Incomplete interval valued fuzzy preference relations

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Feb 12, 2016 - According to Orlovsky [18], P is additive reciprocal if for all i, j it satisfies pij + pji = 1. ii. P is additive transitive if pij = pik + pkj − 0.5 for all i, j, k.
Information Sciences 348 (2016) 15–24

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Incomplete interval valued fuzzy preference relations Asma Khalid∗, Ismat Beg Center for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan

a r t i c l e

i n f o

Article history: Received 7 May 2015 Revised 4 February 2016 Accepted 8 February 2016 Available online 12 February 2016 Keywords: Interval valued fuzzy preference relation Incomplete relation Additive consistency for interval valued relation Interval valued multiplicative preference

a b s t r a c t An interval valued preference relation is a preference structure that is used to describe uncertainty in complex decision making problems. Retrieving complete information from experts is improbable in real life scenarios. Discarding incomplete information leads to loss of important data. In this paper, we introduce an upper bound condition to deal with incomplete interval valued fuzzy preference relations. With the help of this condition, missing preferences are estimated such that they are expressible. Moreover, the resultant complete relation is consistent. In case if an expert is unable to abide by the proposed upper bound condition, an algorithm is formulated to assist the expert in complying to the upper bound condition. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Fuzzy preference relations have successfully modeled decision making problems. To combat vagueness, interval valued preference relations are introduced by Bilgic [7] and Xu [30,33]. Interval valued relations add flexibility to the uncertainty representation problem [3]. Construction of interval valued fuzzy preference relations using ignorance functions and their applications is introduced by Barrenechea et al. [4]. Interval weights were derived by Wang and Kevin [26] using goal programming approach on interval fuzzy preference relations. Application of decision making in finance can be studied in [9]. Literature proposes several methods to incorporate for incompleteness in preference relations [8,10,13]. Some methods discard decision makers providing incomplete information. Others estimate missing values using preferences of other experts. However, methods that consider expert’s own preferences to estimate the missing information are more appropriate [5,25]. Herrera-Viedma et al. [14] proposed methods to complete preference relation of an expert providing (n − 1 ) preference values of the form { p12 , p23 , ..., p(n−1 ),n }. Furthermore, [13] included the case where a complete row or column of preference intensities is given by the expert. Khalid and Awais [16] stressed that in earlier methods, estimated missing preferences surpassed the domain. Transformation functions were introduced to bring such preferences back to the unit interval but at the cost of voiding the originality of the preference values provided by experts. It was further discussed in [16,17] that the existing methods to complete an incomplete preference relation did not focus on consistency of the resultant relation. Therefore, to estimate missing values that do not surpass the unit interval, an upper bound condition was presented in [16]. The completed preference relation with this condition satisfied additive transitivity and Saaty’s consistency in case of incomplete multiplicative fuzzy preference relations. The focus of this paper is on interval valued fuzzy preference and multiplicative fuzzy preference relations. Given a large set of alternatives, it is reasonable to expect incomplete interval valued preferences intensities from decision makers. The ∗

Corresponding author. E-mail address: [email protected] (A. Khalid).

http://dx.doi.org/10.1016/j.ins.2016.02.013 0020-0255/© 2016 Elsevier Inc. All rights reserved.

