International Journal of Computational Intelligence Systems

Volume 14, Issue 1, 2021, Pages 1022 - 1033

Regret Theory-Based Case-Retrieval Method with Multiple Heterogeneous Attributes and Incomplete Weight Information

Authors
Kai Zhang1, Ying-Ming Wang2, *, Jing Zheng3
1College of Information and Intelligent Transportation, Fujian Chuanzheng Communications College, Fujian, 350007, P.R. China
2Business School, Wuchang University of Technology, Hubei, 430223, P.R. China
3College of Electronics and Information Science, Fujian Jiangxia University, Fujian, 350108, P.R. China
*Corresponding author. Email: msymwang@hotmail.com
Corresponding Author
Ying-Ming Wang
Received 27 December 2020, Accepted 18 February 2021, Available Online 3 March 2021.
DOI
10.2991/ijcis.d.210223.002How to use a DOI?
Keywords
Case retrieval; Regret theory; Multiple heterogeneous attributes; Incomplete weight information; Mathematical programming
Abstract

Case retrieval is a crucial step in case-based reasoning (CBR), which is related to decision-making effectiveness. To improve decision support, CBR usually calculates case similarity and evaluates utility. However, the psychological behavior of decision makers is seldom considered in case retrieval. This paper proposes a novel case-retrieval method that deals with multiple heterogeneous attributes and incomplete weight information based on regret theory (RT). First, we define the function of the perceived utility based on attribute similarity and RT. Next, a mathematical programming model is constructed to determine the attribute weights based on linear programming technique for multidimensional analysis of preference (LINMAP). Based on this, we can calculate the perceived utility and determine a set of similar historical cases. Furthermore, the utilities of the evaluated attributes are calculated based on RT and LINMAP. Subsequently, we compute the comprehensive utilities of similar historical cases and obtain the ranking order of similar historical cases. Thus, the most suitable historical case is obtained. Finally, a case study of a gas explosion is conducted to illustrate the use of the proposed method.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Case-based reasoning (CBR) is a type of comparative reasoning, wherein new problems are solved by referring to the solutions of old problems. Because CBR can find solutions quickly, it is widely used in many fields such as emergency decision-making [1,2], fault detection [3,4], and medicine [4,5], and the historical case with the highest similarity is used as a reference. However, some studies have proved that the most similar historical case may not be the most suitable for a target case [68]. To this end, the most suitable historical case is typically retrieved by first evaluating case similarity and then retrieving the most suitable historical case based on the evaluation criteria. Therefore, a good case-similarity evaluation and a good retrieval method are vital to identify the most suitable historical case.

The case information in the decision-making process is often qualitative and quantitative, which are usually described by heterogeneous information [911]. There are two main research directions related to case-similarity evaluation with multiple heterogeneous attributes. The first is to propose a new case-similarity evaluation method for a heterogeneous multi-attribute problem. For example, Fan et al. [12] proposed a hybrid similarity evaluation method with five formats of attribute values: crisp symbols, crisp numbers, interval numbers, fuzzy linguistic variables, and random variables; Zheng et al. [13] presented a hybrid multi-attribute case-similarity evaluation method with four formats of attribute values: crisp numbers, interval numbers, multi-granularity linguistic variables, and intuitionistic fuzzy numbers; and Yu et al. [14] considered crisp numbers, interval numbers, crisp symbols, linguistic terms, and probabilistic linguistic term sets for case-similarity evaluation, and in another study, Yu et al. [15] considered crisp numbers, crisp symbols, interval numbers, and fuzzy linguistic variables for case-similarity evaluation. The second is to determine attribute weights. For example, Zheng et al. [16] proposed a new case-retrieval method based on double-frontier data-envelopment analysis (DEA): DEA models determined the attribute weights automatically without the need to be specified; Yan et al. [17] presented a method of optimizing weights during case retrieval to improve problem-solving; and Wu et al. [18] proposed a weight-determination method using particle swarm optimization. However, existing studies have rarely considered the psychological behavior of decision makers. Various emotions, such as rejoicing, regret, or dislike, may affect the decision-making process [19]. Therefore, it is necessary to consider the decision maker's psychological behavior when evaluating case similarities.

To improve problem-solving, some studies have begun to investigate how the most appropriate historical case can be retrieved from similar historical cases according to evaluation criteria. For example, Qi et al. [6] used the technique for order of preference by similarity to ideal solution (TOPSIS) to calculate the utility of similar historical cases. Zheng et al. [8] proposed a DEA model to obtain the priority vector of the evaluation matrix to determine the most suitable historical cases. Wang et al. [1] also considered the utilities of similar historical cases. However, these selections unrealistically assumed that the decision makers were completely rational.

Recently, several theories, such as prospect theory (PT) [20,21], regret theory (RT) [21,22], and cumulative prospect theory (CPT) [23], have been proposed to model psychological behavior in the decision-making process. PT and CPT are commonly used but have some limitations; for example, reference points must be given in advance but are difficult to determine, and many parameters must be set in the calculation formulas. In contrast, RT does not require determining the reference points in advance and has fewer parameters. RT is a model of psychological behavior that focuses on regret and rejoicing from the selection of one alternative over another. RT has been used in many studies to solve decision-making problems, considering the decision makers' psychological behavior. Zhang et al. [22] integrated RT into group decision-making to consider the regret aversion of decision makers. Zhou et al. [24] proposed a new method based on TOPSIS and RT to solve the gray random multi-attribute decision-making problem. Peng and Yang [21] presented a new method based on RT and PT to solve random multi-attribute decision-making problems. Therefore, RT is an effective tool for solving decision-making problems considering psychological behavior.

