An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations

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Fuzzy Sets and Systems ••• (••••) •••–•••

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An axiomatic approach to approximation-consistency of triangular fuzzy reciprocal preference relations ✩

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Fang Liu

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, Witold Pedrycz

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, Zhong-Xing Wang , Wei-Guo Zhang

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a School of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China

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b Department of Electrical and Computer Engineering, University of Alberta, Edmonton, T6R 2G7, AB, Canada c School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, China d Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia

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e School of Business Administration, South China University of Technology, Guangzhou, Guangdong 510641, China

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Received 20 May 2016; received in revised form 13 February 2017; accepted 13 February 2017

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Abstract Under the assumption of rational economics, the consistency of judgments is one of the important issues in multiple criteria decision making (MCDM) methods. We propose three axiomatic properties of the consistent judgments in relative measurements being considered from the perspective of strict logic and rationality: (A1) the reciprocal property of pairwise comparisons, (A2) the invariance of consistency with respect to permutations of alternatives, and (A3) the robustness of the ranking of alternatives. These properties are further applied to analyze the consistency of the judgments expressed by positive real numbers and triangular fuzzy numbers, respectively. The inconsistency of comparison matrices encountered in the Analytic Hierarchy Process (AHP) can be considered to be the case of weakening the axiomatic property (A3). Fuzzifying the judgments is due to the weakening of axiomatic property (A1). A method of weakening the axiomatic properties for triangular fuzzy reciprocal matrices is proposed and a new concept of approximation-consistency is put forward to distinguish from the typical consistency. By considering the permutations of alternatives, an illustrative example is presented to show the use of the proposed procedures for solving the decision making problems with triangular fuzzy reciprocal preference relations. © 2017 Elsevier B.V. All rights reserved.

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Keywords: Decision analysis; Axiomatic property; Consistent judgment; Relative measurement; Triangular fuzzy number

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The work was supported by the National Natural Science Foundation of China (Nos. 71571054, 71201037), the Natural Sciences and Engineering Research Council of Canada (NSERC), Canada Research Chair Program and the Recruitment Program of Global Experts, the Guangxi Natural Science Foundation (No. 2014GXNSFAA118013), and the Guangxi Natural Science Foundation for Distinguished Young Scholars (No. 2016GXNSFFA380004). * Corresponding author at: School of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China. Fax: +86 771 3232084. E-mail addresses: [email protected] (F. Liu), [email protected] (W. Pedrycz), [email protected] (Z.-X. Wang), [email protected] (W.-G. Zhang). http://dx.doi.org/10.1016/j.fss.2017.02.004 0165-0114/© 2017 Elsevier B.V. All rights reserved.

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1. Introduction

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In daily lives, we are always faced with various decision making pursuits. Some decisions are simple, for example, say how to choose one of the fruits such as apple, banana and grape. Some decisions are complex, for instance, how to purchase a real estate. When one faces a complex problem with the conflicting criteria, an effective decision making method becomes helpful to make an acceptable and reasonable decision. Then the so-called multiple criteria decision making (MCDM) has attracted much attention and many methods have been proposed to solve various practical problems [24,28,55]. The MCDM can be broadly classified as Multiple Attribute Decision Making (MADM) and Multiple Objective Decision Making (MODM) [19]. For example, the Analytic Hierarchy Process (AHP) [48] is one of the MADM methods and it has been studied in a comprehensive way [7,8,27,56,50]. Based on the AHP, one may model a complex decision-making problem through forming and analyzing a hierarchy of criteria, subcriteria and alternatives. The decision maker (DM) may express her/his opinions by using the relative or the absolute measurements over a set of alternatives, then to make the best choice. Under the assumption of rational behavior, the decision maker tries to minimize her/his errors and produce the consistent judgments [54]. The consistency of the judgments is one of the important issues in the typical AHP [48], although the transitivity and the rationality are not included in the axiomatic system [49]. When a large number of alternatives are to be considered, it becomes difficult to realize a sequence of consistent pairwise comparisons. In order to measure the departure from this consistency requirement, the consistency index and the consistency ratio of comparison matrices have been proposed in [48]. The concept of acceptable consistency of multiplicative reciprocal comparison matrices has been given. The inconsistency ratio of 0.10 or lower indicates that the departure from consistency of the judgments to a certain degree in the AHP is tolerated. Moreover, many other inconsistency indices have been proposed for pairwise comparisons [10,14] and the axiomatic properties have been studied recently in [9,11,16]. When the goals and/or the constraints in a decision making problem are not completely defined, fuzzy sets could be used to propose a feasible decision making method in a fuzzy environment [5,13,31]. Furthermore, when the judgments of decision makers are expressed in terms of fuzzy numbers [12,57], the derived fuzzy MCDM became popular [18,19,29,32]. The definition of consistent multiplicative reciprocal matrices was given in [48]. However, when fuzzy numbers are applied to evaluate the judgments of decision makers, it is difficult to define the consistency of the given comparison matrices following the typical definition of consistency. For example, by directly extending the consistency idea in [48], Buckley [12] defined a consistent fuzzy positive reciprocal matrix with trapezoidal fuzzy numbers. It is further considered that the consistency definition in [12] can be used to analyze the consistency of triangular fuzzy reciprocal preference relations [58]. Moreover, the notions of reciprocity and consistency for pairwise comparison matrices with triangular fuzzy elements of Abelian linearly ordered group (Alo-group) over a real interval have been generalized in [46,47]. In addition, the analysis of the consistency properties in [58] has been presented in [43] where the existing shortcoming has been identified. Wang [61] has shown that the consistency definition of triangular fuzzy reciprocal preference relations in [43] depends on the labeling of alternatives. Similarly, there are various methods and some controversies emerged in defining the consistency of interval multiplicative reciprocal comparison matrices [1,36,59,60,62,64]. In particular, Dubois [21] concluded that a fuzzy-valued preference matrix lacks the properties of consistent multiplicative reciprocal comparison matrices, since the condition a˜ ij = 1  a˜ j i does not mean that a˜ ij ⊗ a˜ j i = 1, where a˜ ij are fuzzy intervals. Hereafter the symbols ⊗ and  denote the multiplication and division operators applied to fuzzy intervals, respectively. In virtue of the comparisons of the preference relations with numeric and fuzzy-valued entries, it is seen that we encounter difficulties with the consistency definitions of fuzzy-valued preference relations. Moreover, it is noted that the numeric and the fuzzy-valued entries of preference relations are only the different formats for evaluating the judgments of experts. From the viewpoint of logical and rational behavior, there exist some reasonable properties of consistent judgments and they should be considered and investigated. In the paper, we propose three axioms to describe the strict properties of the consistent judgments in relative measurements. The inconsistency and the fuzzifying measurements are all considered to be a case of weakening the axiomatic properties. Furthermore, one can see that the consistency definition of multiplicative reciprocal matrices satisfies the proposed properties. However, the known definitions of consistent triangular fuzzy reciprocal preference relations do not satisfy all the three axioms. It motivates us to propose the approximation-consistency definition of triangular fuzzy reciprocal preference relations. The structure of the paper is organized as follows. Section 2 discusses the reasonable properties of the consistent judgments in relative measurements and three axioms are provided. According to these axioms, the preference relations with

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numeric and triangular fuzzy numbers entries are analyzed. In Section 3, a method of weakening the axiomatic properties of the consistent judgments in relative measurements is proposed. A new concept of approximation-consistency for triangular fuzzy reciprocal preference relations is proposed and the properties are studied in detail. In Section 4, by considering the randomness in comparing alternatives, an example is presented to illustrate the proposed methods for coping with the decision making problems with triangular fuzzy reciprocal preference relations. Conclusions are covered in Section 5.

