Distribution linguistic preference relations with incomplete symbolic proportions for group decision making

Journal Pre-proof Distribution linguistic preference relations with incomplete symbolic proportions for group decision making Xiaoan Tang, Qiang Zhang, Zhanglin Peng, Witold Pedrycz, Shanlin Yang

PII: DOI: Reference:

S1568-4946(19)30787-2 https://doi.org/10.1016/j.asoc.2019.106005 ASOC 106005

To appear in:

Applied Soft Computing Journal

Received date : 2 January 2019 Revised date : 18 November 2019 Accepted date : 4 December 2019 Please cite this article as: X. Tang, Q. Zhang, Z. Peng et al., Distribution linguistic preference relations with incomplete symbolic proportions for group decision making, Applied Soft Computing Journal (2019), doi: https://doi.org/10.1016/j.asoc.2019.106005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Elsevier B.V. All rights reserved.

Journal Pre-proof

Distribution L inguistic Preference Relations With Incomplete Symbolic Proportions for G roup Decision M aking Xiaoan Tanga,b,c, Qiang Zhanga,b*, Zhanglin Penga,b*, Witold Pedryczc,d, Shanlin Yanga,b a

School of Management, Hefei University of Technology, Hefei, Box 270, Hefei 230009, Anhui, P.R. China;

b

Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education, Hefei, Box 270, Hefei 230009, Anhui, P.R. China; Department of Electrical & Computer Engineering, University of Alberta, Edmonton T6R 2V4 AB Canada; d

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Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

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A bstract Distribution linguistic preference relations (DLPRs) with complete symbolic proportions have been recently investigated to record the comparison information coming from decision makers (DMs) in the context of linguistic decisions. Due to various reasons such as a lack of experience and partial knowledge about the pairs of decision alternatives, it is not always easy for DMs to provide complete symbolic proportions in DLPRs. In this paper, we propose a new style of pairwise comparison called DLPR with incomplete symbolic proportions to represent DMs¶ comparison information. Two aggregation operators for DLPRs with incomplete symbolic proportions and their desirable properties are presented. An expectation-based numerical preference relation (EBNPR) is deduced from a DLPR with incomplete symbolic proportions using numerical scale models. The consistency of DLPR with incomplete symbolic proportions is defined via its associated EBNPR. On the other hand, solving linguistic decision problems implies the need for invoking the principles of computing with words (CW). The key point about CW is that words might exhibit different meaning for different people. Hence, another aim of this paper is to deal with the point about CW by setting personalized numerical scales of linguistic terms for different DMs in group decision making (GDM) with the newly introduced preference relations. Several numerical scale computation models are developed to personalize numerical scales for each DM to show their individual difference in understanding the meaning of words. Finally, we present the applications of the aforesaid theoretical results to GDM situations, which are demonstrated by solving a GDM problem of evaluating and selecting research projects.

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Corresponding author. Tel: +86 0551 62901501; fax: +86 0551 62905263. E-m ail addresses: [email protected] (X. Tang), [email protected] (Q. Zhang), [email protected] (Z. Peng). Keywords: Distribution linguistic preference relation (DLPR); Numerical scale model; Consistency; Numerical scale computation model; Group decision making (GDM) 1. Introduction

Decision-making is a common activity existing all the times in daily life [1, 2]. Due to the complexity of decision making problems and the limited knowledge of decision makers (DMs), many decision making problems are usually conducted through group decision-making (GDM) processes [3-6]. When addressing GDM problems, the DMs usually need to express their judgements over several alternatives, and make the optimal choices [7-9]. The judgements of DMs

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are generally expressed through two forms of information: direct evaluation values of alternatives on multiple criteria, and pairwise comparisons of alternatives. Based on the first form of judgement information, lots of effective techniques have been reported to deal with GDM under uncertainty, such as fuzzy GDM approaches for coping with GDM problems with fuzzy evaluation information [10-16], and linguistic GDM approaches for handling GDM problems with linguistic evaluation information [1, 17-23]. These GDM techniques require DMs to give their evaluation values of alternatives directly. When encountering complex decision problems, it may not be easy for the DMs to do so owing to their lack or incomplete of knowledge and the ambiguity inherent to human thinking. On the other hand, other effective techniques have been developed to address GDM problems with pairwise comparisons of alternatives [24-26]. In this paper, we continue to focus research on GDM with pairwise comparisons of alternatives. In what follows, we first recall existing expressions and then introduce a new way of pairwise comparison called distribution linguistic preference relation (DLPR) with incomplete symbolic proportions. Furthermore, we discuss the applications of the newly introduced preference relations to GDM scenarios keeping the fact that words mean different things for different people. In pairwise comparison decision making, preference relations form an efficient tool for DMs to depict their judgements over alternatives. The preference relations roughly fall into two categories: (i) numerical preference relations, (ii) and linguistic term-based preference relations. Over the past decades, numerous researchers have investigated decision models with different numerical preference relations, including fuzzy preference relations [24], multiplicative preference relations [27, 28], interval-valued fuzzy preference relations [26, 29], intuitionistic fuzzy preference relations [30, 31], interval-valued intuitionistic fuzzy preference relations [32], and hesitant fuzzy preference relations [33, 34]. The DMs use precise numbers to describe their preference information in such decision models. However, the considered problems may present qualitative aspects that are complex to evaluate using exact and precise numbers [35, 36]. That is to say, in some cases, it is impossible for the DMs to get numbers or numerical entries but only the qualitative ones over the decision alternatives [37]. In such a scenario, linguistic variable [38] shows its suitability and superiority in describing the '0V¶ preference information. Linguistic variables whose values are not numbers but words, intuitively relate to the cognitive processes of human beings. Therefore, decision making with linguistic term-based preference relations greatly attracts researchers¶ attention and has become an active field of decision analysis. Representative linguistic term-based preference relations include linguistic preference relations [39, 40], hesitant fuzzy linguistic preference relations [41, 42], probabilistic linguistic preference relations [37], and DLPRs [36]. The hesitant fuzzy linguistic preference relations use several successive linguistic terms instead of using single linguistic terms as linguistic preference relations do to depict DMs¶ linguistic information in decision-making problems. Further, by considering the importance degrees and the different proportions among the possible linguistic terms in hesitant fuzzy linguistic preference relations, the remaining two forms of preference relations come into being. If the probabilistic or proportional information is complete, both the remaining two techniques are indeed mathematically consistent [1]. DLPRs not only can depict each individual outcomes of pairwise comparisons but also can elicit the outcomes of a group using linguistic distribution assessments in which the symbolic proportions are exact numbers and the proportional information is complete. To cope with the situation where DMs cannot provide exact symbolic proportions in linguistic distribution assessments, Dong et al. [18] introduced the concept of

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linguistic distribution assessments with interval symbolic proportions. In this paper, we consider another situation where the proportional information in a linguistic distribution assessment is incomplete. Due to various reasons such as a lack of experience and partial knowledge about the pair of alternatives under consideration, it is not always easy for DMs to provide a complete proportion distribution over the given linguistic term set. In other words, the total summation of symbolic proportions provided by the DMs may be less than 1. In such a situation, using a new form of preference relation which is called DLPR with incomplete symbolic proportions in this paper to record the comparison information of a pair of alternatives therefore becomes much easier and more feasible. When using fuzzy linguistic approaches to deal with decision problems, computing with words (CW) [18, 43-45] is required to capture linguistic information within a formal mathematical framework. Its importance in linguistic decision making has been highlighted in the recent work [46]. For this reason, many feasible linguistic computational models have been developed for supporting computations at the level of linguistic terms, such as 2-tuple linguistic representation model [47, 48], linguistic hierarchy model [19, 49], and proportional 2-tuple linguistic model [50, 51]. Subsequently, a new version, namely, the numerical scale model [52-54] has been proposed by integrating and extending the existing models [19, 48-51]. The numerical scale model can yield the models in [49-51] by setting different numerical scales. One can find the state-of-the-art numerical scale-based techniques for managing CW in [55, 56]. A key point about CW is that words might exhibit different meaning for different people [17, 45]. DMs may have a varying understanding of a given linguistic term in linguistic GDM. Customizing specific semantics of the linguistic terms for each DM becomes a critical task in such linguistic GDM. Up to now, several personalized individual semantics models [53, 57, 58] have been proposed to address this task and successfully applied to support linguistic GDM with preference relations. But a few studies consider the same situation when recording comparison information by means of DLPRs. Especially, using DLPRs with incomplete symbolic proportions to express preferences is originally introduced in this paper. Another objective of this paper is to deal with the point about CW by setting personalized numerical scales of linguistic terms for different DMs in GDM with the newly introduced preference relations. In summary, the study exhibits several facets of originality. First, the use of DLPRs with incomplete symbolic proportions to represent the comparison information coming from DMs in GDM forms an original contribution of this study. Another aspect of novelty is the presentation of some numerical scale computation models to derive personalized numerical scales of linguistic terms from the newly introduced preference relations, and the investigation of their applications to GDM situations. Furthermore, we demonstrate how the aforesaid theoretical results work in practice by solving a GDM problem of evaluating and selecting research projects. The organization of the rest of this paper reads as follows. Section 2 reviews the basic knowledge that will be used in the following discussion. Section 3 introduces the concept of DLPR with incomplete symbolic proportions. Section 4 develops some numerical scale computation models to personalize numerical scales for linguistic term sets in the newly introduced preference relations. Section 5 depicts applications of the preference relations and the numerical scale computation models to GDM situations, which are demonstrated by solving a GDM problem of evaluating and selecting research projects in Section 6. Finally, concluding remarks are included in Section 7.

Journal Pre-proof 2. Basic knowledge In this section, we briefly introduce the basic concepts of DLPRs and numerical scales, which will be used in the discussion that follows.

2.1. DLPR m atrix

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Linguistic decision making [47, 59, 60] models the preferences of DMs by means of linguistic variables [38]. Suppose that L = {l t|t = -T«-« T} is a linguistic label set with odd cardinality (2T + 1), where the label l t denotes a possible value of a linguistic variable and l0 VWDQGV IRU DQ HYDOXDWLRQ RI ³LQGLIIHUHQFH´ The set of linguistic labels often should satisfy the following conditions:

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(1) The set is ordered: lD ! lE if and only if D ! E. (2) There is a negation operator: neg(lD) = l-D. In order to address linguistic decision-making problems with pairwise comparisons of alternatives, linguistic preference relations [39, 40] and hesitant fuzzy linguistic preference relations [41, 42] are widely used to describe DMs¶RXWFomes of pairwise comparisons based on the linguistic label set L. Furthermore, similar to the two commonly accepted forms of preference relations, Zhang et al. [36] originally developed the concept of DLPR as follows. Definition 1. [36] Given an alternative set X = {x1, x2, ..., xn} and a linguistic term set L = {l t|t =

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-T«-« T}, a DLPR matrix on X comes as D  X u X, D = (dij)nun with a linguistic distribution assessment dij = {(l t, pij(l t)), t = -T  « -   « T} that is called a linguistic distribution preference of L, indicating the preference degrees of the alternative xi over xj and satisfying dii = {(l0, 1)} and neg(dij) = {(l t, pij(l-t)), t = - T«-« T} = dji = {(l t, pji(l t)), t = -T«-« T}, namely pij(l t) = pji(l-t) for all i , j  {0, 1, ..., n}, where pij(l t) denotes the symbolic proportion associated with the linguistic term l t in the relation between xi and xj”pij(l t) ”DQG ¦ Tt T pij (lt ) = 1.

Definition 2. [36] Given a collection of E DLPR matrices { D 1, ..., D e, ..., D E } on X, where D e = (d )nun with deij = {(l t, peij (l t)), t = -T « -   « T}, as defined before, and an associated

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e ij

weighting vector W = {w1, ..., we, ..., w E } VXFKWKDW”we ”IRU e = 1, 2, ..., E and ¦ eE 1 we = 1. The collective DLPR matrix obtained by the weighted averaging operator comes as

D c = (dcij )nun with dcij = {(l t, pcij (l t)), t = -T«-«T},

(1)

where pcij (l t) = ¦ eE 1 we ˜ pije (lt ) .

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The basic properties of the above operator and an ordered weighted averaging operator are introduced in Zhang et al.[36].

2.2. Numerical scale model To quantify linguistic terms used in linguistic decision making, Dong et al. [52] introduced the concept of numerical scale. With the help of such a concept, several techniques for managing CW in decision making have been proposed [55, 56]. More specifically, based on numerical scales, Li et al. [55] studied the problem of personalizing individual semantics with comparative linguistic expressions in hesitant linguistic GDM. Li et al. [56] discussed the use of the personalized individual semantics in large scale GDM problems.