The following three challenges exist in applying an RT-based case-retrieval method to select the most suitable historical case: (1) evaluating case similarities based on RT, (2) determining utility according to the evaluation based on RT, and (3) evaluating attribute weights considering the decision makers' psychological behavior. Furthermore, the attributes are often only partially known because of the associated complexity, lack of data, or lack of knowledge [25].

This paper proposes a RT-based case-retrieval method based on case retrieval and RT for scenarios with multiple heterogeneous attributes and incomplete weight information. First, we calculate case similarity considering the psychological behavior of decision makers using RT. Second, we construct a set of similar historical cases according to the perceived utility based on case similarity. Third, we select the most suitable historical case, considering the psychological behavior of decision makers, and RT is incorporated to evaluate utility. Finally, we calculate the comprehensive case utilities according to the perceived utility based on case similarity and the comprehensive perceived utility based on the evaluated information. Moreover, the weights of historical case attributes and evaluated attributes are considered incomplete, and the decision makers assign their preferences for similar historical cases. With the linear programming for multidimensional analysis of preference (LINMAP), we construct a mathematical programming model based on consistency and inconsistency measurements to determine the attribute weights.

The novelties of the developed approach include the following aspects: (1) The proposed method considers the psychological behavior of decision makers by using RT. The proposed method is more consistent with the actual decision and can provide better decision results. (2) The weights of the problem attribute and evaluate attribute are determined by LINMAP. The results of case retrieval are more objective and accurate. (3) The proposed method not only considers the perceived utility based on case similarity between the historical cases and the target case, but also the evaluation utility of the decision makers. Therefore, the results are more suitable for the target case.

The remainder of this paper is organized as follows. Section 2 reviews basic concepts and notations. Section 3 describes the proposed RT-based case-retrieval method based on case retrieval and RT with multidimensional preference information and incomplete weight information. Section 4 presents a case study of a gas explosion to illustrate the use of the proposed method and compares the proposed method with other methods. Section 5 concludes this paper.

2. PRELIMINARIES

This section describes the concepts of fuzzy numbers including intuitionistic fuzzy numbers and single-valued neutrosophic numbers. Next, incomplete weight information is introduced. Finally, RT is described.

2.1. Fuzzy Numbers

Definition 1.

[26] Let U={u1,u2,,um} be a finite universe of discourse. An intuitionistic fuzzy set A in U can be defined as A=<ui,uA(ui),vA(ui)>|uiU, where functions uA:U[0,1] and vA:U[0,1] represent the degree of membership and degree of nonmembership, respectively, and they satisfy the condition 0uA(ui)+vA(ui)1. Let πA(ui)=1uA(ui)vA(ui) be the hesitant degree of the intuitionistic fuzzy set, satisfying the condition 0πA(ui)1.

For convenience, we define the intuitionistic fuzzy number as α=uα,vα, which satisfies the conditions uα[0,1], vα[0,1], 0uα+vα1. The score function can be defined as s(α)=uαvα.

Definition 2.

[27] Let X be a set of objects. A single-valued neutrosophic set can be defined as

A=xTA(x),IA(x),FA(x)|xX,
where TA(x), IA(x), and FA(x) represent the truth-membership function, indeterminacy-membership function, and falsity-membership function, respectively, satisfying the following condition: xX,TA(x),IA(x),FA(x)0,1 and 0TA(x),IA(x),FA(x)3.

For convenience, we define the single-valued neutrosophic number as TA(x),IA(x),FA(x), recorded briefly as A=(Tx,Ix,Fx).

2.2. Incomplete Weight Information

In the real world, case information is often uncertain; thus, the attribute weights may also be incomplete. The incomplete weight information can be described in the following five forms, which are denoted by a subset, Λt(t=1,2,3,4,5), of attribute weights in Λ0 [28,29]. Let W=w1,w2,,wm be a vector of the attribute weights such that i=1mwi=1 and 0wi1, iM, M=1,2,,m.

Form 1 Weak ranking: Λ1=wΛ0|wiwj,for all iI1and jJ1, where I1 and J1 are two disjoint subsets of the subscript index set M of all attributes.

Form 2 Strict ranking: Λ2=wΛ0|uijwiwjλij,for all iI2 and jJ2, where uij>0 and λij>0 are constants, satisfying uij>λij; I2 and J2 are two disjoint subsets of the subscript index set M.

Form 3 Difference ranking: Λ3={wΛ0|wiwjwkwl,for all iI3,jJ3,kK3 and lL3}, where I3, J3, K3, and L3 are four disjoint subsets of the subscript index set M.

Form 4 Multiplication ranking: Λ4=wΛ0|wiηijwj,for all iI4 and jJ4, where ηij>0 is a constant; I4 and J4 are two disjoint subsets of the subscript index set M.

