Incomplete interval fuzzy preference relations and ... - Semantic Scholar
Computers & Industrial Engineering 67 (2014) 93–103
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Incomplete interval fuzzy preference relations and their applications q Yejun Xu a,b,⇑, Kevin W. Li c, Huimin Wang a,b a
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China Business School, Hohai University, Jiangning, Nanjing, Jiangsu 211100, PR China c Odette School of Business, University of Windsor, Windsor, Ontario N9B 3P4, Canada b
a r t i c l e
i n f o
Article history: Received 30 July 2012 Received in revised form 17 August 2013 Accepted 29 October 2013 Available online 11 November 2013 Keywords: Incomplete interval fuzzy preference relation Additive consistent Group decision making
a b s t r a c t This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n 1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process. Crown Copyright 2013 Published by Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy preference relations are one of the most common preference relations for expressing a decision maker’s (DM’s) preference over alternatives. In a decision making process, the DM generally needs to compare a set of n decision alternatives xi (i = 1, 2, . . ., n), thereby constructing a fuzzy preference relation (Herrera-Viedma, Herrera, Chiclana, & Luque, 2004;Kacprzyk, 1986; Orlovsky, 1978;Tanino, 1984; Wang & Fan, 2007;Xu, Da, & Liu, 2009; Xu, Patnayakuni, & Wang, 2013c;Xu, 2005). However, the DM may have vague knowledge about the preference degrees of one alternative over another and cannot estimate his/her preference with an exact numerical value, but with an interval number. In this case, the DM constructs an interval preference relation. Saaty and Vargas (1987) first presented interval judgments as a way to model subjective uncertainty. Afterwards, some methods are proposed to generate weights from interval comparison matrices, such as linear programing (LP) (Arbel, 1989;Kress, 1991), Lexicographic Goal Programming (LGP) (Islam, Biswal, & Alam, 1997;Wang, 2006), fuzzy preference programming (FPP) (Mikhailov, 2002;Mikhailov, 2004), two-stage logarithmic goal programming (TLGP) (Wang, Yang, & Xu, 2005), eigenvector method (EM) (Wang & Chin, 2006), Lambda-Max method (Csutora & Buckley, 2001), goal programming method (GPM) (Wang & Elhag, 2007), etc.
q
This manuscript was processed by Area Editor Imed Kacem.
⇑ Corresponding author at: State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China. Tel.: +86 25 68514612; fax: +86 25 85427972. E-mail address: [email protected] (Y. Xu).
For IFPRs, Xu (2004b) defined the concept of compatibility degree between two IFPRs, and showed the compatibility relationship between individual IFPRs and collective IFPR. Herrera, Martíze, and Sánchez (2005) developed an aggregation process for combining IFPRs with other types of information such as numerical preference relation and linguistic preference relation. Jiang (2007) gave an index to measure the similarity degree between two IFPRs, and employed an error-propagation principle to determine a priority vector for the aggregated IFPRs. Recently, Xu and Chen (2008b) established some linear programming models for deriving priority weights from various IFPRs. Wang and Li (2012) developed goal-programming-based models for deriving interval weights from IFPRs for both individual and group decision-making situations. The aforesaid research focused on preference relations with complete information. A complete preference relation of order n necessitates the completion of all n(n 1)/2 judgments in its entire top triangular portion. Sometimes, however, a DM may develop a preference relation with incomplete information due to a variety of reasons such as time pressure, lack of knowledge, and the DM’s limited expertise related with the problem domain (Chiclana, Herrera-Viedma, Alonso, & Herrera, 2008;Lee, Chou, Fang, Tseng, & Yeh, 2007; Xu & Da, 2008;Xu & Da, 2009; Xu, Da, & Wang, 2010;Xu, Gupta, & Wang, 2013a; Xu, Patnayakuni, & Wang, 2013b;Xu, 2004a; Xu, 2005;Xu & Chen, 2008a); In addition, when the number of the alternatives, n, is large, it may be impractically to require the DM to perform all the n(n 1)/2 required comparisons for a complete the pairwise comparison matrix (Fedrizzi & Silvio, 2007); More over, it is sometimes convenient or necessary to skip some direct comparison between alternatives even if the total number of alternatives is small (Fedrizzi & Silvio, 2007); In some other
0360-8352/$ - see front matter Crown Copyright 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.10.010
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cases, a DM is unable to express any kind of preference between two or more options, which may be due to an expert not possessing a precise or sufficient level of knowledge of part of the problem, or because that expert is unable to discriminate the degree to which one option is preferred to another (Alonso, Chiclana, Herrera, & Herrera-Viedma, 2004;Alonso et al., 2008; HerreraViedma, Alonso, Chiclana, & Herrera, 2007a;Herrera-Viedma, Chiclana, Herrera, & Alonso, 2007b; Xu, 2012); A critical concern for the incomplete fuzzy preference relations is to estimate the missing values. Herrera-Viedma et al. (2007a) proposed an iterative procedure to estimate the missing information in an expert’s incomplete fuzzy preference relation. The procedure is guided by the additive consistency property and only uses the preference values provided by the expert. Fedrizzi and Silvio (2007) put forward a new method for calculating missing elements in an incomplete fuzzy preference relation by maximizing global consistency. Later, Chiclana, Herrera-Viedma, and Alonso (2009) pointed out that the two methods are very similar in calculating missing values. Chiclana et al. (2008) presented a new estimation method based on based on the U-consistency criterion for incomplete fuzzy preference relations. Alonso et al. (2008) presented a procedure to estimate missing preference values for incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Liu, Pan, Xu, and Yu (2012b) developed a method to calculate missing values by minimizing the squared error of an incomplete fuzzy preference relation and its priority weight vector. Xu (2012) devised an approach to extending each incomplete multiplicative preference relation to a complete one by exploiting the multiplicative transitivity properties. Xia, Xu, and Wang (2014) furnished an algorithm to estimate missing values for an incomplete linguistic preference relation based on multiplicative consistency. Recently, research has been extended to IFPRs. For instance, Alonso et al. (2008) put forward a procedure to estimate missing information for the incomplete IFPR. Genç, Boran, Akay, and Xu (2010) examined consistency, missing value (s) and derivation of priority vectors from IFPRs based on multiplicative transitivity. Liu, Zhang, and Wang (2012a) proposed a new method to obtain priority weights from incomplete interval multiplicative preference relations. However, limited research has been devoted to incomplete IFPRs. As such, it is necessary to pay attention to this issue. Another important issue is the consistency of the judgment provided by experts (Chiclana, Herrera, & Herrera-Viedma, 2002; Herrera-Viedma et al., 2004). It is obvious that consistent information is more relevant or important than the information containing contradictions. Consistency is associated with certain transitivity properties. Different properties have been suggested to model transitivity of fuzzy preference relations. One of these properties is the ‘‘additive transitivity’’, which, as shown in (Herrera-Viedma et al., 2004), can be seen as a parallel concept of Saaty’s consistency property for multiplicative reciprocal preference relations. The aim of this paper is to propose some methods for constructing additive consistent IFPRs based on acceptable incomplete IFPRs. We first extend an additive consistency property proposed by Herrera-Viedma et al. (2004) for the fuzzy preference relations to a general case. Then, this property is extended to IFPRs based on the additive transitivity. After further characterizing additive consistent IFPRs, we develop two algorithms for estimating missing elements from acceptable incomplete IFPRs. A procedure is then laid out for handling GDM problems with acceptable incomplete IFPRs. The rest of this paper is organized as follows. Section 2 reviews some properties of fuzzy preference relations. Section 3 first introduces the concepts of interval multiplicative reciprocal preference relations and IFPRs as well as their transformation function. The property of additive consistent fuzzy preference relations in Section 2 is then extended to IFPRs, followed by further additive
consistent IFPRs. In Section 4, we propose two approaches to construct additive consistent IFPRs based on acceptable incomplete IFPRs. A case study is furnished in Section 5 to illustrate how to apply our algorithms. We conclude the paper in Section 6. 2. Additive consistent fuzzy preference relations Let X = {x1, x2, . . ., xn}(n P 2) be a finite set of alternatives, where xi denotes the ith alternative. In multiple attribute decision making problems, a DM needs to rank alternatives x1, x2, . . ., xn from the best to the worst according to preference information. A brief description of multiplicative and fuzzy preference relations is given below. 2.1. Multiplicative preference relations A multiplicative preference relation is a positive preference relation A X X, A = (aij)nn, where aij denotes the relative preference of alternative xi over xj. The measurement of aij is described by a ratio scale and in particular, as shown by Saaty (1980), aij 2 {1/9, 1/8, . . ., 1, 2, . . ., 9}: aij = 1 denotes the DM’s indifference between xi and xj, aij = 9 (or aji = 1/9) denotes that xi is absolutely preferred to xj, and aij 2 {2, 3, . . ., 8} denotes intermediate preference evaluations. This relation is multiplicative reciprocal, i.e., aijaji = 1, "i, j 2 {1, 2, . . ., n} and in particular, aii = 1, "i 2 {1, 2, . . ., n}. Its consistency is defined by Saaty (1980) as follows. Definition 1. Let A = (aij)nn be a multiplicative preference relation, then A is called consistent (Saaty, 1980), if aij = aikakj, for all i, j, k. 2.2. Fuzzy preference relations A fuzzy preference relation R is described as follows: R X X, R = (rij)nn, with membership function uR:X X ? [0, 1], where uR(xi, xj) = rij denotes the preference degree of alternative xi over xj (Kacprzyk, 1986;Tanino, 1984): rij = 0.5 denotes indifference between xi and xj, rij = 1, denotes that xi is definitely preferred to xj, and 0.5 < rij < 1 (or 0 < rji < 0.5) denotes that xi is preferred to xj to a varying degree. Definition 2. Let R = (rij)nn be a preference relation, then R is called a fuzzy preference relation, if
rij 2 ½0; 1;
r ij þ r ji ¼ 1;
r ii ¼ 0:5;
for all i; j 2 N
ð1Þ
Definition 3. Let R = (rij)nn be a fuzzy preference relation, then R is called an additive transitive fuzzy preference relation if the following additive transitivity (Tanino, 1984) is satisfied:
rij ¼ r ik rjk þ 0:5;
for all i; j; k ¼ 1; 2; . . . ; n
ð2Þ
2.3. Characterizing additive consistency of fuzzy preference relations Herrera-Viedma et al. (2004) studied the transformation between multiplicative preference relations with values in the interval scale [1/9, 9] (Alonso et al., 2004) and fuzzy preference relations with values in [0, 1] and furnished the following propositions. Proposition 1 (Herrera-Viedma et al., 2004). Consider a set of alternatives X = {x1, x2, . . ., xn}, associated with a multiplicative reciprocal preference relation A = (aij)nn with aij 2 [1/9, 9]. Then, a corresponding fuzzy preference relation, R = (rij)nn, with rij 2 [0, 1], associated with A is given as follows:
rij ¼ gðaij Þ ¼
1 ð1 þ log9 aij Þ 2
ð3Þ
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Proposition 2 (Herrera-Viedma et al., 2004). For a fuzzy preference relation R = (rij)nn, the following statements are equivalent: (a) rij þ rjk þ r ki ¼ 32 ; 8i; j; k; (b) rij þ rjk þ r ki ¼ 32 ; i < j < k.
