Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model

Applied Soft Computing Journal xxx (xxxx) xxx

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Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model ∗

Peng Wu a , Jinpei Liu b , Ligang Zhou a,c , , Huayou Chen a a

School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China School of Business, Anhui University, Hefei, Anhui, 230601, China c China Institute of Manufacturing Development, Nanjing University of Information Science and Technology, Nanjing, Jiangsu, 210044, China b

article

info

Article history: Received 17 March 2019 Received in revised form 12 October 2019 Accepted 18 November 2019 Available online xxxx Keywords: Group decision making Linguistic preference relation Additive consistency Integer optimization

a b s t r a c t Linguistic preference relation (LPR) composed by linguistic terms can well express decision makers’ (DMs’) qualitative preference opinion by comparing alternatives with each other. The investigation of its consistency becomes an important issue to guarantee the rationality of the decision making solutions. Therefore, it is significant to investigate the consistency measure and the consistency improving approach for LPRs. In this paper we present a new method for group decision making (GDM) with LPRs. First, an additive consistency index is introduced on the basis of the information of the original LPR to check whether a LPR is acceptably additive consistency. For unacceptably additively consistent LPR, an integer optimization model is further developed to obtain the acceptably additively consistent LPR. Moreover, the optimization model can guarantee the integrity of the information of the LPR with acceptably additive consistency. Then, with respect to GDM with LPRs, an entropy weight method is proposed to determine the weights of DMs. Finally, the proposed methods are implemented in two numerical examples including a GDM problem. Meanwhile, the comparative analysis with existing methods are discussed in detail to demonstrate the validity of the proposed methods. © 2019 Elsevier B.V. All rights reserved.

1. Introduction For decision making problems, it often involves multiple alternatives and uncertainty of decision making environment, and it is hard for a decision maker (DM) to give a reliable decision without an overall view. As a result, theory and application of group decision making (GDM) were proposed at that moment. Based on the preference opinions, GDM aims at selecting the optimal alternative(s) from a set of feasible alternatives. Recently, based on different decision environment, research on GDM has been extensively widely studied [1–4] and they are very significant. As one of the effective tools to express DMs’ preference opinion in decision process, preference relations have been received significant level of attention. In recent years, various preference relations are investigated to deal with GDM problems, including multiplicative preference relation [5–7], fuzzy preference relation [8,9], interval valued preference relation [10], triangular fuzzy preference relation [11], trapezoidal fuzzy preference relation [12–14], intuitionistic fuzzy preference relation [15], hesitant fuzzy preference relation [16]. ∗ Corresponding author at: School of Mathematical University, Hefei, Anhui, 230601, China. E-mail address: [email protected] (L. Zhou).

Sciences,

Anhui

As the rapid development of social economy, the uncertain and complex the real-world decision making problems are highlighted. DMs would be inclined to express their preference by using qualitative preference opinion in such decision making environment. Hence, the linguistic preference relation (LPR) is proposed, where the judgments on the LPR (or complete LPR) are expressed by linguistic terms from a linguistic term set. However, sometimes, DMs may give their LPRs with some values missing. It may be that DMs does not have enough knowledge for a specific problem, or because the DMs does not have the ability to discriminate the degree to which some alternatives are better than others. In this situation, LPRs are called incomplete LPRs. Consistency is one of the important characteristics of preference relations. It ensures that preference relations are neither random nor illogical in pairwise comparisons. However, the basic condition to ensure the logical of preference relation is the ordinal consistency. Please see [17–20]. Due to the limitation of human’s cognitive ability and the complexity of the decision making problems, it may be difficult for DMs to give judgments with complete consistency on alternatives in some special cases. Nevertheless, lack of consistency of preference relation in decision making may lead to a misleading solution [21–26]. In recent years, since LPRs and incomplete LPRs were proposed, much research has been devoted to discussing the consistency measure and consistency improving algorithms of them. The research on consistency measure for LPRs and incomplete LPRs are summarized in Table 1.

https://doi.org/10.1016/j.asoc.2019.105955 1568-4946/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

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P. Wu, J. Liu, L. Zhou et al. / Applied Soft Computing Journal xxx (xxxx) xxx

Table 1 Discussion of the related work about consistency measure of LPRs and incomplete LPRs. Types of LPR

Literatures

Advantages (A) and drawbacks (D)

[27–35]

(A) The additive consistency index of an LPR is defined. (D) These method for defining additive consistency index may result in diverse indices.

[29,30,36]

(A) The multiplicative consistency index of LPR is defined [29,36]. The multiplicative consistency measure of multiplicative LPR is defined [30]. (D) These method for defining additive consistency index may result in diverse indices.

[28,36,37]

(A) The order consistency is defined [28,36]. The additive consistency is defined [37]. The ordinal consistency is defined [17,18]. The cardinal consistency is defined [17]. (D) They are unable to quantify the consistency level of LPRs.

[38–43]

(A) Some GDM methods are developed. (D) They did not discuss the consistency of LPRs.

[44–49]

(A) Some methods to estimate the unknown values are proposed. (D) The completing values may be virtual linguistic variables [44,46–48]. Consistency level is not discussed [45,49].

[50,51]

(A) The consistency measure is defined. (D) Consistency level and the method to estimate the unknown values are not discussed.

[52]

(A) Two distinct completing algorithms are presented. (D) The selection of completing values is random.

[53]

(A) The ordinal consistency is defined. (D) The decision process may be complicated.

LPRs

Incomplete LPRs

As can been seen from Table 1, the related work of consistency of LPRs and incomplete LPRs are summarized from two aspects: advantages and drawback. For LPRs, the related work mainly focuses on discussing the consistency measure, including additive consistency, multiplicative consistency, order consistency and ordinal consistency. However, some of them may result in diverse consistency indices. For incomplete LPRs, the related work mainly is devoted to designing the completing methods for unknown values. The research on consistency improving method for LPRs and incomplete LPRs are summarized in Table 2. As can been seen from Table 2, the related work of consistency improving method for LPRs and incomplete LPRs are summarized from two aspects: advantages and drawback. In this aspect, for LPRs, two parts, including automatic iterative methods [28,31,36] and optimization models [27,35], are performed to improve the consistency level of LPR without acceptable consistency. For incomplete LPRs, few investigations have been done to dig into the consistency improving method. This paper focuses on investigate the consistency index and consistency improving method for LPR. Based on Tables 1 and 2, for LPRs, there are still some limitations as follows:

• In the above literatures [28,31,32,36], some consistency indices of LPRs were developed on the basis of the deviation between estimated LPR and its consistent LPR. Different methods generally would provide various consistent LPRs based evaluated LPRs, which may result in diverse consistency indices of LPRs. In reality, the consistency degree of the LPR should not change with its related to completely consistent LPR. Meanwhile, some methods [27,33] used the transformation function and consistency of fuzzy preference relations to define consistency index of LPR. It may cause loss of preference information provided by DMs. Based on numerical scale and interval numerical scale, in [34] the authors first transformed LPR into interval fuzzy preference relation and only discussed the consistency by using optimistic consistency and pessimistic consistency. However, the consistency of LPR was ignored. • Some consistency improving methods [28,36] adopted automatically iterative algorithms to modify the unacceptably consistent LPRs. Other method [31] used an optimization method to modify LPRs with unacceptable consistency. These consistency improving methods are automatic and save time as there is no need for further DMs interactions.