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Definition 3. [52] Given a linguistic term set L = { l t|t = -T«-« T } and a real number set R, the monotonically increasing function N S: L ĺ R is defined as a numerical scale of L, and NS(l t) is called the numerical index of l t. Via choosing a sound scale function, one can connect linguistic terms to real numbers and thereby build the relationship between linguistic pairwise comparisons and numerical pairwise comparisons. Definition 4. [61] Given a linguistic term set L = {lt|t = -T « -   « T} and a monotonically increasing function N S: L ĺ[0, 1] as defined before. The function NS is called an additive scale function if N S(l t) + N S(l-t) = 1, where N S(l t • The linear scale presented in Xu [62] is a commonly encountered additive numerical scale and the corresponding scale function comes as follows. NS(l t) = 0.5 + t/2T, where t = -T«-«T. (2) One can obtain an associated fuzzy preference relation [63] by identifying certain additive scale function to quantify a linguistic preference relation [61, 62]. 3. D L PR matrix with incomplete symbolic proportions

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In this section, we first introduce a new way of pairwise comparison called DLPR with incomplete symbolic proportions. Then, transformations from the newly introduced preference relations into numerical preference relations are discussed. Two aggregation operators for DLPRs with incomplete symbolic proportions as well as their desirable properties are also presented here.

3.1. Concept of a DLPR m atrix with incomplete symbolic proportions

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DLPRs not only can depict each individual outcomes of pairwise comparisons but also elicits the outcomes of a group using distribution assessments of a given linguistic term set L = {l t|t = -T « -   « T} [36, 64]. It is obvious from Definition 1 that symbolic proportions associated to the possible linguistic terms in a distribution assessment are exact numbers and the proportional information is complete, namely, the total summation of such symbolic proportions is equal to 1. To cope with the situation where DMs cannot provide exact symbolic proportions in linguistic distribution assessments, Dong et al. [18] introduced the concept of linguistic distribution assessments with interval symbolic proportions. As analyzed in the introduction section, we herein consider a different situation where the proportional information in a linguistic distribution assessment is incomplete, and we come up with the concept of DLPR with incomplete symbolic proportions to represent the pairwise comparisons of alternatives. Definition 5. Given an alternative set X = {x1, x2, ..., xn} and a linguistic term set L = {l t|t = -T«

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-   « T}, a DLPR matrix on X comes as D  X u X, D = (dij)nun with a linguistic distribution assessment dij = {(l t, pij( l t)), (L, pij(L )), t = -T « -   « T} that is called a linguistic distribution preference of L, indicating the preference degrees of the alternative xi over xj and satisfying dii = {(l0, 1)} and neg(dij) = {(l t, pij(l-t)), ( L, pij(L)), t = -T«-« T} = dji = {(l t, pji(l t)), (L, pji(L )), t = -T«-« T }, namely pij(l t) = pji(l -t) for all i, j  {0, 1, ..., n} and pij(L) = pji(L), where pij(l t) denotes the symbolic proportion associated with the linguistic term l t in the relation between xi and xj VXFKWKDW”pij(l t ” pij(L) denotes the symbolic proportion of global ignorance (uncertainty) in the relation between xi and xj such that  ” pij( L  ”  and T

¦t

T

pij (lt ) + pij(L ) = 1. If

T

¦t

T

pij (lt ) = 1, namely pij(L ) = 0 for all i, j  {0, 1, ..., n}, we call D

Journal Pre-proof a DLPR matrix with complete symbolic proportions of L, and dij a complete linguistic distribution preference of L. Otherwise, we call D a DLPR matrix with incomplete symbolic proportions of L, and dij an incomplete linguistic distribution preference of L. E xample 1. A simple DLPR matrix with incomplete symbolic proportions is provided on four alternatives based on a nine linguistic term set L = {l t|t = -4«-«4}.

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D=

{(l0 ,1)} {(l3 ,0.3),(l2 ,0.6),(L ,0.1)} {(l1 ,0.3),(l 2 ,0.5),(L ,0.2)} {(l1 ,0.4),(l 2 ,0.4),(L ,0.2)} · § ¨ ¸ {(l0 ,1)} {(l2 ,0.3),(l3 ,0.4),(l4 ,0.2),(L,0.1)}{(l2 ,0.2),(l3 ,0.7),(L ,0.1)} ¸ ¨ {(l2 ,0.6),(l3 ,0.3),(L ,0.1)} . ¨ {(l2 ,0.5),(l1 ,0.3),(L ,0.2)}{(l4 ,0.2),(l3 ,0.4),(l2 ,0.3),(L ,0.1)} {(l0 ,1)} {(l2 ,0.4),(l1 ,0.4),(L ,0.2)} ¸ ¨¨ ¸¸ {(l3 ,0.7),(l2 ,0.2),(L ,0.1)} {(l1 ,0.4),(l2 ,0.4),(L ,0.2)} {(l0 ,1)} © {(l2 ,0.4),(l1 ,0.4),(L ,0.2)} ¹

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Note 1. From the above definition, one can find that the linguistic distribution preference in Zhang et al. [36] is a complete linguistic distribution preference. The corresponding preference relation matrix in Zhang et al. [36] is a DLPR matrix with complete symbolic proportions, which is a special case of the DLPR matrix with incomplete symbolic proportions in this paper. Note 2. Apart from the representation [36], there is another similar way of pairwise comparison, namely, the probabilistic linguistic preference relation [37]. The similarity between the probabilistic linguistic preference relation and the proposal presented in this paper is that both use distribution assessments to express preferences. What makes them different will be discussed in the later sections.

3.2. Expectation-based numerical preference relation matrix transformed from a DLPR m atrix with incomplete symbolic proportions

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To facilitate comparison of different linguistic distribution assessments, Zhang et al. [36] defined the computational model of expectation for linguistic distribution assessments. Definition 6. [36] Given an alternative set X = {x1, x2, ..., xn}, a linguistic term set L = {l t|t = -T«-« T}, and a complete linguistic distribution preference of L, namely dij = {(l t,

pij(l t)), t = -T«-« T}, ZKHUH” pij(l t ”DQG ¦ Tt T pij (lt ) = 1, as defined before.

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The expectation of dij comes as

E (dij) = ¦ pij (lt ) ˜ NS (lt ) t T

(3)

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where N S(l t) is the numerical index of l t. For two complete linguistic distribution preferences d1ij and d2ij , if E (d1ij ) < E (d2ij ), then d1ij is smaller than d2ij , and vice versa; if E (d1ij ) = E (d2ij ), then the expectation of d1ij is identical to that of d2ij , namely d1ij = d2ij . In this research, we adhere with this definition. For an incomplete linguistic distribution preference dij, however, pij(L ) > 0 and pij(L) could be assigned to any subset of the given linguistic term set L, thus E (dij) cannot be simply represented by an exact value. It can be inferred from Definition 3 that two extreme situations will happen to E (dij). Assigning pij(L) to pij(l -T) will yield the maximum value of E (dij) and assigning pij(L ) to pij(l T) will generate its minimum value. All other situations are positioned in-between the above two extreme cases. Consequently, E (dij) can be expressed as an interval which is defined in the following manner. Definition 7. Given an alternative set X = {x1, x2, ..., xn}, a linguistic term set L = {l t|t = -T« -« T}, and an incomplete linguistic distribution preference of L, namely dij = {(l t, pij(l t)), (L, pij(L)), t = -T«-«T}, as defined before. The expectation of dij comes as

Journal Pre-proof E (dij) = [ E ij , E ij ],

(4)

T

T

t T

t T

where E ij = ¦ pij (lt ) ˜ NS (l t ) + pij ( L ) ˜ NS (lT ) and E ij = ¦ pij (lt ) ˜ NS (lt ) + pij ( L ) ˜ NS (lT ) .

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For notational simplicity, we denote E (dij) as E ij in this paper. E ij is an interval, so different incomplete linguistic distribution preferences can be compared by using the method that is developed in Dong et al. [18] to compare linguistic distribution assessments with interval symbolic proportions. Based on the expectations of incomplete linguistic distribution preferences, a DLPR matrix with incomplete symbolic proportions on X = {x1, x2, ..., xn} can be transformed into a numerical preference relation matrix such that E  X u X, E = ( E ij)nun with E ij = [ E -ij, E +ij ], herein referred to as expectation-based numerical preference relation (EBNPR) matrix. E xample 2. Based on the linear scale in (2), one can transform the preference relation matrix D provided in Example 1 into the following associated EBNPR matrix by use of (4):

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E

§ [0.5000,0.5000] [0.1875,0.2875] [0.562 5,0.7625] [0.5500,0.7500] · ¨ ¸ [0.7125,0.8125] [0.5000,0.5000] [0.775 0,0.8750] [0.7625,0.8625] ¸ = ¨ . ¨ [0.2375,0.4375] [0.1250,0.2250] [0.500 0,0.5000] [0.2500,0.4500] ¸ ¨¨ ¸¸ © [0.5500,0.7500] [0.1375,0.2375] [0.5500, 0.7500] [0.5000,0.5000] ¹

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As mentioned above, the relationship between linguistic preference relations and fuzzy preference relations can be constructed by identifying certain additive scale function [61, 62]. Similarly, by selecting certain additive scale function to quantify a DLPR matrix with incomplete symbolic proportions, one can obtain an interval fuzzy preference relation [26, 65]. Definition 8. [26, 65] Given an alternative set X = {x1, x2, ..., xn}, an interval fuzzy preference relation matrix on X comes as A  X u X, A = (a ij)nun with a ij = [a-ij, a+ij ] ‘ [0, 1] representing the interval preference degree of the alternative xi over xj and satisfying a-ij + a+ji =1 for all i, j  {0, 1, ..., n}. T heorem 1. Suppose that X = {x1, x2, ..., xn} is a set of alternatives, N S is an additive scale

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function, and E = ( E ij)nun with E ij = [ E -ij, E +ij ] is the EBNPR matrix derived from a DLPR matrix with incomplete symbolic proportions on X via the application of the scale function NS, as defined before. Then, E is an interval fuzzy preference relation. Proof. As per Definitions 4 and 7, one can reason that E ij ‘ [0, 1] and T

T

t T

t T

E -ij + E +ji = ( ¦ pij (lt ) ˜ NS (lt ) + pij ( L ) ˜ NS (lT ) ) + ( ¦ p ji (lt ) ˜ NS (lt ) + p ji ( L ) ˜ NS (lT ) ) T

T

t T

t T

= ( ¦ pij (lt ) ˜ NS (lt ) + pij ( L ) ˜ NS (lT ) ) + ( ¦ pij (lt ) ˜ NS (lt ) + pij ( L) ˜ NS ( lT ) ) T

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= ¦ pij (lt ) ˜  NS (lt )  NS (l t ) + pij ( L ) ˜  NS (lT )  NS (lT ) t T T

= ¦ pij (lt ) + pij(L) = 1 for all i, j  {0, 1, ..., n}. t T

As a consequence, based on Definition 8, one can easily prove that E is an interval fuzzy preference relation. This completes the SURRIRI7KHRUHP     Ƒ Theorem 1 shows that DLPR matrices with incomplete symbolic proportions can be transformed into interval fuzzy preference relations by using additive scale functions. In the same way, one can demonstrate that the EBNPR associated with a DLPR matrix with complete

Journal Pre-proof symbolic proportions is a fuzzy preference relation. This conclusion is obvious, so we omit its demonstration here.

3.3. Aggregation of DLPR matrices with incomplete symbolic proportions Two aggregation operators for DLPR matrices with incomplete symbolic proportions and their desirable properties are presented in this section. Definition 9. Given a collection of E DLPR matrices with incomplete symbolic proportions

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{ D 1, ..., D e, ..., D E } on X, where D e = (deij )nun with deij = {(l t, peij (l t)), (L, peij (L)), t = -T«-1, 0, « T}, as defined before, and an associated weighting vector W = {w1, ..., we, ..., wE } such that ” we ”IRU e = 1, 2, ..., E and ¦ eE 1 we = 1. The collective matrix obtained by the weighted averaging ( DLPRIWA ) operator comes as D c = (dcij )nun with dcij = DLPRIWA (d1ij , ..., deij , ..., dijE ) = {(l t, pcij (l t)), (L, pcij (L)), t = -T«-«T},

(5)

where pcij (l t) = ¦ eE 1 we ˜ pije (lt ) and pcij (L) = ¦ eE 1 we ˜ pije ( L ) .

