 n X > > l > x þ xlj P 1; > i > > > j¼1;j–i > > > n < X xui þ xlj 6 1; s:t: > > j¼1;j–i > > > > > xui xli P 0; > > > : u xi P 0; xli P 0;

i ¼ 1; 2; . . . ; n i ¼ 1; 2; . . . ; n

addition, we can get an error matrix d ¼ ðdij Þnn dij ¼ 12 lij þ pij lij ¼ 12 pij (i; j ¼ 1; 2; . . . ; nÞ as well.

i ¼ 1; 2; . . . ; n i ¼ 1; 2; . . . ; n

The main idea of Model 3 is to minimize the squares values of deviations between the preferences and their corresponding consistent values. In analogous, if minimizing the absolute values of deviations between the preferences and their corresponding consistent values, the following goal programming model can be established to yield the interval-valued priorities [23]: Model 4

Min J 2 ¼

n X n X

eþij þ eij þ cþij þ cij

i¼1 j¼1

8 l x 35ðlij 0:5Þ xuj eþij þ eij ¼ 0; > > > i > > 5ð0:5v ij Þ l u > xj cþij þ cij ¼ 0; > > xi 3 > > > n n > X > l X < xi þ xlj P 1; xui þ xlj 6 1; s:t: j¼1;j–i j¼1;j–i > > > > > xui xli P 0; xui P 0; xli P 0; > > > > > eþij P 0; eij P 0; cþij P 0; cij P 0; > > > : þ eij eij ¼ 0; cþij cij ¼ 0;

where Dyi ði ¼ 1; 2; . . . ; nÞ are the ranges of errors for a set of random variables fy1 ; y2 ; . . . ; yn g and z ¼ f ðy1 ; y2 ; . . . ; yn Þ is a random function. ~ ¼ ð~rij Þ , its corresponding interval-valued comFor an IFPR R nn ^ ¼ ð^rij Þ plementary judgment matrix R can be transformed with nn þ ^rij ¼ ð^r ^ (i; j ¼ 1; 2; . . . ; nÞ. Furthermore, an ij ; r ij Þ ¼ ½lij ; 1 v ij ¼ ðrij Þ expected fuzzy preference relation R with rij ¼ nn 1 1 ¼ l þ l þ p l þ p (i; j ¼ 1; 2; . . . ; nÞ is also obtained. In ij ij ij ij 2 2 ij

i ¼ 1; 2;. .. ;n; j ¼ 1; 2;. ..; n i ¼ 1; 2;. .. ;n; j ¼ 1; 2;. ..; n i ¼ 1; 2;. .. ;n

with

For any fuzzy preference relation B ¼ ðbij Þnn , with bij P 0; bij þ bji ¼ 1; bii ¼ 0:5; i; j ¼ 1; 2; . . . ; n, Xu [60] introduced a simple formula for deriving the crisp priority vector

x ¼ ðx1 ; x2 ; . . . ; xn ÞT of B as:

xi ¼

! n X 1 n cij þ 1 ; nðn 1Þ j¼1 2

i ¼ 1; 2; . . . ; n

ð26Þ

P where xi P 0; i ¼ 1; 2; . . . ; n, and ni¼1 xi ¼ 1. Thus, considering the ¼ ðr ij Þ , we can get the priorexpected fuzzy preference relation R nn ¼ ðr ij Þ ity vector of the expected fuzzy preference relation R nn as x ¼ ðx1 ; x2 ; . . . ; xn ÞT , where

! n X 1 n r ij þ 1 nðn 1Þ j¼1 2 !

n X 1 1 n lij þ pij þ 1 ; ¼ nðn 1Þ j¼1 2 2

xi ¼

i ¼ 1; 2;. .. ;n i ¼ 1; 2;. .. ;n; j ¼ 1; 2;. ..; n

i ¼ 1; 2; . . . ; n

ð27Þ

~ i ¼ f ðri1 ; r i2 ; . . . ; rin Þ (i ¼ 1; 2; . . . ; nÞ, then according to Eqs. Let x (25) and (27), we have The goal programming method 2