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reason for this incompleteness could be lack of information, uncertainty, ambiguity or inability to make a choice. Decision making processes should be modeled in a way such that they incorporate for incompleteness, estimates missing preferences and also promise consistency of the resultant completed relation. An inconsistent preference relation of any nature is less useful as compared to preference expressions that are consistent. Consistency in preference and multiplicative preference relations is addressed in [19–21,23,27]. In interval valued fuzzy preference relations, consistency was introduced by Bilgic [7]. Zeshui [33] introduced incomplete interval fuzzy preference relation and using continuous interval argument ordered weighted average, transformed them into incomplete fuzzy preference relations. Alonso et al. [2] presented a method to estimate missing preference values in fuzzy, multiplicative, interval valued and linguistic fuzzy preference relations. Jiang [15] proposed a similarity index for interval fuzzy preference relations. He checked consistency degree of the group undergoing decision making process by using this similarity index. Genc et al. studied [12,31] and introduced test for consistency of interval valued multiplicative preference relations. They proposed that instead of deriving priority weights from linear programming models to check for consistency, simple formulas can be derived from the concept of interval multiplicative transitivity of an interval fuzzy preference relation. Furthermore, two approaches to estimate missing values in interval valued multiplicative relations were proposed. This paper aims to incorporate incomplete interval valued fuzzy preference and multiplicative fuzzy preference relations in a decision making process. A complete preference relation consists of n(n − 1 )/2 preference intensities in the upper diagonal of the preference relation. If a decision maker is unable to compare two given alternatives then the situation cannot be modeled as that of indifference. In this paper we emphasize that instead of deriving weights by linear programming model [31] or calculating the priority vectors to estimate missing values [12], the missing values can be estimated by implying an upper bound condition on the decision makers. Alonso et al. [1] proposed a general method to estimate missing values in case of interval valued fuzzy preference relations. The drawback of this method is that estimated values may come out to be supersets of the unit interval. Such an estimation will not imply anything about the preference of the decision maker. This work is motivated to estimate missing preferences while ensuring their expressibility. If estimated preferences void the giving domain then the resultant relation will not qualify as an interval valued preference relation. Therefore, such estimated preferences will have no interpretation. Moreover, this paper stresses on the fact that if the surpassed estimated preferences are translated to the appropriate domain using transformation functions, then such a decision will cost originality of the preference intensities provided by the decision makers. Therefore, this paper proposes a method which estimates missing preferences while abiding by the specified domain and does not alter the provided preferences of the decision makers. Moreover, this paper extends the work to cater for situations when decision makers are unable to abide by the upper bound condition. In this situation, an algorithm is designed to revise minimum possible opinions such that the upper bound condition is met and consequently, the estimated preferences are expressible. The paper is organized as follows. Section 1 provides literature review. Section 2 discusses the basic definitions that are used in the sequel. Section 3 introduces the concept of expressible and non-expressible interval valued preferences. This section further defines an upper bound condition, property (ubc) that is imposed on experts if they are to propose incomplete interval valued preference relations. This section proves that if decision makers abide by property (ubc) then the resultant completed relation is expressible and consistent. Section 4 further focuses on the rare case when a decision maker, despite the instructions, is unable to conform to property (ubc). In this section, a rule is defined to carry out minimal possible revisions in the provided preferences so that the incomplete relation satisfies the condition. This rule is defined so that such an interval valued relation is not discarded and so that most of the appropriate information given by the decision maker is used. This section proposes flow chart, Rule 1, to deal with both the situations where experts respect property (ubc) and the case where they fail to do so. Section 5 deals with incompleteness in interval valued multiplicative fuzzy preference relations and state a corresponding upper bound condition (mubc) for such relations. This section also briefly discusses the case when a decision maker fails to satisfy the proposed condition. Section 6 concerns future work and draws conclusion to this work. 2. Preliminaries Consider X = {x1 , x2 , ...., xn }, (n ≥ 2 ) to be a non-empty set of alternatives. Definition 1. [32] A fuzzy set A on X is characterized by a membership function μA : X → [0, 1] where μA (x) is defined as the degree of membership of element x in fuzzy set A for each x ∈ X. Definition 2. [6,11,22,23] A fuzzy preference relation P on X is characterized by a function μP : X × X → [0, 1] where μP (xi , x j ) = pi j indicates the preference intensity with which alternative xi is preferred over xj . i. According to Orlovsky [18], P is additive reciprocal if for all i, j it satisfies pi j + p ji = 1. ii. P is additive transitive if pi j = pik + pk j − 0.5 for all i, j, k. Definition 3. [20–22] Let A ⊂ X × X denote a multiplicative fuzzy preference relation, the intensity of preference aij is measured using a ratio scale, particularly, a 1 − 9 scale. Here, ai j = 1 indicates indifference between xi and xj and ai j = 9 indicates that xi is absolutely preferred to xj .

A. Khalid, I. Beg / Information Sciences 348 (2016) 15–24

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Rule 1.

i. A is multiplicative reciprocal if ai j .a ji = 1 for all i, j. ii. A reciprocal multiplicative preference relation is consistent if it satisfies Saaty’s consistency. That is, if ai j .a jk = aik for all i, j, k. Definition 4. [13] A function f: X → Y is partial when not every element in the set X necessarily maps onto an element in the non empty set Y. When every element from the set X maps onto one element of the set Y, then we have a total function. The incomplete fuzzy preference relation P and incomplete multiplicative fuzzy preference relation A on X is a fuzzy set on the product set X × X that is characterized by a partial membership function. Definition 5. [7,24,32] Let L([0, 1]) denote the set of all closed subintervals of [0, 1]. Let A∗ denote an interval valued fuzzy set on X. Then A∗ : X → L([0, 1]) and the membership of each x ∈ X is given by A∗ (x ) = [A∗ (x ), A∗ (x )] where A∗ (x ), A∗ (x ) ∈ [0, 1] and A∗ (x ) ≤ A∗ (x ). Consider A∗1 , A∗2 ∈ L([0, 1] ), where, A∗1 = [A∗1 , A∗1 ] and A∗2 = [A∗2 , A∗2 ]. Then, i. A∗1 ⊆ A∗2 if and only if A∗1 ≥ A∗2 and A∗1 ≤ A∗2 . ii. A∗1 + A∗1 = [A∗1 + A∗2 , A∗1 + A∗2 ] and iii. A∗1 − A∗2 = [A∗1 − A∗2 , A∗1 − A∗2 ]. Least and greatest elements of L([0, 1]) are 0 = [0, 0] and 1 = [1, 1] respectively. Definition 6. [1,28,30] An interval valued fuzzy preference relation R ⊂ X × X is characterized by a membership function

μ˜ R : X × X → L([0, 1] ) with μ˜ R (xi , x j ) = ri j = [ril j , rirj ] where ril j and rirj are the left and right limits of ri j , respectively.

i. R is said to satisfy additive reciprocity if ri j + r ji = 1 for all i, j. In terms of left and right limit of interval-valued preferences, additive reciprocity is defined as ril j + r rji = r lji + rirj = 1. ii. R is additive consistent if and only if ri j + r jk + rki = rk j + r ji + rki for all i, j, k. In terms of left and right limit of intervall + r l − 0.5 and r r = r r + r r − 0.5. valued preferences, additive transitivity is defined as ril j = rik ij ik kj kj Definition 7. Let L([1/9, 9]) denote the set of all closed subintervals of [1/9, 9]. A multiplicative interval valued fuzzy set A on X is a mapping A: X → L([1/9, 9]) where membership of each x ∈ X is given by A(x ) = [A(x ), A(x )] such that A(x ), A(x ) ∈ [1/9, 9] and A(x ) ≤ A(x ). Consider, a1 , a2 ∈ L([1/9, 9]) where a1 = [a1 , a1 ] and a2 = [a2 , a2 ]. Then, i. a1 ⊆a2 if and only if a 1 ≥ a2 and a1 ≤ a2 . ii. a1 + a2 = [a1 + a2 , a1 + a2 ] iii. a1 − a2 = [a1 − a2 , a1 − a2 ]. Least and greatest elements of L([1/9, 9]) are 1/9 = [1/9, 1/9] and 9 = [9, 9] respectively.