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Proposition 1. Suppose that { D 1, ..., D e, ..., D E } is a collection of E DLPR matrices with incomplete symbolic proportions. Then, the collective matrix obtained by the DLPRIWA operator is still a DLPR matrix with incomplete symbolic proportions. Proof. Based on Definitions 5 and 9, Proposition 1 can be easily proved. Ƒ One can easily prove that the DLPRIWA operator enjoys the following desirable properties, as concluded in Propositions 2-4. Proposition 2. (Boundedness) The DLPRIWA operator is positioned in-between the min and max operators min(d1ij , ..., deij , ..., dijE ) ” DLPRIWA (d1ij , ..., deij , ..., dijE ) ” max(d1ij , ..., deij , ..., dijE ). Proposition 3. (Idempotency) Suppose that deij = dij for all e = 1, 2, ..., E , then DLPRIWA (d1ij , ..., deij , ..., dijE ) = dij. Proposition 4. (Monotonicity) Suppose that {d1ij , ..., deij , ..., dijE } and {m1ij , ..., meij , ..., mijE } are two collections of incomplete linguistic distribution preferences of L as defined before. If deij ” meij for all e = 1, 2, ..., E , then DLPRIWA (d1ij , ..., deij , ..., dijE ) ” DLPRIWA (m1ij , ..., meij , ..., mijE ). Now, we present another aggregation operator for DLPR matrices with incomplete symbolic proportions. Definition 10. Given { D 1, ..., D e, ..., D E } and {w1, ..., we, ..., w E } as defined before. The collective

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matrix obtained by the ordered weighted averaging ( DLPRIOWA ) operator comes as D c = (dcij )nun with dcij = DLPRIOWA (d1ij , ..., deij , ..., dijE ) = {(l t, pcij (l t)), (L, pcij (L)), t = -T«-«T}, (6) where pcij (l t) = ¦ eE 1 we ˜ pijV ( e ) (lt ) , pcij (L) = ¦ eE 1 we ˜ piVj ( e ) ( L ) and {W(1), ..., W(e), ..., W( E )} is a permutation of {1, ..., e, ..., E } such that dWij (e-1) • dWij (e) for e = 2, 3, ..., E . Proposition 5. Suppose that { D 1, ..., D e, ..., D E } is a collection of E DLPR matrices with incomplete symbolic proportions. Then, the collective matrix obtained by the DLPRIOWA operator is still a DLPR matrix with incomplete symbolic proportions. Proof. Based on Definitions 5 and 9, Proposition 5 can be easily proved. Ƒ Note 3. The DLPRIOWA operator also holds the desirable properties, namely Boundedness,

Journal Pre-proof Idempotency and Monotonicity that are presented in Propositions 2-4. 4. Numerical scale computation for the linguistic term set in D L PR matrix with incomplete symbolic proportions

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As analyzed in the previous sections, connecting linguistic pairwise comparisons to their associated numerical pairwise comparisons implies the need for setting numerical scales for the used linguistic term sets. For example, with the linear scale [62], the relationship between linguistic preference relations and fuzzy preference relations can be constructed. As such, the problem of setting a numerical scale becomes an open research issue [66]. It is obvious that words might exhibit different meaning for different people. DMs may have a varying understanding of a given linguistic term in practical fuzzy linguistic decision   scenarios because of their different background. Even for the same DM, his/her understanding of the linguistic term may change with the decision scenario and decision problem, etc. Using a certain numerical scale to character different DMV¶ preferences or to represent the same DM¶s preferences under various situations therefore is not always reasonable. On the other hand, even if the DMs¶ preferences are the same sometimes, they are not always linear as depicted by the linear scale. This inevitably increases the difficulties of DMs in directly assigning numerical values for the linguistic terms. Given all that, computing specific numerical scales for linguistic term sets becomes necessary in the case of fuzzy linguistic decision making. Therefore, in this section, some models for computing personalized numerical scales are developed based on the consistency of DLPRs with incomplete symbolic proportions. Before the models are presented, we first analyze the consistency of the DLPRs with incomplete symbolic proportions.

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4.1. Consistency definition for DLPRs with incomplete symbolic proportions from their EBNPRs perspective

Jo

urn a

Consistency is an important problem in the context of decision making with preference relations. One can find an overview on managing consistency of such reciprocal preference relations as fuzzy preference relations, multiplicative preference relations, interval-valued fuzzy preference relations and hesitant fuzzy preference relations in the recent work [67]. In a way similar to the preference relations proposed by Bilgic [65] and Dong and Herrera-Viedma [53], the DLPRs with incomplete symbolic proportions provided by DMs are logical and rational. Namely, they satisfy a set of pre-established transitive properties (e.g., additive transitivity, multiplicative transitivity, etc.[24]. However, the DMs may offer inconsistent preference relations due to the complexity of the decision environments, which imply the need of consistency analysis for verifying the logicality of the preference relations. In what follows, the concept of additive consistency [68] is recalled as the basis for the consistency analysis associated to the DLPRs with incomplete symbolic proportions. The multiplicative consistency analysis for this form of preference relations can be deduced in a similar manner. Definition 11. [68] Given is an alternative set X = {x1, x2, ..., xn} and an interval fuzzy preference relation matrix A = (a ij)nun with a ij = [a-ij, a+ij ] on X, as defined before. Then, A is considered to be additive consistent if the additive transitivity, namely a ij + a jk + aki = a kj + a ji + a ik  i, j, k  {1, 2, ..., n} is fulfilled, which is given by the following expression: aij

i=j ­°[0.5,0.5] [ aij , aij ] ®     [0.5  0.5( Z  Z ),0.5  0.5( Z  Z )] i z j °¯ i j i j

(7)

Journal Pre-proof where Z = (Z1, Z2, ..., Zn)T = ([Z-1 , Z+1 ], [Z-2 , Z+2 ], ..., [Z-n , Z+n ])T is a normalized interval weight vector [69] satisfying the set of conditions n

n

j 1 j zi

j 1 j zi

0 ” Z-i ” Z+i ” 1, ¦ Z j  Zi d 1 , Zi  ¦ Z j t 1 i = 1, 2, ..., n.

(8)

Similar to the above consistency definition, this study defines the additive consistency of DLPRs with incomplete symbolic proportions via their associated EBNPRs in the following way. Definition 12. Given an alternative set X = {x1, x2, ..., xn}, a DLPR matrix with incomplete

pro of

symbolic proportions D = (dij)nun with dij = {(l t, pij(l t)), (L, pij(L)), t = -T«-« T} on X, and the EBNPR matrix E = ( E ij)nun with E ij = [ E -ij , E +ij ] derived from D using an additive scale function N S, as defined before. Then, D is considered to be additive consistent if the additive transitivity, namely E ij + E jk + E ki = E kj + E ji + E ik  i, j, k  {1, 2, ..., n} is fulfilled, which is given by the following expression: E ij

i=j ­°[0.5,0.5] [ E ij , E ij ] ®     [0.5  0.5( Z  Z ),0.5  0.5( Z  Z )] i z j °¯ i j i j

(9)

re-

where Z = (Z1, Z2, ..., Zn)T = ([Z-1 , Z+1 ], [Z-2 , Z+2 ], ..., [Z-n , Z+n ])T is a normalized interval weight vector satisfying the condition (8). From Definition 7, (9) can be equivalently expressed as: T ª T º pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (l T ), ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ N S ( lT ) » «t ¦ t T ¬ T ¼

lP

i=j °­[0.5,0.5] . ®     [0.5  0.5( Z  Z ),0.5  0.5( Z  Z )] i z j °¯ i j i j

(10)

T ª T º [ E ij , E ij ] « ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (l T ), ¦ pij (l t ) ˜ N S ( lt )  pij ( L ) ˜ NS ( lT ) » t T ¬ t T ¼

Jo

E ij

urn a

Note 4. Zhang et al. [36] defined the consistency of DLPRs with complete symbolic proportions based on the consistency of the corresponding expectation matrices. The consistency definition in this paper differs from the one in Zhang et al. [36]. In the above definition, the EBNPR matrices are interval fuzzy relations, while in the definition of Zhang et al. [36] the expectation matrices are linguistic preference relations. But as illustrated above, linguistic preference relations can be transformed into fuzzy preference relations (special interval fuzzy relations) with an additive scale function, which uncovers the relationship between the two definitions. Proposition 6. Suppose that the following relationship

i=j °­[0.5,0.5] ®     °¯[0.5  0.5(Zi  Z j ),0.5  0.5(Zi  Z j )] i z j

Holds. Then E ji

T ª T º [ E ji , E ji ] « ¦ p ji (lt ) ˜ NS (lt )  p ji ( L ) ˜ NS (l T ), ¦ p ji (lt ) ˜ N S ( lt )  p ji ( L) ˜ NS ( lT ) » t T ¬ t T ¼

j =i ­°[0.5,0.5] . ®     °¯[0.5  0.5(Z j  Zi ),0.5  0.5(Z j  Zi )] j z i

Proof. One can conclude from Theorem 1 that E -ji = 1 - E +ij and E +ji = 1 - E -ij. Therefore,

(11)

Journal Pre-proof E ji

T ª T º [ E ji , E ji ] « ¦ p ji (lt ) ˜ NS (lt )  p ji ( L ) ˜ NS (l T ), ¦ p ji (lt ) ˜ N S ( lt )  p ji ( L) ˜ NS ( lT ) » t T ¬ t T ¼

ª § T · § T ·º «1  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸ ,1  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L) ˜ NS ( lT ) ¸ » ¹ © t T ¹¼ ¬ © t T

j =i ­ °[0.5,0.5] ®ª     1  0.5  0.5(Zi  Z j ) ,1  0.5  0.5(Zi  Z j ) º¼ j z i ° ¯¬







j =i ­°[0.5,0.5] . ®     °¯[0.5  0.5(Z j  Zi ),0.5  0.5(Z j  Zi )] j z i

pro of



This completes the proof of Proposition 6. Ƒ Proposition 6 implies that when testing the consistency for the preference relation D , one can consider only the preferences in the upper triangular matrix of D instead of examining all the preferences. Hence, based on Theorem 1 and Proposition 6, one can prove that Definition 12 is equivalent to Definition 13. Definition 13. Given an alternative set X = {x1, x2, ..., xn}, a DLPR matrix with incomplete

re-

symbolic proportions D = (dij)nun with dij = {(l t, pij(l t)), (L, pij(L)), t = -T«-« T} on X, and the EBNPR matrix E = ( E ij)nun with E ij = [ E -ij , E +ij ] derived from D using an additive scale function N S, as defined before. Then, D is considered to be additive consistent if the additive transitivity, namely E ij + E jk + E ki = E kj + E ji + E ik  i, j, k  {1, 2, ..., n} is satisfied, which is given by the following expression: T ª T º pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ), ¦ pij ( lt ) ˜ NS ( lt )  pij ( L ) ˜ NS ( lT ) » «t ¦ t T ¬ T ¼

lP

E ij

i=j °­[0.5,0.5] ®     [0.5  0.5( Z  Z ),0.5  0.5( Z  Z )] i  j °¯ i j i j

(12)

E ijc

urn a

where Z = (Z1, Z2, ..., Zn)T = ([Z-1 , Z+1 ], [Z-2 , Z+2 ], ..., [Z-n , Z+n ])T satisfying the condition (8). Proposition 7. Suppose that D e (e = 1, 2, ..., E ) are additive consistent and D c is the collective relation obtained by the DLPRIWA operator. Then, D c is additive consistent, which can be given as follows: T ª T c º pij (lt ) ˜ NS (lt )  pijc ( L ) ˜ NS (lT ), ¦ pijc (lt ) ˜ NS ( lt )  picj ( L ) ˜ NS ( lT ) » «t ¦ t T ¬ T ¼

i =j °­[0.5,0.5] ® c c c c [0.5  0.5( Z  Z ),0.5  0.5( Z  Z )] i z j °¯ i j i j

(13)

Jo

where Zc = (Z1c , Z2c , ..., Znc )T with Zic [Zic  , Zic  ] = [¦ eE 1 we ˜ Zie  , ¦ eE 1 we ˜ Zie  ] (i = 1, 2, ..., n) is a normalized interval weight vector of D c satisfying the condition (8). Proof. Since D e (e = 1, 2, ..., E ) are additive consistent, then one can obtain from Definition 12 that

E eij + E ejk + E eki = E ekj + E eji + E eik  i, j, k  {1, 2, ..., n}. Moreover, dcij = DLPRIWA (d1ij , ..., deij , ..., dijE ) = {(l t, pcij (l t)), (L, pcij (L)), t = -T«-«T}, where pcij (l t) = ¦ eE 1 we ˜ pije (lt ) and pcij (L) = ¦ eE 1 we ˜ pije ( L ) , so

Journal Pre-proof E ijc

T ª T § E º · § E · § E · § E · e e e e « ¦ ¨ ¦ we ˜ pij (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pij ( L) ¸ ˜ NS (l T ), ¦ ¨ ¦ we ˜ pij (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pij ( L) ¸ ˜ NS (lT ) » t T © e 1 ¹ ©e 1 ¹ ¹ ©e 1 ¹ ¬ t T © e 1 ¼

ª

º

= «¦ we ˜ §¨ ¦ pije (lt ) ˜ NS (lt )  pije ( L) ˜ NS (lT ) ·¸, ¦ we ˜ §¨ ¦ pije (lt ) ˜ NS (lt )  pije ( L) ˜ NS (lT ) ·¸ » e 1 t T e 1 t T E

T

E

©

¬

T

¹

©

¹¼

E E = ª« ¦ we ˜ E ije  , ¦ we ˜ E ije  º»

¬e

1

¼

e 1

E

= ¦ we ˜ E ije . Then, we have E

E

E

e 1

e 1

e 1

E cij + E cjk + E cki = ¦ we ˜ E ije + ¦ we ˜ E ejk + ¦ we ˜ E kie E

= ¦ we ˜  E ije  E ejk  E kie e 1 E

= ¦ we ˜  E kje  E eji  E ike e 1

E

E

E

e 1

e 1

pro of

e 1

= ¦ we ˜ E kje + ¦ we ˜ E eji + ¦ we ˜ E ike e 1

c ji

c ik

re-

=E +E +E. Consequently, D c is additive consistent. In addition, D e (e = 1, 2, ..., E ) can be represented by c kj

