# Scheduling parallel machine problem under general effects of deterioration and learning with past-sequence-dependent setup time: heuristic and meta-heuristic approaches

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## Abstract

This study investigates an identical parallel machine scheduling problem with past-sequence-dependent setup times and general effects of deteriorating and learning. The actual job processing time on each machine is defined by a two-element function of the normal processing times of the preprocessed jobs and its scheduled position on the same machine. Moreover, the job setup time on each machine is a function of the actual processing times of the preprocessed jobs on the same machine. A novel mixed-integer programming model is developed to satisfy the goal of minimizing total completion time. Due to the NP-hard characteristic and intractability of the problem, three efficient methodologies including a heuristic algorithm (HA), a genetic algorithm (GA) with an enhanced exploration ability and an ant colony optimization (ACO) combined with a new stochastic elitism strategy are designed to find optimal/near-optimal solutions within an appropriate period of time. The effectiveness and efficiency of the presented model and the proposed algorithms are verified by computational experiments. The computational results indicate that the suggested algorithms are effective and executable approaches to generate solutions as good as optimal solution in the small-sized problems. Also, the ACO statistically outperformed the HA and GA in the medium- and large-sized problems.

## Keywords

Scheduling Deteriorating jobs Learning effect Setup times Genetic algorithm Ant colony optimization## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- Afzalirad M, Rezaeian J (2016) Resource-constrained unrelated parallel machine scheduling problem with sequence dependent setup times, precedence constraints and machine eligibility restrictions. Comput Ind Eng 98:40–52Google Scholar
- Alidaee B, Womer NK (1999) Scheduling with time dependent processing times: review and extensions. J Oper Res Soc 50:711–720zbMATHGoogle Scholar
- Azadeh A, Hasani Goodarzi A, Hasannia Kolaee M, Jebreili S (2018) An efficient simulation–neural network–genetic algorithm for flexible flow shops with sequence-dependent setup times, job deterioration and learning effects. Neural Comput Appl. https://doi.org/10.1007/s00521-018-3368-6 Google Scholar
- Bai D, Tang M, Zhang Z-H, Santibanez-Gonzalez EDR (2018) Flow shop learning effect scheduling problem with release dates. Omega 78:21–38Google Scholar
- Biskup D (1999) Single-machine scheduling with learning considerations. Eur J Oper Res 115(1):173–178MathSciNetzbMATHGoogle Scholar
- Biskup D, Herrmann J (2008) Single-machine scheduling against due dates with past-sequence-dependent setup times. Eur J Oper Res 191(2):587–592MathSciNetzbMATHGoogle Scholar
- Chen Z-L (1996) Parallel machine scheduling with time dependent processing times. Discrete Appl Math 70:81–93MathSciNetzbMATHGoogle Scholar
- Cheng TCE, Wang G (2000) Single machine scheduling with learning effect considerations. Ann Oper Res 98(1):273–290MathSciNetzbMATHGoogle Scholar
- Cheng TCE, Ding Q, Lin BMT (2004) A concise survey of scheduling with time-dependent processing times. Eur J Oper Res 152(1):1–13MathSciNetzbMATHGoogle Scholar
- Cheng TCE, Wu C-C, Lee W-C (2008) Some scheduling problems with deteriorating jobs and learning effects. Comput Ind Eng 54(4):972–982Google Scholar
- Cheng TCE, Lee W-C, Wu C-C (2010a) Single-machine scheduling with deteriorating functions for job processing times. Appl Math Model 34(12):4171–4178MathSciNetzbMATHGoogle Scholar
- Cheng TCE, Lee W-C, Wu C-C (2010b) Scheduling problems with deteriorating jobs and learning effects including proportional setup times. Comput Ind Eng 58(2):326–331Google Scholar
