Abstract

In order to improve the efficiency of the recycling of the electric vehicle power batteries and reduce the recycling cost, it is of great importance to select an optimal power battery recycling mode. In this paper, an extended MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus full Multiplicative form) approach which combines with the two-dimension uncertain linguistic variables (TDULVs) and the regret theory, called TDUL-RT-MULTIMOORA method, is developed for solving the power battery recycling mode decision-making (PBRMDM) problem. Firstly, the evaluations of the power battery recycling modes over criteria are given by the experts using the TDULVs, and the evaluations of all experts are aggregated into a group linguistic decision matrix by the TDULDWA operator. On the basis of the regret theory, the perceived utility decision matrix is constructed. And then, in order to avoid the disadvantages of the subjective weighting methods, such as the deviation from the measured data and the dependence on the experience and knowledge of the experts, an objective entropy weighting method is applied. After that, the MULTIMOORA method is introduced to rank the power battery recycling modes. In the end, an illustrative example is given to verify the effectiveness and practicability of the proposed method.

1. Introduction

Compared with the traditional fuel vehicles, the electric vehicles have the characteristics of lower emission, lower noise, and lower pollution. In addition, due to the fact that the energy structure required by electric vehicles can be diversified, it helps to get rid of the dependence on nonrenewable oil resources. Therefore, it has a very important practical significance to develop the electric vehicle industry. Nowadays, in China, the electric vehicle industry has been strongly supported by the government and has played an important role in the reduction of greenhouse gas [1]. However, as the energy source of the whole electric vehicle, the life length of the power battery is limited. By 2020, the accumulative number of the power batteries going to enter the end-of-life period in China will reach 120000-170000 tons. If the wasted power batteries are not properly recovered or reused, this will not only cause the wastes of resources but also cause serious pollution to the environment [2]. Therefore, based on the theory of the sustainable development, the reasonable recycling of power batteries is one of the important factors to promote the development of electric vehicle industry.

Issues related to the development of electric vehicles have been widely studied in China recently [3]. Nowadays, there are several recycling modes of power batteries for the electric vehicle manufacturers to adopt. The best selection of the recycling mode will help the manufacturers improve the efficiency of the recovery and reduce the cost. However, in the process of the power battery recycling, because of the complexity and uncertainty of objective things and the fuzziness of human being’s thinking, it is difficult to describe the vague information by precise values. The multiple criteria decision-making (MCDM) problem proposed by Churchman et al. [4] is a discipline for supporting experts to figure out an optimal choice from all options based on multiple criteria [5]. Since the power battery recycling mode decision-making (PBRMDM) problem involves many qualitative and quantitative evaluation criteria, therefore, it is a feasible way to solve the PBRMDM problem as a MCDM problem. At present, there are few studies on the PBRMDM problem, and most of the related researches about the selection of the power battery recycling mode are mainly under the consideration of the recycling cost control. For example, Yun et al. [6] summarized two main basic aspects of recycling batteries, including mechanical procedure and chemical recycling, and proposed a framework for recycling batteries. Ordonez et al. [2] presented a qualitative analyzing approach for solving the recovery and regeneration technology of lithium batteries, which can recycle the valuable elements in the battery. Liu and Gao [7] put forward some corresponding battery recycling countermeasures based on the analysis of the urgency of power battery recycling in China. Tang et al. [8] proposed a reward-penalty mechanism including some policies for recycling the power battery and the costs of three single recovery modes and three competitive dual recovery modes were also tested by using the Stackelberg game theory.

In the real process of decision-making, it is hard for the decision makers to give their evaluations to the fuzzy or uncertainty information by exact numeric values. Recently, most researchers prefer to represent their opinions by means of the uncertain linguistic information. The uncertain linguistic variables (ULVs) presented by Xu [9] can express the evaluations of decision makers more accurately. An uncertain linguistic variable (ULV) is composed of a lower limited value and an upper limited value [9]. It can be used in more fuzzy and uncertain situation [10]. However, the ULVs do not consider the reliability of the experts’ subject evaluations. Liu and Zhang [11] developed the two-dimension uncertain linguistic variables (TDULVs) to represent the fuzziness of the information on the basis of the ULVs. A two-dimension uncertain linguistic variable (TDULV) is composed of two parts, which includes the class and the class linguistic information, where the class information represents the assessment of decision maker to the evaluated objects, and the class linguistic information denotes the reliability of the class assessment denoted by the decision maker. Until now, the TDULVs have been applied in many areas, such as the technology innovation ability evaluation problem [12], the extraefficient economic industry system selection [13], and the river basin ecosystem health evaluation problem [14].

