Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process

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Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process Huchang Liao, Zeshui Xu ∗ Business School, Sichuan University, Chengdu, Sichuan 610065, China

a r t i c l e

i n f o

Article history: Received 18 September 2014 Received in revised form 4 April 2015 Accepted 5 April 2015 Available online xxx Keywords: Atanassov’s intuitionistic fuzzy set (A-IFS) Intuitionistic fuzzy preference relation Multiplicative consistency Group decision making Group intuitionistic fuzzy analytic hierarchy process (GIFAHP)

a b s t r a c t Intuitionistic fuzzy preference relations (IFPRs), which are based on Atanassov’s intuitionistic fuzzy sets (A-IFS), have turned out to be a useful structure in expressing the experts’ uncertain judgments, and the intuitionistic fuzzy analytic hierarchy process (IFAHP) is a method for solving multiple criteria decision making problems. To provide a theoretical support for group decision making with IFAHP, this paper presents some straightforward and useful results regarding to the aggregation of IFPRs. Firstly, a new type of aggregation operator, namely, simple intuitionistic fuzzy weighted geometric (SIFWG) operator, is developed to synthesize individual IFPRs. It is well known that for traditional comparison matrices, if all individual comparison matrices are of acceptable consistency, then their weighted geometric mean complex judgment matrix is of acceptable consistency. In this paper, we prove that this property holds for IFPRs as well if we use the SIFWG operator to synthesize the individual IFPRs. A numerical example is given to verify the theorems. It is pointed out that the well-known simple intuitionistic fuzzy weighted averaging (SIFWA) operator, the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy weighted geometric (IFWG) operator and the symmetric intuitionistic fuzzy weighted geometric (SYIFWG) operator do not have this property. Finally, the group IFAHP (GIFAHP) procedure is developed to aid group decision making process with IFPRs. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In many practical decision making situations, such as choosing a car to buy or selecting a person for higher managing position, the decision makers or experts may prefer to express their preference information by comparing each pair of objects and construct a preference relation which stores their preference information over a set of alternatives or criteria in a matrix. There are many different types of preference relations, such as multiplicative preference relations (MPRs) [1], fuzzy preference relations (FPRs) [2], and intuitionistic fuzzy preference relations (IFPRs) [3] which are based on Atanassov’s intuitionistic fuzzy sets (A-IFSs). Xu [4] made a survey on different kinds of preference relations and discussed their properties. The goal of establishing a preference relation is to derive the priority weights of objects from the preference relation and then rank the objects according to the priority weights [5]. However, in many cases, the preference relation may not be consistent. Consistency of a preference relation requires that the expert’s judgments yield no contradiction. Due to the fact that the lack of consistency for a preference relation can lead to inconsistent or incorrect conclusion, the consistency of a preference relation turns out to be a very important research topic, which has been attracting more and more scholars’ attention. The earliest work on consistency was done by Saaty [1], who proposed a consistency ratio from a MPR and suggested that a MPR is of acceptable consistency if its consistency ratio is less than 0.1. He also presented that it is difficult to obtain such MPR, especially when the MPR has a high order. For the inconsistent MPR, two ways can be used to deal with such kind of MPR: one is to return such an inconsistent MPR to the decision maker to reconsider constructing new MPR until the acceptable consistency is reached [6]; the other way is to improve the inconsistent MPR automatically by some iterative algorithms [7,8]. The first method is accurate and reliable but wastes a lot of time, and in some settings, if the decision makers do not want to interact with the experts, or if they cannot find the initial experts to re-evaluate

∗ Corresponding author. E-mail addresses: [email protected] (H. Liao), [email protected] (Z. Xu). http://dx.doi.org/10.1016/j.asoc.2015.04.015 1568-4946/© 2015 Elsevier B.V. All rights reserved.

Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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and alter their preferences, or if consistency must be urgently obtained, the feedback mechanism is out of use [5]. The iterative algorithms to improve the consistency of a MPR involve two sorts: modifying a single element [7] and modifying all elements [8]. Transitivity is the most important concept for consistency issue. As for FPR, Tanino [9,10] introduced 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 for a FPR. The weak transitivity is the minimum requirement condition to find out whether a FPR is consistent or not [11]. The max–max transitivity, the max–min transitivity, the restricted max–min transitivity and the restricted max–max transitivity do not imply reciprocity [12]. Both the additive transitivity and the multiplicative transitivity imply reciprocity, and thus have been used widely in practical applications. Although the additive consistency of a FPR is equivalent to Saaty’s consistency property of a MPR [11,13], it is in conflict with the [0,1] scale used for providing the preference values [13]. Therefore, it is an inappropriate property to model the consistency of FPR [14]. Xia and Xu [15] proposed an iterative algorithm based on the multiplicative consistency of a FPR to improve the consistency of a FPR until it is of acceptable consistency. They also proved that if all individual FPRs are multiplicative consistent, then their fused FPR is multiplicative consistent. With respect to IFPR, different kinds of consistency were proposed. Via the transformation between the IFPR and its corresponding interval-valued fuzzy preference relation (IVFPR), Xu [16] introduced a definition of additive consistent IFPR, which is based on the additive consistent IVFPR. Gong et al. [17] proposed another form of definition for additive consistent IFPR. Later, Wang [18] directly employed the membership and non-membership degrees to define the additive consistent IFPR. Based on the corresponding converted interval-valued membership degrees, Gong et al. [19] introduced the definition of multiplicative consistent IFPR. Xu et al. [20] also proposed another definition of multiplicative consistent IFPR based on the membership and non-membership degrees. Due to the fact that Gong et al.’s [19] definition of multiplicative consistent IFPR is not based on the Atanassov’s intuitionistic fuzzy judgments directly, and Xu et al.’s [20] definition is not reasonable in some cases, Liao and Xu [21] introduced an novel form of definition of multiplicative consistent IFPR based on the membership and non-membership degrees of the decision maker’s Atanassov’s intuitionistic fuzzy judgments. It should be noted that the elements in an IFPR are intuitionistic fuzzy values (IFVs) [22] which are composed of a membership degree, a non-membership degree and a hesitancy degree. Due to the powerfulness of A-IFSs in describing fuzziness and uncertainty [23–25], the IFPR is more useful than the MPR and the FPR in expressing comprehensive preference information. Nowadays, more and more scholars and practitioners applied the IFPR into practical decision making problems. Xu [3] developed an approach to group decision making based on IFPRs and then used it to assess the agroecological regions in Hubei Province, China. In order to handle complex decision making problems, Xu and Liao [26] extended the classical AHP method to the IFAHP and then employed it to global supplier development problem which includes both qualitative and quantitative factors. Liao and Xu [21] investigated the intuitionistic fuzzy priority derivation methods for an IFPR and then applied the methods in selecting the flexible manufacturing systems for a company. Later, they [27] also proposed some fractional models to determine the intuitionistic fuzzy priorities from the IFPRs in group decision making, and implemented the methods in evaluating the candidate exchange doctoral students from all over the world. Recently, Xu and Liao [28] made a state of the art survey of approaches to decision making with IFPR. In this paper, we focus our attention on group decision making with IFPRs. Group decision making, which involves diverse decision makers’ or experts’ opinions, takes place commonly in our daily life. In group decision making with any types of preference relations, the most important issue is how to aggregate all the experts’ preference information into reliable collective preference information. Forman and Peniwati [29] described two basic aggregation methods: • Aggregating individual priorities. The aggregation of individual priorities is suitable when the group acts as separate individuals. In such a case, the weighted arithmetic mean (WAM) method is usually used. • Aggregating individual judgments. The aggregation of individual judgments is suitable when the group acts as one individual and the opinions of the decision makers are explicitly exchanged. Aczel and Alsina [30] pointed out that the weighted geometric mean (WGM) operator is the only appropriate method for the aggregation of individual judgments when the weights of the decision makers are not equal. Bernasconi et al. [31] investigated the empirical properties of the various aggregation methods of aggregating individual judgments and individual priorities in group decision making. In the Atanassov’s intuitionistic fuzzy circumstances, different types of aggregation methods and operators have been proposed to fuse the Atanassov’s intuitionistic fuzzy preference information, such as the intuitionistic fuzzy weighted averaging (IFWA) operator [22,32], the intuitionistic fuzzy weighted geometric (IFWG) operator [33], the symmetric intuitionistic fuzzy weighted geometric (SYAIFWG) operator [34] and so on [35,36]. Choosing an appropriate aggregation operator to fuse the group preference information is very important. It is well known that if all the individual MPRs are of perfect consistency, their weighted geometric MPR is of perfect consistency [1,6]. Xu [37] further proved that if all the individual MPRs are of acceptable consistency, their weighted geometric MPR is also of acceptable consistency. Although Lin et al. [38] questioned about Xu’s conclusion, Groˇselj and Stirn [39] further provided another strict proof for this conclusion. This conclusion is very attractive because it implies that once all the individual MPRs pass the consistency test then the group MPR derived by the WGM operator would pass the consistency test as well and there is no need to check it. In fact, this property has been applied widely in group decision making [40–43]. As for group decision making with IFPRs, whether this conclusion still holds or not with the intuitionistic fuzzy aggregation operators is a question. Recently, based on Xu et al.’s [20] definition of multiplicative consistency, Xu and Xia [44] proved that if all individual IFPRs are perfect multiplicative consistent, then the fused IFPR aggregated by the SIFWA operator is perfect multiplicative consistent. This work can be seen as the first attempt to answer the question. However, their study has several flaws: (1) As presented by Liao and Xu [21], Xu et al.’s [20] definition of multiplicative consistent IFPR is unreasonable, and thus Xu and Xia’s [44] conclusion is somehow not reliable. (2) The conclusion of Xu and Xia [44] only reveals that the perfect multiplicative consistency of the aggregated IFPR under the condition that all individual IFPRs are perfect multiplicative consistent. If some of the individual IFPRs are not perfect multiplicative consistent but only acceptable multiplicative consistent, whether the conclusion still holds or not is a question.

