Document Type : Original Article


1 Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran.

2 Department of Industrial Engineering, Karaj Branch, Islamic Azad University, Karaj, Iran.


The purpose of this paper is to optimize the integrated problem of lot-sizing and scheduling in a flexible job-shop environment considering energy efficiency. The main contribution of the paper is simultaneously considering lot-sizing and scheduling decisions, while accounting for energy efficiency.  In order to achieve this objective, a mathematical model has been developed for integrated optimization of scheduling and lot-sizing problems. The developed model uses a big bucket approach and is presented as a mixed integer nonlinear problem (MINLP). The BARON solver in GAMS software has been used to solve the proposed MINLP model. By defining the relative optimality limit (OPTCR) of 0.05 for the termination criterion in BARON solver, GAMS has not been able to solve large problems at a specified time to achieve relative optimality. Therefore, due to the NP-hard nature of the problem, a new genetic-based evolutionary algorithm has been developed to solve the problem of large scale. In the developed algorithm, a different approach (instead of cross-over and mutation operators) is used to generate a new solution. By presenting and solving various problems, the efficiency of this algorithm for solving big problems is shown. Comparing the values of the objective function obtained from the genetic algorithm and the exact method shows that, especially in large problems, the genetic algorithm has been able to achieve a better solution than GAMS software in a limited time. It has also been shown that energy efficiency has a significant effect on the solution of the problem.


