Intuitionistic preference relations and their application ...

Information Sciences 177 (2007) 2363–2379 www.elsevier.com/locate/ins

Intuitionistic preference relations and their application in group decision making q Zeshui Xu

*

Department of Management Science and Engineering, School of Economics and Management, Tsinghua University, Beijing 100084, China Received 7 February 2006; received in revised form 9 December 2006; accepted 15 December 2006

Abstract Intuitionistic fuzzy set, characterized by a membership function and a non-membership function, was introduced by Atanassov [Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87–96]. In this paper, we define the concepts of intuitionistic preference relation, consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and study their various properties. We develop an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Finally, a practical example is provided to illustrate the developed approaches. 2007 Elsevier Inc. All rights reserved. Keywords: Intuitionistic fuzzy set; Intuitionistic preference relation; Transitivity; Group decision making; Aggregation

1. Introduction Atanassov [3,5] introduced the concept of intuitionistic fuzzy set characterized by a membership function and a non-membership function, which is a generalization of fuzzy set [62]. Gau and Buehrer [21] introduced the concept of vague set. But Bustince and Burillo [10] showed that vague sets are intuitionistic fuzzy sets. Over the last decades, intuitionistic fuzzy set theory has been applied to many different fields, such as decision making [8,12,28,45–48,61], logic programming [7], topology [1,2,15,19,38,40], medical diagnosis [17,58], pattern recognition [34,36,39,50], machine learning and market prediction [36], etc. Chen and Tan [12] presented a technique for handling multi-criteria fuzzy decision making problems based on vague sets. They provided a q

The work was supported by the National Natural Science Foundation of China (No. 70571087 and No. 70321001), China Postdoctoral Science Foundation (No. 20060390051), and the National Science Fund for Distinguished Young Scholars of China (No. 70625005). * Tel.: +86 1 62795845. E-mail address: [email protected] 0020-0255/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2006.12.019

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score function to measure the degree of suitability of each alternative with respect to a set of criteria presented by vague values. Later, Hong and Choi [31] proposed an improved technique based on score function and accuracy function for multi-criteria fuzzy decision making with vague sets. They utilized the score function to measure the degree of suitability and utilized the accuracy function to measure the degree of accuracy in the grades of membership of each alternative with respect to a set of criteria represented by vague values. Szmidt and Kacprzyk [45–48] used intuitionistic fuzzy sets to solve group decision making problems. Atanassov et al. [8] proposed an intuitionistic fuzzy interpretation of multi-person multi-criteria decision making. Gabriella et al. [20] constructed a generalized net model of multi-person multi-criteria decision making process based on intuitionistic fuzzy graphs. Atanassov and Georgiev [7] presented a logic programming system, which uses intuitionistic fuzzy set theory to model various forms of uncertainty, and considered the problem of propagating uncertainty through logical inference and various models of interpretation. The framework discussed allows knowledge representation and inference under uncertainty in the form of rules suitable for expert systems. De et al. [17] studied Sanchez’s approach for medical diagnosis using intuitionistic fuzzy sets. Li and Cheng [34], Liang and Shi [36], Hung and Yang [32], and Wang and Xin [50] introduced some similarity measures of intuitionistic fuzzy sets and applied them to pattern recognition. Xu and Yager [61] developed some geometric aggregation operators based on intuitionistic fuzzy sets, and applied them to multiple attribute decision making. Consider that, in some real-life situations, a decision maker (DM) may not be able to accurately express his/her preferences for alternatives due to that (1) the DM may not possess a precise or sufficient level of knowledge of the problem; (2) the DM is unable to discriminate explicitly the degree to which one alternative are better than others [29], in such cases, the DM may provide his/her preferences for alternatives to a certain degree, but it is possible that he/she is not so sure about it [18]. Thus, it is very suitable to express the DM’s preference values with the use of intuitionistic fuzzy values rather than exact numerical values or linguistic variables [28,46,47,61]. In this paper, we will pay attention to this issue, and investigate the approaches to group decision making based on intuitionistic preference relations or incomplete intuitionistic preference relations. In order to do that, the remainder of this paper is organized as follows: Section 2 briefly reviews the existing main preference relations. Section 3 introduces the concepts of intuitionistic preference relation and consistent intuitionistic preference relation, and then studies their various properties. Section 4 develops an approach to group decision making based on intuitionistic preference relations. Section 5 gives the definitions of incomplete intuitionistic preference relation and acceptable incomplete intuitionistic preference relation, and then develops an approach to group decision making based on incomplete intuitionistic preference relations. Section 6 provides a practical example to illustrate the developed approaches, and Section 7 concludes the paper. 2. Review of preference relations For a decision making problem, let X ¼ fx1 ; x2 ; . . . ; xn g be a discrete set of alternatives. In the process of decision making, a DM generally needs to provide his/her preferences for each pair of alternatives, and then constructs a preference relation, which can be defined as follows: Definition 1. [29]. A preference relation P on the set X is characterized by a function lP : X X ! D, where D is the domain of representation of preference degrees. A number of studies have been conducted on decision making problems with preference relations [13,14,24–30,33,37,41,42,44,49,51–57,59,60]. These preference relations can be mainly classed into the following three categories: (1) Multiplicative preference relation [24,44,51]: A multiplicative preference relation P on the set X is represented by a reciprocal matrix P ¼ ðpij Þnn X X with pij > 0; pij pji ¼ 1; pii ¼ 1;

for all i; j ¼ 1; 2; . . . ; n

ð1Þ

where pj is interpreted as the ratio of the preference intensity of the alternative xi to that of xj. In particular, pij ¼ 1 indicates indifference between xi and xj ; pij > 1 indicates that xi is preferred to xj, and pij < 1 indicates that xj is preferred to xi.