Proposition 3 (Herrera-Viedma et al., 2004). For a fuzzy preference relation R = (rij)nn, the following statements are equivalent: (a) rij þ rjk þ r ki ¼ 32 ; i < j < k; (b) riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ r ðj1Þj þ r ji ¼ jiþ1 ; i < j. 2 2.4. A new characterization of additive consistency Herrera-Viedma et al. (2004) showed that Proposition 3 can be used to construct an additive consistent fuzzy preference relation from a set of n 1 values {r12, r23, . . ., rn1n}. The aforesaid propositions were also used by Wang and Chen (2007, 2008). In the following, a more general result is provided. Proposition 4. For a fuzzy preference relation R = (rij)nn, the following statements are equivalent: (a) rij þ rjk þ r ki ¼ 32 ; 8i; j; k 2 N; (b) rij1 þ rj1 j2 þ þ rjt1 jt þ r jt i ¼ tþ1 ; 8i; jl 2 N; l ¼ 1; 2; . . . ; t. 2 Proof. (a) ) (b) Mathematical induction is employed to prove this part of the proposition. It is obviously true for t = 1, as it is reduced to the additive reciprocity property in Definition 2. Next, if the hypothesis is true for t = n
r ij1 þ rj1 j2 þ þ r jn2 jn1 þ rjn1 jn þ r jn i
nþ1 ¼ 2
and an IFPR. The characterizations of additive consistent fuzzy preference relations in Section 2 are subsequently extended to IFPRs followed other results. byþsome useful þ þ 1 ¼ a Let a 1 ; a1 ; a2 ¼ a2 ; a2 ; a ¼ ½a ; a be three positive interval numbers, then
1 a 2 ¼ a1 ; aþ1 a2 ; aþ2 ¼ a1 þ a2 ; aþ1 þ aþ2 ðaÞ a þ þ 1 a 2 ¼ a1 ; a1 a2 ; a2 ¼ a1 aþ2 ; aþ1 a2 ðbÞ a þ þ 1 a 2 ¼ a1 ; a1 a2 ; a2 ¼ a1 a2 ; aþ1 aþ2 ðcÞ a Þ ¼ ½logn a ; logn aþ ðdÞ logn ða
ð7Þ
1 ¼ ½1=aþ ; 1=a ðeÞ a
ð8Þ
ð4Þ ð5Þ ð6Þ
ij Þ is multipliDefinition 4. An interval preference relation A ¼ ða ji ¼ a 1 cative reciprocal if and only if a ij . Definition 5 Xu, 2010. If a positive interval multiplicative reciproij Þ satisfies: cal preference relation A ¼ ða
ij ¼ a ik a kj ; a
for all i; k; j ¼ 1; 2; . . . ; n;
and i 6 k 6 j
ð9Þ
ij Þ is multiplicative consistent. then A ¼ ða Note that if an interval multiplicative reciprocal preference relation is multiplicative consistent, for all i, k, j 2 {1, 2, . . ., n}, as pointed out in Xu (2010), it is necessary to require i 6 k 6 j, otherij Þ would be reduced to a crisp number judgment mawise, A ¼ ða trix (Saaty’s multiplicative reciprocal preference relation). For more detail, readers are referred to (Xu, 2010). That is Eq.(9) holds only for the upper (or lower) triangular of the preference relation. Definition 6 Xu, 2004b. Let R ¼ ðrij Þ be a preference relation, where
h i r ij ¼ r ij ; r þij ;
h i r ij ¼ rji ; r þji ;
r þij P r ij P 0;
for all i; j ¼ 1; 2; . . . ; n
rij þ rþji ¼ r þij þ rji ¼ 1;
then R is called an interval fuzzy preference relation (IFPR).
then it is true for t = n + 1
r ij1 þ rj1 j2 þ þ r jn1 jn þ r jn jnþ1 þ r jnþ1 i ¼ ðr ij1 þ rj1 j2 þ þ r jn2 jn1 þ r jn1 jn Þ þ rjn jnþ1 þ r jnþ1 i ¼ ¼
nþ1 2 n1 2
r jn i þ r jn jnþ1 þ r jnþ1 i
Definition 7. Let R ¼ ðr ij Þ be a IFPR, if
rij rjk ¼ r ik ½0:5; 0:5;
8i < j < k
ð10Þ
then R is called an additive consistent IFPR.
þ r ijn þ r jn jnþ1 þ r jnþ1 i ¼ n1 þ 32 ¼ nþ2 2 2
So the result is established. (b) ) (a).
r ij þ r jk þ r ki ¼ 1 r ji þ 1 r kj þ rki ¼ 2 þ r ij1 þ þ rjt1 j tþ1 þ rjj1 þ þ r jt1 k tþ1 þ rki 2 2 ¼ 1 t þ ðr ij1 þ þ r jt1 j þ r jj1 þ þ r jt1 k þ r ki Þ ¼ 1 t þ 2tþ1 ¼ 32 2 This completes the proof. h Furthermore, in the proof process, it is obvious that the differences of j2 j1, j3 j2, . . ., jt jt1 are not necessarily equal to 1. As a matter of fact, the differences do not have to be identical. Proposition 4 differs from Proposition 3 in that any the sequential values of rij1 ; rj1 j2 ; . . . ; r jt1 jt ; r jt i (for example r31, r15, r57, r73) will work for Proposition 4. But Proposition 3 requires preference values to follow a consecutive order such as r12, r23, . . ., rn1n, rn1. Therefore, Proposition 3 is a special case of Proposition 4. 3. Interval fuzzy preference relations and their characterizations In the following we shall first introduce some operational laws of interval numbers (Hayes, 2003;Moore, 1966). We then present a relationship between an interval multiplicative preference relation
Proposition 5. For a set of alternatives X = {x1, x2, . . ., xn}, and its associated interval multiplicative reciprocal preference relation ij Þ with a ji ¼ a 1 A ¼ ða ij , a corresponding IFPR, R ¼ ðr ij Þ, associated with A is given as
ij Þ ¼ rij ¼ gða
1 ij Þ ð½1; 1 log9 a 2
ð11Þ
such that þ (a) r ij þ r ji ¼ 1, "i, j 2 N; (b) r þ þ r ji ¼ 1, "i, j 2 N. ij
ij Þ is an interval multiplicative reciprocal preferProof. As A ¼ ða ji ¼ a 1 ence relation, by Definition 4, a ij , that is
h
i aji ; aþji ¼
"
1 1 ; aþij aij
#
ð12Þ
Thus
aji ¼
1 ; aþij
aji aþij ¼ 1;
aþji ¼
1 aij
aþji aij ¼ 1
ð13Þ ð14Þ
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By Proposition 1 and Eq. (11) and the operational law (Eq. (7)) of interval numbers, we have
r ij ¼
h i 1 1 ij Þ ¼ ð1 log9 a ½1; 1 log9 aij ; aþij 2 2
Thus
Let i < j, and k = j i. The expression (c) can be rewritten as follows:
riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ rðj1Þj þ rþji ¼
kþ1 ; 2
8i < j
Mathematical induction is used to prove this part. It is clearly true for k = 1. Next if the hypothesis is true for k = n
1 1 þ log9 aij ¼ 2 1 þ r ij ¼ 1 þ log9 aþij 2 r ij
riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ rðiþn1ÞðiþnÞ þ r þðiþnÞi ¼
nþ1 2
Similarly,
then for k = n + 1:
1 1 þ log9 aji 2 1 1 þ log9 aþji r þji ¼ 2
r iðiþ1Þ þ r ðiþ1Þðiþ2Þ þ þ r ðiþn1ÞðiþnÞ þ r ðiþnÞðiþnþ1Þ þ rþðiþnþ1Þi ¼ riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ rðiþn1ÞðiþnÞ þ r ðiþnÞðiþnþ1Þ þ rþðiþnþ1Þi nþ1 ¼ r þðiþnÞi þ r ðiþnÞðiþnþ1Þ þ r þðiþnþ1Þi 2 nþ1 ¼ þ r iðiþnÞ 1 þ r ðiþnÞðiþnþ1Þ þ rþðiþnþ1Þi 2 n1 3 nþ2 ¼ þ ¼ 2 2 2
r ji ¼
Therefore,
1 1 1 þ log9 aij þ 1 þ log9 aþji ¼ 1 þ log9 aij aþji ¼ 1 2 2 1 1 þ þ r ij þ r ji ¼ 1 þ log9 aij þ 1 þ log9 aji ¼ 1 þ log9 aþij aji ¼ 1; 2 2 8i; j 2 f1; 2; . . . ; ng
r ij þ r þji ¼
The proof is thus completed. h Next, we examine the relationship between the multiplicative consistency of an interval multiplicative reciprocal preference relation and the additive consistency of its converted IFPR as per Eq. (11). Proposition 6. If an interval multiplicative reciprocal preference ij Þ is multiplicative consistent, then its corresponding relation A ¼ ða IFPR R ¼ ðrij Þ is additive consistent, and
(a) (b) (c) (d) (e) (f)
þ 3 r ij þ r jk þ r ki ¼ 2 ; 8i < j < k; þ 3 rþ þ r þ r ¼ ; 8i < j < k; ki ij jk 2 þ r þ r þ þ r iðiþ1Þ ðiþ1Þðiþ2Þ ðj1Þj þ r ji þ þ þ riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ r ðj1Þj þ r ji þ tþ1 r ij1 þ r j1 j2 þ þ r jt1 jt þ r jt i ¼ 2 ; þ þ tþ1 rþ ij1 þ r j1 j2 þ þ r jt1 jt þ r jt i ¼ 2 ;
¼ jiþ1 ; 8i < j; 2 ¼ jiþ1 ; 8i < j; 2 8i < j1 < j2 < < jt ; 8i < j1 < j2 < < jt .
thus the expression (c) is confirmed and (d) can be analogously asserted. Similar to Proposition 4, (e) and (f) can be verified. h Proposition 7. An IFPR R ¼ ðrij Þ is additive consistent if and only if þ þ þ þ (1) r ij r ik ¼ r lj r lk ; (2) r ij r ik ¼ r lj r lk ; 8i < j < k, l = 1, 2, . . ., n. Proof. If an IFPR R ¼ ðrij Þ is additive consistent by Definition 7, then
r ij r jk ¼ r ik ½0:5; 0:5; 8i < j < k 1 1 r ij þ r jk ¼ r ik þ ; r þij þ rþjk ¼ r þik þ ; 8i < j < k 2 2 1 1 þ þ þ r ij r ik ¼ rjk ; r ij r ik ¼ rjk ; 8i < j < k 2 2 Similarly,
r lj r jk ¼ r lk ½0:5; 0:5; 8l < j < k 1 1 r lj þ r jk ¼ r lk þ ; r þlj þ rþlk ¼ r þlk þ ; 8l < j < k 2 2 1 1 r lj r lk ¼ rjk ; r þlj r þlk ¼ rþlk ; 8l < j < k 2 2
ij Þ is multiplicative consistent by Definition 5, Proof. Since A ¼ ða ij a jk ¼ a ik , for "i < j < k. Taking a logarithm operation (Eq. then a (7)) on both sides yields
rij r ik ¼ rlj r lk ;
ij log9 a jk ¼ log9 a ik ; log9 a
On the contrary,
8i < j < k
thus
If r ij r ik ¼ r lj r lk ;
Thus
r þij r þik ¼ r þlj r þlk ;
r þij rþik ¼ r þlj rþlk ;
rij r ik ¼ 0:5 r jk ;
By Eq. (11), we have
r ij þ r jk ¼ 0:5 þ rik ;
r ij r jk ¼ rik ½0:5; 0:5 h i h i r ij ; r þij r jk ; r þjk ¼ rik ; r þik ½0:5; 0:5
r ij r jk ¼ r ik ½0:5; 0:5;
þ
r jk
¼
r ik
þ 0:5;
rþij
þ
r þjk
¼
rþik
3 ; 2
rþij þ r þjk þ r ki ¼
þ 0:5
3 ; 2
rþij rþik ¼ 0:5 r þjk ;
8i < j < k
r þij þ r þjk ¼ 0:5 þ rþik ;
8i < j < k
That is
By Proposition 5, we have
r ij þ rjk þ rþki ¼
8i < j < k
þ Let l = j, since r jj ¼ r jj ¼ 0:5, then
1 1 ij Þ ð½1; 1 log9 a jk Þ ð½1; 1 log9 a 2 2 1 ik Þ ½0:5; 0:5 ¼ ð½1; 1 log9 a 2
r ij
8i; l < j < k
8i < j < k
Thus the expressions (a) and (b) are established.
8i < j < k
This completes the proof. h Proposition 7 reveals an important property of an additive consistent IFPR. For upper (or lower) triangular interval values, the difference of the lower bounds between any two columns should be a constant for all rows. The same is true for the upper bounds of the interval preference values. Propositions 6 and 7 will play an important role in devising our algorithms to construct complete IFPRs based on an incomplete relation.
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Y. Xu et al. / Computers & Industrial Engineering 67 (2014) 93–103
Note that, if the primary values are different then we may have obtained a matrix R with entries not in the interval [0, 1], but in an interval [c, 1 + c], where c > 0, c indicates the minimum value of matrix R; 1 þ c gives the maximum value of matrix R. In this case, the obtained values have to be converted using a transformation function that preserves reciprocity and additive consistency, i.e., f: [c, 1 + c] ? [0, 1], verifying
ðaÞ f ðcÞ ¼ 0 ðbÞ f ð1 þ cÞ ¼ 1 ðcÞ f ðx Þ þ f ðxþ Þ ¼ 1;
8x 2 ½c; 1 þ c
3 3 ðdÞ If x þ y þ z ¼ ; f ðx Þ þ f ðy Þ þ f ðzþ Þ ¼ ; 2 2 8x ; y ; zþ 2 ½c; 1 þ c 3 3 ðeÞ If xþ þ yþ þ z ¼ ; f ðxþ Þ þ f ðyþ Þ þ f ðz Þ ¼ ; 2 2 8xþ ; yþ ; z 2 ½c; 1 þ c
ð1Þ
rij ¼ k1 r ij rþij
¼
ð1Þþ k1 r ij ð1Þ
rji ¼ k1 r ji rþji
¼
ð1Þþ k1 r ji
ð2Þ
þ k2 r ij þ
ð2Þþ k2 r ij ð2Þ
þ k2 r ji þ
ð2Þþ k2 r ji
ðmÞ
þ þ km r ij þ þ
ðmÞ
þ þ km r ji þ þ
;
ðmÞþ km r ij ;
8i; j 2 f1; 2; . . . ; ng
;
ðmÞþ km r ji ;
8i; j 2 f1; 2; . . . ; ng
ð15Þ
ð1Þ ð1Þþ ð2Þ ð2Þþ ðmÞ ðmÞþ rij þ r þji ¼ k1 r ij þ r ji þ k2 r ij þ r ji þ þ km rij þ rji
ð16Þ
¼ k1 þ k2 þ þ km ¼ 1
ð17Þ
ð1Þþ ð1Þ ð2Þþ ð2Þ ðmÞþ ðmÞ þ k2 r ij þ r ji þ þ km rij þ rji rþij þ r ji ¼ k1 r ij þ r ji
þ
ð18Þ
¼ k1 þ k2 þ þ km ¼ 1 The proof is thus completed.