Meanwhile, in [31] the authors presented an iterative algorithm to revise LPRs with unacceptable consistency. It allows that the DMs to participate in dealing with unacceptable LPRs. However, the revised linguistic terms derived from consistency improving methods [28,31,36] are virtual terms, so in some actual decision making problems, it may lead to the problem being difficult to understand. 1.1. Motivations and contribution The motivations of this paper primarily come from the above weaknesses. For consistency index, it is meaningful to develop a stable consistency index for investigating the consistency level of LPRs. For consistency improving method, it is necessary to present an effective consistency improving method, which can guarantee revised elements for LRPs are all simple linguistic terms. In this paper, for decision problems with LPRs, a new method is proposed composed by a stable additive consistency, an integer optimization-based model and an entropy weight model. The contribution of this paper mainly are as follows: (1) According to the Hamming distance, the additive consistency index of LPR is developed, which only relies on the preference information of the original LPR. Compared with traditional consistency indices, it is stable and reliable. (2) For improving the additive consistency level of LPR without acceptable additive consistency, a mathematical optimization model is presented. Furthermore, in order to guarantee the revised linguistic terms of modified LPR with unacceptable additive consistency are simple linguistic terms, an integer mathematical programming model is further established. (3) The final solutions should be obtained by aggregating all the individual LPR with acceptable additive consistency in GDM. Therefore, an entropy weight model is presented to derive the DMs’ weights. 1.2. Outline of the paper The remainder of the paper is unfolded as follows. Section 2 mainly reviews several knowledges associated with linguistic term set and LPR. In Section 3, we define the additive consistency measure of LPR. After that, an integer optimization model is constructed to derive an acceptably additively consistent LPR

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

P. Wu, J. Liu, L. Zhou et al. / Applied Soft Computing Journal xxx (xxxx) xxx

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Table 2 Discussion of the related work about consistency improving methods for LPRs and incomplete LPRs. Types of LPR LPRs

Incomplete LPRs

Literatures

Advantages (A) and drawbacks (D)

[27,28,31,35,36]

(A) An optimization method and an iterative algorithm are developed [31]. An iterative algorithm is developed [28,36]. An optimization method is proposed [27,35]. (D) These methods may result in revised LPR contains virtual linguistic terms.

[17,44–53]

(A) An algorithm is proposed to improve incomplete LPR’s ordinal consistency [46,53]. Discussion about consistency of incomplete LPRs [17,44,45,47–52]. (D) The completing values may be virtual linguistic variables [44,46–48]. Consistency level is not discussed [45,49,50]. The completing values may be virtual linguistic variables [46]. The process of algorithm may be complicated [53].

from an unacceptably additively consistent one. In Section 4, an entropy weight approach is proposed to obtain the DMs’ weights. Section 5 presents a novel approach for solving GDM with LPRs. In Section 6, we demonstrate the validity of the proposed models by using some examples. Finally, Section 7 epitomizes the main conclusions of the paper. 2. Preliminaries Some concepts of linguistic preference relation (LPR) and consistent LPR are mainly reviewed. 2.1. Linguistic variable and its operations A subscript-symmetric additive linguistic term set denoted as S = {sα |α = −τ , . . . , 0, . . . , τ } with odd cardinality, in which τ is a positive integer, s−τ and sτ are the lower bound and upper bound of S respectively, and S satisfies the following characteristics [30]: (1) Ordered: α ≥ β ⇔ sα ≥ sβ ; (2) Negation operator: neg(sα ) = s−α . For example, a subscript-symmetric linguistic term set with 9 linguistic variables can be defined as: s−4 = extremely bad, s−3 = v ery bad, s−2 = bad, s−1 = slightly bad, s0 = medium, s1 = slightly good, . s2 = good, s3 = v ery good, s4 = extremely good

{ ′

S =

}

Definition 2 ([30]). Let ˜ A = (˜ aij )n×n be as before. If it satisfies:

˜ aik = ˜ aij ⊕ ˜ ajk , ∀i, j, k, Then ˜ A is a consistent LPR.

(2)

Note that throughout this paper, let Ln be the set of all LPRs. For convenience, we utilize I(sα ) to denote the subscript of linguistic term sα ∈ S, then we have I(sα ) = α . More generally, Chen, Zhou and Han [54] developed the following theorem: Theorem 1 ([54]). If sαi ∈ Sˆ and λi ∈ [0, 1], then

( I

n

⊕ λi sαi

) =

i=1

n ∑

λi I(sαi ).

(3)

i=1

3. Additive consistency of linguistic preference relation This section first develops an additive consistency index of a LPR. Next, an integer optimization model is established to derive an acceptably additively consistent LPR from one with unacceptably additive consistency. 3.1. Additive consistent index of linguistic preference relation

To preserve all the preferences of S, Xu [43] developed a continuous additive linguistic term set Sˆ = {sα |α ∈ [−p, p] }, where p(p > τ ) represents a sufficiently large positive integer. If sk ∈ S, we call sk the original linguistic term, otherwise, we call sk the virtual additive linguistic term. ˆ then we have the Let sα and sβ be two linguistic variables of S, following operations [30]: (1) sα ⊕ sβ = sα+β . (2) λsα = sλα . (3) sα ⊕ sβ = sβ ⊕ sα . (4) λ(sα ⊕ sβ ) = λsα ⊕ λsβ , λ ∈ [0, 1]. (5) (λ + µ)sα = λsα ⊕ µsα , λ, µ ∈ [0, 1]. 2.2. Linguistic preference relation A finite set of alternatives denoted by X = {x1 , x2 , . . . , xn }. Based on S and X , a DM can express his/her opinion using LPR. The LPR can be defined as: Definition 1 ([30]). A LPR on the set X is characterized by a linguistic decision matrix ˜ A = (˜ aij )n×n . For ∀i, j = 1, 2, . . . , n, it has the following properties

˜ aij ∈ S ,˜ aij ⊕ ˜ aji = s0 ,˜ aii = s0 ,

A very important property of LPR is the consistency [30]. It can be defined as Definition 2.

(1)

where ˜ aij represents the preference degree of the alternative xi over xj . Especially, ˜ aij = s0 represents that xi is equivalent to xj , ˜ aij > s0 represents that xi is preferred to xj , and ˜ aij < s0 represents that xj is preferred to xi .

Let sα , sβ ∈ S, Xu [43] defined the distance between sα and sβ by d(sα , sβ ) =

|α − β|

(4) T where T = 2τ is the number of linguistic terms of S. In some specified situations, it is hard for DMs to give LPRs with completely additive consistency. According to Definition 2, Xu [30] presented the concept of additively consistent LPR. It is used to check whether a LPR is additive consistency or not. The consistency level of an inconsistent LPR cannot be measured by Xu [30]. For measuring the consistency level of LPR, the additive consistency index is defined below. In line with Eq. (2), we can utilize the deviation between ˜ aij ⊕ ˜ ajk and ˜ aik to measure the consistency level of LPR ˜ A = (˜ aij )n×n . Based on Hamming distance, the total deviation is determined as D(˜ A) =

n ⏐ 1 ∑⏐ ⏐I(˜ aij ) + I(˜ ajk ) − I(˜ aik )⏐ . 3τ

(5)

i
Then, based on the D(˜ A), the additive consistency index of ˜ A is defined as follows. Definition 3. Let ˜ A = (˜ aij )n×n be as before, then its additive consistency index is defined as ACI(˜ A) =

1

2

τ n(n − 1)(n − 2)

n ∑ ⏐ ⏐ ⏐I(˜ aij ) + I(˜ ajk ) − I(˜ aik )⏐ .