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(2) Fuzzy preference relation [13,14,30,33,37,41,42,49,51,52,56,59,60]: A fuzzy preference relation R on the set X is represented by a complementary matrix R ¼ ðrij Þnn X X with rij P 0; rij þ rji ¼ 1; rii ¼ 0:5

for all i; j ¼ 1; 2; . . . ; n

ð2Þ

where rij denotes the preference degree of the alternative xi over xj. In particular, rij ¼ 0:5 indicates indifference between xi and xj, rij > 0:5 indicates that xi is preferred to xj, and rij < 0:5 indicates that xj is preferred to xi. (3) Linguistic preference relation [22,23,25–28,53,55,57]: Consider a finite and totally ordered discrete linguistic label set S ¼ fsi ji ¼ t; . . . ; tg, where si represents a linguistic variable [43,63] and satisfies the following characteristics: (1) the set is ordered: si > sj if i > j; (2) there is a negation operator: neg ðst Þ ¼ st . The cardinality of S must be small enough so as not to impose useless precision on the DM and it must be rich enough in order to allow a discrimination of the performances of each alternative in a limited number of grades [9]. To preserve all the given information, the discrete label set S should be extended to a continuous label set S ¼ fsa ja 2 ½q; qg, where qðq > tÞ is a sufficiently large positive integer. If sa 2 S, then sa is termed an original linguistic label, otherwise, sa is termed a virtual linguistic label [55]. In general, a DM uses the original linguistic labels to evaluate alternatives, and the virtual linguistic we define their operalabels can only appear in operation. Consider any two linguistic labels sa ; sb 2 S, tional laws as follows [51]: (1) sa sb ¼ saþb ; (2) ksa ¼ ska , k 2 ½0; 1. A linguistic preference relation L on the set X is represented by a linguistic decision matrix L ¼ ðlij Þnn X X with lij lji ¼ s0 ; lii ¼ s0 ; for all i; j ¼ 1; 2; . . . ; n lij 2 S; ð3Þ where lij denotes the preference degree of the alternative xi over xj. In particular, lij ¼ s0 indicates indifference between xi and xj, lij > s0 indicates that xi is preferred to xj, and lij < s0 indicates that xj is preferred to xi. 3. Intuitionistic preference relations Let Y be a universe of discourse, then a fuzzy set A ¼ f< y; lA ðyÞ > jy 2 Y g

ð4Þ

defined by Zadeh [62] is characterized by a membership function lA : Y ! ½0; 1, where lA ðyÞ denotes the degree of membership of the element y to the set A. In [3,5], Atanassov introduced a generalized fuzzy set called intuitionistic fuzzy set, shown as follows: A ¼ f< y; lA ðyÞ; vA ðyÞ > jy 2 Y g

ð5Þ

which is characterized by a membership function lA : Y ! ½0; 1 and a non-membership function vA : Y ! ½0; 1 with the condition 0 6 lA ðyÞ þ vA ðyÞ 6 1 for all y 2 Y

ð6Þ

where the numbers lA ðyÞ and vA ðyÞ represent, respectively, the degree of membership and the degree of nonmembership of the element y to the set A. Definition 2 ([3,16]). If A1 and A2 are two intuitionistic fuzzy sets, then (1) (2) (3) (4) (5)

1 ¼ f< y; vA ðyÞ; lA ðyÞ > jy 2 Y g. A 1 1 A1 þ A2 ¼ f< y; lA1 ðyÞ þ lA2 ðyÞ lA1 ðyÞ lA2 ðyÞ; vA1 ðyÞ vA2 ðyÞ > jy 2 Y g. A1 A2 ¼ f< y; lA1 ðyÞ lA2 ðyÞ; vA1 ðyÞ þ vA2 ðyÞ vA1 ðyÞ vA2 ðyÞ > jy 2 Y g. kA1 ¼ f< y; 1 ð1 lA1 ðyÞÞk ; ðvA1 ðyÞÞk > jy 2 Y g; k > 0. Ak1 ¼ f< y; ðlA1 ðyÞÞk ; 1 ð1 vA1 ðyÞÞk > jy 2 Y g; k > 0.

