## 3 Materials and Methods
### Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)  
TOPSIS is a multi-criteria decision method developed by [PERSON] and [PERSON] [49] for ranking alternatives considering the shortest distance from the positive-ideal solution (PIS) and the farthest separation from the negative-ideal solution (NIS) [50; 51]. The outcome is an indicator called \"closeness coefficient\" (priority index) ranging between 0 and 1, which enables schedulers, production managers, and practitioners to prioritize operations swiftly. The TOPSIS technique also holds the following advantages:  
* It can consider both conflicting quantitative and qualitative features, a context often found in scheduling scenarios where the operations need to be prioritized according to a set of selection criteria [52]. * The FJS algorithms often utilize one prioritization rule and apply others in case of a tie among different operations or jobs; however, the closeness coefficient derived from TOPSIS can be adopted as an overall measure representing these rules simultaneously and thereby emulating the real decision-making scenario addressed by production managers and practitioners. * It allows decision-makers to establish how the operation/job priority may change if its features are modified during the negotiation process with a client or in presence of different market and production configuration conditions. In this regard, sensitivity analysis can be fully underpinned based on separations from PIS and NIS [53; 54; 55; 56]. The general TOPSIS procedure is outlined in Figure 5[57]:  
Figure 5: TOPSIS algorithm. **Step 1:** A decision matrix \(X\) with \"\(m\)\" operations and \"\(n\)\" sub-criteria is created via Equation (8). \(X_{ij}\) is the value of the sub-criterion \(S_{j}\)\((j=1,\,2,\ldots,n)\) in each operation \(O_{l}(i=1,\,2,\ldots,m)\). \[X= \begin{array}{c}O_{1}\\ O_{2}\\ O_{3}\\ \vdots\\ O_{M}\end{array} \left[ \begin{array}{cccc}S_{1}&S_{2}&\ldots&S_{n}\\ x_{11}&x_{12}&\ldots&x_{1n}\\ x_{21}&x_{22}&\ldots&x_{2n}\\ x_{31}&x_{32}&\ldots&x_{3n}\\ \vdots&\vdots&\ldots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{mn}\end{array} \right] \tag{8}\]  
**Step 2:** The normalized decision matrix \(R\) is calculated using Equation (9). \(n_{ij}\) is the norm used by TOPSIS via Equation (10). Additionally, \(r_{ij}\) is defined as the element of this matrix. \[R=X\cdot n_{ij} \tag{9}\]  
\[n_{ij}=\frac{x_{ij}}{\sqrt{\sum_{i=1}^{y}x_{ij}^{2}}} \tag{10}\]  
**Step 3:** The weighted normalized decision matrix \(V\) is obtained using Equation (11). The sub-criteria weights (\(w_{j}\)) are those resulting from the FAHP method. \[V=\left[w_{j}r_{ij}\right]=\left[v_{ij}\right] \tag{11}\]  
**Step 4:** The ideal \((C^{+})\) and anti-ideal \((C^{-})\) scenarios are established according to Equations (12) and (13), respectively:  
\[C^{+}=\left\{\binom{max}{i}c_{ij}\left|j\in J\right.\binom{min}{i}c_{ij}\left| j\in J^{\prime}\right)for\;i=1.2.\;\ldots m\right\}\left\{c_{1}^{+}.c_{2}^{+}.\ldots.c_{j}^{+}.\ldots.c_{n}^{+}\right\} \tag{12}\]  
\[C^{-}=\left\{\binom{min}{i}c_{ij}\left|j\in J\right.\binom{max}{i}c_{ij}\left| j\in J^{\prime}\right)for\;i=1.2.\;\ldots m\right\}=\left\{c_{1}^{-}.c.\ldots.c_{j}^{-}.\ldots.c_{n}^{-}\right\} \tag{13}\]  
Considering that:  
\[J=\left\{j=1.2.\ldots.n|j\;\mbox{associated with the benefit sub $-$ criterion}\right\}\] \[J=\left\{j=1.2.\ldots.n|j\;\mbox{associated with the cost sub $-$ criterion}\right\}\]  
**Step 5**: The separation measures of each operation to \(C^{+}\) and \(C^{-}\) are calculated using the Euclidean separation via Equations (14) and (15). _Euclidean separation from ideal scenario_  
\[d_{i}^{+}=\sqrt{\sum_{j=1}^{n}\left(c_{ij}-c_{j}^{+}\right)^{2}}\qquad i=1.2. \ldots.m \tag{14}\]  
_Euclidean separation from anti-ideal scenario_  
\[d_{i}^{-}=\sqrt{\sum_{j=1}^{n}\left(c_{ij}-c_{j}^{-}\right)^{2}}\qquad i=1.2. \ldots.m \tag{15}\]  
**Step 6:** The relative closeness coefficient \((CC_{i})\) is calculated by using Equation (16).