(6)

i
Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

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The ACI(˜ A) reflects the rationality of DM’ preference information ˜ A. Obviously, the smaller of ACI(˜ A), the more rational and consistent opinions in the LPR ˜ A. As we can see, the additive consistency index ACI(˜ A) satisfies ACI(˜ A) ∈ [0, 1]. If ACI(˜ A) = 0, then ˜ A = (˜ aij )n×n is completely additive consistency. Specially, the additive consistency index ACI can be equal to 1. For example, suppose that ˜ AExample = (˜ aij )3×3 is a LPR on XExample = {x1 , x2 , x3 } which is shown as follows:

( ˜ AExample =

s0 s−4 s4

s4 s0 s−4

s−4 s4 s0

Theorem 2. The objective function of (M-1) is bounded.

∑n ⏐

ij

. 1 1 4 3

(|4 + 4 + 4|) =

Definition 4. Let CI be as before. If ACI(˜ A) ≤ CI, then LPR ˜ A is acceptably additive consistency. Otherwise, ˜ A is unacceptably additive consistency. In particularly, if ACI(˜ A) = 0, LPR ˜ A is completely additive consistency. Generally speaking, the consistency threshold CI is predefined in the interval [0, 1]. For LPRs, Dong et al. [31] discussed the consistency threshold in detail based on statistical perspective. They gave a fixed selection standard to select different threshold with regard to different decision making problems. 3.2. An integer optimization model to derive the acceptably additively consistent linguistic preference relation

aij ). 0 ≤ ⏐I(˜ aij ) − I(˜ a∗ij )⏐ ≤ τ + I(˜

⏐

⏐

0 ≤ obj3 ≤

2 n2 (n − 1)2

Inspired by [27,31,35,55], for an unacceptably additively consistent LPR ˜ A = (˜ aij )n×n , an important mission is to discover a LPR ˜ A∗ = (˜ a∗ij )n×n with acceptably additive consistency. Meanwhile, we make ˜ A∗ as close as possible to ˜ A. We can accomplish this goal by minimizing the distance between ˜ A and ˜ A∗ . Accordingly, an optimization model (M-1) is constructed by:

s.t .

The objective function is to preserve preference information of LPR ˜ A as much as possible. Namely, it can guarantee ˜ A∗ as close as possible to ˜ A. In model (M-1), the first constraint condition

i
I(˜ aij ).

Based on model (M-1), assume that

{ εij , εij ≥ 0 0, εij ≥ 0 , εij− = , 0, εij < 0 −εij , εij < 0 { δijk , δijk ≥ 0 + δijk = I(˜ a∗ij ) + I(˜ a∗jk ) − I(˜ a∗ik ), δijk = 0, δijk < 0, { 0 , δ ≥ 0 ijk − δijk = , −δijk , δijk < 0 ⏐ ⏐ ⏐ ⏐ then ⏐εij ⏐ = ε + +ε − , εij = ε + −ε − , ⏐δijk ⏐ = δ + +δ − , δijk = δ + −δ − . εij = I(˜ aij ) − I(˜ a∗ij ), εij+ =

ij

{

ij

ij

ijk

ijk

ijk

ijk

Hence, the following goal programming (M-2) is equivalent to model (M-1): (M − 2) min

s.t .

(7)

n(n − 1) τ

Remark 2. For the case where multiple solutions exist in model (M-1), any optimal solution is feasible for model (M-1) from a theoretical point of view, and it can be used as the final optimal solution of model (M-1). For actual decision problems, we hope to find all the multiple solutions of model (M-1) as much as possible. Then, according to the actual decision problems, the DM chooses an optimal solution that best suits his or her fundamental interests from all the optimal solutions as the final optimal solution of model (M-1). On the other hand, inspired by Wu et al. [57], we also can utilize the ‘core’ concept used in the dominance-based rough set approach [58] to deal with the case.

i
n ⏐ 2 1 ∑⏐ ∗ ⏐I(˜ aij ) + I(˜ a∗jk ) − I(˜ a∗ik )⏐ ≤ CI n(n − 1)(n − 2) τ i
n 1∑

1

Property 1. Model (M-1) exits at least one optimal solution.

n ⏐ 1 ∑⏐ ⏐I(˜ aij ) − I(˜ a∗ij )⏐

⎧ ⎪ ⎨

+

If there exists a ˜ A∗ = (˜ a∗ij )n×n within each ˜ a∗ij = s0 , it is a feasible solution of (M-1). It means that the (M-1) has feasible solutions. Based on Theorem 2 and Weierstrass’ Theorem [56], the following property is apparent.

ij

n(n − 1) τ

ij

Then, we have

According to a consistency threshold CI ∈ [0, 1], the acceptably additively consistent LPR is defined in Definition 4.

1

ij

Since −τ ≤ I(˜ a∗ij ) ≤ τ , it follows that

Remark 1. Based on the deviation of LPR and the corresponding additively consistent LPR, Jin et al. [28] defined a consistency index of a LPR. Furthermore, Jin et al. [36] defined a multiplicative consistent LPR. Next, the consistency index is developed by calculating the deviation between the LPR and the corresponding multiplicatively consistent LPR. Dong et al. [31] presented a consistency index of a LPR by computing the deviation between the LPR and its consistent LPR. Zhou et al. [32] measured the consistency level of LPR by computing the compatibility between the LPR and any LPR given by a leading DM. Generally speaking, different approaches for solving consistent LPRs would obtain various different consistent LPRs, which would result in various consistency indices for LPRs. In the paper, the additive consistency index with reliability and stability defined in Definition 3 since it only utilizes the original information of LPR.

(M − 1) min

⏐

⏐ ˜ ˜∗ ⏐ Proof. Let J = n(n1−1) τ1 i
)

According to Eq. (6), we have ACI(˜ AExample ) = 1.

guarantees that the obtain matrix ˜ A∗ is acceptably additive consistency. The second condition is to accord with the expression of the DMs.

1

n n 1∑∑

n(n − 1) τ

(εij+ + εij− )

i=1 j=i+1 ⎧ n ⎪ 2 1 ∑ + ⎪ − ⎪ (δijk + δijk ) ≤ CI ⎪ ⎪ ⎪ n(n − 1)(n − 2) τ ⎪ i
(8)

By solving model (M-2), the optimal solutions ˜ a∗ij are derived. Denoting the optimal solution to model (M-2) as ˜ a∗ij = aij (i < j),

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

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then according to Definition 1, the adjusted LPR ˜ A∗ = (˜ a∗ij )n×n is constructed by

{ ∗

˜ aij =

aij , s0 , neg(aji ),

i j.

(9)

Example 1. Let S be as before, and let ˜ A be a LPR shown as follows:

⎛

s0 ⎜ s−1 ˜ A=⎝ s−2 s−2

s1 s0 s−4 s−3

⎞

s2 s4 s0 s−4

s2 s3 ⎟ . s4 ⎠ s0

In the light of Eq. (6), ACI(˜ A) = 0.2917. It means that the LPR ˜ A is unacceptably additive consistency. Using model (M-2), a goal programming model is built by: min

s.t .