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In order to answer the question and circumvent the above mentioned flaws, in this paper, we first propose a novel aggregation operator to fuse individual IFPRs and then prove that with such an operator, if all individual IFPRs are of perfect multiplicative consistency, then their aggregated IFPR is always perfect multiplicative consistent, and if all individual IFPRs are of acceptable multiplicative consistency, then their aggregated IFPR is also of acceptable multiplicative consistency. As the proposed aggregation operator can maintain the consistency property of each individual IFPR but other aggregation operators do not have this characteristic, only the proposed aggregation operator can be used to fuse IFPRs in group decision making. To better aid the decision making process, we then implement the proposed aggregation operator to group decision making with IFAHP. The rest of this paper is organized as follows: Section 2 presents some basic knowledge about A-IFS, IFPR and its consistency. The description of group decision making with IFPRs is conducted in Section 3. The SIFWG operator is also introduced in this section. Afterwards, we propose some useful theorems to show that with the SIFWG operator, the fused IFPR is of multiplicative consistency or acceptable multiplicative consistency under the condition that all individual IFPRs are of multiplicative consistency or acceptable multiplicative consistency. A numerical example is also given to validate the conclusion. In Section 4, we point out that the other kinds of intuitionistic fuzzy aggregation operators do not have this property. The proposed aggregation operator is implemented into group IFAHP in Section 5. The paper ends with some concluding remarks in Section 6. 2. Intuitionistic fuzzy preference relation and its consistency To facilitate our presentation, in what follows, let us review some basic concepts. 2.1. Atanassov’s intuitionistic fuzzy set ˜ in X is an object of Definition 1. [23,24] Let a crisp set X be fixed and A ⊂ X be a fixed set. An Atanassov’s intuitionistic fuzzy set (A-IFS) A the following form: ˜= A





(x, A (x), vA (x)) |x ∈ X ,

(1)

where the functions A : X → [0, 1] and vA : X → [0, 1] define the degree of membership and the degree of non-membership of the element ˜ on X, then x ∈ X to the set A, respectively, and for every x ∈ X, 0 ≤ A (x) + vA (x) ≤ 1 holds. For each A-IFS A A (x) = 1 − A (x) − vA (x)

(2)

is called the degree of non-determinacy (uncertainty) of the membership of element x ∈ X to the set A. In the case of ordinary fuzzy sets, A (x) = 0 for every x ∈ X. For convenience, Xu [22] called ˛ = (˛ , v˛ , ˛ ) an intuitionistic fuzzy value (IFV), where ˛ ∈ [0, 1], v˛ ∈ [0, 1], ˛ ∈ [0, 1] ˛ + v˛ ≤ 1, and let  be the set of all IFVs. Szmidt and Kacprzyk [45] justified that A (x) cannot be omitted when calculating the distance between two A-IFSs. However, considering the relations among all the three components: ˛ = 1 − ˛ − v˛ , in this paper, without any confusion, we here and also thereafter denote ˛ = (˛ , v˛ , ˛ ) only by its two former components ˛ = (˛ , v˛ ) for brevity. 2.2. Intuitionistic fuzzy preference relation For a decision making problem, let X = {x1 , x2 , · · · , xn } be a collection of alternatives under consideration. The experts are asked to evaluate these alternatives and provide their preferences through pairwise comparison. In the case that the experts cannot give crisp membership degrees of their preferences over alternatives because of vague information and incomplete knowledge about the preference degrees between any pair of alternatives, it is suitable to use the A-IFSs to express the affirmative, negative and hesitative information. Motivated by the idea of A-IFS, Szmidt and Kacprzyk [46] firstly proposed the concept of IFPR. Later, Xu [3] gave a simple notion of it. Definition 2.

 

˜ = r˜ij [3] An intuitionistic fuzzy preference relation (IFPR) on the set X = {x1 , x2 , ..., xn } is represented by a matrix R





n×n

,

where r˜ij =< (xi , xj ), (xi , xj ), v(xi , xj ), (xi , xj ) > for all i, j = 1, 2, . . ., n. For convenience, we let r˜ij = ij , vij , ij where ij denotes the degree to which the object xi is preferred to the object xj , vij indicates the degree to which the object xi is not preferred to the object xj , and ij = 1 − ij − vij is interpreted as an indeterminacy degree or a hesitancy degree, with the conditions: ij , ij ∈ [0, 1], ij , ij ∈ [0, 1], ij = vji , ii = vii = 0.5, ij = 1 − ij − vij , for all i, j = 1, 2, ..., n.

(3)

Xu [3] also proposed the concept of incomplete IFPR in which some of the preference values are unknown. Some algorithms were proposed to estimate the missing values for the incomplete IFPR [20]. For convenience, in this paper we assume that the experts can provide complete IFPRs. 2.3. Consistency of IFPR As presented in the introduction, consistency is a very important issue for any kinds of preference relations, and the lack of consistency in preference relations may lead to unreasonable conclusions. With respect to IFPR, several different forms of consistency have been proposed, which mainly involve two sorts: the additive consistency and the multiplicative consistency. Xu [16], Gong et al. [17], and Wang [18] proposed three different definitions of additive consistent IFPR, respectively. However, as pointed out by Liao and Xu [21], the additive consistency is inappropriate in modeling consistency because it is in conflict with the [0,1] scale, while the multiplicative consistency does not have this flaw and thus more suitable in measuring the consistency of an IFPR. Based on the transformation between an IFPR and its corresponding IVFPR, Gong et al. [19] introduced a definition of multiplicative consistent IFPR, but, the multiplicative consistency they gave was not based on the IFVs directly and thus was not easy to use and may not be sufficient in representing the original intuitionistic Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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fuzzy preference information of the experts. Xu et al. [20] proposed another definition of multiplicative consistent IFPR, which was based on the membership and non-membership degrees of IFVs, shown as follows: ˜ = (rij ) [20] An IFPR R with rij = (ij , vij ) (i, j = 1, 2, ..., n) is multiplicative consistent if n×n

Definition 3.

ij =

vij =

⎧ ⎪ ⎨ 0, ⎪ ⎩

(ik , kj ) ∈

(0, 1) , (1, 0)



, for all i ≤ k ≤ j,

ik kj ik kj + (1 − ik )(1 − kj )

⎧ ⎪ ⎨ 0, ⎪ ⎩



,

(4)

otherwise

(vik , vkj ) ∈



(0, 1) , (1, 0)

vik vkj , otherwise vik vkj + (1 − vik )(1 − vkj )

 , for all i ≤ k ≤ j.

(5)

Liao and Xu [21] pointed out that the above definition was not reasonable in some cases because with the above consistency conditions, the relationship ij · jk · ki = ik · kj · ji (for all i, j, k = 1, 2, . . ., n) cannot be derived any more. Then, they introduced a general definition of multiplicative consistent IFPR, shown as follows: Definition 4. satisfied:

˜ = (˜rij ) [21] An IFPR R with r˜ij = (ij , vij ) is called multiplicative consistent if the following multiplicative transitivity is n×n

ij · jk · ki = vij · vjk · vki , for all i, j, k = 1, 2, . . ., n.

(6)

Liao and Xu [21] have further clarified that the conditions in Definition 3 satisfy (6), which implies the consistency measured by the conditions given in Definition 3 is a special case of multiplicative consistency defined as Definition 4 for an IFPR. Hence, in general, Definition 3 is not sufficient and suitable to measure the multiplicative consistency for an IFPR. In this paper, we use Liao and Xu’s [21] definition of multiplicative consistency shown as Definition 4 as a standard to measure the consistency of an IFPR. 3. Consistency of the fused IFPR with SIFWG operator in group decision making In this section, we mainly discuss how to aggregate different IFPRs provided by distinct experts into a fused IFPR without losing the consistency property of each individual IFPR. 3.1. Description of the intuitionistic fuzzy group decision making problem A group decision making problem with intuitionistic fuzzy preference information can be described as follows: Let X = {x1 , x2 , . . ., xn } be the set of alternatives under consideration, and E = {e1 , e2 , . . ., es } be the set of decision makers, who are invited to evaluate the

s alternatives. The weight vector of the decision makers el (l = 1, 2, ..., s) is  = (1 , 2 , ..., s )T , where l > 0, l = 1, 2, ..., s, and  = 1, l=1 l which can be determined subjectively or objectively according to the decision makers’ expertise, experience, judgment quality and related knowledge. In general, they can be assigned equal importance if there is no evidence to show significant differences among the decision makers or specific preference on some decision makers. In the existing literature, many techniques have been developed for determining the decision makers’ weights (for more information, refer to Refs. [47–49]). In this paper, we assume that the weights of experts can always be given. In the process of decision making, it is straightforward for the decision makers to provide their assessments over the alternatives through pairwise comparisons. In many cases, if the problem is very complicated or the decision makers are not familiar with the problem and thus they cannot give explicit preferences over the alternatives, it is suitable to use the IFVs, which express the preference information from three aspects: “preferred”, “not preferred”, and “indeterminate”, to represent their opinions. Suppose that the decision maker ek provides (l) (l) (l) (l) his/her preference values for the alternative xi against the alternative xj as r˜ij = (ij , vij ), (i, j = 1, 2, . . ., n, l = 1, 2, . . ., s) in which ij (l)

denotes the degree to which the object xi is preferred to the object xj , vij indicates the degree to which the object xi is not preferred to the (l) object xj , and ij

=

(l) 1 − ij

(l) − vij

(l) (l) (l) (l) ˜ (k) = ij = vij , ii = vii = 0.5, for all i, j = 1, 2, . . ., n, l = 1, 2, . . ., s. The IFPR R