ð19Þ
A linear function satisfying (a) and (b) takes the form
h
Proposition 9. Let Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be m additive consistent IFPRs, then their weighted average
R ¼ k1 Rð1Þ k2 Rð2Þ . . . km RðmÞ ;
f ðx Þ ¼ ux þ b;
u; b 2 R f ðxþ Þ ¼ uxþ þ b; u; b 2 R
kl 2 ½0; 1;
m X kl ¼ 1
ð21Þ
l¼1
is also an additive consistent IFPR.
These functions are Proof. Since Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are additive consistent IFPRs by Definition 7.
1 c x þ c x þ ¼ 1 þ 2c 1 þ 2c 1 þ 2c 1 c xþ þ c f ðxþ Þ ¼ xþ þ ¼ 1 þ 2c 1 þ 2c 1 þ 2c
f ðx Þ ¼
rðlÞ ðlÞ ðlÞ ij r jk ¼ r ik ½0:5; 0:5; then by Eq. (21), we have
(c) can be easily verified as
rij ¼ k1r ð1Þ ð2Þ ðmÞ ij k2 r ij km r ij
x þ c xþ þ c ðx þ xþ Þ þ 2c f ðx Þ þ f ðx Þ ¼ þ ¼ ¼1 1 þ 2c 1 þ 2c 1 þ 2c
8i < j < k
þ
rjk ¼ k1r ð1Þ ð2Þ ðmÞ jk k2 r jk km r jk
If x þ y þ zþ ¼ 32 ; xþ þ yþ þ z ¼ 32
x þ c y þ c z þ þ c þ þ 1 þ 2c 1 þ 2c 1 þ 2c þ ðx þ y þ z Þ þ 3c 3=2 þ 3c 3 ¼ ¼ ¼ 1 þ 2c 1 þ 2c 2
rik ¼ k1r ð1Þ ð2Þ ðmÞ ik k2 r ik km r ik
f ðx Þ þ f ðy Þ þ f ðzþ Þ ¼
r ij r jk ¼ k1 r ð1Þ ð1Þ k2 r ð2Þ ð2Þ km r ðmÞ ðmÞ ij r jk ij r jk ij r jk ð1Þ ð2Þ ðmÞ ¼ k1 r ik ½0:5; 0:5 k2 r ik ½0:5; 0:5 km r ik ½0:5;0:5
xþ þ c y þ þ c z þ c þ þ 1 þ 2c 1 þ 2c 1 þ 2c ðxþ þ yþ þ z Þ þ 3c 3=2 þ 3c 3 ¼ ¼ ¼ 1 þ 2c 1 þ 2c 2
ð1Þ
f ðxþ Þ þ f ðyþ Þ þ f ðz Þ ¼
ð2Þ
ðmÞ
¼ k1 r ik k2 r ik kmr ik ½ðk1 þ k2 þ þ km Þ=2; ðk1 þ k2 þ þ km Þ=2 ¼ rik ½0:5; 0:5
This completes the proof.
h
(d) and (e) are also confirmed. Proposition 8. Let Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be m IFPRs, then their weighted average
R ¼ k1 Rð1Þ k2 Rð2Þ . . . km RðmÞ ;
kl 2 ½0; 1;
m X kl ¼ 1 l¼1
is also an IFPR, i.e., satisfies
r ij þ r þji ¼ 1;
r þij þ r ji ¼ 1;
8i; j 2 f1; 2; . . . ; ng
Proof. Since Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are IFPRs, it follows that ðlÞ
r ij
ðlÞþ
þ r ji
¼ 1;
ðlÞþ
r ij
ðlÞ
þ rji
l ¼ 1; 2; . . . ; m Then by Eq. (20), we have
¼ 1;
8i; j 2 f1; 2; . . . ; ng;
ð20Þ
4. Procedures for repairing incomplete interval fuzzy preference relations 4.1. Incomplete interval fuzzy preference relation A complete n n preference relation requires n(n 1)/2 judgments in its entire top (or lower) triangular portion. Sometimes, however, a DM may furnish a preference relation with incomplete information due to a variety of reasons. Next, we first present basic concepts of incomplete interval IFPRs, followed by two estimation procedures for missing values. Definition 8. Let R ¼ ðr ij Þ be an n n preference relation, if at least on element in the upper (or lower) triangular part is not given by the DM, denoted by an unknown variable ‘‘x’’, and remaining interval values by the DM satisfy
rij ¼ ½1; 1 r ji ;
r ii ¼ ½0:5; 0:5;
for all r ij 2 X
ð22Þ
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Y. Xu et al. / Computers & Industrial Engineering 67 (2014) 93–103
R is called an incomplete interval IFPR, where X is the set of all the known elements in R. Definition 9. The elements r ij ; r kl of R are called adjacent, if {i, j} \ {k, l} – ;. For a missing element r ij , it can be determined indirectly if there exist a series of known elements rij1 ; rj1 j2 ; . . . ; r jt j . Definition 10. Let R ¼ ðrij Þ be an incomplete IFPR. If all missing elements of R can be obtained by the known elements, then R is called an acceptable incomplete IFPR. Otherwise, R is an unacceptable incomplete IFPR. Next, we extend the necessary condition (Herrera-Viedma et al., 2007b;Xu & Da, 2008) of acceptable incomplete fuzzy preference relations to the case of incomplete IFPRs. Proposition 10. Let R ¼ ðr ij Þ be an incomplete IFPR. If R is an acceptable incomplete IFPR, then there exists at least one known nondiagonal element in each line or each column of R, i.e. there exist at least (n 1) judgments provided by the DM. Definition 11. Let R ¼ ðrij Þ be an incomplete IFPR, if the known elements satisfy
tþ1 ¼ ; for 8i < j1 < j2 < < jt 2 tþ1 r þij1 þ r þj1 j2 þ þ rþjt1 jt þ r jt i ¼ ; for 8i < j1 < j2 < < jt 2
r ij1
þ r j1 j2
þ þ rjt1 jt
þ r þjt i
ð23Þ ð24Þ
then R is called an additive consistent incomplete IFPR.
r31 ¼ ½0:4; 0:7; r35 ¼ ½0:4; 0:7;
r32 ¼ ½0:1; 0:3; r36 ¼ ½0:3; 0:8;
r34 ¼ ½0:3; 0:4; r37 ¼ ½0:4; 0:9
Step 1. By Definition 8 and the aforesaid information provided by the DM, one obtains the following acceptable incomplete IFPR, where ‘‘x’’ denotes the unknown judgment. 3 ½0:5; 0:5 x x x x x x 7 6 x ½0:5;0:5 x x x x x 7 6 7 6 6 ½0:4; 0:7 ½0:1;0:3 ½0:5; 0:5 ½0:3; 0:4 ½0:4; 0:7 ½0:3; 0:8 ½0:4; 0:9 7 7 6 7 6 R¼6 x x x ½0:5; 0:5 x x x 7 7 6 7 6 x x x x x x ½0:5; 0:5 7 6 7 6 x x x x x ½0:5; 0:5 x 5 4 x x x x x x ½0:5; 0:5 2
Step 2. Utilize Propositions 6 and 7 to determine all missing elements in R as follows: r13 ¼ 1 r þ31 ¼ 0:3; r þ13 ¼ 1 r31 ¼ 0:6; r23 ¼ 1 r þ32 ¼ 0:7; r þ23 ¼ 1 r32 ¼ 0:9 3 3 r12 ¼ r 23 r þ31 ¼ 0:1; r þ12 ¼ rþ23 r31 ¼ 0:2; r21 ¼ 1 r þ12 ¼ 0:8; r þ21 ¼ 1 r12 ¼ 0:9 2 2 r23 r33 ¼ r 24 r 34 ) r 24 ¼ r 23 r 33 þ r 34 ¼ 0:5; r þ24 ¼ rþ23 r þ33 þ r þ34 ¼ 0:8 r25 ¼ r 23 r 33 þ r 35 ¼ 0:6; rþ25 ¼ r þ23 r þ33 þ r þ35 ¼ 1:1 r26 ¼ r 23 r 33 þ r 36 ¼ 0:5; rþ26 ¼ r þ23 r þ33 þ r þ36 ¼ 1:2 r27 ¼ r 23 r 33 þ r 37 ¼ 0:6; rþ27 ¼ r þ23 r þ33 þ r þ37 ¼ 1:3 r14 ¼ r 13 r 33 þ r 34 ¼ 0:1; rþ14 ¼ r þ13 r þ33 þ r þ34 ¼ 0:5 r15 ¼ r 13 r 33 þ r 35 ¼ 0:2; rþ15 ¼ r þ13 r þ33 þ r þ35 ¼ 0:8 r16 ¼ r 13 r 33 þ r 36 ¼ 0:1; rþ16 ¼ r þ13 r þ33 þ r þ36 ¼ 0:9 r17 ¼ r 13 r 33 þ r 37 ¼ 0:2; rþ17 ¼ r þ13 r þ33 þ r þ37 ¼ 1 r45 ¼ r 35 r 34 þ r 44 ¼ 0:6; rþ45 ¼ r þ35 r þ34 þ r þ44 ¼ 0:8 r46 ¼ r 36 r 34 þ r 44 ¼ 0:5; rþ46 ¼ r þ36 r þ34 þ r þ44 ¼ 0:9 r47 ¼ r 37 r 34 þ r 44 ¼ 0:6; rþ47 ¼ r þ37 r þ34 þ r þ44 ¼ 1
4.2. An estimation procedure for acceptable incomplete IFPRs with fewest number of judgments Next, by exploiting Propositions 5 and 6 or Proposition 7, a simple and practical method is developed for constructing a complete additive consistent IFPR based on an acceptable incomplete IFPR with fewest number of judgment data (i.e., n 1 preference values):
r56 ¼ r 36 r 35 þ r 55 ¼ 0:4; rþ56 ¼ r þ36 r þ35 þ r þ55 ¼ 0:6 r57 ¼ r 37 r 35 þ r 55 ¼ 0:5; rþ57 ¼ r þ37 r þ35 þ r þ55 ¼ 0:7 r67 ¼ r 37 r 36 þ r 66 ¼ 0:6; rþ67 ¼ r þ37 r þ36 þ r þ66 ¼ 0:6 r41 ¼ 1 r þ14 ¼ 0:5; r þ41 ¼ 1 r14 ¼ 0:9; r42 ¼ 1 r þ24 ¼ 0:2; r þ42 ¼ 1 r24 ¼ 0:5 r43 ¼ 1 r þ34 ¼ 0:6; r þ43 ¼ 1 r34 ¼ 0:7 r51 ¼ 1 r þ15 ¼ 0:2; r þ51 ¼ 1 r15 ¼ 0:8; r52 ¼ 1 r þ25 ¼ 0:1; r þ52 ¼ 1 r25 ¼ 0:4 r53 ¼ 1 r þ35 ¼ 0:3; r þ53 ¼ 1 r35 ¼ 0:6; r54 ¼ 1 r þ45 ¼ 0:2; r þ54 ¼ 1 r45 ¼ 0:4 r61 ¼ 1 r þ16 ¼ 0:1; r þ61 ¼ 1 r16 ¼ 0:9; r62 ¼ 1 r þ26 ¼ 0:2; r þ62 ¼ 1 r26 ¼ 0:5 r63 ¼ 1 r þ36 ¼ 0:2; r þ63 ¼ 1 r36 ¼ 0:7; r64 ¼ 1 r þ46 ¼ 0:1; r þ64 ¼ 1 r46 ¼ 0:5 r65 ¼ 1 r þ56 ¼ 0:4; r þ65 ¼ 1 r56 ¼ 0:6
Algorithm 1 Step 1. For a decision problem, let X = {x1, x2, . . ., xn} be a discrete set of alternatives. The DM conducts pairwise comparisons among the alternatives and furnished his/her assessment as an acceptable incomplete IFPR R ¼ ðrij Þnn (if the DM provides his/her evaluation as an acceptable incomplete interval multiplicative reciprocal preference relation ij Þnn , then A ¼ ða ij Þnn can be converted to a correA ¼ ða sponding incomplete IFPR R ¼ ðr ij Þnn by Propositions 5), with only n 1 judgments. Step 2. Utilizing Proposition 6 or Proposition 7 to determine all unknown elements in R, and yield an interval additive consistent IFPR R_ ¼ ðr_ ij Þnn . If this preference relation contains any values falling outside the unit interval [0, 1], but within the interval [c, 1 + c], then a transformation funcxþc tion f ðxÞ ¼ 1þ2c can be applied to preserve the reciprocity and additive transitivity, resulting in an additive consistent IFPR. Step 3. End.