Form 5 Interval ranking: Λ5=wΛ0|λiwijλi+εi,for all iI5, where λi>0 and εi>0 are constants; I5 is a subset of M.

2.3. Regret Theory

In the case-retrieval process, decision makers are often not completely rational because of the uncertainty and limitations of their knowledge. They generally make rational choices; however, they have some emotions (such as regret and rejoicing, rewards, and costs) when making decisions. Emotions should be considered in the decision-making process because they tend to influence decisions. RT considers not only the utility of the selected alternative but also the effects of the other alternatives on the decision-making process [30,31]. Therefore, RT consists of two parts: the utility function of the current result and the regret/rejoice function in comparison with other results.

Let A={A1,A2,,An} be a set of alternative, where Ai represents the ith alternative. Let xi be the possible result obtained after selecting alternative Ai. Then, the decision maker's perceived utility for alternative Ai is defined as

ui=v(xi)+Rvxivx,(1)
where x=maxxi|i=1,2,,n. This implies that the decision maker will regret selecting alternative xi rather than the best alternative. Here, v(xi) is the utility function of the current result, which satisfies the conditions v(xi)0 and v(xi)0. In this paper, we use a power function to define the utility function, and it can be defined as
v(xi)=(xi)α,(2)
where α is the risk aversion coefficient of decision makers, 0α1; a smaller α indicates a greater degree of risk aversion of decision makers. Ruxiux represents the regret value, which satisfies the condition Ruxiux<0. In this paper, we define the regret/rejoice function as follows [22]:
Ruxiux=1expδuxiux,(3)
where δ is the decision maker's risk aversion coefficient. A smaller δ indicates a greater degree of risk aversion of decision makers. Figure 1 shows the effect of δ on the regret/rejoice function Ruxiux.

Figure 1

Regret/rejoice function of regret theory (RT).

3. PROPOSED METHOD

This section presents the proposed case-retrieval method based on RT with heterogeneous attributes (such as crisp numbers, interval numbers, intuitionistic fuzzy numbers, and single-valued neutrosophic numbers). Moreover, the weights of case attributes and evaluated attributes are incomplete. Figure 2 shows the basic procedure of the proposed method.

Figure 2

Flowchart of the proposed RT-based case-retrieval method.

3.1. Calculation of the Perceived Utility Based on Attribute Similarity

The following notations are used. C={C1,C2,,Cm} is a set of historical cases, where Ci represents the ith historical case, and C0 is the target case. The problem attribute vector with regard to the historical cases and the target case is X={X1,X2,,Xn}, where Xj represents the jth problem attribute. The problem attribute value of the historical case is xij, where i{1,2,,m} and j{1,2,,n}, and the problem attribute value of the target case is x0j. WP=w1P,w2P,,wnP is the vector of attribute weights, where wjP represents the jth attribute weight, such that j=1nwjP=1 and 0wjP1, wjPΛ0. Moreover, dj(C0,Ci) denotes the attribute distance between the target case C0 and historical case Ci with regard to the problem attribute Xj. The problem attribute distance dj(C0,Ci) is defined in four scenarios, depending on the attribute type, as follows:

  1. When attribute Xj is a crisp number, then

    djC0,Ci=|xijx0j|djmax,(4)
    where djmax=max|xijx0j|i{1,2,,m}.

  2. When attribute Xj is an interval number, i.e., xij=xij,xij+, x0j=x0j,x0j+, then

    djC0,Ci=xijx0j2+xij+x0j+2djmax,(5)
    where djmax=maxxijx0j2+xij+x0j+212uiju0j2+vijv0j2+πijπ0j2i{1,2,,m}

  3. When attribute Xj is an intuitionistic fuzzy number, i.e., xij=<uij,vij>, x0j=<u0j,v0j>, then

    djC0,Ci=12uiju0j2+vijv0j2+πijπ0j2djmax,(6)
    where πij=1uijvij, π0j=1u0jv0j, djmax=max12uiju0j2+vijv0j2+πijπ0j212uiju0j2+vijv0j2+πijπ0j2i{1,2,,m}.

  4. When attribute Xj is a single-valued neutrosophic number, i.e., xij=<Tij,Iij,Fij>, x0j=<T0j,I0j,F0j> then

    djC0,Ci=13TijT0j2+IijI0j2+FijF0j2djmax,(7)
    where djmax=max13TijT0j2+IijI0j2+FijF0j212uiju0j2+vijv0j2+πijπ0j2i{1,2,,m}.