Based on the multiplicative consistency deﬁnition of an IFPR shown as Deﬁnition 10, and similar to Model 4, the following goal programming model was constructed to produce the interval-valued priorities [24]: Model 5

Min J 3 ¼

n X n X

þ þ ij þ ij þ ij þ ij

e

e

c

c

~ iÞ ¼ ðDx ¼

1 2

ðnðn 1ÞÞ 1

ðnðn 1ÞÞ2

n X ðDdij Þ2 j¼1

n

X 1 j¼1

2

2

pij ; i ¼ 1; 2; . . . ; n

ð28Þ T

~ ¼ ðDx ~ 1 ; Dx ~ 2 ; . . . ; Dx ~ n Þ , where by which we calculate Dx

i¼1 j¼1

8 lij ðxli þ xuj Þ xli eþij þ eij ¼ 0; i ¼ 1;2;...;n; j ¼ 1;2;...;n > > > > þ u l u > ð1 v ij Þðxi þ xj Þ xi cij þ cij ¼ 0; i ¼ 1;2;...;n; j ¼ 1;2;...;n > > > > > n n X X > > < xl þ xlj P 1; xui þ xlj 6 1; i ¼ 1;2;...;n i s:t: j¼1;j–i j¼1;j–i > > > > xui xli P 0; xui P 0; xli P 0; i ¼ 1;2;...;n > > > > þ þ > e P 0; e P 0; c P 0; c P 0; > ij ij ij ij > > : þ eij eij ¼ 0; cþij cij ¼ 0; i ¼ 1;2;...;n; j ¼ 1;2;...;n

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uX u n 2 1 t p ; ~i ¼ Dx 2nðn 1Þ j¼1 ij

i ¼ 1; 2; . . . ; n

ð29Þ

Therefore, the interval-valued priority vector of the intuitionis~ ¼ ð~rij Þ R tic preference relation is obtained as nn ~ ¼ ðx ~ 1; x ~ 2; . . . ; x ~ n ÞT , where x ~ i ¼ ½xi Dx ~ i ; xi þ Dx ~ i . It has also x ~ i ¼ ½xi Dx ~ i ; xi þ Dx ~ i ½0; 1. been proven that x 4.3. Methods to derive the intuitionistic fuzzy priorities from an IFPR Many scholars have also proposed some approaches to derive the intuitionistic fuzzy priorities from an IFPR:

The error-analysis-based method The normalizing rank summation method Besides the above programming methods, Xu [61] introduced an error-analysis-based method to derivate the interval-valued priorities. This method is based on the following error propagation formula [74]:

ðDzÞ2 ¼

2 n

X @f i¼1

@yi

ðDyi Þ2

ð25Þ

This method was proposed by Xu and Liao [67]. It is an indirect method which requires to ﬁrstly transform the IFPR into its corre ^ ¼ ½l ; 1 v ik sponding interval-valued preference relation R . ik nn Then, based on the operational laws of intervals [71], the crisp priorities can be derived as follows [67]:

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Pn ^rik ^ i ¼ Pn k¼1 Pn x

Pn ½l ; 1 v ik Pn ik ¼ Pn k¼1 ^ i¼1 k¼1 r ik i¼1 k¼1 ½lik ; 1 v ik

Pn Pn k¼1 lik ; k¼1 ð1 v ik Þ Pn Pn ¼ Pn Pn i¼1 k¼1 lik ; i¼1 k¼1 ð1 v ik Þ Pn Pn l ð1 v ik Þ P ; ; Pk¼1 ¼ Pn Pnk¼1 ik n n i¼1 k¼1 ð1 v ik Þ i¼1 k¼1 lik