⎛

(l)

r˜11

(l)

r˜12

(l)

· · · r˜1n

⎞

⎜ (l) (l) ⎟ (l) ⎟ ⎜ r˜21 r˜22 · · · r˜2n ⎜ ⎟ R =⎜ ⎟. .. .. ⎟ ⎜ .. .. ⎝ . . . . ⎠ ˜ (l)

(l)

r˜n1

(l)

r˜n2

···

(l)

(l)

(l)

(l)

is interpreted as an indeterminacy degree or a hesitancy degree, subject to ij , vij ∈ [0, 1], ij + vij ≤ 1,



(k)

r˜ij



for the lth decision maker can be written as: n×n

(7)

(l)

r˜nn

For any a group decision making problem with s decision makers, we can obtain s individual IFPRs with the form of (7). In order to find the final result of the problem, it is needed to aggregate all these s individual IFPRs into a collective one. Before doing this, as presented in the introduction, we shall first check the consistency of each IFPR and make sure that all of them are consistent; otherwise the unreasonable results may be produced. Consistency checking is a highly important step in IFAHP [26] and has been addressed by many scholars (for more details, refer to Refs. [16–21,26,27]). As to those inconsistent IFPRs, we can return them to the experts to reevaluate and construct new ones until the consistency is reached or acceptable, or repair the inconsistent IFPRs automatically by some iterative algorithms. After all the individual IFPRs are consistent or acceptable consistent, we can aggregate them into a collective one. However, two questions Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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arise immediately: how to aggregate the individual IFPRs into an overall one? Does the aggregation methodology make sense? These two questions are fundamental in any group decision making problem and thus should be answered clearly in advance. In the following, we first propose a simple intuitionistic fuzzy weighted geometric (SIFWG) operator to aggregate the individual IFPRs into an overall one, and then give some theorems to show that our SIFWG operator makes sense in fusing the intuitionistic fuzzy preference information in group decision making. 3.2. Simple intuitionistic fuzzy weighted geometric (SIFWG) operator Up to now, quite a large number of aggregation operators have been proposed to fuse the Atanassov’s intuitionistic fuzzy information (for more information, refer to Refs. [22,32–36]). The most popular used intuitionistic fuzzy aggregation operators are the simple intuitionistic fuzzy weighted averaging (SIFWA) operator [32], the intuitionistic fuzzy weighted averaging (IFWA) operator [22], the intuitionistic fuzzy weighted geometric (IFWG) operator [33], and the symmetric intuitionistic fuzzy weighted geometric (SYIFWG) operator [34]. All these aggregation operators can be used to fuse the Atanassov’s intuitionistic fuzzy information for a single person, but they may not be appropriate in fusing the individual Atanassov’s intuitionistic fuzzy preference information into the collective information. In the next section, we would give some theoretical analysis to show that the above four representational aggregation operators are not reasonable in fusing the individual IFPRs into the collective one when the weights of the decision makers are not equal. But, as the departure of our discussion, let us first propose a new simple intuitionistic fuzzy weighted geometric (SIFWG) operator, which is very simple but different to those existing operators. Definition 5. Let ˛l = (˛l , v˛l ) (l = 1, 2, . . ., s) be a collection of IFVs,  = (1 , 2 , . . ., s )T be the aggregation-associated vector such that

s 0 ≤ l ≤ 1,  = 1, then a simple intuitionistic fuzzy weighted geometric (SIFWG) operator is a mapping SIFWG :  n →  , where l=1 l SIFWG (˛1 , ˛2 , . . ., ˛s ) =

s

⊗ ˛l l l=1

=

 s 

s 

l=1

l=1

˛ll ,



T

In the case where  = 1/n, 1/n, . . ., 1/n which is defined as follows: SIFG (˛1 , ˛2 , . . ., ˛s ) =

s

1/n ⊗ ˛l l=1

=



v˛ll

.

(8)

, the SIFWA operator reduces to a simple intuitionistic fuzzy geometric (SIFG) operator,

 s 

s 

l=1

l=1

1/n ˛l ,



v1/n ˛l

.

(9)

Let ˛l = (˛l , ˛l )(l = 1, 2, . . ., s) be a collection of IFVs,  = (1 , 2 , . . ., s )T be the aggregation-associated vector such that 0 ≤ l ≤ 1,

s  = 1. The SIFWG operator has the following properties. and l=1 l Property 1.

(Idempotency) If all the IFVs ˛l (l = 1, 2, . . ., s) are equal, i.e., ˛l = ˛, l = 1, 2, . . ., s, then

SIFWG (˛1 , ˛2 , . . ., ˛s ) = ˛. Property 2.

(10)

(Boundedness) For any , we have

−

˛ ≤ SIFWG (˛1 , ˛2 , . . ., ˛s ) ≤ ˛+ ,

(11)

where ˛− = (min{˛l }, max{˛l }), ˛+ = (max{˛l }, min{˛l }). l

Property 3.

l

l

l

(Monotonicity) Let ˛∗l = (˛∗ , ˛∗ )(l = 1, 2, . . ., s) be a collection of IFVs, if for any l, ˛l ≤ ˛∗ and ˛l ≥ ˛∗ . Then l

SIFWG (˛1 , ˛2 , . . ., ˛s ) ≤

l

SIFWG (˛∗1 , ˛∗2 , . . ., ˛∗s ).

l

l

(12)

The proofs of these three properties are very easy so we omit the proof process. It is noted that in the above monotonicity property, the partial order of A-IFS is used. The partial order of A-IFS is defined as ˛i ≺ - AIFS ˛j ⇔ ˛i ≤ ˛j ∧ v˛i ≥ v˛j , for two IFVs ˛i = (˛i , v˛i ) and ˛j = (˛j , v˛j ) [50]. The top and bottom elements of A-IFS corresponding to the partial order are 1AIFS = (1, 0) and 0AIFS = (0, 1), respectively. Definition 6. [36]. A function fAIFS :  n →  is an aggregation function on A-IFSs if it is monotone with respect to the partial order ≺ - AIFS and satisfies fAIFS (0AIFS , . . ., 0AIFS ) = 0AIFS and fAIFS (1AIFS , . . ., 1AIFS ) = 1AIFS .

From Property 3, the following property holds immediately. Property 4.

The SIFWG operator is an aggregation function over A-IFSs with respect to the partial order.

When an aggregation function requires the sort operation, such as in the OWA function and the discrete Choquet integral, the partial order is not sufficient to rank any A-IFSs, and thus choosing a linear (total) order between A-IFSs turns out to be an important yet difficult is called a linear order [21]. problem [51,52]. If any two IFVs ˛i and ˛j in an A-IFS are comparable, i.e., either ˛i ≺ - ˛j or ˛j ≺ - ˛i , the order ≺ Many scholars have proposed different linear orders for A-IFSs [22,53,54]. Based on the score function S(˛) = ˛ − v˛ and the accuracy function H(˛) = ˛ + v˛ of an IFV ˛(˛ , v˛ ), Xu [22] introduced a linear order to rank any two IFVs ˛i (˛i , v˛i ) and ˛j (˛j , v˛j ): Scheme 1: (1) if S(˛i ) < S(˛j ), then ˛i < ˛j , which indicates ˛i is smaller than ˛j ; (2) if S(˛i ) = S(˛j ), then

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(i) if H(˛i ) < H(˛j ), then ˛i <˛j ; (ii) if H(˛i ) = H(˛j ), then ˛i =˛j , which means that ˛i is equal to ˛j . Beliakov et al. [36] pointed out that the above mentioned ordering has one undesirable property; it is not preserved under multiplication by a scalar: ˛i < ˛j does not necessarily imply ˛i < ˛j where  is a scalar. Szmidt and Kacprzyk [53] also proposed a function (˛) = 0.5(1 + ˛ )(1 − ˛ ) to rank IFVs, and the smaller the value of (˛), the greater the IFV ˛ in the sense of the amount of positive information included and reliability of information. However, only based on this single function, we cannot yield a linear order between IFVs. Afterwards, based on the idea of the TOPSIS method, Zhang and Xu [54] introduced a similarity function L(˛) = (1 − v˛ )/(1 + ˛ ) for an IFV ˛ = (˛ , v˛ ), and then proposed a total order method to rank any two IFVs ˛i = (˛i , v˛i ) and ˛j = (˛j , v˛j ): Scheme 2: (1) If L(˛i ) > L(˛j ), then ˛i > ˛j . (2) If L(˛i ) = L(˛j ), then (i) If H(˛i ) > H(˛j ), then ˛i > ˛j ; (ii) If H(˛i ) = H(˛j ), then ˛i = ˛j . Liao and Xu [21] have made an in-depth comparison over these ranking methods from numerical and theoretical points of view. In addition, motivated by the idea of the admissible order of intervals [52], Liao and Xu [21] also introduced the concept of admissible order for A-IFSs, which is a special linear order and refines the partial order ≺ - AIFS of A-IFS. Besides the SIFWG operator, there are some other geometric aggregation operator for A-IFSs, such as the intuitionistic fuzzy weighted geometric (IFWG) operator [33] and the symmetric intuitionistic fuzzy weighted geometric (SYIFWG) operator [34]. The IFWG is defined as: IFWG (˛1 , ˛2 , ..., ˛s ) =

s

⊗ ˛l l l=1

=

 s 

s 

l=1

l=1



s

˛ll , 1 −

The SYIFWG operator is given as: SYIFWG (˛1 , ˛2 , ..., ˛s ) =

s

⊗ ˛l l l=1

=

s l=1



(1 − v˛l )



˛ll +

l

.

  l l=1 ˛l m (1 − ˛l )l l=1



(13)

s ,

s

 v l l=1 ˛l

+

 v l l=1 ˛l s (1 − v˛l )l l=1



 .

