Some methods to deal with unacceptable incomplete 2-tuple fuzzy linguistic preference relations in group decision making

Knowledge-Based Systems 56 (2014) 179–190

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Some methods to deal with unacceptable incomplete 2-tuple fuzzy linguistic preference relations in group decision making Yejun Xu a,b,⇑, Feng Ma a,b, Feifei Tao b, Huimin Wang a,b a b

State Key Laboratory of Hydrology–Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China Research Institute of Management Science, Business School, Hohai University, Nanjing 211100, PR China

a r t i c l e

i n f o

Article history: Received 22 June 2013 Received in revised form 23 October 2013 Accepted 9 November 2013 Available online 21 November 2013 Keywords: Incomplete 2-tuple FLPR Unacceptable FLPR Consistency Group decision making Additive transitivity

a b s t r a c t In this paper, we first propose a four-way procedure to estimate missing preference values when dealing with acceptable incomplete 2-tuple fuzzy linguistic preference relations (FLPRs). The proposed revised procedure can estimate more missing elements in the first iteration and also has more advantages than the existing methods. Then, we address situations in which an expert do not provide any comparison information for a particular alternative with other alternatives, and thus provides an unacceptable incomplete 2-tuple FLPR. We propose some methods to deal with the unacceptable situations. The main idea of these methods is to set some seed values for the unknown alternative and then to complete the unacceptable incomplete 2-tuple FLPRs by the revised procedure. We also give a simple algorithm to select the best alternative for the group decision making (GDM) problem with incomplete 2-tuple FLPRs. The analysis of advantages and drawbacks for each method are also presented. Crown Copyright  2013 Published by Elsevier B.V. All rights reserved.

1. Introduction Group decision making (GDM) is participatory process in which multiple individuals, often experts, together formulate problems, develop alternatives and eventually select among the alternatives to reach a decision. Each decision maker (DM) may have unique motivations or goals and may approach the decision process from a different angle, but they have a common interest in reaching eventual agreement on selecting the best option(s) for the problem to be solved [43]. Preference relations are popular and powerful tools used by DMs to provide their preference information on alternatives in GDM. According to the nature of the information expressed for each pair of alternatives, there exist many different formats of preference relations: fuzzy preference relations [15,34, 35,54], fuzzy linguistic preference relations (FLPRs) [16,18,19,46, 55,57,64], multiplicative preference relations [8,20,26,52], intuitionistic preference relations [63] and interval-valued preference relations [29,65]. FLPRs appear when it could be very difficult for the DMs to provide precise numerical preferences, and they can only provide qualitative descriptions for the comparison about every pair of alternatives. In [68], Zadeh described an imprecise logical system, Fuzzy Logic, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, and not very true and not very false. ⇑ Corresponding author at: State Key Laboratory of Hydrology–Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China. Tel.: +86 25 68514612. E-mail address: [email protected] (Y. Xu).

Tong and Bonissone [42] presented a technique for fuzzy decision making that is based on linguistic approximation and truth qualification, which generated a linguistic assessment of the decision. Kacprzyk [30] employed a fuzzy majority rule specified by a fuzzy linguistic quantifier, e.g., ‘most’, ‘much more than 50%’, etc. FLPRs [10,16,18,19,22–25,58,60,64] provide very good results to deal with the problems which presented qualitative aspects. Since each DM has his/her own experience concerning the problem being studied they could have some difficulties in giving all their preferences. This may be due to an expert not possessing a precise or sufficient level of knowledge of the problem, or because that expert is unable to discriminate the degree to which some options are better than others. In such situations, experts are only able to provide incomplete FLPRs with some of their values missing or unknown. Over the past decades, incomplete fuzzy preference relations [11,13,31,47–51,53,56,61,67] and incomplete FLPRs [1,5–7,45,59, 60,62] have received great attention. Xu [59] proposed an approach to GDM based on incomplete FLPRs, he utilized the extended arithmetic averaging (EAA) operator and the extended weighted arithmetic averaging (EWAA) operator to deal with the incomplete FLPRs. Xu [66] presented the linguistic geometric averaging (LGA) operator and the linguistic weighted geometric averaging (LWGA) to fill up the incomplete multiplicative linguistic preference relations. Xu [62] developed a simple and practical method for constructing a consistent complete FLPR by using the additive transitivity property [40]. Alonso et al. [1], Cabrerizo et al. [6] proposed a procedure to estimate missing preference values when dealing with incomplete FLPRs assessed using a 2-tuple fuzzy linguistic approach. The procedure is guided by the linguistic additive

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Y. Xu et al. / Knowledge-Based Systems 56 (2014) 179–190

consistency property and only used the preference values provided by the experts. Wang and Chen [44] proposed a novel method that used fuzzy linguistic assessment variables instead of crisp values of incomplete fuzzy preference relations to ensure comparison consistency. Cabrerizo et al. [5] presented a consensus model to help experts in all phases of the consensus reaching process in GDM problems in an unbalanced fuzzy linguistic context with incomplete information. The main novelty of the consensus model is that it supports the management of incomplete unbalanced fuzzy linguistic information and it allows to achieve consistent solutions with a great level of agreement. Chang et al. [7] applied an analytic hierarchical prediction model based on the GDM with incomplete FLPRs to help the organizations become aware of the essential factors affecting the Enterprise Resource Planning (ERP). This approach not only improves the efficiency of pairwise comparisons compared with the traditional AHP, but also avoids the checking of the consistency of FLPRs when the DMs undertake the pairwise comparison processes. Xia et al. [45] proposed a new characterization about multiplicative consistency of the FLPR, and presented an algorithm to estimate missing values from an incomplete FLPR. They also established a decision support system for aiding the experts to complete their FLPRs in a more consistent way. However, in all the above researches, the incomplete FLPRs which DMs provide are assumed to be acceptable, i.e, all the missing values of the incomplete FLPRs can be estimated by the known values. When dealing with GDM problems, the experts who may have not enough knowledge and give us unacceptable FLPRs, that is, he/she provides information in which at least one alternative is not compared with any one of the rest of alternatives. The above procedures would not be applicable in a successful way to deal with this unacceptable situation. How to fill up the unacceptable FLPRs is a common and very important issue. Up to now, there is no researcher who considers the unacceptable situations in the GDM for the fuzzy linguistic information. Therefore, it is necessary to pay attention to this issue. In this paper, we will present a revised estimation procedure to estimate missing information for incomplete FLPRs, which is slightly different from the existing methods. It is based on the linguistic extension of Tanino’s consistency principle. We assume FLPRs assessed on a 2-tuple fuzzy linguistic modeling [27] because it provides some advantages with respect to the ordinal fuzzy linguistic modeling [1,3,4,6,28,36]. We will introduce some methods to estimate missing information in the unacceptable situations. To do this, the rest of this paper is set out as follows. Section 2 presents some basic concepts necessary throughout the paper, that is, the definition of incomplete 2-tuple FLPR, unacceptable 2-tuple FLPR and linguistic additive consistency property. In Section 3, we present a general consistency based revised procedure to estimate the missing preference values in an incomplete 2-tuple FLPR. In Section 4, some methods are provided to estimate missing information for the unacceptable 2-tuple FLPRs. Furthermore, we propose a procedure to select the best alternative for GDM problems with unacceptable FLPRs. Finally, in Section 5, discussions on the advantages and drawbacks of each method are given and conclusions are furnished. 2. Preliminaries In this section, we introduce the definitions of incomplete 2-tuple FLPR, unacceptable 2-tuple FLPR and linguistic additive consistency property. 2.1. Incomplete and unacceptable 2-tuple FLPRs Experts may use many different representation formats to express their opinions. There may be situations where it could be

very difficult for the experts to provide precise numerical preferences, that is it cannot be assessed in a quantitative form, but in a qualitative one, and therefore linguistic assessments could be used instead [18,21,32,64]. As aforementioned, we use the 2-tuple fuzzy linguistic model [27] to represent experts’ preference relations. The main advantage of this representation is to be continuous in its domain, therefore it can express any counting of information in the universe of the discourse. The 2-tuple fuzzy linguistic model represents the linguistic information by means of a pair of values called linguistic 2-tuple, (s, a), where s is a linguistic term and a is a numeric value representing the symbolic translation. Definition 1 [27]. Let S = {s0, s1, . . . , sg1, sg} be a linguistic term set and b 2 [0, g] be the result of an aggregation of the indexes of set S, i.e., being g + 1 the cardinality of S, then the 2-tuple that expresses the equivalent information to b is obtained with the following function:

D : ½0; g ! S  ½0:5; 0:5Þ  si ; i ¼ roundðbÞ DðbÞ ¼ ðsi ; aÞ; with a ¼ b  i; a 2 ½0:5; 0:5Þ where ‘‘round’’ is the usual round operation, si has the closest index label to ‘‘b’’, and ‘‘a’’ is the value of the symbol translation. Generally, g + 1 is an odd number. Definition 2 [27]. Let S = {s0, s1, . . . , sg1, sg} be a linguistic term set and (si, a) be a 2-tuple, there exists a function, D1, such that given a 2-tuple it returns its equivalent numerical value b 2 ½0; g  R.