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Definition 8. [12,29] An interval valued multiplicative fuzzy preference relation R⊆X × X is characterized by a membership function μR : X × X → L([1/9, 9]) with μR (xi , x j ) = ai j = [ali j , ari j ] where ali j and ari j are the left and right limits of ai j , respectively. i. R is said to be multiplicative reciprocal if ai j .a ji = 1 for all i, j. In terms of left and right limits of the interval-valued preferences, multiplicative reciprocity is defined as ali j .arji = alji .ari j = 1. ii. R is multiplicative consistent if and only if ai j .a jk .aki = ak j .a ji .aki for all i, j, k. In terms of left and right limits of the interval-valued preferences, multiplicative consistency is defined as ali j .aljk .alki = alik .alk j .alji and ari j .arjk .arki = arik .ark j .arji . 3. Interval valued fuzzy preference relations The assumption that all decision makers can express preferences over each two tuple of alternatives provided to them is superfluous. Such a case builds on the assumption that all experts possess ample knowledge of the entire decision making problem. Specifically, when cardinality of X is large and the source of information is complex, then retrieving complete information is unrealistic. Moreover, in such cases, it is relatively convenient to use interval valued fuzzy preference relations. Literature proposes different approaches to deal with incomplete information. Based on the assumption that an appropriate decision can only be attained with complete information, some of the methods proposed in literature only process pieces of complete information while rectifying the incomplete information. In many consensus problems, preference relations with incomplete information are discarded and only complete relations are considered to form a collective relation. Moreover, there are methods in literature that negatively rate experts who provide incomplete preferences. These methods undervalue some very useful information present in data. It needs to be noted that incomplete information is not equivalent to low quality information. Therefore, the method proposed in this paper does not fall in the categories mentioned above. In this paper, incomplete information is not rejected or penalized but more focus is on consistency of the information provided. In this paper, an upper bound rule is provided to expert for the reason that as long as the rule is followed, the resultant relation will be consistent. As suggested by Khalid and Awais [16], we refer to an estimated preference value as expressible if it belongs to L([0, 1]). Other than expressibility of estimated preference values, this paper focuses on consistency of the completed relation. Moreover, we refer to a missing preference as crucial if it is estimated with the help of two intervals that contain the lowest and greatest limit. It is called crucial because it may or may not be expressible. We assert that if the crucial interval is expressible then the remaining missing preferences will certainly be expressible. In this section, we aim to identify situations where crucial preferences are not expressible. Example 1. Consider a 5 × 5 incomplete interval valued fuzzy preference relation where intensities of alternative x3 over X = {x1 , x2 , x3 , x4 , x5 } are provided by a decision maker as follows:



[0 . 5 , 0 . 5 ]

[−, −]

[−, −]

[−, −]

⎢ [−, −] ⎢ ⎢[0.78, 0.83] ⎢ ⎣ [−, −]

[0 . 5 , 0 . 5 ]

[−, −]

[−, −]

[0 . 1 , 0 . 4 ]

[0 . 5 , 0 . 5 ]

[0.8, 0.9]

[−, −]

[−, −]

[0.5, 0.5]

[−, −]

[−, −]

[−, −]

[−, −]



[−, −]

[−, −]⎥



[0.35, 0.48]⎥ ⎥ [−, −]⎦

[0 . 5 , 0 . 5 ]

The first step is to identify crucial preference because if the crucial preference is expressible then the remaining missing preferences can be estimated. However, if the crucial preference is not expressible, it implies that there are other missing preferences that are non-expressible. Least and greatest limit is present in the preference interval r32 and r34 . Therefore, with the help of additive transitivity property mentioned in Definition 6, the crucial preference r24 can be determined. Accordingly, r24 = [0.9, 1.3] ∈ / L([0, 1] ). Therefore, the crucial preference is not expressible which implies existence of other non-expressible missing preferences. Theorem 1 states a condition that needs to be satisfied by decision makers providing incomplete interval valued preference relations. Suppose that the preferences provided are rk j , where k is fixed and j ∈ {1, 2, . . . , n}. Let δ represent the greatest right end limit and  < 0.5 represent the smallest left end limit in rk j . The provided preferences are said to satisfy property (ubc) if and only if δ ≤ 0.5 +  . Violation of Theorem 1 results in the case that is depicted in Example 1. Theorem 1. Assume that n preferences of a decision maker in an n × n incomplete interval valued preference relation are given by rk j = [rkl j , rkr j ] where k is fixed and j ∈ {1, 2, … , n}. If rk j satisfies property (ubc) then the missing preferences can be estimated. Moreover, the estimated preferences are expressible. Proof. Since experts are considered to be consistent therefore, rk j = [0.5, 0.5] for k = j. Suppose that from the non diagonal l and r r belong to the intervals (n − 1 ) interval valued preferences provided by the expert, the lowest and greatest limit rki kj l =  < 0.5 and r r = δ . As property (ubc) is satisfied, therefore, rki and rk j respectively, where 0 ≤ rki kj l 0 ≤ rkr j ≤ 0.5 + rki