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fC : (xi , xj ) → pij ,

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In particular, the elements pij can be chosen as positive real numbers [49], fuzzy numbers [45,68], intuitionistic fuzzy values [2,3,38], linguistic terms [67,20] and others. When the set P forms the set of positive real numbers + , it is convenient to replace pij by aij . In the correspondence with the consistent judgments, it is noted that a consistent multiplicative reciprocal preference relation has been defined as follows:

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It is supposed that I is the identity in P. If the judgments are consistent, then the preference intensities pij in relative measurements can be indirectly obtained by the preference intensities pik and pkj as pij = pik ⊗ pkj (∀i, j, k = 1, 2, · · · , n), where ⊗ stands for the multiplication operator defined on the space P. It can be further computed that fC (xi , xi ) = I for ∀i ∈ {1, 2, · · · , n}. In particular, one has pij ⊗ pj i = I, which reflects the reciprocal property of the consistent judgments in relative measurements. Therefore, the first axiom characterizing reciprocity of the consistent judgments can be stated as follows

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A 1. (Reciprocal) For a set of alternatives X = {x1 , x2 , . . . , xn }, the preference intensities pij satisfy the reciprocal property of pij ⊗ pj i = I for ∀i, j = 1, . . . , n.

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2.1. The axiomatic properties

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In what follows, we carefully analyze the axiomatic properties of the consistent preference intensities pij in relative measurements from the viewpoint of logic and by engaging some aspects of rationality.

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• -transitivity: aij ≥ e, aj k ≥ e ⇒ aik ≥ e.

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where  is the group operation and e stands for the identity of the Alo-group. Additionally, the transitivity can be generally expressed as [15]

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• -reciprocity: aij  aj i = e, • -consistency: aij = aik  akj ,

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Furthermore, in the context of pairwise comparison matrices defined over a real Abelian linearly ordered group (Alo-group), the properties of reciprocity and consistency have been generalized as [16]

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Definition 1. [48] A multiplicative reciprocal preference relation A = (aij )n×n is consistent if aij = aik akj for ∀i, j, k = 1, . . . , n.

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It is assumed that the decision maker expresses her/his opinions as the relative preference intensities of alternatives through a process of pairwise comparisons. A set of alternatives X = {x1 , x2 , . . . , xn } is considered and the decision maker compares any pair of alternatives with respect to a criterion C. Let P be the set of the generalized preference information pij of the alternative xi over the alternative xj , and a mapping from X × X to P is defined as follows:

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2. Reasonable properties of the consistent judgments in relative measurements

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One can see that the axiom A1 reflects the basic property of judgments in relative measurements [16,35]. Moreover, when the decision maker faces a set of alternatives, she/he may randomly choose two alternatives to be compared

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with respect to a given criterion. There are n! possible cases for n alternatives to form a comparison matrix. Owing to the rational judgments, the consistency of the judgments should be independent of the randomness of the decision maker’s choices. In other words, the consistent judgments should not depend on the permutations of alternatives from the viewpoint of rational judgments. For convenience, let us define the function σ such that σ : i → k,

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with σ (i) = σ (j ) for i = j . Then the permutations of X = {x1 , x2 , . . . , xn } can be written as Xσ = {xσ (1) , xσ (2) , . . . , xσ (n) }. The invariance of the consistency of the judgments with respect to the permutations of alternatives can be described as the following axiom

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2.2. The case of real comparison ratios

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The axiom A3 shows that if the judgments are consistent with respect to a criterion, the action of adding or deleting an alternative cannot change the ranking of the old or remaining alternatives. However, if the ranking of alternatives is invariant when adding or deleting an alternative, the given pairwise comparison matrices are not necessarily consistent. It is seen that the reciprocal property, the invariance and the robustness reflect the strict logic and rationality of the judgments in relative measurements. They form the fundamentals of the consistent judgments when pairwise comparisons of alternatives with respect to a criterion are made by using relative measurements. Moreover, as an axiomatic system, the axioms should be independent and without logical contradiction, which is in agreement with the axiomatic properties of inconsistency indexes [9,11]. In what follows, based on the axiomatic properties, we analyze the cases when the preference intensities pij are positive real numbers [48] and fuzzy numbers [12,45], respectively. It is noticeable that the consistent multiplicative reciprocal preference relation defined in Definition 1 satisfies all the axioms A1–A3, implying that the axiomatic system is not logically contradictory. At the same time, each axiom is not derivable from the remaining one or two axioms, which indicates that the axioms are not redundant.

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A 3. (Robustness) For the set of alternatives Xσ = {xσ (1) , xσ (2) , . . . , xσ (n) } with any permutation σ , the ranking of the old or remaining alternatives with respect to a criterion is invariant when adding or deleting an alternative.

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The axiom A2 stresses that the judgments are consistent for any permutation of alternatives, if and only if there exists a permutation of alternatives such that the judgments are consistent. It is interesting to find that the axiom A2 is linked to Axiom 1 and Axiom 2 in [11] concerning the properties of inconsistency indexes. Under any permutation of alternatives, if an inconsistency index is invariant and it is used to identify all the consistent matrices, then the consistency of the judgments is invariant. To the end, it is considered that for some reasons, the decision maker may add or delete an alternative in the process of decision making. The ranking of the old or the remaining alternatives should be held under fully rational and logical behavior. In the typical AHP, many methods of evaluating the weight vector have been proposed such as the eigenvector method [48], the geometric mean (GM) method [4], the linear programming method [17] and others. Based on the different methods, the ranking of alternatives is invariant for the consistent judgments with respect to a criterion. It is convenient to call the above consideration as the robustness of the ranking of alternatives. That is, one has the following axiom

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A 2. (Invariance) For the set of alternatives Xσ = {xσ (1) , xσ (2) , . . . , xσ (n) } with permutation σ , the consistency of the judgments is invariant for any permutation σ and ∀i, j, k = 1, . . . , n.

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As mentioned before, it is convenient to write the comparison ratio of the alternative i over the alternative j in the set of positive real numbers as aij . According to Definition 1, the consistency condition aij = aik · akj means aii = aij · aj i . One further has the reciprocal property aij · aj i = 1 in virtue of aii = 1, meaning that the axiom A1 is satisfied. Moreover, it is noted that the reciprocal property aij = 1/aj i is a generic one of the axioms as the foundations of the AHP [49]. One can see that aij · aj i = 1 is equivalent to aij = 1/aj i when aij ∈ + . However, when the preference degrees pij are not positive real numbers, pij ⊗ pj i = I may not imply that pij = I  pj i . For example, when pij are interval numbers a˜ ij [51], the reciprocal relation a˜ ij = 1  a˜ j i does not mean that a˜ ij ⊗ a˜ j i = 1 [21].

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In addition, under the consideration of the axiom A2, we have an interesting theorem for the typical AHP:

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Theorem 1. Let A = (aij )n×n be a positive multiplicative reciprocal matrix and Aσ = (aσ (i)σ (j ) )n×n is obtained in virtue of A by considering a permutation σ of alternatives. One has

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Proof. The positive multiplicative reciprocal matrix A is written as ⎞ ⎛ x 1 x 2 · · · xn ⎜ x1 1 a12 · · · a1n ⎟ ⎟ ⎜ ⎟ ⎜ A = (aij )n×n = ⎜ x2 a21 1 · · · a2n ⎟ . ⎜ .. .. .. .. .. ⎟ ⎝ . . . . . ⎠ xn an1

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Aσ

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aσ (1)σ (2) 1 .. .