D1 : S  ½0:5; 0:5Þ ! ½0; g; D1 ðsi ; aÞ ¼ i þ a ¼ b: It is obvious that the conversion of a linguistic term into a linguistic 2-tuple consist of adding to it the value 0 as a symbolic translation, si 2 S  (si, 0). Definition 3 [1]. A 2-tuple FLPR Ph (the h-th expert’s preference relation P) on a set of alternatives X = {x1, . . . , xn} is a set of 2-tuple on the product set X  X, i.e., which is characterized by a 2-tuple linguistic membership function

lph : X  X ! S  ½0:5; 0:5Þ when the cardinality of X is small, the preference relation   may be conveniently represented by a n  n matrix Ph ¼ phik , being phik ¼ lPh ðxi ; xk Þ; 8i; k 2 f1; . . . ; ng and phik 2 ðS  ½0:5; 0:5ÞÞ. In the models of solving GDM problems, we also assume that experts are always able to provide all the preferences required. However, this situation is not always possible to achieve. And there will be missing information appeared. It may be due to experts’ lack of knowledge about part of the problems, or simply because they may not be able to quantify the degree preference of one alternative over another. It must be clear then that when an expert eh is not able to express the particular value pik, because he/she does not have a clear idea of how better alternative xi is over alternative xk, this does not mean that he/she prefers both options with the same intensity. There also will be a situation that some experts do not provide any information about a particular alternative xj. In order to model these situations, in the following we introduce the definitions of the incomplete 2-tuple FLPR and unacceptable 2-tuple FLPR.

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Definition 4 [1]. A function f: X ? Y is partial when not every element in the set X necessarily maps to an element in the set Y. When every element from the set X maps to one element in the set Y then we have a total function. h

Definition 5. A 2-tuple FLPR P on a set of alternatives X with a partial 2-tuple linguistic membership function is a partial incomplete 2-tuple FLPR. If the missing elements of Ph can be determined by the known elements, then Ph is called an acceptable incomplete FLPR, otherwise, Ph is not an acceptable incomplete FLPR. The necessary condition of acceptable incomplete FLPR Ph is that there exists at least one known element in each row or column of Ph except for the diagonal elements (pii, i = 1, 2, . . . , n), i.e., there needs at least (n  1) judgments. Therefore, an unacceptable FLPR is that there exists at least one row and its corresponding column elements are unknown, i.e., there exists at least one alternative that the expert does not provide any comparison information for the alternative to the others. Now given an incomplete FLPR Ph, the following sets are defined:

A ¼ fði; jÞji; j 2 f1; . . . ; ngg  n o  MV h ¼ ði; jÞ 2 Aphij is unknown  n o  KV h ¼ A n MV h ¼ ði; jÞ 2 Aphij is known KV hi ¼ fða; bÞjða; bÞ 2 KV h ^ ða ¼ i _ b ¼ iÞ:g   9i; hKV hi – /

D½D1 ðpik Þ  D1 ðsg=2 ; 0Þ;

8i; j; k 2 f1; . . . ; ng

ð2Þ

Linguistic additive transitivity implies linguistic additive reciprocity. Indeed, because pii = (sg/2, 0), "i, if k = i in Eq. (2), then we have: D (D1(pij) + D1(pji) = (sg, 0), " i, j 2 {1, . . . , n}, it also can be rewritten as:

pik ¼ DðD1 ðpij Þ þ D1 ðpjk Þ  D1 ðsg=2 ; 0ÞÞ;

8i; j; k 2 f1; . . . ; ng ð3Þ

In this paper, additive consistency is the only considered property for 2-tuple FLPR, and also pii = (sg/2, 0). Although Alonso et al. [1] assumed that pii is always equal to (sg/2, 0) and denoted as ‘‘–’’, pii is not considered as a known value in their estimation procedure. This difference is depicted in Example 1. 3. A revised estimation procedure of missing values for incomplete 2-tuple FLPRs Because experts are not always able to provide preference degrees between each pair of possible alternatives, missing information will appear. Therefore, it is necessary to estimate the missing values before the application of a selection model. In this section we use an iterative complete procedure to estimate the missing values for an incomplete 2-tuple FLPR, which is based on the linguistic additive consistency property. 3.1. Estimating linguistic values based on the linguistic additive consistency

where MVh is the set of pairs of alternatives for which the preference degrees are unknown or missing, KVh is the set of pairs of alternatives for which preference degrees are given by the expert eh, and KV hi is the set of pairs of alternatives involving alternative xi for which expert eh provides preference values. Definition 6. Let E = {e1, . . . , em} be a group of experts, and each expert gives his/her preferences on a set of alternatives X = {x1, . . . , xn} by means of an incomplete FLPR Ph. An unacceptable incomplete 2-tuple FLPR in GDM is defined that when at least one of the experts, eh, does not provide any preference values involving one alternative xi 2 X:

  9i; hKV hi ¼ /

xi is called the unknown or ignored alternative for eh. Obviously, a 2-tuple FLPR is complete when its membership function is totally defined. Clearly, when incomplete or unacceptable incomplete 2-tuple FLPRs are provided, we should develop some methods to deal with these situations.

Eq. (3) could be used to estimate missing values pik. However, three other possible ways to estimate missing values can be derived from Eq. (2), in fact, the preference value pik (i – k) can be estimated using an intermediate alternative xj in four different ways: 1. Since pik = D(D1(pij) + D1(pjk)  D1(sg/2, 0)), we can estimate pik by

ðcpik Þj1 ¼ DðD1 ðpij Þ þ D1 ðpjk Þ  D1 ðsg=2 ; 0ÞÞ 2. Since pjk = D(D pik by

1

(pji) + D

1

(pik)  D

(sg/2, 0)), we can estimate

ðcpik Þj2 ¼ DðD1 ðpjk Þ  D1 ðpji Þ þ D1 ðsg=2 ; 0ÞÞ 3. Since pij = D(D pik by

1

(pik) + D

1

ð4Þ

1

(pkj)  D

1

ð5Þ

(sg/2,0)), we can estimate

ðcpik Þj3 ¼ DðD1 ðpij Þ  D1 ðpkj Þ þ D1 ðsg=2 ; 0ÞÞ

ð6Þ

4. Since D (D1(pik) + D1(pkj)  D1(sg/2, 0)) + D (D1(pji)  D1 (sg/2, 0)) = pii = sg/2, we have

2.2. Linguistic additive consistency Consistency is usually characterized by transitivity. There are several possible characterizations for the transitivity property [15]. In this paper, we adopt the additive transitivity property, which for fuzzy preference relations can be seen as the parallel concept of Saaty’s consistency property for multiplicative preference relations. The mathematical formulation of the additive transitivity was given by Tanino [40]:

ðpij  0:5Þ þ ðpjk  0:5Þ ¼ ðpik  0:5Þ;

D½ðD1 ðpij Þ  D1 ðsg=2 ; 0ÞÞ þ ðD1 ðpjk Þ  D1 ðsg=2 ; 0ÞÞ ¼

8i; j; k 2 f1; . . . ; ng

ð1Þ

DðD1 ðpik Þ þ D1 ðpkj ÞÞ ¼ Dð3D1 ðsg=2 Þ  D1 ðpji ÞÞ

ð7Þ

Hence we can estimate pik by

ðcpik Þj4 ¼ Dð3D1 ðsg=2 ; 0Þ  D1 ðpji Þ  D1 ðpkj ÞÞ

ð8Þ

The overall estimated value cpik of pik is obtained by all possible (cpik)j1, (cpik)j2, (cpik)j3 and (cpik)j4 values.

or equivalently,

pik ¼ pij þ pjk  0:5;

8i; j; k 2 f1; . . . ; ng

When the preference values are expressed by fuzzy linguistic, we can define the linguistic additive transitivity property for 2tuple FLPR by using the transformation functions D and D1 as follows [1]:

3.2. A revised procedure to estimate missing values for incomplete 2-tuple FLPRs Based on Eqs. (4)–(6), Alonso el al..[1] proposed a procedure to estimate missing preference values for incomplete 2-tuple FLPRs. In the following, we present a revised procedure to estimate

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missing values for incomplete 2 FLPRs based on Eqs. (4)–(6) and (8), it is showed that the revised procedure has some advantages compared with Alonso el al. [1]’s. (1) Estimate the missing values in each iteration of the procedure Given an incomplete 2-tuple FLPR Ph, we define the sets h1 h3 h4 Hik ; Hh2 ik ; H ik and H ik , respectively, which are used to estimate the missing preference value pik. Then the subset of missing values MVh that can be estimated in step t of our procedure is denoted by EMV ht (estimated missing values) and defined as follows:

(

EMV ht ¼

ði; kÞ 2 MV h n

t1 [ l¼0

  t  t  EMV l i – k ^ 9j 2 Hh1 [ Hh2 ik ik

 t  t  h4 [ H [ Hh3 ik ik

f ðyÞ ¼

8 > ðs0 ; 0Þ if D1 ðyÞ < 0 > > < > > > :

ðsg ; 0Þ

if D1 ðyÞ > g

y

otherwise

Then, the complete iterative estimation procedure pseudo-code is as follows:

ð9Þ

with

 t Hh1 ik  t Hh2 ik  t Hh3 ik  t Hh4 ik

When we use the function to compute the final estimated value of missing value phik , we should point out that some estimated values might lie the interval  outside    [0, g], i.e., for some (i, k) we may have D1 cphik < 0 or D1 cphik > g. In order to normalize the expression domains in the decision model, the following function is employed:

¼ fjjði; jÞ; ðj; kÞ 2 KV and ði; kÞ 2 MVg

ð10Þ

¼ fjjðj; iÞ; ðj; kÞ 2 KV and ði; kÞ 2 MVg

ð11Þ

¼ fjjði; jÞ; ðk; jÞ 2 KV and ði; kÞ 2 MVg

ð12Þ

¼ fjjðj; iÞ; ðk; jÞ 2 KV and ði; kÞ 2 MVg

ð13Þ

0. EMV h0 ¼ / 1. t = 1   2. while EMV ht – / { 3. for ev eryði; kÞ 2 EMV ht { 4. estimate_p (h,i,k,t) 5. } 6. t++ 7.}



t  t  t  t and EMV h0 ¼ / (by definition); Hh1 ; Hh2 ; Hh3 ; Hh4 are the ik ik ik ik sets of intermediate alternatives xj that can be used to estimate the preference value phik ði – kÞ. When EMV hmaxIter ¼ / with maxIter > 0, the procedure will stop as there will not be any more missing

In step t, we can estimate the missing preference value pik by

0 00

1 0 1 11 , ,     X X     j1 j2 C B C B BB CC D1 cphik A # Hh1 D1 cphik A # Hh2 þ@ B B@ CC ik ik B B j2 Hh1 CC h2 j2ðHik Þ ð ik Þ B1 B CC CC B 1 0 1 cpik ¼ DB Bj B 0 CC , , B B CC     X X  h j3 C  h j4 C B B B B h3 h4 CC 1 1 @ @ þ@ D cpik A # Hik þ @ D cpik A # Hik AA j2ðHh3 j2ðHh4 ik Þ ik Þ values to be estimated. Furthermore, if [maxIter EMV hl ¼ MV h , then all l¼0 missing values are estimated, and consequently, the procedure is said to be successful in the completion of the incomplete 2-tuple FLPR. (2) Estimate a particular value phik in the step t In iteration t, to estimate a particular value phik with ði; kÞ 2 EMV ht , the following function estimate p(h, i, k, t) is established:

ð14Þ

Remark 1. In [1,6], the authors only used the Eqs. (4)–(6) to estimate the missing values in an incomplete 2-tuple FLPR. Actually, in Eq. (4), if D1(pij) = g  D1(pji), we have Eq. (5), if D1(pkj) = g  D1(pjk), we have Eq. (6), if D1(pij) = g  D1(pji) and D1(pkj) = g  D1(pjk) simultaneously, we have Eq. (8). Therefore, Eq. (8) is also a way which can be used to estimate the missing values. Furthermore, we have the following result. Theorem 1. If an incomplete 2-tuple FLPR can be completed by Eqs. (4)–(6) and (8), then D1(pik) + D1(pki) = g.

function estimate_p (h,i,k,t) 1. j = 0 8

 t

 t  h j1 P < 1 tD D cpik –0 =# Hh1 ; j þ þ if # Hh1 h1 ik ik j2ðHh1 2. cpik ¼ ik Þ : otherwise ðs0 ; 0Þ 8

 t

 t  h j2 P < 1 tD D cpik –0 =# Hh2 ; j þ þ if # Hh2 ik ik j2ðHh2 ¼ Þ 3. cph2 ik ik : otherwise ðs0 ; 0Þ 8

 t

 t  h j3 P < 1 D cpik –0 =# Hh3 ; j þ þ if # Hh3 h3 t D ik ik j2ðHik ¼ Þ 4. cph3 ik : otherwise ðs0 ; 0Þ 8

 t

 t  h j4 P < 1 tD D cpik –0 =# Hh4 ; j þ þ if # Hh4 ik ik j2ðHh4 ¼ 5. cph4 Þ ik ik : otherwise ðs0 ; 0Þ 6. Calculate (      h2     h4  1 1 D j1 D1 cph1 cpik þ D1 cph3 cpik ;j – 0 ik þ D ik þ D cpik ¼ x; j ¼ 0 end function

Proof. If there exists j that satisfies Eq. (4), then it also satisfies Eq. (8), and we can get D1((cpik)j1) + D1((cpki)j4) = g. Similarly, for each j which satisfies Eqs. (5) and (6), there will be D1((cpik)j2) + D1((cpki)j2) = g, D1((cpik)j3) + D1((cpki)j3) = g, D1((cpik)j4) + D1((cpki)j1) = g. D1(pik) + D1(pki) is the average value of all the estimated value by Eqs. (4)–(6) and (8), thus, D1(pik) + D1(pki) = g holds. h Remark 2. Theorem 1 shows that if Eq. (8) is added to estimate the missing values, the reciprocity property holds for the missing values, while Alonso el al. [1]’s method could not, which also can be seen from their examples. In the following, we illustrate an example to show the property of the revised estimate procedure.

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Y. Xu et al. / Knowledge-Based Systems 56 (2014) 179–190

Example 1. Let X = {x1, x2, x3, x4} be a set of four alternatives and S = {s0, s1, s2, s3, s4, s5, s6}, g = 6. Suppose the following incomplete FLPR provided by an expert:

0

1 x x x ðs3 ; 0Þ B x ðs3 ; 0Þ ðs2 ; 0:4Þ ðs4 ; 0:4Þ C B C P¼B C @ ðs4 ; 0:4Þ A x ðs3 ; 0Þ x x ðs3 ; 0Þ x ðs2 ; 0:2Þ

Iteration 1. The set of elements that can be estimated is: EMV1 #3 = {(2, 1), (3, 4), (4, 3)}.  To estimate p21, the procedure is as follows:

   H121 ¼ f3g ) cp121 ¼ D D1 cp31 21         ¼ D D1 D D1 p123 þ D1 p131  D1 ðsg=2 ; 0Þ  1   1  ¼ D D Dð2:4 þ 3:6  3Þ ¼ D D Dð3Þ ¼ ðs3 ; 0Þ

     D1 cp121 þ D1 cp221 þ D1 cp321 1

3þ0þ0 ¼D ¼ ðs3 ; 0Þ 1

!

j ¼ 1 ) cp21 ¼ D

 To estimate p34, the procedure is as follows:



1



cp22 34



¼ f2g ) ¼D D  1   1  2     ¼ D D D D p24  D1 p223 þ D1 ðsg=2 ; 0Þ   ¼ D D1 Dð3:6  2:4 þ 3Þ ¼ DðD1 Dð4:2ÞÞ ¼ ðs4 ; 0:2Þ

     ! D1 cp134 þ D1 cp234 þ D1 cp334 ¼D 1

0 þ 4:2 þ 0 ¼D ¼ ðs4 ; 0:2Þ 1

 To estimate p43, the procedure is as follows:

   H143 ¼ f2g ) cp143 ¼ D D1 cp21 43         ¼ D D1 D D1 p142 þ D1 p123  D1 ðsg=2 ; 0Þ ¼ DðD1 Dð1:8 þ 2:4  3ÞÞ ¼ DðD1 Dð1:2ÞÞ ¼ ðs1 ; 0:2Þ H243



 ¼ f2g ) cp243 ¼ D D cp22 43         ¼ D D1 D D1 p223  D1 p224 þ D1 ðsg=2 ; 0Þ

 1

¼ DðD1 Dð2:4  3:6 þ 3ÞÞ ¼ DðD1 Dð1:8ÞÞ ¼ ðs2 ; 0:2Þ H343 ¼ / ) cp343 ¼ ðs0 ; 0Þ

j ¼ 2 ) cp43 ¼ D

D

 1



cp143 þ D

 1

cp243 2

x



ðs2 ; 0:4Þ

x



x

1

C ðs4 ; 0:4Þ C C C ðs4 ; 0:2Þ C A

ðs2 ; 0:2Þ ðs2 ; 0:5Þ



Case 2. If we use Eqs. (4)–(6) and (8) to estimate the missing elements proposed in this paper where i – j – k. The estimation procedure is as follows: Iteration 1. The set of elements that can be estimated is: EMV1 #4 = {(1, 2), (2, 1), (3, 4), (4, 3)}.  To estimate p12, the procedure is as follows:

H112 ¼ H212 ¼ H312 ¼ / ) cp112 ¼ cp212 ¼ cp312 ¼ ðs0 ; 0Þ    H412 ¼ f3g ) cp412 ¼ D D1 cp34 12  1   1  ¼ D D D 3D ðsg=2 ; 0Þ  D1 ðp31 Þ  D1 ðp23 Þ ¼ DðD1 Dð9  3:6  2:4ÞÞ ¼ DðD1 Dð3ÞÞ ¼ ðs3 ; 0Þ        ! D1 cp112 þ D1 cp212 þ D1 cp312 þ D1 cp412 ¼D 1

0þ0þ0þ3 ¼ ðs3 ; 0Þ ¼D 1

 To estimate p21, the procedure is as follows:

  H121 ¼ f3g ) cp121 ¼ DðD1 cp31 21 Þ         ¼ D D1 D D1 p123 þ D1 p131  D1 ðsg=2 ; 0Þ ¼ DðD1 Dð2:4 þ 3:6  3ÞÞ ¼ DðD1 Dð3ÞÞ ¼ ðs3 ; 0Þ H221 ¼ H321 ¼ H421 ¼ / ) cp221 ¼ cp321 ¼ cp421 ¼ ðs0 ; 0Þ

H134 ¼ H334 ¼ / ) cp134 ¼ cp334 ¼ ðs0 ; 0Þ

j ¼ 1 ) cp34

x

j ¼ 1 ) cp12

H221 ¼ H321 ¼ / ) cp221 ¼ cp321 ¼ ðs0 ; 0Þ

cp234

 B B ðs3 ; 0Þ B P#3 ¼ B B ðs4 ; 0:4Þ @ x

Case 1. If we use Eqs. (4)–(6) to estimate the missing elements which was proposed by Alonso el al. [1] where i – j – k. The estimation procedure is as follows:

H234

0



 ! þ D1 cp343



1:2 þ 1:8 þ 0 ¼ ðs2 ; 0:5Þ ¼D 2 After these elements have been estimated, we have

j ¼ 1 ) cp21

       ! D1 cp121 þ D1 cp221 þ D1 cp321 þ D1 cp421 ¼D 1

3þ0þ0þ0 ¼ ðs3 ; 0Þ ¼D 1

 To estimate p34, the procedure is as follows:

   H234 ¼ f2g ) cp234 ¼ D D1 cp22 34         ¼ D D1 D D1 p224  D1 p223 þ D1 ðsg=2 ; 0Þ ¼ DðD1 Dð3:6  2:4 þ 3ÞÞ ¼ DðD1 Dð4:2ÞÞ ¼ ðs4 ; 0:2Þ H134 ¼ H334 ¼ / ) cp134 ¼ cp334 ¼ ðs0 ; 0Þ    H434 ¼ f2g ) cp412 ¼ D D1 cp24 12     ¼ D D1 D 3D1 ðsg=2 ; 0Þ  D1 ðp23 Þ  D1 ðp42 Þ ¼ DðD1 Dð9  2:4  1:8ÞÞ ¼ DðD1 Dð4:8ÞÞ ¼ ðs5 ; 0:2Þ

j ¼ 2 ) cp34

       ! D1 cp134 þ D1 cp234 þ D1 cp334 þ D1 cp434 ¼D 2

0 þ 4:2 þ 0 þ 4:8 ¼ ðs5 ; 0:5Þ ¼D 2

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   H432 ¼ f2g ) cp413 ¼ D D1 cp24 13     ¼ D D1 D 3D1 ðsg=2 ; 0Þ  D1 ðp23 Þ  D1 ðp22 Þ

 To estimate p43, the procedure is as follows:

   H143 ¼ f2g ) cp143 ¼ D D1 cp21 43         ¼ D D1 D D1 p142 þ D1 p123  D1 ðsg=2 ; 0Þ

¼ DðD1 Dð9  2:4  3ÞÞ ¼ DðD1 Dð3:6ÞÞ ¼ ðs4 ; 0:4Þ

¼ DðD1 Dð1:8 þ 2:4  3ÞÞ ¼ DðD1 Dð1:2ÞÞ ¼ ðs1 ; 0:2Þ H243



cp243

1



cp22 43

j ¼ 1 ) cp32

       ! D1 cp132 þ D1 cp232 þ D1 cp332 þ D1 cp432 ¼D 1

0 þ 0 þ 0 þ 3:6 ¼ ðs4 ; 0:4Þ ¼D 1



¼ f2g ) ¼D D  1   1  2     ¼ D D D D p23  D1 p224 þ D1 ðsg=2 ; 0Þ ¼ DðD1 Dð2:4  3:6 þ 3ÞÞ ¼ DðD1 Dð1:8ÞÞ ¼ ðs2 ; 0:2Þ

H343 ¼ H443 ¼ / ) cp343 ¼ cp443 ¼ ðs0 ; 0Þ After these values are estimated, we obtain

j ¼ 2 ) cp43

       ! D cp143 þ D1 cp243 þ D1 cp343 þ D1 cp443 ¼D 2

1:2 þ 1:8 þ 0 þ 0 ¼ ðs2 ; 0:5Þ ¼D 2 1

After these values have been estimated, we obtain

0

 B ðs ; 0Þ 3 B P#4 ¼ B @ ðs4 ; 0:4Þ x

ðs3 ; 0Þ

x



ðs2 ; 0:4Þ

x



x

1

ðs4 ; 0:4Þ C C C ðs5 ; 0:5Þ A

ðs2 ; 0:2Þ ðs2 ; 0:5Þ



Compared with the above results, it is showed that the estimated value p34 in the first iteration by Alonso et al.’s method and our method is different. Our estimated missing values could preserve the reciprocity property of the FLPR (i.e., D(D1(p34) + D1(p43)) = (s6, 0)), which meets Theorem 1. Furthermore, in the first iteration, we could estimate the value p12 by Eq. (8) while Alonso et al.’s method could not. Alonso et al.’s method would estimate the value p12 in the second iteration, which would be lack of accuracy, because the value p12 would be estimated by the estimated value in the first iteration, while our method could estimate p12 by the known value directly provided by the expert. Case 3. In the above cases, both procedures require the condition i – j – k, actually, if any two of i, j, k could be equal (i.e., i = j, i = k or j = k), the estimation procedure is as follows: Iteration 1. The set of elements that can be estimated is: EMV 1 #40 ¼ fð1; 2Þ; ð1; 3Þ; ð2; 1Þ; ð3; 2Þ; ð3; 4Þ; ð4; 3Þg.

0

ðs3 ; 0Þ ðs3 ; 0Þ B ðs ; 0Þ ðs3 ; 0Þ 3 B P#40 ¼ B @ ðs4 ; 0:4Þ ðs4 ; 0:4Þ x

ðs2 ; 0:2Þ

ðs2 ; 0:4Þ ðs2 ; 0:4Þ ðs3 ; 0Þ ðs2 ; 0:5Þ

1

x

ðs4 ; 0:4Þ C C C ðs5 ; 0:5Þ A ðs3 ; 0Þ

Obviously, if any two of i,j,k could be equal, we could estimate more values in the first iteration. As we know, if the missing values are estimated directly by the known values, they are more accurately than they are estimated in the second or later iterations, because in the second or later iterations the missing elements are estimated based on the former estimated values. Our revised procedure can estimate the missing elements more quickly. Therefore, in this paper, we use Eqs. (4)–(6) and (8) to estimate the missing elements and any two of i, j, k could be equal in Eqs. (4)–(6) and (8). This procedure is able to estimate all the missing values for a given incomplete FLPR where each alternative is compared at least once [2]. When we meet these partial incomplete 2-tuple FLPR problems, they can be successfully solved by this procedure. However, this procedure will be unavailable when dealing with unacceptable incomplete 2-tuple FLPRs. In the following section, we present several methods to deal with these situations. 4. Some methods to deal with unacceptable incomplete 2-tuple FLPRs

H113 ¼ H213 ¼ H313 ¼ / ) cp113 ¼ cp213 ¼ cp313 ¼ ðs0 ; 0Þ

As defined in Definition 6, when at least one of the experts eh does not provide any preference value involving a particular alternative, xi, then it is called the ‘‘unknown alternative’’. We cannot get any missing values pik or pki for the special alternative xi by the above estimation procedure, we should investigate other methods to deal with the unacceptable situations. First, we give an example of GDM which includes partial incomplete and unacceptable incomplete 2-tuple FLPRs.