T ª T e º pij (lt ) ˜ NS (lt )  pije ( L ) ˜ NS (l T ), ¦ pije (lt ) ˜ N S (lt )  pije ( L ) ˜ NS (lT ) » «t ¦ t T ¬ T ¼

lP

i =j °­[0.5,0.5] ® e e e e [0.5  0.5( Z  Z ),0.5  0.5( Z  Z )] i z j °¯ i j i j

where Ze = (Z1e , Z2e , ..., Zne )T = ([Z1e- , Z1e+ ], [Z2e- , Z2e+ ], ..., [Zne- , Zne+ ])T satisfies (8). Thus, we have

urn a

ªE § T e · E § T e ·º e e «¦ we ˜ ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸, ¦ we ˜ ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L) ˜ NS (lT ) ¸ » e 1 t  T e 1 t  T © ¹ © ¹¼ ¬

i=j ­[0.5,0.5] ° E E ®ª º e e e e °« ¦ we ˜ 0.5  0.5(Zi  Z j ) , ¦ we ˜ 0.5  0.5(Zi  Z j ) » i z j e 1 ¼ ¯¬ e 1









i =j ­[0.5,0.5] ° E E E E . ®ª § § e e · e e  ·º ° «0.5  0.5 ¨ ¦ we ˜ Zi  ¦ we ˜ Z j ¸ ,0.5  0.5 ¨ ¦ we ˜ Zi  ¦ we ˜ Z j ¸ » i z j e 1 e 1 ©e 1 ¹ ©e 1 ¹¼ ¯¬

Jo

Suppose that Zic  ¦ eE 1 we ˜ Zie  , Zic  ¦ eE 1 we ˜ Zie  , Z cj  ¦ eE 1 we ˜ Z ej  and Z cj  ¦ eE 1 we ˜ Z ej  , then we can get

i =j ­[0.5,0.5] ° E E E E ®ª § § e e · e e  ·º ° «0.5  0.5 ¨ ¦ we ˜ Zi  ¦ we ˜ Z j ¸ ,0.5  0.5 ¨ ¦ we ˜ Zi  ¦ we ˜ Z j ¸ » i z j e 1 e 1 ©e 1 ¹ ©e 1 ¹¼ ¯¬ i =j ­°[0.5,0.5] . ® c c c c °¯[0.5  0.5(Zi  Z j ),0.5  0.5(Zi  Z j )] i z j

Finally, we can easily verify that Zc = (Z1c , Z2c , ..., Znc )T satisfy the condition (8), which

Journal Pre-proof completes the proof of Proposition 7. Ƒ Proposition 7   demonstrates that the collective DLPR with incomplete symbolic proportions obtained by a DLPRIWA operator is additive consistent if individual preference relations are additive consistent.

4.2. Consistency-based optim ization models to compute numerical scales for linguistic term sets

re-

pro of

In this section, we establish an optimization model to compute numerical scales   of linguistic terms in DLPR matrix with incomplete symbolic proportions through the application of the aforesaid consistency definition. Before the optimization model is presented, we first recall one related study [53] to introduce the basic idea of the model. In [53], Dong and Herrera-Viedma developed a consistency-driven methodology to set personalized interval numerical scales of 2-tuple linguistic term sets in the decision making problems with linguistic preference relations. The consistency-driven methodology is based on two natural premises regarding the consistency of preference relations. Namely, the linguistic preference relations are of acceptable   consistency, and the corresponding numerical preference relations transformed from the linguistic preference relations by the interval numerical scales are also of acceptable consistency. Accordingly, motivated by the study in [53], the consistency-driven optimization-based model in this study is proposed to compute numerical scales of linguistic terms in DLPR matrix with incomplete symbolic proportions. Given that the DLPRs with incomplete symbolic proportions provided by DMs are logical and rational, the numerical scales for linguistic term sets can be derived by maximizing the consistency of such preference relations. Meanwhile, in this study, we set the range of numerical

min f ij

lP

scales for linguistic term sets as: N S(l t)  [(t+ T-1)/2T, (t+ T+1)/2T], t = -T+1«-« T-1. As a result, the following multi-objective programming model is built: 0.5  0.5(Zi  Z j )  E ij  0.5  0.5(Zi  Z j )  E ij § T · = 0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (l T ) ¸ © t T ¹

urn a

§ T · + 0.5  0.5(Zi  Z j )  ¨ ¦ pi j (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸ © t T ¹

Jo

­ NS (l t )  NS (lt ) 1, t T , }, 1, 0, 1, }, T . ° ° NS (lT ) 0 and NS (lT ) 1 ° NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 °° s.t. ®(t  T  1) / 2T d NS (lt ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 n n °       °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi ° °¯i , j 1, 2, ..., n

(14)

where į is a constant and 0 < į < 1. In the above model, the conditions in the first four rows guarantee N S is an additive scale function; and the conditions in the last two rows ensure Z = (Z1, Z2, ..., Zn)T = ([Z-1 , Z+1 ], [Z-2 , Z+2 ], ..., [Z-n , Z+n ])T is a normalized interval weight vector. From Theorem 1 and Proposition 6, we have 0.5  0.5(Zi  Z j )  §¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L) ˜ NS (lT ) ·¸ T

©t

= 0.5  0.5(Z j  Zi )  §¨ ¦ p ji (lt ) ˜ NS (lt )  p ji ( L) ˜ NS (lT ) ·¸ T

©t

T

¹

T

¹

for i, j = 1, 2, ..., n, i  j; and

Journal Pre-proof § T · 0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸ © t T ¹

= 0.5  0.5(Zi  Z j )  §¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L) ˜ NS (lT ) ·¸ T

©t

¹

T

= 0 if i = j. Therefore, the above model can be simplified into the following model: 0.5  0.5(Zi  Z j )  E ij  0.5  0.5(Zi  Z j )  E ij

min f ij

§ T · = 0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (l T ) ¸ © t T ¹ § T · + 0.5  0.5(Zi  Z j )  ¨ ¦ pi j (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸ © t T ¹

pro of

­ NS (l t )  NS (lt ) 1, t T , }, 1, 0, 1, }, T . ° ° NS (lT ) 0 and NS (lT ) 1 ° NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 °° s.t. ®(t  T  1) / 2T d NS (lt ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 n n °       °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi ° °¯i , j 1, 2, ..., n, j ! i

(15)

Suppose that J ij 0.5  0.5(Zi  Z j )  E ij = 0.5  0.5(Zi  Z j )  §¨ ¦ pij (lt ) ˜ NS (lt )  pij (L ) ˜ NS (lT ) ·¸ , T

©t

J ij  J ij 2

J ij  J ij

, J ij

2

can get J ij J ij  J ij , K ij ˜ K ij

0 for i,

§ · 0.5  0.5(Zi  Z j )  ¨ ¦ pij (l t ) ˜ NS (lt )  pi j (L ) ˜ NS (lT ) ¸ © t T ¹

= , Kij J ij

Kij  Kij 2

re-

J ij

0.5  0.5(Zi  Z j )  E ij

T

and Kij

Kij  Kij

J ij  J ij , Kij Kij  Kij

2

and

,

and

let

for i, j = 1, 2, ..., n, j > i. Then, we

Kij

Kij  Kij , where J ij ˜ J ij 0

and

lP

Kij

¹

T

j = 1, 2, ..., n, j > i. For this reason, the solution to the above programming

problem can be obtained by solving the optimization model below: n 1

n

min F = ¦ ¦ [ ij ˜ J ij  J ij  Kij  Kij

urn a

i 1 j i 1

Jo

­ § T ·     °0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸  J ij  J ij 0 t T © ¹ ° ° § T ·     °0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸  Kij  Kij 0 © t T ¹ ° ° NS (l t )  NS (lt ) 1, t T , }, 1, 0, 1, }, T ° ° NS (lT ) 0 and NS (lT ) 1 ° s.t. ® NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 °(t  T  1) / 2T d NS (l ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 t ° n n °       °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi °     °J i j t 0, J ij t 0,Kij t 0,K ij t 0 ° °i , j 1, 2, ..., n, j ! i ¯

(16)

where ȟij is the weight factor associated with the goal function f ij. Unless otherwise specified, we usually assume that all goal functions f ij (i = 1, 2, ..., n-1, j = i + 1, ..., n) are fair. In this case, we can set ȟij = 1 ( i = 1, 2, ..., n-1, j = i + 1, ..., n), and the optimization model (16) can be equivalently expressed as:

Journal Pre-proof n 1

n

min F = ¦ ¦ J ij  J ij  Kij  Kij i 1 j i 1

(17)

pro of

­ § T ·     °0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸  J ij  J ij 0 t T © ¹ ° ° § T ·     °0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS (lT ) ¸  Kij  Kij 0 t  T © ¹ ° ° NS (l t )  NS (lt ) 1, t T , }, 1, 0, 1, }, T ° ° NS (lT ) 0 and NS (lT ) 1 ° s.t. ® NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 . °(t  T  1) / 2T d NS (l ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 t ° n n °       °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi °     °J i j t 0, J ij t 0,Kij t 0,K ij t 0 ° °i , j 1, 2, ..., n, j ! i ¯

re-

This optimization model herein is referred to as a numerical scale computation model, by solving which we can compute the numerical scale  of the linguistic term set in DLPR matrix with incomplete symbolic proportions. If F = 0 in the optimal solution, the DLPR with incomplete symbolic proportions is additive consistent. In this case, Ȗij+ = Ȗ-ij = Șij+ =Ș-ij =0, which means the preference relation can be represented by (10), so it is additive consistent based on Definition 12.

lP

Furthermore, we can obtain the corresponding normalized interval weight vector Z = (Z1, Z2, ..., Zn)T, according to which the pair-wise comparison alternatives can be ranked in individual decision-making with this form of preference relation. E xample 3. Let us proceed with the earlier preference relation D provided in Example 1. Based on (17), we can build the following model:     min F = J 12  J 12  K12  K12  J 13  J 13  K13  K13  J 14  J 14  K14  K14  J 23  J 23  K 23  K 23      J 24  J 24  K 24  K24  J 34  J 34  K34  K34

Jo

urn a

­0.5  0.5(Z1  Z2 )  (0.3 NS (l3 )  0.6 NS (l2 )  0.1 u 0)  J 12  J 12 0 °     °0.5  0.5(Z1  Z2 )  (0.3 NS (l3 )  0.6 NS (l2 )  0.1 u 1)  K12  K12 0 °     °0.5  0.5(Z1  Z3 )  (0.3 NS (l1 )  0.5 NS (l2 )  0.2 u 0)  J 13  J 13 0 °0.5  0.5(Z   Z  )  (0.3 NS (l )  0.5 NS (l )  0.2 u 1)  K   K  0 1 3 1 2 13 13 ° °0.5  0.5(Z1  Z4 )  (0.4 NS (l1 )  0.4 NS (l2 )  0.2 u 0)  J 14  J 14 0 °     °0.5  0.5(Z1  Z4 )  (0.4 NS (l1 )  0.4 NS (l2 )  0.2 u 1)  K14  K14 0 °     °0.5  0.5(Z2  Z3 )  (0.3 NS (l2 )  0.4 NS (l3 )  0.2 u 1  0.1 u 0)  J 23  J 23 0 °0.5  0.5(Z   Z  )  (0.3 NS (l )  0.4 NS (l )  0.2 u 1  0.1 u 1)  K   K  0 2 3 2 3 23 23 °   °0.5  0.5(Z2  Z4 )  (0.2 NS (l2 )  0.7 NS (l3 )  0.1 u 0)  J 24  J 24 0 ° s.t. ®0.5  0.5(Z   Z  )  (0.2 NS (l )  0.7 NS (l )  0.1 u 1)  K   K  0 2 4 2 3 24 24 °     °0.5  0.5(Z3  Z4 )  (0.4 NS (l2 )  0.4 NS (l1 )  0.2 u 0)  J 34  J 34 0 °     °0.5  0.5(Z3  Z4 )  (0.4 NS (l2 )  0.4 NS (l1 )  0.2 u 1)  K34  K34 0 ° NS (l )  NS (l ) 1, t 4, }, 1, 0, 1, }, 4 t t ° ° NS (l4 ) 0 and NS (l4 ) 1 ° NS (l )  NS (l ) t 0.05, t 4, }, 1, 0, 1, },3 t 1 t ° °(t  3) / 8 d NS (lt ) d (t  5) / 8, t 3, }, 1, 0, 1, },3 ° 4 4 °0 d Z  d Z  d 1, Z   Z  d 1, Z  + Z  t 1, i 1, 2, ..., 4 . ¦ j i ¦ j i i i ° j 1 j 1 ° j zi j zi °                         J , J , K , K , J , J , K , K , J , J , K ¯ 12 12 12 12 13 13 13 13 14 14 14 ,K14 , J 23 , J 23 ,K 23 ,K 23 , J 24 , J 24 ,K 24 ,K 24 , J 34 , J 34 ,K34 ,K34 t 0

(18)

Journal Pre-proof By solving (18), we get J 12 =J 12 =K12 =K12 =J 13 =K13 =K13 =J 14 =K14 =J 23 =J 23 =K23 =K23 =J 24 =J 24 =K24 =K24 =J 34 =K34 =0 , Ȗ13+ =0.0200, Ȗ-14 =

0.0100, Ș-14 = 0.0300, Ȗ34- = 0.0425, Ș-34 = 0.0225, F = 0.1250;