Model 7

Min Z 2 ¼

Pn

i¼1

lik

k¼1 ð1

mik Þ

i ¼ 1; 2; . . . ; n

Pn ð1 mik Þ P ; ; 1 Pk¼1 n n i¼1

k¼1

ð30Þ

lik

i ¼ 1; 2; . . . ; n ð31Þ

from

which

the

intuitionistic

fuzzy

priority

vector

~ ¼ ðx ~ 1; x ~ 2; . . . ; x ~ n ÞT of the IFPR R ¼ ðr ik Þnn is obtained. x

rþij þ rij þfþij þfij

8 l 0:5xi þ0:5xvj lij rþij þ rij ¼0; i¼1;2;...;n1; j¼iþ1;...;n > > > > > > > > 0:5xv þ0:5xu v ij fþ þf ¼0; i¼1;2;...;n1; j¼iþ1;...;n > ij ij j i > > > > > > l l v v > > i¼1;2;...;n1 > xi ; xi 2½0;1; xi þ xi 61; > > > > > n > X > > < xlj 6 xvi ; j¼1;j–i

> > > > > n X > > > > xli þn2P xvj ; > > > > j¼1;j–i > > > > > þ > þ > > > rij P0; rij P0; fij P0;fij P0; > > > > : þ rij rij ¼0; fþij fij ¼0;

i¼1;2;...;n1

i¼1;2;...;n1; j¼iþ1;...;n

where

rij ¼ 0:5xli þ 0:5xvj lij ; fij ¼ 0:5xvi þ 0:5xuj v ij ; jfij j þ fij jfij j fij jr j þ rij jrij j rij rþij ¼ ij ; rij ¼ ; fþij ¼ ; fij ¼ ; 2 i; j ¼ 1; 2; . . . ; n; i – j

ij

ij

ij

i ¼ 1;2;...;n1; j ¼ iþ1;...;n i ¼ 1;2;...;n1

i ¼ 1;2;...;n1

i ¼ 1;2;...;n1; j ¼ iþ1;...;n

l

eij ¼

l

xi

2xi

lij ; l xvi þ xj xvj þ 2 l

nij ¼ nþij ¼

2xj

xli xvi þ xlj xvj þ 2 jnij j þ nij ; 2

nij ¼

v ij ;

jnij j nij ; 2

eþij ¼

jeij j þ eij ; 2

eij ¼

jeij j eij ; 2

i; j ¼ 1; 2; . . . ; n; i – j

i¼1 j¼iþ1

s:t:

ij

i ¼ 1;2;...;n1; j ¼ iþ1;...;n

where

This method is based on the additive consistency of IFPR shown as Deﬁnition 8. The main idea of this method is to minimize the distance between the given IFPR and its corresponding additive consistent IFPR constructed by Eq. (14). The linear goal programming model can be developed as below [52]: Model 6 n1 X n X

8 l 2xi þ > > l v þxl xv þ2 lij eij þ eij ¼ 0; > x x > i i j j > > > l > 2xj > > > v ij nþij þnij ¼ 0; l v > > xi xi þxlj xvj þ2 > > > > > xli ; xvi 2 ½0;1; xli þ xvi 6 1; > > > > n < X xlj 6 xvi ; s:t: > > j¼1;j–i > > > n > X > > > xli þn2 P xvj ; > > > > j¼1;j–i > > > > þ > > eþij P 0; eij P 0; nij P 0; > > > : n P 0; eþ e ¼ 0; nþ n ¼ 0; ij

The linear goal programming method

Min Z 1 ¼

eþij þ eij þnþij þnij

i¼1 j¼iþ1

Transforming each interval-valued priority into its corresponding IFN, then we have

xi ¼ Pn Pnk¼1

n1 X n X

2

2

2

The fractional programming method This method is based on the multiplicative consistency deﬁnition of IFPR (see Deﬁnition 12). The main idea of this method is to minimize the distance between the given IFPR and its corresponding multiplicative consistent IFPR constructed by Eq. (22). Hence, the underlying intuitionistic fuzzy priority ~ can be derived by the following fractional programvector x ming model [29]:

5. Review on the consensus measures for group decision making with IFPRs This section mainly focuses on the group decision making (GDM) with IFPRs. Recently, Liao et al. [33] established a framework of GDM with IFPRs, which divides the GDM process into three parts, i.e., the consistency checking process of each IFPR, the consensus checking process of the group and the selection process. In the process of GDM with IFPRs, the most crucial issue is to ﬁnd a consensus solution for the problem because the consensus reaching process is a pathway to a true group decision being supported by all the experts despite their different opinions [44,56]. Strictly speaking, consensus is meant as a full and unanimous agreement among all the experts regarding all the possible alternatives [19]. However, such a strict concept of consensus often is a utopia [30]. Ness and Hoffman [35] introduced the softened concept of consensus as ‘‘a decision that has been reached when most members of the team agree on a clear option and the few who oppose it think they have had a reasonable opportunity to inﬂuence that choice. All team members agree to support the decision.’’ The consensus is the state of mutual agreement among members of a group where all opinions have been heard and addressed to the satisfaction of the group [38]. Such a consensus is not enforced through negotiations or bargaining process, but emerges after the exchanges of opinions among the members of a group [3]. In the consensus reaching process, the consensus measures play a critical role in ﬁnding an agreeable solution for a group decision making problem. In what follows, let us review the existing consensus measures for the GDM with IFPRs. 5.1. The a-cut based consensus measure Szmit and Kacprzyk [42] ﬁrstly proposed a consensus measure for the GDM with IFPRs, which is inspired by the idea of Spillman et al. [39] on consensus of fuzzy relations. This approach ﬁrst divides each IFPR into two separate fuzzy preference relations, then uses Spillman et al.’s method over these two kinds of fuzzy

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preference relations and derives two consensus degrees, which are taken as the upper and lower bounds of the ﬁnal interval-valued consensus degree of the IFPRs. With the a-cut based consensus measure, the ﬁnal consensus degree is represented by an interval. The algorithm for calculating the a-cut based consensus measure can be summarized as follows: ~ ðlÞ with ~r ðlÞ ¼ ðlðlÞ ; v ðlÞ ; pðlÞ Þ into two Step 1: Divide each IFPR R ij ij ij ij ðlÞ

ðlÞ

CM 2 ¼

ðlÞ

Step 2: Reduce the matrices U ðlÞ and DðlÞ into the a-cut preference relations U

aðlÞ

8 < 1; if : 0;

if

aðlÞ

and D

, where

aðlÞ

lij P a aðlÞ

lij < a

;

aðlÞ

dij

¼

8 < 1; if : 0;

if

aðlÞ

Pa

aðlÞ

< 0:5; > :

ðlÞ dð~rij ;~rcij Þ

ðlÞ ðlÞ dð~r ij ;~rij Þþdð~r ij ;~r cij Þ

ðlÞ if ~r ij ¼ ~r ij ¼ ~rcij

ð43Þ

; otherwise

Thus, the consensus of the group is deﬁned as:

K min ¼ 0:1ð0:1 þ 2K min;0:2 þ 2K min;0:4 þ 2K min;0:6 þ 2K min;0:8 þ K 1 Þ K max ¼ 0:1ð0:1 þ 2K max;0:2 þ 2K max;0:4 þ 2K max;0:6 þ 2K max;0:8 þ K 1 Þ ð36Þ 5.2. The similarity based consensus measure Besides the a-cut based consensus measure, Szmit and Kacprzyk [45,46] also proposed a similarity based consensus measure, which is based on the similarity between two IFSs. For any IFN x ¼ ðlx ; v x ; px Þ, its complement set xc can be deﬁned as xc ¼ ðv x ; lx ; px Þ. Then, the similarity measure for two IFNs x ¼ ðlx ; v x ; px Þ and y ¼ ðly ; v y ; py Þ can be introduced as:

ð37Þ

where dðx; yÞ is the hamming distance given as:

1 ðjl ly j þ jv x v y j þ jpx py jÞ 2 x

pij ¼

pii ¼ 0:5; i; j ¼ 1; 2; . . . ; n

nðn 1Þ

Simðx; yÞ ¼ dðx; yC Þ dðx; yÞ

s X ðlÞ kl v ij ; l¼1

ðlÞ Simð~r ij ; ~rij Þ

Go to the next step. Step 5: Use the trapezoidal rules for numerical integration to ﬁnd the total consensus K min and K min , and the ﬁnal consensus of the group is CM1 ¼ ½K min ; K max , where

dðx; yÞ ¼

ð40Þ

ðlÞ

with dij ¼ lij þ pij , then go to next step.

