# General Lower Bounds for the Total Completion Time in a Flowshop Scheduling Problem - MaxPlus Approach

### Nhat-Vinh Vo, Pauline Fouillet, Christophe Lenté

#### Abstract

The flowshop scheduling problem has been largely studied for 60 years. As a criterion, the total completion time also receives a great amount of attention. Many studies have been carried out in the past but they are limited in the number of machines or constraints. MaxPlus algebra is also applied to the scheduling theory but the literature focuses on some concrete constraints. Therefore, this study presents a new method to tackle a general permutation flowshop problem, with additional constraints, to elaborate on lower bounds for the total completion time. These lower bounds can take into account several constraints, like delays, blocking or setup times, but they imply to solve a Traveling Salesman Problem. The theory is developed, based on a MaxPlus modeling of flowshop problems and experimental results are presented.

#### References

- Allahverdi, A. and Al-Anzi, F. S. (2006). A branch-andbound algorithm for three-machine flowshop scheduling problem to minimize total completion time with separate setup times. European Journal of Operational Research, 169(3):767-780.
- Augusto, V., Lenté, C., and Bouquard, J.-L. (2006). Résolution d'un flowshop avec delais minimaux et maximaux. In MOSIM.
- Aydilek, H. and Allahverdi, A. (2010). Two-machine flowshop scheduling problem with bounded processing times to minimize total completion time. Computers & Mathematics with Applications, 59(2):684-693.
- Blazewicz, J., Ecker, K.-H., Pesch, E., Schmidt, G., and Weglarz, J. (1996). Scheduling in computer and manufacturing processes. Springer Verlag, Berlin.
- Bouquard, J.-L. and Lenté, C. (2006). Two-machine flow shop scheduling problems with minimal and maximal delays. 4or, 4(1):15-28.
- Bouquard, J.-L., Lenté, C., and Billaut, J.-C. (2006). Application of an optimization problem in Max-Plus algebra to scheduling problems. Discrete Applied Mathematics, 154(15):2064-2079.
- Brucker, P. (2006). Scheduling Algorithms. Springer, 5 edition.
- Carpaneto, G., Dell'amico, M., and Toth, P. (1995). Exact solution of large asymmetric traveling salesman problems. ACM Transactions on Methematical Software, 21(4):394-409.
- Cohen, G., Dubois, D., Quadrat, J.-P., and Viot, M. (1985). A linear system-theoretic view of discret-event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automatic Control, 30:210- 220.
- Della Croce, F., Narayan, V., and Tadei, R. (1996). The twomachine total completion time flow shop problem. European Journal Of Operational Research, 90:227- 237.
- Emmons, H. and Vairaktarakis, G. (2013). Flow Shop Scheduling. Springer US, New York, 182 edition.
- Gaubert, S. (1992). Théorie des systèmes linéaires dans les dioïdes. PhD thesis.
- Gaubert, S. and Maisresse, J. (1999). Modeling and analysis of timed Petri nets using heaps of pieces. IEEE Trans. Automatic Control, 44(4):683-698.
- Giffler, B. (1963). Schedule algebras and their use in formulating general systems simulations. In Industrial schduling. Prentice Hall, New Jersey.
- Graham, R. L., Lawler, E. L., Lenstra, J. K., and Rinnooy Kan, A. H. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5(2):287-326.
- Gunawardena, J. (1998). Idempotency. Publications of the Newton Institute.
- Hanen, C. and Munier, A. (1995). Cyclic scheduling on parallel processors: an overview. John Wiley.
- Ignall, E. and Schrage, L. (1965). Application of branchand-bound technique to some flow shop problems. Operations Research, 13(3):400-412.
- Johnson, S. M. (1954). Optimal two- and three-stage production schedules with setup times included. Naval Research Logistics, 1:61-68.
- Lenté, C. (2001). Analyse Max-Plus de problèmes d'ordonnancement de type Flowshop. PhD thesis, Université Franc¸ois Rabelais de Tours.
- Lenté, C. (2011). Mathématiques, Ordonnancement et Santé. Habilitation à diriger des recherches, Université Franc¸ois Rabelais de Tours.
- Pan, Q.-K. and Ruiz, R. (2013). A comprehensive review and evaluation of permutation flowshop heuristics to minimize flowtime. Computers & Operations Research, 40(1):117-128.
- Smith, W. E. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3(1- 2):59-66.
- Su, L.-H. and Lee, Y.-Y. (2008). The two-machine flowshop no-wait scheduling problem with a single server to minimize the total completion time. Computers & Operations Research, 35(9):2952-2963.
- Trabelsi, W., Sauvey, C., and Sauer, N. (2012). Heuristics and metaheuristics for mixed blocking constraints flowshop scheduling problems. Computers & Operations Research, 39(11):2520-2527.
- Vo, N. V. and Lenté, C. (2013). Equivalence between Two Flowshop Problems - MaxPlus Approach. In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems, pages 174- 177, Barcelona. SciTePress - Science and and Technology Publications.

#### Paper Citation

#### in Harvard Style

Vo N., Fouillet P. and Lenté C. (2014). **General Lower Bounds for the Total Completion Time in a Flowshop Scheduling Problem - MaxPlus Approach** . In *Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-017-8, pages 382-389. DOI: 10.5220/0004833403820389

#### in Bibtex Style

@conference{icores14,

author={Nhat-Vinh Vo and Pauline Fouillet and Christophe Lenté},

title={General Lower Bounds for the Total Completion Time in a Flowshop Scheduling Problem - MaxPlus Approach},

booktitle={Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2014},

pages={382-389},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004833403820389},

isbn={978-989-758-017-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - General Lower Bounds for the Total Completion Time in a Flowshop Scheduling Problem - MaxPlus Approach

SN - 978-989-758-017-8

AU - Vo N.

AU - Fouillet P.

AU - Lenté C.

PY - 2014

SP - 382

EP - 389

DO - 10.5220/0004833403820389