Furthermore, according to [12], the inverse exponential function is used to calculate the attribute similarity. Specifically, attribute similarity Simj(C0,Ci) is defined as

Simj(C0,Ci)=expdj(C0,Ci).(8)

When calculating the case similarity, RT considers emotions. If the value of the attribute similarity is not the largest, the decision maker will regret; otherwise, they will rejoice. Therefore, RT is introduced to calculate the decision maker's psychological behavior toward case similarity. The perceived utility, based on case similarity, is calculated as follows:

Step 1: Suppose that the ideal attribute similarity with regard to problem attribute Xj is Simj+, given by

Simj+=maxSimj(C0,Ci)|i{1,2,,m}.(9)

Step 2: Calculate the perceived utility based on the attribute similarity. According to Section 2.3, the perceived utility consists of two parts: the utility value and the regret/rejoice value. Let uij be the utility based on attribute similarity, given by

uij=Simj(C0,Ci)α.(10)

Step 3: Suppose that the regret/rejoice value based on attribute similarity is rij, given by

rij=1expδSimj(C0,Ci)Simj+,Simj(C0,Ci)<Simj+0,Simj(C0,Ci)Simj+(11)

Based on this, the perceived utility based on attribute similarity, vij, can be defined as

vij=rij+uij.(12)

Step 4: The perceived utility based on case similarity, Φi, can be calculated as

Φi=j=1nwjPΦij.(13)

Obviously, Φi[0,1]. A higher value of Φi corresponds to a more suitable historical case.

3.2. Perceived Utility Value of Case-Similarity Consistency and Inconsistency Measurements

Let Ω=(k,l)|CkCl,k,l=1,2,,m be the multidimensional preference information given by the decision maker. If ΦkΦl for the pair of historical cases (k,l)Ω, then historical case Ck has more perceived utility than historical case Cl. Thus, the ranking order of historical cases Ck and Cl, determined by Φk and Φl based on wjP,Simj+, is consistent with the preferences given by the decision maker. Conversely, if Φk<Φl, then wjP,Simj+ is not chosen because the ranking order determined by Φk and Φl is inconsistent with the preferences given by the decision maker. Subsequently, we define an index to measure the degree of consistency between the ranking order of historical cases Ck and Cl determined by Φk and Φl. The consistency index is defined as

ΦlΦk+=ΦlΦk,ΦlΦk0,Φl<Φk.(14)

Next, the consistency index can be rewritten as

ΦlΦk+=max0,ΦlΦk.(15)

Furthermore, the total consistency index is defined as

G=k,lΩΦlΦk+=k,lΩmax0,ΦlΦk.(16)

Clearly, a bigger G corresponds to a higher degree of total consistency.

Similarly, an index to measure the degree of inconsistency between the ranking order of historical cases Ck and Cl determined by Φk and Φl is defined as

ΦlΦk=ΦkΦl,Φl<Φk0,ΦlΦk.(17)

Subsequently, the inconsistency index can be rewritten as

ΦlΦk=max0,ΦkΦl.(18)

Furthermore, the total inconsistency index is defined as

B=k,lΩΦlΦk=k,lΩmax0,ΦkΦl.(19)

Similarly, a bigger B corresponds to a higher degree of total inconsistency.

3.3. Mathematical Programming Model Based on LINMAP

For case retrieval, it is crucial to determine attribute weights because they are linked to the perceived utility values. Recent research has mainly used two approaches: development of an optimization model [13,32] and machine learning [17,33]. However, these two approaches are not suitable for solving our problem because we consider fuzzy reference information. Moreover, our weight information is incomplete and contains subjective multidimensional attributes. The LINMAP method is based on a pairwise comparison of the alternatives given by the decision maker. It generates the best alternative when the solution is close to the ideal solution. Moreover, the LINMAP method determines the attribute weights by constructing a mathematical programming model. Therefore, we use LINMAP to determine the attribute weights. According to [22,34], the mathematical programming model should ensure that the consistency degree of the perceived utility is as large as possible, and the inconsistency degree of the perceived utility is as small as possible. Based on this, we construct the following model to determine the attribute weights:

Min B=(k,l)Ωmax0,ΦkΦl
s.t.  GBη,(20)
wjPε,
WPΛ,
where η>0 is a priori threshold, which is given by the decision makers. Moreover, ε>0 is a sufficiently small positive number to ensure wjP>0.

For each pair of (k,l)Ω, let λkl=max{0,ΦkΦl}. Hence, λkl0 and λklΦkΦl; i.e., λkl0 and ΦlΦk+λkl0. We can obtain

GB=(k,l)ΩΦkΦl+ΦkΦl=(k,l)ΩΦkΦl.(21)

Thus, (20) can be rewritten as the following mathematical programming model:

min(k,l)Ωλkl
s.t.  (k,l)Ω(ΦkΦl)ρ,
λklΦk+Φl0,(22)
λkl0,
wjPε,
WPΛ.

Optimal attribute weights wjP=w1P,w2P,,wmP can be obtained by solving this problem.

3.4. Set of Similar Historical Cases

After determining the attribute weights wjP=w1P,w2P,,wmP, we can obtain the perceived utility based on case similarity Φi. Subsequently, we can retrieve similar historical cases. According to [35,36], a set of similar historical cases can be constructed using Φi. We select historical cases with a high perceived utility based on case similarities. Therefore, we set a perceived utility based on the case-similarity threshold. Let ξ denote the perceived utility, ξmin{Φi},max{Φi}, which is given by the decision maker based on their knowledge and experience. A large value of ξ indicates that the decision maker has high expectations of perceived utility.

A historical case Ci is selected if it satisfies the condition Φi>ξ. Furthermore, the selected historical cases form a similar case set Sv, v{1,2,,h}, such that {Sv={Ci}|iNv}, which represents the index set of all similar historical cases.