(14)

Motivated by the idea of representable aggregation function on interval-valued fuzzy sets [36], the representable aggregation function fAIFS on AIFSs can be defined as fAIFS (˛1 , ˛2 , . . ., ˛s ) = ˇ with ˇ = (f1 (˛1 , ˛2 , . . ., ˛s ), f2 (v˛1 , v˛2 , . . ., v˛s ), ) and ˛l = (˛l , v˛l ), l = 1, 2, . . ., s. From this point of view, we can see that the SIFWG operator and the SYIFWG operator are representable aggregation operators in which the SIFWG operator is a natural extension of the weighted geometric (WG) operator and the SYIFWG is a natural extension of the symmetric weighted geometric (SWG) operator, but the IFWG operator is not a representable aggregation function. However, the following example shows that the SYIFWG operator is not consistent with the aggregation operators on the ordinary fuzzy sets when  = 1 − v: Example 1. Consider two IFVs ˛1 = (0.9, 0.1) and ˛2 = (0.5, 0.5) with their weights  = (0.8, 0.2), then for the classical fuzzy values which are only membership values, the weighted geometric (WG) aggregation value over 0.9 and 0.1 is WG(0.9, 0.5) = 0.90.8 · 0.50.2 = 0.8002. As for the IFWG operator, we have IFWG (˛1 , ˛2 ) = (0.90.8 · 0.50.2 , 1 − 0.90.8 · 0.50.2 ) = (0.8002, 0.1380). As for the SIFWG operator, we have SIFWG (˛1 , ˛2 ) = (0.90.8 · 0.50.2 , 1 − 0.90.8 · 0.50.2 ) = (0.8002, 0.1998). As for the SYIFWG operator, we have SYIFWG (˛1 , ˛2 ) = (

0.90.8 · 0.50.2 0.9

0.8

· 0.5

0.2

+ 0.1

0.8

· 0.5

0.2

,

0.10.8 · 0.50.2 0.1

0.8

· 0.50.2 + 0.90.8 · 0.50.2

) = (0.8529, 0.1471).

From the results, we can see that the IFWG operator and the SIFWG operator yield the same result as that of the weighted geometric operator on the ordinal fuzzy sets. Hence, the SIFWG operator is not only a representable aggregation function but also consistent with aggregation operation on the ordinary fuzzy sets. As this paper is mainly focused on the consistency property of IFPRs in the process of aggregation with the operators, in what follows, we shall not pay our attention on the mathematical properties of these operators but investigate the advantages of the SIFWG operator over the other intuitionistic fuzzy aggregation operators from the point of maintaining the consistency of original individual IFPRs. 3.3. Consistency of the fused IFPR with SIFWG operator After we have established the intuitionistic fuzzy aggregation operators, the following thing we have to do is to check whether the aggregation operators are reasonable or not in synthesizing the individual Atanassov’s intuitionistic fuzzy information into the overall preference information. It should be noted that in the setting of traditional MPRs, if all the individual MPRs are of perfect consistency, their weighted geometric preference relation is of perfect consistency [1,6]; in addition, Xu [37] have further pointed out that if all the individual MPRs are of acceptable consistency, their weighted geometric preference relation is also of acceptable consistency. That is to say, the aggregation operator should not change the consistency properties of the preference relations. Once the individual preference relations have passed the consistency checking process, there is no need to check the consistency of the aggregated preference relation. This property is intuitive and significant. As to IFPRs, if the aggregation operator is reasonable, then the aggregated IFPR should also have this property, i.e., if we Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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make sure that each individual IFPR is of consistency or acceptable consistency, then their aggregated IFPR should be also of consistency or acceptable consistency. This is a very important property for group decision making with IFPRs. In the following, we shall prove that with the SIFWG operator, this property holds for any IFPRs. Lemma 1.

[34] If xl > 0, l > 0, l = 1, 2, . . ., s, and (l)

Let R(l) = (rij )

Theorem 1.

(l)

n×n

s

(l)

(l)

s

xl l=1 l

= 1, then

 l=1 l

≤

s

x, l=1 l l

with equality if and only if x1 = x2 = · · · = xs .

(l)

with rij = (ij , vij , ij )(l = 1, 2, . . ., s) be s individual IFPRs given by the decision makers el (l = 1, 2, . . ., s)

respectively, and  = (1 , 2 , . . ., s )T be the weight vector of the decision makers with 0 ≤ l ≤ 1, by the SIFWG operator is also an IFPR, where ¯ ij ),  ¯ ij , v¯ ij ,  ¯ ij = r¯ ij = (

s  

(l)

ij

l

s  

, v¯ ij =

l=1 (l)

(l) v(l) = 0.5, ij = ii

l

= 1, then their fusion R¯ = (¯rij )n×n

 l=1 l

¯ ij = 1 −  , ¯ ij − v¯ ij , i, j = 1, 2, ..., n.

(15)

l=1 (l)

Since R(l) = (rij )

Proof.

v(l) ij

s

(l)

(l)

(l)

(l)

(l)

(l)

(l = 1, 2, . . ., s) are IFPRs, according to Definition 2, we have ij , vij ∈ [0, 1], ij + vij ≤ 1, ij = vji , ii =

n×n (l) (l) 1 − ij − vij ,

for all i, j = 1, 2, . . ., n. From Lemma 1, it follows that 0 ≤

 ¯ ij ∈ [0, 1]. Similarly, v¯ ij ∈ [0, 1]. s  

In addition,  ¯ ij + v¯ ij =

(l)

ij

l

s  

+

l=1 v(l) = ii

v(l) ij

l

≤

l=1

s 

(l)

l ij +

s 

l=1

(l)

l vij =

s  

l=1

(l)

s  l=1

ij



s 

(l)

l ij + vij

(l)

l

≤

l=1

≤

s

(l)  l=1 l ij

≤

s

 l=1 l

= 1, i.e.,

l = 1.

l=1

(l) (l) (l) 0.5 hold for each l = 1, 2, . . ., s, it is clear that  ¯ ij = v¯ ji and  ¯ ii = v¯ ii = 0.5 for all l = 1, 2, . . ., s. Hence, R¯ = (¯rij )n×n Since ij = vji , ii = is an IFPR, which completes the proof. 

Theorem 1 shows that the SIFWG operator can be used to aggregate the IFPRs. (l)

(l)

If all individual IFPRs R(l) = (rij )

Theorem 2.

n×n s )T

(l)

(l)

(l)

with rij = (ij , vij , ij ) (l = 1, 2, . . ., s) given by the decision makers el (l = 1, 2, . . ., s) are

multiplicative consistent, and  = (1 , 2 , . . ., is the weight vector of the decision makers with 0 ≤ l ≤ 1, IFPR R¯ = (¯rij )n×n by SIFWG operator is also multiplicative consistent. (l)

Since R(l) = (rij )

Proof.

(l)

(l)

(l)

n×n

(l)

s

 l=1 l

= 1, then the fused

(l = 1, 2, . . ., s) are multiplicative consistent, then from Definition 4, we have (l)

(l)

ij · jk · ki = vij · vjk · vki , l = 1, 2, . . ., s.

(16)

Similarly, to prove that the fused IFPR R¯ = (¯rij )n×n is multiplicative consistent, is equivalent to prove  ¯ ij ·  ¯ jk ·  ¯ ki = v¯ ij · v¯ jk · v¯ ki , for all i, j, k = 1, 2, . . ., n, where  ¯ ij =

s  l=1

Since

¯ jk ·  ¯ ki =  ¯ ij · 

(l)

ij

s  

l

(l)

ij

, ¯ jk =

v¯ ij · v¯ jk · v¯ ki =

s  

(l)

(l)

(l)

jk

s  l 

·

, ¯ ki =

·

(l)

ki



s

l

=

v(l) jk

s  l 

·

v(l) ki

(l)

ki

l=1

s  

l=1

l=1 (l)

l

s  l 

l=1

v(l) ij

(l)

(l)

jk

l=1

·

l=1 (l)

s

s  l 

l=1



l

, v¯ ij =

(l)



s l=1

(l)

(l)

ij · jk · ki

l

v(l) ij

l

(17) , v¯ jk =

s l=1

(l) l

(vjk ) , and v¯ ki =

s l=1

(l) l

(vki ) .

,

l=1

l

l=1

=

s  

(l) (l) v(l) · vjk · vki ij

l

,

l=1

(l)

¯ ij ·  ¯ jk ·  ¯ ki = v¯ ij · v¯ jk · v¯ ki , for all i, j, k = 1, 2, . . ., n. This completes and ij · jk · ki = vij · vjk · vki hold for all l = 1, 2, . . ., s, it is obvious that  the proof of Theorem 2.  



T



˜= ˜ = (ω ˜ 1, ω ˜ 2 , . . ., ω ˜ n )T = ((ω1 , ω1v ), (ω2 , ω2v ), . . ., (ωn , ωnv )) be a underlying intuitionistic fuzzy priority weight vector of the IFPR R Let ω     ˜ i = (ωi , ωiv )(i = 1, 2, . . ., n) is an IFV, which satisfies ωi , ωiv ∈ [0, 1] and ωi + ωiv ≤ 1. ωi and ωiv indicate the membership (˜rij )n×n , where ω and non-membership degrees of the alternative xi as per a fuzzy concept of “importance”, respectively. ω ˜ is said to be normalized if it satisfies the following conditions [21]: n 





ωj ≤ ωiv , ωi + n − 2 ≥

j=1,j = / i

n 

ωjv , for all i = 1, 2, . . ., n.

