2.1. Fuzzy Numbers

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

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 0≤wi≤1, i∈M, M=1,2,…,m.

Form 1 Weak ranking: Λ1=w∈Λ0|wi≥wj,for all i∈I1and j∈J1, where I1 and J1 are two disjoint subsets of the subscript index set M of all attributes.

Form 2 Strict ranking: Λ2=w∈Λ0|uij≥wi−wj≥λij,for all i∈I2 and j∈J2, 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={w∈Λ0|wi−wj≥wk−wl,for all i∈I3,j∈J3,k∈K3 and l∈ L3}, where I3, J3, K3, and L3 are four disjoint subsets of the subscript index set M.

Form 4 Multiplication ranking: Λ4=w∈Λ0|wi≥ηijwj,for all i∈I4 and j∈J4, where ηij>0 is a constant; I4 and J4 are two disjoint subsets of the subscript index set M.

Form 5 Interval ranking: Λ5=w∈Λ0|λi≤wi j≤λi+εi,for all i∈I5, 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

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 where α is the risk aversion coefficient of decision makers, 0≤α≤1; a smaller α indicates a greater degree of risk aversion of decision makers. Ruxi−ux∗ represents the regret value, which satisfies the condition Ruxi−ux∗<0. In this paper, we define the regret/rejoice function as follows [22]: 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 Ruxi−ux∗.

Figure 1

Regret/rejoice function of regret theory (RT).

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 0≤wjP≤1, wjP∈Λ0. 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=|xij−x0j|djmax′,(4)
   where djmax′=max|xij−x0j|i∈{1,2,…,m}.

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

   djC0,Ci=xij−−x0j−2+xij+−x0j+2djmax″,(5)
   where djmax″=maxxij−−x0j−2+xij+−x0j+212uij−u0j2+vij−v0j2+πij−π0j2i∈{1,2,…,m}

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

   djC0,Ci=12uij−u0j2+vij−v0j2+πij−π0j2djmax‴,(6)
   where πij=1−uij−vij, π0j=1−u0j−v0j, djmax‴=max12uij−u0j2+vij−v0j2+πij−π0j212uij−u0j2+vij−v0j2+πij−π0j2i∈{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=13Tij−T0j2+Iij−I0j2+Fij−F0j2djmax′′′′,(7)
   where djmax″″=max13Tij−T0j2+Iij−I0j2+Fij−F0j212uij−u0j2+vij−v0j2+πij−π0j2i∈{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

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

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

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

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

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

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)|Ck≥Cl,k,l=1,2,…,m be the multidimensional preference information given by the decision maker. If Φk≥Φl 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

Next, the consistency index can be rewritten as

Furthermore, the total consistency index is defined as

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

Subsequently, the inconsistency index can be rewritten as

Furthermore, the total inconsistency index is defined as

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:

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,Φk−Φl}. Hence, λkl≥0 and λkl≥Φk−Φl; i.e., λkl≥0 and Φl−Φk+λkl≥0. We can obtain

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

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, ξ∈min{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}|i∈Nv}, 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 0≤wvsR≤1, wvsR∈Λ0. 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, yq∈Y can be converted into a triangular fuzzy number as follows:

where yqa, yqb, and yqc are real numbers, and yqa≤yqb≤yqc.

Subsequently, we defuzzify the triangular numbers as follows:

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:

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:

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

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:

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

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:

where λβγ=max0,Vγp−Vβp. Attribute weights wvsR∗ are evaluated by evaluating (31).

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

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 (i∈Nv), i.e., Φv, and the comprehensive perceived utility based on evaluation criteria Uv. Therefore, the comprehensive case utility Γv can be defined as

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

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.

Step 1: We calculate the attribute similarities using Eqs. (4–8), 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={w∈Λ0|w1>2w2,0.01≤w2−w3≤0.2,0.1≤w4≤0.2,w4−w5≥w1−w2,w5≥w6,0.1≤w7≤0.1,0.1≤w7≤0.1,w8≥2w7}.

Step 2: We calculate the perceived utility based on attribute similarity vij using Eqs. (9–12). 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. (14–19), 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.

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}.

Step 5: According to Eqs. (23–24), we convert the linguistic variables into crisp numbers. According to Eqs. (25–28), the perceived utility based on evaluation criteria Vvp can be obtained, and the results are presented in Table 5.

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. (29–31), 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 Γ3≥Γ1≥Γ4≥Γ2. 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.