Z = (Z1, Z2, Z3, Z4)T = ([Z-1 , Z+1 ], [Z-2, Z+2 ], [Z-3, Z+3 ], [Z-4, Z+4 ])T = ([0.175, 0.355], [0.555, 0.575],

E

§ [0.5000,0.5000] ¨ [0.6064,0.7064] = ¨ ¨ [0.3225,0.5225] ¨¨ © [0.3300,0.5300]

pro of

[0, 0.180], [0.075, 0.255])T; And N S(l-4) = 0, NS(l-3) = 0.25, N S(l-2) = 0.375; N S(l-1) = 0.45, N S(l0) = 0.5, N S(l1) = 0.55, N S(l2) = 0.625, N S(l3) = 0.75, N S(l4) = 1. Thus, the numerical scale   of the linguistic term set L = {l t|t = -4 « -   « 4} in the earlier preference relation D can be obtained, and the corresponding EBNPR matrix is as follows: [0.2936,0.3936] [0.477 5,0.6775] [0.4700,0.6700] · ¸ [0.5000,0.5000] [0.696 1,0.7961] [0.6650,0.7650] ¸ . [0.2039,0.3039] [0.500 0,0.5000] [0.3300,0.5300] ¸ ¸¸ [0.2350,0.3350] [0.4700, 0.6700] [0.5000,0.5000] ¹

5. G D M based on D L PRs with incomplete symbolic proportions

re-

In many real decision scenarios, sometimes it is difficult for a single DM or expert to consider all related aspects of the given problems due to the limited computational ability of human thinking and the complexity and uncertainty of such decision problems. This section therefore considers GDM problems, where the preferences provided over multiple alternatives by DMs are expressed in the form of DLPRs with incomplete symbolic proportions.

lP

5.1. G DM with unknown general numerical scale

urn a

In this type of GDM problems, we assume that the numerical scales of the given linguistic term set for all DMs are completely unknown. As such, we may encounter two situations. In one situation the DMs have the same preference for the numerical scale of the given linguistic term set, where we can use a certain general numerical scale to character their preference. Whereas in the other situation the DMs may have different preferences for the numerical scale of the linguistic term set. In this situation, the numerical scale is personalized for each DM. The former situation is discussed in this section followed by the latter one discussed in Section 5.2. Suppose that D e = (deij )nun with deij = {(l t, peij (l t)), (L, peij (L)), t = -T«-« T} (e = 1, 2, ..., E ) are individual DLPRs with incomplete symbolic proportions provided by DMs de (e = 1, 2, ..., E ) to express their preferences over a set of alternatives X = {x1, x2, ..., xn}. W = (w1, ..., we, ..., w E)T is the corresponding weighting vector for DMs VXFKWKDW” we ”IRU e = 1, 2, ..., E

Jo

and ¦ eE 1 we = 1. Since the DMs have the same preference for the numerical scale of the given linguistic term set L = {l t|t = -T«-« T}, the individual preference relations can be aggregated by the DLPRIWA (or DLPRIOWA ) operator to generate a collective preference relation. Here we use the DLPRIWA operator to aggregate D e (e = 1, 2, ..., E ) and let D c = (dcij )nun with dcij = {(l t, pcij (l t)), (L, pcij (L)), t = -T « -   « T}, where pcij (l t) = ¦ eE 1 we ˜ pije (lt ) and pcij (L) = E ¦ e 1 we ˜ pije ( L ) be the collective preference relation obtained by this operator. Of course, the

collective preference relation obtained by the DLPRIOWA operator can be similarly extended.

Journal Pre-proof We can reason from Proposition 7  that the collective preference relation is additive consistent if each individual preference relation is additive consistent. Given that D e (e = 1, 2, ..., E ) are logical and rational, D c is therefore logical and rational. Accordingly, a general numerical scale for the

min f ijc

pro of

linguistic term set can be derived by maximizing the consistency of D c. Assume that E c = ( E cij )nun with E cij = [ E cij- , E cij+ ] is the associated EBNPR of D c and Zc = (Z1c , Z2c , ..., Znc )T with Zci = [Zci - , Zci + ] (i = 1, 2, ..., n) is a normalized interval weight vector satisfying the condition (8). In what follows, based on the consistency of D c, a multi-objective programming model is constructed to derive a general numerical scale: 0.5  0.5(Zic   Z cj  )  E ijc   0.5  0.5(Zic   Z cj  )  E ijc 

§ T · = 0.5  0.5(Zic   Z cj  )  ¨ ¦ pijc (lt ) ˜ NS (lt )  pijc ( L ) ˜ NS (l T ) ¸ © t T ¹ § T · + 0.5  0.5(Zic   Z cj  )  ¨ ¦ pijc (lt ) ˜ NS (lt )  pijc ( L ) ˜ NS (lT ) ¸ © t T ¹

§ T § E · · § E · = 0.5  0.5(Zic   Z cj  )  ¨ ¦ ¨ ¦ we ˜ pije (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pije ( L ) ¸ ˜ NS (l T ) ¸ t  T e 1 e 1 © ¹ © ¹ © ¹

§ T § E · · § E · + 0.5  0.5(Zic   Z cj  )  ¨ ¦ ¨ ¦ we ˜ pije (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pije ( L ) ¸ ˜ NS (lT ) ¸ . ¹ ©e 1 ¹ © t T © e 1 ¹

(19)

lP

re-

­ NS (l t )  NS (lt ) 1, t T , }, 1, 0, 1, }, T ° ° NS (lT ) 0 and NS (lT ) 1 ° NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 °° s.t. ®(t  T  1) / 2T d NS (lt ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 n n ° c c c c c c °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi ° °¯i , j 1, 2, ..., n

where į is a constant and 0 < į < 1. The conditions in the first four rows insure N S is an additive

min f ijc

urn a

scale function; and the conditions in the last two rows guarantee Zc = (Z1c , Z2c , ..., Znc )T with Zci = [Zci - , Zci + ] is a normalized interval weight vector. Similar to the earlier process in Section 4.2, (19) can be simplified into the following model: 0.5  0.5(Zic   Z cj  )  E ijc   0.5  0.5(Zic   Z cj  )  E ijc  § T · = 0.5  0.5(Zic   Z cj  )  ¨ ¦ pijc (lt ) ˜ NS (lt )  pijc ( L ) ˜ NS (l T ) ¸ © t T ¹ § T · + 0.5  0.5(Zic   Z cj  )  ¨ ¦ pijc (lt ) ˜ NS (lt )  pijc ( L ) ˜ NS (lT ) ¸ © t T ¹ § T § E · · § E · = 0.5  0.5(Zic   Z cj  )  ¨ ¦ ¨ ¦ we ˜ pije (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pije ( L ) ¸ ˜ NS (l T ) ¸ t  T e 1 e 1 © ¹ © ¹ © ¹

Jo

§ T § E · · § E · + 0.5  0.5(Zic   Z cj  )  ¨ ¦ ¨ ¦ we ˜ pije (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pije ( L ) ¸ ˜ NS (lT ) ¸ . ¹ ©e 1 ¹ © t T © e 1 ¹

­ NS (l t )  NS (lt ) 1, t T , }, 1, 0, 1, }, T ° ° NS (lT ) 0 and NS (lT ) 1 ° NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 °° s.t. ®(t  T  1) / 2T d NS (lt ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 n n ° c c c c c c °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi ° °¯i , j 1, 2, ..., n, j ! i

(20)

Journal Pre-proof Assume that J ijc 0.5  0.5(Zic   Z cj  )  E ijc  , Kijc 0.5  0.5(Zic   Z cj  )  E ijc  , and let J ijc  J ijc 

J ijc

J ijc  J ijc 2

2

J ijc

J ijc   J ijc  ,

Kijc  ˜ Kijc 

Kijc  Kijc

, Kijc 

0 for i,

and Kijc 

J ijc   J ijc  ,

Kijc  Kijc 2

Kijc Kijc   Kijc 

2

,

for i, j = 1, 2, ..., n, j > i. Then, we can get and

Kijc

Kijc   Kijc  , where

J ijc  ˜ J ijc 

0

and

j = 1, 2, ..., n, j > i. As such, (20) can be rewritten as the following linear

pro of

program: n 1

J ijc  J ijc

n

min F c = ¦ ¦ [ ijc ˜ J ijc   J ijc   Kijc   Kijc  i 1 j i 1

(21)

lP

re-

­ § T § E · · § E · c c e e c c 0 °0.5  0.5(Zi  Z j )  ¨ ¦ ¨ ¦ we ˜ pij (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pij ( L ) ¸ ˜ NS (l T ) ¸  J ij  J ij t  T e 1 e 1 © ¹ © ¹ © ¹ ° ° E E § T · °0.5  0.5(Zic   Z cj  )  ¨ ¦ §¨ ¦ we ˜ pije (lt ) ·¸ ˜ NS (lt )  §¨ ¦ we ˜ pije ( L ) ·¸ ˜ NS ( lT ) ¸  Kijc   Kicj  0 t  T e 1 e 1 ° © ¹ © ¹ © ¹ ° NS (l )  NS (l ) 1, t T , }, 1, 0, 1, }, T  t t ° ° NS (l T ) 0 and NS (lT ) 1 ° s.t. ® NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 °(t  T  1) / 2T d NS (l ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 t ° n n ° c c c c c c °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi ° c c c c J t 0, J t 0, K t 0, K t 0 ° ij ij ij ij ° i , j 1, 2, ..., n , j ! i ° ¯

where ȟcij is the weight factor associated with the goal function fcij . In the case where all goal functions fcij (i = 1, 2, ..., n-1, j = i + 1, ..., n) are equally important, (21) can be equivalently expressed as: n 1

n

min F c = ¦ ¦ J ijc   J ijc   Kijc   Kijc 

urn a

i 1 j i 1

Jo

­ § T § E · · § E · c c e e c c 0 °0.5  0.5(Zi  Z j )  ¨ ¦ ¨ ¦ we ˜ pij (lt ) ¸ ˜ NS (lt )  ¨ ¦ we ˜ pij ( L ) ¸ ˜ NS (l T ) ¸  J ij  J ij ¹ ©e 1 ¹ © t T © e 1 ¹ ° ° E E § T · °0.5  0.5(Zic   Z cj  )  ¨ ¦ §¨ ¦ we ˜ pije (lt ) ·¸ ˜ NS (lt )  §¨ ¦ we ˜ pije ( L ) ·¸ ˜ NS ( lT ) ¸  Kijc   Kicj  0 ° ¹ ©e 1 ¹ © t T © e 1 ¹ ° NS (l )  NS (l ) 1, t T , }, 1, 0, 1, }, T  t t ° ° NS (l T ) 0 and NS (lT ) 1 ° s.t. ® NS (lt 1 )  NS (lt ) t G , t T , }, 1, 0, 1, }, T  1 . (22) °(t  T  1) / 2T d NS (l ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 t ° n n ° c c c c c c °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi + ¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi ° c c c c °J ij t 0, J ij t 0,Kij t 0,Kij t 0 ° °i , j 1, 2, ..., n, j ! i ¯

By solving the above model, we can obtain a general numerical scale for the given linguistic term set. Meanwhile, a normalized interval weight vector Zc = (Z1c , Z2c , ..., Znc )T with Zci = [Zci - , Zci + ] for the collective DLPR with incomplete symbolic proportions D c can be derived. Particularly, if F c = 0, i.e., Ȗcij+ = Ȗcij- = Șcij+ = Șcij- =0 in the optimal solution, D c is consistent and can be expressed

Journal Pre-proof by (10). Subsequently, using the comparison method for interval numbers developed in Dong et al. [18] to compare Zc, a ranking order of the alternatives can be generated.

5.2. G DM with unknown personalized numerical scale In this section, we discuss the situation where DMs have different preferences for the numerical scale of the given linguistic term set. Given the numerical scale is personalized for each DM in this situation, we assume N Se (e = 1, 2, ..., E ) are additive scale functions associated to DMs de (e

n 1

pro of

= 1, 2, ..., E ). Suppose that Ze = (Z1e , Z2e , ..., Zne )T with Zei = [Zei - , Zei + ] (i = 1, 2, ..., n) is a normalized interval weight vector derived from the preference relation D e. For each DM, similar to the numerical scale computation model (17), then the following optimization model is established to compute his/her personalized numerical scale. n

min F e = ¦ ¦ J ije   J ije   Kije   Kije  i 1 j i 1

lP

re-

­ § T e · e e e e e e e 0 °0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS ( lT ) ¸  J ij  J ij © t T ¹ ° ° § T e · e e e e e e e 0 °0.5  0.5(Zi  Z j )  ¨ ¦ pij (lt ) ˜ NS (lt )  pij ( L ) ˜ NS ( lT ) ¸  Kij  Kij t  T © ¹ ° ° NS e (l )  NS e (l ) 1, t T , }, 1, 0, 1, }, T t t ° ° NS e (l T ) 0 and NS e (lT ) 1 ° s.t. °® NS e (lt 1 )  NS e (lt ) t G , t T , }, 1, 0, 1, }, T  1 ° e °(t  T  1) / 2T d NS (lt ) d (t  T  1) / 2T , t T  1, }, 1, 0, 1, }, T  1 n n ° e e e e e e °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi +¦ Z j t 1, i 1, 2, ..., n j 1 j 1 ° j zi j zi ° °J iej  t 0, J ije  t 0,Kije  t 0,Kije  t 0 ° °i , j 1, 2, ..., n, j ! i °¯

(23)

where J ije   J ije  0.5  0.5(Zie   Z ej  )  §¨ ¦ pije (lt ) ˜ NS e (lt )  pije ( L ) ˜ NS e (lT ) ·¸ , T

©t

¹

J ije   J ije 

§ · 0.5  0.5(Zie   Z ej  )  ¨ ¦ pije (lt ) ˜ NS e (lt )  pije ( L) ˜ NS e (lT ) ¸ , © t T ¹

Kije   Kije 

§ T · 0.5  0.5(Zie   Z ej  )  ¨ ¦ pije (lt ) ˜ NS e (lt )  pije ( L ) ˜ NS e ( lT ) ¸ , © t T ¹

T

urn a

and Kije   Kije  Kije  ˜ Kije 

T

§ T · 0.5  0.5(Zie   Z ej  )  ¨ ¦ pije (lt ) ˜ NS e (lt )  pije ( L) ˜ NS e (lT ) ¸ © t T ¹

such that J ije  ˜ J ije  0 and

0 for i, j = 1, 2, ..., n, j > i.