1 + − + − + − + − + − (ε + ε12 + ε13 + ε13 + ε23 + ε23 + ε24 + ε24 + ε34 + ε34 ) 48 ⎧ 12 1 ⎪ + − + − + − + − ⎪ ⎪ (δ123 + δ123 + δ124 + δ124 + δ134 + δ134 + δ234 + δ234 ) ≤ CI , ⎪ ⎪ ⎪ 48

⎪ ⎪ ⎪ + − + − ⎪ I(˜ a12 ) − I(˜ a∗12 ) − ε12 + ε12 = 0, I(˜ a13 ) − I(˜ a∗13 ) − ε13 + ε13 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ + − + − ∗ ∗ ⎪ I(˜ a14 ) − I(˜ a14 ) − ε14 + ε14 = 0, I(˜ a23 ) − I(˜ a23 ) − ε23 + ε23 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ + − + − ∗ ⎪ ⎪I(˜ a24 ) − I(˜ a24 ) − ε24 + ε24 = 0, I(˜ a34 ) − I(˜ a∗34 ) − ε34 + ε34 = 0, ⎪ ⎪ ⎪ ⎨ ∗ + − ∗ ∗ I(˜ a12 ) + I(˜ a23 ) − I(˜ a13 ) − δ123 + δ123 = 0, ⎪ ⎪ + − ∗ ∗ ⎪ I(˜ a ) + I(˜ a24 ) − I(˜ a∗14 ) − δ124 + δ124 = 0, ⎪ ⎪ ⎪ 12 ⎪ ⎪ + − ∗ ∗ ∗ ⎪ a13 ) + I(˜ a34 ) − I(˜ a14 ) − δ134 + δ134 = 0, ⎪ ⎪ I(˜ ⎪ ⎪ ⎪ + − ⎪ ∗ ∗ ⎪ I(˜ a23 ) + I(˜ a34 ) − I(˜ a∗24 ) − δ234 + δ234 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a∗12 ≤ s4 , s−4 ≤ ˜ a∗13 ≤ s4 , s−4 ≤ ˜ a∗14 ≤ s4 , ⎪s−4 ≤ ˜ ⎪ ⎪ ⎪ ⎩ s−4 ≤ ˜ a∗23 ≤ s4 , s−4 ≤ ˜ a∗24 ≤ s4 , s−4 ≤ ˜ a∗34 ≤ s4 .

Solving the goal programming model, the optimal solutions ˜ a∗ij (i ∗ < j) are obtained. Then, by Eq. (9), LPR ˜ A is obtained as

⎛

s0 ⎜ s0 ∗ ˜ A =⎝ s−2 s−2

s0 s0 s−2 s−3

s2 s2 s0 s−3.2328

⎞

s2 s3 ⎟ . s3.2328 ⎠ s0

It must be mentioned that the modified linguistic terms s3.2328 a∗43 ∈ / S. However, and s−3.2328 in ˜ A∗ are virtual terms, namely, ˜ a∗34 , ˜ in actual decision problem, it may be hard to persuade the DM to adopt ˜ A∗ in this way as their new preference. To overcome this drawback, we hope to add some constraint conditions to ensure the optimal adjusted linguistic terms belong to S, i.e. I(˜ a∗ij ) ∈ {−τ , . . . , 0, . . . , τ }. Based on model (M-2), an integer programming model, which is to manage the additive consistency of LPR ˜ A, is established: (M − 3) min

s.t .

1

From the fifth constraint condition in model (M-3), model (M-3) is an integer programming model, which can be solved by LINGO Software or MATLAB Optimization Toolbox. Example 2. This example is the continuation of Example 1. Based on model (M-3), we obtain 1

+ − + − + − + − + − (ε12 + ε12 + ε13 + ε13 + ε23 + ε23 + ε24 + ε24 + ε34 + ε34 ) ⎧48 1 + ⎪ − + − + − + − ⎪ + δ123 + δ124 + δ124 + δ134 + δ134 + δ234 + δ234 ) ≤ CI , ⎪ (δ123 ⎪ ⎪ 48 ⎪ ⎪ ⎪ ⎪ + − + − ⎪ ⎪I(˜ a12 ) − I(˜ a∗12 ) − ε12 + ε12 = 0, I(˜ a13 ) − I(˜ a∗13 ) − ε13 + ε13 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ + − + − ⎪I(˜ ⎪ a ) − I(˜ a∗14 ) − ε14 + ε14 = 0, I(˜ a23 ) − I(˜ a∗23 ) − ε23 + ε23 = 0, ⎪ ⎪ 14 ⎪ ⎪ ⎪ + − + − ∗ ∗ ⎪ ⎪ I(˜ a24 ) − I(˜ a24 ) − ε24 + ε24 = 0, I(˜ a34 ) − I(˜ a34 ) − ε34 + ε34 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ + − ⎪ ⎪I(˜ a∗12 ) + I(˜ a∗23 ) − I(˜ a∗13 ) − δ123 + δ123 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ + − ∗ ∗ ∗ ⎪ ⎪ I(˜ a12 ) + I(˜ a24 ) − I(˜ a14 ) − δ124 + δ124 = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ∗ + − ˜ ˜∗ ˜∗ s.t . I(a13 ) + I(a34 ) − I(a14 ) − δ134 + δ134 = 0, ⎪ ⎪ ⎪I(˜ + − ∗ ⎪ a∗34 ) − I(˜ a∗24 ) − δ234 + δ234 = 0, ⎪ ⎪ a23 ) + I(˜ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ⎪ a12 ≤ s4 , s−4 ≤ ˜ a13 ≤ s4 , ⎪s−4 ≤ ˜ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s−4 ≤ ˜ a∗14 ≤ s4 , s−4 ≤ ˜ a∗23 ≤ s4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s−4 ≤ ˜ a∗24 ≤ s4 , s−4 ≤ ˜ a∗34 ≤ s4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪˜ a∗ ∈ {−4, . . . , 0, . . . , 4} ,˜ a∗13 ∈ {−4, . . . , 0, . . . , 4} , ⎪ ⎪ ⎪ 12 ⎪ ⎪ ⎪ ⎪ ˜ a∗14 ∈ {−4, . . . , 0, . . . , 4} ,˜ a∗23 ∈ {−4, . . . , 0, . . . , 4} , ⎪ ⎪ ⎪ ⎪ ⎩ ∗ ˜ a24 ∈ {−4, . . . , 0, . . . , 4} ,˜ a∗34 ∈ {−4, . . . , 0, . . . , 4} .

min

Solving the above model, the optimal solutions ˜ a∗ij are obtained. ∗ Then, ˜ A is obtained based on Eq. (9) as

⎛

s0 s ⎜ ∗ ˜ A =⎝ 0 s−2 s−2

⎛

˜ AITER

s0 ⎜ s−0.5000 =⎝ s−1.7500 s−3.5000

i
(10)

s2 s2 s0 s−3

⎞

s2 s3 ⎟ . s3 ⎠ s0

s0.5000 s0 s−2.5000 s−3.7500

s1.7500 s2.5000 s0 s−1.7500

⎞

s3.5000 s3.7500 ⎟ . s1.7500 ⎠ s0

Meanwhile, according to the optimization method, we also obtain the revised LPR ˜ AOPTI

⎛

˜ AOPTI

n(n − 1) τ

s0 s0 s−2 s−3

In [31], both iterative and optimal methods were developed, which are now utilized to improve the additive consistency of LPR ˜ A. By the iterative algorithm, we obtain the revised LPR ˜ AITER

n

1∑ + (εij + εij− )

5

s0 ⎜ s−0.7602 =⎝ s−1.9521 s−2.2877

s0.7602 s0 s−3.6164 s−3.1439

s1.9521 s3.6164 s0 s−3.5685

⎞

s2.2877 s3.1439 ⎟ . s3.5685 ⎠ s0

In [28], the automatic iterative algorithm was presented. Now, we utilize it to improve the additive consistency of LPR ˜ A. Then, the modified LPR ˜ AAUTO is

⎛

˜ AAUTO

s0 ⎜ s−0.0625 =⎝ s−1.8125 s−3.5625

s0.0625 s0 s−2.5000 s−3.1439

s1.8125 s2.5000 s0 s−2.3125

⎞

s3.5625 s3.1439 ⎟ . s2.3125 ⎠ s0

It should be observed that in ˜ AITER , ˜ AOPTI and ˜ AAUTO , the revised linguistic terms are virtual terms. The internal mechanism of

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

6

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these consistency improving algorithms will lead to this phenomenon. However, using the integer optimization model, the modified linguistic terms are not virtual linguistic terms. It may be easy for DMs to accept the modified linguistic terms as their new preference information. Therefore, compared with existing consistency improving algorithms, the integer optimization model and the revised LPR ˜ A∗ are easier to describe to DM. 4. Derive the weights of decision makers for group decision making

(t ) Theorem 4. Let ˜ A(t ) = (˜ aij )n×n and Aˆ = (aˆ ij )n×n be as before.

