Abstract
This paper investigates a practical scheduling problem for radiotherapy patients, who are to be scheduled on different devices at different times. The treatment duration is increasing with time because of the continuously decaying effect of the radiation source, which also results in the decline of the serving ability. Therefore, the maintenance activity (replacing radiation source) is necessary to maintain the serving ability of medical institutions. The problem is to determine the schedule of all treatments and also when to have a maintenance activity, so as to minimize the maximum completion time of all the treatments on all devices. The lower bound of the problem is given in this paper. We prove that the optimal solution of the subproblem, i.e., scheduling patients on a single device, is independent of the sequence of the patients and is only related to the division of patients who are assigned before and after the maintenance, and thus, the subproblem can be converted to a two-partition problem. An improved dynamic programming algorithm is proposed to obtain an optimal scheme for this subproblem and its performance is better than other approaches. For multiple-device problem, an effective hybrid algorithm Gaussian crow search algorithm (GCSA) combined with crow search algorithm (CSA) and Gaussian distribution is proposed to assign all patients to different treatment devices. Finally, computational experiments demonstrate the effectiveness and stability of the proposed GCSA which is compared with CSA, simulated annealing (SA) and particle swarm optimization (PSO). The comparison results show that GCSA outperforms other algorithms in a feasible time.
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References
Askarzadeh A (2016) A novel metaheuristic method for solving constrained engineering optimization problems: crow search algorithm. Comput Struct 169:1–12. https://doi.org/10.1016/j.compstruc.2016.03.001
Cheng M, Xiao S, Luo R, Lian Z (2018) Single-machine scheduling problems with a batch-dependent aging effect and variable maintenance activities. Int J Prod Res 56(23):7051–7063. https://doi.org/10.1080/00207543.2017.1398424
Conforti D, Guerriero F, Guido R (2010) Non-block scheduling with priority for radiotherapy treatments. Eur J Oper Res 201(1):289–296. https://doi.org/10.1016/j.ejor.2009.02.016
Ela AAAEL, El-Sehiemy RA, Shaheen AM, Shalaby AS (2018) Application of the crow search algorithm for economic environmental dispatch. In: 2017 19th international middle-east power systems conference, MEPCON 2017—proceedings, 2018–Febru (December), pp 78–83. https://doi.org/10.1109/MEPCON.2017.8301166
Eremeev AV, Kovalyov MY, Kuznetsov PM (2020) Lot-size scheduling of a single product on unrelated parallel machines. Optim Lett 14(3):557–568. https://doi.org/10.1007/s11590-018-1307-1
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):147–163. https://doi.org/10.1007/s10898-017-0536-7
Gocgun Y (2018) Simulation-based approximate policy iteration for dynamic patient scheduling for radiation therapy. Health Care Manag Sci 21(3):317–325. https://doi.org/10.1007/s10729-016-9388-9
Goeke D (2019) Granular tabu search for the pickup and delivery problem with time windows and electric vehicles. Eur J Oper Res 278(3):821–836. https://doi.org/10.1016/j.ejor.2019.05.010
Graham RL, Lawler EL, Lenstra JK, Kan AHGR (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann Discrete Math 5:287–326. https://doi.org/10.1016/S0167-5060(08)70356-X
Gu M, Lu X, Gu J, Zhang Y (2016) Single-machine scheduling problems with machine aging effect and an optional maintenance activity. Appl Math Model 40(21–22):8862–8871. https://doi.org/10.1016/j.apm.2016.01.038
Guido DCFGR (2008) Optimization models for radiotherapy patient scheduling. 4OR. https://doi.org/10.1007/s10288-007-0050-8
Guo P, Cheng W, Wang Y (2015) Parallel machine scheduling with step-deteriorating jobs and setup times by a hybrid discrete cuckoo search algorithm. Eng Optim 47(11):1564–1585. https://doi.org/10.1080/0305215X.2014.982634
Habachi R, Touil A, Charkaoui A, Echchatbi A (2018) Eagle strategy based crow search algorithm for solving unit commitment problem in smart grid system. Indones J Electr Eng Comput Sci 12(1):17–29. https://doi.org/10.11591/ijeecs.v12.i1.pp17-29