Jo

Solving this model, a personalized numerical scale N Se for the given linguistic term set L = {l t|t = -T«-« T} can be derived from the preference relation D e. Similar to the argument in the last paragraph in Section 4.2, if F e = 0 in the optimal solution, then D e is of additive consistency. Meanwhile, the corresponding normalized interval weight vector Ze = (Z1e , Z2e , ..., Z e T e ee+ n ) with Z i = [Zi , Zi ] ( i = 1, 2, ..., n) can be obtained. Therefore, a collective priority vector can be calculated in the following manner: Zi

[Zi , Zi ] =

ª º ¦ we ˜ Zie = « ¦ we ˜ Zie  , ¦ we ˜ Zie  » . E

E

e 1

¬e

E

1

e 1

¼

(24)

Furthermore, comparison results of the alternatives can be produced according to the values of

Journal Pre-proof their priorities.

5.3. An approach for G DM based on   DLPRs with incomplete symbolic proportions

re-

pro of

Based on the analysis made in the above sub-sections, as a whole, we present an approach for addressing GDM problems where DMs¶ preferences over alternatives are expressed in terms of DLPRs with incomplete symbolic proportions. The steps of the approach are summarized below. Step 1: Prepare for analyzing the GDM problem. First, the moderator determines the linguistic term set L = {l t|t = -T«-« T} and the weighting vector of the DMs W. Step 2: Classify the DMs de (e = 1, 2, ..., E ). The moderator classifies the DMs de (e = 1, 2, ..., E ) based on their preferences for the numerical scale of L, and puts them into three categories: 1) S1 = {d1e|e =1, 2«#d1} in which the DMs share a general numerical scale that is completely unknown; 2) S2 = {d2e|e =1, 2«#d2} where the numerical scale of L for each DM is personalized but completely unknown; 3) S3 = {d3e|e =1, 2«#d3} where the numerical scale of L for each DM is known. Here, #d1, #d2, and #d3 represent the numbers of the DMs belonging to the above three categories respectively, and #d1 + #d2 + #d3 = E . Step 3: Collect individual preference relations from the DMs.

lP

The DMs provide D e = (deij )nun with deij = {(l t, peij (l t)), (L, peij (L)), t = -T«-« T} (e = 1, 2, ..., E ) over the set of alternatives X = {x1, x2, ..., xn}. After that, the moderator collects all the individual preference relations { D 1, ..., D e, ..., D E }. Step 4: Derive normalized interval weight vectors from preference relations. (1) If S1 is nonempty, then apply (5) or (6) to aggregate the #d1 preference relations provided by DMs from S1, and use (22) to compute a general numerical scale N Sc and derive a normalized interval weight vector Zc = (Z1c , Z2c , ..., Znc )T with Zci = [Zci - , Zci + ] (i = 1, 2, ..., n) from the obtained collective preference relation D c. (2) If S2 is nonempty, then use (23) to compute personalized numerical scales N Se (e =1, 2«

urn a

#d2) and derive the corresponding Ze = (Z1e , Z2e , ..., Zne )T with Zei = [Zei - , Zei + ] (e =1, 2«#d2, i = 1, 2, ..., n) from preference relations provided by the DMs in S2. (3) If S3 is nonempty, then use (17) to derive Ze = (Z1e , Z2e , ..., Zne )T with Zei = [Zei - , Zei + ] (e =1, 2«#d3, i = 1, 2, ..., n) from preference relations provided by the DMs in S3. Step 5: Calculate a collective priority vector of alternatives. Using (24) to aggregate all the normalized interval weight vectors obtained in Step 4 to

Jo

calculate Z. Step 6: Generate a solution to the problem. Comparison results of the alternatives can be generated based on the values of their priorities. As a result, a solution to the GDM problem is obtained. 6. A n application of the proposed approach in G D M In this section, we first consider a research project evaluation and selection problem to illustrate the implementation process of the proposed approach. Then, we describe briefly two similar methodologies and compare them with our proposal.

6.1. Application to the research project evaluation and selection problem

Journal Pre-proof

urn a

lP

re-

pro of

Research project evaluation and selection is a common and significant task for many research funding agencies. The research project evaluation process mainly includes two stages: preliminary examination and peer review. After that, the agencies make final decisions based on the project review information from multiple peer reviewers, namely, they choose some of the projects for implementation. Suppose that one research funding agency receives four applications, {x1, x2, ..., x4}, which are successfully submitted for  peer review after preliminary examination. In the peer review stage, the projects are rated by four peer reviewers {d1, d2, d3, d4}. Supported by the review histories about the four reviewers, the agency uses the direct assignment method in [12] to determine their weights. First, the most important reviewer set {d2, d3} is identified from {d1, d2, d3, d4} and weight one is assigned to them. Then, the remaining reviewer set {d1, d4} is compared with the reviewer set {d2, d3} and a relative weight for {d1, d4}, i.e., 2/3, is obtained. By normalizing the two relative weights 1 and 2/3, one can get that w2 + w3 = 0.6 and w1 + w4 = 0.4. With respect to {d2, d3}, repeating the above steps and identifying the most important reviewer set {d3} whose relative weight is 1, and a relative weight for {d2},  i.e., 1/4, one can get that w2 = 0.12 and w3 = 0.48. Similarly, one gets that w1 = 0.15 and w4 = 0.25. Finally, each reviewer has been assigned with a weight, i.e., W = (0.15, 0.12, 0.48, 0.25)T. In order to provide a fair evaluation concerning the comprehensive level of the projects, the reviewers need to compare them in terms of the following ten aspects, such as capacity of research team, rationality, feasibility, research content, research objective, research foundation, academic novelty,  potential, scientific value and project budget, and finally express their review information by means of DLPRs with incomplete symbolic proportions, using the following linguistic evaluation term set L = {l-5 = absolutely poorer, l-4 = very much poorer, l-3 = much poorer, l-2 = moderately poorer, l-1 = m arginally poorer, l0 = indifferent, l1 = m arginally better, l2 = moderately better, l3 = much better, l4 = very much better, l5 = absolutely better}. Step 1 is completed. Assume that Reviewer 1 shares an identical but completely unknown numerical scale with Reviewer 2, i.e., S1 = {d1, d2}, while the numerical scale function of L for Reviewer 3 is personalized and Reviewer 4 prefers it to be the common additive numerical scale, i.e., S2 = {d3} and S3 = {d4}. Step 2 is completed. The reviewers supply the pairwise review information of the projects, which are presented in

Jo

Matrices D e = (deij )nun with deij = {(l t, peij (l t)), (L, peij (L)), t = -5«-«5} (i, j = 1, 2, ..., 4, e = 1, 2, ..., 4). For example, when comparing project x1 with x2, Reviewer 1 is sure that x1 is much poorer than x2 on one of the above ten aspects, and moderately poorer than x2 on eight of those aspects. But she is not sure what the relation between x1 and x2 is on the remaining aspect due to her partial knowledge about these two projects. Then, the pairwise review information of x1 and x2 coming from Reviewer 1 can be recorded as {(l-3, 0.1), (l-2, 0.8), (L, 0.1)}, which is denoted by entry12 in Matrix D 1. Step 3 is completed. {(l0 ,1)} {(l3 ,0.1),(l2 ,0.8),(L ,0.1)} {(l1 ,0.8),(l0 ,0.05),(L ,0.15)} {(l1 ,0.75),(l2 ,0.15),(L ,0.1)} · § ¨ ¸ {(l2 ,0.8),(l3 ,0.1),(L ,0.1)} {(l0 ,1)} {(l3 ,0.2),(l4 ,0.6),(l5 ,0.1),(L ,0.1)}{(l2 ,0.3),(l3 ,0.6),(L ,0.1)}¸ D 1= ¨¨ {(l ,0.05),(l1 ,0.8),(L ,0.15)}{(l5 ,0.1),(l4 ,0.6),(l3 ,0.2),(L ,0.1)} {(l0 ,1)} {(l1 ,0.55),(l2 ,0.35),(L ,0.1)} ¸ ¨¨ 0 ¸¸ {(l2 ,0.35),(l1 ,0.55),(L ,0.1)} {(l0 ,1)} © {(l2 ,0.15),(l1 ,0.75),(L ,0.1)} {(l3 ,0.6),(l2 ,0.3),(L ,0.1)} ¹

D

{(l0 ,1)} {(l3 ,0.4),(l2 ,0.5),(L ,0.1)} {(l 1 ,0.5),(l0 ,0.3),(L ,0.2)} {(l 2 ,0.3),(l3 ,0.6),(L ,0.1)} · § ¨ ¸ {( l ,0.5),( l ,0.4),( L ,0.1)} {(l0 ,1)} {(l0 ,0.6),(l3 ,0.2),(L ,0.2)} {(l2 ,0.25),(l3 ,0.65),(L ,0.1)} ¸ 3 = ¨¨ 2 {(l ,0.3),(l1 ,0.5),(L ,0.2)} {(l3 ,0.2),(l0 ,0.6),(L ,0.2)} {(l0 ,1)} {(l1 ,0.4),(l4 ,0.5),(L ,0.1)} ¸ ¨¨ 0 ¸¸ {( l ,0.6),( l ,0.3),( L ,0.1)} {( l ,0.65),( l ,0.25),( L ,0.1)} {( l ,0.5),( l ,0.4),( L ,0.1)} {(l0 ,1)} 2 3 2 4 1 © 3 ¹

2

Journal Pre-proof {(l0 ,1)} {(l2 ,0.3),(l3 ,0.35),(l 4 ,0.25),(L ,0.1)} {(l1 ,0.3),(l 2 ,0.6),(L ,0.1)} {(l 2 ,0.4),(l0 ,0.5),(L ,0.1)} · § ¨ ¸ {( l ,0.25), ( l ,0.35),( l ,0.3),( L ,0.1)} {(l0 ,1)} {(l1 ,0.2),(l1 ,0.65),(L ,0.15)}{(l0 ,0.45),(l1 ,0.4),(L ,0.15)} ¸ 3 2 D 3= ¨¨ 4 {(l2 ,0.6),(l1 ,0.3),(L ,0.1)} {(l1 ,0.65),(l1 ,0.2),(L ,0.1)} {(l0 ,1)} {(l1 ,0.4),(l2 ,0.5),(L ,0.1)} ¸ ¨¨ ¸¸ {( l ,0.5),( l ,0.4),( L ,0.1)} {( l ,0.4),( l ,0.45),( L ,0.15)} {( l ,0.5),( l ,0.4),( L ,0.1)} {(l0 ,1)} 0 2 1 0 2 1 © ¹

D

{(l0 ,1)} {(l1 ,0.3),(l2 ,0.6),(L ,0.1)} {(l3 ,0.6),(l4 ,0.3),(L ,0.1)} {(l1 ,0.25),(l2 ,0.6),(L ,0.15)} · § ¨ ¸ {( l ,0.6),( l ,0.3),( L ,0.1)} {( l ,1)} {( l ,0.8),( l ,0.1)( L ,0.1)} {( l1 ,0.25),(l2 ,0.5),(L ,0.25)} ¸  2  1 0 1 2 = ¨¨ {(l ,0.3),(l3 ,0.6),(L ,0.1)} {(l2 ,0.1),(l1 ,0.8),(L ,0.1)} {(l0 ,1)} {(l1 ,0.3),(l0 ,0.3),(l1 ,0.3),(L ,0.1)}¸ ¨¨ 4 ¸¸ {(l0 ,1)} © {(l2 ,0.6),(l1 ,0.25),(L ,0.15)}{(l2 ,0.5),(l1 ,0.25),(L ,0.25)}{(l1 ,0.3),(l0 ,0.3),(l1 ,0.3),(L ,0.1)} ¹