   H413 ¼ f3g ) cp413 ¼ D D1 cp34 13     ¼ D D1 D 3D1 ðsg=2 ; 0Þ  D1 ðp31 Þ  D1 ðp33 Þ

Example 2. Let X = {x1, x2, x3, x4} be a set of four alternatives and S = {N, MW, W, E, B, MB, T} be a set of seven linguistic labels with the following meaning:

 To estimate p13, the procedure is as follows:

¼ DðD1 Dð9  3:6  3ÞÞ ¼ DðD1 Dð2:4ÞÞ ¼ ðs2 ; 0:4Þ

j ¼ 1 ) cp13

       D1 cp113 þ D1 cp213 þ D1 cp313 þ D1 cp413 1

0 þ 0 þ 0 þ 2:4 ¼ ðs2 ; 0:4Þ ¼D 1 ¼D

 To estimate p32, the procedure is as follows:

H132 ¼ H232 ¼ H332 ¼ / ) cp132 ¼ cp232 ¼ cp332 ¼ ðs0 ; 0Þ

N ¼ Null; MW ¼ MuchWorse; W ¼ Worse; E ¼ EquallyPreferred; B ¼ Better; MB ¼ MuchBetter; T ¼ Total:

!

That is, s0 = N, s1 = MW, s2 = W, s3 = E, s4 = B, s5 = MB, s6 = T, g = 6. Let us suppose that three different experts E = {e1, e2, e3} provide the following FLPRs using the linguistic expression domain S:

0

s3 x s5 x

1

0

s3 s2 s4 s5

1

0

s3 s1 x s2

1

Bx s x s C Bs s s s C Bs s x s C 3 1C 4C B B 6 3 4 6C B 5 3 2 3 P1 ¼ B C; P ¼ B C; P ¼ B C @ s2 x s3 s3 A @ s2 s2 s3 s1 A @ x x s3 x A x s4 s3 s3

s0 s1 s5 s3

s4 s3 x s3

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Then, they can be transformed into the corresponding 2-tuple FLPRs as follows respectively: 1

0

0

x ðs5 ; 0Þ x ðs3 ; 0Þ ðs3 ; 0Þ C B B C B B B x B ðs6 ; 0Þ x ðs1 ;0Þ C ðs3 ; 0Þ C B B 2 P1 ¼ B C; P ¼ B C B ðs ; 0Þ B ðs ; 0Þ x ðs ; 0Þ ðs ;0Þ 3 3 C B 2 B 2 A @ @ x ðs4 ; 0Þ ðs3 ; 0Þ ðs3 ;0Þ ðs0 ; 0Þ

0

ðs3 ; 0Þ ðs1 ; 0Þ

x

ðs2 ; 0Þ

ðs2 ; 0Þ ðs4 ; 0Þ ðs5 ; 0Þ

1

C C ðs3 ; 0Þ ðs4 ; 0Þ ðs6 ; 0Þ C C C ðs2 ; 0Þ ðs3 ; 0Þ ðs1 ; 0Þ C C A ðs1 ; 0Þ ðs5 ; 0Þ ðs3 ; 0Þ

1

C B B ðs ; 0Þ ðs ; 0Þ x ðs4 ; 0Þ C 3 C B 5 C P ¼B C B B x x ðs3 ; 0Þ x C A @ 3

ðs4 ; 0Þ ðs3 ; 0Þ

x

EMV 11 ¼ fð1; 4Þ; ð2; 3Þ; ð3; 2Þ; ð4; 1Þg After these elements have been estimated, we have:

ðs3 ; 0Þ

B B x B 1 P ¼B B B ðs2 ; 0Þ @ ðs2 ; 0:5Þ

x ðs3 ; 0Þ

ðs2 ; 0:5Þ

ðs5 ; 0:5Þ

ðs3 ; 0Þ

ðs4 ; 0Þ

ðs3 ; 0Þ

As an example, to estimate

H11 14

ðs5 ; 0Þ

p114 ,

ðs5 ; 0:5Þ

1

C ðs1 ; 0Þ C C C C ðs3 ; 0Þ C A ðs3 ; 0Þ

the procedure is as follows:

  31  ¼ f3g ) ¼ D D1 cp114         ¼ D D1 D D1 p113 þ D1 p134  D1 ðsg=2 ; 0Þ cp11 14

¼ DðD1 Dð5 þ 3  3ÞÞ ¼ DðD1 Dð5ÞÞ ¼ ðs5 ; 0Þ H12 14

  1 32  1 ¼ f3g ) cp12 cp14 14 ¼ D D  1   1  1     ¼ D D D D p34  D1 p131 þ D1 ðsg=2 ; 0Þ ¼ DðD1 Dð3  2 þ 3ÞÞ ¼ DðD1 Dð4ÞÞ ¼ ðs4 ; 0Þ

  1 33  1 13 H13 cp14 14 ¼ f3g ) cp14 ¼ D D  1   1  1     ¼ D D D D p13  D1 p143 þ D1 ðsg=2 ; 0Þ ¼ DðD1 Dð5  3 þ 3ÞÞ ¼ DðD1 Dð5ÞÞ ¼ ðs5 ; 0Þ   1 34  1 14 H14 cp14 14 ¼ f3g ) cp14 ¼ D D  1   1    ¼ D D D 3D ðsg=2 ; 0Þ  D1 p131  D1 ðp143 Þ ¼ DðD1 Dð9  2  3ÞÞ ¼ DðD1 Dð4ÞÞ ¼ ðs4 ; 0Þ cp14

ðs3 ; 0Þ B B ðs0 ; 0:25Þ B P1 ¼ B B ðs2 ; 0Þ @

ðs6 ; 0:25Þ

ðs5 ; 0Þ

ðs3 ; 0Þ

ðs2 ; 0:5Þ

ðs5 ; 0:5Þ

ðs3 ; 0Þ

ðs4 ; 0Þ

ðs3 ; 0Þ

ðs2 ; 0:5Þ

ðs5 ; 0:5Þ

1

C ðs1 ; 0Þ C C C ðs3 ; 0Þ C A ðs3 ; 0Þ

By this way, we can get the complete 2-tuple FLPR. Obviously,   the  estimated missing elements are reciprocal, i.e., D1 cp1ik þ  D1 cp1ki ¼ g. It is easy to complete the incomplete 2-tuple FLPR P1, but when we use the procedure to deal with unacceptable incomplete 2-tuple FLPR P3, it is not enough. In the following subsections, we present some methods to deal with unacceptable situations. 4.1. Ad-Hoc methods to deal with unacceptable situations

ðs3 ; 0Þ

Now, we use the estimation procedure proposed in Section 3 to estimate the missing values. As we observe two 2-tuple FLPRs are incomplete {P1, P3}. P1 is a partial incomplete 2-tuple FLPR, so we can complete it using the consistency based procedure to estimate missing information presented in Section 3.1. Iteration 1. The set of elements of P1 that can be estimated are:

0

0



   13   14  1  1  11  1 1 ¼D D cp14 þ D1 cp12 D cp D cp þ þ 14 14 14 4

5þ4þ5þ4 ¼ ðs5 ; 0:5Þ: ¼D 4

Iteration 2. The set of elements that can be estimated are:

EMV 12 ¼ fð1; 2Þ; ð2; 1Þg After these elements have been estimated, we have the following complete 2-tuple FLPR:

Method 1. Assume indifference values in the missing values As the expert does not provide any comparison information on the alternative xi to the rest ones, we may set each missing value for the ignored alternative with sg/2, which means that the expert thinks the alternative is indifference to the others. That is: Estimation Procedure 1. If an incomplete 2-tuple FLPR Ph has an ignored alternative xi, this method will set all its associated missing values as:

D1 ðpik Þ ¼ g=2;

D1 ðpki Þ ¼ g=2 8k 2 f1; . . . ; ng;

k–i

In Example 2, the expert e3 gives us the following 2-tuple FLPR

0

ðs3 ; 0Þ ðs1 ; 0Þ

x

ðs2 ; 0Þ

1

B ðs ; 0Þ ðs ; 0Þ x ðs4 ; 0Þ C 3 C B 5 P3 ¼ B C @ x x ðs3 ; 0Þ x A ðs4 ; 0Þ ðs3 ; 0Þ

x

ðs3 ; 0Þ

The expert e3 gives no information about alternative x3, so we are in unacceptable situation. Estimation Procedure 1 assumes that D1(p3k) = D1(pk3) = g/2, and all the missing elements will be replaced by (s3, 0), i.e.,

0

ðs3 ; 0Þ ðs1 ; 0Þ ðs3 ; 0Þ ðs2 ; 0Þ

1

B ðs ; 0Þ ðs ; 0Þ ðs ; 0Þ ðs ; 0Þ C 3 3 4 C B 5 P3 ¼ B C @ ðs3 ; 0Þ ðs3 ; 0Þ ðs3 ; 0Þ ðs3 ; 0Þ A ðs4 ; 0Þ ðs3 ; 0Þ ðs3 ; 0Þ ðs3 ; 0Þ Method 2. Assume random values in the missing values In this case, the method estimates the missing values for an ignored alternative as random values, and an unknown preference value will be computed randomly between the maximum and minimum preference degrees of its corresponding column and row, the estimation procedure is as follows: Estimation Procedure 2. If an 2-tuple FLPR Ph has an ignored alternative xi, this method will compute all the missing values as:

D1 ðpik Þ ¼ g  randðminfD1 ðpjk Þ=gg; max DfD1 ðpjk Þ=ggÞ; D1 ðpki Þ ¼ g  randðminfD1 ðpkj Þ=gg; maxfD1 ðpkj Þ=ggÞ8j; k 2 ð1; . . . ; nÞ: where rand(a, b) means a random value between a and b, max(. . .) and min(. . .) are the usual maximum and minimum operators. We can use the Estimation Procedure 2 to reconstruct the missing values with random values between the maximum and minimum preference degrees provided by the expert e3. For example,

D1 ðp13 Þ 2 g  randðminfD1 ðp1k Þ=gg; max DfD1 ðp1k Þ=ggÞ; that is,

D1 ðp13 Þ 2 6  randð1=6; 1=3Þ

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An example of a possible reconstructed preference relation is:

0

ðs1 ; 0Þ ðs3 ; 0Þ B B ðs ; 0Þ ðs3 ; 0Þ 5 B P3 ¼ B B B ðs4 ; 0:19Þ ðs2 ; 0:33Þ @ ðs4 ; 0Þ

ðs3 ; 0Þ

ðs1 ; 0:43Þ ðs5 ; 0:3Þ ðs3 ; 0Þ ðs4 ; 0:26Þ

ðs2 ; 0Þ

1

C ðs4 ; 0Þ C C C C ðs3 ; 0:1Þ C A ðs3 ; 0Þ

4.2. Individual methods to deal with unacceptable situations As aforementioned, we need at least one preference value for each alternative to estimate the rest missing values. However, when there is not any comparison information for one alternative, in order to estimate the missing values, we must initial at least one value for the ignored alternative, and we refer to the initial preference values as the seed values. The presented method in this part consists of two different phases: (1) We should define the particular seed values which will be used. (2) We will utilize the estimation procedure proposed in Section 3 to estimate the missing values with the seed values. Depending on the computation of that seed values we propose the following two consistency based methods. Method 3. Consistency individual method with the indifference seed values In this case, we can assume indifference on the preference values for the ignored alternative. And the estimation procedure of missing values is as follows: Estimation Procedure 3. Suppose an incomplete FLPR with an ignored alternative xi (pij = x and pji = x for all j 2 {1, 2, . . ., n}), and assume D1 ðpij Þ ¼ D1 ðpji Þ ¼ 2g for a particular j. With this assumption we apply the consistency-based estimation procedure to obtain a final estimated value for every missing value pik via Eqs. (4)–(6) and (8):

ðcpik Þj1 ¼ Dðg=2 þ D1 ðpjk Þ  D1 ðsg=2 ; 0ÞÞ ) ðcpik Þj1 ¼ DðD1 ðpjk ÞÞ ðcpik Þj2 ¼ DðD1 ðpjk Þ  g=2 þ D1 ðsg=2 ; 0ÞÞ ) ðcpik Þj2 ¼ DðD1 ðpjk ÞÞ j3

1

1

ðcpik Þ ¼ Dðg=2  D ðpkj Þ þ D ðsg=2 ; 0ÞÞ ) ðcpik Þ

j3

¼ Dðg  D1 ðpkj ÞÞ

Now let us apply this method to P3. Let j = 1, that is, D ðp31 Þ ¼ D1 ðp13 Þ ¼ 2g , we can obtain the following estimated values for p23 and p34: 1

cp23

!

g D1 ðp21 Þ D1 ðp12 Þ g 5 1 ¼ ðs5 ; 0Þ ¼D þ  þ  ¼D 2 2 2 2 2 2

and

cp34

!

g D1 ðp42 Þ D1 ðp24 Þ g 3 4 ¼ ðs3 ; 0:5Þ ¼D þ  þ  ¼D 2 2 2 2 2 2

Then the estimated complete 2-tuple linguistic preference relation is:

0

ðs3 ; 0Þ ðs1 ; 0Þ

ðs3 ; 0Þ

B ðs ; 0Þ ðs ; 0Þ 3 B 5 P3 ¼ B @ ðs3 ; 0Þ ðs1 ; 0Þ

ðs5 ; 0Þ ðs3 ; 0Þ

ðs4 ; 0Þ ðs3 ; 0Þ ðs4 ; 0:5Þ

ðs2 ; 0Þ

1

ðs4 ; 0Þ C C C: ðs3 ; 0:5Þ A ðs3 ; 0Þ

Method 4. Consistency individual method with random seed values This method is similar to the second method, it is based on just one seed value which is random and then we can estimate the rest missing values for the ignored alternative. In this case the estimation procedure of missing values is as follows: Estimation Procedure 4. Suppose an incomplete 2-tuple FLPR Ph with an ignored alternative xi. The estimation procedure is drawn in the following scheme: 1. do{ 2. k = irand(1,n)//Choose random k 3.}while(k – i) 4. if(rand(0,1) < 0.5){//Placeitin missing row 5. pik ¼ Dðg  ðrandðminðfD1 ðpjk Þ=ggÞ; maxðfD1 ðpjk Þ=ggÞÞÞÞ "j 2 {1, . . . ,n},j – i 6.}else{//Placeitin missing column 7. pki ¼ Dðg  ðrandðminðfD1 ðpkj Þ=ggÞ; maxðfD1 ðpkj Þ=ggÞÞÞÞ "j 2 {1, . . . ,n},j – i 8.} 9. Apply the estimation procedure

ðcpik Þj4 ¼ Dð3D1 ðsg=2 ; 0Þ  g=2  D1 ðpkj ÞÞ ) ðcpik Þj4 ¼ Dðg  D1 ðpkj ÞÞ: It is obvious that any preference value can be assumed to be indifference for any j 2 {1, . . . ,n},j – i. Then the final estimated value of the missing element pik is:

       1 D1 ðcpik Þj1 þ D1 ðcpik Þj2 þ D1 ðcpik Þj3 þ D1 ðcpik Þj4 A cpik ¼ D@ 4 ! D1 ðpjk Þ þ D1 ðpjk Þ þ ðg  D1 ðpkj ÞÞ þ ðg  D1 ðpkj ÞÞ ¼D 4 ! 1 1 g D ðpjk Þ D ðpkj Þ  ¼D þ 2 2 2 0

The parallel application of the above assumption for the preference value pki provides the following estimation of the values: 1 1 g D ðpkj Þ D ðpjk Þ cpki ¼ D þ  2 2 2

!

where irand(a, b) results to an integer random value between a and b. Now we apply this method to solve P3 and estimate the missing values. First of all, we obtain a random value k – i. For example k = 1, we will obtain a rand value between [0, 1] to determine a seed value for p31 or p13. Suppose that rand(0, 1) = 0.68, we will obtain a random value for p13 2 D(6  (rand(1/6, 2/6))), for example, p13 = (s2, 0.4). Then, we apply the estimation procedure proposed in Section 3 and have: 0

1 0 1 ðs3 ; 0Þ ðs1 ; 0Þ ðs2 ; 0:4Þ ðs2 ; 0Þ ðs3 ; 0Þ ðs1 ; 0Þ ðs2 ; 0:4Þ ðs2 ; 0Þ B ðs ; 0Þ ðs ; 0Þ C B ðs ; 0Þ C x ðs ; 0Þ ðs ; 0Þ ðs ; 0:4Þ ðs ; 0Þ 3 4 5 3 4 4 B 5 C B C P3 ¼ B C!B C @ x x ðs3 ; 0Þ x A @ ðs4 ; 0:4Þ ðs2 ; 0:4Þ ðs3 ; 0Þ ðs3 ; 0:4Þ A ðs4 ; 0Þ ðs3 ; 0Þ ðs4 ; 0Þ x ðs3 ; 0Þ ðs3 ; 0Þ ðs3 ; 0:4Þ ðs3 ; 0Þ

4.3. Social methods to deal with unacceptable situations The above proposed methods are simple and could be used to solve unacceptable situations, these method do not take any other experts’ information into account. Now we will introduce two

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methods which are in consideration of some social criteria. The first method uses only the consensus preference values provided by those experts nearest to the expert whose preference relation we try to complete, while the second method makes use of the aggregated information provided by the set of experts, that is, using consensus preference values of the collective preference relation, which is computed by aggregating all the rest experts’ individual preference relations [12,14,39,41,62]. As both methods use consensus preference values, we refer to them as consensusguided social methods. Method 5. Consensus social method based on an expert proximity seed values This approach computes the seed values from the preference values provided by those experts nearest to the expert whose preference relation we try to complete. In this case, if an unacceptable incomplete 2-tuple FLPR Ph given by expert eh has an unknown alternative xi, and Pv is an complete FLPR give by ev (if Pv is an incomplete and acceptable FLPR, we can apply the aforesaid revised procedure to complete it), we apply the following scheme:

Both methods can be used to calculate the distances, here we adopt the 2-tuple linguistic normalized Hamming distance. For the above estimated FLPRs P1,P2 and P3, we have n X

dNH ðP1 ; P3 Þ ¼

j¼1;k¼1;ðj;kÞ2KV h

For every expert ev 2 E,ev – eh { Compute dv = dist(ev,eh) } NE = {evjmin{dv}} n o vh vh 5. NV ¼ minj–i dj ¼ dl ; v 2 fv jev 2 NEg  v v 6. The seed values ¼ pil ; pli 7. Apply the estimation procedure

where dist(ev, eh) is a distance function that is used to find the vh distance between ev and eh. dj are the distances between the preferences on each alternative xj of ev and eh. After we get the complete P1 and P2, we compute the distances between the complete FLPRs and unacceptable FLPR. Drawing on the well-known Hamming distance and the Euclidean distance, we propose 2-tuple linguistic normalized Hamming distance as follows:

dNH ðP v ; Ph Þ ¼

1 #phjk

 n      X  1 v  pjk =g  D1 phjk =g  D

ð15Þ

j ¼ 1; k ¼ 1 ðj; kÞ 2 KV h

dNH ðP2 ; P3 Þ ¼ 0:22 Because dNH(P1,P3) > dNH(P2, P3), then NE = {e2}, we can get seed values from P2, then we compute the distance for each alternative between the experts e2 and e3. For example, Pn jD1 ðp21k Þ=gD1 ðp31k Þ=gj 23 k¼1;k–3 d1 ¼ ¼ 0:22. The minimum distance is 3 23

d2 ¼ 0:11, so the seed values are p323 ¼ ðs4 ; 0Þ and p332 ¼ ðs2 ; 0Þ, then we apply the estimation procedure and have: 0

1 0 x ðs2 ;0Þ ðs3 ;0Þ ðs1 ;0Þ ðs3 ;0Þ ðs1 ;0Þ B ðs ;0Þ ðs ;0Þ ðs ; 0Þ ðs ;0Þ C B ðs ;0Þ ðs ;0Þ 5 3 4 4 5 3 B C B P3 ¼ B C!B @ x x A @ ðs4 ;0Þ ðs2 ;0Þ ðs2 ;0Þ ðs3 ; 0Þ x

B 1 dNE ðPv ; Ph Þ ¼ B @#ph

ðs3 ;0Þ

ðs2 ;0Þ ðs4 ;0Þ ðs3 ;0Þ

1 ðs2 ; 0Þ ðs4 ; 0Þ C C C ðs3 ;0:5Þ A

ðs4 ;0Þ ðs3 ;0Þ ðs4 ;0:5Þ

ðs3 ; 0Þ

Method 6. Consensus social method based on collective seed values The method is based on the use of seed values chosen among the consensus preference values of a collective FLPR. We should aggregate all the individual preference relations which are given by the rest experts, that is, when aggregating the preference relations, the unacceptable FLPRs are not involved. 0 0 Firstly, we define the collective preference relation Pc . Pc is the collective FLPR without aggregating the unacceptable FLPR Ph. Then we complete the incomplete Ph which with an ignored alternative xi. We apply the following scheme, let a pair of the symmetrical values are still missing, other original missing elements will be replaced by the corresponding position values in 0 0 0 0 0 Pc as seed values, that is, phik ¼ pcik ; phki ¼ pcki ; k – 1 or k – 2 or k – 4, then we can get the missing values phik and phki ; k ¼ 1; 2; 4. 0 In order to get the collective preference relation Pc , the 2-tuple weighted average (WA) operator [27] is an simple and effective method. It is as follows:

pcik ¼ D

m X   D1 plik wl

!

l¼1

We suppose all the experts are equally important, i.e., wl ¼ m1 , and 0 we can get PcWA with the complete 2-tuple FLPRs’ average values. For example,

and 2-tuple linguistic normalized Euclidean distance:

0

¼ 0:31

10

ðs4 ;0Þ ðs3 ;0Þ

1. 2. 3. 4.

       1 1  pjk =g  D1 p3jk =g  D

11=2  n     2 C X  1 v  pjk =g  D1 phjk =g  C D A

ð16Þ

jk j ¼ 1; k ¼ 1

0 pc13

!   D1 p113 þ D1 ðp213 Þ ¼ Dð4:5Þ ¼ ðs5 ; 0:5Þ ¼D 2

ðj; kÞ 2 KV h 0

where Pv and Ph are FLPRs provided by the experts ev and eh ; #phjk is the number of known values in Ph. We can further extend Eqs. (15) and (16) into generalized 2-tuple linguistic normalized distance:

0

n      k C X  1 v  pjk =g  D1 phjk =g  C D A

ðs3 ; 0Þ

B ðs ; 0:13Þ 0 B 3 PcWA ¼ B @ ðs2 ; 0Þ ðs1 ; 0:25Þ

11=k

0 B 1 dNG ðP v ; Ph Þ ¼ B @#ph jk

and the collective preference relation P cWA is

ð17Þ

where k > 0. Especially, if k = 1, then the generalized 2-tuple linguistic normalized distance is reduced to the 2-tuple linguistic normalized Hamming distance; if k = 2, then it is reduced to the 2-tuple linguistic normalized Euclidean distance.

ðs5 ; 0:5Þ

ðs5 ; 0:25Þ

1

ðs3 ; 0Þ ðs3 ; 0:25Þ

ðs3 ; 0:25Þ ðs3 ; 0Þ

ðs4 ; 0:5Þ ðs2 ; 0Þ

C C C A

ðs3 ; 0:5Þ

ðs4 ; 0Þ

ðs3 ; 0Þ

If we utilize the 2-tuple ordered weighted average (OWA) operator [27]:

j ¼ 1; k ¼ 1 ðj; kÞ 2 KV h

ðs4 ; 0:12Þ

pcik

m X ¼D wl bl

!

l¼1

where bl is the l-th largest value of the D1 ðplik Þ values. Here we assume w1 = 0.6 and w2 = 0.4 for simplicity. The collective preference 0 relation P cOWA will be:

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Y. Xu et al. / Knowledge-Based Systems 56 (2014) 179–190

1 ðs4 ; 0:25Þ ðs5 ; 0:4Þ ðs5 ; 0:2Þ ðs3 ; 0Þ B ðs ; 0:3Þ ðs3 ; 0Þ ðs3 ; 0Þ ðs4 ; 0Þ C C B 4 ¼B C: @ ðs2 ; 0Þ ðs4 ; 0:5Þ ðs3 ; 0Þ ðs2 ; 0:2Þ A 0

0 P cOWA

ðs1 ; 0:1Þ ðs3 ; 0:2Þ

ðs4 ; 0:2Þ

ðs3 ; 0Þ

Remark 3. When 2-tuple linguistic WA operator is adopted,      h  h P P P 1 1 D1 pcik þ D1 pcki ¼ m pki wh þ m pik wh ¼ m h¼1 D h¼1 D h¼1       P m D1 phki þ D1 phik wh ¼ h¼1 wh g ¼ g, and the aggregated FLPR Pc is still reciprocal. But if 2-tuple linguistic OWA operator is adopted, the reciprocal property does not always maintained.