(18)

j=1,j = / i

˜ = (ω ˜ 1, ω ˜ 2 , . . ., ω ˜ n )T , a multiplicative consistent IFPR R∗ = (rij∗ ) With the underlying intuitionistic fuzzy priority weight vector ω be established by the following formula [21]:

rij∗

=



(∗ij , v∗ij )

=

⎧ ⎪ ⎨ ⎪ ⎩ 

(0.5, 0.5)  2ωi   v ωi − ωi + ωj

where ωi , ωiv ∈ [0, 1], ωi + ωiv ≤ 1,

n





2ωj

,   − ωjv + 2 ωi − ωiv + ωj − ωjv + 2

/ i j=1,j =





ωj ≤ ωiv , and ωi + n − 2 ≥

if

i=j

if

i= / j

n / i j=1,j =

,

n×n

can

(19)

ωjv , for all i = 1, 2, . . ., n.

Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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Due to the complexity of the problem and the limited knowledge of the decision makers, we often obtain some multiplicative inconsistent IFPR. Perfect multiplicative consistent IFPR is somehow too strict for the decision makers to construct especially when the number of objects is too large. Since in practical cases, it is impossible to get the multiplicative consistent IFPRs, we introduce the concept of acceptable multiplicative consistent IFPR. Definition 7. Let R = (rij )n×n be an IFPR with rij = (ij , vij , ij ), i, j = 1, 2, . . ., n, then we call R an acceptable multiplicative consistent IFPR, if d(R, R∗ ) ≤ ,

(20)

where d(R, R*) is the distance measure between the given IFPR R and its corresponding underlying multiplicative consistent IFPR can be calculated by d(R, R∗ ) =

1 (n − 1)(n − 2)

R* ,

which

n        ij − ∗  + vij − v∗  + ij − ∗  , ij

ij

(21)

ij

1≤i
and ␰ is the consistency threshold. Note: In the distance measure (21), we take the denominator as (n − 1)(n − 2) but not n2 even though the summation goes over n2 elements of two compared matrices. The reason for this is that we only need to calculate the differences over the upper triangular elements, and the number of the upper triangular elements is (n − 1)(n − 2). (l)

Theorem 3. If all individual IFPRs R(l) = (rij )

(l)

n×n

(l)

(l)

(l)

(l = 1, 2, . . ., s) with rij = (ij , vij , ij ) (l = 1, 2, · · · , s) given by the decision makers el (l = 1,

2, . . ., s) are with acceptable multiplicative consistency, and  = (1 , 2 , . . ., s )T is the weight vector of the decision makers with 0 ≤ l ≤ 1,

s  l=1 l

Proof.

= 1, then the fused IFPR R¯ = (¯rij )n×n is also of acceptable multiplicative consistency.

Since each individual IFPRs R(l) is with acceptable multiplicative consistency, it follows that d(R(l) , R*) ≤ , i.e., 1 (n − 1)(n − 2)

n         (l)   (l)   (l)  ij − ∗ij  + vij − v∗ij  + ij − ij∗  ≤ , l = 1, 2, . . ., s.

(22)

1≤i
Multiplying (22) by I , we have 1 (n − 1)(n − 2)

n         (l)   (l)   (l)  l ij − l ∗ij  + l vij − l v∗ij  + l ij − l ij∗  ≤ l , l = 1, 2, . . ., s. 1≤i
Summing up all the s inequalities shown as (23) and noting that

s

 l=1 l

= 1, it follows that

      s n s s        1    (l) (l) (l) ∗ ∗ ∗ l ij − ij  +  l vij − vij  +  l ij − ij  ≤

 (n − 1)(n − 2)       1≤i
l=1

l=1

(23)

(24)

l=1

Thus, from Lemma 1, we have

      s s  s   l      l l       (l) (l) (l) ij − ∗ij  +  vij − v∗ij  +  ij − ij∗         1≤i
1 ¯ R∗ ) = d(R, (n − 1)(n − 2)

n 

1≤i
l=1

l=1

l=1

which implies that the fused IFPR R¯ = (¯rij )n×n is with acceptable multiplicative consistency. This completes the proof of Theorem 3.  Theorem 2 reveals that if all individual IFPRs are multiplicative consistent, then their fused IFPR is also multiplicative consistent. This result is reasonable however not very practical because in general cases, the decision makers always determine inconsistent IFPRs. When some of the IFPRs are not multiplicative consistent, this conclusion does not surely hold. Theorem 3 further guarantees that if all individual IFPRs are acceptable multiplicative consistent, then their fused IFPR is also acceptable multiplicative consistent. This theorem is very important in group decision making with IFPRs because they shows that once we make sure that each individual IFPR is with multiplicative consistent or acceptable multiplicative consistent, we can employ the SIFWG operator to obtain the fused IFPR, and then use the aggregated IFPR to rank the alternatives directly without checking its consistency. In the following, we present a numerical example to validate the conclusion. Example 2. The current globalized market trend identifies the necessity of the establishment of long term business relationship with competitive global suppliers spread around the world [26]. This can lower the total cost of supply chain; lower the inventory of enterprises; enhance information sharing of enterprises; improve the interaction of enterprises and obtain more competitive advantages for enterprises. Thus, how to select different unfamiliar international suppliers according to the broad evaluation is very critical and has a direct impact on the performance of an organization. Global supplier development is a complex multiple criteria decision making problem which includes much qualitative information, especially when the experts do not have exact information about the suppliers. In such a case, the decision maker asks several experts from different areas to evaluate the candidate global suppliers. Herein the main objective is the selection of the best global supplier for manufacturing firm and the criteria considered in achieving the objective are:

Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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(1) C1 : Overall cost of the product, which consists of three sub-criteria: S1 : product price; S2 : freight cost, and S3 : tariff; (2) C2 : Quality of the product, which consists of four sub-criteria: S4 : rejection rate of the product, S5 : increased lead time; S6 : quality assessment; and S7 : remedy for quality problems; (3) C3 : Service performance of supplier, which consists of four sub-criterion: S8 : delivery schedule; S9 : technological and R&D support; S10 : response to changes; and S11 : ease of communication; (4) C4 : Supplier’s profile, which consists of four sub-criteria: S12 : financial status; S13 : customer base, S14 : performance history; S15 : production facility and capacity; In such a multiple criteria group decision making problem, the first thing we should do is to determine the weight of each criteria. Hence, the experts made pairwise comparisons over these criteria and then construct their individual preference relations to express their preferences. Since the experts are not very familiar with the global suppliers and each criterion has several subcriteria, it is reasonable for them to use the A-IFSs to describe their preferences. Suppose that three experts e1 , e2 and e3 evaluate four criteria c1 , c2 , c3 and c4 , and the three experts provide their preference information over the four criteria with the following IFPRs:

⎛

˜ (1) R

(0.5, 0.5) ⎜ (0.2, 0.5) =⎝ (0.1, 0.7) (0.3, 0.5)

⎛

˜ (2) R

(0.5, 0.5) ⎜ (0.1, 0.6) =⎝ (0.2, 0.8) (0.3, 0.6)

⎞

(0.5, 0.2) (0.5, 0.5) (0.2, 0.6) (0.6, 0.3)

(0.7, 0.1) (0.6, 0.2) (0.5, 0.5) (0.6, 0.3)

(0.5, 0.3) (0.3, 0.6) ⎟ , (0.3, 0.6) ⎠ (0.5, 0.5)

(0.6, 0.1) (0.5, 0.5) (0.1, 0.5) (0.7, 0.3)

(0.8, 0.2) (0.5, 0.1) (0.5, 0.5) (0.6, 0.4)

(0.6, 0.3) (0.3, 0.7) ⎟ , (0.4, 0.6) ⎠ (0.5, 0.5)

⎛

⎞

⎞

(0.5, 0.5) (0.6, 0.2) (0.8, 0.1) (0.7, 0.2) ⎜ (3) ˜ = ⎝ (0.2, 0.6) (0.5, 0.5) (0.6, 0.1) (0.2, 0.7) ⎟ R . (0.1, 0.8) (0.1, 0.6) (0.5, 0.5) (0.2, 0.3) ⎠ (0.2, 0.7) (0.7, 0.2) (0.3, 0.2) (0.5, 0.5) Using the fractional programming models constructed by Liao and Xu [21], the underlying intuitionistic fuzzy weights for these three individual IFPRs are (1)

(1)

(1)

(1) T

(2)

(2)

(2)

(2) T

(3)

(3)

(3)

(3) T

ω ˜ (1) = (ω ˜1 ,ω ˜2 ,ω ˜3 ,ω ˜ 4 ) = ((0.3951, 0.4221), (0.1354, 0.8397), (0.0451, 0.8894), (0.2370, 0.6298))T ,

ω ˜ (2) = (ω ˜1 ,ω ˜2 ,ω ˜3 ,ω ˜ 4 ) = ((0.4137, 0.5517), (0.1552, 0.7069), (0.0862, 0.9138), (0.2069, 0.6897))T .