ˆ ≤ CI, in which ACI(A) ˆ is If ACI(˜ A(t ) ) ≤ CI for all t, then ACI(A) called group additive consistency index and ACI(˜ A(t ) ) is called the individual additive consistency index. Proof. Since ACI(˜ A(t ) ) ≤ CI for all t, we have 2

ACI(˜ A(t ) ) =

τ n(n − 1)(n − 2)

n ⏐ ⏐ ∑ ⏐ (t ) (t ) (t ) ⏐ aij ) + I(˜ ajk ) − I(˜ aik )⏐ ≤ CI . ⏐I(˜

i
m

In this section, the properties of group linguistic preference relation have been discussed. To determine DMs’ weights, a entropy weight model is developed.

The set of DMs and weighting vector of DMs are denoted by T D = {d1 , d2 , . . . , dm }, and v = (v∑ 1 , v2 , . . . , vm ) respectively, m v = 1. Assume the where vt ≥ 0, t = 1, 2, . . . , m, t t =1 DM dt (t = 1, 2, . . . , m) provides his/her linguistic preferences (t ) (t ) aij ∈ S(i, j = 1, 2, . . . , n), then denoted as ˜ A(t ) = (˜ aij )n×n , where ˜ we obtain the group LPR of ˜ A(1) , ˜ A(2) , . . . , ˜ A(m) by the following definition. Definition 5 ([43]). Let ˜ A(t ) ∈ Ln (t = 1, 2, . . . , m). If the following equation holds: m

(t )

aij , aˆ ij = ⊕ vt˜

k=1

m

(t )

t =1

=

2

τ n(n − 1)(n − 2)

Definition 6. Let ˜ A(t ) ∈ Ln , Aˆ ∈ Ln be as before. ACI(˜ A(t ) ) =

2

τ n(n − 1)(n − 2)

n ⏐ ⏐ ∑ ⏐ (t ) (t ) (t ) ⏐ aij ) + I(˜ ajk ) − I(˜ aik )⏐ ⏐I(˜

i
is the individual additive consistency index of dt , and

ˆ = ACI(A)

n ∑ ⏐ ⏐ ⏐I(aˆ ij ) + I(aˆ jk ) − I(aˆ ik )⏐

2

τ n(n − 1)(n − 2)

i
i
According to Theorem 1, we have

( I

m

(t ) aij ⊕ vk˜

) =

t =1

=

Theorem 3. Let ˜ A(1) , ˜ A(2) , . . . , ˜ A(m) and Aˆ = (aˆ ij )n×n be as before. If ˜ A(t ) is consistent, then Aˆ is also consistent. Proof. According to Definition 5, we have m

m

(t )

t =1

aˆ ij ⊕ aˆ jk =

m

(t ) ⊕ vt˜ aij ⊕

(

t =1 m

m

(t ) ⊕ vt˜ ajk

t =1

)

)

t =1 m

(t ) aik ⊕ vt˜

t =1

) =

m ∑

(t ) vt I(˜ aik ).

k=1

It follows that

⏐ n ⏐ m ∑ ⏐∑ (t ) vt I(˜ aij ) ⏐ ⏐ τ n(n − 1)(n − 2) i
ˆ = ACI(A)

≤

m ∑

i
vt CI

k=1

= CI . (12) Corollary 1. Let ˜ A(t ) , v and Aˆ be as before. Then t

t =1

(t )

)

(t ) vt I(˜ ajk ), I

(

m

(t ) ajk ⊕ vt˜

ˆ ≤ max ACI(˜ ACI(A) A(t ) ).

(t )

(t )

(t )

Since ˜ A(t ) is consistent for all t, we have ˜ aij ⊕˜ ajk = ˜ aik . It follows that

(

(

t =1

aˆ ij = ⊕ vt˜ aij , aˆ jk = ⊕ vt˜ ajk , aˆ ik = ⊕ vt˜ aik . t =1

∑

(t ) vt I(˜ aij ), I

t =1

Next, we further discuss the consistency properties of group ˆ LPR A.

(t )

m ∑ t =1 m

is called the group additive consistency index.

m

i
τ n(n − 1)(n − 2) ⏐ n ⏐ ∑ ⏐ m ⏐ m m (t ) (t ) (t ) ⏐ ⏐I( ⊕ vt˜ ˜ ˜ × a ) + I( ⊕ v a ) − I( ⊕ v a ) t t jk ik ⏐ ⏐ t =1 ij t =1 k=1

t =1

Individual additive consistency index and group additive consistency index can be formally defined as in Definition 6.

n ∑ ⏐ ⏐ ⏐I(aˆ ij ) + I(aˆ jk ) − I(aˆ ik )⏐

2

(11)

ˆ ˜(t ) then ∑m A = (aˆ ij )n×n is called group LPR of A , where vt ≥ 0 and v = 1. t =1 t

(t )

k=1

and aˆ ik = ⊕ vt˜ aik . It follows that

ˆ = ACI(A)

4.1. The properties of group linguistic preference relation

m

(t )

In line with Definition 5, we obtain aˆ ij = ⊕ vt˜ aij , aˆ jk = ⊕ vt˜ ajk

Proof. From Theorem 3, we have

ˆ ≤ ACI(A)

( ) m (t ) (t ) = ⊕ vt ˜ aij ⊕ ˜ ajk t =1

(t )

= ⊕ vt˜ aik = aˆ ik . t =1

Based on Definition 2, we obtain group LPR Aˆ is consistent.

m ∑

vt × ACI(˜ A(t ) )

t =1

≤

m ∑ t =1

vt × max ACI(˜ A(t ) ) = max ACI(˜ A(t ) ). t

t

From Corollary 1, we know that the group additive consistency index is less than the maximum individual additive consistency

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P. Wu, J. Liu, L. Zhou et al. / Applied Soft Computing Journal xxx (xxxx) xxx

index in GDM. It means that the individual additive consistency index affects the group additive consistency index. 4.2. Entropy weight model to derive weighting vector of DMs Definition 7. Let ˜ A = (˜ aij )n×n be as before and PDV be a vecn(n−1)

. If the vector PDV consists of the tor with dimension of 2 elements above the main diagonal line of LPR ˜ A: PDV = (˜ a12 ,˜ a13 , . . . ,˜ a(n−1)n )T .

