1. Introduction

Case-based reasoning (CBR) is good at solving new problems by referring to the solution of similar past experience¹. It can help decision maker (DM) to generate alternatives quickly. Therefore, it has been widely used in many fields, such as environment preparedness system², emergency decision making³, business failure prediction⁴, medicine⁵, fault diagnosis⁶, wastewater treatment⁷. CBR usually includes four steps¹, i.e., retrieval, adaptation, revision and retaining. Among the four steps, retrieval is regarded as the first and core step. If the retrieved historical case(s) is the most desirable case, the solution will be effective; otherwise, the result would not be good. Hence, it is essential to study case retrieval methods.

Up to now, a number of case retrieval methods have been proposed in the practical CBR applications. There are mainly two kinds. One kind is to propose a similarity function to retrieve the similar historical case(s) and directly use it in the applications. For example, Kwong et al.⁸ proposed a similarity measure based on Euclidean distance to concurrent design of low power transformers. Yu et al.⁹ developed a hybridization of both symbolic and numeric reasoning techniques for mining of scarce construction databases. Li et al.⁴ proposed a similarity computation method, which transferred the attribute distance into Gaussian distance and solved the nonlinear data, to improve the prediction accuracy in business failure prediction. Sun et al.¹⁰ developed a similarity function using grey theory to improve the ability of similar case retrieval and prediction accuracy. Li et al.¹¹ proposed a similarity measure by combining four independent CBR models to amplify advantages of individual techniques and minimize their limitations. Apparently, it is useful to mix several similarity measure methods. The other kind of methods proposed case similarity measurements and selected the appropriate historical case(s) according to the evaluation of historical cases. To select the most desirable historical case, some studies attempted to introduce multi-criteria decision making (MCDM) to select the most effective historical case. For example, Qi et al.¹² proposed a case retrieval method which used the algorithm of order preference by similarity to an ideal solution (TOPSIS) to evaluate the most similar cases in terms of product criteria to pick out the most suitable case. Li et al.¹³ developed a CBR forecasting method based on the similarities to positive and negative ideal cases. Fan et al.³ generated the desirable response alternative by evaluating the retrieved historical case(s).

The existing studies have made significant contributions to decision making based on CBR. These studies provided various retrieval methods for DMs to solve problems. However, among the existing studies, the similarity calculation methods have a premise, that is, the DM is perfectly rational. In fact, the DM has some emotions such as rejoicing regret, or dislike, in decision making¹⁴, and they would affect the DM's decision. In other words, people are often limitedly rational rather than perfectly rational when facing decision making. Therefore, it is necessary to investigate the case retrieval methods considering human behavior for the purpose of providing effective decision support to DMs. Furthermore, the existing case selection methods can only deal with the evaluation given by one DM, yet the process of decision making may have several formats such as in group decision-making. Therefore, case retrieval methods should consider the psychological behavior in the similarity calculation and several evaluation formats in the most suitable historical case selection.

Since Tversky and Kaheneman proposed cumulative prospect theory (CPT)¹⁵, which is a descriptive model of decision making under conditions of risk, many scholars have employed it to solve various decision-making problems considering DM’s behavior, such as, emergency decision making¹⁶, MCDM¹⁷. This is because CPT describes the DM’s behavior characteristics well and gives the calculation formulas on values and weights of potential outcome. In the case retrieval step, the DM usually has some emotions, i.e., loss aversion, diminishing sensitivity, and reference dependence. For instance, the DM has a thought that if the attribute value of one historical case is very different from the target case, even the case similarity is higher than the other historical cases, the historical case is not similar with the target case. Therefore, how to incorporate CPT into case retrieval deserves more attention.

Extensive studies of MCDM techniques have been undertaken over the past decades, where methods of analytic hierarchy process (AHP), elimination and choice translating reality (ELECTRE), and technique for TOPSIS have been proved to be effective approaches. TOPSIS method has been integrated into CBR to generate a more suitable alternative. But TOPSIS has a defect that cannot handle large set of alternatives and criteria¹⁸. AHP is a simple and effective MCDM aid tool, and it can evaluate several similar cases and identify a suitable design alternative. Rammanathan¹⁹ developed a method which combined data envelopment analysis (DEA) with AHP to form a DEAHP method, but it has a significant drawback that there is no guarantee that the DEAHP method can produce rational weight vectors for inconsistent pairwise comparison matrices (PCM). Afterwards, Wang et al.²⁰ proposed a new DEA model to determine the priority, which can derive logical priorities for PCM. Hence, how to derive the priority vector in the AHP for the similar case set is worth attention.