ω ˜ (3) = (ω ˜1 ,ω ˜2 ,ω ˜3 ,ω ˜ 4 ) = ((0.4686, 0.4143), (0.1406, 0.7891), (0.0586, 0.9414), (0.1538, 0.6700))T . With these underlying intuitionistic fuzzy weight vectors, by (19), the corresponding multiplicative consistent IFPRs can be generated:

⎛

(0.5000, 0.5000)

(0.6228, 0.2134)

(0.7001, 0.0799)

(0.5001, 0.3000)

(0.3000, 0.5001)

(0.5250, 0.2999)

(0.6213, 0.1182)

(0.5000, 0.5000)

(0.5000, 0.5000)

(0.6315, 0.2369)

(0.7999, 0.1667)

(0.5999, 0.3000)

(0.3000, 0.5999)

(0.4286, 0.3215)

(0.6001, 0.2500)

(0.5000, 0.5000)

(0.5000, 0.5000)

(0.6667, 0.2000)

(0.8000, 0.1000)

(0.6093, 0.2000)

(0.2000, 0.6093)

(0.3683, 0.3366)

(0.5118, 0.1950)

(0.5000, 0.5000)

⎞

⎜ (0.2134, 0.6228) (0.5000, 0.5000) (0.5999, 0.1998) (0.2999, 0.5250) ⎟ ⎟ ⎝ (0.0799, 0.7001) (0.1998, 0.5999) (0.5000, 0.5000) (0.1182, 0.6213) ⎠ ,

˜ (1)∗ = ⎜ R

⎛

⎞

⎜ (0.2369, 0.6315) (0.5000, 0.5000) (0.5001, 0.2778) (0.3215, 0.4286) ⎟ ⎟ ⎝ (0.1667, 0.7999) (0.2778, 0.5001) (0.5000, 0.5000) (0.2500, 0.6001) ⎠ ,

˜ (2)∗ = ⎜ R

⎛

⎞

⎜ (0.2000, 0.6667) (0.5000, 0.5000) (0.6000, 0.2501) (0.3366, 0.3683) ⎟ ⎟ ⎝ (0.1000, 0.8000) (0.2501, 0.6000) (0.5000, 0.5000) (0.1950, 0.5118) ⎠ .

˜ (3)∗ = ⎜ R

Thus, via the distance measure (21), we can obtain d(R(1) , R(1)* ) = 0.1382, d(R(2) , R(2)* ) = 0.2569, and d(R(3) , R(3)* ) = 0.2837. Suppose = 0.3, then all these three individual IFPRs are of acceptable multiplicative consistency. If Theorem 3 holds, the aggregated ¯ R∗ ) ≤ should hold. IFPR R¯ by the SIFWG operator should also of acceptable multiplicative consistency, i.e., d(R,

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(1) To verify this truth, we first consider the case that the experts have equal weights, i.e., 1 = (1/3, 1/3, 1/3)T . Then, by (15), the fuse IFPR is

⎛

(0.5000, 0.5000)

(0.5646, 0.1587)

(0.7652, 0.1260)

(0.5944, 0.2621)

(0.2621, 0.5944)

(0.6649, 0.2621)

(0.4762, 0.2884)

(0.5000, 0.5000)

⎞

⎜ (0.1587, 0.5646) (0.5000, 0.5000) (0.5646, 0.1260) (0.2621, 0.6649) ⎟ ⎟ ⎝ (0.1260, 0.7652) (0.1260, 0.5646) (0.5000, 0.5000) (0.2884, 0.4762) ⎠

R¯ 1 = ⎜

Using the fraction model given by Liao and Xu [21] to derive the underlying intuitionistic fuzzy weights, we obtain ω ˜¯ 1 = ((0.4588, 0.4108), (0.1330, 0.8130), (0.0755, 0.9245), (0.2023, 0.7067))T . Thus the corresponding multiplicative consistent IFPR is generated via (19):

⎛

(0.5000, 0.5000)

(0.6708, 0.1944)

(0.7653, 0.1259)

(0.5945, 0.2621)

(0.2621, 0.5945)

(0.4961, 0.3261)

(0.6257, 0.2335)

(0.5000, 0.5000)

⎞

⎜ (0.1944, 0.6708) (0.5000, 0.5000) (0.5648, 0.3206) (0.3261, 0.4961) ⎟ ⎟ ⎝ (0.1259, 0.7653) (0.3206, 0.5648) (0.5000, 0.5000) (0.2335, 0.6257) ⎠ ,

˜∗ = ⎜ R 1

and the distance between the fused IFPR and its corresponding multiplicative consistent IFPR is d(R¯ 1 , R1∗ ) = 0.2184 ≤ , which implies that the fused IFPR is of acceptable multiplicative consistency. This verifies the conclusion of Theorem 3. (2) If the experts’ weight vector 2 = (0.3, 0.4, 0.3)T , in analogous, the fused IFPR is

⎛

(0.5000, 0.5000)

(0.5681, 0.1516)

(0.7686, 0.1320)

(0.5950, 0.2656)

(0.2656, 0.5950)

(0.6684, 0.2656)

(0.4874, 0.2980)

(0.5000, 0.5000)

⎞

⎜ (0.1516, 0.5681) (0.5000, 0.5000) (0.5578, 0.1231) (0.2656, 0.6684) ⎟ ⎟ ⎝ (0.1320, 0.7686) (0.1231, 0.5578) (0.5000, 0.5000) (0.2980, 0.4874) ⎠ ,

R¯ 2 = ⎜

and its corresponding intuitionistic fuzzy weight vector is ω ˜¯ 2 = ((0.4560, 0.4261), (0.1443, 0.7836), (0.0783, 0.9217), (0.2035, 0.7008))T . Then the corresponding multiplicative consistent IFPR is computed via (19):

⎛

(0.5000, 0.5000)

(0.6558, 0.2075)

(0.7686, 0.1320)

(0.5951, 0.2656)

(0.2656, 0.5951)

(0.4714, 0.3343)

(0.6173, 0.2375)

(0.5000, 0.5000)

⎞

⎜ (0.2075, 0.6558) (0.5000, 0.5000) (0.5579, 0.3027) (0.3343, 0.4714) ⎟ ⎟ ⎝ (0.1320, 0.7686) (0.3027, 0.5579) (0.5000, 0.5000) (0.2375, 0.6173) ⎠ .

˜∗ = ⎜ R 2

Thus, we can calculate the distance between R¯ 2 and R2∗ : d(R¯ 2 , R2∗ ) = 0.2168 ≤ , which implies that the fused IFPR R¯ 2 is also of acceptable multiplicative consistency and thus verifies the conclusion of Theorem 3. (3) If the experts’ weight vector 3 = (0.2, 0.3, 0.4)T , the fused IFPR is

⎛

(0.5000, 0.5000)

(0.6088, 0.1908)

(0.7965, 0.1550)

(0.6476, 0.2877)

(0.2877, 0.6476)

(0.7034, 0.2877)

(0.4785, 0.3137)

(0.5000, 0.5000)

⎞

⎜ (0.1908, 0.6088) (0.5000, 0.5000) (0.5978, 0.1446) (0.2877, 0.7034) ⎟ ⎟ ⎝ (0.1550, 0.7965) (0.1446, 0.5978) (0.5000, 0.5000) (0.3137, 0.4785) ⎠ ,

R¯ 3 = ⎜

and its corresponding intuitionistic fuzzy weight vector is ω ˜¯ 3 = ((0.4454, 0.5546), (0.1268, 0.7299), (0.0867, 0.8591), (0.1979, 0.7131))T . Then, via (19), the corresponding multiplicative consistent IFPR is derived:

⎛

(0.5000, 0.5000)

(0.6918, 0.1969)

(0.7965, 0.1550)

(0.6476, 0.2877)

(0.2877, 0.6476)

(0.4489, 0.2876)

(0.5556, 0.2434)

(0.5000, 0.5000)

⎞

⎜ (0.1969, 0.6918) (0.5000, 0.5000) (0.4061, 0.2777) (0.2876, 0.4489) ⎟ ⎟ ⎝ (0.1550, 0.7965) (0.2777, 0.4061) (0.5000, 0.5000) (0.2434, 0.5556) ⎠ .

˜∗ = ⎜ R 3

With (21), we obtain d(R¯ 3 , R3∗ ) = 0.2041 ≤ , which means the fused IFPR R¯ 3 is of acceptable multiplicative consistency. This further verifies Theorem 3. Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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(4) If the experts’ weight vector 4 = (0.6, 0.3, 0.1)T , the fused IFPR is

⎛

(0.5000, 0.5000)

(0.5378, 0.1625)

(0.7384, 0.1231)

(0.5462, 0.2881)

(0.2881, 0.5462)

(0.6382, 0.2881)

(0.5598, 0.3140)

(0.5000, 0.5000)

⎞

⎜ (0.1625, 0.5378) (0.5000, 0.5000) (0.5681, 0.1516) (0.2881, 0.6382) ⎟ ⎟ ⎝ (0.1231, 0.7384) (0.1516, 0.5681) (0.5000, 0.5000) (0.3140, 0.5598) ⎠ ,

R¯ 4 = ⎜

and its corresponding intuitionistic fuzzy weight vector is ω ˜¯ 4 = ((0.3600, 0.5526), (0.1501, 0.6188), (0.0600, 0.8924), (0.1899, 0.6792))T .

The corresponding multiplicative consistent IFPR is:

⎛

(0.5000, 0.5000)

(0.5378, 0.2242)

(0.7385, 0.1231)

(0.5462, 0.2881)

(0.2881, 0.5462)

(0.3645, 0.2881)

(0.5599, 0.1769)

(0.5000, 0.5000)

⎞

⎜ (0.2242, 0.5378) (0.5000, 0.5000) (0.4295, 0.1717) (0.2881, 0.3645) ⎟ ⎟ ⎝ (0.1231, 0.7385) (0.1717, 0.4295) (0.5000, 0.5000) (0.1769, 0.5599) ⎠ .