Example 1. Assume that a decision problem involves evaluating seven faculties xi (i = 1, 2, . . ., 7) at a university. The DM assesses these seven faculties (alternatives) by pairwise comparison and provides his/her judgment as follows:
r71 ¼ 1 r þ17 ¼ 0; rþ71 ¼ 1 r 17 ¼ 0:8; r 72 ¼ 1 rþ27 ¼ 0:3; rþ72 ¼ 1 r 27 ¼ 0:4 r73 ¼ 1 r þ37 ¼ 0:1; r þ73 ¼ 1 r37 ¼ 0:6; r74 ¼ 1 r þ47 ¼ 0; rþ74 ¼ 1 r 47 ¼ 0:4 r75 ¼ 1 r þ57 ¼ 0:3; r þ75 ¼ 1 r57 ¼ 0:5; r76 ¼ 1 r þ67 ¼ 0:4; r þ76 ¼ 1 r67 ¼ 0:4
Thus, 2
½0:5;0:5 6 ½0:8;0:9 6 6 6 ½0:4;0:7 6 _R ¼ 6 ½0:5;0:9 6 6 6 ½0:2;0:8 6 6 4 ½0:1;0:9 ½0; 0:8
½0:1; 0:2 ½0:3;0:6 ½0:1; 0:5 ½0:2; 0:8 ½0:1; 0:9
½0:2;1
3
½0:5; 0:5 ½0:7;0:9 ½0:5; 0:8 ½0:6;1:1 ½0:5;1:2 ½0:6; 1:3 7 7 7 ½0:1; 0:3 ½0:5;0:5 ½0:3; 0:4 ½0:4; 0:7 ½0:3; 0:8 ½0:4;0:9 7 7 7 ½0:2; 0:5 ½0:6;0:7 ½0:5; 0:5 ½0:6; 0:8 ½0:5; 0:9 ½0:6;1 7 7 ½0:1;0:4 ½0:3;0:6 ½0:2; 0:4 ½0:5; 0:5 ½0:4; 0:6 ½0:5;0:7 7 7 7 ½0:2;0:5 ½0:2;0:7 ½0:1; 0:5 ½0:4; 0:6 ½0:5; 0:5 ½0:6;0:6 5 ½0:3;0:4 ½0:1;0:6 ½0;0:4 ½0:3; 0:5 ½0:4; 0:4 ½0:5;0:5
As the preference relation contains values falling outside the interxþc val [0, 1] and c = 0.3, a transformation function f ðxÞ ¼ 1þ2c is applied _ yielding to the lower and upper bounds of each interval value in R, 2
½0:5;0:5 6 ½0:69;0:75 6 6 6 ½0:44;0:63 6 € ¼ 6 ½0:5;0:75 R 6 6 6 ½0:31;0:69 6 6 4 ½0:25;0:75 ½0:19;0:69
½0:25;0:31 ½0:5;0:5 ½0:25;0:38 ½0:31;0:5 ½0:13;0:44 ½0:06;0:5 ½0;0:44
½0:38;0:56 ½0:63;0:75 ½0:5;0:5 ½0:56;0:63 ½0:38;0:56 ½0:31;0:68 ½0:25;0:56
½0:25;0:5 ½0:5;0:69 ½0:38;0:44 ½0:5;0:5 ½0:31;0:44 ½0:25;0:5 ½0:19;0:44
½0:31;0:69 ½0:56;0:88 ½0:44;0:63 ½0:56;0:69 ½0:5;0:5 ½0:44;0:56 ½0:38;0:5
½0:25; 0:75 ½0:5; 0:94 ½0:38; 0:69 ½0:5; 0:75 ½0:44; 0:56 ½0:5; 0:5 ½0:44; 0:44
3 ½0:31;0:81 ½0:56;1 7 7 7 ½0:44;0:75 7 7 7 ½0:56;0:81 7 7 ½0:5;0:63 7 7 7 ½0:56;0:56 5 ½0:5;0:5
Alonso et al. (2008) proposed a different procedure to estimate missing in an incomplete IFRP (see Appendix). For a comparison with our approach, their procedure is employed to determine the missing judgments in this example.
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Y. Xu et al. / Computers & Industrial Engineering 67 (2014) 93–103
By Eq. (32), RL and RR are obtained as follows: 2
3
2
3
0:5 x x x x x x 0:5 x x x x x x 6 x 0:5 x 6 x 0:5 x x x x x 7 x x x x 7 6 6 7 7 6 6 7 7 6 0:4 0:1 0:5 0:3 0:4 0:3 0:4 7 6 0:7 0:3 0:5 0:4 0:7 0:8 0:9 7 6 6 7 7 6 6 7 7 x x 0:5 x x x 7; RR ¼ 6 x x x 0:5 x x x 7 RL ¼ 6 x 6 6 7 7 6 x 6 7 x x x 0:5 x x 7 x x x 0:5 x x 7 6 6 x 7 6 6 7 7 x x x x 0:5 x 5 x x x x 0:5 x 5 4 x 4 x x x x x x x 0:5 x x x x x x 0:5
Step 1. In RL and RR, ‘‘x’’ denotes an unknown value, and by Eq. (35), EV of the known values are determined as:
by the DM and estimated values during earlier iterations. In contrast, our method can estimate missing values in one step and all the missing values are estimated based on the preference values furnished by the DM. Thirdly, in the final results, we get rr27 = 1 and rr72 = 0 as this is based on Eqs. (47) and (48). This constraints make the final estimated matrix RR do not have the additive consistent property. Finally, the derived R by Alonso et al. (2008)’s method is no longer an IFPR, because it does not satisfy the condition (a), (b) of Proposition 5, or Proposition 7, while our estimated matrix R is an additive consistent IFPR. 4.3. An estimation procedure for acceptable incomplete IFRP with more known judgments
EV ¼ fð3; 1Þ; ð3; 2Þ; ð3; 4Þ; ð3; 5Þ; ð3; 6Þ; ð3; 7Þg Step 2. By applying Eqs. (36)–(46), one has: 1
H11 12 ¼ /;
) crl12 ¼ 0;
crr112 ¼ 0
H12 12 ¼ f3g 2
) crl12 ¼ rl32 rl31 þ 0:5 ¼ 0:1 0:4 þ 0:5 ¼ 0:2; crr112 ¼ rr32 rr 31 þ 0:5 ¼ 0:3 0:7 þ 0:5 ¼ 0:1 H13 12 ¼ /;
j¼1
3
) crl12 ¼ 0; crr312 ¼ 0 0 þ 0:2 þ 0 ) crl12 ¼ ¼ 0:2; 1
crr 12 ¼
0 þ 0:1 þ 0 ¼ 0:1 1
Similarly, we can get crl14 ¼ 0:4; crr 14 ¼ 0:2; crl15 ¼ 0:5; crr15 ¼ 0:5; crl16 ¼ 0:4; crr16 ¼ 0:6 crl17 ¼ 0:5; crr 17 ¼ 0:7
Next, we consider an acceptable incomplete IFRP R ¼ ðr ij Þ with more known elements (on top of the minimum n 1 values). In this case, by Proposition 6, each missing element rjk1 jk in R ¼ ðrij Þ can be estimated. First, find a sequence of values rij1 ; rj1 j2 ; . . . ; r jt i (i < j1 < j2 < . . . < jt) that include one and only one unknown element r jk1 jk . If jk1 < jk, r jk1 jk is located in the middle of the sequence, and h i this missing element can be estimated as r_ jk1 jk ¼ r_ _þ by Eqs. (23) and (24), where jk1 jk ; r jk1 jk X t þ 1 1 r ij1 þ r j1 j2 þ r jk2 jk1 þ r jk jkþ1 þ r jt1 jt þ r þjt i #H i<j <j :
j2Hh1 ik
2 crlik
¼
8 X j2 > crlik ; if #Hh2 < #H1h2 ik – 0 ik
> :
j2Hh1 ik
;
crr 2ik
¼
0; otherwise
8 X j3 > crlik ; if #Hh3 < #H1h3 ik – 0 ik
> :
j2Hh1 ik
; crr 3ik ¼
0; otherwise
j2Hh1 ik
0; otherwise
8 X j2 1 > crrik ; if #Hh2 < #Hh2 ik – 0 ik
> :
j2Hh1 ik
0; otherwise
8 X j3 1 > crrik ; if #Hh3 < #Hh3 ik – 0 ik
> :
j2Hh1 ik
0; otherwise ð42Þ
1
crlik ¼ (
j¼
2
3
crlik þ crlik þ crlik
j
;
crr ik ¼
h2 h3 #Hh1 ik þ #H ik þ #Hik ;
1;
crr 1ik þ crr 2ik þ crr 3ik
j
h2 h3 if #Hh1 ik þ #Hik þ #H ik – 0
otherwise
crlik ¼ rlij þ rljk 0:5;
j2
j3
crrj1 ik ¼ rr ij þ rr jk 0:5;
ð34Þ
EV ¼ B n MV
ð35Þ
where MV is the set of pairs of alternatives for which the preference degree of the first alternative over the second one is unknown or missing; EV is the set of pairs of alternatives for which the expert provides preference values, and the symbol ‘‘n’’ denotes the exclusion relation. In Alonso et al. (2008)’s method, it does not take into account the preference value of one alternative over itself as xi xi is always assumed.
then crlik ¼ 0 or crrik ¼ 0
else if crlik > 1 or crrik > 1;
References
ð33Þ
ð46Þ
Step 3.
Step 4. End.
B ¼ fði; jÞji; j 2 f1; . . . ; ng ^ i – jg
crr j2 ik ¼ rr jk rr ji þ 0:5;
crr j3 ik ¼ rr ij rr kj þ 0:5
Step 1. Determine the set B, the set MV and the set EV, as follows:
MV ¼ fði; jÞ 2 Bjr ij is unknowng
ð45Þ
and
if crlik < 0 or crr ik < 0; ð32Þ
ð44Þ
crlik ¼ rljk rlji þ 0:5;
crlik ¼ rlij rlkj þ 0:5
Alonso et al. (2008) proposed the following procedure to estimate missing values for IFPRs. An IFPR R ¼ ðr ij Þ can be viewed as two ‘‘independent’’ fuzzy preference relations, the first one RL corresponding to the left bounds of the intervals and the second one RR corresponding to the right bounds of the intervals, respectively,
ð43Þ
where the symbol ‘‘#’’ denotes the number of elements in a given set, j1
R ¼ ðr ij Þ ¼ ð½rlij ; rrij Þ with RL ¼ ðrlij Þ and RR ¼ ðrr ij Þ and rlij 6 rrij 8i; j
ik
> :
ð41Þ
3
Appendix A
; crr 1ik ¼
0; otherwise
8 X j1 1 > crrik ; if #Hh1 < #Hh1 ik – 0
ð40Þ
crlik ¼
The authors are very grateful to the Associate Editor and the two anonymous reviewers for their constructive comments and suggestions that have further helped to improve the quality and presentation of this paper. Yejun Xu would like to acknowledge the financial support of a Grant (No. 71101043) from National Natural Science of China. Kevin W. Li would like to acknowledge the financial support of a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant, and Grants (Nos. 71272129 and 71271188) from National Natural Science Foundation of China. Huimin Wang would like to acknowledge the financial support of a Grant (No. 12&ZD214) from Major Program of the National Social Science Foundation of China.
8 X j1 > crlik ; if #Hh1 < #H1h1 ik – 0
then crlik ¼ 1 or crr ik ¼ 1
ð47Þ ð48Þ
Alonso, S., Chiclana, F., Herrera, F., & Herrera-Viedma, E. (2004). A learning procedure to estimate missing values in fuzzy preference relations based on additive consistency. In V. Torra & Y. Narukawa (Eds.), MDAI 2004, LNAI 3131 (pp. 227–238). Berlin Heidelberg: Springer-Verlag. Alonso, S., Chiclana, F., Herrera, F., Herrera-Viedma, E., Alcalá-Fdez, J., & Porcel, C. (2008). A consistency-based procedure to estimate missing pairwise preference values. International Journal of Intelligent Systems, 23, 155–175. Arbel, A. (1989). Approximate articulation of preference priority derivation. European Journal of Operational Research, 43, 317–326. Chen, C. T., Lin, C. T., & Huang, S. F. (2006). A fuzzy approach for supplier evaluation and selection in supply chain management. International Journal of Production Economics, 102, 289–301.