The regret theory was firstly introduced by Loomes and Sugden [15] with the intention of depicting intuitive judgments simply and consistently. It is an important behavioural decision-making theory by considering the outcomes of the alternative choices and the possible results of unselected alternatives. In the regret theory, the perceived utility values are used to measure the expected value of satisfaction by choosing one alternative and rejecting another. Recently, the regret theory has been applied to solve different kinds of problems, such as the selection of the human-agent collaborative teams [16], trip distribution and traffic assignment [17], environmentally friendly supplier selection [18], and the selection of charging facility design for electric vehicles [19].

Entropy weighting method was originally proposed by Shannon [20]. The entropy can measure the probability of objective, and it can show the direct reflection of the information size and its uncertainty [21], and it is also a method with precise calculation process. Due to the characteristic of entropy weighting method, it has been widely applied in many different fields. For example, Liu and Li [22] proposed the comprehensive forecasting model by using the entropy weighting method. Delgado and Reyes [23] used the entropy weighting method to select the best alternative plants. Zhang et al. [24] proposed a novel ship detection method by using the entropy weighting method to extract the features of the synthetic aperture radar images.

Brauers and Zavadskas [25] proposed a MOORA method in 2006. In 2011, Chakrabory checked the robustness of six common MCDM methods, including the MOORA method [25], the AHP method [26], the TOPSIS method [27], the VIKOR method [28], the ELECTRE method [29], and the PROMETHEE method [30], as nonsubjectively as possible, and the results showed that only MOORA method satisfied all conditions of robustness of the MCDM [31]. Inspired by the MOORA method, Brauers and Zavadskas [32] developed a MULTIMOORA method by improving and synthesizing the MOORA method, and the results of the MULTIMOORA method show more robustness and accuracy compared with the MOORA method. The decision process of the MULTIMOORA method includes three parts: the ratio system, the reference point method, and the full multiplicative form of multiple objectives. Until recently, no other method is known as meeting all conditions of robustness for the multiple objects optimization; therefore, the MULTIMOORA method is regarded as the most robust technique for solving the MCDM problem [33]. So far, the MULTIMOORA method has been applied in many areas, such as the materials selection of power gears [34], the biomaterials selection [35], the pharmacological therapy selection [36], the selection of sites for ammunition depots [37], the supplier selection [38], and the risk evaluation problem [39].

Based on the above discussions, in this paper, to solve the PBRMDM problem, an extended MULTIMOORA method with two-dimension uncertain linguistic variables and the regret theory, called the TDUL-RT-MULTIMOORA method, is developed. The remainder of this paper is organized as follows: the preliminaries of this work are introduced in Section 2. Section 3 presents the framework of the TDUL-RT-MULTIMOORA method. An illustrative instance is conducted to demonstrate the effectiveness and practicality of the proposed method in Section 4. In the end, some conclusions are drawn in Section 5.

2. Preliminaries

2.1. Linguistic Variables

Suppose is a predefined linguistic term set with finite and totally ordered elements, where is an odd number, and then is called a linguistic variable [40].

2.2. Uncertain Linguistic Variables

Definition 1 (see [9]). Let be a continuous linguistic term set, and , where are the lower and upper limit value of , respectively, and then is called an uncertain linguistic variable (ULV) of .

2.3. Two-Dimension Uncertain Linguistic Variables

Definition 2 (see [11, 41]). Let be a TDULV, where is the first class of , which expresses the assessment of the decision maker to an evaluated object, while is the second class of , which denotes the decision maker’s subjective evaluation on the reliability of the first class result. are the lower and upper limit value of the first class, and are the lower and upper limit value of the second class, respectively, and then is called a two-dimension uncertain linguistic variable (TDULV).

Definition 3 (see [42, 43]). Suppose and are any two TDULVs, and then the operational rules between and are given as below:

Definition 4 (see [10]). Suppose is a TDULV, and then the expectation value of is

Definition 5 (see [10]). Suppose and are any two TDULVs, and if , then , or vice versa.

2.4. Aggregation Operators of the TDULVs

Definition 6 (see [44]). Let be a collection of 2DULVs and is the weights associated with and . Let be a set of 2DULVs, then, the WAA: , and ifthen, the WAA operator is called a weighted arithmetic averaging operator.

Definition 7 (see [45, 46]). Let be a TDULV, be a clustering class of the TDULVs, be the set of all TDULVs, and TDULDWA be , ifwherewhere is the weights of , satisfying . is the density influence index, and ; generally, ; is the number of the elements in , and ; represents the scale information of the cluster group ; is the weight vector of . Then, the TDULDWA operator is called a two-dimension uncertain linguistic density weighted averaging operator.