˜∗ = ⎜ R 4

Then we get d(R¯ 4 , R4∗ ) = 0.2037 ≤ , which shows the fused IFPR R¯ 4 is of acceptable multiplicative consistency. This verifies the conclusion of Theorem 3. The above example illustrates that once all the individual IFPRs are of acceptable consistency, their fused IFPR, derived by the SIFWG operator, is also of acceptable consistency no matter how the experts’ weight vector changes. On the other hand, if we set the consistency threshold ’ = 0.21, then in Example 1, d(R(1) , R(1)* ) = 0.1382 < ’ , d(R(2) , R(2)* ) = 0.2569 > , d(R(3) , R(3)* ) = 0.2837 > , which means, R(1) is of acceptable consistency but both R(2) and R(3) are not of acceptable consistency. In such a case, from the computational results, we can see that with different weights of the experts, the fused IFPR also may be of acceptable consistency. T For example, when 1 = (1/3, 1/3, 1/3) and 2 = (0.3, 0.4, 0.3)T , d(R¯ 1 , R1∗ ) = 0.2184 > , and d(R¯ 2 , R2∗ ) = 0.2168 > , which implies that T the fused IFPR R¯ 1 and R¯ 2 are of acceptable consistency, but when 3 = (0.2, 0.3, 0.4) and 4 = (0.6, 0.3, 0.1)T , d(R¯ 3 , R3∗ ) = 0.2041 ≤ , and ∗ d(R¯ 4 , R4 ) = 0.2037 ≤ , which shows that the fused IFPR R¯ 3 and R¯ 4 are not of acceptable consistency. Hence, when some of the individual IFPRs are not of acceptable consistency, whether the fused IFPR R¯ is of acceptable consistency or not depends not only on the individual IFPRs, but also on the weights of the experts. 4. Consistency of the fused IFPR with other geometric aggregation operators The above section shows a desirable property of the SIFWG operator and the underlying relationships between the individual IFPRs and their fused IFPR regarding consistency: if all individual IFPRs are of acceptable multiplicative consistency, the fused IFPR by SIFWG operator is of acceptable multiplicative consistency. This property is fundamental and useful in aggregating different opinions of distinct decision makers in group decision making with IFPRs. As pointed out by Aczel and Alsina [30], the weighted geometric operator is the only appropriate method for the aggregation of individual judgments when the weights of the decision makers are not equal. Hence, the SIFWA operator [32] and the IFWA operator [22] cannot be employed to aggregate individual IFPRs. In this subsection, we shall discuss the question: if we use some other well-known geometric aggregation operators to fuse the multiplicative consistent or acceptable multi(l) plicative consistent individual IFPRs R(l) = (rij ) (l = 1, 2, . . ., s), whether the fused IFPR R¯ = (¯rij )n×n is still of multiplicative consistency or acceptable multiplicative consistency.

n×n

4.1. The IFWG operator (l)

The intuitionistic fuzzy weighted geometric (IFWG) operator was proposed by Xu and Yager [33]. For s individual IFPRs R(l) = (rij )

n×n

(l =

1, 2, . . ., s) given by the decision makers el (l = 1, 2, . . ., s), and  = (1 , 2 , . . ., s )T be the weight vector of the decision makers with 0 ≤ l ≤ 1,

s  = 1, according to the IFWG operator, the fused IFPR R¯ IFWG = (¯rij )n×n satisfies l=1 l ¯ ij ),  r¯ ij = ( ¯ ij , v¯ ij ,  ¯ ij =

s   l=1

(l)

ij

l

, v¯ ij = 1 −

s  

(l)

1 − vij

l

¯ ij = 1 −  , ¯ ij − v¯ ij , i, j = 1, 2, . . ., n.

(25)

l=1

With the IFWG operator, it is very hard to prove that the fused IFPRs is still of consistency under the condition that each individual IFPR is of consistency. Below we give some simple theoretical analysis. (l) Supposing R(l) = (rij ) (l = 1, 2, . . ., s) are of multiplicative consistency, then from Definition 4, we have n×n

(l) ij

(l) · jk

(l) · ki

=

v(l) ij

(l)

(l)

· vjk · vki , i, j, k = 1, 2, · · ·, n, l = 1, 2, . . ., s.

(26)

According to Definition 4 and (25), to prove that the fused IFPR R¯ IFWG = (¯rij )n×n is multiplicative consistent is equivalent to prove  ¯ ij ·  ¯ jk ·  ¯ ki = v¯ ij · v¯ jk · v¯ ki , for all i, j, k = 1, 2, . . ., n,

(27)

Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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where  ¯ ij =

s  l=1

(l)

1 − vki

s 

l

l=1

(l)

ij

l

, ¯ jk =

s  l=1

(l)

jk

l

s 

, ¯ ki =

l=1

(l)

ki

l

, v¯ ij = 1 −

s  l=1

(l)

1 − vij

l

, v¯ jk = 1 −

s  l=1

(l)

1 − vjk

l

, and v¯ ki = 1 −

.

Since s  

 ¯ ij ·  ¯ jk ·  ¯ ki =

(l)

ij

s  l 

·

l=1

s  l 

·

l=1

 v¯ ij · v¯ jk · v¯ ki =

(l)

jk

l

s  

=

l=1

 

s 

(l) l (1 − vij )

1−

(l)

ki

·

1−

l=1

(l)

(l)

(l)

ij · jk · ki

l

s  

=

l=1

(l) l (1 − vjk )

l

,

l=1

 

s 

(l) (l) v(l) · vjk · vki ij

·

1−

l=1



s 

(l) l (1 − vki )

,

l=1

to prove (27) is equivalent to prove s  

v(l) ij

(l) · vjk

(l) · vki

l

 =

 

s 

(l) l (1 − vij )

1−

l=1

·

1−

 

s 

(l) l (1 − vjk )

l=1

·

1−

l=1



s 

(l) l (1 − vki )

.

(28)

l=1

Since (28) does not surely hold, it is unreasonable to say that the aggregated IFPR derived by the IFWA operator is still multiplicative consistent. 4.2. The SYIFWG operator Xia and Xu [34] introduced another kind of geometric aggregation operator, named symmetric intuitionistic fuzzy weighted geometric (l) (SYIFWG) operator. Let R(l) = (rij ) (l = 1, 2, . . ., s) be s individual IFPRs, then their fusion R¯ SYIFWG = (¯rij )n×n by the SYIFWG operator is n×n

also an IFPR, where

⎛ r¯ ij = ⎝ 

s

s

(l) l

(ij ) l=1

s (l) l (ij ) l=1

m

+

l=1

(l) l

(1 − ij )

,

⎞

(l) l

(v ) l=1 ij

s

(l) l

l=1

(vij )

+

s

(l) l

l=1

⎠ , i, j = 1, 2, . . ., n

(29)

(1 − vij )

Recently, Xu and Xia [44] proved that with this SYIFWG operator, the fused IFPR R¯ SYAIFWG = (¯rij )n×n is still of multiplicative consistency under the condition that each individual IFPR is multiplicative consistent. However, they used Definition 3 as the condition of multiplicative consistency. Since Liao and Xu [21] pointed out that Definition 3 is sometimes unreasonable in measuring the consistency of an IFPR, it is needed to check that the conclusion to the SYIFWG operator again using the more convincing Definition 4 to measure the multiplicative consistency of an IFPR. In fact, with the SYIFWG operator, we have

s  ¯ ij ·  ¯ jk ·  ¯ ki =

s

(l) l

l=1

=



s (l) l (ij ) l=1

+



s (l) l (v ) l=1 ij

+

(l)

Since R(l) = (rij )

m

(l)

s

(vij )

m

+

(l) l

l=1

(1 − ij ) l=1

(l) l

l=1

(ij )

m

(1 − ij )



 l s ·

·

s

(l)

(1 − vij ) l=1

·

(l) l

l=1

·



+



s

·

s

(l) l

l=1

(ki )

(l) l

(ki ) l=1

+

(vjk )

·



m

(l) l

l=1

(1 − ki ) (l) l

(1 − ki ) l=1

(vki )













(l)

n×n

;

(l) l

l=1



+

(ki )

m

s

(l) l

l=1

s



(1 − jk ) l=1

·

(l) l

l=1

s s s (l) l (l) l (l) l (l) l (v ) + l=1 (1 − vjk ) (v ) + l=1 (1 − vki ) l=1 jk l=1 ki s (l) (l) (l) l (v · vjk · vki ) l=1 ij . s m s m (l) l (l) l (l) l (l) l ( v ) + (1 − v ) · ( v ) + (1 − v ) l=1 jk l=1 l=1 ki l=1 jk ki

(1 − vij )

 l

(jk )

m (l) l (l) l (jk ) + l=1 (1 − jk ) l=1 s (l) (l) (l) l (ij · jk · ki ) l=1 m s (l) l (l) l

(jk ) l=1

(vij )

s

(l) l

l=1

s

(l) l

l=1

v¯ ij · v¯ jk · v¯ ki =  s =

+

(ij )

s

(l) l

l=1

(l)

(l)

(l)

(l)

(l)

(l = 1, 2, . . ., s) are of multiplicative consistency, then from Definition 4, we have ij · jk · ki = vij · vjk · vki , i, j,

k = 1, 2, . . ., n, l = 1, 2, . . ., s. If the fused IFPR is multiplicative consistent, then,  ¯ ij ·  ¯ jk ·  ¯ ki = v¯ ij · v¯ jk · v¯ ki , that is to say, the following equation

 =

s (l) l (ij ) l=1



+

s (l) l (v ) l=1 ij

m

+

(l) l

l=1

(1 − ij )

m

l=1

 s

(l) l

(1 − vij )

·

(l) l

l=1

 s ·

(jk )

l=1

(l) l

(vjk )

+ +

m

(l) l

l=1

(1 − jk )

l=1

(1 − vjk )

m

(l) l

 s ·

 s ·

(l) l

l=1

l=1

(ki )

(l) l

(vki )

+

+

m

(l) l

l=1

m

l=1

(1 − ki ) (l) l

(1 − vki )



(30)

should hold. However, this equation does not surely stand. Hence, the aggregated IFPR derived by the SYIFWG operator is not surely multiplicative consistent. Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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Based on the above discussion, the following conclusions are drawn immediately: (1) In group decision making with IFPRs, when the weights of experts are not equal, as pointed out by Aczel and Alsina [30], only the weighted geometric mean (WGM) operator is appropriate in aggregating individual judgments. That is to say, all the IFWA operator [22], the SIFWA operator [32] and their extensions cannot be employed to aggregate individual IFPRs. (2) Although the geometric aggregation operators, such as the IFWG operator and the SYIFWG operator, can be used to fuse the individual IFPRs, as discussed above, the aggregated IFPR derived by the IFWG operator and the SYIFWG operator is not surely multiplicative consistent. That is to say, the IFWG operator and the SYIFWG operator cannot maintain the consistency property of each individual IFPR in the process of aggregation. Hence, these types of geometric aggregation operators are not adequate in aggregating individual preference information of experts in intuitionistic group decision making. (3) By contrast, we have proven and illustrated in Section 3 that the proposed SIFWG has a very important property that once we make sure that each individual IFPR is of consistency or acceptable consistency, then their aggregated IFPR is also of consistency or acceptable consistency. In other words, the SIFWG operator can maintain the consistency property of the individual IFPRs. Hence, from this point of view, the SIFWG operator is the most suitable one in aggregating group intuitionistic fuzzy preferences.