The objective of this paper is to develop a case retrieval method based on similarity measurement and DEA model for generating a desirable alternative. In similarity measurement stage, a similarity measurement based on CPT is proposed, which considers DM’s psychological behavioral. It is more consistent with decision-making process. Meanwhile, we mix three classic similarity methods with the similarity measurement based on CPT to get proper similar historical cases set. The mix can expand the advantages of the similarity measurements and make the decision result more effective. In alternative generation stage, a DEA model is constructed to get the priority vector in AHP, and generate a proper alternative. The DEA model can deal with various forms of evaluations and make the method more applicable.

The rest of the paper is organized as follows. In section 2, we give a brief review of the classic similarity measurements and CPT. In section 3, we develop a case retrieval method combined with similarity measurement and DEA model for alternative generation. In section 4, numerical examples are provided to illustrate the use of the proposed method. In section 5, the discussion of this study is presented. In section 6, conclusions of this study are provided.

2. Preliminaries

This section provides a brief introduction about concepts related to similarity measurement and CPT that are used in the later proposed method.

3. The proposed method

In this section, we present a method for case retrieval which combines similarity measurement and DEA model for alternative generation as Fig. 1. Firstly a hybrid similarity measurement is proposed, which mixes the similarity measurement based on CPT with three classic similarity measurements. Then alternative evaluation is got using PCM. Furthermore, a DEA model is constructed to gain the priority of the similar historical cases. Finally we gain the ranking order of the similar historical case and the desirable historical case will be determined. The method is introduced as follows.

Fig. 1.

Framework of the case retrieval model

3.1. Similarity measurement

In order to amplify the advantages of similarity measurements and minimize their limitations¹¹, the combination of several similarity measurements is employed to gain the similarities between historical cases and target case. The case similarity measurement includes two aspects and is shown in Fig. 2. The first aspect is to calculate the case similarity based on prospect theory, the other aspect is to calculate the case similarity based on three classic similarity measurements.

Fig. 2.

The process of the hybrid similarity calculation

3.1.1. Similarity measurement based on cumulative prospect theory

Suppose there are g historical cases denoted by C_g(g = 1, …, h) and one target case denoted by C₀. Let P = {P₁, P₂, …, P_m} represent the vector of m attributes with regard to the problem of both the historical cases and the target case, where P_l denotes the lth attribute, l ∈ {1, 2, …, m} Let Wp={w1p,w2p,...,wmp} be a vector of attribute weights, where wlp denotes the weight of the attribute P_l, such that ∑l=1mwlp=1 and 0≤wlp≤1 . Let X_g = {x_g1, x_g2, …, x_gm} represent a vector of attribute value with regard to the historical case C_g, where x_gl denotes the attribute value concerning attribute P_l with regard to the historical case C_g. Let X₀ = {x₀₁, x₀₂, …, x_0m} represent a vector of attribute value with regard to the target case C₀, where x_0l denotes the attribute value concerning attribute P_l with regard to the target case C₀. Then we can define the attribute distance d_0gl between historical case C_g and target case C₀ concerning the attribute P_l as follows:

(8)d0gl=|x0l−xgl|max0≤g≤h{xgl}−min0≤g≤h{xgl}

where 0 ≤ d_0gl ≤ 1. Thus, the case similarity can be expressed as 1 – d_0gl²⁵.