(1)

A. Khalid, I. Beg / Information Sciences 348 (2016) 15–24

19

l ≤ r r and r l ≤ r r . It further implies the following equations: Also, rki ki kj kj r 0 ≤ rkr j ≤ 0.5 + rki

(2)

r 0 ≤ rkl j ≤ 0.5 + rki

(3)

Now, the proof is stated in two steps. We first prove that if crucial preferences are expressible then the remaining missing preferences are expressible. We then prove that as property (ubc) is satisfied thus the crucial value is always estimated to be expressible. According to the given assumption, the crucial preference is identified to be ri j = [ril j , rirj ]. For the first part,

assume that ri j is expressible. Then let rs j = [rsl j , rsr j ], s = j, s, j ∈ {1, 2, .., n} be a missing preference other than the crucial preference. l + r l − 0.5 < r l + r l − 0.5 = (1 − r r ) + r l . We know that r l ≤ r r ≤ r r . Equivalently, 1 − r r ≥ 1 − r l ≥ 1 − Thus, rsl j = rsk ks ks kj ki kj sk kj ki ki sk r ) + r l ≤ 1 − r r + r l = r l + r l . Since r l < 0.5, hence 2 (r l ) < 1. Which proves that r l ∈ [0, 1]. r ljk . Therefore, rsl j < (1 − rks sj ki ki ki ki ki ki ki

r + r r − 0.5 < r r + r r − 0.5 = (1 − r l ) + r r . Also, r r ≤ r l ≤ r l . Equivalently, 1 − r l ≥ 1 − r r ≥ 1 − r r . Similarly, rsr j = rsk kj sk kj ki ki sk jk ks ks kj ki

l ) + r r ≤ 1 − r l + r r = r r + r r . Since r r < 0.5, hence 2 (r r ) < 1. Which proves that if the crucial prefTherefore, rsr j < (1 − rks ki ki ki ki ki ki ki erence is expressible, then so are other estimated preferences. We now prove that as property (ubc) is satisfied by the provided preferences thus the crucial preference ri j is estimated to be expressible. That is, ri j ∈ L([0, 1] ). l + r l − 0.5 = Using additive consistency for interval valued preference relations from Definition 6, we state that ril j = rik kj

r ) + r l − 0.5 = (r l − r r ) + 0.5. According to Eq. (3), r l − r r ≤ 0.5 it implies that r l ≤ 1. (1 − rki ij ki ki kj kj kj r + r r − 0.5 = (1 − r l ) + r r − 0.5 = (r r − r l ) + 0.5. According to Eq. (1), r r − r l ≤ 0.5 it implies that Similarly, rirj = rik kj kj kj kj ki ki ki rirj ≤ 1. That is, ri j ∈ L([0, 1] ) and hence the crucial value is expressible. Hence, the statement is proved. 

It needs to be noted that property (ubc) is based on the condition  < 0.5. The reason for which is that if the left and right limits of each interval provided are greater than 0.5 then there will be no missing value that is non expressible. However, assuming an interval valued preference relation where each alternative is strongly preferred over the rest of the alternatives is a superficial assumption. A more realistic case, worthy of consideration, is when some of the decision makers have lower preference of an alternative over another in which case upper limit of the corresponding interval will be less than 0.5. This case is considered in Example 1. Corollary 1. If preferences provided in an incomplete interval valued fuzzy preference relation satisfy property (ubc) then the estimated preferences are consistent. Proof. Follows from proof of Theorem 1.



Corollary 2. Preferences in each row of a complete interval valued additive transitive relation satisfies property (ubc). Proof. Assume that in the k-th row, the largest and smallest preference limits present in the interval valued preference l respectively, where k ∈ {1, 2, ..., n}, k = j = i. relation are rkr j and rki l + Suppose that on the contrary, the largest upper limit of the interval does not respect property (ubc). That is, rkr j > rki

l − 0.5 > 0. That is, r r − (1 − r r ) − 0.5 > 0. Then r r + r r − 0.5 > 1. It implies that r r > 1. This proves that 0.5. Then, rkr j − rki ij kj ki kj ik ri j ∈ / L([(0, 1 )] ) and therefore it is not expressible. 

Corollary 3. An incomplete Interval valued preference relation completed using Theorem 1 satisfies additive reciprocity. Proof. The interval valued preference relation completed using Theorem 1 satisfies l + r l − 0.5 = (1 − r r ) + (1 − r r ) − 0.5 = 1 − (r r + r r − 0.5 ) = 1 − r r . Similarly, ril j = rik ji ki jk ki jk kj

(1

− r ljk )

− 0.5 = 1 −

l (rki

+

r ljk

− 0.5 ) =

1 − r lji .