3.5. Evaluation of Utility of Similar Historical Cases

According to the similar case set, the pth decision maker Dp, p={1,2,,h} evaluates the effects of the alternative solutions applied to the target case. Suppose that the evaluated attributes are denoted by R={R1,R2,,Rt}, where Rs is an evaluated attribute, and s={1,2,,t}. Let rvsp denote the evaluated attribute value of Dp for a similar historical case Sv. Let WvR=wv1R,wv2R,,wvtR be a vector of the evaluated attribute weights wvsR, such that s=1twvsR=1 and 0wvsR1, wvsRΛ0. In the case retrieval, decision makers mainly consider linguistic variables and crisp numbers to express the evaluated attribute values. Then, the steps of evaluating the utilities of similar historical cases are shown as follows.

Step 1: Convert linguistic variables into triangular fuzzy numbers and defuzzify them. Let Y be a linguistic term set with odd cardinalities; i.e., Y=yq|q{1,2,,T}. According to [37], the linguistic variable yq, yqY can be converted into a triangular fuzzy number as follows:

y˜q=yqa,yqb,yqc=max(y1)/T,0,y/T,min(y+1)/T,1,(23)
where yqa, yqb, and yqc are real numbers, and yqayqbyqc.

Subsequently, we defuzzify the triangular numbers as follows:

ηq=yqa+4yqb+yqc/6.(24)

Step 2: Normalize the evaluated attributes. When the linguistic variables are transformed into crisp numbers, all the evaluated attributes are crisp. Next, we normalize the evaluated attributes as follows:

r¯vsp=rvspgsp/kspgsp,rvspNbksprvsp/kspgsp,rvspNc,(25)
where ksp=maxrvsp|v{1,2,,h}, gsp=minrvsp|v{1,2,,h}, Nb is the benefit attribute value, and Nc is the cost attribute value.

Step 3: Calculate the utility of the evaluated attribute πvsp. According to Eq. (2), πvsp can be obtained as follows:

πvsp=r¯vspα.(26)

Step 4: Calculate the regret/rejoice value based on the evaluation attribute utility Rvsp. According to Eq. (3), Rvsp can be obtained as follows:

Rvsp=1expδπvspπsp,(27)
where πsp=maxπvsp|v{1,2,,h}.

Step 5: Calculate the perceived utility based on the evaluation criteria Vvp. According to Eq. (1), Vvp can be obtained as follows:

Vvp=s=1twvsRπvsp+Rvsp.(28)

Step 6: Calculate the consistency index G and the inconsistency index B based on Vvsp (according to Section 3.2), as follows:

G=p=1h(β,γ)Ωpmax0,VβpVγp,(29)
B=p=1h(β,γ)Ωpmax0,VγpVβp,(30)
where (β,γ)Ωp represents the multidimensional preference information given by decision maker Dp.

Step 7: Construct the mathematical programming model to determine the attribute weights (according to Section 3.3), as follows:

min(β,γ)Ωpλβγp
s.t.  p=1h(β,γ)ΩpVβpVγpρ,
λβγpVβp+Vγp0,(31)
λβγp0,
wvsRε,
WRΛ,
where λβγ=max0,VγpVβp. Attribute weights wvsR are evaluated by evaluating (31).

Step 8: Calculate the comprehensive perceived utility Uv based on the evaluation criteria, as follows:

Uv=p=1hVvp.(32)

3.6. Comprehensive Case Utility and Ranking Historical Cases

To retrieve the most suitable historical case, we need to measure the comprehensive case utility and rank similar historical cases. Thus, we use a simple additive method to aggregate the perceived utility based on case similarity Φi (iNv), i.e., Φv, and the comprehensive perceived utility based on evaluation criteria Uv. Therefore, the comprehensive case utility Γv can be defined as

Γv=ΦvUv.(33)

Clearly, a larger value of Γv indicates that the historical case Cv is a better alternative for the target case.

4. ILLUSTRATIVE EXAMPLE

This section presents a case retrieval for a gas explosion as a case study to demonstrate the applicability and practicability of the proposed method.

4.1. Case Study

In recent years, various gas-explosion emergencies have occurred in China, which have led to huge losses for society. Such emergencies involve similar problems that may be solved by similar solutions. Thus, CBR can be used to quickly generate alternative solutions for the target case.

Company A is a coal company in Fujian Province, China. When a new gas-explosion emergency occurs, the company uses CBR to retrieve the most similar historical case to generate an alternative. To this end, this company collects 10 historical cases (C1,C2,,C10), containing 8 attributes (X1,X2,,X8), namely the number of underground personnel (X1), area of impact of the explosion (X2), degree of damage to the ventilation system (X3), degree of landslide (X4), scope of the fire (X5), O2 concentration (X6), CO concentration (X7), and CH4 concentration (X8). Among them, X1, X6, X7, and X8 are crisp numbers; X2 is an interval number; X3 is an intuitionistic number; and X4 and X5 are single-valued neutrosophic numbers. Table 1 describes the historical cases and the target case (C0). The objective of this study is to retrieve the most suitable historical case and help decision makers to generate alternative solutions for the target case. Subsequently, we describe the computation process and the results obtained using the proposed method. The computation processes and results are presented as follows.