Let Sim_0gl denote the attribute similarity between historical case C_g and target case C₀ concerning the attribute P_l and d_l denote a reference point with regard to the lth attribute distance. Because the DM’s preference for the attribute distance is d_l, it means that if the attribute distance is less than d_l, the DM would feel it is “gain”, otherwise, if the attribute distance is more than d_l, the DM would feel it is “loss”. Based on CPT, we can define the attribute similarity Sim_0gl between historical case C_g and target case C₀ concerning the attribute P_l as follows:

(9)sim0gl={(1−d0gl)α,d0gl≤dl−λ(1−d0gl)β,d0gl>dl

where (1 – d_0gl)^α represents the attribute similarity when the DM feels “gain”, –λ(1 – d_0gl)^β represents the attribute similarity when the DM feels “loss”. According to the [21], let α = 0.89, β = 0.92, λ = 2.25.

Since different attribute similarities are usually incommensurate, Sim_0gl needs to be normalized as Sim¯0gl by using the following formula:

(10)Sim¯0gl=Sim0gl|Sim0g|max

where |Sim0g|max=max{|Sim0g1|,|Sim0g2|,...,|Simogm|} , and it is obvious that 0≤Sim¯0gl≤1 .

Finally, by using the simple additive weighting (SAW) method, the overall prospect of each historical case similarity Sim″_g can be calculated as follows:

(11)Simg″=∑l=1mwlSim¯0gl

The CBR model built on prospect theory is called PCBR.

The following example illustrates the feasibility and effectiveness for the similarity measurement based on CPT.

Example 1

A high-rise building fire took place in City F. The emergency decision center considers four attributes mainly, which are fire rating (P₁), fire area (P₂, unit: m²), casualties (P₃, unit: person) and economic losses (P₄, unit: ten thousand RMB). Table 1 shows the attribute values with regard to the target case C₀ and historical cases C_i (i = 1, 2, 3). The attribute weight vector provided by the emergency management center is W = (0.25, 0.25, 0.25, 0.25), the DM preference attribute distance vector is (0.4, 0.3, 0.3, 0.3).

	P1	P2	P3	P4
C1	1	16	18	7
C2	3	19	20	10
C3	4	25	25	11
C0	2	18	20	8

Table 1.

Attribute values with regard to target case and historical cases

According to Eqs. (1) – (2) , the attribute distance d_0gl and case similarity Sim_g are gained, and the results can be found in Table 2.

	d0g1	d0g2	d0g3	d0g4	Simg
C1	0.3333	0.2222	0.2857	0.2500	0.7272
C2	0.3333	0.1111	0.0000	0.5000	0.7639
C3	0.6667	0.7778	0.7143	0.7500	0.3562

Table 2.

The computation result of attribute distance and case similarity using Euclidean distance

According to Eqs. (8) – (11) , the attribute similarity sim_0gl and case similarity Sim″_g are gained, and the results can be found in Table 3.

	sim0g1	sim0g2	sim0g3	sim0g4	Sim″g
C1	0.6971	0.7996	0.7412	0.7741	0.7530
C2	0.6971	0.9005	1.0000	−1.1891	0.3521
C3	−0.8189	−0.5639	−0.7106	−0.6285	−0.6805

Table 3.

The computation result of attribute similarity and case similarity using similarity measurement based on CPT

In Table 2, historical case C₂ is the most similar case with the target case C₀, but the attribute distance d₀₂₄ is very large. It means that the target case C₀ and the historical case C₂ have significant differences on attribute P₄ and it would lead to different alternatives for the historical case C₂ and the target case C₀. So, DMs have their preference in the attribute distance. DMs prefer to choose the historical case with a lower similarity and lower attribute distance as the most similar case. The similarity measurement based on CPT can well express this kind of psychological behavior of DMs. The attribute similarity sim₀₂₄ in the third row and fifth column of Table 3 is very small, because its attribute distance is larger than the DM preference attribute distance 0.3. The most similar case is C_1. Therefore, the similarity measurement based on CPT is in accordance with the DMs’ behavior.

