## 3 Materials and Methods  
### Problem Definition and Modeling Phase  
The suggested dispatching-fuzzy AHP-TOPSIS (DFT) methodology entails the combination of three methods for solving the FJSP in smart manufacturing: the dispatching algorithm, the fuzzy analytic hierarchy process, and TOPSIS. Such a combination is developed through a four-phase methodology outlined in Figure 4. The proposed approach is an adaptation of the algorithm presented by [PERSON] and [PERSON] [43]. A more detailed explanation of each phase is provided below:  
**Phase 1:** In this step, the production system is characterized by defining the route of each product reference, the available machines associated with each sub-process, the setup times (\(SUI_{f}\)), and the processing times (\(p_{j,k,l,j}\)), whereas the jobs are fully detailed in terms of their features. This information defines the N subset, which covers all the operations to be scheduled. Following this, the E subset, comprised of all operations eligible for immediate execution (the first operation of each job), is arranged while specifying their \(r_{1,k,l}\) values. The \(fp_{j}\) is then defined in each machine, and \(fp_{min}\) is finally identified from the list of m machines.  
**Phase 2:** In the next step, the machine with \(fp_{min}\) is selected. Then, all the candidate operations associated with the selected machine are enlisted. In the case of various machines with \(fp_{min}\), the operation to be scheduled must be chosen by implementing the operation prioritization framework depicted in Phase 3. A \(fp_{min}=\alpha\) denotes that all the operations have been programmed.  
**Phase 3:** This stage involves choosing the operation to be scheduled or enlisted in the subset P. In this regard, there are two possible scenarios. In the case of a set of eligible operations, the DFT (Dispatching-Fuzzy AHP-TOPSIS) approach must be implemented. To this end, an operation prioritization model has to be first defined considering the criteria or job features that are of special interest to the company. FAHP (see Section 3.2) is then employed to estimate the relative importance of these criteria under uncertainty. After this, TOPSIS (see Section 3.3) calculates a priority index (closeness coefficient) for each eligible operation. The operation with the highest index is finally selected to be accordingly  
Figure 4: The proposed DFT approach for solving the FJSP.  
included in the subset P. The remaining scenario is the one in which there is only one candidate operation; in such a case, the scheduler must pass it immediately to the subset P without implementing the operation prioritization model here proposed.  
**Phase 4:** The final step of this methodology is to enlist the selected operation while outlining its start and end times. The end time is calculated by using the formula specified in Figure 3 considering whether \(j=j^{\prime}\). Following this, the \(fp_{j}\) value of the machine \(j\) associated with the selected operation is updated. \(rp_{i,k,l,j}\) and \(fp_{min}\) are then computed according to Figure 3. Ultimately, the scheduler must return to Phase 1 until all the operations are scheduled (\(fp_{min}=\alpha\)).