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Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Incomplete interval fuzzy preference relations and their applications q Yejun Xu a,b,⇑, Kevin W. Li c, Huimin Wang a,b a
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China Business School, Hohai University, Jiangning, Nanjing, Jiangsu 211100, PR China c Odette School of Business, University of Windsor, Windsor, Ontario N9B 3P4, Canada b
a r t i c l e
i n f o
Article history: Received 30 July 2012 Received in revised form 17 August 2013 Accepted 29 October 2013 Available online 11 November 2013 Keywords: Incomplete interval fuzzy preference relation Additive consistent Group decision making
a b s t r a c t This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n 1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process. Crown Copyright 2013 Published by Elsevier Ltd. All rights reserved.
1. Introduction Fuzzy preference relations are one of the most common preference relations for expressing a decision maker’s (DM’s) preference over alternatives. In a decision making process, the DM generally needs to compare a set of n decision alternatives xi (i = 1, 2, . . ., n), thereby constructing a fuzzy preference relation (Herrera-Viedma, Herrera, Chiclana, & Luque, 2004;Kacprzyk, 1986; Orlovsky, 1978;Tanino, 1984; Wang & Fan, 2007;Xu, Da, & Liu, 2009; Xu, Patnayakuni, & Wang, 2013c;Xu, 2005). However, the DM may have vague knowledge about the preference degrees of one alternative over another and cannot estimate his/her preference with an exact numerical value, but with an interval number. In this case, the DM constructs an interval preference relation. Saaty and Vargas (1987) first presented interval judgments as a way to model subjective uncertainty. Afterwards, some methods are proposed to generate weights from interval comparison matrices, such as linear programing (LP) (Arbel, 1989;Kress, 1991), Lexicographic Goal Programming (LGP) (Islam, Biswal, & Alam, 1997;Wang, 2006), fuzzy preference programming (FPP) (Mikhailov, 2002;Mikhailov, 2004), two-stage logarithmic goal programming (TLGP) (Wang, Yang, & Xu, 2005), eigenvector method (EM) (Wang & Chin, 2006), Lambda-Max method (Csutora & Buckley, 2001), goal programming method (GPM) (Wang & Elhag, 2007), etc.
q
This manuscript was processed by Area Editor Imed Kacem.
⇑ Corresponding author at: State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China. Tel.: +86 25 68514612; fax: +86 25 85427972. E-mail address: [email protected] (Y. Xu).
For IFPRs, Xu (2004b) defined the concept of compatibility degree between two IFPRs, and showed the compatibility relationship between individual IFPRs and collective IFPR. Herrera, Martíze, and Sánchez (2005) developed an aggregation process for combining IFPRs with other types of information such as numerical preference relation and linguistic preference relation. Jiang (2007) gave an index to measure the similarity degree between two IFPRs, and employed an error-propagation principle to determine a priority vector for the aggregated IFPRs. Recently, Xu and Chen (2008b) established some linear programming models for deriving priority weights from various IFPRs. Wang and Li (2012) developed goal-programming-based models for deriving interval weights from IFPRs for both individual and group decision-making situations. The aforesaid research focused on preference relations with complete information. A complete preference relation of order n necessitates the completion of all n(n 1)/2 judgments in its entire top triangular portion. Sometimes, however, a DM may develop a preference relation with incomplete information due to a variety of reasons such as time pressure, lack of knowledge, and the DM’s limited expertise related with the problem domain (Chiclana, Herrera-Viedma, Alonso, & Herrera, 2008;Lee, Chou, Fang, Tseng, & Yeh, 2007; Xu & Da, 2008;Xu & Da, 2009; Xu, Da, & Wang, 2010;Xu, Gupta, & Wang, 2013a; Xu, Patnayakuni, & Wang, 2013b;Xu, 2004a; Xu, 2005;Xu & Chen, 2008a); In addition, when the number of the alternatives, n, is large, it may be impractically to require the DM to perform all the n(n 1)/2 required comparisons for a complete the pairwise comparison matrix (Fedrizzi & Silvio, 2007); More over, it is sometimes convenient or necessary to skip some direct comparison between alternatives even if the total number of alternatives is small (Fedrizzi & Silvio, 2007); In some other
0360-8352/$ - see front matter Crown Copyright 2013 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2013.10.010
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Y. Xu et al. / Computers & Industrial Engineering 67 (2014) 93–103
cases, a DM is unable to express any kind of preference between two or more options, which may be due to an expert not possessing a precise or sufficient level of knowledge of part of the problem, or because that expert is unable to discriminate the degree to which one option is preferred to another (Alonso, Chiclana, Herrera, & Herrera-Viedma, 2004;Alonso et al., 2008; HerreraViedma, Alonso, Chiclana, & Herrera, 2007a;Herrera-Viedma, Chiclana, Herrera, & Alonso, 2007b; Xu, 2012); A critical concern for the incomplete fuzzy preference relations is to estimate the missing values. Herrera-Viedma et al. (2007a) proposed an iterative procedure to estimate the missing information in an expert’s incomplete fuzzy preference relation. The procedure is guided by the additive consistency property and only uses the preference values provided by the expert. Fedrizzi and Silvio (2007) put forward a new method for calculating missing elements in an incomplete fuzzy preference relation by maximizing global consistency. Later, Chiclana, Herrera-Viedma, and Alonso (2009) pointed out that the two methods are very similar in calculating missing values. Chiclana et al. (2008) presented a new estimation method based on based on the U-consistency criterion for incomplete fuzzy preference relations. Alonso et al. (2008) presented a procedure to estimate missing preference values for incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Liu, Pan, Xu, and Yu (2012b) developed a method to calculate missing values by minimizing the squared error of an incomplete fuzzy preference relation and its priority weight vector. Xu (2012) devised an approach to extending each incomplete multiplicative preference relation to a complete one by exploiting the multiplicative transitivity properties. Xia, Xu, and Wang (2014) furnished an algorithm to estimate missing values for an incomplete linguistic preference relation based on multiplicative consistency. Recently, research has been extended to IFPRs. For instance, Alonso et al. (2008) put forward a procedure to estimate missing information for the incomplete IFPR. Genç, Boran, Akay, and Xu (2010) examined consistency, missing value (s) and derivation of priority vectors from IFPRs based on multiplicative transitivity. Liu, Zhang, and Wang (2012a) proposed a new method to obtain priority weights from incomplete interval multiplicative preference relations. However, limited research has been devoted to incomplete IFPRs. As such, it is necessary to pay attention to this issue. Another important issue is the consistency of the judgment provided by experts (Chiclana, Herrera, & Herrera-Viedma, 2002; Herrera-Viedma et al., 2004). It is obvious that consistent information is more relevant or important than the information containing contradictions. Consistency is associated with certain transitivity properties. Different properties have been suggested to model transitivity of fuzzy preference relations. One of these properties is the ‘‘additive transitivity’’, which, as shown in (Herrera-Viedma et al., 2004), can be seen as a parallel concept of Saaty’s consistency property for multiplicative reciprocal preference relations. The aim of this paper is to propose some methods for constructing additive consistent IFPRs based on acceptable incomplete IFPRs. We first extend an additive consistency property proposed by Herrera-Viedma et al. (2004) for the fuzzy preference relations to a general case. Then, this property is extended to IFPRs based on the additive transitivity. After further characterizing additive consistent IFPRs, we develop two algorithms for estimating missing elements from acceptable incomplete IFPRs. A procedure is then laid out for handling GDM problems with acceptable incomplete IFPRs. The rest of this paper is organized as follows. Section 2 reviews some properties of fuzzy preference relations. Section 3 first introduces the concepts of interval multiplicative reciprocal preference relations and IFPRs as well as their transformation function. The property of additive consistent fuzzy preference relations in Section 2 is then extended to IFPRs, followed by further additive
consistent IFPRs. In Section 4, we propose two approaches to construct additive consistent IFPRs based on acceptable incomplete IFPRs. A case study is furnished in Section 5 to illustrate how to apply our algorithms. We conclude the paper in Section 6. 2. Additive consistent fuzzy preference relations Let X = {x1, x2, . . ., xn}(n P 2) be a finite set of alternatives, where xi denotes the ith alternative. In multiple attribute decision making problems, a DM needs to rank alternatives x1, x2, . . ., xn from the best to the worst according to preference information. A brief description of multiplicative and fuzzy preference relations is given below. 2.1. Multiplicative preference relations A multiplicative preference relation is a positive preference relation A X X, A = (aij)nn, where aij denotes the relative preference of alternative xi over xj. The measurement of aij is described by a ratio scale and in particular, as shown by Saaty (1980), aij 2 {1/9, 1/8, . . ., 1, 2, . . ., 9}: aij = 1 denotes the DM’s indifference between xi and xj, aij = 9 (or aji = 1/9) denotes that xi is absolutely preferred to xj, and aij 2 {2, 3, . . ., 8} denotes intermediate preference evaluations. This relation is multiplicative reciprocal, i.e., aijaji = 1, "i, j 2 {1, 2, . . ., n} and in particular, aii = 1, "i 2 {1, 2, . . ., n}. Its consistency is defined by Saaty (1980) as follows. Definition 1. Let A = (aij)nn be a multiplicative preference relation, then A is called consistent (Saaty, 1980), if aij = aikakj, for all i, j, k. 2.2. Fuzzy preference relations A fuzzy preference relation R is described as follows: R X X, R = (rij)nn, with membership function uR:X X ? [0, 1], where uR(xi, xj) = rij denotes the preference degree of alternative xi over xj (Kacprzyk, 1986;Tanino, 1984): rij = 0.5 denotes indifference between xi and xj, rij = 1, denotes that xi is definitely preferred to xj, and 0.5 < rij < 1 (or 0 < rji < 0.5) denotes that xi is preferred to xj to a varying degree. Definition 2. Let R = (rij)nn be a preference relation, then R is called a fuzzy preference relation, if
rij 2 ½0; 1;
r ij þ r ji ¼ 1;
r ii ¼ 0:5;
for all i; j 2 N
ð1Þ
Definition 3. Let R = (rij)nn be a fuzzy preference relation, then R is called an additive transitive fuzzy preference relation if the following additive transitivity (Tanino, 1984) is satisfied:
rij ¼ r ik rjk þ 0:5;
for all i; j; k ¼ 1; 2; . . . ; n
ð2Þ
2.3. Characterizing additive consistency of fuzzy preference relations Herrera-Viedma et al. (2004) studied the transformation between multiplicative preference relations with values in the interval scale [1/9, 9] (Alonso et al., 2004) and fuzzy preference relations with values in [0, 1] and furnished the following propositions. Proposition 1 (Herrera-Viedma et al., 2004). Consider a set of alternatives X = {x1, x2, . . ., xn}, associated with a multiplicative reciprocal preference relation A = (aij)nn with aij 2 [1/9, 9]. Then, a corresponding fuzzy preference relation, R = (rij)nn, with rij 2 [0, 1], associated with A is given as follows:
rij ¼ gðaij Þ ¼
1 ð1 þ log9 aij Þ 2
ð3Þ
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Proposition 2 (Herrera-Viedma et al., 2004). For a fuzzy preference relation R = (rij)nn, the following statements are equivalent: (a) rij þ rjk þ r ki ¼ 32 ; 8i; j; k; (b) rij þ rjk þ r ki ¼ 32 ; i < j < k.
