Abstract
In this study, an energy-efficient unrelated parallel machine scheduling problem is discussed. The speed scaling mechanism has been taken into account as an energy-efficient strategy. Unrelated parallel machine scheduling with speed scaling is generalized by considering machine-sequence dependent setup times and learning effect features. A multiobjective mixed-integer linear programming (MILP) model has been proposed for the problem. Due to the NP-hard nature of the problem, a multiobjective evolutionary algorithm, the NSGA-II-based memetic algorithm, is proposed. An encoding scheme, decoding algorithm, and local search algorithms are proposed for the problem. Speed tuning heuristic and job-machine switch heuristic algorithms are proposed as local search algorithms. A restarting strategy has been applied to ensure the diversification of the algorithm. The classical NSGA-II algorithm and the proposed memetic algorithm were compared over the generated test problems. As a result, the proposed memetic algorithm is more successful according to performance metrics.
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Abbreviations
- I :
-
Machine indices and \(i \in \{1,\dots ,m\}\)
- h :
-
Position indices and \(h \in \{1,\dots ,n\}\)
- k, j :
-
Job indices and \(k, j \in \{1,\dots ,n\}\)
- o :
-
Speed indices and \(o \in \{1,\dots ,S\}\)
- a :
-
Learning index
- \({s}_{ijk}\) :
-
Setup time of job k processing after job j on machine i
- \({P}_{ij}\) :
-
Normal processing time of job j on machine i
- \({v}_{o}\) :
-
Alternative speed values
- \({q}_{ijo}\) :
-
Energy consumption rate on the speed of for job j on machine i
- M :
-
Very big number
- \({C}_{j}\) :
-
Completion time of job j
- \({C}_{\rm max}\) :
-
Maximum completion time (makespan)
- E :
-
Total energy consumption
- \({x}_{ijho}\) :
-
\(\left\{ {\begin{array}{*{20}c} {1;\quad {\text{If}}\,{\text{job}}\,{\text{j}}\,{\text{is}}\,{\text{processing}}\,{\text{on}}\,{\text{machine}}\,{\text{i}}\,{\text{with}}\,{\text{speed}}\,{\text{o}}\,{\text{and}}\,{\text{position}}\,{\text{h}}} \\ {0;\quad o.w.} \\ \end{array} } \right.\)
References
Zandi, A.; Ramezanian, R.; Monplaisir, L.: Green parallel machines scheduling problem: A bi-objective model and a heuristic algorithm to obtain Pareto frontier. J. Oper. Res. Soc. 71(6), 967–978 (2020)
Wu, X.; Che, A.: A memetic differential evolution algorithm for energy- efficient parallel machine scheduling. Omega 82, 155–165 (2019)
Yao, F.; Demers, A.; Shenker, S.: A scheduling model for reduced CPU energy. In: Proceedings of IEEE symposium on foundations of computer science. 374−382 (1995)
Fang, K.; Uhan, N.; Zhao, F.; Sutherland, J.W.: A new approach to scheduling in manufacturing for power consumption and carbon footprint reduction. J. Manuf. Syst. 30, 234–240 (2011)
Yüksel, D.; Taşgetiren, M. F.; Kandiller, L.; Gao, L.: An energy- efficient bi- objective no- wait permutation flowshop scheduling problem to minimize total tardiness and total energy consumption. Comput. Ind. Eng. 145, 106431 (2020)
Fang, K.; Uhan, N.; Zhao, F.; Sutherland, J.W.: Scheduling on a single machine under time- of- use electricity tariffs. Ann. Oper. Res. 238, 199–227 (2016)
Öztop, H.; Tasgetiren, M. F.; Eliiyi, D. T.; Pan, Q.; Kandiller, L.: An energy- efficient permutation flowshop scheduling problem. Expert Syst. Appl. 150, 113279 (2020)
Bektur, G.: A hybrid heuristic solution based on simulated annealing algorithm for energy efficient single machine scheduling problem with sequence dependent setup times. J. Faculty Eng Arch Gazi Univ. 36(1), 407–420 (2021)
Jia, Z.; Zhang, Y.; Leung, J.; Li, K.: Bi- criteria ant colony optimization algorithm for minimizing makespan and energy consumption on parallel batch machines. Appl. Soft Comput. 55, 226–237 (2017)
Zhou, S.; Li, X.; Du, N.; Pang, Y.; Chen, H.: A multi- objective differential evolution algorithm for parallel batch processing machine scheduling considering electricity consumption cost. Comput. Oper. Res. 96, 55–68 (2018)
Zheng, Y.; Che, A.; Wu, X.: Bi- objective scheduling on uniform parallel machines considering electricity cost. Eng. Optim. 50(1), 19–36 (2018)
Fang, K.; Lin, B.: Parallel-machine scheduling to minimize tardiness penalty and power cost. Comput. Ind. Eng. 64, 224–234 (2013)
Jin, X.; Zhang, F.; Fan, L.; Song, Y.; Liu, Z.: Scheduling for energy minimization on restricted parallel processors. J. Parallel Distrib Comput. 81–82, 36–46 (2015)