- Ding J, Shen L, Lü Z, Peng B (2019) Parallel machine scheduling with completion-time-based criteria and sequence-dependent deterioration. Comput Oper Res 103:35–45MathSciNetzbMATHGoogle Scholar
- Dorigo M, Stützle T (2010) Ant colony optimization: overview and recent advances. Int Seri Oper Res Manag Sci 146:227–263Google Scholar
- Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Syst 26(1):29–41Google Scholar
- Expósito-Izquierdo C, Angel-Bello F, Melián-Batista B, Alvarez A, Báez S (2019) A metaheuristic algorithm and simulation to study the effect of learning or tiredness on sequence dependent setup times in a parallel machine scheduling problem. Expert Syst Appl 117(1):62–74Google Scholar
- Fan W, Pei J, Liu X, Pardalos PM, Kong M (2018) Serial-batching group scheduling with release times and the combined effects of deterioration and truncated job-dependent learning. J Glob Optim 71(1):147–163MathSciNetzbMATHGoogle Scholar
- Gawiejnowicz S (2008) Time-dependent scheduling. Springer, BerlinzbMATHGoogle Scholar
- Geyik F, Elibal K (2017) A linguistic approach to non-identical parallel processor scheduling with fuzzy processing times. Appl Soft Comput 55:63–71Google Scholar
- Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, ReadingzbMATHGoogle Scholar
- Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 5:287–326MathSciNetzbMATHGoogle Scholar
- Holland HJ (1975) Adaptation in natural and artificial systems. The University of Michigan Press, Ann ArborGoogle Scholar
- Huang X, Wang M-Z (2011) Parallel identical machines scheduling with deteriorating jobs and total absolute differences penalties. Appl Math Model 35(3):1349–1353MathSciNetzbMATHGoogle Scholar
- Huang X, Wang M-Z, Ji P (2014) Parallel machines scheduling with deteriorating and learning effects. Optim Lett 8:493–500MathSciNetzbMATHGoogle Scholar
- Ji M, Tang X, Zhang X, Cheng TCE (2016) Machine scheduling with deteriorating jobs and DeJong’s learning effect. Comput Ind Eng 91:42–47Google Scholar
- Jou C (2005) A genetic algorithm with sub-indexed partitioning genes and its application to production scheduling of parallel machines. Comput Ind Eng 48(1):39–54Google Scholar
- Koulamas C, Kyparisis GJ (2008) Single-machine scheduling problems with past-sequence-dependent setup times. Eur J Oper Res 187(3):1045–1049MathSciNetzbMATHGoogle Scholar
- Kuo W-H, Yang D-L (2007) Single-machine scheduling with past-sequence-dependent setup and learning effects. Inf Process Lett 102(1):22–26MathSciNetzbMATHGoogle Scholar
- Lai P-J, Lee W-C (2011) Single-machine scheduling with general sum-of-processing-time-based and position-based learning effects. Omega 39(5):467–471Google Scholar
- Lee W-C (2004) A note on deteriorating jobs and learning in single-machine scheduling problems. Int J Bus Econ 3(1):83–89Google Scholar
- Lee W-C (2011) Scheduling with general position-based learning curves. Inf Sci 181(24):5515–5522MathSciNetzbMATHGoogle Scholar
- Lee W-C (2014) Single-machine scheduling with past-sequence-dependent setup times and general effects of deterioration and learning. Optim Lett 8:135–144MathSciNetzbMATHGoogle Scholar
- Lee W-C, Lai P-J (2011) Scheduling problems with general effects of deterioration and learning. Inf Sci 181(6):1164–1170MathSciNetzbMATHGoogle Scholar
- Li L, Wang J-J (2018) Scheduling jobs with deterioration effect and controllable processing time. Neural Comput Appl 29(11):1163–1170Google Scholar
- Li X, Yang Z, Ruiz R, Chen T, Sui S (2018) An iterated greedy heuristic for no-wait flow shops with sequence dependent setup times, learning and forgetting effects. Inf Sci 453:408–425MathSciNetGoogle Scholar
- Liao B, Song Q, Pei J, Yang S, Pardalos PM (2018) Parallel-machine group scheduling with inclusive processing set restrictions, outsourcing option and serial-batching under the effect of step-deterioration. J Glob Optim. https://doi.org/10.1007/s10898-018-0707-1 Google Scholar