2.5. Regret Theory

Loomes and Sugden [15] raised the regret theory for the first time. In the regret theory, the perceived utility function is constructed by accumulating the realized utility of the selected alternative and the regret or rejoice utility between the best alternative and the selected one [15]. The perceived utility function of the regret theory is composed of two parts: the utility function and the regret or rejoice function . Let be a set of alternatives and be the results of all alternatives, where is the final evaluation value of the alternative . Then, is constructed as follows [47–49]:wherewhere , which indicates that the expert feels regret after choosing the alternative rather than the alternative ; is the risk aversion coefficient, and the smaller value of , the greater risk aversion of the expert, or vice versa [50]; is the regret aversion coefficient; the larger value of , the greater regret aversion tendency of the expert [51]. Tversky and Kahneman [50] gave the value of which is equal to 0.88 and equals 0.3 after experimental verification.

2.6. The MULTIMOORA Method

The MULTIMOORA method was first developed by Brauers and Zavadskas [32] on the basis of the MOORA method. It is a powerful tool for dealing with the MCDM problem. The process of MULTIMOORA method is made up of three parts: the ratio system, the reference point method, and the full multiplicative form of multiple objectives. The MULTIMOORA method is the most robust system of multiple objectives optimization than other multiple criteria decision-making methods [33]. Suppose is a decision matrixwhere is the evaluation value of alternative which refers to criteria , . Then, the steps of MULTIMOORA method are as below.

Step 1 (calculate the importance coefficients of criteria). The importance coefficients of criteria for criteria with reference to the criteria are calculated bywhere (j=1,2, n) is the weight of criterion , satisfying and ;

Step 2 (normalize the decision matrix into ). The decision matrix is normalized into by

Step 3 (the ratio system). In order to obtain the optimization, based on the ratio system, the best alternative is obtained bywhere the overall evaluation value of alternative refers to all criteria and is added in the circumstances of maximization and subtracted in the circumstances of minimization for every alternative [52]:where are the benefit criteria; are the cost criteria.

Step 4 (the reference point approach). The best alternative is got byThen the absolute value between the reference point and the normalized evaluation value of alternative refers to criteria calculated bywhere

Step 5 (the full multiplicative form). The preferred alternative is obtained bywhere

Step 6 (rank the alternatives). Firstly, the overall evaluations are ranked in descending order, the absolute values are ranked in ascending order, and the overall utility values are ranked in descending order. Then, after the calculation of the subordinate rank results, the above three rankings of alternatives are integrated into a final MULTIMOORA ranking on the basis of the generalized dominance relations of the dominance theory. The dominance theory [52] is a tool for ranking the subordinate alternatives by the MULTIMOORA method, which includes the plurality rule assisted with a kind of lexicographic method and the method of correlation of ranks.

3. The TDUL-RT-MULTIMOORA Approach for the PBRMDM Problem

3.1. Description of the PBRMDM Problem

Suppose is a set of the power battery recycling modes, is a set of criteria, and is the weights of criteria, where and . Let be a set of experts, and is the weights of the experts, where and . Expert gives his/her evaluation to the mode with respect to the criteria by TDULVs, where and . The flowchart of the TDUL-RT-MULTIMOORA method is demonstrated in Figure 1.

Figure 1 

The flowchart of the TDUL-RT-MULTIMOORA method.

3.2. TDUL-RT-MULTIMOORA Method

To solve the PBRMDM problem, an extended MULTIMOORA method with the TDULVs and the regret theory, called the TDUL-RT-MULTIMOORA method, is put forward. The decision processes of the TDUL-RT-MULTIMOORA method are described as follows.

Stage 1. Construct the perceived utility decision matrix.

Step 1.1. The evaluations of experts are aggregated bywherewhere ; ; is the density weighted vector, ; is the density influence index and ; is the number of the elements in , ; is the scale information of the clustered group ; is the weight vector of .

Step 1.2. Normalize the decision matrix into .

The TDULV group evaluation decision matrix is normalized by

Step 1.3. The expectation value of all evaluation information is calculated by

Step 1.4 (calculate the perceived utility values). According to the regret theory, the perceived utility value is got bywhereThus, the perceived utility decision matrix is established.

Stage 2 (calculate the weights of criteria). The entropy weighting method was produced by Shannon [20]. It is a method that employs probability theory to measure the uncertainty of information, which can avoid the negative effect of subjective elements. The entropy weighting method is a useful method to measure the uncertainty in the decision-making problem [53]. The steps of the entropy weighting method are as follows.

Step 2.1. Calculate the entropy value by

Step 2.2. Calculate the difference degree by

Step 2.3. Calculate the entropy weight of each criterion by