In some real-life situations, a DM may provide his/her preferences for alternatives to a certain degree, but it is possible that he/she is not so sure about it [18]. Thus, it is very suitable to express the DM’s preference values

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with intuitionistic fuzzy values rather than exact numerical values or linguistic variables [28,46,47,61]. Szmidt and Kacprzyk [47] generalized the fuzzy preference relation to the intuitionistic fuzzy preference relation, and defined the concepts of intuitionistic fuzzy core and consensus winner. They aggregated the individual intuitionistic fuzzy preference relations into a social fuzzy preference relation on the basis of fuzzy majority equated with a fuzzy linguistic quantifier. In the following, we introduce the concept of intuitionistic preference relation: Definition 3. An intuitionistic preference relation B on the set X is represented by a matrix B ¼ ðbij Þnn X X with bij ¼< ðxi ; xj Þ; lðxi ; xj Þ; vðxi ; xj Þ > for all i; j ¼ 1; 2; . . . ; n. For convenience, we let bij ¼ ðlij ; vij Þ, for all i; j ¼ 1; 2; . . . ; n, where bij is an intuitionistic fuzzy value, composed by the certainty degree lij to which xi is preferred to xj and the certainty degree vij to which xi is non-preferred to xj, and pij ¼ 1 lij vij is interpreted as the uncertainty degree to which xi is preferred to xj. Furthermore, lij and vij satisfy the following characteristics: 0 6 lij þ vij 6 1;

lji ¼ vij ;

vji ¼ lij ;

lii ¼ vii ¼ 0:5

for all i; j ¼ 1; 2; . . . ; n

ð7Þ

To aggregate intuitionistic preference information, we introduce the following relations and operations: Definition 4. If bij ¼ ðlij ; vij Þ and bkl ¼ ðlkl ; vkl Þ are two intuitionistic fuzzy values, then (1) (2) (3) (4) (5)

bij ¼ ðvij ; lij Þ. bij þ bkl ¼ ðlij þ lkl lij lkl ; vij vkl Þ. bij bkl ¼ ðlij lkl ; vij þ vkl vij vkl Þ. k kbij ¼ ð1 ð1 lij Þ ; vkij Þ; k > 0. k k k bij ¼ ðlij ; 1 ð1 vij Þ Þ; k > 0:

Based on Definition 4, we have Theorem 1. Let bij ¼ ðlij ; vij Þ and bkl ¼ ðlkl ; vkl Þ be two intuitionistic fuzzy values, and let crs ¼ bij þ bkl , c_ rs ¼ bij bkl , d ij ¼ kbij and d_ ij ¼ bkij (k > 0Þ, then all crs , c_ rs , d ij and d_ ij are also intuitionistic fuzzy values, and (1) (2) (3) (4) (5) (6)

bij þ bkl ¼ bkl þ bij . bij bkl ¼ bkl bij . kðbij þ bkl Þ ¼ kbij þ kbkl , k > 0. k ðbij bkl Þ ¼ bkij bkkl ; k > 0: k1 bij þ k2 bij ¼ ðk1 þ k2 Þbij , k1 ; k2 > 0. k þk bkij1 bkij2 ¼ ðbij Þ 1 2 , k1 ; k2 > 0.

Definition 5. Let B ¼ ðbij Þnn be an intuitionistic preference relation, if it satisfies the multiplicative transitivity: bij ¼ bik bkj

for all i; j; k ¼ 1; 2; . . . ; n

ð8Þ

then we call B a consistent intuitionistic preference relation. A consistent intuitionistic preference relation can be interpreted as follows: For all i; j; k, the alternative xi is preferred to xj with a intuitionistic fuzzy value bij should be equal to the product of the intensities of preferences when using an intermediate alternative xk. An intuitionistic preference relation has the following properties: Theorem 2. Let BT be the transposition of an intuitionistic preference relation B, then BT ¼ B. Proof. It follows immediately from Definitions 3 and 4.

h

Theorem 3. Let B ¼ ðbij Þnn be an intuitionistic preference relation, if we remove the ith row and ith column from B, then the preference relation composed by the remainder ðn 1Þ rows and ðn 1Þ columns of B is also an intuitionistic preference relation.


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Proof. It follows immediately from Definition 3. Similar to [12], we introduce the score function D of an intuitionistic fuzzy value bij ¼ ðlij ; vij Þ, shown as follows: Dðbij Þ ¼ lij vij

ð9Þ

where Dðbij Þ is the score of bij, and Dðbij Þ 2 ½1; 1. The larger the score Dðbij Þ, the greater the intuitionistic fuzzy value bij. Similar to [31] we define an accuracy function: H ðbij Þ ¼ lij þ vij