X1 X2 X3 X4 X5 X6 X7 X8
C1 45 [26, 29] <0.8, 0.16> (0.7, 0.3, 0.1) (0.7, 0.3, 0.1) 12 31 7
C2 68 [25, 30] <0.8, 0.14> (0.9, 0.1, 0.1) (0.8, 0.2, 0.1) 13 29 9
C3 41 [22, 27] <0.6, 0.3> (0.9, 0.1, 0.2) (0.85, 0.15, 0.05) 28 32 12
C4 75 [20, 28] <0.7, 0.2> (0.8, 0.2, 0.2) (0.85, 0.15, 0.05) 27 42 13
C5 70 [29, 38] <0.75, 0.2> (0.9, 0.1, 0.2) (0.7, 0.3, 0.1) 22 23 10
C6 42 [32, 40] <0.8, 0.15> (0.75, 0.25, 0.1) (0.75, 0.25, 0.2) 28 28 8
C7 43 [26, 32] <0.9, 0.05> (0.8, 0.2, 0.2) (0.65, 0.35, 0.3) 31 27 15
C8 37 [15, 23] <0.75, 0.2> (0.95, 0.05, 0.1) (0.8, 0.2, 0.2) 30 32 13
C9 60 [17, 23] <0.7, 0.25> (0.65, 0.35, 0.1) (0.75, 0.25, 0.2) 25 29 10
C10 65 [25, 34] <0.6, 0.35> (0.95, 0.05, 0.15) (0.75, 0.25, 0.2) 24 31 11
C0 61 [23, 32] <0.75, 0.2> (0.8, 0.2, 0.15) (0.8, 0.2, 0.15) 21 30 12
Table 1

Information of historical cases and target cases.

Step 1: We calculate the attribute similarities using Eqs. (48), and the computation results are listed in Table 2. We set the multidimensional preference information to Ω={(2,1),(3,4),(5,4),(6,3),(6,10),(7,4),(10,9)}. Moreover, we assume that the attribute weights are incomplete. Λ={wΛ0|w1>2w2,0.01w2w30.2,0.1w40.2,w4w5w1w2,w5w6,0.1w70.1,0.1w70.1,w82w7}.

Sim1(C0, Ci) Sim2(C0, Ci) Sim3(C0, Ci) Sim4(C0, Ci) Sim5(C0, Ci) Sim6(C0, Ci) Sim7(C0, Ci) Sim8(C0, Ci)
C1 0.5134 0.7030 0.6065 0.5025 0.5614 0.4066 0.9200 0.3679
C2 0.7470 0.7907 0.3679 0.5025 0.8249 0.4493 0.9200 0.5488
C3 0.4346 0.6548 1.0000 0.5025 0.6241 0.4966 0.8465 1.0000
C4 0.5580 0.6602 0.6065 0.5025 0.6241 0.5488 0.3679 0.8187
C5 0.6873 0.4943 1.0000 0.5025 0.5614 0.9048 0.5580 0.6703
C6 0.4531 0.3679 0.6065 0.6721 0.7165 0.4966 0.8465 0.4493
C7 0.4724 0.7795 0.6065 0.7950 0.3679 0.3679 0.7788 0.5488
C8 0.3679 0.3679 0.6065 0.3679 0.8249 0.4066 0.8465 0.8187
C9 0.9592 0.4073 0.6065 0.3679 0.7165 0.6703 0.9200 0.6703
C10 0.8465 0.7907 0.6065 0.3778 0.7165 0.7408 0.9200 0.8187
Table 2

The attribute similarity Simj(C0, Ci).

Step 2: We calculate the perceived utility based on attribute similarity vij using Eqs. (912). The computation results are listed in Table 3. Next, we set ρ=0.01, ε=0.05, α=0.88, and δ=0.3. Then, we construct the mathematical programming model according to Eqs. (1419), and the attribute weights can be obtained as WP={0.1200,0.0600,0.0500,0.0200,0.1400,0.1300,0.1000,0.2000.

vi1 vi2 vi3 vi4 vi5 vi6 vi7 vi8
C1 0.4131 0.7068 0.5187 0.4540 0.5194 0.2917 0.9293 0.2060
C2 0.7079 0.8133 0.2060 0.4540 0.8442 0.3482 0.9293 0.4448
C3 0.3099 0.6473 1.0000 0.4540 0.5984 0.4098 0.8413 1.0000
C4 0.4706 0.6540 0.5187 0.4540 0.5984 0.4771 0.2346 0.7827
C5 0.6339 0.4449 1.0000 0.4540 0.5194 0.9158 0.4838 0.5993
C6 0.3343 0.2796 0.5187 0.6674 0.7127 0.4098 0.8413 0.3150
C7 0.3596 0.7998 0.5187 0.8172 0.2678 0.2400 0.7592 0.4448
C8 0.2207 0.2796 0.5187 0.2781 0.8442 0.2917 0.8413 0.7827
C9 0.9640 0.3317 0.5187 0.2781 0.7127 0.6304 0.9293 0.5993
C10 0.8292 0.8133 0.5187 0.2913 0.7127 0.7175 0.9293 0.7827
Table 3

The perceived utility based on attribute similarity vij.