It implies that ri j = 1 − r ji .

r + r r − 0.5 = ( 1 − r l ) + rirj = rik kj ki



Example 2. Consider the following 4 × 4 incomplete interval valued preference relation where a decision maker is successful in expressing preferences of the first alternative over the set of alternatives X = {x1 , x2 , x3 , x4 } through pairwise comparison. The incomplete interval valued fuzzy preference relation is as follows:



[0 . 5 , 0 . 5 ]

[0.15, 0.3]

[0.2, 0.4]

[−, −]

[0 . 5 , 0 . 5 ]

[−, −]

[−, −]

[−, −]

[0.5, 0.5]

[−, −]



[−, −]

⎢ ⎢ ⎣



[0.35, 0.6]

∗⎥



−⎦ [0 . 5 , 0 . 5 ]

Note that the crucial preference is r24 . According to Theorem 1, if this crucial value is expressible then the remaining missing preferences are expressible as well. Using additive consistency, r24 = [0.55, 0.95] and therefore, r42 = [0.05, 0.45]. The

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remaining preferences are estimated and the completed relation is formulated as follows:



[0 . 5 , 0 . 5 ]

[0.15, 0.3]

[0 . 2 , 0 . 4 ]



[0.35, 0.6]

⎢[0.7, 0.85] ⎢ ⎣ [0 . 6 , 0 . 8 ]

[0.5, 0.5]

[0.4, 0.75]

[0.25, 0.6]

[0 . 5 , 0 . 5 ]

[0.55, 0.95]⎥ [0.45, 0.9]⎦

[0.4, 0.65]

[0.05, 0.45]

[0.1, 0.55]

[0 . 5 , 0 . 5 ]



Note that the completed relation is additive transitive and hence reciprocal ri j = 1 − r ji for all i, j ∈ {1, 2, 3, 4} as stated in corollary 1 and 3. Moreover, each row in the completed relation satisfies property (ubc).

4. Non-expressible interval valued preferences The main assumption in Section 3 is that if a decision maker provides incomplete interval valued fuzzy preferences, then she satisfies property (ubc) before presenting the incomplete information. However, if interval valued preferences provided by experts do not respect the defined condition then instead of discarding the preference relation altogether, the problematic left and right limit of preferences may be modified. Revision of provided preferences is discouraged in Section 2 and that is the reason why property (ubc) was proposed to the experts at the first place. But for unavoidable circumstances, the following algorithm is generated which ensures minimum possible revisions of the provided preference intensities. Since, rii = [0.5, 0.5] because of consistency, it has no role to play in Theorem 1, therefore, (n − 1 ) preferences need to be tested. Rule 1. Suppose n preferences ri j for fixed i and j ∈ {1, 2, .., n} are provided by a decision maker. Consider a set of left

and right end limit of each interval provided by the decision maker as  = {ril 1 , rir 1 , .., ril k , rir k , ..., ril n , rir n }, k = i . Let , l ⊂  such that = {rir 1 , rir 2 , ..., rir n } and l = {ril 1 , ril 2 , ..., ril n }. Also, let γ = {rir k : rir k satisfy property (ubc)} ⊂ and γ l = {ril k : ril k satisfy property (ubc)} ⊂ . Case (a): If | γ |≥ n−1 2 Step 1: Consider sup( ) = δ1 . If δ 1 ∈ γ then revise δ1 = δ1 where δ1 = inf( ) + 0.5. Now, test the corresponding left end limit of the interval that contains δ 1 , denote it as left(δ 1 ).





If le f t (δ1 ) ≤ δ1 revision is not required. Otherwise, if left(δ 1 ) ∈ γ l then le f t (δ1 ) = le f t (δ1 ) where le f t (δ1 ) = δ1 − min j {| r l ri j − ri j |}. Step 2: Consider sup( /δ1 ) = δ2 .



If δ 2 ∈ γ then revise δ2 = δ2 where δ2 = inf( ) + 0.5. Now, if le f t (δ2 ) ≤ δ2 then revision is not required. Otherwise, if





left(δ 2 ) ∈ γ l then le f t (δ2 ) = le f t (δ2 ) where le f t (δ2 ) = δ2 − min j {| rir j − ril j |}.  Step | | − | γ |: Consider sup( /δ| |−|γ |−1 ) = δ| |−|γ | .





If δ| |−|γ | ∈ / γ then δ| |−|γ | = δ| |−|γ | where δ| |−|γ | =  + 0.5. Also, if le f t (δ| |−|γ | ) ≤ δ| |−|γ | revision is not required. Oth



erwise, le f t (δ| |−|γ | ) = le f t (δ| |−|γ | ) where le f t (δ| |−|γ | ) = δ| |−|γ | − min j {| rir j − ril j |}. Case (b): On the other hand, If | γ l |< n−1 2 then Step 1: Consider inf( l ) = 1 .



If  1 ∈ γ l then 1 = 1 where 1 = sup( l ) − 0.5.





Now, if right (1 ) ≥ 1 then no revision is required. Otherwise, if right (1 ) < 1 then right (1 ) = 1 + min j {| rir j − ril j |}.



Step 2: Consider, inf( l /1 ) = 2



If  2 ∈ γ l then 2 = 2 where 2 = sup( l ) − 0.5. If right (2 ) ≥ 2 then no revision is required. Otherwise, if right (2 ) <



2 then right (2 ) = 2 + min j {| rir j − ril j |}.

 Step |γ l |: Consider, inf( l /|γ l |−1 ) = |γ l | .