Aliabad, H.; Nafchi, M.; Moslehi, G.: Energy-efficient scheduling in an unrelated parallel- machine environment under time-of-use electricity tariffs. J. Clean. Prod. 249, 119393 (2020)
Ding, J.; Song, S.; Zhang, R.; Chiong, R.; Wu, C.: Parallel machine scheduling under time- of- use electricity prices: new models and optimization approaches. IEEE Trans. Autom. Sci. Eng. 13(2), 1138–1154 (2016)
Che, A.; Zhang, S.; Wu, X.: Energy- conscious unrelated parallel machine scheduling under time-of-use electricity tariffs. J. Clean. Prod. 156, 688–697 (2017)
Abikarram, J.B.; McConcky, K.; Proano, R.: Energy cost minimization for unrelated parallel machine scheduling under real time and demand charge pricing. J. Clean. Prod. 208, 232–242 (2019)
Liang, P.; Yang, H.; Liu, G.; Guo, J.: An ant optimization model for unrelated oarallel machine scheduling with energy consumption and total tardiness. Math. Prob. Eng. 907034 (2015)
Li, Z.; Yang, H.; Zhang, S.; Liu, G.: Unrelated parallel machine scheduling problem with energy and tardiness cost. Int. J. Adv. Manuf. Technol. 84, 213–226 (2016)
Cota, L. P.; Guimaraes, F. G.; Ribeiro, R. G.; Meneghini, I. R.; Oliveira, F.; Souza, M.; Siarry, P.: An adaptive multi- objective algorithm based on decomposition and large neighborhood search for a green machine scheduling problem. Swarm Eval. Comput. 51, 100601 (2019)
Zhang, H.; Wu, Y.; Pan, R.; Xu, G.: Two- stage parallel speed- scaling machine scheduling under time-of-use tariffs. J. Intell. Manuf. 32, 91–112 (2021)
Biskup, D.: Single- machine scheduling with learning considerations. Eur. J. Oper. Res. 115, 173–178 (1999)
Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)
Yahui, W.; Ling, S.; Cai, Z.; Liuqiang, F.; Xiangjie, J.: NSGA- II algorithm and application for multi- objective flexible workshop scheduling. J. Algo. Comput. Technol. 14, 1–9 (2020)
Han, Y.; Gong, D.; Sun, X.; Pan, Q.: An improved NSGA- II algorithm for multi-objective lot-streaming flow shop scheduling problem. Int. J. Prod. Res. 52181, 2211–2231 (2014)
Benlic, U.; Hao, J.: Memetic search for the quadratic assignment problem. Exp. Syt. Appl. 42(1), 584–595 (2015)
Mei, Y.; Tang, K.; Yao, X.: Decomposition-based memetic algorithm for multiobjective capacitated arc routing problem. IEEE Trans. Evol. Comput. 15(2), 151–165 (2011)
Mencia, R.; Sierra, M.R.; Mencia, C.; Varela, R.: Memetic algorithms for the job shop scheduling problem with operators. Appl. Soft Comput. 34, 94–105 (2015)
Pan, Q.; Ruiz, R.: An estimation of distribution algorithm for lot- streaming flow shop problems with setup times. Omega 40(2), 166–180 (2012)
Wang, H.; Fu, Y.; Huang, M.; Huang, G.Q.; Wang, J.: A NSGA- II based memetic algorithm for multiobjective parallel flowshop scheduling problem. Comput. Ind. Eng. 113, 185–194 (2017)
Zhang, W.; Wang, Y.; Yang, Y.; Gen, M.: Hybrid multiobjective evolutionary algorithm based on differential evolution for flow shop scheduling problems. Comput. Ind. Eng. 130, 661–670 (2019)
Gong, G.; Deng, Q.; Chiong, R.; Gong, X.; Huang, H.: An effective memetic algorithm for multi- objective job- shop scheduling. Knowledge Based Syst. 182, 104840 (2019)
Arnaout, J.: A worm optimization algorithm to minimize the makespan on unrelated parallel machines with sequence- dependent setup times. Ann. Oper. Res. 285, 273–293 (2020)
Mavrotas, G.: Effective implementation of the ε- constraint method in multi-objective mathematical programming problems. Appl. Math. Comput. 213, 455–465 (2009)
Bektur, G.; Saraç, T.: A mathematical model and heurisitc algorithms for an unrelated parallel machine scheduling problem with sequence- dependent setup times, machine eligibility restrictions and a common server. Comput. Oper. Res. 103, 46–63 (2019)
Mustu, S.; Eren, T.: The single machine scheduling problem with sequence- dependent setup times and a learning effect on processing times. Appl. Soft Comput. 71, 291–306 (2018)
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Bektur, G. An NSGA-II-Based Memetic Algorithm for an Energy-Efficient Unrelated Parallel Machine Scheduling Problem with Machine-Sequence Dependent Setup Times and Learning Effect. Arab J Sci Eng 47, 3773–3788 (2022). https://doi.org/10.1007/s13369-021-06114-4
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DOI: https://doi.org/10.1007/s13369-021-06114-4