3.2.2. TOPSIS

TOPSIS, an analysis method that compares and selects multiple alternatives based on multiple criteria, was firstly proposed by Hwang and Yoon (1981) [52]. Fundamental for TOPSIS is to determine the positive ideal solution and negative ideal solution of each attribute. The positive ideal solution is the optimal solution among the alternatives, and its attribute values reach the best value, while the negative ideal solution is the worst solution. After calculating the Euclidean distance between each scheme, the positive ideal solution, the distance between each scheme and the negative ideal solution, the approximate degree of each alternative to the optimal solution can be obtained, which can be used as the basis for evaluating the merits of the alternatives. The TOPSIS method has many advantages. It has no strict constraints on the data distribution and sample content. It is applicable to the analysis of small samples as well as large systems with multiple evaluation units and indexes. It is flexible and convenient to use and has universal applicability. The steps of TOPSIS are as follows (Figure 4):

**Figure 4.** Process of technique for order of preference by similarity to ideal solution (TOPSIS).

**Step 1:** Obtain and normalize the original decision matrix.

Assuming that there are *N* criteria and *M* alternatives, the decision matrix formed by them is

$$X = \begin{bmatrix} \ \mathbf{x}\_{11} & \ \cdots & \ \mathbf{x}\_{1n} \\ \vdots & \ddots & \vdots \\ \mathbf{x}\_{m1} & \cdots & \ \mathbf{x}\_{mn} \end{bmatrix} \tag{2}$$

To eliminate the effects of various scales of criteria, the normalized decision matrix (*B*) is given by

$$b\_{ij} = \frac{\mathbf{x}\_{ij}}{\sqrt{\sum\_{i=1}^{m} \mathbf{x}\_{ij}^{2}}}, i = 1, 2, \dots, m; j = 1, 2, \dots, n \tag{3}$$

**Step 2:** Obtain the weighted normalized matrix. The weighted normalized matrix (*Z*) is obtained by

$$z\_{i\bar{j}} = b\_{i\bar{j}} \cdot w\_{\bar{j}}, i = 1, \ 2, \dots, m; j = 1, \ 2, \dots, n \tag{4}$$

where *wj* is the weight of the criterion.

**Step 3:** Determine the positive ideal solution (*V*+) and the negative ideal solution (*V*−) using the following equations:

$$V^{+} = \left(z\_1^{+}, z\_2^{+}, \dots, z\_n^{+}\right) \text{ where } z\_j^{+} = \max\left\{z\_{i\bar{j}}\right\}, i = 1, 2, \dots, m; j = 1, 2, \dots, n \tag{5}$$

$$V^- = \left(z\_1^-, z\_2^-, \dots, z\_n^-\right) \text{ where } z\_j^- = \min\left\{z\_{i\bar{j}}\right\}, i = 1, 2, \dots, m; j = 1, 2, \dots, n \tag{6}$$

**Step 4:** Calculate the distance of each alternative from *V*<sup>+</sup> and *V*<sup>−</sup> using Equations (7) and (8):

$$D^{+} = \sqrt{\sum\_{j=1}^{n} \left(z\_{ij} - z\_{j}^{+}\right)^{2}}, i = 1, 2, \dots, m \tag{7}$$

$$D^{-} = \sqrt{\sum\_{j=1}^{n} \left(z\_{ij} - z\_j^{-}\right)^2}, i = 1, 2, \dots, m \tag{8}$$

**Step 5:** Obtain the closeness coefficient (CC) of each alternative as follows:

$$\text{CC}\_{i} = \frac{D\_{i}^{-}}{D\_{i}^{+} + D\_{i}^{-}}, i = 1, 2, \dots, m \tag{9}$$

**Step 6:** Rank the alternatives.

The alternatives are ranked according to the value of CC. The best alternative has the highest closeness coefficient.

#### **4. Application in Case Study**

A case study with data from a Chinese engine manufacturer was conducted to verify the proposed approach. In recent years, increasing attention has been paid to the research and development of automobile remanufacturing in China. Compared with traditional manufacturing, remanufacturing has the characteristics of energy saving and material saving, which are important in the circular economy. With the introduction of relevant policies and the increase in public awareness of environmental issues, more and more automobile enterprises are actively engaging in remanufacturing activities. By June 2020, the motor vehicle population in China has reached 360 million, among which more than 270 million are automobiles [6]. The annual maintenance and replacement parts output value is nearly 1 trillion yuan, which represents a huge potential market for remanufacturing [6]. As an automobile part with high standardization and strong versatility, mainly made of metal materials, the engine is the automotive part that is commonly remanufactured.

For engine remanufacturing, the main process is the dismantling and cleaning of faulty or obsolete engine parts. This is followed by technical processing and transformation, according to the size modification requirements. Finally, after being reassembled and tested, it will be made into a new engine [53]. Before the remanufacturing processes, the reverse logistics of waste products should be completed first, which is a process requiring systematic operation. The company's production equipment level is above the average level of the same industry in China. At present, its products include remanufactured engines from various brands which have more than ten variations, including Steyr, Cummins and Chaochai 6102, which are used as maintenance parts for the after-sale market. This paper takes the problem of this company as an example and applies the proposed framework to select a suitable RL provider for it.

#### *4.1. Application of AHP*

First, an online AHP questionnaire was created according to the hierarchy shown in Figure 2. The questionnaire was sent to 340 respondents between September 14 and October 5, and 200 responses were collected after it was issued. Then, pairwise comparison matrices for each level were obtained according to the collected data. The weights of the four dimensions after data screening and processing are shown in Table 4 (this matrix passed the consistency test). "Technology" had the highest weight, followed by "circularity" and "society" at similar importance, and "economy" was the least important factor.

**Table 4.** The weights and ranking of the four dimensions.


The weights of each criterion are listed in Table 5. Global weights were obtained by multiplying the relative weights between each criterion and the weights of each dimension. The CR of each matrix was less than 0.1 and could be used for evaluation and selection. Compared with "operating cost (E1)", "RL cost (E2)" is considered to be a more important indicator. "Resource utilization (E4)" and "quality management (E6)" are considered as the most important indicators in the social dimension and the technical dimension, respectively. "Eco-friendly raw materials (E11)" is the most important criterion and "environmental standards (E10)" is the least important criterion in the circularity dimension.

**Table 5.** The weights and ranking of criteria.


#### *4.2. Application of TOPSIS*

The decision-making team composed of eight experts scored three different modes (based on Figure 2, the three different modes were TPT, MT and RT) against the evaluation criteria, as shown in Table 6. For each mode, better performance for a criterion is reflected by a higher score. The range of the score is 1 to 9. Each final score was based on the sum of the average scores of the eight experts.


**Table 6.** Original decision matrix of alternatives.

After normalization (Table 7), the weight of each index was calculated to obtain the weighted normalized matrix shown in Table 8. The positive ideal solutions *V*<sup>+</sup> and negative ideal solutions *V*− are listed at the bottom of the table.

**Table 7.** Normalized decision matrix.




The distance of each alternative was then calculated using Equations (7) and (8). Finally, Equation (9) was used to calculate the closeness coefficient, and the results are shown in Table 9. The higher the closeness coefficient, the higher the alternative is ranked. The results show that TPT is the best solution, and its CC*<sup>i</sup>* is 0.7565. MT has the furthest distance from the negative ideal solution, ranking the second.

**Table 9.** Distances to ideal solution and closeness coefficient.