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Clearly, in the position (i, j ) of Aσ , the entry is aσ (i)σ (j ) , and the entry in the position (j, i) is aσ (j )σ (i) . Since σ (i), σ (j ) ∈ {1, 2, · · · , n} and the matrix A is multiplicative reciprocal, one obtains that aσ (i)σ (j ) · aσ (j )σ (i) = 1, and Aσ is a positive multiplicative reciprocal matrix. On the other hand, it is assumed that A is consistent and the condition of aij = aik · akj for ∀i, j, k ∈ {1, 2, · · · , n} ¯ j¯, k¯ ∈ {1, 2, · · · , n} such that i = σ (i), ¯ j = σ (j¯) is satisfied. For any permutation σ , there exist the integral numbers i, ¯ according to the definition of the function σ . Since the numbers i, j, k are arbitrary, one has a ¯ ¯ = and k = σ (k) σ (i)σ (j ) ¯ j¯, k¯ ∈ {1, 2, · · · , n}. That is, Aσ is consistent for any permutation σ . aσ (i)σ ¯ (k) ¯ · aσ (k)σ ¯ (j¯) for ∀i,

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Definition 2. A multiplicative reciprocal preference relation with permutations Aσ = (aσ (i)σ (j ) )n×n is consistent if aσ (i)σ (j ) = aσ (i)σ (k) · aσ (k)σ (j ) for ∀i, j, k = 1, . . . , n.

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In other words, if Aσ = (aσ (i)σ (j ) )n×n is a positive multiplicative reciprocal matrix and there exists a permutation σ0 such that Aσ0 is consistent, Aσ is consistent for any permutation σ . Theorem 1 reveals that the consistency of the judgments in the AHP satisfies the axiom A2. Furthermore, Definition 1 can be rewritten as follows:

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• Aσ is a positive multiplicative reciprocal matrix for each permutation σ . • If A is consistent, then Aσ is consistent for any permutation σ .

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Obviously, Definition 2 is a modification and an extension of Definition 1 under the consideration of the permutations of alternatives. For example, we consider the consistent multiplicative reciprocal matrix A1 . One can check that by applying any permutation σ to A1 , the given Aσ1 is consistent (that could be checked through some direct computations). By the way, it is found that when a multiplicative reciprocal matrix A is acceptably consistent according to the definition presented in [48], Aσ is still acceptably consistent. In other words, the acceptable consistency of multiplicative reciprocal matrices in [48] exhibits the invariance with respect to the permutations of alternatives. ⎞ ⎛ x1 x2 x3 x4 ⎜ x1 1 4 5 3 ⎟ ⎟ ⎜ ⎟. ⎜ 1/4 1 5/4 3/4 x A1 = ⎜ 2 ⎟ ⎝ x3 1/5 4/5 1 3/5 ⎠ x4 1/3 4/3 5/3 1

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Fig. 1. The variations of the weights ω1add (i) of xi (i = 1, 2, 3, 4, 5) versus a15 .

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Now we apply the axiom A3 to analyze the consistent multiplicative reciprocal preference relations. It is noted that the important problem of rank preservation and reversal in the AHP has been widely investigated [6,52,44]. One can prove that if the comparison matrices are kept to be consistent either by adding or deleting an alternatives, then the ranking of the old or remaining alternatives is not changed with respect to a single criterion [51,63]. On the contrary, if the ranking of the old or remaining alternatives is not changed when deleting or adding an alternative, the comparison matrices may not be consistent. The result reveals that the robustness of the ranking of alternatives in the axiom A3 indeed reflects the property of the ideal, strictly logical and strictly rational behavior of the decision maker. For instance, by deleting x3 and adding x5 in the consistent matrix A1 , one should produce the consistent multiplicative reciprocal comparison matrices with consistency as follows ⎞ ⎛ x1 x2 x4 ⎜ x1 1 4 3 ⎟ ⎟, ⎜ Adel 1 = ⎝ x 1/4 1 3/4 ⎠ 2 x4 1/3 4/3 1 ⎛

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and

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x5 a15 ⎟ ⎟ a15 /4 ⎟ ⎟. a15 /5 ⎟ ⎟ a15 /3 ⎠ 1

According to A1 , we obtain the weight vector as ω1 = (0.9077, 0.2269, 0.1815, 0.3026) by using the eigenvector del method [48] and arrive at x1 x4 x2 x3 . Application of Adel 1 leads to ω1 = (0.9231, 0.2308, 0.3077) and gives x1 x4 x2 , meaning that the ranking of x1 , x2 and x4 holds. Moreover, Fig. 1 shows the variations of the weights ω1add (i) of xi (i = 1, 2, 3, 4, 5) versus a15 . It is seen from Fig. 1 that the ranking of x1 x4 x2 x3 holds for any fixed a15 . In addition, we rewrite the comparison ratio of x1 over x4 as 2.8, and give the new comparison matrix A2 . If we still delete x3 and add x5 in the inconsistent matrix A2 respectively, the computing results show that the ranking of the remaining alternatives x1 , x2 , x4 holds and that of the old alternatives x1 , x2 , x3 , x4 is still not changed for a fixed a15 . The above phenomenon reveals that the invariance of the ranking of alternatives by adding or deleting an alternative does not mean the consistency of the judgments. From the above analysis, one can conclude that the consistent multiplicative reciprocal matrix in [48] satisfies the three axioms, which shows that the axiomatic system is not logically contradictory. Moreover, the three axioms reflect the reasonable properties of consistent judgments viewed from different aspects and they are not derivable from each other. When a comparison matrix is inconsistent, one can consider that it does not satisfy the axiom A3. The reason is that when an inconsistent comparison matrix is given, the phenomenon of rank reversal may appear [63]. When a multiplicative reciprocal preference relation is allowed to be acceptably consistent in the typical AHP, it can be considered that one has weakened the requirement of the axiom A3 to a certain degree. Finally, it is noted that the

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variance under inversion of multiplicative reciprocal matrices is considered as an axiomatic property of inconsistency indexes in [9]. From the consistency condition aij = aik · akj , one has 1/aij = 1/aik · 1/akj , meaning that the inversion of a consistent multiplicative reciprocal matrix is consistent. The invariance under preference relations by means of transposition is mainly based on the mathematical intuition and it is suitable to be considered as a derivable property of consistent multiplicative reciprocal matrices. Here since we investigate the basic properties of giving consistent judgments in relative measurements, the inversion of preference relations is not considered.

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In order to capture the uncertainty experienced by the decision maker in making pairwise comparisons, interval numbers, triangular fuzzy numbers and trapezoidal fuzzy numbers have been proposed to quantify such judgments [57, 12,51,45,39,41]. Without loss of the generality, here we consider that the judgments of decision makers are evaluated by triangular fuzzy numbers. ˜ on the set of real numbers  is Let us briefly recall the definition of triangular fuzzy numbers. A fuzzy number Q said to be a triangular fuzzy number if its membership function μQ˜ (x) :  → [0, 1] is defined as [23] ⎧ x−l ⎪ ⎪ ⎪ ⎨ m−l, l ≤x ≤m μQ˜ (x) = u − x , m ≤ x ≤ u , (5) ⎪ ⎪ u − m ⎪ ⎩ 0, otherwise ˜ respectively, while m is the median where l and u represent the lower and upper bounds of the fuzzy number Q, ˜ = (l, m, u). Q˜ = (l, m, u) and its membership value. For convenience, the triangular fuzzy number is denoted as Q function μQ˜ (x) are equivalent. Furthermore, let us recall the algebraic operations for triangular fuzzy numbers [34,57]. ˜ 2 = (l2 , m2 , u2 ), one has ˜ 1 = (l1 , m1 , u1 ) and Q Considering two triangular fuzzy numbers Q

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• addition ⊕:

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where li > 0, mi > 0, ui > 0, i = 1, 2, • division :

39 40

μQ˜ 1 Q˜ 2 (z) = sup (min(μQ˜ 1 (x), μQ˜ 2 (z × x)),

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x∈

42

where li > 0, mi > 0, ui > 0, i = 1, 2.