If |γ l | ∈ γ l then |γ l | =  l and  l = sup( l ) − 0.5. If right (|γ l | ) ≥  l then no revision is required. Otherwise, |γ | |γ | |γ |



right ( l ) =  l + min j {| rir j − ril j |}. |γ |

|γ |

Note that this rule suggests minimum possible revisions. Example 3. Consider the following example where rector of City University, requests the registrar to define a criteria for hiring of new maintenance staff. However, the set of alternatives X = {x1 : dedication , x2 : punctuality , x3 : young age, x4 : physically fitness, x5 : matriculation degree , x6 : experience, x7 : criminal record } is complex. Registrar intends to propose an incomplete interval valued fuzzy preference relation. At which, the rector suggests the registrar to abide by property (ubc).

A. Khalid, I. Beg / Information Sciences 348 (2016) 15–24

21

Information retrieved by the registrar is represented as follows.



[0 . 5 , 0 . 5 ]

[0 . 6 , 0 . 7 ]

[0.68, 0.92]

[0.31, 0.4]

[0 . 2 , 0 . 3 ]

[0.3, 0.45]

[−, −]

[0 . 5 0 . 5 ]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[0 . 5 , 0 . 5 ]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[0 . 5 , 0 . 5 ]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[0 . 5 , 0 . 5 ]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[0.5, 0.5]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

[−, −]

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣



[0.92, 0.98]

[−, −]⎥ ⎥

⎥ ⎥ [−, −]⎥ ⎥ [−, −]⎥ ⎥ [−, −]⎦ [−, −]⎥

[0 . 5 , 0 . 5 ]

Clearly, registrar’s preferences do not satisfy property (ubc). Therefore, Rule 1 is used to revise minimal possible preferences. According to the information provided,  = {0.6, 0.7, 0.68, 0.92, 0.31, 0.4, 0.2, 0.3, 0.45, 0.92, 0.98}, = {0.7, 0.92, 0.4, 0.3, 0.45, 0.98}, l = {0.6, 0.68, 0.31, 0.2, 0.3, 0.92}, γ = {0.7, 0.4, 0.3, 0.45} and γ l = {0.6, 0.68, 0.31, 0.3, 0.2} Since | γ |= 4 ≥ 6/2 , therefore, case (a) of Rule 1 is to be followed: Step 1: Consider sup( ) = 0.98 = δ1 . Note that δ 1 ∈ γ , so δ1 = δ1 where δ1 = 0.2 + 0.5 = 0.7. Note that le f t (δ1 ) > δ1 , therefore, le f t (δ1 ) = 0.7 − min j={2,3,...,7} {0.1, 0.24, 0.15, 0.06}. So, le f t (δ1 ) = 0.64. Step 2: Consider sup( /δ1 ) = 0.92 = δ2 . Note that δ 2 ∈ γ , So δ2 = δ2 where δ2 = 0.7. Now, since le f t (δ2 ) ≤ δ2 therefore no further revision is required. Revised preferences are stated as follows in bold. Moreover, missing preferences are estimated to complete the interval valued preference relation.



[0 . 5 , 0 . 5 ]

⎢ [0 . 3 , 0 . 4 ] ⎢ ⎢[0.3, 0.32] ⎢ ⎢ ⎢[0.6, 0.69] ⎢ ⎢ [0 . 7 , 0 . 8 ] ⎢ ⎣[0.55, 0.7] [0.3, 0.36]

[0 . 6 , 0 . 7 ]

[0.68, 0.7]

[0.31, 0.4]

[0 . 2 , 0 . 3 ]

[0.3, 0.45]

[0 . 5 0 . 5 ]

[0.48, 0.6]

[0.1, 0.3]

[0 , 0 . 2 ]

[0.1, 0.35]

[0.4, 0.52]

[0 . 5 , 0 . 5 ]

[0.11, 0.22]

[0, 0.12]

[0.1, 0.27]

[0 . 7 , 0 . 9 ]

[0.78, 0.89]

[0.5, 0.5]

[0.3, 0.49]

[0.4, 0.64] [0.5, 0.85]



[0.64, 0.7]

[0.44, 0.6]⎥ ⎥

[0 . 8 , 1 ]

[0.5, 0.62]

[0.51, 0.7]

[0 . 5 , 0 . 5 ]

[0.55, 0.7]

[0.73, 0.9]

[0.36, 0.6]

[0.15, 0.5]

[0 . 5 , 0 . 5 ]

⎥ ⎥ [0.74, 0.89]⎥ ⎥ [0.84, 1]⎥ ⎥ [0.69, 0.9]⎦

[0.3, 0.36]

[0.48, 0.56]

[0.11, 0.26]

[0, 0.16]

[0.1, 0.31]

[0 . 5 , 0 . 5 ]

[0.44, 0.52]⎥

Note that with the revised preferences, the crucial values r57 = [0.84, 1] and r53 = [0.5, 0.62] are expressible. Moreover, the completed relation is consistent. Consider the following flow chart for incomplete interval valued multiplicative preference relations. There may be more than one right and left end limit of preference intensity in the interval valued preference relations which voids property (ubc). The above flow chart explains the revision process that is mandatory in case when property (ubc) is not satisfied. 5. Interval valued multiplicative fuzzy preference relations Suppose that incomplete interval valued multiplicative preference ai j where i is fixed and j ∈ {1, 2, . . . , n}, i = j is provided by a decision maker. Then this incomplete multiplicative fuzzy preference relation R can be completed with expressible preferences provided that experts follow the condition portrayed in the form of Theorem 2. An estimated preference is expressible in this case, if it is a subset of L([1/9, 9]). Moreover, a preference is crucial if it is to be estimated with the assistance of least and greatest right and left limit present in the provided intervals. Consider a multi-valued bijective function F : F ([1/9, 9] ) → L([0, 1] ) such that

pi j = F ( ai j ) =

1 2

1 2

(1 + log9 ali j ), (1 + log9 ari j )