Step 3: We obtain the perceived utility based on case similarity according to Eq. (13), as follows:

Φ1=0.4535, Φ2=0.5802, Φ3=0.6380, Φ4=0.5382, Φ5=0.6036, Φ6=0.5165, Φ7=0.5141, Φ8=0.5216, Φ9=0.6117, Φ10=0.6750.

Step 4: The decision makers set the perceived utility threshold ξ=0.6, and a set of similar historical cases can be obtained as S={S1,S2,S3,S4}={C3,C5,C9,C10}. Subsequently, three decision makers evaluate three attributes: the effect of emergency rescue (R1), casualty reduction rate (R2), and property loss reduction rate (R3). Table 4 lists the evaluation criteria. The evaluated attribute R1 is determined from the linguistic variable set, s={very good:VG,good:G,normal:N,bad:B,very bad:VB}.

E1
E2
E3
R1 R2 R3 R1 R2 R3 R1 R2 R3
s1 G 8.5 12 N 9 11 G 9 11
s2 N 9 9.6 G 10 10 N 10 9
s3 VG 11 7 VG 11 9 VG 12 9
s4 B 7 8.7 N 9 12 N 9 10
Table 4

Evaluate information.

Step 5: According to Eqs. (2324), we convert the linguistic variables into crisp numbers. According to Eqs. (2528), the perceived utility based on evaluation criteria Vvp can be obtained, and the results are presented in Table 5.

E1
E2
E3
R1 R2 R3 R1 R2 R3 R1 R2 R3
s1 0.7403 0.7342 1.0000 0.4674 0.7884 0.9039 0.7403 0.7069 1.0000
s2 0.4674 0.7884 0.7668 0.7403 0.8951 0.8063 0.4674 0.8063 0.7884
s3 1.0000 1.0000 0.5023 1.0000 1.0000 0.7069 1.0000 1.0000 0.7884
s4 0.1760 0.5684 0.6768 0.4674 0.7884 1.0000 0.4674 0.7069 0.8951
Table 5

Perceived utility based on evaluate information.

Step 6: For the three decision makers, we set the multidimensional preference information as follows: Ω1={(1,2),(1,3),(2,3),(3,4)}, Ω2={   (3,    1),  (2,  4),  (3,4)}, and Ω3={(1,2),(2,4),(3,1)}. According to Eqs. (2931), the mathematical programming model was constructed to determine the attribute weights, and the results were obtained as follows: WR={0.2180,0.4360,0.3460}. Subsequently, we calculated the comprehensive perceived utility based on information Uv evaluated by Eq. (32). The results were U1=0.8005,U2=0.7559,U3=0.8844,U4=0.6773.

Step 7: The comprehensive case utility Γv was calculated using Eq. (33). The results were Γ1=0.5107, Γ2=0.4562, Γ3=0.5409, and Γ4=0.4572. Subsequently, we ranked similar historical cases according to their comprehensive case utility as Γ3Γ1Γ4Γ2. As a result, the most suitable historical case was C9.

4.2. Comparative Analysis and Advantages of the Proposed Approach

This section compares some existing case-retrieval methods with the proposed approach.

4.2.1. Comparative analysis

To illustrate the characteristics and effectiveness of the proposed method, we compare it with some existing case-retrieval methods including (1) the traditional case-retrieval method [12], namely CBR-F; (2) the case-retrieval method without considering decision makers' psychological behavior, namely CBR-NPB; and (3) the case-retrieval method based on PT [8] without considering the evaluation information, namely CBR-PT; and (4) the case-retrieval method based on PT considering the evaluation information, namely CBR-PTT. The most suitable historical cases based on the four case-retrieval methods for the above case study are detailed in Table 6. Figure 3 shows the ranking of historical cases by case-retrieval methods without considering the evaluation of decision makers, i.e., the proposed method, CBR-F, and CBR-PT. Figure 4 shows the ranking of similar historical cases by case-retrieval methods, i.e., the proposed method, CBR-NPB and CBR-PTT.

Figure 3

The ranking of the historical cases.

Figure 4

The ranking of similar historical cases.

Approach The Most Suitable Historical Case
CBR-F C3
CBR-NPB C3
CBR-PT C10
The proposed method C9
Table 6

Comparison with different approaches.

Based on Table 6 and Figure 3, we analyzed the four case-retrieval methods as follows:

First, CBR-F obtains the most suitable historical case based on case similarities. The ranking of the case similarities is C3>C10>C5>C9>C2>C8>C7>C4>C6>C1, and the most suitable historical case was C3. However, in reality, the most suitable historical case should be selected while considering the case similarity and the evaluation criteria for similar historical cases [8]. In addition, decision makers are not completely rational during the decision-making process [13]. Therefore, selecting the most suitable historical case considering the evaluation criteria and the decision makers' psychological behavior is more appropriate in real-world decision--making scenarios.

Next, CBR-NPB does not consider the decision makers' psychological behavior when calculating case similarities and utilities. A set of similar historical cases is {C3,C5,C9,C10}. We calculate the utility of the evaluation criteria according to [20]. Subsequently, we calculate the comprehensive utilities of similar historical cases using Eq. (31). The ranking obtained using the CBR-NPB method is C3>C9>C5>C10. This is different from the results of the proposed method because CBR-NPB does not consider the psychological behavior of decision makers. However, in reality, decision-making is affected by personal preferences and psychological behavior. Therefore, it is more reasonable to include psychological behavior in the model.