3.1.2. Classic similarity measurement

Classic similarity measurements are proved to be useful for the case retrieval, so we consider their effectiveness. We assemble ECBR, GCBR and RCBR to get the classic case similarity. When the ECBR is got using Eqs. (1) – (2) , GCBR is got using Eqs. (3) – (4) , RCBR is got using Eqs. (5) – (6) , we use weighted mean for aggregating three classic similarity measurements. Let Sim′_g be the average similarity of the above three classic similarities. The aggregation formula is given by

(12)Simg′=Simg1+Simg2+Simg33

In order to aggregate the advantages of these similarity measurements, it is necessary to mix the similarities based on the CPT and the average similarity based on three classic similarities measurements. Let Sim_g denote the hybrid similarity between the historical case C_g and the target case C₀. The calculation of Sim_g is given by

(13)Simg=θSimg′+γSimg″

where θ and γ represent the preference of the two similarities, and 0 ≤ θ, γ ≤ 1, θ + γ = 1.

Let ξ be the threshold for the similarity between target case and historical cases, such that ξ ∈ [min{Sim_g ∣ g ∈ {1, 2, …, h}}, max{Sim_g ∣ g ∈ {1, 2, …, h}}]. The bigger the value of ξ is, the higher requirement of the case similarity the DM has. The value of ξ usually is given by the DM according to his (her) experience and the realistic data. When Sim_g ≥ ξ, the historical case C_g would be extracted, and it would constitute the similar historical cases set Z^Sim, i.e., Z^Sim = {C_k∣k ∈ M^Sim}, where M^Sim = {g ∣ Sim_g≥ξ, g = 1, 2, …, h} = {1, 2, …, s}. If the similar case set has only one case, the case would be selected as the most suitable case. Otherwise, we would select the most suitable case by a DEA model.

3.2. Multi-criteria decision making

The DEA model proposed by Wang et al.²⁰ can produce true weights for PCM. Here, we use this DEA model to evaluate the similar cases.

3.2.1. DEA model for alternatives priority

Based on the similar case set, the DM evaluates the alternatives by using a PCM denoted as A¹ = (a_ij)_s×s, which satisfies a_ii = 1 and a_ji = 1/a_ij for j ≠ i. Let W = (w₁, w₂, …, w_s) be the priority vector of the matrix A¹.

We use a DEA model to get the priority vector of the matrix A¹ and get the most suitable historical case. According to [20], we view each row of the matrix A¹ as a decision making unit (DMU), each column as an output and assume a dummy input value of one for all the DUMs. Each DMU has s outputs and one dummy constant input, based on which the DEA model for relative score can be formulates as

(14)Maximize w0=∑j=1sa0jzjSubject to {∑j=1n(∑i=1naij)zj=1,∑j=1naijzj≥nzi, i=1,...,s,zj≥0, j=1,...,s.

where DMU₀ represents the similar historical case under evaluation, w0* represents the DEA efficiency of DMU₀ and is used as its priority. Linear programming model (14) is solved for all the similar historical cases in Z^Sim to obtain the priority vector W*=(w1*,w2*,...,ws*) of the PCM A¹.

3.2.2. DEA model for alternatives priority concerning group decision making

In decision analysis, group decision making is usually used to assemble several experts’ wisdom. In order to select the most effective alternative, sometimes DMs use group decision making. We consider several DMs to give their preference on the similar case set Z^Sim by PCM. Let A(k)=(aij(k))s×s be a PCM provided by the kth DM_k, (k = 1, …, f), r_k > 0 be its relative importance weight satisfying ∑k=1frk=1 . The selection steps for the most suitable historical case are introduced as follows.

First, we use SAW method to aggregate several DMs’ preferences, which means that we integrate the k A^(k) into a PCM B = (b_ij)_s×s. The formula is given as follows:

(15)bij=∑k=1frkaij(k)

Second, we use the following DEA model²⁰ to gain the priority vector:

(16)Maximize w0=∑j=1sbijzj,Subject to {∑j=1s(∑i=1sbij)zj=1,∑j=1sbijzj≥szi, i=1,…,s,zj≥0, j=1,…,s.

By solving the above model (16) for each w_i (i = 1, …, s), we will get the best priority vector W^*.