4

pro of

Following Step 4 in Section 5.3, the review information from different categories of reviewers should be respectively addressed. (1) For the first category, i.e., S1 = {d1, d2}, the pairwise review information of the projects in Matrices D 1 and D 2 are first aggregated into a collective preference relation using (5), which is presented in D c. 29 31 4 1 {(l0 ,1)} {(l3 , 307 ),(l2 , 23 ),(L, 101 )} {(l1 , 23 ),(l0 , 180 ),(L , 180 )} {(l1 , 125 ),(l2 , 13 § 60 ),(l3 , 15 ),(L , 10 )} · ¨ ¸ 7 13 5 28 2 1 4 1 1 1 1 {(l2 , 3 ),(l3 , 30 ),(L , 10 )} {(l0 ,1)} {(l0 , 15 ),(l3 , 5 ),(l4 , 3 ),(l5 , 18 ),(L, 90 )}{(l2 , 18 ),(l3 , 45 ),(L , 10 )} ¸ c ¨ D =¨ 29 31 29 {(l0 , 180 ),(l1 , 23 ),(L , 180 )} {(l5 , 181 ),(l4 , 13 ),(l3 , 15 ),(l0 , 154 ),(L , 13 {(l1 , 60 ),(l2 , 367 ),(l4 , 92 ),(L , 101 )} ¸¸ 90 )} {(l0 ,1)} ¨ ¨ {(l , 4 ),(l , 13 ),(l , 5 ),(L, 1 )}{(l , 28 ),(l , 5 ),(L, 1 )} {(l , 2 ),(l , 7 ),(l , 29 ),(L, 1 )} ¸ {(l0 ,1)} 3 45 2 18 4 9 2 36 1 60 10 10 10 © 3 15 2 60 1 12 ¹

re-

From the results in D c, we can establish the following optimization model using (22): c c c c min F c = J 12c   J 12c   K12c   K12c   J 13c   J 13c   K13c   K13c   J 14c   J 14c   K14c   K14c   J 23  J 23  K 23  K 23 c c c c c c c c  J 24  J 24  K 24  K 24  J 34  J 34  K34  K34

Jo

urn a

lP

­0.5  0.5(Z1c   Z2c  )  ( 307 NS (l3 )  23 NS (l2 )  101 u 0)  J 12c   J 12c  0 ° c c c c °0.5  0.5(Z1  Z2 )  ( 307 NS (l3 )  23 NS (l2 )  101 u1)  K12  K12 0 ° c c c c 29 31 2 °0.5  0.5(Z1  Z3 )  ( 3 NS (l1 )  180 NS (l0 )  180 u 0)  J 13  J 13 0 ° c c c c 29 31 2 °0.5  0.5(Z1  Z3 )  ( 3 NS (l1 )  180 NS (l0 )  180 u1)  K13  K13 0 c  c  °0.5  0.5(Z  Z )  ( 5 NS (l )  13 NS (l )  5 NS (l )  1 u 0)  J c   J c  0 1 4 1 2 3 14 14 12 60 15 10 ° c c 5 1 °0.5  0.5(Z1c   Z4c  )  ( 125 NS (l1 )  13 N S ( l )  N S ( l )  u 1)  K  K 0 2 3 14 14 60 15 10 ° c c c c 13 4 1 1 1 °0.5  0.5(Z2  Z3 )  ( 15 NS ( l0 )  5 NS (l3 )  3 NS (l4 )  18 NS (l5 )  90 u 0)  J 23  J 23 0 ° c c c  c  13 4 1 1 1 0.5  0.5( Z  Z )  ( N S ( l )  N S ( l )  N S ( l )  N S ( l )  u 1)  K  K 0 ° 2 3 0 3 4 5 23 23 15 5 3 18 90 ° c c c c 5 28 0.5  0.5( Z  Z )  ( N S ( l )  N S ( l )  0.1 u 0)  J  J 0 2 4 2 3 24 24 18 45 °° s.t. ®0.5  0.5(Z c   Z c  )  ( 5 NS (l )  28 NS (l )  0.1u1)  K c   K c  0 2 4 2 3 24 24 18 45 ° 29 °0.5  0.5(Z3c   Z4c  )  ( 60 NS (l1 )  367 NS (l2 )  92 NS (l4 )  101 u 0)  J 34c  J 34c 0 ° c c 29 NS (l1 )  367 NS (l2 )  92 NS (l4 )  110 u1)  K34c  K34c 0 °0.5  0.5(Z3  Z4 )  ( 60 ° ° NS (l t )  NS (lt ) 1, t 5, }, 1, 0, 1, },5 ° NS (l5 ) 0 and NS (l5 ) 1 ° ° NS (lt 1 )  NS (lt ) t 0.05, t 5, }, 1, 0, 1, }, 4 °(t  4) /10 d NS (l ) d (t  6) /10, t 4, }, 1, 0, 1, }, 4 t ° 4 4 ° c c c c c c °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi +¦ Z j t 1, i 1, 2, ..., 4 j 1 j 1 ° j zi j zi ° c c c c c c c c c c c c c c c c c c c c c c c c °¯J 12 , J 12 ,K12 ,K12 , J 13 , J 13 ,K13 ,K13 , J 14 , J 14 ,K14 ,K14 , J 23 , J 23 ,K 23 ,K 23 , J 24 , J 24 ,K 24 ,K 24 , J 34 , J 34 ,K34 ,K34 t 0

(25)

Then, by solving this model, we obtain cJ 12c  =J 12c  =K12c  =K12c  =J 13c  =J 13c  K13c  K13c  =J 14c  =K14c  =J 23c  =K23c  =J 24c  =K24c  =J 34c  =J 34c  =K34c  =K34c  =0 , Ȗ14 = 0.0204,

Ș14c- = 0.0482, Ȗ23c- = 0.0186, Ș23c- = 0.0352, Ȗ24c+ = 0.1065, Ș24c+ =0.0343, F c = 0.2632;

Zc = (Z1c , Z2c , Z3c , Z4c )T = ([Z1c- , Z1c+ ], [Z2c- , Z2c+ ], [Z3c- , Z3c+ ], [Z4c- , Z4c+ ])T = ([0.1161, 0.2605], [0.5289, 0.5845], [0.1550, 0.3550], [0.0000, 0.0000])T;

Journal Pre-proof And N S(l-5) = 0, NS(l-4) = 0.2, N S(l-3) = 0.25; NS(l-2) = 0.3112; N S(l-1) = 0.45, NS(l0) = 0.5, NS(l1) = 0.55, N S(l2) = 0.6888, N S(l3) = 0.75, NS(l4) = 0.8, N S(l5) = 1. (2) For the second category, i.e., S2 = {d3}, based on the preference relation D 3, we can build the following optimization model using (23): 3 3 3 3 min F 3 = J 123  J 123  K123  K123  J 133  J 133  K133  K133  J 143  J 143  K143  K143  J 23  J 23  K 23  K 23 3 3 3 3 3 3 3 3  J 24  J 24  K 24  K 24  J 34  J 34  K34  K34

lP

re-

pro of

­0.5  0.5(Z13  Z23 )  (0.3 NS 3 (l2 )  0.35 NS 3 (l3 )  0.25 NS 3 (l 4 )  0.1u 0)  J 123  J 123 0 ° 3 3 3 3 3 3 3 °0.5  0.5(Z1  Z2 )  (0.3 NS (l2 )  0.35 NS (l3 )  0.25 NS ( l 4 )  0.1 u1)  K12  K12 0 ° 3 3 3 3 3 3 °0.5  0.5(Z1  Z3 )  (0.3 NS (l1 )  0.6 NS (l2 )  0.1u 0)  J 13  J 13 0 °0.5  0.5(Z 3  Z 3 )  (0.3 NS 3 (l )  0.6 NS 3 (l )  0.1u1)  K 3  K 3 0 1 3 1 2 13 13 ° °0.5  0.5(Z13  Z43 )  (0.4 NS 3 (l2 )  0.5 NS 3 (l0 )  0.1u 0)  J 143  J 143 0 ° °0.5  0.5(Z13  Z43 )  (0.4 NS 3 (l2 )  0.5 NS 3 (l0 )  0.1u1)  K143  K143 0 ° 3 3 3 3 3 3 °0.5  0.5(Z2  Z3 )  (0.2 NS (l1 )  0.65 NS (l1 )  0.15 u 0)  J 23  J 23 0 ° 3 3 3 3 3 3 °0.5  0.5(Z2  Z3 )  (0.2 NS (l1 )  0.65 NS (l1 )  0.15 u1)  K23  K23 0 °0.5  0.5(Z 3  Z 3 )  (0.45 NS 3 (l )  0.4 NS 3 (l )  0.15 u 0)  J 3  J 3 0 2 4 0 1 24 24 °° 3 3  K24 0 s.t. ®0.5  0.5(Z23  Z43 )  (0.45 NS 3 (l0 )  0.4 NS 3 (l1 )  0.15 u1)  K24 ° 3 3 3 3 3 3 °0.5  0.5(Z3  Z4 )  (0.4 NS (l1 )  0.5 NS (l 2 )  0.1u 0)  J 34  J 34 0 °0.5  0.5(Z 3  Z 3 )  (0.4 NS 3 (l )  0.5 NS 3 (l )  0.1u1)  K 3  K 3 0 3 4 1 2 34 34 ° ° NS 3 (l t )  NS 3 (lt ) 1, t 5, }, 1, 0, 1, },5 ° 3 3 ° NS (l5 ) 0 and NS (l5 ) 1 ° 3 3 ° NS (lt 1 )  NS (lt ) t 0.05, t 5, }, 1, 0, 1, }, 4 °(t  4) /10 d NS 3 (l ) d (t  6) /10, t 4, }, 1, 0, 1, }, 4 t ° 4 4 ° 3 3 3 3 3 3 °0 d Zi d Zi d 1, ¦ Z j  Zi d 1, Zi +¦ Z j t 1, i 1, 2, ..., 4 j 1 j 1 ° j zi j zi ° 3 3 3 3 3 3 3 3 3 3 3 3 °¯J 123 , J 123 ,K132 ,K123 , J 133 , J 133 ,K133 ,K133 , J 143 , J 143 ,K143 ,K143 , J 23 , J 23 ,K 23 ,K 23 , J 24 , J 24 ,K 24 ,K 24 , J 34 , J 34 ,K 34 ,K 34 t0

(26)

Solving the above model, we get J 123 =K123 =J 133 =J 133 K133 =K133

3 3 3 3 J 143 =K143 =J 23 =K23 =J 24 =J 24 K243 =K243

J 343 =K343 =K343 =0 , Ȗ 3-12

= Ș 3-12

=

urn a

3 3+ 333+ 0.0750, Ȗ3+ 14 = Ș14 = 0.1600, Ȗ23 = Ș23 = 0.0475, Ȗ34 = 0.0500, F = 0.6150;

Z3 = (Z31 , Z32 , Z33 , Z34 )T = ([Z3-1 , Z3+1 ], [Z3-2 , Z3+2 ], [Z3-3 , Z3+3 ], [Z3-4 , Z3+4 ])T = ([0.3475, 0.3975], [0.0975, 0.2475], [0.1475, 0.2975], [0.0575, 0.2075])T; And N S3(l-5) = 0, N S3(l-4) = 0.2, N S3(l-3) = 0.3; N S3(l-2) = 0.4; NS3(l-1) = 0.45, NS3(l0) = 0.5, NS3(l1) = 0.55, NS3(l2) = 0.6, NS3(l3) = 0.7, NS3(l4) = 0.8, NS3(l5) = 1. (3) For the third category, i.e., S3 = {d4}, the corresponding preference relation D 4 can be transformed into its corresponding EBNPR by use of (2) and (4), which is presented in E 4. [0.60,0.70] [0.50,0.50] [0.35,0.45] [0.25,0.50]

Jo

§ [0.50,0.50] ¨ [0.30,0.40] E4 = ¨ ¨ [0.15,0.25] ¨¨ © [0.28,0.43]

[0.75,0.85] [0.55,0.65] [0.50,0.50] [0.45,0.55]

[0.57,0.72] · ¸ [0.50,0.75]¸ . [0.45,0.55] ¸ ¸ [0.50,0.50] ¸¹

From the results in E 4, we can construct the following optimization model using (17):