Proposition 3 (Herrera-Viedma et al., 2004). For a fuzzy preference relation R = (rij)nn, the following statements are equivalent: (a) rij þ rjk þ r ki ¼ 32 ; i < j < k; (b) riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ r ðj1Þj þ r ji ¼ jiþ1 ; i < j. 2 2.4. A new characterization of additive consistency Herrera-Viedma et al. (2004) showed that Proposition 3 can be used to construct an additive consistent fuzzy preference relation from a set of n 1 values {r12, r23, . . ., rn1n}. The aforesaid propositions were also used by Wang and Chen (2007, 2008). In the following, a more general result is provided. Proposition 4. For a fuzzy preference relation R = (rij)nn, the following statements are equivalent: (a) rij þ rjk þ r ki ¼ 32 ; 8i; j; k 2 N; (b) rij1 þ rj1 j2 þ þ rjt1 jt þ r jt i ¼ tþ1 ; 8i; jl 2 N; l ¼ 1; 2; . . . ; t. 2 Proof. (a) ) (b) Mathematical induction is employed to prove this part of the proposition. It is obviously true for t = 1, as it is reduced to the additive reciprocity property in Definition 2. Next, if the hypothesis is true for t = n
r ij1 þ rj1 j2 þ þ r jn2 jn1 þ rjn1 jn þ r jn i
nþ1 ¼ 2
and an IFPR. The characterizations of additive consistent fuzzy preference relations in Section 2 are subsequently extended to IFPRs followed other results. byþsome useful þ þ 1 ¼ a Let a 1 ; a1 ; a2 ¼ a2 ; a2 ; a ¼ ½a ; a be three positive interval numbers, then
1 a 2 ¼ a1 ; aþ1 a2 ; aþ2 ¼ a1 þ a2 ; aþ1 þ aþ2 ðaÞ a þ þ 1 a 2 ¼ a1 ; a1 a2 ; a2 ¼ a1 aþ2 ; aþ1 a2 ðbÞ a þ þ 1 a 2 ¼ a1 ; a1 a2 ; a2 ¼ a1 a2 ; aþ1 aþ2 ðcÞ a Þ ¼ ½logn a ; logn aþ ðdÞ logn ða
ð7Þ
1 ¼ ½1=aþ ; 1=a ðeÞ a
ð8Þ
ð4Þ ð5Þ ð6Þ
ij Þ is multipliDefinition 4. An interval preference relation A ¼ ða ji ¼ a 1 cative reciprocal if and only if a ij . Definition 5 Xu, 2010. If a positive interval multiplicative reciproij Þ satisfies: cal preference relation A ¼ ða
ij ¼ a ik a kj ; a
for all i; k; j ¼ 1; 2; . . . ; n;
and i 6 k 6 j
ð9Þ
ij Þ is multiplicative consistent. then A ¼ ða Note that if an interval multiplicative reciprocal preference relation is multiplicative consistent, for all i, k, j 2 {1, 2, . . ., n}, as pointed out in Xu (2010), it is necessary to require i 6 k 6 j, otherij Þ would be reduced to a crisp number judgment mawise, A ¼ ða trix (Saaty’s multiplicative reciprocal preference relation). For more detail, readers are referred to (Xu, 2010). That is Eq.(9) holds only for the upper (or lower) triangular of the preference relation. Definition 6 Xu, 2004b. Let R ¼ ðrij Þ be a preference relation, where
h i r ij ¼ r ij ; r þij ;
h i r ij ¼ rji ; r þji ;
r þij P r ij P 0;
for all i; j ¼ 1; 2; . . . ; n
rij þ rþji ¼ r þij þ rji ¼ 1;
then R is called an interval fuzzy preference relation (IFPR).
then it is true for t = n + 1
r ij1 þ rj1 j2 þ þ r jn1 jn þ r jn jnþ1 þ r jnþ1 i ¼ ðr ij1 þ rj1 j2 þ þ r jn2 jn1 þ r jn1 jn Þ þ rjn jnþ1 þ r jnþ1 i ¼ ¼
nþ1 2 n1 2
r jn i þ r jn jnþ1 þ r jnþ1 i
Definition 7. Let R ¼ ðr ij Þ be a IFPR, if
rij rjk ¼ r ik ½0:5; 0:5;
8i < j < k
ð10Þ
then R is called an additive consistent IFPR.
þ r ijn þ r jn jnþ1 þ r jnþ1 i ¼ n1 þ 32 ¼ nþ2 2 2
So the result is established. (b) ) (a).
r ij þ r jk þ r ki ¼ 1 r ji þ 1 r kj þ rki ¼ 2 þ r ij1 þ þ rjt1 j tþ1 þ rjj1 þ þ r jt1 k tþ1 þ rki 2 2 ¼ 1 t þ ðr ij1 þ þ r jt1 j þ r jj1 þ þ r jt1 k þ r ki Þ ¼ 1 t þ 2tþ1 ¼ 32 2 This completes the proof. h Furthermore, in the proof process, it is obvious that the differences of j2 j1, j3 j2, . . ., jt jt1 are not necessarily equal to 1. As a matter of fact, the differences do not have to be identical. Proposition 4 differs from Proposition 3 in that any the sequential values of rij1 ; rj1 j2 ; . . . ; r jt1 jt ; r jt i (for example r31, r15, r57, r73) will work for Proposition 4. But Proposition 3 requires preference values to follow a consecutive order such as r12, r23, . . ., rn1n, rn1. Therefore, Proposition 3 is a special case of Proposition 4. 3. Interval fuzzy preference relations and their characterizations In the following we shall first introduce some operational laws of interval numbers (Hayes, 2003;Moore, 1966). We then present a relationship between an interval multiplicative preference relation
Proposition 5. For a set of alternatives X = {x1, x2, . . ., xn}, and its associated interval multiplicative reciprocal preference relation ij Þ with a ji ¼ a 1 A ¼ ða ij , a corresponding IFPR, R ¼ ðr ij Þ, associated with A is given as
ij Þ ¼ rij ¼ gða
1 ij Þ ð½1; 1 log9 a 2
ð11Þ
such that þ (a) r ij þ r ji ¼ 1, "i, j 2 N; (b) r þ þ r ji ¼ 1, "i, j 2 N. ij
ij Þ is an interval multiplicative reciprocal preferProof. As A ¼ ða ji ¼ a 1 ence relation, by Definition 4, a ij , that is
h
i aji ; aþji ¼
"
1 1 ; aþij aij
#
ð12Þ
Thus
aji ¼
1 ; aþij
aji aþij ¼ 1;
aþji ¼
1 aij
aþji aij ¼ 1
ð13Þ ð14Þ
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Y. Xu et al. / Computers & Industrial Engineering 67 (2014) 93–103
By Proposition 1 and Eq. (11) and the operational law (Eq. (7)) of interval numbers, we have
r ij ¼
h i 1 1 ij Þ ¼ ð1 log9 a ½1; 1 log9 aij ; aþij 2 2
Thus
Let i < j, and k = j i. The expression (c) can be rewritten as follows:
riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ rðj1Þj þ rþji ¼
kþ1 ; 2
8i < j
Mathematical induction is used to prove this part. It is clearly true for k = 1. Next if the hypothesis is true for k = n
1 1 þ log9 aij ¼ 2 1 þ r ij ¼ 1 þ log9 aþij 2 r ij
riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ rðiþn1ÞðiþnÞ þ r þðiþnÞi ¼
nþ1 2
Similarly,
then for k = n + 1:
1 1 þ log9 aji 2 1 1 þ log9 aþji r þji ¼ 2
r iðiþ1Þ þ r ðiþ1Þðiþ2Þ þ þ r ðiþn1ÞðiþnÞ þ r ðiþnÞðiþnþ1Þ þ rþðiþnþ1Þi ¼ riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ rðiþn1ÞðiþnÞ þ r ðiþnÞðiþnþ1Þ þ rþðiþnþ1Þi nþ1 ¼ r þðiþnÞi þ r ðiþnÞðiþnþ1Þ þ r þðiþnþ1Þi 2 nþ1 ¼ þ r iðiþnÞ 1 þ r ðiþnÞðiþnþ1Þ þ rþðiþnþ1Þi 2 n1 3 nþ2 ¼ þ ¼ 2 2 2
r ji ¼
Therefore,
1 1 1 þ log9 aij þ 1 þ log9 aþji ¼ 1 þ log9 aij aþji ¼ 1 2 2 1 1 þ þ r ij þ r ji ¼ 1 þ log9 aij þ 1 þ log9 aji ¼ 1 þ log9 aþij aji ¼ 1; 2 2 8i; j 2 f1; 2; . . . ; ng
r ij þ r þji ¼
The proof is thus completed. h Next, we examine the relationship between the multiplicative consistency of an interval multiplicative reciprocal preference relation and the additive consistency of its converted IFPR as per Eq. (11). Proposition 6. If an interval multiplicative reciprocal preference ij Þ is multiplicative consistent, then its corresponding relation A ¼ ða IFPR R ¼ ðrij Þ is additive consistent, and
(a) (b) (c) (d) (e) (f)
þ 3 r ij þ r jk þ r ki ¼ 2 ; 8i < j < k; þ 3 rþ þ r þ r ¼ ; 8i < j < k; ki ij jk 2 þ r þ r þ þ r iðiþ1Þ ðiþ1Þðiþ2Þ ðj1Þj þ r ji þ þ þ riðiþ1Þ þ rðiþ1Þðiþ2Þ þ þ r ðj1Þj þ r ji þ tþ1 r ij1 þ r j1 j2 þ þ r jt1 jt þ r jt i ¼ 2 ; þ þ tþ1 rþ ij1 þ r j1 j2 þ þ r jt1 jt þ r jt i ¼ 2 ;
¼ jiþ1 ; 8i < j; 2 ¼ jiþ1 ; 8i < j; 2 8i < j1 < j2 < < jt ; 8i < j1 < j2 < < jt .
thus the expression (c) is confirmed and (d) can be analogously asserted. Similar to Proposition 4, (e) and (f) can be verified. h Proposition 7. An IFPR R ¼ ðrij Þ is additive consistent if and only if þ þ þ þ (1) r ij r ik ¼ r lj r lk ; (2) r ij r ik ¼ r lj r lk ; 8i < j < k, l = 1, 2, . . ., n. Proof. If an IFPR R ¼ ðrij Þ is additive consistent by Definition 7, then
r ij r jk ¼ r ik ½0:5; 0:5; 8i < j < k 1 1 r ij þ r jk ¼ r ik þ ; r þij þ rþjk ¼ r þik þ ; 8i < j < k 2 2 1 1 þ þ þ r ij r ik ¼ rjk ; r ij r ik ¼ rjk ; 8i < j < k 2 2 Similarly,
r lj r jk ¼ r lk ½0:5; 0:5; 8l < j < k 1 1 r lj þ r jk ¼ r lk þ ; r þlj þ rþlk ¼ r þlk þ ; 8l < j < k 2 2 1 1 r lj r lk ¼ rjk ; r þlj r þlk ¼ rþlk ; 8l < j < k 2 2
ij Þ is multiplicative consistent by Definition 5, Proof. Since A ¼ ða ij a jk ¼ a ik , for "i < j < k. Taking a logarithm operation (Eq. then a (7)) on both sides yields
rij r ik ¼ rlj r lk ;
ij log9 a jk ¼ log9 a ik ; log9 a
On the contrary,
8i < j < k
thus
If r ij r ik ¼ r lj r lk ;
Thus
r þij r þik ¼ r þlj r þlk ;
r þij rþik ¼ r þlj rþlk ;
rij r ik ¼ 0:5 r jk ;
By Eq. (11), we have
r ij þ r jk ¼ 0:5 þ rik ;
r ij r jk ¼ rik ½0:5; 0:5 h i h i r ij ; r þij r jk ; r þjk ¼ rik ; r þik ½0:5; 0:5
r ij r jk ¼ r ik ½0:5; 0:5;
þ
r jk
¼
r ik
þ 0:5;
rþij
þ
r þjk
¼
rþik
3 ; 2
rþij þ r þjk þ r ki ¼
þ 0:5
3 ; 2
rþij rþik ¼ 0:5 r þjk ;
8i < j < k
r þij þ r þjk ¼ 0:5 þ rþik ;
8i < j < k
That is
By Proposition 5, we have
r ij þ rjk þ rþki ¼
8i < j < k
þ Let l = j, since r jj ¼ r jj ¼ 0:5, then
1 1 ij Þ ð½1; 1 log9 a jk Þ ð½1; 1 log9 a 2 2 1 ik Þ ½0:5; 0:5 ¼ ð½1; 1 log9 a 2
r ij
8i; l < j < k
8i < j < k
Thus the expressions (a) and (b) are established.
8i < j < k
This completes the proof. h Proposition 7 reveals an important property of an additive consistent IFPR. For upper (or lower) triangular interval values, the difference of the lower bounds between any two columns should be a constant for all rows. The same is true for the upper bounds of the interval preference values. Propositions 6 and 7 will play an important role in devising our algorithms to construct complete IFPRs based on an incomplete relation.