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2.3. The case of fuzzy comparison ratios

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It is noted that the multiplication and division operators lead to a polynomial form of the membership functions. In order to simplify the computations and enhance the efficiency, an approximation formula is feasible. Dubois and Prade [22] proposed the standard approximation formula for the multiplication operator in the form ˜ 1 ⊗ Q˜ 2  (l1 l2 , l1 m2 + l2 m1 , l1 u2 + l2 u1 ). Q

45 46 47 48

(6)

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˜ 1 ⊗Q ˜ 2 is not a triangular fuzzy number and the approximation formula reads as follows Moreover, as shown in [57], Q

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˜ 1 ⊗ Q˜ 2  (l1 l2 , m1 m2 , u1 u2 ). Q

(7)

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The error of the standard approximations was further analyzed and a new approximation formula was developed by adding a modification term to the standard approximation formulae [26]. Here for the sake of simplicity, the approximation formula (7) will be used in the following computations. Then a comparison matrix with triangular fuzzy numbers is defined as follows

5 6 7 8 9 10 11 12 13 14

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

49 50 51 52

4

7 8

(8)

9 10 11

where a˜ ij is interpreted as the triangular fuzzy degree of the alternative xi over the alternative xj . lij and uij represent the lower and upper bounds of the triangular fuzzy number a˜ ij , respectively and mij is the median value. lij , mij and uij are positive real numbers with lij ≤ mij ≤ uij .

12 13 14 15

Now making use of the axiomatic property A1, we analyze the consistency of triangular fuzzy reciprocal preference relations. When the triangular fuzzy reciprocal matrix A˜ is consistent with the condition a˜ ij = a˜ ik ⊗ a˜ kj , one has a˜ ij ⊗ a˜ j i = I according to the axiom A1. This means that lij · lj i = 1,

mij · mj i = 1,

uij · uj i = 1,

(9)

where the triangular fuzzy number multiplication (7) has been used. On the other hand, when one considers the preference relation A˜ under the reciprocity requirement, the property lij uj i = mij mj i = uij lj i = 1,

16 17 18 19 20 21 22

(10)

23

is satisfied. The combination of (9) and (10) leads to lij = uij and lj i = uj i , implying that the triangular fuzzy reciprocal matrix A˜ degenerates to the multiplicative reciprocal preference relation. The above result reveals that generally one cannot define the consistency of triangular fuzzy reciprocal preference relations to satisfy the axiom A1. The observation is in accordance with that in [21], where it has been pointed out that the consistency property of all entries does not adhere to the requirement a˜ ij = a˜ ik ⊗ a˜ kj , where a˜ ij are fuzzy intervals. Furthermore, here we elaborate why the consistency of triangular fuzzy reciprocal preference relations cannot be defined as the requirement in the axiom A1. In fact, fuzzy set theory is designed to model the vagueness of the objectives and it has decreased the requirements of strict rationality. Otherwise, the consistency of the judgments reflects the strict logic and rationality. Therefore the idea of fuzzy sets is incompatible with that of the consistency, and in essence the fuzzy judgments are inconsistent. Then based on the typical consistency idea, all the attempts to define the consistency of comparison matrices with fuzzy numbers are unfeasible. It is seen that the definitions of consistent comparison matrices with fuzzy numbers given in the literature [12,53,59,36,43,60–62] violate the reciprocal property presented in the axiom A1. For example, the following triangular fuzzy reciprocal matrix ⎛ ⎞ (1, 1, 1) (2/3, 1, 3/2) (2/3, 1, 3/2) (2/3, 1, 3/2) ⎜ (2/3, 1, 3/2) (1, 1, 1) (2/3, 1, 3/2) (2/3, 1, 3/2) ⎟ ⎟, A˜ 1 = ⎜ ⎝ (2/3, 1, 3/2) (2/3, 1, 3/2) (1, 1, 1) (2/3, 1, 3/2) ⎠ (2/3, 1, 3/2) (2/3, 1, 3/2) (2/3, 1, 3/2) (1, 1, 1)

24

is consistent by using the definition in [61]. However, it is easy to compute that a˜ 12 ⊗ a˜ 21 = (4/9, 1, 9/4) = (1, 1, 1), meaning that A˜ 1 does not satisfy the axiom A1. In Section 3, we focus on the methods of weakening the axiomatic properties A1–A3 for the consistency of triangular fuzzy reciprocal preference relations.

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3. The method of weakening the axiomatic properties

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Definition 3. [57] A preference relation with triangular fuzzy numbers is in the following form ⎞ ⎛ (1, 1, 1) (l12 , m12 , u12 ) · · · (l1n , m1n , u1n ) ⎜ (l21 , m21 , u21 ) (1, 1, 1) · · · (l2n , m2n , u2n ) ⎟ ⎟ ⎜ A˜ = (a˜ ij )n×n = ⎜ ⎟, .. .. .. .. ⎠ ⎝ . . . . (1, 1, 1) (ln1 , mn1 , un1 ) (ln2 , mn2 , un2 ) · · ·

15 16

1

46 47

It is worth noting that triangular fuzzy reciprocal preference relations do not satisfy the axiomatic property A1. Since the judgments with triangular fuzzy numbers exhibit more flexibility than the numeric ones, it is natural to weaken the axiomatic properties A1–A3 for triangular fuzzy reciprocal comparison matrices. In what follows, we propose a method that weakens the requirement and delivers the new concept of approximation-consistency of triangular fuzzy reciprocal comparison matrices.

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3.1. The weakened axiomatic properties

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Although the judgments expressed by triangular fuzzy numbers do not satisfy the axiom A1, the reciprocal property ˜ Similar of lij uj i = mij mj i = uij lj i = 1 should be considered when forming the triangular fuzzy reciprocal matrix A. to the allowance of the departure from the consistency requirement in the typical AHP, we soften the requirement of consistent triangular fuzzy reciprocal matrixes. In the case, the reciprocal property presented in the axiom A1 is weakened and it is rewritten as the form

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

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29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

5 6 7

A 1. (Reciprocal) For a set of alternatives X = {x1 , x2 , . . . , xn }, the comparison ratios a˜ ij = (lij , mij , uij ) satisfy the reciprocal property of lij uj i = mij mj i = uij lj i = 1 for ∀i, j = 1, . . . , n.

9

Under the consideration of the axiom A 1, the consistency of triangular fuzzy reciprocal matrices is different from the typical one in [48] and the condition in the axiom A1. In order to distinguish from the typical consistency, we propose the terminology of approximation-consistency to characterize the weak consistency of triangular fuzzy reciprocal matrices, which is similar to that presented in [37] for interval multiplicative reciprocal comparison matrices. Furthermore, the axioms associated with the invariance and the robustness may be modified. It should be pointed out that various methods may be used to weaken the axioms A2 and A3. Here we propose a possible way and for the sake of clarity, the corresponding theorems are given as follows:

12

A 2. (Invariance) For the set of alternatives Xσ = {xσ (1) , xσ (2) , . . . , xσ (n) } with permutation σ , the approximationconsistency of the judgments with triangular fuzzy numbers is invariant for any permutation σ and ∀i, j, k = 1, . . . , n.