ð10Þ

to evaluate the degree of accuracy of the intuitionistic fuzzy value bij, where H ðbij Þ 2 ½0; 1. The larger the value of H ðbij Þ, the more the degree of accuracy of the intuitionistic fuzzy value bij. As described above, the score function D and the accuracy function H are, respectively, defined as the difference and the sum of the certainty degree to which one alternative is preferred to another and the certainty degree to which one alternative is non-preferred to another. The relation between the score function D and the accuracy function H is similar to the relation between mean and variance in statistics. Based on the score function D and the accuracy function H, in the following, we introduce an order relation between any pair of intuitionistic fuzzy values. h Definition 6. Let bij ¼ ðlij ; vij Þ and bkl ¼ ðlkl ; vkl Þ be two intuitionistic fuzzy values, Dðbij Þ ¼ lij vij and Dðbkl Þ ¼ lkl vkl be the scores of bij and bkl, respectively, and let H ðbij Þ ¼ lij þ vij and H ðbkl Þ ¼ lkl þ vkl be the accuracy degrees of bij and bkl, respectively, then • If Dðbij Þ < Dðbkl Þ, then bij is smaller than bkl, denoted by bij < bkl ; • If Dðbij Þ ¼ Dðbkl Þ, then (1) If H ðbij Þ ¼ H ðbkl Þ, then bij and bkl represent the same information, denoted by bij ¼ bkl ; (2) If H ðbij Þ < H ðbkl Þ, then bij is smaller than bkl , denoted by bij < bkl . Based on Definitions 3, 4, and 6, we have Properties (I). Let B ¼ ðbij Þnn be an intuitionistic preference relation, where bij ¼ ðlij ; vij Þ; i; j ¼ 1; 2; . . . ; n, then (1) If bik þ bkj P bij for all i; j; k ¼ 1; 2; . . . ; n, then we say B satisfies the triangle condition.This condition can be explained geometrically, that is, if we take the alternatives xi, xk and xj for the vertices of a triangle with length sides bik, bkj and bij, then the length corresponding to the vertices xi, xj should not exceed the sum of the lengths corresponding to the vertices xi, xk, and xk, xj. (2) If bik P ð0:5; 0:5Þ; bkj P ð0:5; 0:5Þ ) bij P ð0:5; 0:5Þ, for all i; j; k ¼ 1; 2; . . . ; n, then we say B satisfies the weak transitivity property.This property can be interpreted as follows: If the alternative xi is preferred to xk, and xk is preferred to xj, then xi should be preferred to xj. (3) If bij P minfbik ; bkj g, for all i; j; k ¼ 1; 2; . . . ; n, then we say B satisfies max–min transitivity property.The max–min transitivity property is that the intuitionistic fuzzy value derived from a direct comparison between two alternatives should be equal to or greater than the minimum partial values derived from comparing both alternatives with an intermediate one. (4) If bij P maxfbik ; bkj g, for all i; j; k ¼ 1; 2; . . . ; n, then we say B satisfies max–max transitivity property.The max–max transitivity property can be described as follows: The intuitionistic fuzzy value derived from a direct comparison between two alternatives should be equal to or greater than the maximum partial values derived from comparing both alternatives with an intermediate one. (5) If bik P ð0:5; 0:5Þ, bkj P ð0:5; 0:5Þ ) bij P minfbik ; bkj g, for all i; j; k ¼ 1; 2; . . . ; n, then we say B satisfies the restricted max–min transitivity property.The restricted max–min transitivity property can be interpreted in the following way: When the alternative xi is preferred to xk with an intuitionistic fuzzy value bik, and xk

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is preferred to xj with a value bkj, then xi should be preferred to xj with at least an intuitionistic fuzzy value bij equal to the minimum of the above values. With the equality only when there is indifference between at least two of the three alternatives. (6) If bik P ð0:5; 0:5Þ, bkj P ð0:5; 0:5Þ ) bij P maxfbik ; bkj g, for all i; j; k ¼ 1; 2; . . . ; n, then we say B satisfies the restricted max–max transitivity property.The restricted max–max transitivity property implies that when the alternative xi is preferred to xk with an intuitionistic fuzzy value bik, and xk is preferred to xj with an intuitionistic fuzzy value bkj, then xi should be preferred to xj with at least an intuitionistic fuzzy value bij equal to the maximum of the above values. The equality holds only when there is indifference between at least two of the three alternatives. By Theorem 1 and mathematical induction on m, we have ð1Þ

ð2Þ

ðmÞ

ðkÞ

ðkÞ

ðkÞ

Theorem 4. Let bij ; bij ; . . . ; bij be m intuitionistic fuzzy values, where bij ¼ ðlij ; vij Þ, k ¼ 1; 2; . . . ; m; and let Pm ð1Þ ð2Þ ðmÞ T w ¼ ðw1 ; w2 ; . . . ; wm Þ be the weight vector of bij ; bij ; . . . ; bij , wk > 0; k ¼ 1; 2; . . . ; m; k¼1 wk ¼ 1, then the ð2Þ ðmÞ ij of bð1Þ aggregated value b ij ; bij ; . . . ; bij is also an intuitionistic fuzzy value, where bij is obtained by using the intuitionistic fuzzy weighted arithmetic averaging operator: m X ðkÞ wk bij ; i; j ¼ 1; 2; . . . ; n ð11Þ bij ¼ k¼1

or by using the intuitionistic fuzzy weighted geometric averaging operator: m wk Y ðkÞ bij ; i; j ¼ 1; 2; . . . ; n bij ¼

ð12Þ

k¼1

In particular, if w ¼ ð1=m; 1=m; . . . ; 1=mÞT , then (11) and (12) are, respectively, reduced to the intuitionistic fuzzy arithmetic averaging operator: m 1 X ðkÞ b ; i; j ¼ 1; 2; . . . ; n ð13Þ bij ¼ m k¼1 ij and the intuitionistic fuzzy geometric averaging operator !m1 m Y ðkÞ bij ; i; j ¼ 1; 2; . . . ; n bij ¼