Ahmadi, E., Zandieh, M., Farrokh, M., & Emami, S. M. (2016). A multi objective optimization approach for flexible job shop scheduling problem under random machine breakdown by evolutionary algorithms. Computers & Operations Research, 73, 56-66.
Al-Turki, U. M., Arifusalam, S., El-Seliaman, M., & Khan, M. (2011). Resource allocation, batching and dispatching in a stochastic flexible job shop. In Advanced Materials Research (Vol. 264, pp. 1758-1763). Trans Tech Publications Ltd.
Baki, M. F., Chaouch, B. A., & Abdul-Kader, W. (2014). A heuristic solution procedure for the dynamic lot sizing problem with remanufacturing and product recovery. Computers & Operations Research, 43, 225-236.
Bitran, G. R., & Yanasse, H. H. (1982). Computational complexity of the capacitated lot size problem. Management Science, 28(10), 1174-1186.
Brandimarte, P. (1993). Routing and scheduling in a flexible job shop by tabu search. Annals of Operations research, 41(3), 157-183.
Brucker, P., & Schlie, R. (1990). Job-shop scheduling with multi-purpose machines. Computing, 45(4), 369-375.
Buschkühl, L., Sahling, F., Helber, S., & Tempelmeier, H. (2010). Dynamic capacitated lot-sizing problems: a classification and review of solution approaches. Or Spectrum, 32(2), 231-261.
Chan, F. T. S., Wong, T. C., & Chan, L. Y. (2006). Flexible job-shop scheduling problem under resource constraints. International journal of production research, 44(11), 2071-2089.
Chan, F. T., & Choy, K. L. (2011). A genetic algorithm-based scheduler for multiproduct parallel machine sheet metal job shop. Expert systems with Applications, 38(7), 8703-8715.
Chen, J. C., Chen, K. H., Wu, J. J., & Chen, C. W. (2008). A study of the flexible job shop scheduling problem with parallel machines and reentrant process. The International Journal of Advanced Manufacturing Technology, 39(3), 344-354.
Choi, I. C., & Choi, D. S. (2002). A local search algorithm for jobshop scheduling problems with alternative operations and sequence-dependent setups. Computers & Industrial Engineering, 42(1), 43-58.
Drexl, A., & Haase, K. (1995). Proportional lotsizing and scheduling. International Journal of Production Economics, 40(1), 73-87.
Drexl, A., & Kimms, A. (1997). Lot sizing and scheduling—survey and extensions. European Journal of operational research, 99(2), 221-235.
Gao, J., Sun, L., & Gen, M. (2008). A hybrid genetic and variable neighborhood descent algorithm for flexible job shop scheduling problems. Computers & Operations Research, 35(9), 2892-2907.
Gao, K. Z., Suganthan, P. N., Tasgetiren, M. F., Pan, Q. K., & Sun, Q. Q. (2015). Effective ensembles of heuristics for scheduling flexible job shop problem with new job insertion. Computers & Industrial Engineering, 90, 107-117.
Garey, M. R., Johnson, D. S., & Sethi, R. (1976). The complexity of flowshop and jobshop scheduling. Mathematics of operations research, 1(2), 117-129.
Giglio, D., Paolucci, M., & Roshani, A. (2017). Integrated lot sizing and energy-efficient job shop scheduling problem in manufacturing/remanufacturing systems. Journal of cleaner production, 148, 624-641.
Golany, B., Yang, J., & Yu, G. (2001). Economic lot-sizing with remanufacturing options. IIE transactions, 33(11), 995-1004.
Gomez Urrutia, E. D., Aggoune, R., & Dauzère-Pérès, S. (2014). Solving the integrated lot-sizing and job-shop scheduling problem. International Journal of Production Research, 52(17), 5236-5254.
Hajibabaei, M., & Behnamian, J. (2021). Flexible job-shop scheduling problem with unrelated parallel machines and resources-dependent processing times: a tabu search algorithm. International Journal of Management Science and Engineering Management, 16(4), 242-253.
Hurink, J., Jurisch, B., & Thole, M. (1994). Tabu search for the job-shop scheduling problem with multi-purpose machines. Operations-Research-Spektrum, 15(4), 205-215.
Jans, R., & Degraeve, Z. (2008). Modeling industrial lot sizing problems: a review. International Journal of Production Research, 46(6), 1619-1643.
Kacem, I., Hammadi, S., & Borne, P. (2002). Approach by localization and multiobjective evolutionary optimization for flexible job-shop scheduling problems. IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 32(1), 1-13.
Karimi, B., Ghomi, S. F., & Wilson, J. M. (2003). The capacitated lot sizing problem: a review of models and algorithms. Omega, 31(5), 365-378.
Mahdavi, I., Shirazi, B., & Solimanpur, M. (2010). Development of a simulation-based decision support system for controlling stochastic flexible job shop manufacturing systems. Simulation Modelling Practice and Theory, 18(6), 768-786.
Mehdizadeh, E., & Fatehi-Kivi, A. (2017). A Vibration Damping Optimization Algorithm for Solving the Single-item Capacitated Lot-sizing Problem with Fuzzy Parameters. International Journal of Industrial Engineering & Production Research, 28(1), 33-45.
Pezzella, F., Morganti, G., & Ciaschetti, G. (2008). A genetic algorithm for the flexible job-shop scheduling problem. Computers & operations research, 35(10), 3202-3212.
Piñeyro, P., & Viera, O. (2009). Inventory policies for the economic lot-sizing problem with remanufacturing and final disposal options. Journal of Industrial & Management Optimization, 5(2), 217.
Pineyro, P., & Viera, O. (2010). The economic lot-sizing problem with remanufacturing and one-way substitution. International Journal of Production Economics, 124(2), 482-488.
Richter, K., & Sombrutzki, M. (2000). Remanufacturing planning for the reverse Wagner/Whitin models. European Journal of Operational Research, 121(2), 304-315.
Richter, K., & Weber, J. (2001). The reverse Wagner/Whitin model with variable manufacturing and remanufacturing cost. International Journal of Production Economics, 71(1-3), 447-456.
Rohaninejad, M., Kheirkhah, A., & Fattahi, P. (2015). Simultaneous lot-sizing and scheduling in flexible job shop problems. The International Journal of Advanced Manufacturing Technology, 78(1), 1-18.
Rohaninejad, M., Sahraeian, R., & Nouri, B. V. (2016). Multi-objective optimization of integrated lot-sizing and scheduling problem in flexible job shops. RAIRO-Operations Research, 50(3), 587-609.
Roshani, A., Giglio, D., & Paolucci, M. (2016). A simulated annealing approach for the capacitated dynamic lot sizing problem in a closed remanufacturing system. IFAC-PapersOnLine, 49(12), 1496-1501.
Sahraeian, R., Rohaninejad, M., & Fadavi, M. (2017). A new model for integrated lot sizing and scheduling in flexible job shop problem. Journal of Industrial and Systems Engineering, 10(3), 72-91.
Sifaleras, A., Konstantaras, I., & Mladenović, N. (2015). Variable neighborhood search for the economic lot sizing problem with product returns and recovery. International Journal of Production Economics, 160, 133-143.
Teunter, R. H., Bayindir, Z. P., & Den Heuvel, W. V. (2006). Dynamic lot sizing with product returns and remanufacturing. International Journal of Production Research, 44(20), 4377-4400.
Teunter, R., Tang, O., & Kaparis, K. (2009). Heuristics for the economic lot scheduling problem with returns. International Journal of Production Economics, 118(1), 323-330.
Wang, N., He, Z., Sun, J., Xie, H., & Shi, W. (2011). A single-item uncapacitated lot-sizing problem with remanufacturing and outsourcing. Procedia Engineering, 15, 5170-5178.
Xing, L. N., Chen, Y. W., Wang, P., Zhao, Q. S., & Xiong, J. (2010). A knowledge-based ant colony optimization for flexible job shop scheduling problems. Applied Soft Computing, 10(3), 888-896.
Xiong, W., & Fu, D. (2018). A new immune multi-agent system for the flexible job shop scheduling problem. Journal of Intelligent Manufacturing, 29(4), 857-873.
Yang, J., Golany, B., & Yu, G. (2005). A concave‐cost production planning problem with remanufacturing options. Naval Research Logistics (NRL), 52(5), 443-458.
Zarrouk, R., Bennour, I. E., & Jemai, A. (2019). A two-level particle swarm optimization algorithm for the flexible job shop scheduling problem. Swarm Intelligence, 13(2), 145-168.
Zhang, F., Mei, Y., Nguyen, S., & Zhang, M. (2020). Evolving scheduling heuristics via genetic programming with feature selection in dynamic flexible job-shop scheduling. ieee transactions on cybernetics, 51(4), 1797-1811.
Zhang, G., Gao, L., & Shi, Y. (2011). An effective genetic algorithm for the flexible job-shop scheduling problem. Expert Systems with Applications, 38(4), 3563-3573.