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Note that, if the primary values are different then we may have obtained a matrix R with entries not in the interval [0, 1], but in an interval [c, 1 + c], where c > 0, c indicates the minimum value of matrix R; 1 þ c gives the maximum value of matrix R. In this case, the obtained values have to be converted using a transformation function that preserves reciprocity and additive consistency, i.e., f: [c, 1 + c] ? [0, 1], verifying
ðaÞ f ðcÞ ¼ 0 ðbÞ f ð1 þ cÞ ¼ 1 ðcÞ f ðx Þ þ f ðxþ Þ ¼ 1;
8x 2 ½c; 1 þ c
3 3 ðdÞ If x þ y þ z ¼ ; f ðx Þ þ f ðy Þ þ f ðzþ Þ ¼ ; 2 2 8x ; y ; zþ 2 ½c; 1 þ c 3 3 ðeÞ If xþ þ yþ þ z ¼ ; f ðxþ Þ þ f ðyþ Þ þ f ðz Þ ¼ ; 2 2 8xþ ; yþ ; z 2 ½c; 1 þ c
ð1Þ
rij ¼ k1 r ij rþij
¼
ð1Þþ k1 r ij ð1Þ
rji ¼ k1 r ji rþji
¼
ð1Þþ k1 r ji
ð2Þ
þ k2 r ij þ
ð2Þþ k2 r ij ð2Þ
þ k2 r ji þ
ð2Þþ k2 r ji
ðmÞ
þ þ km r ij þ þ
ðmÞ
þ þ km r ji þ þ
;
ðmÞþ km r ij ;
8i; j 2 f1; 2; . . . ; ng
;
ðmÞþ km r ji ;
8i; j 2 f1; 2; . . . ; ng
ð15Þ
ð1Þ ð1Þþ ð2Þ ð2Þþ ðmÞ ðmÞþ rij þ r þji ¼ k1 r ij þ r ji þ k2 r ij þ r ji þ þ km rij þ rji
ð16Þ
¼ k1 þ k2 þ þ km ¼ 1
ð17Þ
ð1Þþ ð1Þ ð2Þþ ð2Þ ðmÞþ ðmÞ þ k2 r ij þ r ji þ þ km rij þ rji rþij þ r ji ¼ k1 r ij þ r ji
þ
ð18Þ
¼ k1 þ k2 þ þ km ¼ 1 The proof is thus completed.
ð19Þ
A linear function satisfying (a) and (b) takes the form
h
Proposition 9. Let Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be m additive consistent IFPRs, then their weighted average
R ¼ k1 Rð1Þ k2 Rð2Þ . . . km RðmÞ ;
f ðx Þ ¼ ux þ b;
u; b 2 R f ðxþ Þ ¼ uxþ þ b; u; b 2 R
kl 2 ½0; 1;
m X kl ¼ 1
ð21Þ
l¼1
is also an additive consistent IFPR.
These functions are Proof. Since Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are additive consistent IFPRs by Definition 7.
1 c x þ c x þ ¼ 1 þ 2c 1 þ 2c 1 þ 2c 1 c xþ þ c f ðxþ Þ ¼ xþ þ ¼ 1 þ 2c 1 þ 2c 1 þ 2c
f ðx Þ ¼
rðlÞ ðlÞ ðlÞ ij r jk ¼ r ik ½0:5; 0:5; then by Eq. (21), we have
(c) can be easily verified as
rij ¼ k1r ð1Þ ð2Þ ðmÞ ij k2 r ij km r ij
x þ c xþ þ c ðx þ xþ Þ þ 2c f ðx Þ þ f ðx Þ ¼ þ ¼ ¼1 1 þ 2c 1 þ 2c 1 þ 2c
8i < j < k
þ
rjk ¼ k1r ð1Þ ð2Þ ðmÞ jk k2 r jk km r jk
If x þ y þ zþ ¼ 32 ; xþ þ yþ þ z ¼ 32
x þ c y þ c z þ þ c þ þ 1 þ 2c 1 þ 2c 1 þ 2c þ ðx þ y þ z Þ þ 3c 3=2 þ 3c 3 ¼ ¼ ¼ 1 þ 2c 1 þ 2c 2
rik ¼ k1r ð1Þ ð2Þ ðmÞ ik k2 r ik km r ik
f ðx Þ þ f ðy Þ þ f ðzþ Þ ¼
r ij r jk ¼ k1 r ð1Þ ð1Þ k2 r ð2Þ ð2Þ km r ðmÞ ðmÞ ij r jk ij r jk ij r jk ð1Þ ð2Þ ðmÞ ¼ k1 r ik ½0:5; 0:5 k2 r ik ½0:5; 0:5 km r ik ½0:5;0:5
xþ þ c y þ þ c z þ c þ þ 1 þ 2c 1 þ 2c 1 þ 2c ðxþ þ yþ þ z Þ þ 3c 3=2 þ 3c 3 ¼ ¼ ¼ 1 þ 2c 1 þ 2c 2
ð1Þ
f ðxþ Þ þ f ðyþ Þ þ f ðz Þ ¼
ð2Þ
ðmÞ
¼ k1 r ik k2 r ik kmr ik ½ðk1 þ k2 þ þ km Þ=2; ðk1 þ k2 þ þ km Þ=2 ¼ rik ½0:5; 0:5
This completes the proof.
h
(d) and (e) are also confirmed. Proposition 8. Let Rð1Þ ; Rð2Þ ; . . . ; RðmÞ be m IFPRs, then their weighted average
R ¼ k1 Rð1Þ k2 Rð2Þ . . . km RðmÞ ;
kl 2 ½0; 1;
m X kl ¼ 1 l¼1
is also an IFPR, i.e., satisfies
r ij þ r þji ¼ 1;
r þij þ r ji ¼ 1;
8i; j 2 f1; 2; . . . ; ng
Proof. Since Rð1Þ ; Rð2Þ ; . . . ; RðmÞ are IFPRs, it follows that ðlÞ
r ij
ðlÞþ
þ r ji
¼ 1;
ðlÞþ
r ij
ðlÞ
þ rji
l ¼ 1; 2; . . . ; m Then by Eq. (20), we have
¼ 1;
8i; j 2 f1; 2; . . . ; ng;
ð20Þ
4. Procedures for repairing incomplete interval fuzzy preference relations 4.1. Incomplete interval fuzzy preference relation A complete n n preference relation requires n(n 1)/2 judgments in its entire top (or lower) triangular portion. Sometimes, however, a DM may furnish a preference relation with incomplete information due to a variety of reasons. Next, we first present basic concepts of incomplete interval IFPRs, followed by two estimation procedures for missing values. Definition 8. Let R ¼ ðr ij Þ be an n n preference relation, if at least on element in the upper (or lower) triangular part is not given by the DM, denoted by an unknown variable ‘‘x’’, and remaining interval values by the DM satisfy
rij ¼ ½1; 1 r ji ;
r ii ¼ ½0:5; 0:5;
for all r ij 2 X
ð22Þ
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Y. Xu et al. / Computers & Industrial Engineering 67 (2014) 93–103
R is called an incomplete interval IFPR, where X is the set of all the known elements in R. Definition 9. The elements r ij ; r kl of R are called adjacent, if {i, j} \ {k, l} – ;. For a missing element r ij , it can be determined indirectly if there exist a series of known elements rij1 ; rj1 j2 ; . . . ; r jt j . Definition 10. Let R ¼ ðrij Þ be an incomplete IFPR. If all missing elements of R can be obtained by the known elements, then R is called an acceptable incomplete IFPR. Otherwise, R is an unacceptable incomplete IFPR. Next, we extend the necessary condition (Herrera-Viedma et al., 2007b;Xu & Da, 2008) of acceptable incomplete fuzzy preference relations to the case of incomplete IFPRs. Proposition 10. Let R ¼ ðr ij Þ be an incomplete IFPR. If R is an acceptable incomplete IFPR, then there exists at least one known nondiagonal element in each line or each column of R, i.e. there exist at least (n 1) judgments provided by the DM. Definition 11. Let R ¼ ðrij Þ be an incomplete IFPR, if the known elements satisfy
tþ1 ¼ ; for 8i < j1 < j2 < < jt 2 tþ1 r þij1 þ r þj1 j2 þ þ rþjt1 jt þ r jt i ¼ ; for 8i < j1 < j2 < < jt 2
r ij1
þ r j1 j2
þ þ rjt1 jt
þ r þjt i
ð23Þ ð24Þ
then R is called an additive consistent incomplete IFPR.
r31 ¼ ½0:4; 0:7; r35 ¼ ½0:4; 0:7;
r32 ¼ ½0:1; 0:3; r36 ¼ ½0:3; 0:8;
r34 ¼ ½0:3; 0:4; r37 ¼ ½0:4; 0:9
Step 1. By Definition 8 and the aforesaid information provided by the DM, one obtains the following acceptable incomplete IFPR, where ‘‘x’’ denotes the unknown judgment. 3 ½0:5; 0:5 x x x x x x 7 6 x ½0:5;0:5 x x x x x 7 6 7 6 6 ½0:4; 0:7 ½0:1;0:3 ½0:5; 0:5 ½0:3; 0:4 ½0:4; 0:7 ½0:3; 0:8 ½0:4; 0:9 7 7 6 7 6 R¼6 x x x ½0:5; 0:5 x x x 7 7 6 7 6 x x x x x x ½0:5; 0:5 7 6 7 6 x x x x x ½0:5; 0:5 x 5 4 x x x x x x ½0:5; 0:5 2
Step 2. Utilize Propositions 6 and 7 to determine all missing elements in R as follows: r13 ¼ 1 r þ31 ¼ 0:3; r þ13 ¼ 1 r31 ¼ 0:6; r23 ¼ 1 r þ32 ¼ 0:7; r þ23 ¼ 1 r32 ¼ 0:9 3 3 r12 ¼ r 23 r þ31 ¼ 0:1; r þ12 ¼ rþ23 r31 ¼ 0:2; r21 ¼ 1 r þ12 ¼ 0:8; r þ21 ¼ 1 r12 ¼ 0:9 2 2 r23 r33 ¼ r 24 r 34 ) r 24 ¼ r 23 r 33 þ r 34 ¼ 0:5; r þ24 ¼ rþ23 r þ33 þ r þ34 ¼ 0:8 r25 ¼ r 23 r 33 þ r 35 ¼ 0:6; rþ25 ¼ r þ23 r þ33 þ r þ35 ¼ 1:1 r26 ¼ r 23 r 33 þ r 36 ¼ 0:5; rþ26 ¼ r þ23 r þ33 þ r þ36 ¼ 1:2 r27 ¼ r 23 r 33 þ r 37 ¼ 0:6; rþ27 ¼ r þ23 r þ33 þ r þ37 ¼ 1:3 r14 ¼ r 13 r 33 þ r 34 ¼ 0:1; rþ14 ¼ r þ13 r þ33 þ r þ34 ¼ 0:5 r15 ¼ r 13 r 33 þ r 35 ¼ 0:2; rþ15 ¼ r þ13 r þ33 þ r þ35 ¼ 0:8 r16 ¼ r 13 r 33 þ r 36 ¼ 0:1; rþ16 ¼ r þ13 r þ33 þ r þ36 ¼ 0:9 r17 ¼ r 13 r 33 þ r 37 ¼ 0:2; rþ17 ¼ r þ13 r þ33 þ r þ37 ¼ 1 r45 ¼ r 35 r 34 þ r 44 ¼ 0:6; rþ45 ¼ r þ35 r þ34 þ r þ44 ¼ 0:8 r46 ¼ r 36 r 34 þ r 44 ¼ 0:5; rþ46 ¼ r þ36 r þ34 þ r þ44 ¼ 0:9 r47 ¼ r 37 r 34 þ r 44 ¼ 0:6; rþ47 ¼ r þ37 r þ34 þ r þ44 ¼ 1
4.2. An estimation procedure for acceptable incomplete IFPRs with fewest number of judgments Next, by exploiting Propositions 5 and 6 or Proposition 7, a simple and practical method is developed for constructing a complete additive consistent IFPR based on an acceptable incomplete IFPR with fewest number of judgment data (i.e., n 1 preference values):
r56 ¼ r 36 r 35 þ r 55 ¼ 0:4; rþ56 ¼ r þ36 r þ35 þ r þ55 ¼ 0:6 r57 ¼ r 37 r 35 þ r 55 ¼ 0:5; rþ57 ¼ r þ37 r þ35 þ r þ55 ¼ 0:7 r67 ¼ r 37 r 36 þ r 66 ¼ 0:6; rþ67 ¼ r þ37 r þ36 þ r þ66 ¼ 0:6 r41 ¼ 1 r þ14 ¼ 0:5; r þ41 ¼ 1 r14 ¼ 0:9; r42 ¼ 1 r þ24 ¼ 0:2; r þ42 ¼ 1 r24 ¼ 0:5 r43 ¼ 1 r þ34 ¼ 0:6; r þ43 ¼ 1 r34 ¼ 0:7 r51 ¼ 1 r þ15 ¼ 0:2; r þ51 ¼ 1 r15 ¼ 0:8; r52 ¼ 1 r þ25 ¼ 0:1; r þ52 ¼ 1 r25 ¼ 0:4 r53 ¼ 1 r þ35 ¼ 0:3; r þ53 ¼ 1 r35 ¼ 0:6; r54 ¼ 1 r þ45 ¼ 0:2; r þ54 ¼ 1 r45 ¼ 0:4 r61 ¼ 1 r þ16 ¼ 0:1; r þ61 ¼ 1 r16 ¼ 0:9; r62 ¼ 1 r þ26 ¼ 0:2; r þ62 ¼ 1 r26 ¼ 0:5 r63 ¼ 1 r þ36 ¼ 0:2; r þ63 ¼ 1 r36 ¼ 0:7; r64 ¼ 1 r þ46 ¼ 0:1; r þ64 ¼ 1 r46 ¼ 0:5 r65 ¼ 1 r þ56 ¼ 0:4; r þ65 ¼ 1 r56 ¼ 0:6
Algorithm 1 Step 1. For a decision problem, let X = {x1, x2, . . ., xn} be a discrete set of alternatives. The DM conducts pairwise comparisons among the alternatives and furnished his/her assessment as an acceptable incomplete IFPR R ¼ ðrij Þnn (if the DM provides his/her evaluation as an acceptable incomplete interval multiplicative reciprocal preference relation ij Þnn , then A ¼ ða ij Þnn can be converted to a correA ¼ ða sponding incomplete IFPR R ¼ ðr ij Þnn by Propositions 5), with only n 1 judgments. Step 2. Utilizing Proposition 6 or Proposition 7 to determine all unknown elements in R, and yield an interval additive consistent IFPR R_ ¼ ðr_ ij Þnn . If this preference relation contains any values falling outside the unit interval [0, 1], but within the interval [c, 1 + c], then a transformation funcxþc tion f ðxÞ ¼ 1þ2c can be applied to preserve the reciprocity and additive transitivity, resulting in an additive consistent IFPR. Step 3. End.