(4)

Also, the inverse function G : L([0, 1] ) → F ([1/9, 9] ) such that

ai j = G ( pi j ) = [92 pi j −1 , 92 pi j −1 ] l

r

(5)

Given an interval valued fuzzy preference relation, the transformation function F assists in stating the corresponding interval valued multiplicative preference relation. Similarly, the function G produces an interval valued multiplicative fuzzy preference relation corresponding to a fuzzy preference relation. In this section, we use Eq. (5) to construct an upper bound condition, property (mubc) for incomplete multiplicative fuzzy preference relations. Although, consistency property discussed in Definition 8 has been used in literature to estimate missing preferences in an incomplete multiplicative interval valued fuzzy preference relation but without the upper bound condition proposed in Theorem 2, the estimated values will not be expressible as depicted in the following example.

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A. Khalid, I. Beg / Information Sciences 348 (2016) 15–24

Example 4. Consider the incomplete interval valued multiplicative fuzzy preference relation (ai j ), i, j ∈ {1, 2, 3, 4} provided by a decision maker where a12 = [2/3, 4/5], a13 = [7, 8] and a14 = [1/8, 1/7]. Accordingly, the crucial preference is a43 .



[1, 1]

⎢[5/4, 3/2] ⎢ ⎣[1/8, 1/7]



[2 / 3 , 4 / 5 ]

[7 , 8 ]

[1 , 1 ]

[−, −]

[−, −]

[1 , 1 ]

[−, −]⎦

[−, −]

[−, −]

[1, 1]

[7, 8]

[1 / 8 , 1 / 7 ]

[−, −]⎥



Accordingly, a43 = [49, 64] ∈ / L([1/9, 9] ). Therefore, using multiplicative consistency alone to estimate missing preferences may not produce expressible preferences. Theorem 2 proposes a condition that needs to be satisfied by the decision makers providing incomplete interval valued preference relations. Suppose that the preferences provided are ak j , where k is fixed and j ∈ {1, 2, . . . , n}. Let δ mult represent the greatest right end limit and  mult < 1 represent the smallest left end limit in ak j . The provided preferences are said to satisfy upper bound condition for incomplete interval valued multiplicative fuzzy preference relations, or property (mubc) if and only if δ mult ≤ 9 mult . Theorem 2. Assume that n preferences of a decision maker in an n × n incomplete interval valued multiplicative fuzzy preference relation are given by ak j = [alk j , ark j ] where k is fixed and j ∈ {1, 2, . . . , n}. If ak j satisfies property (mubc) then the missing preferences can be estimated. Moreover, the estimated preferences are expressible. Proof. Similar to proof of Theorem 1 using (4) and (5).



Corollary 4. If preferences provided in an incomplete interval valued multiplicative fuzzy preference relation satisfy property (mubc) then the estimated preferences are consistent. Example 5. Consider the incomplete interval valued multiplicative fuzzy preference relation (ai j ), i, j ∈ {1, 2, 3, 4} provided by an expert where a12 = [17/20, 1], a13 = [2, 3] and a14 = [7/2, 19/5]. Accordingly, the crucial preference a24 is estimated to be a24 = [7/20, 76/17] ∈ L([1/9, 9] ). Since the crucial value is expressible the remaining missing preferences are estimated and the complete relation is represented as follows:-



[1 , 1 ]

[17/20, 1]

[2 , 3 ]

⎢ [1, 20/17] ⎢ ⎣ [1 / 3 , 1 / 2 ]

[1 , 1 ]

[2, 60/17]

[17/60, 1/2]

[1 , 1 ]

[5/19, 2/7]

[2/7, 17/76]

[10/19, 6/7]



[7/2, 19/5]

[7/2, 76/17]⎥



[7/6, 19/10]⎦ [1 , 1 ]

This completed interval valued fuzzy preference relation is expressible and consistent because it satisfies a1 j .a jk .ak1 = a1k .ak j .a j1 for all j, k ∈ {2, 3, 4}. The main assumption in this paper is that the decision makers respect the upper bound condition if they are to provide incomplete preferences. In the rare case, when expert is unable to do so, we state a rule for revision so that the provided incomplete relation is refrained from being discarded and that least possible revisions are carried out in the process. Rule 2. Suppose n preferences ai j for fixed i and j ∈ {1, 2, . . . , n} provided by a decision maker. Consider a set of left and

right end limits of each interval provided by a decision maker as mult = {ali 1 , ari 1 , . . . , ali k , ari k , . . . , ali n , ari n }, k = i . Let mult , mult(l) ⊂  such that mult = {ari 1 , ari 2 , . . . , ari n } and mult (l ) = {ali 1 , ali 2 , . . . , ali n }. Also, let γ mult = {ari k : ari k satisfy property (mubc)} ⊂ mult and γ mult (l ) = {ali k : ali k satisfy property (mubc)} ⊂ . Case a: If | γ mult |≥ n−1 2 then Step 1: Consider sup( mult ) = δ1mult .