ð14Þ

k¼1

4. An approach to group decision making based on intuitionistic preference relations In this section, we develop an approach to group decision making based on intuitionistic preference relations, which can be described as follows: Algorithm (I) Step 1. Consider a group decision making problem, let X ¼ fx1 ; x2 ; . . . ; xn g be a discrete set of alternatives, T and let D ¼ fd 1 ; d 2 ; . . . ; d m g be the setPof DMs. Let x ¼ ðx1 ; x2 ; . . . ; xm Þ be the weight vector of m DMs, where xk > 0; k ¼ 1; 2; . . . ; m; k¼1 xk ¼ 1. The DM d k 2 D provides his/her intuitionistic preference for each pair of alternatives, and constructs an intuitionistic preference relation BðkÞ ¼ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðbij Þnn , where bij ¼ ðlij ; vij Þ, 0 6 lij þ vij 6 1, lji ¼ vij , vji ¼ lij , lii ¼ vii ¼ 0:5 for all i; j ¼ 1; 2; . . . ; n. Step 2. Utilize the intuitionistic fuzzy arithmetic averaging operator: ðkÞ

bi ¼

n 1X ðkÞ b ; n j¼1 ij

i ¼ 1; 2; . . . ; n

ð15Þ


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ðkÞ

to aggregate all bij ðj ¼ 1; 2; . . . ; nÞ corresponding to the alternative xi, and then get the averaged ðkÞ intuitionistic fuzzy value bi of the alternative xi over all the other alternatives. Step 3. Utilize the intuitionistic fuzzy weighted arithmetic averaging operator: m X ðkÞ xk bi ; i ¼ 1; 2; . . . ; n ð16Þ bi ¼ k¼1 ðkÞ

to aggregate all bi ðk ¼ 1; 2; . . . ; mÞ corresponding to m DMs into a collective intuitionistic fuzzy value bi of the alternative xi over all the other alternatives. Step 4. Rank all bi ði ¼ 1; 2; . . . ; nÞ by means of the score function (9) and the accuracy function (10), and then rank all the alternatives xi ði ¼ 1; 2; . . . ; nÞ and select the best one in accordance with the values of bi ði ¼ 1; 2; . . . ; nÞ. 5. Incomplete intuitionistic preference relations Considered that, sometimes, a DM may provide incomplete judgments in an intuitionistic preference relation because of time pressure, lack of knowledge, and the DM’s limited expertise related with problem domain, in the following, we investigate the decision making problem with incomplete intuitionistic preference relations. Definition 7. Let B ¼ ðbij Þnn be an intuitionistic preference relation, where bij ¼ ðlij ; vij Þ, for all i; j ¼ 1; 2; . . . ; n, then B is called an incomplete intuitionistic preference relation, if some of its elements cannot be given by the DM, which we denote by the unknown variable ‘‘x’’, and the others can be provided by the DM, which satisfy 0 6 lij þ vij 6 1;

lji ¼ vij ;

vji ¼ lij ;

lii ¼ vii ¼ 0:5 for all i; j ¼ 1; 2; . . . ; n

ð17Þ

Similar to Properties (I) of intuitionistic preference relation, we have Properties (II). Let B ¼ ðbij Þnn be an incomplete intuitionistic preference relation, where bij ¼ ðlij ; vij Þ, i; j ¼ 1; 2; . . . ; n, and let X be the set of all the known elements, then (1) If bik þ bkj P bij , for all bik ; bkj ; bij 2 X, then we say B satisfies the triangle condition. (2) If bik P ð0:5; 0:5Þ, bkj P ð0:5; 0:5Þ ) bij P ð0:5; 0:5Þ for all bik ; bkj ; bij 2 X, then we say B satisfies the weak transitivity property. (3) If bij P minfbik ; bkj g for all bik ; bkj ; bij 2 X, then we say B satisfies max–min transitivity property. (4) If bij P maxfbik ; bkj g for all bik ; bkj ; bij 2 X, then we say B satisfies max–max transitivity property. (5) If bik P ð0:5; 0:5Þ, bkj P ð0:5; 0:5Þ ) bij P minfbik ; bkj g for all bik ; bkj ; bij 2 X, then we say B satisfies the restricted max–min transitivity property. (6) If bik P ð0:5; 0:5Þ, bkj P ð0:5; 0:5Þ ) bij P maxfbik ; bkj g for all bik ; bkj ; bij 2 X; then we say B satisfies the restricted max–max transitivity property. Definition 8. Let B ¼ ðbij Þnn be an incomplete intuitionistic preference relation, then B is called a consistent incomplete intuitionistic preference relation, if it satisfies the multiplicative transitivity: bij ¼ bik bkj

for all bik ; bkj ; bij 2 X

ð18Þ

Definition 9. Let B ¼ ðbij Þnn be an incomplete intuitionistic preference relation, if ði; jÞ \ ðk; lÞ 6¼ /, then the elements bij and bkl are called adjoining. For the unknown element bij, if there exist two adjoining known elements bik and bkj, then bij is called available. Indeed, the element bij can be obtained indirectly by using bij ¼ bik bkj , which means that the estimated element bij should be taken according to the known elements bik and bkj. Definition 10. Let B ¼ ðbij Þnn be an incomplete intuitionistic preference relation, if each unknown element can be derived from its adjoining known elements, then B is called acceptable, otherwise, B is called unacceptable.