(13)

Then the vector PDV is called preference degree vector of LPR ˜ A. In information theories, entropy value is used to measure the level of disorder in a system. The larger the entropy value of the system, the less information is contained. Otherwise, the smaller the entropy value of the system, the more information is contained. Inspired by this view, we can derive the DMs’ weights by (t ) measuring the entropy of LPRs. Let ˜ A(t ) = (˜ aij )n×n be as before. By Definition 7, the corresponding preference degree vector PDV t (t = 1, 2, . . . , m) is presented below: PDV t = (˜ at12 ,˜ at13 , . . . ,˜ atn(n−1) )T Further, let J =

n(n−1) , 2

(14)

On this basis, the entropy of each LPR ˜ At provided by DM dt can be defined as follow: Et = −

J 1 ∑

ln J

NS(˜ ati )

NS(˜ ati ) ln ∑J ˜t ˜t i=1 NS(ai ) i=1 NS(ai )

∑J i=1

) ,

(15)

where NS is the numerical scale of S, NS(˜ ati ) is the numerical index of linguistic term ˜ ati and 0 ≤ Et ≤ 1. The numerical scale builds a mapping with one-to-one of the S and real number set R [59], namely, NS: S → R. In this paper, I(˜ at )

we have NS(˜ ati ) = 8i + 12 [60], where I(˜ ati ) denotes the subscript of linguistic term ˜ ati . Let v = (v1 , v2 , . . . , vm )T be as before. For ∀t = 1, 2, . . . , m, we have

vt =

1 m−1

m ( ∑ t =1

Step 4. By Eq. (11), group LPR Aˆ is obtained. Step 5. Utilize the extended arithmetical averaging (EAA) operator [30] to compute the expected value aˆ i : aˆ i = EAA(aˆ i1 , aˆ i2 , . . . , aˆ in ) =

1 n

n

⊕ (aˆ ij ).

(17)

j=1

Step 6. Based on descending order, we rank the expected value aˆ i . Step 7. In line with the ranking of aˆ i , all the alternatives xi are ranked and select the best alternative(s). Step 8. End. The above procedure for GDM under LPR environment is shown in Fig. 1. Remark 3. The above process is developed to address the GDM problems with LPRs. For a decision-making problem involving one DM, the weights of DMs and group LPR are unnecessary. To deal with these problems, the Steps 3–4 should be omitted. Consequently, the proposed method can also solve decision making problem with one DM. In summary, the presented method can not only address the GDM problems with LPRs, but also can handle the decision making problems with a LPR.

and

PDV t = (˜ at1 ,˜ at2 , . . . ,˜ atJ )T .

(

7

1 − Et 1 − ∑m t =1 (1 − Et )

)

,

(16)

∑m

where vt ≥ 0 and t =1 vt = 1. The weighting vector v = (v1 , v2 , . . . , vm )T of DMs is derived by utilizing Eqs. (15) and (16). 5. Method for GDM with linguistic preference relations Consider a GDM problem. Let X = {x1 , x2 , . . . , xn } be the set of alternatives, D and S be as before. By utilizing S, DMs express their preference information denoted by ˜ A(1) , ˜ A(2) , . . . , ˜ A(m) , respectively. In the following, a new method of GDM with LPRs is proposed. Step 1. By Eq. (6), calculate the additive consistency index ACI(˜ A(t ) ) of individual LPR ˜ A(t ) . When ACI(˜ A(t ) ) ≤ CI, LPR ˜ A(t ) is acceptably additive consistency. When ACI(˜ A(t ) ) > CI, LPR ˜ A(t ) is unacceptably additive consistency. If all LPRs are of acceptable additive consistency, then go to Step 3; otherwise, go to the next Step. ∗(t) Step 2. By solving (M-3), the ˜ aij are obtained from ˜ A(t ) without acceptably additive consistency. Then, the acceptably additively consistent LPR ˜ A∗(t) is generated by using Eq. (9). Step 3. Using the entropy weight model, we can obtain DMs’ weights vt .

6. Numerical examples Our proposed models are implemented in two examples including a decision making problem with a LPR and a GDM problem. Some comparative analyses between our proposed methods and other existing methods [28,30–32,36,61] are also carried out. 6.1. A decision making problem with a LPR In this subsection, an emergency operating center selection problem for individual decision making with a LPR is addressed to interpret the advantages of the method proposed in Section 5. Then, some comparisons with other existing approaches are provided. 6.1.1. Emergency operating center selection example Example 3. This example is takes from Jin et al. [28]. It is a decision making problem about the selection of emergency operating center (EOC), which is used to protect people from the community risks [36]. Suppose that there are four EOCs X = {x1 , x2 , x3 , x4 }, the DM utilizes the linguistic terms to express evaluations over the four EOCs. When all the pairwise comparisons have been done, a LPR ˜ L = (˜ lij )4×4 is constructed as follows:

⎛

s0 ⎜ s−2 ˜ L=⎝ s−1 s−1

s2 s0 s− 4 s− 2

s1 s4 s0 s−4

⎞

s1 s2 ⎟ . s4 ⎠ s0

Step 1. Let CI = 0.1347 [31]. By Eq. (6), the additive consistency index is calculated as: 1 ACI(˜ L) = 48 [|I(˜ l12 ) + I(˜ l23 ) − I(˜ l13 )| + |I(˜ l12 ) + I(˜ l24 ) − I(˜ l14 )| +|I(˜ l13 ) + I(˜ l34 ) − I(˜ l14 )| + |I(˜ l23 ) + I(˜ l34 ) − I(˜ l24 )|] = 0.3750.

Since ACI(˜ L) > CI = 0.1347, ˜ L = (˜ lij )4×4 is an unacceptably additively consistent LPR and go to Step 2.

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

8

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Fig. 1. Decision making procedure for GDM with LPRs.

Step 2. Based on (M-3), an integer optimization model is constructed as follows: 1

− + − + − + − + − (ε + + ε12 + ε13 + ε13 + ε23 + ε23 + ε24 + ε24 + ε34 + ε34 ) ⎧48 12 1 ⎪ − + − + − + − ⎪ ⎪ (δ + + δ123 + δ124 + δ124 + δ134 + δ134 + δ234 + δ234 ) ≤ CI , ⎪ ⎪ 48 123 ⎪ ⎪ ⎪ ⎪ + − + − ∗ ∗ ⎪ ⎪ ⎪I(˜l12 ) − I(˜l12 ) − ε12 + ε12 = 0, I(˜l13 ) − I(˜l13 ) − ε13 + ε13 = 0, ⎪ ⎪ ⎪ ⎪ + − + − ⎪ I(˜ l14 ) − I(˜ l∗14 ) − ε14 + ε14 = 0, I(˜ l23 ) − I(˜ l∗23 ) − ε23 + ε23 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + − + − ⎪ I(˜ l24 ) − I(˜ l∗24 ) − ε24 + ε24 = 0, I(˜ l34 ) − I(˜ l∗34 ) − ε34 + ε34 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ˜∗ + − ∗ ∗ ⎪ ⎪ ⎪I(l12 ) + I(˜l23 ) − I(˜l13 ) − δ123 + δ123 ⎪ ⎪ ⎪ ⎪ + − ⎪ = 0, I(˜ l∗12 ) + I(˜ l∗24 ) − I(˜ l∗14 ) − δ124 + δ124 = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ∗ + − ˜ ˜∗ ˜∗ s.t . I(l13 ) + I(l34 ) − I(l14 ) − δ134 + δ134 ⎪ ⎪ + − ⎪ ⎪ = 0, I(˜ l∗23 ) + I(˜ l∗34 ) − I(˜ l∗24 ) − δ234 + δ234 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s−4 ≤˜ l∗12 ≤ s4 , s−4 ≤˜ l∗13 ≤ s4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s−4 ≤˜ l∗14 ≤ s4 , s−4 ≤˜ l∗23 ≤ s4 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪s−4 ≤˜l∗ ≤ s4 , s−4 ≤˜l∗ ≤ s4 , ⎪ 24 34 ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ˜ ⎪ l12 ∈ {−4, . . . , 0, . . . , 4} ,˜ l∗13 ∈ {−4, . . . , 0, . . . , 4} , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˜l∗14 ∈ {−4, . . . , 0, . . . , 4} ,˜l∗23 ∈ {−4, . . . , 0, . . . , 4} , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩˜∗ l24 ∈ {−4, . . . , 0, . . . , 4} ,˜ l∗34 ∈ {−4, . . . , 0, . . . , 4} .