lijaðlÞ ¼

ð39Þ

Thus, the consensus degree of a group of experts whose preferences are given as IFPRs is deﬁned as:

separate preference relations U ðlÞ ¼ ðlij Þnn and DðlÞ ¼ ðdij Þnn ðlÞ

n1 X n X 1 ðlÞ ðmÞ Simð~rij ; ~r ij Þ nðn 1Þ i¼1 j¼iþ1

SimðEl ; Em Þ ¼

ð38Þ

With (28), the similarity between any two experts El and Em is given as:

CM 3 ¼

s 1X ~ ðlÞ ; RÞ ~ SimðR s l¼1

ð44Þ

5.4. The distance based consensus measure When introducing the framework of intuitionistic fuzzy GDM, Liao et al. [33] also proposed a novel consensus measure. This consensus measure is based on the underlying normalized intuitionisT

~ ðlÞ ¼ ðx ~ ðlÞ ~ ðlÞ ~ ðlÞ tic fuzzy priority vector x 1 ; x2 ; . . . ; xn Þ of each expert El . For the experts El and Em , the distance measure between these two experts is n 1X ~ ðlÞ ~ ðmÞ d x i ; xi n i¼1 n 1X ~ lðlÞ ~ lðmÞ ~ v ðlÞ ~ v ðmÞ ~ pðlÞ ~ pðmÞ ¼ xi xi þ xi xi þ xi xi 2n i

dD ðEl ; Em Þ ¼

ð45Þ

~ iðlÞ ; x ~ iðmÞ is the normalized Hamming distance between where d x ~ iðlÞ and x ~ iðmÞ . Based on this distance measure, the the IFNs x consensus degree among a group can be deﬁned as:

CM 4 ¼ 1

max fdD ðEl ; Em Þg

l;m¼1;2;...;n

ð46Þ

140

Z. Xu, H. Liao / Knowledge-Based Systems 80 (2015) 131–142

It should be noted that there are some other types of distance measures for IFSs, such as the Spherical distance measures [72,73]. With different types of distance measures, the distance based consensus measures are quite different, but the underlying theoretical foundations are the same.

derived by the group. Then, the ordinal consensus measure for each expert can be deﬁned as: ðlÞ

CM6 ¼

n X i¼1

1

! ðlÞ joGi oi j n1

ð52Þ

Moreover, the consensus measure of the group is

5.5. The outranking ﬂow based consensus measure

s s X n joG oi j 1X 1X ðlÞ CM6 ¼ CM 6 ¼ 1 i s l¼1 s l¼1 i¼1 n1 ðlÞ

This type of consensus measure is motivated by the outranking ~ ðlÞ furnished by the ﬂow of a preference relation [4,28]. For an IFPR R expert El , the intuitionistic fuzzy negative outranking ﬂow of the alternative Ai is deﬁned as:

~ ðlÞþ ðAi Þ ¼ u

n 1 ðlÞ r n 1 j¼1;j–i ij

ð47Þ

and the intuitionistic fuzzy negative outranking ﬂow of the alternative Ai is

~ ðlÞ ðAi Þ ¼ u

n 1 ðlÞ r n 1 j¼1;j–i ji

ð48Þ

The intuitionistic fuzzy positive outranking ﬂow describes how an alternative Ai is outranking all the others, which is its power ~ ðlÞþ ðAi Þ, the better the alternative Ai . The character. The higher u intuitionistic fuzzy negative outranking ﬂow shows how an alternative Ai is outranked by all the others, which is its weakness char~ ðlÞ ðAi Þ, the better the alternative Ai . From this acter. The lower u point of view, we can use the outranking ﬂows to represent the overall value of each alternative. Therefore, the distance between any two experts El and Em can be deﬁned as:

dO ðEl ; Em Þ ¼

n 1 X ðlÞþ ðmÞþ ðlÞ ðmÞ d ui ; ui þ d ui ; ui 2n i¼1

ð49Þ

ðlÞþ ðmÞþ is the distance between the intuitionistic where d ui ; ui fuzzy positive outranking ﬂow given by the experts El and Em . With Eqs. (38), (49) turns out to be