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Obviously, for an incomplete intuitionistic preference relation B ¼ ðbij Þnn , if B is acceptable, then there exists at least one known element (except diagonal elements) in each line or each column of B, i.e., there exist at least (n 1Þ judgments provided by the DM (that is to say, each one of the alternatives is compared at least once). Let B ¼ ðbij Þnn be an acceptable incomplete intuitionistic preference relation, then, based on (18), each unknown element bij can be estimated indirectly by using: !z1 ij Y ij ¼ ðbik bkj Þ ð19Þ b k2Z ij

where Z ij ¼ fkjbik ; bkj 2 Xg, zij is the number of the elements in Z ij . Then we get an improved intuitionistic preference relation B ¼ ð bij Þnn , where bij ; bij 2 X bij ¼ ð20Þ bij ; bij 62 X Clearly, an unknown element bij can be estimated if there exists at least one k so that the elements bik and bkj are known. The improved intuitionistic preference relation B contains both the direct intuitionistic preference information given by the DM and the indirect intuitionistic preference information derived from the known intuitionistic preference information. In the following, we develop an approach to group decision making based on incomplete intuitionistic preference relations: Algorithm (II) Step 1. Consider a group decision making problem, let X ; D and x be defined as in Section 2. The DM d k 2 D provides his/her intuitionistic preference for each pair of alternatives, and constructs an incomplete ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ intuitionistic preference relation BðkÞ ¼ ðbij Þnn , where bij ¼ ðlij ; vij Þ, 0 6 lij þ vij 6 1, lji ¼ ðkÞ

Step Step Step Step

ðkÞ

ðkÞ

vij ; lii ¼ vii ¼ 0:5, for all i; j 2 X. If B(k) is an acceptable incomplete intuitionistic preference relation, then go to the next step. However, there may be cases where some missing elements in B(k) cannot be estimated through their adjoining known elements, i.e., B(k) is an unacceptable incomplete intuitionistic preference relation (this kind of situation is not very common in real problems), then the DM d(k) needs to construct a new incomplete intuitionistic preference relation, and following this procedure until it is acceptable. ðkÞ 2. Utilize (19) to construct the improved intuitionistic preference relations BðkÞ ¼ ðbij Þnn ðk ¼ 1; ðkÞ ðkÞ 2; . . . ; mÞ of B ¼ ðbij Þnn ðk ¼ 1; 2; . . . ; mÞ. 3. See Step 2 of Algorithm (I). 4. See Step 3 of Algorithm (I). 5. See Step 4 of Algorithm (I).

6. Illustrative example In this section, we will utilize a practical example (adapted from [35]) involving the assessment of a set of agroecological regions in Hubei Province, China, to illustrate the developed approaches. Located in Central China and the middle reaches of the Changjiang (Yangtze) River, Hubei Province is distributed in a transitional belt where physical conditions and landscapes are on the transition from north to south and from east to west. Thus, Hubei Province is well known as ‘‘a land of rice and fish’’ since the region enjoys some of the favorable physical conditions, with a diversity of natural resources and the suitability for growing various crops. At the same time, however, there are also some restrictive factors for developing agriculture such as a tight man–land relation between, a constant degradation of natural resources and a growing population pressure on land resource reserve. Despite cherishing a burning desire to promote their


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standard of living, people living in the area are frustrated because they have no ability to enhance their power to accelerate economic development because of a dramatic decline in quantity and quality of natural resources and a deteriorating environment. Based on the distinctness and differences in environment and natural resources, Hubei Province can be roughly divided into seven agroecological regions: x1 – Wuhan–Ezhou– Huanggang; x2 – Northeast of Hubei; x3 – Southeast of Hubei; x4 – Jianghan region; x5 – North of Hubei; x6 – Northwest of Hubei; x7 – Southwest of Hubei. In order to prioritize these agroecological regions xi ði ¼ 1; 2; . . . ; 7Þ with respect to their comprehensive functions, a committee comprised of three DMs d k ðk ¼ T 1; 2; 3Þ (whose weight vector is x ¼ ð0:5; 0:2; 0:3Þ Þ has been set up to provide assessment information on xi ði ¼ 1; 2; . . . ; 7Þ. The DMs d k ðk ¼ 1; 2; 3Þ provide intuitionistic preferences for each pair of agroecological regions with respect to their comprehensive functions and construct the intuitionistic preference relations ðkÞ ðkÞ ðkÞ ðkÞ BðkÞ ¼ ðbij Þ77 (bij ¼ ðlij ; vij Þ; i; j ¼ 1; 2; . . . ; 7; k ¼ 1; 2; 3Þ as follows, respectively: 2

Bð1Þ

Bð2Þ

Bð3Þ

ð0:5; 0:5Þ 6 ð0:2; 0:5Þ 6 6 6 6 ð0:1; 0:7Þ 6 ¼ 6 ð0:3; 0:5Þ 6 6 ð0:4; 0:6Þ 6 6 4 ð0:1; 0:9Þ ð0:1; 0:8Þ 2 ð0:5; 0:5Þ 6 6 ð0:1; 0:6Þ 6 6 6 ð0:2; 0:8Þ 6 ¼ 6 ð0:3; 0:6Þ 6 6 ð0:2; 0:7Þ 6 6 4 ð0:1; 0:8Þ ð0:2; 0:8Þ 2 ð0:5; 0:5Þ 6 6 ð0:2; 0:6Þ 6 6 ð0:1; 0:8Þ 6 6 ¼ 6 ð0:2; 0:7Þ 6 6 ð0:2; 0:8Þ 6 6 4 ð0:1; 0:9Þ ð0:1; 0:7Þ