Hsu CJ, Ji M, Guo JY, Yang DL (2013) Unrelated parallel-machine scheduling problems with aging effects and deteriorating maintenance activities. Inf Sci 253:163–169. https://doi.org/10.1016/j.ins.2013.08.053
Huang X, Wang MZ (2011) Parallel identical machines scheduling with deteriorating jobs and total absolute differences penalties. Appl Math Model 35(3):1349–1353. https://doi.org/10.1016/j.apm.2010.09.013
Huang X, Wang MZ (2014) Single machine group scheduling with time and position dependent processing times. Optim Lett 8(4):1475–1485. https://doi.org/10.1007/s11590-012-0535-z
Jacquemin Y, Eric M, Pascal P (2010) Towards an improved resolution of radiotherapy, scheduling. In: 2010 IEEE workshop on health care management (WHCM). IEEE. https://doi.org/10.1109/WHCM.2010.5441263
Ji M, Cheng TCE (2009) Parallel-machine scheduling of simple linear deteriorating jobs. Theoret Comput Sci 410(38–40):3761–3768. https://doi.org/10.1016/j.tcs.2009.04.018
Johnson DS, Garey MR (1979) Computers and intractability: a guide to the theory of NP-completeness. WH Freeman, New York
Kong M, Pei J, Xu J, Liu X, Yu X, Pardalos PM (2019) A robust optimization approach for integrated steel production and batch delivery scheduling with uncertain rolling times and deterioration effect. Int J Prod Res. https://doi.org/10.1080/00207543.2019.1693659
Lalla-Ruiz E, Voß S (2016) Modeling the parallel machine scheduling problem with step deteriorating jobs. Eur J Oper Res 255(1):21–33. https://doi.org/10.1016/j.ejor.2016.04.010
Lee HT, Yang DL, Yang SJ (2013) Multi-machine scheduling with deterioration effects and maintenance activities for minimizing the total earliness and tardiness costs. Int J Adv Manuf Technol 66(1–4):547–554. https://doi.org/10.1007/s00170-012-4348-0
Legrain A, Fortin MA, Lahrichi N, Rousseau LM (2015) Online stochastic optimization of radiotherapy patient scheduling. Health Care Manag Sci 18(2):110–123. https://doi.org/10.1007/s10729-014-9270-6
Lei D, Guo X (2016) Hybrid flow shop scheduling with not-all-machines options via local search with controlled deterioration. Comput Oper Res 65:76–82. https://doi.org/10.1016/j.cor.2015.05.010
Li S, Yuan J (2010) Parallel-machine scheduling with deteriorating jobs and rejection. Theoret Comput Sci 411(40–42):3642–3650. https://doi.org/10.1016/j.tcs.2010.06.008
Li J-Q, Song M-X, Wang L, Duan P-Y, Han Y-Y, Sang H-Y, Pan Q-K (2019) Hybrid artificial bee colony algorithm for a parallel batching distributed flow-shop problem with deteriorating jobs. IEEE Trans Cybern. https://doi.org/10.1109/tcyb.2019.2943606
Liu X, Lu S, Pei J, Pardalos PM (2018) A hybrid VNS-HS algorithm for a supply chain scheduling problem with deteriorating jobs. Int J Prod Res 56(17):5758–5775. https://doi.org/10.1080/00207543.2017.1418986
Lu S, Liu X, Pei J, Thai MT, Pardalos P (2018) A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity. Appl Soft Comput J 66:168–182. https://doi.org/10.1016/j.asoc.2018.02.018
Lv M, Li Y, Kou B, Zhou Z (2017) Integer programming for improving radiotherapy treatment efficiency. PLoS ONE 12(7):1–9. https://doi.org/10.1371/journal.pone.0180564
Mor B, Mosheiov G (2020) Minimizing total load on parallel machines with linear deterioration. Optim Lett 14(3):771–779. https://doi.org/10.1007/s11590-019-01526-6
Oliveira D, Pessoa A (2019) An improved branch-cut-and-price algorithm for parallel machine scheduling problems an improved branch-cut-and-price algorithm for parallel machine scheduling problems. INFORMS J Comput 32:90–100
Pei J, Liu X, Pardalos PM, Fan W, Yang S, Wang L (2014) Application of an effective modified gravitational search algorithm for the coordinated scheduling problem in a two-stage supply chain. Int J Adv Manuf Technol 70(1–4):335–348. https://doi.org/10.1007/s00170-013-5263-8
Pei J, Pardalos PM, Liu X, Fan W, Yang S (2015) Serial batching scheduling of deteriorating jobs in a two-stage supply chain to minimize the makespan. Eur J Oper Res 244(1):13–25. https://doi.org/10.1016/j.ejor.2014.11.034