Example 1. Assume that a decision problem involves evaluating seven faculties xi (i = 1, 2, . . ., 7) at a university. The DM assesses these seven faculties (alternatives) by pairwise comparison and provides his/her judgment as follows:
r71 ¼ 1 r þ17 ¼ 0; rþ71 ¼ 1 r 17 ¼ 0:8; r 72 ¼ 1 rþ27 ¼ 0:3; rþ72 ¼ 1 r 27 ¼ 0:4 r73 ¼ 1 r þ37 ¼ 0:1; r þ73 ¼ 1 r37 ¼ 0:6; r74 ¼ 1 r þ47 ¼ 0; rþ74 ¼ 1 r 47 ¼ 0:4 r75 ¼ 1 r þ57 ¼ 0:3; r þ75 ¼ 1 r57 ¼ 0:5; r76 ¼ 1 r þ67 ¼ 0:4; r þ76 ¼ 1 r67 ¼ 0:4
Thus, 2
½0:5;0:5 6 ½0:8;0:9 6 6 6 ½0:4;0:7 6 _R ¼ 6 ½0:5;0:9 6 6 6 ½0:2;0:8 6 6 4 ½0:1;0:9 ½0; 0:8
½0:1; 0:2 ½0:3;0:6 ½0:1; 0:5 ½0:2; 0:8 ½0:1; 0:9
½0:2;1
3
½0:5; 0:5 ½0:7;0:9 ½0:5; 0:8 ½0:6;1:1 ½0:5;1:2 ½0:6; 1:3 7 7 7 ½0:1; 0:3 ½0:5;0:5 ½0:3; 0:4 ½0:4; 0:7 ½0:3; 0:8 ½0:4;0:9 7 7 7 ½0:2; 0:5 ½0:6;0:7 ½0:5; 0:5 ½0:6; 0:8 ½0:5; 0:9 ½0:6;1 7 7 ½0:1;0:4 ½0:3;0:6 ½0:2; 0:4 ½0:5; 0:5 ½0:4; 0:6 ½0:5;0:7 7 7 7 ½0:2;0:5 ½0:2;0:7 ½0:1; 0:5 ½0:4; 0:6 ½0:5; 0:5 ½0:6;0:6 5 ½0:3;0:4 ½0:1;0:6 ½0;0:4 ½0:3; 0:5 ½0:4; 0:4 ½0:5;0:5
As the preference relation contains values falling outside the interxþc val [0, 1] and c = 0.3, a transformation function f ðxÞ ¼ 1þ2c is applied _ yielding to the lower and upper bounds of each interval value in R, 2
½0:5;0:5 6 ½0:69;0:75 6 6 6 ½0:44;0:63 6 € ¼ 6 ½0:5;0:75 R 6 6 6 ½0:31;0:69 6 6 4 ½0:25;0:75 ½0:19;0:69
½0:25;0:31 ½0:5;0:5 ½0:25;0:38 ½0:31;0:5 ½0:13;0:44 ½0:06;0:5 ½0;0:44
½0:38;0:56 ½0:63;0:75 ½0:5;0:5 ½0:56;0:63 ½0:38;0:56 ½0:31;0:68 ½0:25;0:56
½0:25;0:5 ½0:5;0:69 ½0:38;0:44 ½0:5;0:5 ½0:31;0:44 ½0:25;0:5 ½0:19;0:44
½0:31;0:69 ½0:56;0:88 ½0:44;0:63 ½0:56;0:69 ½0:5;0:5 ½0:44;0:56 ½0:38;0:5
½0:25; 0:75 ½0:5; 0:94 ½0:38; 0:69 ½0:5; 0:75 ½0:44; 0:56 ½0:5; 0:5 ½0:44; 0:44
3 ½0:31;0:81 ½0:56;1 7 7 7 ½0:44;0:75 7 7 7 ½0:56;0:81 7 7 ½0:5;0:63 7 7 7 ½0:56;0:56 5 ½0:5;0:5
Alonso et al. (2008) proposed a different procedure to estimate missing in an incomplete IFRP (see Appendix). For a comparison with our approach, their procedure is employed to determine the missing judgments in this example.
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Y. Xu et al. / Computers & Industrial Engineering 67 (2014) 93–103
By Eq. (32), RL and RR are obtained as follows: 2
3
2
3
0:5 x x x x x x 0:5 x x x x x x 6 x 0:5 x 6 x 0:5 x x x x x 7 x x x x 7 6 6 7 7 6 6 7 7 6 0:4 0:1 0:5 0:3 0:4 0:3 0:4 7 6 0:7 0:3 0:5 0:4 0:7 0:8 0:9 7 6 6 7 7 6 6 7 7 x x 0:5 x x x 7; RR ¼ 6 x x x 0:5 x x x 7 RL ¼ 6 x 6 6 7 7 6 x 6 7 x x x 0:5 x x 7 x x x 0:5 x x 7 6 6 x 7 6 6 7 7 x x x x 0:5 x 5 x x x x 0:5 x 5 4 x 4 x x x x x x x 0:5 x x x x x x 0:5
Step 1. In RL and RR, ‘‘x’’ denotes an unknown value, and by Eq. (35), EV of the known values are determined as:
by the DM and estimated values during earlier iterations. In contrast, our method can estimate missing values in one step and all the missing values are estimated based on the preference values furnished by the DM. Thirdly, in the final results, we get rr27 = 1 and rr72 = 0 as this is based on Eqs. (47) and (48). This constraints make the final estimated matrix RR do not have the additive consistent property. Finally, the derived R by Alonso et al. (2008)’s method is no longer an IFPR, because it does not satisfy the condition (a), (b) of Proposition 5, or Proposition 7, while our estimated matrix R is an additive consistent IFPR. 4.3. An estimation procedure for acceptable incomplete IFRP with more known judgments
EV ¼ fð3; 1Þ; ð3; 2Þ; ð3; 4Þ; ð3; 5Þ; ð3; 6Þ; ð3; 7Þg Step 2. By applying Eqs. (36)–(46), one has: 1
H11 12 ¼ /;
) crl12 ¼ 0;
crr112 ¼ 0
H12 12 ¼ f3g 2
) crl12 ¼ rl32 rl31 þ 0:5 ¼ 0:1 0:4 þ 0:5 ¼ 0:2; crr112 ¼ rr32 rr 31 þ 0:5 ¼ 0:3 0:7 þ 0:5 ¼ 0:1 H13 12 ¼ /;
j¼1
3
) crl12 ¼ 0; crr312 ¼ 0 0 þ 0:2 þ 0 ) crl12 ¼ ¼ 0:2; 1
crr 12 ¼
0 þ 0:1 þ 0 ¼ 0:1 1
Similarly, we can get crl14 ¼ 0:4; crr 14 ¼ 0:2; crl15 ¼ 0:5; crr15 ¼ 0:5; crl16 ¼ 0:4; crr16 ¼ 0:6 crl17 ¼ 0:5; crr 17 ¼ 0:7
Next, we consider an acceptable incomplete IFRP R ¼ ðr ij Þ with more known elements (on top of the minimum n 1 values). In this case, by Proposition 6, each missing element rjk1 jk in R ¼ ðrij Þ can be estimated. First, find a sequence of values rij1 ; rj1 j2 ; . . . ; r jt i (i < j1 < j2 < . . . < jt) that include one and only one unknown element r jk1 jk . If jk1 < jk, r jk1 jk is located in the middle of the sequence, and h i this missing element can be estimated as r_ jk1 jk ¼ r_ _þ by Eqs. (23) and (24), where jk1 jk ; r jk1 jk X t þ 1 1 r ij1 þ r j1 j2 þ r jk2 jk1 þ r jk jkþ1 þ r jt1 jt þ r þjt i #H i<j <j :
j2Hh1 ik
2 crlik
¼
8 X j2 > crlik ; if #Hh2 < #H1h2 ik – 0 ik
> :
j2Hh1 ik
;
crr 2ik
¼
0; otherwise
8 X j3 > crlik ; if #Hh3 < #H1h3 ik – 0 ik
> :
j2Hh1 ik
; crr 3ik ¼
0; otherwise
j2Hh1 ik
0; otherwise
8 X j2 1 > crrik ; if #Hh2 < #Hh2 ik – 0 ik
> :
j2Hh1 ik
0; otherwise
8 X j3 1 > crrik ; if #Hh3 < #Hh3 ik – 0 ik
> :
j2Hh1 ik
0; otherwise ð42Þ
1
crlik ¼ (
j¼
2
3
crlik þ crlik þ crlik
j
;
crr ik ¼
h2 h3 #Hh1 ik þ #H ik þ #Hik ;
1;
crr 1ik þ crr 2ik þ crr 3ik
j
h2 h3 if #Hh1 ik þ #Hik þ #H ik – 0
otherwise
crlik ¼ rlij þ rljk 0:5;
j2
j3
crrj1 ik ¼ rr ij þ rr jk 0:5;
ð34Þ
EV ¼ B n MV
ð35Þ
where MV is the set of pairs of alternatives for which the preference degree of the first alternative over the second one is unknown or missing; EV is the set of pairs of alternatives for which the expert provides preference values, and the symbol ‘‘n’’ denotes the exclusion relation. In Alonso et al. (2008)’s method, it does not take into account the preference value of one alternative over itself as xi xi is always assumed.
then crlik ¼ 0 or crrik ¼ 0
else if crlik > 1 or crrik > 1;
References
ð33Þ
ð46Þ
Step 3.
Step 4. End.
B ¼ fði; jÞji; j 2 f1; . . . ; ng ^ i – jg
crr j2 ik ¼ rr jk rr ji þ 0:5;
crr j3 ik ¼ rr ij rr kj þ 0:5
Step 1. Determine the set B, the set MV and the set EV, as follows:
MV ¼ fði; jÞ 2 Bjr ij is unknowng
ð45Þ
and
if crlik < 0 or crr ik < 0; ð32Þ
ð44Þ
crlik ¼ rljk rlji þ 0:5;
crlik ¼ rlij rlkj þ 0:5
Alonso et al. (2008) proposed the following procedure to estimate missing values for IFPRs. An IFPR R ¼ ðr ij Þ can be viewed as two ‘‘independent’’ fuzzy preference relations, the first one RL corresponding to the left bounds of the intervals and the second one RR corresponding to the right bounds of the intervals, respectively,
ð43Þ
where the symbol ‘‘#’’ denotes the number of elements in a given set, j1
R ¼ ðr ij Þ ¼ ð½rlij ; rrij Þ with RL ¼ ðrlij Þ and RR ¼ ðrr ij Þ and rlij 6 rrij 8i; j
ik
> :
ð41Þ
3
Appendix A
; crr 1ik ¼
0; otherwise
8 X j1 1 > crrik ; if #Hh1 < #Hh1 ik – 0
ð40Þ
crlik ¼
The authors are very grateful to the Associate Editor and the two anonymous reviewers for their constructive comments and suggestions that have further helped to improve the quality and presentation of this paper. Yejun Xu would like to acknowledge the financial support of a Grant (No. 71101043) from National Natural Science of China. Kevin W. Li would like to acknowledge the financial support of a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant, and Grants (Nos. 71272129 and 71271188) from National Natural Science Foundation of China. Huimin Wang would like to acknowledge the financial support of a Grant (No. 12&ZD214) from Major Program of the National Social Science Foundation of China.
8 X j1 > crlik ; if #Hh1 < #H1h1 ik – 0
then crlik ¼ 1 or crr ik ¼ 1
ð47Þ ð48Þ
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