5. The group intuitionistic fuzzy analytic hierarchy process to multiple criteria group decision making In the above sections, we have proven that the SIFWG operator can maintain the consistency property of each individual IFPR in the aggregation process. This is a very important finding because with such characteristic, there is no need to check the consistency of the fused IFPR with the SIFWG operator under the condition that each individual IFPR is consistent or acceptable consistent. In the following, let us give a step by step procedure, named the group intuitionistic fuzzy analytic hierarchy process (GIFAHP), to demonstrate how to implement the SIFWG operator in group decision making with IFPRs. 5.1. Procedure of GIFAHP Step 1: Identify the experts, objective, criteria, and alternatives of the multiple criteria group decision making problem and then construct the hierarchy of the considered problem. Here we consider a multiple criteria group decision making problem with s experts {e1 , e2 , . . ., es } who are asked to evaluate n alternatives {x1 , x2 , . . ., xn } over m criteria

s {c1 , c2 , . . ., cm }. The weight vector of the decision makers el (l = 1, 2, . . ., s) is given as  = (1 , 2 , . . ., s )T , where l > 0, l = 1, 2, . . ., s, and  = 1. Each expert determines an IFPR via pairwise comparisons l=1 l over the criteria. Meanwhile, the individual experts also give their assessments on the alternatives under each criterion in IFVs and then construct their Atanassov’s intuitionistic fuzzy judgment matrices, respectively. Then go to the next step. Step 2: Check the consistency of each IFPRs over criteria by using Definition 6. If all of the IFPRs are of acceptable consistency, go to step 4; Otherwise, go to Step 3. Step 3: Repair the inconsistent IFPRs according to Algorithm II in Ref. [26] (or return the inconsistent intuitionistic preference relations to the decision makers for re-evaluation until they are acceptable). Then go to the next step. Step 4: Use the SIFWG operator to fuse all the individual IFPRs into an overall IFPR, then go to the next step. (l)

(l) T

(l)

Step 5: Calculate the underlying priority vector ω ˜ (l) = (ω ˜1 ,ω ˜ 2 , . . ., ω ˜ m ) (l = 1, 2, . . ., s) for each criterion from the fused IFPR according to the fractional programming models in Ref. [21]. Go the next step. (l) Step 6: Aggregate all the judgments aij (j = 1, 2, . . ., m) on different criteria cj (j = 1, 2, . . ., m) given by the expert el into an overall (l)

assessment ai by the enhanced IFWA (EIFWA) operator:

⎛

ai = EIFWAω˜ (l) (ai1 , ai2 , . . ., aim ) = ⎝1 − (l)

(l)

(l)

(l)

m  

(l)

1−ω ˜j

(l)

· aij



m  

,

j=1

v(l)

ω ˜j

v(l)

+ aij

v(l)

−ω ˜j

v(l)

· aij



⎞ ⎠

(31)

j=1

Afterward, using the IFWA operator [22]:

 ai =

(l) (l) (s) IFWA (ai , ai , . . ., ai )

=

1−

s   l=1

(l) 1 − ai

 ,

s  

v(l)

ai

l

 (32)

l=1

(l)

to aggregate all the overall assessments ai (l = 1, 2, . . ., s) given by different experts into the collective IFVs ai (i = 1, 2, . . ., n). Go to the next step. Step 7: Rank the collective IFVs ai (i = 1, 2, . . ., n) by using Scheme 1 or Scheme 2 in Section 3.2, and then choose the best alternative. Go to the next step. Step 8: End. In the following, let us use Example 2 to further validate the above procedure. Example 3. (Continue with Example 2) In Example 2, the three experts gave their preferences over the four criteria and then constructed T three IFPRs. Suppose that the experts have equal weights, i.e.,  = (1/3, 1/3, 1/3) , then according to the result of Example 1, the weight vec¯ tor of these four criteria is ω ˜ = ((0.4588, 0.4108), (0.1330, 0.8130), (0.0755, 0.9245), (0.2023, 0.7067))T . Suppose that the three experts gave their judgment matrices on three candidate global suppliers over different criteria shown as in Tables 1–3. Then, we can use our GIFAHP procedure to solve this problem.

Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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Table 1 The Atanassov’s intuitionistic fuzzy judgment matrix A(1) given by the expert e(1) .

x1 x2 x3

c1

c2

c3

c4

(0.6, 0.1) (0.2, 0.7) (0.8, 0.1)

(0.5, 0.2) (0.3, 0.4) (0.5, 0.3)

(0.6, 0.3) (0.6, 0.2) (0.2, 0.7)

(0.3, 0.5) (0.7, 0.1) (0.6, 0.3)

Table 2 The Atanassov’s intuitionistic fuzzy judgment matrix A(2) given by the expert e(2) .

x1 x2 x3

c1

c2

c3

c4

(0.6, 0.3) (0.8, 0.1) (0.5, 0.3)

(0.5, 0.3) (0.2, 0.5) (0.6, 0.3)

(0.4, 0.5) (0.6, 0.3) (0.3, 0.4)

(0.2, 0.7) (0.5, 0.4) (0.6, 0.2)

Table 3 The Atanassov’s intuitionistic fuzzy judgment matrix A(3) given by the expert e(3) .

x1 x2 x3

c1

c2

c3

c4

(0.6, 0.3) (0.4, 0.5) (0.6, 0.3)

(0.5, 0.4) (0.6, 0.3) (0.3, 0.5)

(0.9, 0.1) (0.2, 0.7) (0.4, 0.5)

(0.7, 0.2) (0.6, 0.4) (0.5, 0.3)

In Example 2, we have calculated the underlying priority weight of each criterion. Thus, we go to the step 6 directly. Using the EIFWA operator shown as Eq. (31) to aggregate all the judgments on different alternatives into overall assessments, we have (1)

(1)

(1)

(2)

(2)

(2)

(3)

(3)

(3)

a1 = (0.3933, 0.3229), a2 = (0.2854, 0.5055), a3 = (0.4887, 0.3171); a1 = (0.3705, 0.4481), a2 = (0.4713, 0.3323), a3 = (0.3911, 0.3731); a1 = (0.4587, 0.3721), a2 = (0.3498, 0.4937), a3 = (0.3935, 0.4073). Then, we use the IFWA operator to fuse different overall assessments given by different experts into the overall judgments for the alternatives, which are a1 = (0.4087, 0.3776),

a2 = (0.3737, 0.4361),

a3 = (0.4263, 0.3639).

After that, we use Scheme 2 to rank these IFVs, then we have L(a1 ) = 0.5128, L(a2 ) = 0.4738, and L(a3 ) = 0.5258. Thus, it follows a3 a1 a2 , in which ’ ’ ’ ’ means “prior to”. That is to say, the third global supplier is the most desirable supplier for the manufacturing firm. 6. Concluding remarks It is well known that for traditional multiplicative preference relations, if all individual comparison matrices are of acceptable consistency, then the weighted geometric mean complex judgment matrix is of acceptable consistency. Due to the effectiveness of IFPR in representing fuzziness and uncertainty, in this paper, we have studied the consistency of the fused IFPR in group decision making. Firstly, we have introduced a new type of simple intuitionistic fuzzy weighted geometric (SIFWG) operator to synthesize individual IFPRs. Then we have proven that if all individual IFPRs are multiplicative consistent, then their fused IFPR by the SIFWG operator is also multiplicative consistent. We have also proven that if all individual IFPRs are of acceptable multiplicative consistency, their fused IFPR by the SIFWG operator is also of multiplicative consistency no matter how the decision makers’ weights change. The numerical example has verified the result. We have also demonstrated that when some of the individual IFPRs are not of acceptable consistency, whether the fused IFPR is of acceptable consistency or not depends not only on the individual IFPRs, but also on the weights of the decision makers. Furthermore, it has been pointed out that the simple intuitionistic fuzzy weighted averaging (SIFWA) operator, the intuitionistic fuzzy weighted averaging (IFWA) operator, the intuitionistic fuzzy weighted geometric (IFWG) operator and the symmetric intuitionistic fuzzy weighted geometric (SYIFWG) operator do not have these properties and thus cannot be used to fuse the individual IFPRs into a collective one. The conclusion drawn in this paper is fundamental and significant in group decision making with IFPRs. Based on this finding, we have developed a procedure of GIFAHP to aid the group decision making process. The numerical example has shown the applicability of the proposed GIFAHP method. In the future, we would make in-depth research on group decision making with IFPRs and develop some other techniques to aid the decision makers. We will apply the proposed GIFAHP method into other practical group decision making problems. Acknowledgments The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. The work was supported by the National Natural Science Foundation of China (No. 61273209).

Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015

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Please cite this article in press as: H. Liao, Z. Xu, Consistency of the fused intuitionistic fuzzy preference relation in group intuitionistic fuzzy analytic hierarchy process, Appl. Soft Comput. J. (2015), http://dx.doi.org/10.1016/j.asoc.2015.04.015