10 11

13 14 15 16 17 18 19 20 21 22 23

A 3. (Robustness) For the set of alternatives Xσ = {xσ (1) , xσ (2) , . . . , xσ (n) } with permutation σ , the judgments with triangular fuzzy numbers for the old or remaining alternatives have approximation-consistency when one adds or deletes an alternative for any permutation σ .

27 28

4

8

23 24

3

24 25 26 27

A 2

As compared with the axioms A2 and A3, the main modification coming in the axiom is that the terminology of consistency is changed to that of approximation-consistency. The invariance of the ranking of alternatives in A3 is replaced by using that of approximation-consistency of the judgments with triangular fuzzy numbers. It is seen that the axiom A 3 is a weakening version of the axiom A3. The basic reason is that the invariance of the approximation-consistency of the judgments with triangular fuzzy numbers does not imply the invariance of the ranking of alternatives. Based on the axioms A 1–A 3, it has been found that the consistency definition of triangular fuzzy reciprocal preference relations in [43] has been incorporated with the axiom A 1, but it does not satisfy the axiom A 2. The consistency definition of the triangular fuzzy reciprocal matrix in [61] satisfies the axiom A 2, but it is not related to the axiom A 1. For example, in virtue of the definition in [61], the following matrix with triangular fuzzy numbers ⎛ ⎞ (1, 1, 1) (2/7, 1, 7/2) (2/3, 1, 3/2) (2/3, 1, 3/2) ⎜ (2/3, 1, 3/2) (1, 1, 1) (2/3, 1, 3/2) (2/3, 1, 3/2) ⎟ ⎟, A˜ 2 = ⎜ ⎝ (2/3, 1, 3/2) (2/3, 1, 3/2) (1, 1, 1) (2/3, 1, 3/2) ⎠ (2/3, 1, 3/2) (2/3, 1, 3/2) (2/3, 1, 3/2) (1, 1, 1) is consistent. However, it is obvious that A˜ 2 does not satisfy A 1 due to the fact that l12 · u21 = 2/7 · 3/2 = 3/7 = 1. Therefore, in order to satisfy all three axiomatic properties A 1–A 3, the definition of triangular fuzzy reciprocal preference relations with approximation-consistency should be still studied and a new one has to be developed. To the end, it is requisite to summarize the findings of the above analysis. Fig. 2 visualizes the development structure of pairwise comparisons. It presents the axioms being applied to describe the properties of the consistent judgments in relative measurements. Then the decision maker can evaluate her/his opinions by using positive real numbers and fuzzy numbers. In order to allow for the departure from the consistency requirement, the concepts of acceptable consistency and approximation-consistency are proposed by weakening the axiomatic properties of the consistent judgments. The development structure reveals that the axiomatic properties can be considered as a logical starting point when giving various preference intensities of alternatives.

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Fig. 2. The developing structure of pairwise comparisons with respect to a criterion.

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3.2. Approximation-consistency of triangular fuzzy reciprocal matrices

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21

In the section, we focus on the approximation-consistency of triangular fuzzy reciprocal matrices. Similar to Definition 2, a triangular fuzzy reciprocal preference relation with permutations is generally defined as follows:

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33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

xσ (n)

σ a˜ n1

σ a˜ n2

25 26 27 28

(11)

31

a˜ ijσ

where = (lσ (i)σ (j ) , mσ (i)σ (j ) , uσ (i)σ (j ) ) stands for the triangular fuzzy degree of the alternative xσ (i) over the alternative xσ (j ) . lσ (i)σ (j ) , mσ (i)σ (j ) and uσ (i)σ (j ) are positive real numbers with lσ (i)σ (j ) ≤ mσ (i)σ (j ) ≤ uσ (i)σ (j ) and lσ (i)σ (j ) uσ (j )σ (i) = mσ (i)σ (j ) mσ (j )σ (i) = uσ (i)σ (j ) lσ (j )σ (i) = 1. A˜ σ

Definition 4 entails that the entry in the position (i, j ) of is = (lσ (i)σ (j ) , mσ (i)σ (j ) , uσ (i)σ (j ) ), which is changing with the permutation σ . As compared to Definition 3, the main difference in Definition 4 is the introduction of permutation σ . In what follows, we present a definition of triangular fuzzy reciprocal preference relations with approximationσ M consistency property. Based on the analysis in [43], it is intuitively appealing to assume that AL σ = (Lij )n×n , Aσ = σ (Mijσ )n×n and AR σ = (Rij )n×n , where ⎧ ⎧ ⎨ lσ (i)σ (j ) , i < j ⎨ uσ (i)σ (j ) , i < j i=j , i=j , Rijσ = 1, (12) Lσij = 1, ⎩ ⎩ uσ (i)σ (j ) , i > j lσ (i)σ (j ) , i > j

a˜ ijσ

and Mijσ = mσ (i)σ (j ) , ∀i, j = 1, 2, . . . , n. It is further assumed that D

σ

(α, β, γ ) = (dijσ (α, β, γ ))n×n

29 30

· · · (1, 1, 1)

= ((Lσij )α (Mijσ )β (Rijσ )γ )n×n ,

σ M σ R with α + β + γ = 1, ∀α, β, γ ∈ [0, 1]. Then one has D σ (1, 0, 0) = AL σ , D (0, 1, 0) = Aσ , D (0, 0, 1) = Aσ .

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

(13)

48 49 50 51

51 52

23 24

Definition 4. A triangular fuzzy reciprocal preference relation with permutation σ is expressed as ⎞ ⎛ xσ (1) xσ (2) ··· xσ (n) σ σ ⎟ ⎜ xσ (1) (1, 1, 1) a˜ 12 ··· a˜ 1n ⎟ ⎜ σ σ ⎜ a˜ 21 (1, 1, 1) · · · a˜ 2n ⎟ A˜ σ = (a˜ ijσ )n×n = ⎜ xσ (2) ⎟, ⎟ ⎜ .. .. .. .. .. ⎠ ⎝ . . . . .

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22

Remark 1. Considering the permutations of alternatives, we obtain

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• lσ (i)σ (j ) ≤ dijσ (α, β, γ ) ≤ uσ (i)σ (j ) for ∀i, j = 1, 2, · · · , n, and • D σ (α, β, γ ) are multiplicative reciprocal preference relations.

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3

Now applying Definition 1, we formulate a definition of triangular fuzzy reciprocal preference relations with approximation-consistency.