min

The optimal solutions ˜ l∗ij are obtained by solving (M-3). Next, the acceptably additively consistent LPR ˜ L∗ is obtained by Eq. (9):

⎛

s0 ⎜ s0 ∗ ˜ L =⎝ s−1 s−1

s0 s0 s−1 s−2

s1 s1 s0 s−2

⎞

s1 s2 ⎟ . s3 ⎠ s0

It must be mentioned that the EOC selection example is an individual decision making problem with LPR. Therefore, Steps 3–4 should be omitted. Step 5. By Eq. (13), we obtain the expected value aˆ i : aˆ 1 = s0.5 , aˆ 2 = s0.75 , aˆ 3 = s0.25 , aˆ 4 = s−1.25 . Step 6. Ranking expected value aˆ i (i = 1, 2, 3, 4) with descending order way, we have: aˆ 2 > aˆ 1 > aˆ 3 > aˆ 4 . Step 7. Rank xi in accordance with aˆ i , we have x2 ≻ x1 ≻ x3 ≻ x4 . Obviously, the most desirable EOC is x2 . 6.1.2. Comparison with other existing approaches Based on EAA operator and extended ordered weighted averaging (EOWA) operator, Xu [30] developed a method to address decision making problems under linguistic information environment. In line with an optimization method and an iterative algorithm, Dong et al. [31] presented two approaches to manage the LPRs’ consistency. Two approaches, which include two automatic and iterative algorithms, are presented in [28]. Lan et al. [61] proposed an approach is based on a linguistic aggregation operator and ordering relation. Based on multiplicative consistency and an automatic iterative algorithm, Jin et al. [36] developed a method for dealing with decision making problems under linguistic information. To illustrate further the advantage of the proposed methods under LPR environment, some existing approaches are used to solve the EOC selection example. In line with these methods developed by this paper, Xu [30], Dong et al. [31], Jin et al. [28,36] and Lan et al. [61], the sorted results of all EOCs and the best alternative(s) are provided and listed in Table 3. Based on Table 3, the differences of these approaches are analyzed as follows.

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

P. Wu, J. Liu, L. Zhou et al. / Applied Soft Computing Journal xxx (xxxx) xxx Table 3 The decision making results of these different methods.

6.2.1. Schools of university evaluation

Method

Ranking results of EOCs

Best alternative(s)

Method in this paper Jin et al. [28]’s method Jin et al. [28]’s method Jin et al. [36]’s method Xu [30]’s method Dong et al. [31]’s method Lan et al. [61]’s method

x2 x2 x2 x1 x2 x2 x2

x2 x2 x2 x1 x1 , x2 x2 x2 , x3

≻ x1 ≻ x1 ≻ x1 ≻ x2 ∼ x1 ≻ x1 ∼ x3

≻ x3 ≻ x3 ≻ x3 ≻ x3 ≻ x3 ≻ x3 ≻ x1

≻ x4 ≻ x4 ≻ x4 ≻ x4 ≻ x4 ≻ x4 ∼ x4

9

• Comparison with Xu [30]: The approach in this paper and the method in Xu [30] produce the different ranking results of the four EOCs. In [30], Xu’s method directly utilized the EAA operator to fuse all linguistic preference information and did not measure whether the LPR is acceptably additive consistency. However, our method not only defines a new additive consistency index but also develops an integer optimization model to obtain the acceptably additively consistent LPR derived from the consistency of original LPR with unacceptably additive consistency. • Comparison with Dong et al. [31]: The approach proposed by this paper and the method provided by Dong et al. [31] produce the same ranking results of the four EOCs. Our additive consistency index directly uses the original LPR’s information and the integer optimization approach can ensure the revised linguistic terms are not virtual linguistic terms. It may be easy for DMs to accept the optimal adjusted linguistic terms as their new preference information. However, the consistency index in [31] is based on the distance between the LPR and the corresponding consistent LPR. In addition, with the consistency improving algorithms in Dong et al. [31], corresponding consistent LPR and interval-valued LPR are derived from original LPR. Two consistency improving algorithms may provide virtual linguistic terms for DMs as the new preference of adjusted LPR with acceptable consistency. • Comparison with Jin et al. [28,36]: Similarity with [31], the consistency index of LPR is defined by computing the deviation between LPR and its consistent LPR. Meanwhile, the consistency improving algorithms in [28,36] are iterative and convergent. However, the modified linguistic terms derived from these consistency improving algorithms [28,36] may be virtual linguistic terms. It may lead to the problem being hard to understand. Furthermore, the modified LPR depends on the adjusted parameter θ , which makes the process of decision making have certain randomness. • Comparison with Lan et al. [61]: Our approach method and the method in Lan et al. [61] produce the different ranking results of the four EOCs. With the method in Lan et al. [61], it directly calculates the deviation without checking the consistency level of LPR. However, our proposed method can guarantee the LPR is acceptably additive consistency. Thus, the rankings derived from our proposed method are rational.

Example 4. This example is takes from Xu [30]. Five schools of a university are evaluated by a group DMs. Let X = {x1 , x2 , x3 , x4 , x5 } be a set of five schools. One main criterion is the research. The group of DMs is constituted by three DMs. Each DM compares five schools based on the main criterion of research using S ′ :

⎧ ⎪ s−5 = Extremely poor , s−4 = Demonstrately poor , ⎪ ⎪ ⎪ ⎪ ⎪ s−3 = Strongly poor , s−2 = Moderately poor , ⎪ ⎪ ⎨ s−1 = Weakly poor , s0 = Equally good, S′ = ⎪ s1 = Weakly good, s2 = Moderately good, ⎪ ⎪ ⎪ ⎪ ⎪ s 3 = Strongly good, s4 = Demonstrately good, ⎪ ⎪ ⎩ s5 = Extremely good

In this subsection, the method proposed in Section 5 is applied to a GDM problem about schools of university evaluation for verifying its advantages. Then, some comparisons with other existing approaches are provided.

.

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(t ) Each DM gives his/her own LPR ˜ L(t ) = (˜ lij )5×5 (t = 1, 2, 3), respectively.

