## 4 Multi-Objective Solution
### Analytic Hierarchy Process (AHP)-Based Criteria Weights  
This section discusses how the weight parameters (\(\alpha_{i}\)) are obtained from a decision-maker. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) indicated that future applications of AHP include decision-making and other complex issues involving an integrated application of AHP and other techniques. More details regarding recent AHP-based techniques and applications can be found in the literature (e.g., [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018); [PERSON] and [PERSON] (2018)). AHP works by transforming empirical comparisons into numerical values, allowing a decision-maker to incorporate their experience and judgment into an evaluation. In this study, the relative importance scale of 1-9 was used for pairwise comparisons, where indifference corresponds to 1; weak preference to 3; definite preference to 5; strong preference to 7; very strong preference to 9; and 2, 4, 6, and 8 are used as intermediate values for comparisons. These pairwise comparisons are then recorded in a matrix, as shown in (60). The diagonal elements are always 1, illustrating the comparisons of the same element. Only the upper triangular matrix needs to be evaluated, as the reciprocal value in the lower triangular matrix can be simply converted. Thus, there are \(\left[n\times(n-1)\right]/2\) judgments required to solve the matrix, where \(n\) is the number of evaluated criteria.  
\[A=\left[\begin{array}{ccccc}1&a_{12}&\ldots&a_{1n}\\ a_{21}&1&\ldots&a_{2n}\\ \ldots&a_{ji}=\
icefrac{{1}}{{a_{ij}}}&1&\ldots\\ a_{n1}&\ldots&\ldots&1\end{array}\right] \tag{60}\]  
where \(a_{ij}\) is the comparison between \(i\) and \(j\) based on the importance scale.  
Once the comparison matrix is normalized by dividing each value by the total value in each column, the normalized principal eigenvector is calculated, which provides the criteria weights by computing the row average. To ensure that a decision-maker has been consistent, the consistency index (\(CI\)) (61) and consistency ratio (\(CR\)) (62) can be calculated, based on the maximum eigenvalue (\(\lambda_{\text{max}}\)) and random consistency index (\(RI\)), respectively. The \(RI\) values can be obtained based on the following (\(n\), \(RI\)) pairs: (1, 0.00), (2, 0.00), (3, 0.58), (4, 0.90), (5, 1.12), (6, 1.24), (7, 1.32), (8, 1.41), (9, 1.45), and (10, 1.49). The \(CR\) value is used to interpret the consistency, in which a \(CR\) of less than 10% implies an acceptable consistency.  
\[CI=\frac{\lambda_{\text{max}}-n}{n-1} \tag{61}\]  
\[CR=\frac{CI}{RI} \tag{62}\]  
Following the AHP approach, the objective functions and criteria weight were computed, as shown in Table 2. Initially, an AM expert rated the criteria preference using the AHP's scale for all pairwise comparisons. For example, the total cost criterion was indifferent to weakly preferred (i.e., 2) for both the load balance and total lateness criteria, whereas the total unassigned parts criterion was strongly to very strongly preferred to the total cost criterion (i.e., 1/8). The pairwise-comparison matrix was then normalized, the eigenvector was calculated, the maximum eigenvector (\(\lambda_{\text{max}}\)) was obtained, and the \(CI\) and \(CR\) were computed: \(\lambda_{\text{max}}=4.17\), \(CI=(4.17-4)/(4-1)=0.06\). An \(RI\) of 0.90 was then chosen; as such, \(CR=0.06/0.90=0.06\) or 6%. Since the value of \(CR\) was less than 10% (i.e., 0.1), the judgments were found to be acceptably consistent. The obtained criteria weights for the total cost (\(Z_{1}\)), load balance (\(Z_{2}\)), total lateness (\(Z_{3}\)), and total unassigned parts (\(Z_{4}\)) were thus 0.135, 0.078, 0.082, and 0.705, respectively (Table 2). That is, the criteria were ranked from most to least important as total unassigned parts, total cost, total lateness, and load balance, respectively. These obtained criteria weights were then used in the non-preemptive method introduced earlier.