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35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

5 6

Definition 5. If there is a permutation σ such that = = and σ ˜ sistent based on Definition 1, A is considered to be of approximation-consistency. AL σ

(Lσij )n×n , AM σ

(Mijσ )n×n

AR σ

=

(Rijσ )n×n

are all con-

7 8 9

One can find that the axioms A 1–A 3 are satisfied for triangular fuzzy reciprocal matrices with approximationconsistency, which is based on the following several reasons. First, when A˜ σ is with approximation-consistency, there σ¯ M σ¯ R σ¯ exists a permutation saying σ¯ , such that AL σ¯ = (Lij )n×n , Aσ¯ = (Mij )n×n and Aσ¯ = (Rij )n×n are all consistent. Then one has the reciprocal property Lσij¯ · Lσj¯ i = Mijσ¯ · Mjσ¯i = Rijσ¯ · Rjσ¯i = 1. According to Definition 4, it is seen that the reciprocal property is independent of the permutations. Thus the axiom A 1 is satisfied. Additionally, it is seen that M R L M R the consistency of AL σ¯ , Aσ¯ and Aσ¯ does not mean that Aσ , Aσ and Aσ are consistent for any permutation σ . Second, σ M R since the approximation-consistency of A˜ only requires that there is a permutation to make AL σ , Aσ and Aσ to be  consistent, the axiom A 2 is satisfied automatically. Third, when one adds or deletes an alternative, the consistency of M R  AL σ¯ , Aσ¯ and Aσ¯ for the old or the remaining alternatives will be held. That is to say, the axiom A 3 is satisfied. As compared to those in [43], Definition 5 is an extension of that in [43] by considering the permutations of alternatives to overcome the so-called shortcoming given in [61]. For instance, let us consider a triangular fuzzy reciprocal matrix expressed as ⎡ ⎤ (1, 1, 1) (1/3, 2/5, 1/2) (1/3, 2/5, 2/3) (5/4, 8/5, 3) ⎢ (2, 5/2, 3) (1, 1, 1) (2/3, 1, 2) (5/2, 4, 9) ⎥ ⎥. A˜ 3 = ⎢ ⎣ (3/2, 5/2, 3) (1/2, 1, 3/2) (1, 1, 1) (15/4, 4, 9/2) ⎦ (1/3, 5/8, 4/5) (1/9, 1/4, 2/5) (2/9, 1/4, 4/15) (1, 1, 1)

10

One can see that A˜ 3 is inconsistent according to that in [43], but it is of approximation-consistency by using Definition 5. σ σ M Moreover, let us consider the limit case of Definition 5 that for any permutation σ , AL σ = (Lij )n×n , Aσ = (Mij )n×n σ and AR σ = (Rij )n×n are all consistent. It is interesting to arrive to the following theorem:

28

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4

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

29 30 31 32 33

σ σ σ M R Theorem 2. If and only if AL σ = (Lij )n×n , Aσ = (Mij )n×n and Aσ = (Rij )n×n are all consistent for any permutaσ tion σ , A˜ is degenerated to a consistent multiplicative reciprocal matrix (Definition 1).

34

Proof. When A˜ σ is degenerated to a multiplicative reciprocal matrix, we have lσ (i)σ (j ) = mσ (i)σ (j ) = uσ (i)σ (j ) (i, j = 1, 2, · · · , n) for any permutation σ . Based on Theorem 1, the consistency of A˜ σ is independent of permutation σ . That σ σ σ M R is, AL σ = (Lij )n×n , Aσ = (Mij )n×n and Aσ = (Rij )n×n are all consistent for any permutation σ . Inversely, it is easy to see that the consistency of AM σ is independent of permutation σ according to Theorem 1. R However, when the consistency of AL σ and Aσ does not depend on the permutations, one has lσ (i)σ (j ) = uσ (i)σ (j ) (∀σ, ∀i, j = 1, 2, · · · , n). In fact, without loss of generality, it is assumed that i < j < k. For a permutation σ0 , application of the consistency of AL σ0 and (12) leads to

37

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lσ0 (i)σ0 (j ) = lσ0 (i)σ0 (k) · uσ0 (k)σ0 (j ) .

46

There must be another permutation σ1 and the integers i1 , j1 , k1 such that uσ0 (i)σ0 (j ) = uσ1 (i1 )σ1 (j1 ) = lσ1 (i1 )σ1 (k1 ) · uσ1 (k1 )σ1 (j1 ) = lσ0 (i)σ0 (k) · uσ0 (k)σ0 (j ) , so it gives lσ (i)σ (j ) = mσ (i)σ (j ) = uσ (i)σ (j ) consistent multiplicative reciprocal matrix.

35

(i, j = 1, 2, · · · , n) for any permutation σ and A˜ σ is degenerated to a

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It is seen from Theorem 2 that Definition 5 is feasible to define the approximation-consistency of triangular fuzzy reciprocal preference relations. On the other hand, it is difficult to produce a consistent multiplicative reciprocal preference relation in a practical decision making problem and assure that the departure from consistency is allowed to some degree [48]. Along this line, we give the definition of triangular fuzzy reciprocal preference relations with acceptable approximation-consistency as follows

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We investigate the properties of triangular fuzzy reciprocal preference relations with acceptable approximationconsistency. It is noted that some relevant results for the spectral radius of nonnegative matrices were given in [25]. Here we consider the special case of α + β + γ = 1 and formulate the following corollary

R S T λ max ≤ max{λmax , λmax , λmax }.

(14)

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Now we provide the following theorem to characterize the property of triangular fuzzy reciprocal preference relations with acceptable approximation-consistency. Theorem 3. Suppose that A˜ σ is a triangular fuzzy reciprocal preference relation with a permutation σ . A˜ σ has acceptable approximation-consistency if and only if there exists a permutation σ such that D σ (α, β, γ ) determined by (13) are acceptably consistent. σ M Proof. When A˜ σ has acceptable approximation-consistency, there is a permutation such that AL σ = (Lij )n×n , Aσ = σ (Mijσ )n×n and AR σ = (Rij )n×n are all acceptably consistent according to Definition 6. In other words, we have [48]

≤ 0.1,

λM max − n (n − 1)R.I.

≤ 0.1,

λR max − n (n − 1)R.I.

M where R.I. denotes the average of random index of multiplicative reciprocal comparison matrices. λL max , λmax , and L , AM and AR , respectively. Furthermore, it is assumed that λD λR are the greatest eigenvalues of A stands for max σ σ σ max the greatest eigenvalue of D σ (α, β, γ ). In virtue of Corollary 1, one has

λD max − n

≤

M R max{λL max , λmax , λmax } − n

28 29 30 31 32 33 34 35 36 37

≤ 0.1,

38 39 40 41 42 43

≤ 0.1.

(n − 1)R.I. (n − 1)R.I. σ D (α, β, γ ) are acceptably consistent for the permutation σ . On the other hand, if there exists a permutation such that D σ (α, β, γ ) determined by (13) have acceptable consisσ M σ R tency, D σ (1, 0, 0) = AL σ , D (0, 1, 0) = Aσ , D (0, 0, 1) = Aσ are all acceptably consistent. By using Definition 6, σ we note that A˜ has acceptable approximation-consistency property.

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13

25

Proof. The proof follows the discussion in [25] and the details are omitted here.

(n − 1)R.I.

43

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S T Corollary 1. Let R = (rij )n×n , S = (sij )n×n and T = (tij )n×n be three positive matrices. λR max , λmax and λmax are the β γ greatest eigenvalues of R, S and T , respectively. It is assumed that (α, β, γ ) = (φij (α, β, γ ))n×n = (rijα sij tij )n×n with α + β + γ = 1(∀α, β, γ ∈ [0, 1]) and λ max are the greatest eigenvalue of (α, β, γ ). Then we have

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3.3. The properties of acceptable approximation-consistency

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= Usually, it is feasible to apply Definition 6 to carry out decision analysis in a practical situation. When σ = (Mijσ )n×n and AR σ = (Rij )n×n are not acceptably consistent for all permutations, a consistency imM R proving method should be used to adjust AL σ , Aσ and Aσ for a permutation with acceptable consistency.

(Lσij )n×n , AM σ

19 20

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9

AL σ

15 16

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σ σ σ M R Definition 6. If there is a permutation σ such that AL σ = (Lij )n×n , Aσ = (Mij )n×n and Aσ = (Rij )n×n are all with acceptable consistency, A˜ σ is said to be with acceptable approximation-consistency.