3 ð0:5; 0:2Þ ð0:7; 0:1Þ ð0:5; 0:3Þ ð0:6; 0:4Þ ð0:9; 0:1Þ ð0:8; 0:1Þ ð0:5; 0:5Þ ð0:6; 0:2Þ ð0:3; 0:6Þ ð0:7; 0:1Þ ð0:8; 0:2Þ ð0:6; 0:3Þ 7 7 7 ð0:2; 0:6Þ ð0:5; 0:5Þ ð0:3; 0:6Þ ð0:4; 0:5Þ ð0:7; 0:1Þ ð0:7; 0:2Þ 7 7 7 ð0:6; 0:3Þ ð0:6; 0:3Þ ð0:5; 0:5Þ ð0:6; 0:1Þ ð0:8; 0:1Þ ð0:7; 0:3Þ 7 7 ð0:1; 0:7Þ ð0:5; 0:4Þ ð0:1; 0:6Þ ð0:5; 0:5Þ ð0:5; 0:2Þ ð0:4; 0:1Þ 7 7 7 ð0:2; 0:8Þ ð0:1; 0:7Þ ð0:1; 0:8Þ ð0:2; 0:5Þ ð0:5; 0:5Þ ð0:3; 0:7Þ 5 ð0:3; 0:6Þ ð0:2; 0:7Þ ð0:3; 0:7Þ ð0:1; 0:4Þ ð0:7; 0:3Þ ð0:5; 0:5Þ 3 ð0:6; 0:1Þ ð0:8; 0:2Þ ð0:6; 0:3Þ ð0:7; 0:2Þ ð0:8; 0:1Þ ð0:8; 0:2Þ 7 ð0:5; 0:5Þ ð0:5; 0:1Þ ð0:3; 0:7Þ ð0:6; 0:1Þ ð0:7; 0:2Þ ð0:6; 0:2Þ 7 7 ð0:1; 0:5Þ ð0:5; 0:5Þ ð0:4; 0:6Þ ð0:3; 0:5Þ ð0:6; 0:2Þ ð0:5; 0:1Þ 7 7 7 ð0:7; 0:3Þ ð0:6; 0:4Þ ð0:5; 0:5Þ ð0:7; 0:3Þ ð0:8; 0:2Þ ð0:6; 0:2Þ 7 7 ð0:1; 0:6Þ ð0:5; 0:3Þ ð0:3; 0:7Þ ð0:5; 0:5Þ ð0:6; 0:2Þ ð0:4; 0:3Þ 7 7 7 ð0:2; 0:7Þ ð0:2; 0:6Þ ð0:2; 0:8Þ ð0:2; 0:6Þ ð0:5; 0:5Þ ð0:3; 0:6Þ 5 ð0:2; 0:6Þ ð0:1; 0:5Þ ð0:2; 0:6Þ ð0:3; 0:4Þ ð0:6; 0:3Þ ð0:5; 0:5Þ 3 ð0:6; 0:2Þ ð0:8; 0:1Þ ð0:7; 0:2Þ ð0:8; 0:2Þ ð0:9; 0:1Þ ð0:7; 0:1Þ 7 ð0:5; 0:5Þ ð0:6; 0:1Þ ð0:2; 0:7Þ ð0:6; 0:2Þ ð0:8; 0:1Þ ð0:8; 0:2Þ 7 7 ð0:1; 0:6Þ ð0:5; 0:5Þ ð0:2; 0:3Þ ð0:3; 0:4Þ ð0:9; 0:1Þ ð0:6; 0:1Þ 7 7 7 ð0:7; 0:2Þ ð0:3; 0:2Þ ð0:5; 0:5Þ ð0:6; 0:2Þ ð0:8; 0:1Þ ð0:7; 0:2Þ 7 7 ð0:2; 0:6Þ ð0:4; 0:3Þ ð0:2; 0:6Þ ð0:5; 0:5Þ ð0:7; 0:2Þ ð0:7; 0:3Þ 7 7 7 ð0:1; 0:8Þ ð0:1; 0:9Þ ð0:1; 0:8Þ ð0:2; 0:7Þ ð0:5; 0:5Þ ð0:2; 0:8Þ 5 ð0:2; 0:8Þ ð0:1; 0:6Þ ð0:2; 0:7Þ ð0:3; 0:7Þ ð0:8; 0:2Þ ð0:5; 0:5Þ

We first use Szmidt and Kacprzyk’ approach [47] to derive the decision result, which involves the following steps: Step 1. Based on BðkÞ ðk ¼ 1; 2; 3Þ, 2 – 0 0 6 61 – 0 6 61 1 – 6 6 ð1Þ ð1Þ H ¼ ðhij Þ77 ¼ 6 1 0 0 6 61 1 0 6 6 41 1 1 1

1

we construct the following matrices, respectively: 3 0 0 0 0 7 1 0 0 0 7 7 1 1 0 0 7 7 7 – 0 0 0 7 7 1 – 0 ½0 7 7 7 1 1 – 1 5 1 1 ½0 0 –

