## Abstract

The non-waiting constraint is a crucial factor in industries with continuous production flows. Recognizing the significance of this constraint in the manufacturing process, we propose several approaches to minimize the maximum tardiness in the no-wait manufacturing flow shop scheduling problem. This research presents two main approaches: an exact method that utilizes mixed integer linear programming and an approximate method based on constructive heuristics and hybrid meta-heuristics. To tackle the problem, we introduce multiple heuristics and hybrid improved meta-heuristics based Nawaz Enscore Ham, greedy randomized adaptive search procedure and genetic algorithm. Furthermore, we propose three hybrid meta-heuristics based on the simulated annealing approach. To validate the effectiveness and robustness of our methods, we conducted experiments on multiple instances of the flow shop problem proposed by Taillard. The results demonstrate that the hybrid algorithm, which combines a greedy randomized adaptive search procedure with the insertion procedure and simulated annealing, exhibits strong performance. In fact, this algorithm achieved a success rate of \(72\%\) across 200 test instances. It outperformed the other two meta-heuristics, with a minimum average relative percentage deviation of only \(0.0039\%\).

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No datasets were generated or analysed during the current study.

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### Contributions

Omar Nejjarou: Conceptualization, Methodology, Software, Validation. Said Aqil: Conceptualization, Methodology, Writing—review and editing. Mohamed Lahby: Writing—review and editing.

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## Appendix: Numerical illustration

### Appendix: Numerical illustration

Our contribution focuses on improving the GRASP algorithm by integrating the Total Permutation Procedure (TPP) and the NEH heuristic. To our knowledge, this is the first time such an improvement has been applied to GRASP algorithms. First, we’ll present the results obtained with TPP. Then, we’ll apply the NEH approach to these same GRASP algorithm results, in order to further evaluate the effectiveness of our proposed enhancement. We’ll compare these results to those found by MILP. For this, we have chose to apply this approach in a production workshop with six jobs \(\{ J_1, J_2, J_3, J_4, J_5, J_6 \}\) to produce three machines \(\{ M_1, M_2, M_3 \}\) in serial workshop. the following matrices show job processing times on the machines, as well as lead times, we will classify these jobs in descending order of the sum of the operating times of each job (\(TP_j= \sum _{i=1}^{m} p_{ij}\)), this allowed us to have the initial sequence \(S_0=\{ J_6, J_3, J_1, J_5, J_4, J_2 \}\). From this initial solution we will start our approach we apply the GRASP algorithm to each iteration and improve the partial solution by the total permutation method.

On this first iteration we have the first job is \(J_6\) we will combine it with the other jobs \(\{J_3, J_1, J_5, J_4, J_2 \}\) and calculated the value of the objective function and deduce the best partial sequence:

*IGRN solution*

During this phase we have as solution for the first iteration \(S^{gr}_{1} = \{ J_6,J_5 \}\), and we start to insert the other jobs and update the solution; the following figure shows well the first three iterations as well as the results obtained. The Fig. 17 show the last step insertion process using the NEH algorithm in the improved GRASP procedure.

The Fig. 18 illustrate the graphical solution given in IGRN, where the best constructive solution in improved GRASP by NEH is \(S=\{J_4,J_1,J_5,J_3,J_2,J_6\}\). The minimum value of the objective function, denoted as \(T_{max}=38\), is achieved by inserting job \(J_1\) into the previously best sub-sequence.

*IGRP solution*

The Fig. 19 depict last iterations of the improved GRASP with TPP, initiated with the \({J_6, J_5}\) job combination. In each iteration, the generated solution is obtained by considering the total permutations within the selected sub-sequence from the RCL set.

In this approach, we utilized the IGRP algorithm to generate a set of solutions by exhaustively permuting the right sub-sequence. This algorithm facilitates extensive exploration of the solution’s neighborhood by examining various job positions. The final step yielded the best solution, denoted as \(S=\{J_6,J_1,J_5,J_4,J_3,J_2\}\). The graphical Gantt chart in Fig. 20 illustrates the resulting scheduling plan, with a maximum tardiness of \(T{max}=43\) attributed to job \(J_2\).

*MILP solution*

Using the LINGO solver, we found the optimal solution with a value of \(T_{\text {max}} = 34\). This solution is obtained by the sequence \({J_2, J_4, J_3, J_5, J_1, J_6}\), as shown in the Gantt diagram in Fig. 21.

In Figs. 22 and 23 presents the initial solution for instances 20 jobs ans 5 machines of Taillard’s dataset, attained by the two NEH and IGRN, providing visual evidence of the efficacy of these approaches.

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Nejjarou, O., Aqil, S. & Lahby, M. Hybrid meta-heuristic solving no-wait flow shop scheduling minimizing maximum tardiness.
*Evol. Intel.* (2024). https://doi.org/10.1007/s12065-024-00965-0

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DOI: https://doi.org/10.1007/s12065-024-00965-0