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In particular, if we consider the approximation-consistency of triangular fuzzy reciprocal preference relations as the limit case of acceptable approximation-consistency, in light of Definition 5, the following corollary can be formulated.

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Corollary 2. Assume that A˜ σ is a triangular fuzzy reciprocal preference relation with a permutation σ . If and only if there is a permutation σ such that D σ (α, β, γ ) determined by (13) are consistent, A˜ σ has approximation-consistency.

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The above corollary can be proved by directly using Theorem 3; the details are omitted here. As compared to the findings in [43], the main difference is that here the permutations of alternatives are considered. In the end, it is worth to stress that here we only give a possible way to weaken the axioms A1–A3 and other methods may be still remaining to be further studied. Moreover, one can see that the consistency of the collective preference relations in group decision making is important, and the consistency indexes may be also dependent on the permutations of alternatives such as those in [40,42]. The approximation-consistency of the collective preference relations with fuzzy numbers should be addressed in the future. In the next section, we offer a practical example to illustrate the definitions and theorems for the present study.

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In practical applications, it is important that the consistency of pairwise comparison matrices is allowed to be made weaker to some extent [52]. According to the proposed definitions outlined in the previous subsection, it is suitable to consider the acceptable approximation-consistency of triangular fuzzy reciprocal preference relations. Then two important issues should be dealt with. One is how to check the acceptable approximation-consistency of triangular fuzzy reciprocal preference relations, and the other is how to derive the weight vector of alternatives by considering the randomness of comparing alternatives. According to Definition 6, one should check n! times for the acceptable approximation-consistency of a matrix with n alternatives. The reason is that the number of permutations is n! and all the corresponding matrices have to be enumerated. Fortunately, the methods of checking and emulating matrices have been investigated comprehensively in [37] in case of interval multiplicative reciprocal matrices. The proposed methods in [37] can be extended and used directly in the following computing. In addition, it is noted that many methods of determining the weight vector of alternatives have been proposed [48,59,36,35]. Hereafter, since we consider the randomness of comparing alternatives, the method of obtaining the weight vector should be given under the consideration of all the permutations. As shown in [37], the expected value is feasible to evaluate the final weight vector. For a permutation σ , it is assumed that the weight of the ith alternative/criterion is expressed as wiσ (i = 1, 2, · · · , n). Under the consideration of the randomness of pairwise comparisons, the expected value of wiσ can be computed as 1  σ wi = w . n! σ i

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ωkσ (α, β, γ ) = ωσσ (i) (α, β, γ ) = ⎝

n 

⎞1

n

⎛

dijσ (α, β, γ )⎠ = ⎝

j =1

⎡⎛

n 

⎞1

α  β  γ · ωk (AM · ωk (AR . σ ) σ)

39 40 41 42

44

(Lσij )α (Mijσ )β (Rijσ )γ ⎠

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j =1

j =1

ωk (AL σ)

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n

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⎞ 1 ⎤α ⎡⎛ ⎞ 1 ⎤β ⎡⎛ ⎞ 1 ⎤γ n n n n n n    ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎣⎝ Lσij ⎠ ⎦ · ⎣⎝ Mijσ ⎠ ⎦ · ⎣⎝ Rijσ ⎠ ⎦ j =1

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⎛

=

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34

Furthermore, since we consider that the decision maker expresses her/his opinions by using triangular fuzzy numbers, the triangular-fuzzy-number weights are feasible. Let us assume that σ is a permutation of {1, 2, · · · , n} with k = σ (i) (k, i ∈ {1, 2, · · · , n}). ωkσ (α, β, γ ) is the weight of the alternative k derived from D σ (α, β, γ ) (α, β, γ ∈ [0, 1], α + β + γ = 1). Applying the geometric mean [48], we have



16

33

1  σ σ w= (w1 , w2 , · · · , wnσ ). n! σ

51

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31

(15)

Then the weight vector of alternatives w = (w1 , w2 , · · · , wn ) is given as

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4. An illustrative example

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Fig. 3. The hierarchical structure of the illustrative example.

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In virtue of Remark 1, it is supposed that

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wk (L) = min

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wk (M) =

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ωk (AL σ) n , L k=1 ωk (Aσ )

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 ωk (AR σ) n , R k=1 ωk (Aσ )

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ωk (AM σ ) n , ω (AM k σ ) k=1

22 23 24

 ωk (AR ωk (AL σ) σ) wk (R) = max n , n , L R k=1 ωk (Aσ ) k=1 ωk (Aσ ) 

25 26

25 26

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27

and one obtains the normalized triangular fuzzy number weights as

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wkσ = {wk (L), wk (M), wk (R)},

30

k = 1, 2, · · · , n.

(18)

31 32 33 34 35 36 37

31

In what follows, based on the proposed methods of checking comparison matrices and obtaining weights, we investigate a practical decision making problem. The considered example is coming from the one given in [33] and reexamined in [58] and [43], respectively. In order to select a new factory to invest, a Turkish Motors Company, NEKYEK considers three serious alternatives after the first selection. They are designated as x1 (Istanbul), x2 (Ankara) and x3 (Izmir). A committee with three members has been invited to give advice on the alternatives with respect to the following four decision criteria:

38 39 40 41 42

45 46 47 48 49 50 51 52

32 33 34 35 36 37 38

• • • •

environmental regulation (C1 ), host community (C2 ), competitive advantage (C3 ), political risk (C4 ).

39 40 41 42

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30

43

The hierarchical structure is displayed in Fig. 3. By using the scale of 1/9 ∼ 9 [51], the committee first compares the four criteria and the comparison matrix with triangular fuzzy numbers are given as follows ⎛

⎞

C1 C2 C3 C4 ⎜ C1 (1, 1, 1) (3/2, 2, 5/2) (2/7, 1/3, 2/5) (5/2, 3, 7/2) ⎟ ⎟ ⎜ (1, 1, 1) (2/7, 1/3, 2/5) (7/2, 4, 9/2) ⎟ A˜ C = ⎜ ⎟. ⎜ C2 (2/5, 1/2, 2/3) ⎝ C3 (5/2, 3, 7/2) (5/2, 3, 7/2) (1, 1, 1) (5/2, 3, 7/2) ⎠ (1, 1, 1) C4 (2/7, 1/3, 2/5) (2/9, 1/4, 2/7) (2/7, 1/3, 2/5)

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Different from those in [33], here one checks the acceptable approximation-consistency of A˜ C . We construct the matrix AM by using the medium numbers in A˜ C as ⎞ ⎛ C1 C2 C3 C4 ⎜ C1 1 2 1/3 3 ⎟ ⎟ ⎜ 1/2 1 1/3 4 ⎟ C AM = ⎜ ⎟. ⎜ 2 ⎝ C3 3 3 1 3 ⎠ C4 1/3 1/4 1/3 1 CR(AM ) = 0.11 > 0.1,

AM

AM

The consistency ratio of is calculated as meaning that is unacceptable and it should be modified to that with acceptable consistency. For example, we choose the method in [65] to adjust AM as ⎞ ⎛ C1 C2 C3 C4 ⎜ C1 1 1.9555 0.3406 2.9892 ⎟ ⎟ ⎜ C 0.5114 1 0.3365 3.8813 ⎟ AM = ⎜ ⎟, ⎜ 2 ⎝ C3 2.9362 2.9721 1 3.0908 ⎠ 1 C4 0.3345 0.2576 0.3235