Journal Pre-proof 4 4 4 4 min F 4 = J 124  J 124  K124  K124  J 134  J 134  K134  K134  J 144  J 144  K144  K144  J 23  J 23  K 23  K 23 4 4 4 4 4 4 4 4  J 24  J 24  K24  K 24  J 34  J 34  K34  K34

pro of

­0.5  0.5(Z14  Z24 )  0.6  J 124  J 124 0 ° 4 4 4 4 °0.5  0.5(Z1  Z2 )  0.7  K12  K12 0 ° 4 4 4 4 °0.5  0.5(Z1  Z3 )  0.75  J 13  J 13 0 °0.5  0.5(Z 4  Z 4 )  0.85  K 4  K 4 0 1 3 13 13 ° °0.5  0.5(Z14  Z44 )  0.57  J 144  J 144 0 ° °0.5  0.5(Z14  Z44 )  0.72  K144  K144 0 ° 4 4 4 4 °0.5  0.5(Z2  Z3 )  0.55  J 23  J 23 0 ° 4 4  K 23 0 s.t. ®0.5  0.5(Z24  Z34 )  0.65  K 23 ° 4 4 4 4 . 0.5  0.5( Z  Z )  0.5  J  J 0 2 4 24 24 ° °0.5  0.5(Z 4  Z 4 )  0.75  K 4  K 4 0 2 4 24 24 ° 4 4 °0.5  0.5(Z34  Z44 )  0.45  J 34  J 34 0 ° 4 4 4 4 °0.5  0.5(Z3  Z4 )  0.55  K34  K34 0 ° 4 4 °0 d Zi4 d Zi4 d 1, ¦ Z 4j   Zi4 d 1, Zi4 +¦ Z 4j  t 1, i 1, 2, ..., 4 ° j 1 j 1 j zi j zi ° °J 4 , J 4 ,K 4 ,K 4 , J 4 , J 4 ,K 4 ,K 4 , J 4 , J 4 ,K 4 ,K 4 , J 4 , J 4 ,K 4 ,K 4 , J 4 , J 4 ,K 4 ,K 4 , J 4 , J 4 ,K 4 ,K 4 t 0 ¯ 12 12 12 12 13 13 13 13 14 14 14 14 23 23 23 23 24 24 24 24 34 34 34 34

By solving (27), we obtain

J 144  =K144  J 234  =J 234  =K234  K234  =J 244  =J 244  =K244  =J 344  =K344  =0 , Ȗ4-13 = 0.0042, Ș4-13

re-

J 124  =J 124  =K124  =K124  =J 134  =K134 

(27)

4 4+ 444= 0.0958, Ȗ4+ 14 = 0.1258, Ș14 = 0.0675, Ș24 = 0.0667, Ȗ34 = 0.0042, Ș34 = 0.0125, F = 0.3767; and

Z4 = (Z41 , Z42 , Z43 , Z44 )T = ([Z4-1 , Z4+1 ], [Z4-2 , Z4+2 ], [Z4-3 , Z4+3 ], [Z4-4 , Z4+4 ])T = ([0.5667, 0.5750], [0.1750,

Jo

urn a

lP

0.3667], [0.0667, 0.0750], [0, 0.1750])T. To clearly show the individual differences of the four reviewers in understanding the used linguistic term set, their personalized numerical scales are plotted in Figure 1. One can see here that the following linguistic terms: very much poorer, much poorer, moderately poorer, m arginally poorer, m arginally better, moderately better, much better, and very much better exhibit different meaning for different reviewers. Step 4 is completed.

Figure 1. Personalized numerical scales of linguistic terms for different reviewers. By using (24), all the normalized interval weight vectors obtained in Step 4 are aggregated into form a collective priority weight vector as Z = (Z1, Z2, Z3, Z4)T = ([0.3398, 0.4049], [0.2334, 0.3683], [0.1293, 0.2574], [0.0276, 0.1434])T. Step 5 is completed.

Journal Pre-proof The resulting priority weights of the projects are interval numbers, so we refer to the method of ranking interval numbers proposed by Xu and Da [70] to compare Zi (i =1, 2« 4). As per the procedure of this representative method, the following likelihood matrix is deduced.

P

1 1 · § 0.5 0.8577 ¨ ¸ 0.1423 0.5 0.9086 1 ¸ = ¨ ¨ 0 0.0914 0.5 0.9426 ¸ ¨¨ ¸ 0 0.0574 0.5 ¸¹ © 0

degrees x1

0.8577

x2

0.9086

x3

0.9426

pro of

From the results in P, we can generate a ranking order of the four projects with possibility

x4, indicating that project x1 is superior to x2 to the degree of

6.2. Discussion and comparative studies

re-

85.77%, x2 is superior to x3 to the degree of 90.86%, and x3 is superior to x4 to the degree of 94.26%. As such, a solution to this research project evaluation and selection problem is obtained. That is, the four projects should be preferentially chosen for implementation in the above ranking order. Step 6 is completed. This section illustrates the entire implementation process of decision-making based on DLPRs with incomplete symbolic proportions by analyzing a research project evaluation and selection problem. In this process, the DMs express their preference information in the form of DLPRs with incomplete symbolic proportions. The personalized numerical scale functions of linguistic terms for each DM can be derived and the priority weights of all alternatives can be deduced.

lP

In the GDM problem shown above, Reviewer 1 is supposed to share an identical numerical scale with Reviewer 2. In what follows, we discuss the situation where the numerical scale functions of L for Reviewers 1 and 2 are personalized and completely unknown, i.e., S1 = ĭ and S2 = {d1, d2, d3}. In such a situation, the personalized numerical scales of L for the four reviewers are plotted in Figure 2. The resulting collective priority weight vector can also be deduced, which is Z = (Z1, Z2, Z3, Z4)T = ([0.3648, 0.4237], [0.1983, 0.3634], [0.1163, 0.2177], [0.0536, 0.1782])T.

x3

0.726

urn a

Similarly, a ranking order of the four projects with possibility degrees is generated x1

1

x2

0.9272

x4, meaning that project x1 is superior to x2 to the degree of 100%, x2 is superior to x3 to

Jo

the degree of 92.72%, and x3 is superior to x4 to the degree of 72.6%.

pro of

Journal Pre-proof

Jo

urn a

lP

re-

Figure 2. Personalized numerical scales of linguistic terms for different reviewers. As analyzed in Section 3, there are two similar methodologies which use linguistic distribution assessments to recall comparison information. (1) Zhang et al. [36] introduced the concept of DLPRs based on linguistic distribution assessments. The DLPRs provide DMs with a suitable method to describe their subjective preference information in decision contexts. The proportional information over linguistic terms in a linguistic distribution assessment is complete. For example, when comparing project x1 with x2 in the GDM problem investigated in Section 6.1, if Reviewer 2 is 40% sure project x1 is much poorer than x2, and 60% sure x1 is moderately poorer than x2, then his preference information about (x1, x2) can be recalled as {(much poorer, 0.4), (moderately poorer, 0.6)}. However, as the reviewers do in the research project evaluation and selection problem, DMs may offer incomplete symbolic proportions over the given linguistic term set in practical decision situations. The proposal in this paper addresses this issue by extending DLPRs. As such, the DLPR introduced in [36] is a special case of the one in this paper. In addition, the numerical preferences of linguistic terms for each DM in [36] are the same. (2) Zhang et al. [37] presented another similar way of pairwise comparison, namely, the probabilistic linguistic preference relation. The probabilistic linguistic preference relation records the comparison information using probabilistic linguistic term sets [1]. About the differences between the probabilistic linguistic term set and the linguistic distribution assessment, Pang et al. performed a detailed discussion in [1]. If a probabilistic linguistic preference is in the upper triangular matrix, the linguistic terms associated with their probabilities are arranged in ascending order based on the products of those linguistic WHUPV¶VXEVFULSWV and the corresponding probabilities, and vice versa if the probabilistic linguistic preference is in the lower triangular matrix. In our proposal, this condition is relaxed. Moreover, the incomplete probabilistic information in a probabilistic linguistic preference is averagely assigned to the possible linguistic terms appeared in the preference during its normalization process, while the incomplete proportional information in an incomplete linguistic distribution preference could be assigned to any subset of the given linguistic term set. What¶s more, the approach in [37] can only be suitable to handle the linguistic GDM problems where DMs have an identical

Journal Pre-proof understanding of the given linguistic term set. Based on the above analysis, we   summarize the comparison results in Table 1. From the comparison, one can find that the proposal in this paper not only incorporates such a situation as the linguistic distribution preference with incomplete symbolic proportions, but also reflects the different understanding of linguistic terms for different DMs. Table 1 A summary on the comparison of different approaches Proportional

The way of handling

Numerical

of preferences

information over

incomplete

linguistic terms for different

linguistic terms

proportions

DMs

Complete

--

Same

Incomplete

Averagely assigned to

Linguistic distribution assessments

Probabilistic

Linguistic

linguistic

distribution

several

preference

assessments

linguistic terms

relations [37] Linguistic

Incomplete

distribution

of

Assigned

Same

possible

to

re-

Our proposal

preferences

pro of

DLPRs [36]

The expression

any

Personalized

subset of the given

assessments

linguistic term set

7. Concluding remar ks

Jo

urn a

lP

Being different from the situations investigated by [18] and [36], in this paper, we consider another situation where the proportional information in a linguistic distribution assessment is incomplete, and put forward the concept of DLPR with incomplete symbolic proportions. The main contributions in this paper are outlined as follows. (1) This paper proposes the concept of DLPR with incomplete symbolic proportions, in which the proportional information is incomplete, namely, the total summation of such symbolic proportions is less than 1. Two aggregation operators for DLPR matrices with incomplete symbolic proportions and their desirable properties are presented as well. (2) This paper defines the expectations of incomplete linguistic distribution preferences to facilitate their comparison, and connects incomplete linguistic distribution preferences to interval fuzzy preferences by numerical scale models. (3) The consistency of DLPR matrices with incomplete symbolic proportions is studied through the corresponding EBNPRs. Based on the consistency definition, some goal-programming models to compute numerical scales  of the linguistic term set in DLPR matrix with incomplete symbolic proportions are developed for individual and GDM situations. (4) An approach for addressing GDM problems with incomplete linguistic distribution preference information is proposed. A GDM problem of evaluating and selecting research projects is analyzed to illustrate the implementation process of the proposed approach. The validity of application of DLPR matrices with incomplete symbolic proportions to GDM is also demonstrated in the process. When addressing GDM problems, an important process is that of consensus reaching [56, 71, 72]. In the future, novel consensus models for solving GDM with the newly introduced preference

Journal Pre-proof relations will be studied. A cknowledgements This research was supported by the Foundation for Innovative Research Groups of the Natural Science Foundation of China (No. 71521001), and the National Natural Science Foundation of China (Nos. 71690230, 71690235, 71501056, 71601066, 71501055, 71571060, 71501054 and 71571166).

pro of

References [1] Q. Pang, H. Wang, and Z. S. Xu, "Probabilistic linguistic linguistic term sets in multi-attribute group decision making," Information Sciences, vol. 369, pp. 128-143, Nov 10 2016.

[2] X. A. Tang, C. Fu, D. L. Xu, and S. L. Yang, "Analysis of fuzzy Hamacher aggregation functions for uncertain multiple attribute decision making," Information Sciences, vol. 387, pp. 19-33, May 2017.

[3] F. Mata, L. Martinez, and E. Herrera-Viedma, "An Adaptive Consensus Support Model for Group Decision-Making Problems in a Multigranular Fuzzy Linguistic Context," Ieee Transactions on

Fuzzy Systems, vol. 17, pp. 279-290, Apr 2009.

re-

[4] R. M. Rodriguez, L. Martinez, and F. Herrera, "A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets," Information

Sciences, vol. 241, pp. 28-42, Aug 20 2013.

[5] P. Wang, X. H. Xu, S. Huang, and C. G. Cai, "A Linguistic Large Group Decision Making Method Based on the Cloud Model," Ieee Transactions on Fuzzy Systems, vol. 26, pp. 3314-3326, Dec 2018.

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[6] Z. Zhang and W. Pedrycz, "A Consistency and Consensus-Based Goal Programming Method for Group Decision-Making With Interval-Valued Intuitionistic Multiplicative Preference Relations,"

IE E E Trans Cybern, pp. 1-15, Jun 19 2018.

[7] F. Liu and W. G. Zhang, "TOPSIS-Based Consensus Model for Group Decision-Making with Incomplete Interval Fuzzy Preference Relations," Ieee Transactions on Cybernetics, vol. 44, pp.

urn a

1283-1294, Aug 2014.

[8] N. Capuano, F. Chiclana, H. Fujita, E. Herrera-Viedma, and V. Loia, "Fuzzy Group Decision Making With Incomplete Information Guided by Social Influence," Ieee Transactions on Fuzzy

Systems, vol. 26, pp. 1704-1718, Jun 2018. [9] X. A. Tang, S. L. Yang, and W. Pedrycz, "Multiple attribute decision-making approach based on dual hesitant fuzzy Frank aggregation operators," Applied Soft Computing, vol. 68, pp. 525-547, Jul 2018.

[10] S.-M. Chen, "Fuzzy group decision making for evaluating the rate of aggregative risk in software

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Systems with Applications, vol. 99, pp. 83-92, Jun 1 2018.

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*Highlights (for review)

Journal Pre-proof Highlights (1) Introduce the concept of distribution linguistic preference relation (DLPR) with incomplete symbolic proportions to record comparison information. (2) Present two aggregation operators for DLPRs with incomplete symbolic proportions and investigate their desirable properties. (3) Study the consistency of DLPRs with incomplete symbolic proportions.

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(4) Develop several goal-programming models to personalize numerical scales of linguistic terms for individual and GDM situations.

(5) Propose a GDM approach with incomplete linguistic distribution preference information and

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analyze its application to a problem of evaluating and selecting research projects.

Journal Pre-proof *Declaration of Interest Statement

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

*Author Contributions Section

Journal Pre-proof Author Contributions

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Xiaoan Tang: Writing- Original draft preparation, Investigation; Qiang Zhang: Methodology, Software; Zhanglin Peng: Conceptualization, Project administration; Witold Pedrycz: Supervision, Language improvement; Shanlin Yang: Supervision, Funding acquisition.