3.2.3. DEA model for alternative priority concerning multi-criteria evaluation

In decision making, the DM cannot evaluate the alternative directly sometimes, and what she/he can do is to evaluate the alternative from a few criteria. In this situation, a hierarchical structure often exists as shown in Fig. 3. In order to select the most suitable historical case, we can evaluate the similar historical cases set from different criteria. Between the historical case selection level and the p decision criteria level, we can get a priority vector {v₁, v₂, …, v_p} by solving the model (14) and between the p criteria level and s similar historical cases level, we can get p priority vectors {v_1j, v_2j, …, v_sj} (j = 1, …, p) by solving the model (14). Based on the above priority vectors, we use the SAW to aggregate them and get a global priority vector {w₁, w₂, …, w_s} as follows:

Fig. 3.

A hierarchical structure for multi-criteria historical case selection

(17)wi=∑j=1pvijvj

4. Example

In this section, we provide a numerical example from three aspects to illustrate the advantages of the proposed method and its potential applications in case retrieval. Consider an application of the proposed method to parametric car design. A car design company E constructs a data base which includes 15 historical cases (C₁, C₂, …, C₁₅) concerning parametric car design. According to parametric car design, the car design company considers four attributes mainly, namely, hundred kilometers acceleration (P₁, unit: second), braking distance (P₂, unit: m), horsepower (P₃, unit: hp) and hundred kilometers fuel consumption (P₄, unit: 1). Now, a new kind of car needs to be designed, and it is regarded as the target case C₀. Table 4 shows the values of the attributes with regard to the historical cases C_i and the target case C₀. Then we use the case retrieval method combined with similarity measurement and DEA model to generate an alternative. The computation processes and results are presented as follows.

Step 1: The case similarity based on CPT Sim″_g can be calculated by using Eqs. (8) – (11) , and the computation results are shown in Table 5.

Step 2: The case similarities by using three classic similarity measurements are calculated. The case similaritiy Simg1 is calculated by using Eq. (1) and Eq. (2) , and the results are shown in Table 5. The case similarity Simg2 is calculated by using Eq. (3) and Eq. (4) , and the results are shown in Table 5. The case similarity Simg3 is calculated by using Eq. (5) and Eq. (6) , and the results are shown in Table 5. Based on these case similarities, we use Eq. (12) to get the average similarity Sim′_g, and the results are shown in Table 5. Considering the importance of the two kinds of similarity measurements is equal, let γ = θ = 0.5. Then, the hybrid similarity Sim_g is calculated by using Eq. (13) , the results are shown in Table 5.

Step 3: The DMs give the threshold of the case similarity according to his (her) experience, i.e., ξ = 0.5, and we gain the similar cases set, i.e.,

Step 4: The DM gives a PCM A concerning the similar cases set according to his (her) experience as follows:

Step 5: The priority of the similar cases set can be gained by solving model (14), the result is w = (0.0304, 0.0261, 0.0609, 0.0609, 0.5478, 0.1826, 0.0913).

Step 6: Finally, the comprehensive coefficient CC of the similar cases set can be got by using Eq. (18) , the result is Sim = (0.0159, 0.0142, 0.0317, 0.0398, 0.4246, 0.1411, 0.0648). The most suitable historical case is got by ranking CCs and is determined to be C₁₁.

It is indicated from the computational results obtained by using the proposed case retrieval method that the retrieved historical case C₁₁ is the most suitable one to the target case C₀. Thus, the design plan for the case C₁₁ can be considered as that for the target case C₀. Further, designer of company E can make improvements based on the design plan of C₁₁.

In order to further illustrate the proposed method, alternative selection based on group decision making is described below. Three experts from three different departments are invited to make comparisons about the eight similar historical cases, and the three PCMs provided by them are as follows:

Suppose the relative importance weights of the three DMs are (0.5, 0.3, 0.2), and we use Eq. (15) to transform the three matrices into a matrix B. Then, by solving model (16) for each similar historical case, we get the best priorities shown in the first row of Table 6. If the weights of the three DMs are different, it would get a different matrix B. Then a different priority vector can be derived by model (16). Table 6 shows the best priority vectors derived under different weights of the three DMs. Furthermore, we get the comprehensive coefficient CC of the similar cases set by using Eq. (18) , and the results are shown in Table 7, from which we can see that the most suitable historical case is C₁₁. So we select the alternative of C₁₁ as the finally alternative.