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R ˜ with the consistency ratio CR(AM ) = 0.0990 < 0.1. Moreover, we can check that AL σ and Aσ made from AC are unacceptably consistent at the same time for any permutation of criteria. Consequently, we choose a permutation such R as σ = (1, 2, 3, 4) to adjust AL σ and Aσ as follows ⎞ ⎛ C1 C2 C3 C4 ⎜ C1 1 1.4607 0.2944 2.4590 ⎟ ⎟ ⎜ L ⎜ 1 0.2926 3.3314 ⎟ Aσ = ⎜ C2 0.6846 ⎟, ⎝ C3 3.3971 3.4171 1 2.6382 ⎠ 1 C4 0.4067 0.3002 0.3790

27

and

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2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

⎛ ⎜ C1 ⎜ R Aσ = ⎜ ⎜ C2 ⎝ C3 C4

⎞

C1 C2 C3 C4 1 2.4470 0.4075 3.5079 ⎟ ⎟ 0.4087 1 0.4014 4.3976 ⎟ ⎟. 2.4539 2.4916 1 3.5718 ⎠ 0.2851 0.2274 0.2800 1 CR(AL σ ) = 0.0984 < 0.1

CR(AσR ) = 0.0972 < 0.1,

The consistency ratios are and respectively. From the above analysis, the normalized weights of criteria in triangular fuzzy numbers can be calculated as wc1 = (0.2245, 0.2484, 0.2719),

wc2 = (0.1761, 0.1892, 0.2082),

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wc4 = (0.0809, 0.0856, 0.0925),

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where the matrices and have been used. Moreover, with respect to the criteria C1 , C2 , C3 , and C4 , the judgments of alternatives are given as follows: ⎛ ⎞ C1 x1 x2 x3 ⎜ x1 (1, 1, 1) (2/5, 1/2, 2/3) (2/5, 1/2, 2/3) ⎟ ⎟, A˜ C1 = ⎜ ⎝ x2 (3/2, 2, 5/2) (1, 1, 1) (1/2, 2/3, 1) ⎠ (1, 3/2, 2) (1, 1, 1) x3 (3/2, 2, 5/2)

41

wc3 = (0.4541, 0.4768, 0.4918), AM ,

⎛

AL σ

AσR

⎞ C2 x1 x2 x3 ⎜ x1 (1, 1, 1) (2/5, 1/2, 2/3) (1/2, 2/3, 1) ⎟ ⎟, A˜ C2 = ⎜ ⎝ x2 (3/2, 2, 5/2) (1, 1, 1) (1, 3/2, 2) ⎠ (1/2, 2/3, 1) (1, 1, 1) x3 (1, 3/2, 2)

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Table 1 Weights of alternatives with respect to criteria, global weights and final scores.

1

2 3

Criterion

3

Weights of alternatives

4 5 6 7 8 9 10 11

x1

x2

x3

w1j (C1 ) w2j (C2 ) w3j (C3 ) w4j (C4 )

(0.1757, 0.1985, 0.2332) (0.1946, 0.2211, 0.2659) (0.1935, 0.2238, 0.2659) (0.1946, 0.2211, 0.2659)

(0.2979, 0.3469, 0.4077) (0.3909, 0.4600, 0.5035) (0.2679, 0.3695, 0.4294) (0.3909, 0.4600, 0.5035)

(0.3852, 0.4546, 0.5003) (0.2679, 0.3189, 0.3772) (0.3397, 0.4067, 0.5035) (0.2679, 0.3189, 0.3772)

wGj

(0.1773, 0.2168, 0.2741)

(0.2890, 0.3888, 0.4734)

(0.3096, 0.3945, 0.4971)

Sj

0.2227

0.3837

0.4004

⎛

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2

⎞

⎞ x1 x2 x3 C4 ⎜ x1 (1, 1, 1) (2/5, 1/2, 2/3) (1/2, 2/3, 1) ⎟ ⎟. A˜ C4 = ⎜ ⎝ x2 (3/2, 2, 5/2) (1, 1, 1) (1, 3/2, 2) ⎠ (1/2, 2/3, 1) (1, 1, 1) x3 (1, 3/2, 2) ⎛

As compared with the results reported in [33], the triangular fuzzy reciprocal matrices have been corrected and one has A˜ C2 = A˜ C4 . Similarly, the acceptable approximation-consistency of A˜ Ck (k = 1, 2, 3, 4) are checked. It has been found that A˜ Ck (k = 1, 2, 3, 4) all have acceptable approximation-consistency. One can further compute the weights wkj of alternatives with respect to criteria Ck . They are given in Table 1, where the formula (16) and the normalized geometric mean method have been used. Similar to the approach in [58,43], the global weights wGj and the final scores Sj of alternatives are shown in Table 1, where the following aggregation formulae have been used wGj =

k=1

wck ⊗ wkj = (lj , mj , uj ), Sj =

lj + mj + uj , 3

j = 1, 2, 3.

One can find that the raking of alternatives is x3 x2 x1 , which is in agreement with the results reported in [33], [58] and [43]. But it is seen that the present result is more convincing. There are several reasons behind that. One is that the weights in W and WER given by Kahraman et al. [33] exhibit zero elements, meaning that a criterion and an alternative in a situation have been neglected. The phenomenon is a limit case and it goes against the common sense. The other is that triangular fuzzy reciprocal matrices with consistency are constructed in [58] and [43] by using the (n − 1) paired comparisons. It is only an ideal case and many information data have been neglected. Here we allow for the departure from consistency to some extent and check the acceptable approximation-consistency of triangular fuzzy reciprocal matrices. The randomness of the decision maker experiencing in comparing alternatives is considered and the final weights are evaluated unbiasedly by using the expected value. The proposed methods are in accordance with the common sense yielding a reasonable ranking of alternatives.

42 43

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In multiple criteria decision making (MCDM), the judgments of decision makers are always expressed by using relative measurements. In the paper, from the viewpoint of strict logic and rationality, we have proposed three axiomatic properties of the consistent judgments in relative measurements as follows:

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5. Conclusions

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C3 x1 x2 x3 ⎜ x1 (1, 1, 1) (1/2, 2/3, 1) (2/5, 1/2, 2/3) ⎟ ⎟, A˜ C3 = ⎜ ⎝ x2 (1, 3/2, 2) (1, 1, 1) (1/2, 1, 3/2) ⎠ (1, 1, 1) x3 (3/2, 2, 5/2) (2/3, 1, 2)

4 

4

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• the reciprocal property, which is the basic requirement of constructing the consistent judgments in relative measurements; • the variance of consistency with respect to permutations of alternatives, which is under the consideration of the randomness experienced by the decision maker in making comparisons;

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• the robustness of the ranking of alternatives, which focuses on the invariance of ranking when adding or deleting an alternative.

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References

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[30] [66]

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Uncited references

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The authors would like to thank the anonymous referees for their valuable comments and suggestions for improving the paper.

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Acknowledgements

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The axiomatic properties have been used to analyze the cases with real and fuzzy comparison ratios, respectively. It is found that these axiomatic properties are satisfied for the case with real comparison ratios and unsatisfied for the case with fuzzy comparison ratios. One observes that the three axioms are independent and without logical contradiction. The inconsistency and the fuzzifying judgments are considered to be the cases of relaxing the corresponding axiomatic property. A method of softening the axiomatic properties for triangular fuzzy reciprocal preference relations has been proposed and a new concept of approximation-consistency has been given to distinguish from the typical consistency. It attempts to clarify the controversy behind the application of fuzzy set theory to the AHP and reconstruct the logical relation between real and fuzzy comparison ratios from the viewpoint of strict logic and rationality.

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49

50

50

51

51

52

52