## 3 Materials and Methods
Generally speaking, the quality sorting model based on AHP-entropy grey clustering is divided into three stages, namely, problem structuring, evaluation, and class assignment. The method of clustering based on grey possibility functions is primarily used to verify whether or not the observational objects belong to pre-determined classes [17]. Overall, the decision model can be described as follows. First, the structuring problem focuses on the development of effective structuring for the problem situation into a hierarchical structure of decisions, thereby defining the criteria to be evaluated for the goal when using AHP and entropy weighting. The purpose of problem structuring in the methodology is to help decision-makers better understand the problem, better explain and justify the conclusions, and finally ease the validation process. The development of an effective structure for the sorting problem hierarchy requires criteria to evaluate the incoming core quality in order to divide the cores into several classes and determine core alternatives. With this problem, \"criteria\" refers to the quality conditions of products, i.e., the technological, physical, and usage conditions. To classify the incoming core quality, \(J\) quality classes are considered, and we refer to the quality class \(j\) in subscript to differentiate between the different qualities of cores, i.e., \(j=1,2,\ldots,J\). Moreover, classifying the \(s^{th}\) core into the \(j^{th}\) grey class based on the observed value \(X_{sr}\) of the \(s^{th}\) core judged against the \(r^{th}\) criterion is referred to as grey clustering. The smaller \(j\) is, the better the quality class of the core. The quality levels of the cores in the same quality class indicate the same remanufacturing process requirements. Secondly, the evaluation process is a systematic process of collecting data, comparing preferences, and weighting to determine the clustering coefficient. Once the data are obtained, the next step is to assess the relative importance between criteria and sub-criteria at each level. For this reason, the AHP uses pairwise comparison judgement and entropy weighting to evaluate the strengths of importance. The following process can be carried out via adjustment of the whitening weight functions. The classification of \(S\) incoming cores into \(J\) grey classes using the \(r^{th}\) criterion is known as the \(r^{th}\) criterion sub-classification by the whitening weight function of the \(r^{th}\) criterion \(j^{th}\) sub-class referred as \(f^{j}_{r}(\cdot)\). When the whitening weight function \(f^{j}_{r}(\cdot)\) takes the typical form as can be seen as Figure 4, then \(X^{j}_{r}(1),\ X^{j}_{r}(2),\ X^{j}_{r}(3),\ X^{j}_{r}(4)\) denotes the turning points of \(f^{j}_{r}(\cdot)\). The second step in the evaluation process is determining the whitening weight functions of each sub-criterion. Moreover, in the grey weight clustering process, the whitening weight function can be grouped into subclass \(j\) of criterion \(r\). Then, the third step in the evaluation process is determining a clustering weight for each criterion \(r\) (\(\eta_{r}\)). When assessing the quality levels of incoming cores, it is important to understand how important each criteria and sub-criteria are. In general, the weights between criteria have to be calculated by experts or inspectors. To obtain the weights, the AHP and Shannon entropy methods are used. Pairwise comparisons are generated using a nine-point scale with a standard AHP system that transforms expert preferences into available alternatives, such as equal, moderately, strongly, very strongly, or extremely preferred. Once the estimation of the relative preferences of the pairwise comparison matrix and entropy weights are obtained, it is necessary to compromise between them to determine the clustering weights of each criterion.