- Liu F, Yang J, Lu Y-Y (2018) Solution algorithms for single-machine group scheduling with ready times and deteriorating jobs. Eng Optim. https://doi.org/10.1080/0305215x.2018.1500562 Google Scholar
- Loyola P, Román PE, Velásquez JD (2012) Predicting web user behavior using learning-based ant colony optimization. Eng Appl Artif Intell 25:889–897Google Scholar
- Lu S, Liu X, Pei J, Thai MT, Pardalos PM (2018) A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity. Appl Soft Comput 66:168–182Google Scholar
- Mustu S, Eren T (2018) The single machine scheduling problem with sequence-dependent setup times and a learning effect on processing times. Appl Soft Comput 71:291–306Google Scholar
- Nembhard DA, Osothsilp N (2002) Task complexity effects on between-individual learning/forgetting variability. Int J Ind Ergon 29(5):297–306Google Scholar
- Pei J, Cheng B, Liu X, Pardalos PM, Kong M (2017) Single-machine and parallel-machine serial-batching scheduling problems with position-based learning effect and linear setup time. Ann Oper Res 10:15. https://doi.org/10.1007/s10479-017-2481-8 zbMATHGoogle Scholar
- Pinedo M (2008) Scheduling: theory, algorithms, and systems, 3rd edn. Prentice Hall, New JerseyzbMATHGoogle Scholar
- Przybylski B (2018) A new model of parallel-machine scheduling with integral-based learning effect. Comput Ind Eng 121:189–194Google Scholar
- Rostami M, Ebrahimzadeh Pilerood A, Mahdavi Mazdeh M (2015) Multi-objective parallel machine scheduling problem with job deterioration and learning effect under fuzzy environment. Comput Ind Eng 85:206–215Google Scholar
- Salehi Mir MS, Rezaeian J (2016) A robust hybrid approach based on particle swarm optimization and genetic algorithm to minimize the total machine load on unrelated parallel machines. Appl Soft Comput 41:488–504Google Scholar
- Shahvari O, Logendran R (2018) A comparison of two stage-based hybrid algorithms for a batch scheduling problem in hybrid flow shop with learning effect. Int J Prod Econ 195:227–248Google Scholar
- Soroush HM (2014) Scheduling in bicriteria single machine systems with past-sequence-dependent setup times and learning effects. J Oper Res Soc 65:1017–1036Google Scholar
- Sun L (2009) Single-machine scheduling problems with deteriorating jobs and learning effects. Comput Ind Eng 57(3):843–846Google Scholar
- Voutsinas TG, Pappis CP (2010) A branch and bound algorithm for single-machine scheduling with deteriorating value of jobs. Math Comput Model 52(2):55–61MathSciNetzbMATHGoogle Scholar
- Wang J-B (2006) A note on scheduling problems with learning effect and deteriorating jobs. Int J Syst Sci 37(12):827–833MathSciNetzbMATHGoogle Scholar
- Wang J-B (2007) Single-machine scheduling problems with the effects of learning and deterioration. Omega 35(4):397–402Google Scholar
- Wang JB (2008) Single-machine scheduling with past-sequence-dependent setup times and time-dependent learning effect. Comput Ind Eng 55(3):584–591Google Scholar
- Wang X, Cheng TCE (2007) Single-machine scheduling with deteriorating jobs and learning effects to minimize the makespan. Eur J Oper Res 178(1):57–70MathSciNetzbMATHGoogle Scholar
- Wang J-B, Li J-X (2011) Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects. Appl Math Model 35(3):1388–1395MathSciNetzbMATHGoogle Scholar
- Wang X-Y, Wang J-J (2013) Scheduling problems with past-sequence-dependent setup times and general effects of deterioration and learning. Appl Math Model 37:4905–4914MathSciNetzbMATHGoogle Scholar
- Wang X-Y, Wang J-J (2014) Scheduling deteriorating jobs with a learning effect on unrelated parallel machines. Appl Math Model 38(21–22):5231–5238MathSciNetzbMATHGoogle Scholar
- Wang J-B, Huang X, Wang X-Y, Yin N, Wang L-Y (2009a) Learning effect and deteriorating jobs in the single machine scheduling problems. Appl Math Model 33(10):3848–3853MathSciNetzbMATHGoogle Scholar
- Wang J-B, Jiang Y, Wang G (2009b) Single-machine scheduling with past-sequence-dependent setup times and effects of deterioration and learning. Int J Adv Manuf Technol 41:1221–1226Google Scholar