dO ðEl ; Em Þ ¼

n 1 X lðlÞþ lðmÞþ v ðlÞþ v ðmÞþ ðjui ui j þ ju i ui j 4n i¼1

pðlÞþ

þ ju i

v ðmÞ

ui

pðmÞþ

ui

pðlÞ

j þ ju i

lðlÞ

j þ jui

pðmÞ

ui

lðmÞ

ui jÞ

v ðlÞ

j þ ju i

ð50Þ

Similar to Eq. (46), the consensus degree of such a group can be deﬁned as:

CM 5 ¼ 1

max fdO ðEl ; Em Þg

l;m¼1;2;...;n

ð51Þ

where dO ðEl ; Em Þ is the distance between the experts El and Em deﬁned as Eq. (50). 5.6. The ordinal consensus measure All the above consensus measures are based on the preferences given by the experts. However, it is also reasonable to deﬁne the consensus measure based on the ranking of alternatives. As there are many methods, shown in Section 4, to derive the priorities from each IFPR given by each individual expert, it is easy to obtain the ranks from every expert. Meanwhile, we can also derive the overall IFPR by some aggregation operators [27,65] and then yield the group ranks for alternatives. In such a situation, the consensus can be deﬁned as the difference between the orders of alternatives derived by the group IFPR and that of the individual IFPR. ðlÞ Mathematically, suppose that oi is the rank of the ith alternaG tive based on the lth expert, oi is the rank of the ith alternative

! ð53Þ

This kind of consensus measure considers the relative positions of alternatives derived by a single expert and those by the group. There is no need to calculate the distance or similarity degrees between any pair of alternatives. It is easy to be understood by the experts and other relevant persons. All the priority vector derivation methods can be used to calculate the individual ranks of the alternatives. Meanwhile, all the aggregation operators can be used to aggregate all individual IFPRs into an overall IFPR and then the priority vector derivation methods can be employed to derive the ranks of the group. 6. Summary and future directions As the IFPR can represent vague and imprecise preference information from preferred, not preferred and indeterminate points of view, it has shown deﬁnite advantages over the traditional multiplicative preference relation and fuzzy preference relation and has attracted many scholars’ attentions. In this paper, we have addressed a survey of IFPRs. Firstly, we have discussed the originality of IFPR, which is in line with the multiplicative preference relation and the fuzzy preference relation. Afterward, based on the different types of transitivity properties, different kinds of consistency deﬁnitions have been reviewed. All of these deﬁnitions can be classiﬁed into two sorts, i.e., the additive consistency and the multiplicative consistency. Based on the different deﬁnitions of consistency, many methods have been proposed to yield different kinds of priority vectors from an IFPR. We have recalled all of these priority vector derivation methods. Considering that group decision making is common in practice, a comprehensive survey has been given on distinct consensus measures for IFPRs. This survey gives an overview on the state of the art of IFPRs. Based on this review, we can ﬁnd that there are some directions that should be considered for future research: 1. Although we have reviewed different forms of consistency definitions for the IFPR, there is no one deﬁnition that can be taken as a standard. The consistency of the IFPR still needs to be investigated. As we can see, all the existing deﬁnitions of consistent IFPR are motivated by the transitivity properties of fuzzy preference relation or multiplicative preference relation. However, as the IFPR is a new type of preference relation, it should have some unique properties with respect to its consistency. 2. As for the IFPR which is not consistent, we should provide a much more reasonable way to improve its consistency or repair its consistency. Much more conditions should be explored to help an expert to furnish a consistent IFPR. 3. Much attention should be focused on the priority derivation methods. Although there are several different methodologies to yield the priorities from an IFPR, these methodologies are all based on some programming models, which are quite different from the priorities derivation method used in AHP. In addition, different methods have different advantages and disadvantages, it is very hard for us to decide which one is the best.

Z. Xu, H. Liao / Knowledge-Based Systems 80 (2015) 131–142

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