Furthermore, we compare the proposed method with CBR-PT. We set the parameters in PT as α=0.89, β=0.92, and λ=2.25. The reference point for all attribute distances is set at 0.5. The ranking of the case similarities is C10>C9>C8>C3>C6>C7>C5>C2>C4>C1. Therefore, the most similar historical case is C10. This result is the same as that of the proposed method when calculating the case similarity. However, CBR-PT does not consider the evaluation criteria, and the final result is different. In the selection of the most suitable case, it cannot be determined by the similarity alone, and the historical case with a slightly lower similarity may be more consistent with the target case. Therefore, it is necessary to determine the appropriate historical case through the evaluation of the decision maker, and the results would be more accurate.

Finally, the ranking of the similar historical cases obtained using the method based on CBR-PTT is C9>C10>C3>C5. The most suitable historical case is C9, which is the same as that obtained using the proposed method. However, first, PT requires the determination of the reference point. Furthermore, there are several parameters that need to be determined. However, RT determines fewer parameters. The determination of these parameters is generally done by decision makers. With the increase in the number of parameters, the subjective decision of decision maker will lead to great fluctuation of the results, so that the difference between the calculated results will be large, which will lead to the reduction of the accuracy of the results. Therefore, the case retrieval based on RT is more suitable for the method based on PT.

In addition, the weights of case attributes and evaluated attributes are determined by LINMAP, which sets up a mathematical programming model, considering the case information and incomplete multidimensional preference information to determine the weights. Compared with the methods that construct models to determine weights [13,20], this method considers not only objective information, but also the subjective preferences of decision makers. Therefore, it is more suitable for the real-world decision-making process.

4.2.2. Advantages of the proposed approach

In our comparison with the existing case-retrieval methods, based on case retrieval, we identified the following advantages of the proposed approach:

  1. The existing case-retrieval methods seldom consider psychological behavior in the calculation of case similarity. Although PT-based case retrieval considers psychological behavior, many parameters should be set in the calculation process. Moreover, a change in the parameters changes the results. In contrast, the proposed method not only considers the psychological behavior, but also has fewer parameters and simpler calculations.

  2. When selecting the most suitable historical case, the proposed method considers an evaluation of alternatives from historical cases similar to the target case.

  3. We use the LINMAP method to determine the incomplete weights in case attributes and evaluated attributes. This approach can provide more objective and accurate results when calculating case similarities and comprehensive utilities.

5. CONCLUSION

In this paper, we propose an RT-based case-retrieval method based on case retrieval and RT, to retrieve the most suitable historical case in scenarios with multiple heterogeneous attributes and incomplete weight information. The proposed method has the following three characteristics: (1) Heterogeneous multi-attribute information is considered in uncertain cases. (2) Case similarities and comprehensive utilities are calculated considering the psychological behavior of decision makers, which is a more realistic scenario. (3) To improve the accuracy of the results, the LINMAP method is used to determine the weights of case attributes and evaluated attributes. Furthermore, LINMAP aggregates three types of information: case similarity or evaluated utility, incomplete weight information, and multidimensional preference information.

The proposed method helps decision makers to select the most suitable historical case. However, some limitations also exist. For example, the process of case retrieval only considers one state, but, in reality, there may be constant changes in emergencies. Therefore, future research can focus on a case-retrieval method for scenarios with a dynamic evolution of emergencies.

CONFLICTS OF INTEREST

The authors declare no conflicts of interest regarding the publication for the paper.

AUTHORS' CONTRIBUTIONS

Zhang Kai and Wang Ying-Ming proposed the methodology, Zhang Kai and Zheng Jing conducted the validation and formal analysis, Zhang Kai wrote the original draft preparation, Wang Ying-Ming reviewed the writing, Zhang Kai edited the writing. All authors read and approved the final manuscript.

Funding Statement

This work was partly supported by the National Natural Science Foundation of China under the Grant No. 71371053, Humanities and Social Sciences Foundation of Chinese Ministry of Education, No. 20YJC630229, Humanities and Social Science Foundation of Fujian Province, No. FJ2019B079, and Special project of school-level science and technology service team of Fujian Chuanzheng Communications College, No. 20200205.

ACKNOWLEDGMENTS

The authors would like to thank to the anonymous reviewers and editor for their insightful comments.

REFERENCES

Journal
International Journal of Computational Intelligence Systems
Volume-Issue
14 - 1
Pages
1022 - 1033
Publication Date
2021/03/03
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.d.210223.002How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Kai Zhang
AU  - Ying-Ming Wang
AU  - Jing Zheng
PY  - 2021
DA  - 2021/03/03
TI  - Regret Theory-Based Case-Retrieval Method with Multiple Heterogeneous Attributes and Incomplete Weight Information
JO  - International Journal of Computational Intelligence Systems
SP  - 1022
EP  - 1033
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210223.002
DO  - 10.2991/ijcis.d.210223.002
ID  - Zhang2021
ER  -