Let X = {x1, x2, . . . , xn} be a set of alternatives, E = {e1, e2, . . . , em} be a set of DMs, and w = (w1, w2, . . . , wm)T be the weight vector of DMs, P such that wh P 0; h ¼ 1; 2; . . . ; m; m h¼1 wh ¼ 1. Suppose that the m DMs provide their preference relations over the n decision alterna  tives with FLPRs P h ¼ phik nn ; h ¼ 1; 2; . . . ; m. These FLPRs may be complete, incomplete, unacceptable incomplete or even not reciprocal. For the acceptable incomplete FLPRs, we can complete them by the revised procedure in Section 3. For the unacceptable incomplete FLPRs, we can complete them by any of the above proposed methods. A simple solution for such a GDM problem is provided in Algorithm 1. Algorithm 1.

c0

In the following, PWA is used as the collective FPLR. We will estimate a pair of missing elements and the left missing elements are replaced by the corresponding elements in the collective relation 0 PcWA as the seed values. For example, to estimate the missing values 0 0 0 0 3 p13 and p331 , we set p33k ¼ pc3k ; p3k3 ¼ pck3 , k – 1, then we apply the 3 estimation procedure, we have p13 ¼ ðs2 ; 0:12Þ; p331 ¼ ðs4 ; 0:12Þ, i.e., 0

1 0 1 ðs3 ; 0Þ ðs1 ; 0Þ x ðs2 ; 0Þ ðs1 ;0Þ ðs2 ;0:12Þ ðs2 ;0Þ ðs3 ;0Þ B C B C B C B C ðs3 ;0Þ ðs3 ; 0:25Þ ðs4 ;0Þ C B ðs5 ; 0Þ ðs3 ; 0Þ ðs3 ;0:25Þ ðs4 ; 0Þ C B ðs5 ;0Þ B C B C B C!B C B x B ðs3 ; 0:25Þ ðs3 ; 0Þ ðs2 ; 0Þ C ðs3 ;0Þ ðs2 ;0Þ C B C B ðs4 ; 0:12Þ ðs3 ;0:25Þ C @ A @ A ðs4 ; 0Þ

ðs3 ; 0Þ

ðs4 ; 0Þ

ðs3 ; 0Þ

ðs4 ;0Þ

ðs3 ;0Þ

ðs4 ;0Þ

ðs3 ;0Þ

Similarly, we can estimate other missing elements and finally complete the preference relation P3 as follows:

1

0

ðs1 ; 0Þ ðs2 ; 0:12Þ ðs2 ; 0Þ ðs3 ; 0Þ C B B ðs ; 0Þ ðs3 ; 0Þ ðs5 ; 0:13Þ ðs4 ; 0Þ C 5 C B 3 C P ¼B C B B ðs4 ; 0:12Þ ðs1 ; 0:13Þ ðs3 ; 0Þ ðs3 ; 0:12Þ C A @ ðs4 ; 0Þ

ðs3 ; 0Þ

ðs3 ; 0:12Þ

ðs3 ; 0Þ

Remark 4. Methods 5 and 6 are called consensus social based methods. In the GDM problems, we assume that there are two sets of FLPRs. The first set includes the complete and acceptable incomplete FLPRs, and there has at least one FLPR. The second one includes only the unacceptable incomplete FLPRs. If there are some acceptable incomplete FLPRs in the first set, we could use the revised estimation procedure proposed in this paper to complete them. Thus, the first set can be deemed to include only the complete FLPRs. Then Methods 5 and 6 could be used to deal with the second set which includes only the unacceptable incomplete FLPRs. The main idea of Method 5 is to calculate the distances between an unacceptable FLPR and the complete FLPRs, then to find a FLPR which has the minimum distance to an unacceptable FLPR. Subsequently, the approach determines a pair of seed values from the FLPR. Method 6 is first to aggregate the set of complete FLPRs to a collective FLPR, the seed values are then chosen among the collective FLPR. After the seed values are obtained, both Methods 5 and 6 use the revised estimation procedure to estimate the unknown values. If there are two or more experts do not give any information on a particular alternative, or there is one expert do not give any information on two or more particular alternatives, both the two methods could be used to estimate the known values, as the two methods divide the FLPRs into two sets, these cases can be seen general ones in the unacceptable incomplete FLPRs of the second set.

4.4. Application to GDM with incomplete 2-tuple FLPRs Following the general solution process of GDM, a procedure for GDM with unacceptable incomplete 2-tuple FLPRs can be developed.

Step 1. Complete all the incomplete FLPRs (whether acceptable or unacceptable) by the above revised procedure and methods. Step 2. Aggregation phase: 1. Weighted Average Operator [27]

pcik

m X   ¼D D1 plik wl

! ð18Þ

l¼1

to aggregate all the individual FLPRs to a collective one. 2. Ordered Weighted Average (OWA) Operator

pcik ¼ D

m X wl bl

! ð19Þ

l¼1

where bl is the l-th largest value of the D1 ðplik Þ values. Step 3. Exploitation phase: Utilize the 2-tuple arithmetic mean operator [27]

zi ¼ D

n X 1 k¼1

n

  D1 pcik

! ð20Þ

to get the preference degree zi of the ith alternative over all the other alternatives. Step 4. Rank all the alternatives and select the optimal one(s) in accordance with the values of zi (i = 1, 2, . . ., n). Step 5. End.

Example 3. The GDM problem in Example 2. In Example 2, all the preference values of P2 are known, P1 can be completed by the revised procedure in Section 3, P3 is an unacceptable FLPR, which can be completed by the above 6 methods. In Method 6, P3 is completed by the collective FLPR, this would lead to P3 is more consistent to the other FLPRs. Thus, we adopt the completed P3 by Method 6. In this paper, assume that each expert has the same weight. By Eq. (18), the collective 2-tuple FLPR Pc is

0

ðs3 ; 0Þ

B ðs ; 0:25Þ B 4 Pc ¼ B @ ðs3 ; 0:29Þ

ðs3 ; 0:08Þ ðs4 ; 0:37Þ ðs4 ; 0:17Þ ðs3 ; 0Þ

1

ðs4 ; 0:46Þ ðs4 ; 0:33Þ C C C: ðs3 ; 0Þ ðs2 ; 0:29Þ A

ðs2 ; 0:46Þ ðs2 ; 0:17Þ ðs3 ; 0:33Þ ðs4 ; 0:29Þ

ðs3 ; 0Þ

Afterwards, we utilize Eq. (20) to get the overall preference degree zi of the ith alternative over all the other alternatives:

z1 ¼ Dð3:345Þ ¼ ðs3 ; 0:345Þ; z3 ¼ Dð2:615Þ ¼ ðs3 ; 0:385Þ;

z2 ¼ Dð3:49Þ ¼ ðs3 ; 0:49Þ; z4 ¼ Dð2:8025Þ ¼ ðs3 ; 0:1975Þ:

Rank the alternatives in accordance with the values of zi (i = 1, 2, 3, 4):

x2 x1 x4 x3

Y. Xu et al. / Knowledge-Based Systems 56 (2014) 179–190

Thus, the best alternative is x2.

5. Discussion and conclusion We have presented a revised procedure to estimate missing values based on linguistic additive consistency to deal with GDM problems with incomplete fuzzy linguistic information. The revised procedure is a four-way estimation method while Alonso et al.’s is a three-way method. From the illustrative example, we know that the revised procedure can estimate more missing values in the first iteration in some cases, and thus need less iterations. It also can preserve the reciprocity property for the estimated missing values. Then, we propose some methods to deal with the unacceptable incomplete FLPRs in which there exist at least one alternative without any comparison information. The alternative is called ignorance alternative. The main idea of these methods is to set some seed values for the ignorance alternative, then to complete the unacceptable incomplete 2-tuple FLPRs based on the revised procedure. We also give a simple algorithm to select the best alternative for the group decision making problem with incomplete 2-tuple FLPRs. The proposed methods have the following advantages and drawbacks: Method 1 is a very simple approach to solve unacceptable situations, which only simply set all the unknown values equivalent to indifference. This method is useful when there is no source of information about the problem. Method 2 is also a simple approach, but it can produce a higher level of diversity in the opinions given by the experts. However, if we use this method several times for solving a problem, the solution obtained could be different in each time due to that it uses the random values. It should be pointed out that this method will decrease the consistency of FLPRs, because the random values will not always fulfill the additive consistency property. If there are a high number of experts with a high consistency, this method can be a good one. Method 3 improves the Method 1, because it sets some unknown values equivalent to indifference firstly, then uses the revised procedure to estimate the missing values. This method is particularly useful when there is no additional information about the problem and when a high consistency level is required in the experts’ preference relations. Method 4 tries to unify the advantages of Methods 2 and 3. Our opinion of the missing value is not equivalent to difference, so this method will be more appropriate than Method 3 and it can maintain a higher consistency degree than Method 2. However, the seed values of this method are still random values between the minimum and maximum values from the known values, the results may be inaccurate. Method 5 is more reliable than former methods, because the seed value comes from the nearest expert’s lowest distance. And the nearest expert’s judgments may be able to better represent the missing information, so it can produce more accurate collection result. This method also can obtain a high consistency level. However, in some situations, it would lead to unreliable result as it is based on just the information of the nearest expert rather than the information from the whole group. Method 6 is appropriate for GDM problems, but it requires that there is at least one expert provides the acceptable FLPR. It can help to reach a solution of consensus more easily, making the opinions of the experts closer to each other, because the missing values are completed from global information. Thus, this method could be useful in GDM problems where a fast and converging consensus process is needed. In the future, we will address the extension of the management procedure of unacceptable incomplete information to the

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unbalanced FLPRs [5] or dynamic GDM problems [36,37] and its application to model users’ preferences or desires in different problems like internet business [33], technology selection [9], web quality evaluation [17] and university digital libraries [38].

Acknowledgements The authors are very grateful to the Associate Editor and the anonymous reviewers for their constructive comments and suggestions that have helped to improve the quality of this paper. This work was partially supported by the National Natural Science Foundation of China (No. 71101043), and a major project of the National Social Science Foundation of China (No. 12&ZD214).

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