Abstract
We study a multi-objective scheduling problem on two dedicated processors. The aim is to minimize simultaneously the makespan, the total tardiness and the total completion time. This NP-hard problem requires the use of well-adapted methods. For this, we adapted genetic algorithms to multi-objective case. Four methods are presented to solve this problem. The first is an aggregative genetic algorithm (GA), the second is a Pareto GA, the third is a non-dominated sorting GA (NSGA-II) and the fourth is a constructive algorithm based on lower bounds (CABLB). We proposed some adapted lower bounds for each criterion to evaluate the quality of the found results on a large set of instances. Indeed, these bounds also make it possible to determine the dominance of one algorithm over another based on the different results found by each of them. We used two metrics to measure the quality of the Pareto front: the hypervolume indicator (HV) and the number of solutions in the Pareto front (ND). The obtained results show the effectiveness of the proposed algorithms.
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References
Afrati F, Milis I (2006) Designing ptass for min-sum scheduling problems. Discr Appl Math 154(4):622–639
Afrati F, Bampis E, Chekuri C, Karger D, Kenyon C, Khanna S, Milis I, Queyranne M, Sartinutella M, Stein C, Sviridenko M (1999) Approximation schemes for minimizing average weighted completion time with release dates. In: IEEE symposium on foundations of computer science, pp 32–44
Alberto I, Mateo P (2011) A crossover operator that uses pareto optimality in its definition. Top 19(1):67–92
Alhadi G, Kacem I, Laroche P, Osman I (2020) Approximation algorithms for minimizing the maximum lateness and makespan on parallel machines. Ann Oper Res 285:369–395
Berrichi A, Amodeo L, Yalaoui F, Chatelet E, Mezghiche M (2007) Biobjective optimization algorithms for joint production and maintenance scheduling: application to the parallel machine problem. J Intell Manuf 20(4):389–400
Bianco L, Blazewicz J, Dell’Olmo P, Drozdowski M (1997) Preemptive multiprocessor task scheduling with release times and time windows. Ann Oper Res 70(1):43–55
Blazewicz J, Dell’Olmo P, Drozdowski M (2002) Scheduling multiprocessor tasks on two parallel processors. RAIRO-Oper Res 36(1):37–51
Blazewicz J, Ecker K, Pesch E, Schmidt G, Weglarz J (2019) Handbook on scheduling. Springer, New York
Bradstreet L (2011) The hypervolume indicator for multi-objective optimisation: calculation and use. PhD thesis, University of Western Australia
Chu C (1992) A branch and bound algorithm to minimize the total of tardness with different release date. Naval Res Log 39(2):256–283
Coffman E, Garey M, Johnson D, LaPaugh AS (1985) Scheduling file transfers. SIAM J Comput 14(4):743–780
Craig G, Kime CR, Saluja K (1988) Test scheduling and control for vlsi built-in self-test. IEEE Trans Comput 37(9):1099–1109
Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: Nsga ii. IEEE Trans Evol Comput 6(2):182–197
Drozdowski M (1996) Scheduling multiprocessor tasks an overview. Eur J Oper Res 94(2):215–230
Emmons H (1969) Some new uniform design. Oper Res 17(4):701–715
Fang K, Li J (1994) Multi-objective genetic algorithms made easy: selection, sharing and mating restrictions. Hong Kong, Baptist Univ., Hong Kong, Tech. Rep. Math-042
Fonseca LC, Mand P, López-Ibñáez M, Guerreiro AP (2018) Computation of the hypervolume indicator. http://lopez-ibanez.eu/hypervolume. Accessed 3 Feb 2019
Holland J (1975) Adaptation in natural and artificial systems. Michigan Press, University of Michigan Press, USA
Hoogeveen J, de Velde SLV, Veltman B (1994) Complexity of scheduling multiprocessor tasks with prespecified processors allocations. Discr Appl Math 55(3):259–272
Kacem A, Dammak A (2017) A genetic algorithm to minimize the total of tardiness multiprocessing tasks on two dedicated processors. In: IEEE control, decision and information technologies, Barcelona, Spain 5-7 April. 2017:85–90
Kacem A, Dammak A (2019) Bi-objective scheduling on two dedicated processors. Eur J Ind Eng 5:681–700
Kacem I (2007) Lower bounds for tardiness minimization on a single machine with family setup times. Int J Oper Res 4(1):18–31
Karasakal E, Silav A (2016) A multi-objective genetic algorithm for a bi-objective facility location problem with partial coverage. Top 24(1):206–232
Leung Y, Wang Y (2000) Multiobjective programming using uniform design and genetic algorithm. IEEE Trans Syst Man Cybern 30(C):293–303
Li G (1997) Single machine earliness and tardiness scheduling. Eur J Oper Res 96(3):546–558
Lopez-Ibanez M, Stutzle T (2014) Automatically improving the anytime behaviour of optimisation algorithms. Eur J Oper Res 235(3):569–582
Manaa A, Chu C (2010) Scheduling multiprocessor tasks to minimise the makespan on two dedicated processors. Eur J Ind Eng 4(3):265–279
Oguz C, Ercan M (2005) A genetic algorithm for hybrid flow-shop scheduling with multiprocessor tasks. J Sched 8(4):323–351
Pareto V (1896) Cours économie politique. Lausane Switzerland, Switzerland
Rebai M, Kacem I, Adjallah K (2010) Earliness tardiness minimization on a single machine to schedule preventive maintenance tasks: metaheuristic and exact methods. J Intell Manuf 23(4):1207–1224
Vallada E, Ruiz R (2011) A genetic algorithm for the unrelated parallel machine scheduling problem with sequence dependent setup times. Eur J Oper Res 211(3):612–622
Vallada E, Ruiz R (2012) Scheduling unrelated parallel machines with sequence dependent setup times and weighted earliness–tardiness minimization. In: Just-in-time systems, Springer, pp 67–90
Venkata P, Usha M, Viswanath K (2018) Order acceptance and scheduling in a parallel machine environment with weighted completion time. Eur J Ind Eng 12(4):535–557
While L, Hingston P, Barone L, Husband S (2006) A faster algorithm for calculating hypervolume. IEEE Trans Evol Comput 10(1):29–38
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Kacem, A., Dammak, A. Multi-objective scheduling on two dedicated processors. TOP 29, 694–721 (2021). https://doi.org/10.1007/s11750-020-00588-5
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DOI: https://doi.org/10.1007/s11750-020-00588-5