According to the study of [26], we evaluate the alternatives of the similar historical cases from three criteria, such as safety (G₁), cost (G₂) and accident loss (G₃). In order to improve the requirement of similarity, we set the threshold of the similarity as ξ = 0.54, and the similar historical cases set is (C₅, C₁₀, C₁₁, C₁₂, C₁₃). Fig 4 shows the hierarchical structure for this alternative selection problem. Table 8 shows the PCM for three criteria. We get the priority weights by solving Eq. (14) , the results are shown in Table 8. Table 9 shows the PCM for the five alternatives with respect to the three criteria. The priority weights are gained by solving Eq. (14) , respectively the results are shown in Table 9. Based on the above results, we get the global priority using Eq.(17) , that is w^g = (0.0973, 0.0639, 0.4072, 0.2695, 0.1700). Finally we use Eq. (18) to get the comprehensive coefficient, and the result is that Sim = (0.0530, 0.0418, 0.3156, 0.2083, 0.1207), based on which, we select alternative C₁₁ as the response alternative.

Fig. 4.

Hierarchy of selection for the most suitable historical case

5. Comparative analysis and discussion

To further verify the validity of the proposed method, the results derived from the proposed method are further compared with other existing case retrieval methods.

First, PCBR is compared with the classic case similarity measurements, such as, ECBR, GCBR RCBR. As can be seen from Example 1, the similarity measurement based on CPT can express the psychological behavior of DM’s preference on distance well. From Table 5, the case similarities and the historical ranking based on CPT is a little different from the three classic case similarity measurements, because it considers the DM’s psychological behavior and can distinguish the DM’s preference strictly. This is more in line with the actual decision situations.

Furthermore, from Table 5, the classic case similarity measurements and PCBR are different, because ECBR considers the distance between two points, GCBR uses the Gaussian function to change the Euclidean distance into nonlinear form, RCBR represents the case similarity using the case correlation, PCBR considers the DM’s psychological behavior. The proposed method aggregates all the advantages of these methods.

The case retrieval method combined with similarity measurement and MCDM has been proposed by Qi et al.¹² and Li et al.¹³ They used the technique for TOPSIS to evaluate the similar historical case set. But [13] pointed out that the TOPSIS has a drawback that the number of cases under evaluation is limited. However, DEA has no limit on the number of DMU and the DEA model has no such restriction. Furthermore, the DEA model not only can solve the situation where one DM gives his/her evaluation for the alternatives by using PCM, but also can solve group decision making and hierarchical structure in the AHP. From Tables 6, 8 and 9, it can be seen that the DEA model can give a best priority vector from a PCM and can help DM to make more accurate decision making.

6. Conclusion

This paper develops a new case retrieval method from a comprehensive view of combining the similarity measurement and DEA model. The primary contributions are summarized as follows:

1. (1)

   A similarity measurement based on CPT is proposed. This method considers the DM’s psychological behavior and makes the decision result more realistic.

2. (2)

   The case similarity is gained through the aggregation of four case similarities. This can amplify the advantage of case similarity measurements and overcome the one-sidedness brought about by a single approach.

3. (3)

   In the case selection, a DEA model is constructed to get the priority vector. It can produce true priority vector for PCM and the DEA model can deal with three formats of evaluations, such as a DM evaluates the alternatives, several DMs evaluate the alternatives and the DM evaluates the alternative from a few criteria. These evaluation formats are usually used very often. So, this DEA model has good usability.

An example about the high-rise fire has shown the necessity of the similarity measurement based on CPT, which considers the DM’s psychological behavior. In addition, the example about the car design has shown the feasibility and validity of the proposed method, which can help DM to find the suitable historical case and to make decision.

For future work, a promising research direction is the use of artificial intelligence method^27,28 to improve the validity of the retrieval results. In addition, the case adaptation seems a promising and fruitful research line.

Acknowledgments

We thank two anonymous reviewers for their valuable comments and suggestions which have helped to improve the paper. This work was partly supported by the National Natural Science Foundation of China under the Grant Nos. 71371053 and 61773123, Humanities and Social Science Foundation of Chinese Ministry of Education under the Grant No. 16YJC630008, Fujian Natural Science Foundation of China, No. 2017J01513.