# Two-stage hybrid flowshop scheduling with simultaneous processing machines

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## Abstract

Simultaneous processing machines, common in processing industries such as steel and food production, can process several jobs simultaneously in the first-in, first-out manner. However, they are often highly energy-consuming. In this paper, we study a new two-stage hybrid flowshop scheduling problem, with simultaneous processing machines at the first stage and a single no-idle machine with predetermined job sequence at the second stage. A mixed integer programming model is proposed with the objective of minimizing the total processing time to reduce energy consumption and improve production efficiency. We give a sufficient and necessary condition to construct feasible sequencing solutions and present an effective approach to calculate the time variables for a feasible sequencing solution. Based on these results, we design a list scheduling heuristic algorithm and its improvement. Both heuristics can find an optimal solution under certain conditions with complexity O(*n*log*n*), where *n* is the number of jobs. Our experiments verify the efficiency of these heuristics compared with classical heuristics in the literature and investigate the impacts of problem size and processing times.

## Keywords

Scheduling Hybrid flowshop Simultaneous processing machine Heuristic List scheduling## Notes

### Acknowledgements

This work was financially supported by the Beijing Natural Science Foundation (No. 9174038), the Humanity and Social Science Youth Foundation of Ministry of Education of China (No. 17YJC630143), the National Natural Science Foundation of China (No. 71701016), the China Scholarship Council, the China Postdoctoral Science Foundation (No. 2012M510324) and the Discovery Grant of the Natural Science and Engineering Research Council of Canada (NSERC).

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