If δ1mult ∈ / γ mult then revise δ1mult = δ1mult where δ1mult = 9 inf(mult ). Now, test the corresponding left end limit of the

interval that contains δ1mult , denote it as le f t (δ1mult ). If le f t (δ1mult ) ≤ δ1mult then no revision

mult mul t ( l ) mult mult Otherwise, if le f t (δ1 ) ∈ /γ then le f t (δ1 ) = le f t (δ1 ) where



le f t (δ1mult ) = δ1mult / min j {(ari j /ali j )}. Step 2: Consider sup( mult /δ1mult ) = δ2mult .



If δ2mult ∈ / γ mult then revise δ2mult = δ2mult where δ2mult = 9 inf(mult ). Now, if le f t (δ2mult )

quired. Otherwise, if le f t (δ2mult ) ∈ / γ mult (l ) then le f t (δ2mult ) = le f t (δ2mult ) where

le f t (δ2mult ) = δ2mult / min j {(ari j /ali j )}.

 Step | mult | − | γ mult |: Consider sup( mult /δ mult mult |



required.



≤ δ2mult then revision is not re-



|−|γ mult |−1

) = δ|mult . If δ mult ∈ / γ mult then δ mult = δ mult where mult |−|γ mult | | mult |−|γ mult | | mult |−|γ mult | | mult |−|γ mult |

δ|mult = 9 inf(mult ). Also, if le f t (δ mult ) ≤ δ|mult then no revision is required. mult |−|γ mult | | mult |−|γ mult | mult |−|γ mult | Otherwise, le f t (δ mult mult |



|−|γ mult |

) = le f t (δ|mult ) where mult |−|γ mult |

A. Khalid, I. Beg / Information Sciences 348 (2016) 15–24

23

Incomplete interval valued multiplicative fuzzy relation

Condition (mubc) satisfied

No

Use Rule 2

Estimate Missing Values

Yes

Rule 2.

le f t (δ mult mult |



|−|γ mult |

) = δ|mult / min j {(ari j /ali j )}. mult |−|γ mult |

Case b: On the other hand, If | γ mult (l ) |< n−1 2 then Step 1: Consider inf( mult (l ) ) = 1mult .





If 1mult ∈ γ mult (l ) then 1mult = 1mult where 1mult = sup( mult (l ) )/9. Now, if right (1mult ) ≥ 1mult then no revision is re-





quired. Otherwise, if right (1mult ) < 1mult then right (1mult ) = min j {ari j /ali j }1mult . Step 2: Consider, inf( mult (l ) /1mult ) = 2mult



If 2mult ∈ γ mult (l ) then 2mult = 2mult where 2mult = sup( mult (l ) )/9. If right (2mult )



Otherwise, if right (2mult ) < 2mult then right (2mult ) = min j {ari j /ali j }2mult .



≥ 2mult then no revision is required.

 Step |γ mult(l) |: Consider, inf( mult (l ) /|γ mult (l ) |−1 ) = |γ mult (l ) | .



If |γ mult (l ) | ∈ γ mult (l ) then |γ mult (l ) | =  mult and  mult = sup( mult (l ) )/9. |γ mult (l ) | |γ mult (l ) |

mult If right (|γ mult (l ) | ) ≥  mult (l ) then no revision is required. |γ

|





Otherwise, right ( mult ) = min j {ari j /ali j } mult mul t (l ) mul t (l ) |γ

|



|

Consider the following flow chart for incomplete multiplicative fuzzy preference relations which incorporates for the two cases when expert respects the upper bound condition and when the condition is ignored. According to the algorithm, when incomplete multiplicative preference relation is presented by an expert, it needs to be tested for property (mubc). If the incomplete relation satisfies property (mubc) then the missing preferences are estimated using transitivity property for interval valued multiplicative preference relations. If on the other hand, property (mubc) is not satisfied then Rule 2 is used to revise minimum possible preferences. This results in expressible missing preferences. Moreover, the resultant relation is consistent. 6. Conclusion Interval valued preference relation are used to model decision making processes where uncertainty and ambiguity persist. If experts are restricted to choose a particular value among the interval [0, 1], like in fuzzy preference relations, then they might find it difficult to express the exact preference intensity of one alternative over the other. However, giving them the flexibility to express their preference in terms of an interval facilitates them in many ways. Incompleteness is an inevitable situation specifically in decision making problems where questions addressed are complex. Rejecting such relations may lead to a collective decision that does not best represent choices of the panel of experts. Moreover, this option may lead to loss of some important information that is present in an incomplete interval valued fuzzy preference relation. This paper supports the method where incomplete preference relations are not discarded and in fact the missing preferences are estimated with the help of expert’s own choices on other alternatives. Using additive and multiplicative

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A. Khalid, I. Beg / Information Sciences 348 (2016) 15–24

transitivity, missing preferences can be estimated but it needs to be noted that L([0, 1]) and L([1/9, 9]) are not closed under the operation of addition and subtraction. Therefore, an upper bound condition is presented for experts to obey if they are presenting incomplete relations. Upper bound condition promises the expressibility of estimated values and moreover the resultant relations are consistent as well. In case that an expert does not conform to the upper bound condition, a rule is employed to revise some of his preferences. This rule assures the least possible revisions. For future directions, preference relations based on trapezoidal fuzzy numbers or hesitant fuzzy sets can be considered and their incompleteness can be studied. In addition to incomplete preference intensities, hesitancy of a decision making to provide a preference degree needs to be incorporated in the same model. Such theoretical work will have important applications in real world decision making problems. 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