- Wang JB, Sun LH, Sun LY (2010a) Scheduling jobs with an exponential sum-of-actual- processing-time-based learning effect. Comput Math Appl 60(9):2673–2678MathSciNetzbMATHGoogle Scholar
- Wang J-B, Wang D, Zhang G-D (2010b) Single-machine scheduling with learning functions. Appl Math Comput 216(4):1280–1286MathSciNetzbMATHGoogle Scholar
- Wang C, Liu C, Zhang Z-H, Zheng L (2016) Minimizing the total completion time for parallel machine scheduling with job splitting and learning. Comput Ind Eng 97:170–182Google Scholar
- Wang T, Baldacci R, Lim A, Hu Q (2018) A branch-and-price algorithm for scheduling of deteriorating jobs and flexible periodic maintenance on a single machine. Eur J Oper Res 271(3):826–838MathSciNetzbMATHGoogle Scholar
- Woo Y-B, Jung S, Soo Kim B (2017) A rule-based genetic algorithm with an improvement heuristic for unrelated parallel machine scheduling problem with time-dependent deterioration and multiple rate-modifying activities. Comput Ind Eng 109:179–190Google Scholar
- Wu Y-B, Wang J-J (2016) Single-machine scheduling with truncated sum-of-processing-times-based learning effect including proportional delivery times. Neural Comput Appl 27(4):937–943Google Scholar
- Wu Y-B, Wang X-Y, Ji P (2012) A note on single-machine scheduling problems with both deteriorating jobs and learning effects. Appl Math Model 36(12):6341–6344MathSciNetzbMATHGoogle Scholar
- Wu C-H, Lee W-C, Lai P-J, Wang J-Y (2016) Some single-machine scheduling problems with elapsed-time-based and position-based learning and forgetting effects. Discrete Optim 19:1–11MathSciNetzbMATHGoogle Scholar
- Wu W-H, Yin Y, Cheng TCE, Lin W-C, Chen J-C, Luo S-Y, Wu C-C (2017) A combined approach for two-agent scheduling with sum-of-processing-times-based learning effect. J Oper Res Soc 68(2):111–120Google Scholar
- Xu J, Wu C-C, Yin Y, Zhao C, Chiou Y-T, Lin W-C (2016) An order scheduling problem with position-based learning effect. Comput Oper Res 74:175–186MathSciNetzbMATHGoogle Scholar
- Yang D-L, Kuo W-H (2009) Single-machine scheduling with both deterioration and learning effects. Ann Oper Res 172(1):315–327MathSciNetzbMATHGoogle Scholar
- Yang S-H, Wang J-B (2011) Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration. Appl Math Comput 217(9):4819–4826MathSciNetzbMATHGoogle Scholar
- Yeh W-C, Lai P-J, Lee W-C, Chuang M-C (2014) Parallel-machine scheduling to minimize makespan with fuzzy processing times and learning effects. Inf Sci 269:142–158MathSciNetzbMATHGoogle Scholar
- Yin N, Wang J-B, Wang D, Wang L-Y, Wang X-Y (2010) Deteriorating jobs and learning effects on a single-machine scheduling with past-sequence-dependent setup times. Int J Adv Manuf Technol 46(5):707–714MathSciNetGoogle Scholar
- Yin Y, Xu D, Cheng S-R, Wu C-C (2012) A generalisation model of learning and deteriorating effects on a single-machine scheduling with past-sequence-dependent setup times. Int J Comput Integr Manuf 25(9):804–813Google Scholar
- Yin Y, Wu W-H, Cheng TCE, Wu C-C (2015) Single-machine scheduling with time-dependent and position-dependent deteriorating jobs. Int J Comput Integr Manuf 28(7):781–790Google Scholar
- Yin Y, Wang Y, Cheng TCE, Liu W, Li J (2017) Parallel-machine scheduling of deteriorating jobs with potential machine disruptions. Omega 69:17–28Google Scholar
- Zhang X, Yin Y, Wu C-C (2017) Scheduling with non-decreasing deterioration jobs and variable maintenance activities on a single machine. Eng Optim 49(1):84–97MathSciNetGoogle Scholar
- Zhao C, Tang H (2010) Single machine scheduling with past-sequence-dependent setup times and deteriorating jobs. Comput Ind Eng 59(4):663–666Google Scholar
- Zhao C, Hsu C-J, Cheng S-R, Yin Y, Wu C-C (2014) Due date assignment and single machine scheduling with deteriorating jobs to minimize the weighted number of tardy jobs. Appl Math Comput 248:503–510MathSciNetzbMATHGoogle Scholar