⎛ ⎜ ⎜ ˜ L(1) = ⎜ ⎝ ⎛ ⎜ ⎜ ˜ L(2) = ⎜ ⎝ ⎛ ⎜ ⎜ ˜ L(3) = ⎜ ⎝

s0 s−1 s2 s−3 s1

s1 s0 s−4 s−1 s2

s−2 s4 s0 s−2 s−3

s3 s1 s2 s0 s−4

s−1 s−2 s3 s4 s0

⎞

s0 s−2 s1 s−2 s2

s2 s0 s−4 s−2 s1

s−1 s4 s0 s−3 s−2

s2 s2 s3 s0 s−4

s−2 s−1 s2 s4 s0

⎞

s0 s−3 s3 s−1 s3

s3 s0 s−3 s−2 s2

s−3 s3 s0 s−3 s−1

s1 s2 s3 s0 s−4

s−3 s−2 s1 s4 s0

⎞

⎟ ⎟ ⎟, ⎠

⎟ ⎟ ⎟, ⎠

⎟ ⎟ ⎟. ⎠

Step 1. By Eq. (6), the additive consistency index of individual LPR

˜ L(t ) are calculated as follows:

ACI(˜ L(1) ) = 0.3417, ACI(˜ L(2) ) = 0.3917, ACI(˜ L(3) ) = 0.3917. L(t ) are Obviously, for ∀t, ACI(˜ L(t ) ) > CI, which means that all ˜ unacceptably additive consistency. Then, go to the next step. ∗(t) L(t ) by Step 2. The optimal solutions ˜ lij are obtained from LPRs ˜ solving (M-3). Next, according to Eq. (9), the acceptably additively consistent LPRs ˜ L∗(t) (t = 1, 2, 3) are obtained as

⎛ ⎜ ⎜ ˜ L∗(1) = ⎜ ⎝ ⎛ ⎜ ⎜ ˜ L∗(2) = ⎜ ⎝ ⎛

6.2. Application to GDM with LPRs

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎜ ⎜ ˜ L∗(3) = ⎜ ⎝

s1 s2 s3 s1 s0

⎞

s2 s2 s3 s0 s0

s−2 s0 s2 s0 s0

⎞

s1 s2 s2 s0 s0

s−3 s2 s1 s0 s0

⎞

s0 s−1 s0 s−2 s−1

s1 s0 s−4 s−1 s−2

s0 s4 s0 s−2 s−3

s2 s1 s2 s0 s−1

s0 s−2 s0 s−2 s2

s2 s0 s−4 s−2 s0

s0 s4 s0 s−3 s−2

s0 s−3 s0 s−1 s3

s3 s0 s− 2 s− 2 s−2

s0 s2 s0 s−2 s−1

⎟ ⎟ ⎟, ⎠

⎟ ⎟ ⎟, ⎠

⎟ ⎟ ⎟. ⎠

Step 3. Using the entropy weight model, weights vt of DMs are determined as

v1 = 0.3970, v2 = 0.3214, v3 = 0.2816.

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

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P. Wu, J. Liu, L. Zhou et al. / Applied Soft Computing Journal xxx (xxxx) xxx

7. Conclusions

Table 4 The decision making results with respect to different methods. Method

Ranking results

Method in this paper Xu [30]’s method Zhou et al. [32]’s method Zhou et al. [32]’s method

x2 x3 x3 x3

≻ x1 ≻ x2 ≻ x2 ≻ x2

≻ x3 ≻ x1 ≻ x1 ≻ x1

≻ x5 ≻ x5 ≻ x4 ≻ x4

Best alternative(s)

≻ x4 ≻ x4 ≻ x5 ≻ x5

x2 x3 x3 x3

Step 4. In line with Eq. (11), the group LPR Lˆ is determined:

⎛ ⎜ ⎜ ⎝

Lˆ = ⎜

s0 s−1.8845 s0 s−1.7184 s1.0905

s1.8845 s0 s−3.4368 s−1.6030 s−1.3573

s0 s3.4368 s0 s−2.3214 s−2.1155

s1.7184 s1.6030 s2.3214 s0 s−0.3970

s−1.0905 s1.3573 s2.1155 s0.3970 s0

⎞ ⎟ ⎟ ⎟. ⎠

Step 5. Utilizing Eq. (17), the expected values ˆli (i = 1, 2, . . . , 5) are derived as

ˆl1 = 2.5125, ˆl2 = 4.5125, ˆl3 = 1.0000, ˆl4 = −5.2457, ˆl5 = −2.7793. Step 6. Rank the expected values ˆli (i = 1, 2, . . . , 5) by descending order

ˆl2 > ˆl1 > ˆl3 > ˆl5 > ˆl4 . Step 7. By descending the expected values ˆli , the ranking order is x2 ≻ x1 ≻ x3 ≻ x5 ≻ x4 . It means that x2 is the optimal alternative. 6.2.2. Comparison with other existing approaches According to our proposed method, Xu [30] and Zhou et al. [32], the ranking orders of five schools and the best school are summarized in Table 4. Form Table 4, the ranking results derived from methods [30, 32] and our proposed method are different. More details of the differences between our method and these methods [30,32] are analyzed as:

• Comparison with Xu [30]: In [30], Xu’s method directly utilized the EAA operator and EOWA operator to fuse all linguistic preferences. Nevertheless, the approach only focuses on the aggregation of linguistic information and leaves the consistency of LPRs out of consideration. It causes the ranking results of five schools is inconsistent. However, based on the additive consistency index, our method develops an integer optimization-based model to obtain the acceptably additively consistent LPR derived from LPR with unacceptably additive consistency. Therefore, the decision making results provided by our method are more reliable. • Comparison with Zhou et al. [32]: In [32], two approaches for GDM with LPRs were developed. One approach is based on relative consensus degree induced linguistic ordered weighted average (RCD-ILOWA) operator, the other is based on compatibility index induced linguistic ordered weighted average (CI-ILOWA) operator. The two approaches neglected an important issue that consistency plays a vital role in the decision making process. However, our proposed models not only define additive consistency index based on the information of the original LPR, but also develop an integer optimization-based model, which is used to derive the acceptably additively consistent LPR. Therefore, our method is more reliable and rational than Zhou et al. [32]’s method.

In this paper, we present a new GDM method with LPRs via an integer optimization model. Firstly, according to the information of LPR, the additive consistency index of LPR is defined to check the consistency level of LPR. Then, in order to manage the unacceptably additively consistent LPR, an integer optimization model is constructed to obtain modified LPR with acceptably additive consistency derived from the LPR without acceptably additive consistency. This model can help the DMs to obtain reasonable decision making results. In addition, entropy weight method is further proposed to determine the DMs’ weights. Finally, considering the application of these models, two examples are analyzed. It is utilized to show the feasibility of our proposed methods. In line with our proposed methods, further research would be focused on the following studies: (1) According to drawbacks and advantages for incomplete LPR analyzed in Introduction, some methods for them will be proposed from two aspects, including completing method for unknown values and consistency improving algorithm. (2) Based on the proposed integer optimization model, we will discuss the completing methods for incomplete hesitant fuzzy linguistic preference relation [62], incomplete fuzzy preference relation [63] and incomplete reciprocal intuitionistic preference relation [64], incomplete uncertain linguistic preference relation [65]. (3) In GDM, the consensus process is necessary to obtain a final solution with a certain level of agreement between the DMs [66]. In recent years, a lot of achievements have been achieved on consensus in GDM [67–69]. In the future, inspired by these works, we will design consensus reaching process for GDM with various decision opinions. (4) Ordinal consistency and cardinal consistency are two vital consistency measure for preference relations [19]. Therefore, inspired by [70], our future work mainly focuses on address the ordinal consistency and cardinal consistency of incomplete linguistic preference relations. Meanwhile, we will discuss the intrinsic relationship between additive consistency and ordinal consistency (or cardinal consistency) of incomplete linguistic preference relations.

Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105955.

Acknowledgments The authors would like to thank the Editor, Prof. M. Köppen, Associate Editor, Prof. Enrique Herrera-Viedma and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. The work was supported by National Natural Science Foundation of China (Nos. 71771001, 71701001, 71501002, 71871001, 71901001, 71901088), The Natural Science Foundation for Distinguished Young Scholars of Anhui Province (No. 1908085J03), The Academic and Technical Leaders Reserve Talents Research Activities Funding Project of Anhui Province (No. 2018H179).

Please cite this article as: P. Wu, J. Liu, L. Zhou et al., Algorithm for improving additive consistency of linguistic preference relations with an integer optimization model, Applied Soft Computing Journal (2019) 105955, https://doi.org/10.1016/j.asoc.2019.105955.

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