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2

ð2Þ

H ð2Þ ¼ ðhij Þ77

0 0

0

0

0

0

61 6 6 61 6 6 ¼ 61 6 61 6 6 41

– 0

1

0

0

0 7 7 7 0 7 7 7 0 7 7 ½0 7 7 7 1 5

2

ð3Þ

H ð3Þ ¼ ðhij Þ77

P

¼

ð1Þ ðpij Þ77

P

¼

ð2Þ ðpij Þ77

0:1 –

6 0:3 6 6 6 0 6 6 ¼ 6 0:1 6 6 0:1 6 6 4 0:1 0 –

2

ð3Þ

Pð3Þ ¼ ðpij Þ77

1 –

1

1

0

0 0

–

0

0

1 0

1

–

0

1 1

1

1

–

1 1 1 ½0 0 – 3 0 0 0 0 0 0 – 0 1 0 0 07 7 7 1 – ½0 ½0 0 0 7 7 7 0 ½0 – 0 0 07 7 1 ½0 1 – 0 07 7 7 1 1 1 1 – 15

1 1 1 1 – 0:3 0:2

6 0:3 6 6 6 0:2 6 6 ¼ 6 0:2 6 6 0 6 6 4 0 2

ð2Þ

1 –

61 6 6 61 6 6 ¼ 61 6 61 6 6 41 2

ð1Þ

6 0:2 6 6 6 0:1 6 6 ¼ 6 0:1 6 6 0 6 6 4 0 0:2

1 0 – 0:2 0 0

ðkÞ

hij

ðkÞ

3

0:2

0:1 0:3

0:1 0:1 0:3 0

0 0:5 0:1 0:1

0 0:1

0:2

0:1 0:2

0:2

–

0:1 0:1

0:1 0:1

–

0:3

0:2 0:1

0:3

–

0

–

0:4

0

0:3

0:4

–

0

0:2

0

0

–

0

0:3 0:2

0

–

0:1 0:2

0

0:2

0:2 0:4 0:2 0:1

0:2 0:3 0:1 0

–

0:3

0:1 0:2

0:3

–

0:5 0:3

0:1 0:5

–

0:2

0:2 0:3

0:2

–

0:1

0

0:1 0:1

0

0:3

ðkÞ

ðkÞ

0:1

for bii ¼ ðlii ; vii Þ ðkÞ

ðkÞ

for lij < 0:5 and vij < 0:5 ðkÞ

0:1

0:1 7 7 7 0:2 0:1 7 7 7 0:1 0 7 7 0:3 0:5 7 7 7 – 0 5

–

where 8 – > > > > > 0 > > : ½0

3

–

for lij P 0:5 otherwise

0

0

– 3 0 0:1 0:2 7 7 7 0:2 0:4 7 7 7 0 0:2 7 7 0:2 0:2 7 7 7 – 0:1 5 0:1 – 3 0 0:2 0:1 0 7 7 7 0 0:3 7 7 7 0:1 0:1 7 7 0:1 0 7 7 7 – 0 5 0

–

Z. Xu / Information Sciences 177 (2007) 2363–2379 ðkÞ

ðkÞ

ðkÞ

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ðkÞ

ðkÞ

and pij ¼ 1 lij vij , i; j ¼ 1; 2; . . . ; 7; k ¼ 1; 2; 3. hij ¼ \ " means that bii does not matter, ðkÞ ðkÞ hij ¼ 1 means that the DM d(k) prefers xj over xi, hij ¼ 0 means that the DM d(k) prefers xi over ðkÞ xj, and hij ¼ ½0 means no option is preferred. Step 2. Use the formula: m n X X 1 ðkÞ h ; m ¼ 3; n ¼ 7 hj ¼ mðn 1Þ k¼1 i¼1;i6¼j ij to calculate the extent that all the DMs d k ðk ¼ 1; 2; 3Þ are not against xj: h1 ¼ 1;

h2 ¼

12 ; 18

h3 ¼

6 ; 18

Step 3. Use the formula: m n X X 1 ðkÞ pj ¼ p ; mðn 1Þ k¼1 i¼1;i6¼j ij

h4 ¼

14 ; 18

h5 ¼

6 ; 18

h6 ¼ 0;

h7 ¼

3 18

m ¼ 3; n ¼ 7

to calculate the hesitation margin related to xj: p1 ¼

2 ; 18

p2 ¼

31 ; 180

p3 ¼

36 ; 180

p4 ¼

22 180

p5 ¼

33 ; 180

p6 ¼

16 ; 180

p7 ¼

25 180

Step 4. Add the value pj to hj, which gives the upper bound of the interval h0j ¼ ½hj ; hj þ pj , and thus get the following ranges: 20 12 151 6 96 14 162 6 93 ; ; ; ; h01 ¼ 1; ; h02 ¼ ; h03 ¼ ; h04 ¼ h05 ¼ ; 18 18 180 18 180 18 180 18 180 16 3 55 ; ; h07 ¼ h06 ¼ 0; 180 18 180 Step 5. Assume a fuzzy majority given as a fuzzy linguistic quantifier Q = ‘‘most’’: 8 for x P 0:8 >