## 3 AHP/ANP with Shapley Values
### Methods for AHP and ANP  
The AHP, introduced in [21], serves as a decision-making methodology for intricate, multi-criteria problems. The AHP hinges on pairwise comparisons within a hierarchical structure, thereby streamlining decision-making by deconstructing it into smaller, more manageable components [15]. The mathematical model of the AHP encompasses the construction of the PCM, the calculation of criteria or alternative weights, and the computation of the consistency ratio [18; 20]. Since its inception, the AHP has witnessed significant advancements in both theory and application. Researchers have concentrated on enhancing  
Figure 1: The framework of the proposed method. the mathematical foundations, broadening their applicability, and amalgamating them with other methods to address various decision-making problems. To quantify the PCMs, Ref. [21] introduced a scale spanning from 1 to 9, where 1 signifies equal importance, and 9 denotes extreme importance. Let \(\mathbf{A}=\left[a_{ij}\right]\) be a PCM matrix, where \(a_{ij}\) represents the importance of element \(i\) relative to element \(j\). The decision-maker allocates values from this scale to convey the relative significance of one element over another. This culminates in a reciprocal PCM that is employed to calculate the eigenvector corresponding to the largest eigenvalue. This eigenvector embodies the normalized weights of the criteria or alternatives [21], which can be represented as solving the subsequent eigenvalue problem:  
\[\mathbf{A}\mathbf{w}=\lambda_{max} \tag{1}\]  
where \(\mathbf{A}\) is the PCM, \(\mathbf{w}\) denotes the weight vector of criteria, and \(\lambda_{max}\) denotes the maximum eigenvalue. The methodology of the AHP also accommodates the consistency of the decision-maker's assessments. [PERSON]. [21] proposed the consistency ratio (CR), which juxtaposes the consistency index (CI) with the random index (RI)--an average index derived from randomly generated matrices. A CR less than or equal to 0.10 is generally deemed acceptable, signifying that the decision-makers' judgments exhibit a reasonable degree of consistency. The formulas for CR and CI are as follows:  
\[CR=\frac{\text{CI}}{\text{RI}}=\frac{\lambda_{max}}{n-1}, \tag{2}\]  
where \(n\) denotes the number of criteria, and RI is used to estimate the average consistency of a randomly generated pairwise comparison matrix, which can be obtained from Table 2[21]. Concerning theoretical foundations, consistency and inconsistency measures have been proposed to appraise the reliability of pairwise comparisons, such as the CR in [21; 22] and the geometric consistency index (GCI) in [23]. Alternative weight derivation methods have been devised, including the row geometric mean method (RGMM) in [24], the logarithmic least squares method (LLSM) in [25], and the least square priorities method (LSM) in [26]. Furthermore, group decision-making has been integrated into the AHP through aggregation approaches like the weighted arithmetic mean (WAM) in [27] and the geometric aggregation operator (GAO) in [28]. With respect to practical applications, AHP has been employed in supply chain management for supplier selection and evaluation, incorporating green supplier evaluation [16] and assessing supply chain resilience [29]. In environmental decision-making, the AHP has been applied to prioritize climate change adaptation strategies and renewable energy source prioritization in Turkey [30]. The ANP was proposed by [PERSON] [8] to generalize the AHP for considering interdependencies and feedback between criteria. Initially, we must depict the relationship between the criteria to form the structured graph of a problem. It is noteworthy that the graph should satisfy the property of the irreducible Markov chain, as previously explained, to ensure that the steady state exists. Subsequently, we derive the relative weights of criteria through the AHP to indicate how much a column criterion is more important than the row