Pei J, Dražić Z, Dražić M, Mladenović N, Pardalos PM (2019a) Continuous variable neighborhood search (C-VNS) for solving systems of nonlinear equations. INFORMS J Comput 31(2):235–250. https://doi.org/10.1287/ijoc.2018.0876
Pei J, Liu X, Fan W, Pardalos PM, Lu S (2019b) A hybrid BA-VNS algorithm for coordinated serial-batching scheduling with deteriorating jobs, financial budget, and resource constraint in multiple manufacturers. Omega (UK) 82:55–69. https://doi.org/10.1016/j.omega.2017.12.003
Pei J, Wei J, Liao B, Liu X, Pardalos PM (2019c) Two-agent scheduling on bounded parallel-batching machines with an aging effect of job-position-dependent. Ann Oper Res. https://doi.org/10.1007/s10479-019-03160-y
Petrovic S, Leite-Rocha P (2008) Constructive and GRASP approaches to radiotherapy treatment scheduling. In: Proceedings—advances in electrical and electronics engineering—IAENG special edition of the world congress on engineering and computer science 2008, WCECS 2008. IEEE, New York, pp 192–200. https://doi.org/10.1109/WCECS.2008.31
Petrovic D, Morshed M, Petrovic S (2009) Genetic algorithm based scheduling of radiotherapy treatments for cancer patients. In: Lecture notes in computer science (including subseries lecture notes in artificial intelligence and lecture notes in bioinformatics), 5651 LNAI, pp 101–105. https://doi.org/10.1007/978-3-642-02976-9_14
Petrovic D, Morshed M, Petrovic S (2011) Multi-objective genetic algorithms for scheduling of radiotherapy treatments for categorised cancer patients. Expert Syst Appl 38(6):6994–7002. https://doi.org/10.1016/j.eswa.2010.12.015
Rahmani Hosseinabadi AA, Vahidi J, Saemi B, Sangaiah AK, Elhoseny M (2018) Extended genetic algorithm for solving open-shop scheduling problem. Soft Comput 23(13):1–18. https://doi.org/10.1007/s00500-018-3177-y
Riff MC, Cares JP, Neveu B (2016) RASON: a new approach to the scheduling radiotherapy problem that considers the current waiting times. Expert Syst Appl 64:287–295. https://doi.org/10.1016/j.eswa.2016.07.045
Sauré A, Patrick J, Tyldesley S, Puterman ML (2012) Dynamic multi-appointment patient scheduling for radiation therapy. Eur J Oper Res 223(2):573–584. https://doi.org/10.1016/j.ejor.2012.06.046
Taherkhani M, Safabakhsh R (2016) A novel stability-based adaptive inertia weight for particle swarm optimization. Appl Soft Comput 38(4):281–295. https://doi.org/10.1016/j.asoc.2011.01.037
Vogl P, Braune R, Doerner KF (2019) Scheduling recurring radiotherapy appointments in an ion beam facility: considering optional activities and time window constraints. J Sched 22(2):137–154. https://doi.org/10.1007/s10951-018-0574-0
Wang D, Wang JB (2010) Single-machine scheduling with simple linear deterioration to minimize earliness penalties. Int J Adv Manuf Technol 46(1–4):285–290. https://doi.org/10.1007/s00170-009-2086-8
Wang JB, Wang JJ (2015) Single-machine scheduling problems with precedence constraints and simple linear deterioration. Appl Math Model 39(3–4):1172–1182. https://doi.org/10.1016/j.apm.2014.07.028
Wang D, Huo Y, Ji P (2014) Single-machine group scheduling with deteriorating jobs and allotted resource. Optim Lett 8(2):591–605. https://doi.org/10.1007/s11590-012-0577-2
Xu Y, Tian Y, Dai J (2014) A simple method to prolong the service life of radioactive sources for external radiotherapy. J Appl Clin Med Phys 15(4):161–167. https://doi.org/10.1120/jacmp.v15i4.4789
Xue Y, Jiang J, Zhao B, Ma T (2018) A self-adaptive artificial bee colony algorithm based on global best for global optimization. Soft Comput 22(9):2935–2952. https://doi.org/10.1007/s00500-017-2547-1
Yang SJ (2013) Unrelated parallel-machine scheduling with deterioration effects and deteriorating multi-maintenance activities for minimizing the total completion time. Appl Math Model 37(5):2995–3005. https://doi.org/10.1016/j.apm.2012.07.029
Yeh WC, Lai PJ, Lee WC, Chuang MC (2014) Parallel-machine scheduling to minimize makespan with fuzzy processing times and learning effects. Inf Sci 269:142–158. https://doi.org/10.1016/j.ins.2013.10.023
Yoan J, Eric M, Pascal P (2011) A pattern-based approach of radiotherapy scheduling. In: IFAC proceedings volumes (IFAC-PapersOnline), vol 44. IFAC. https://doi.org/10.3182/20110828-6-IT-1002.00502
Zhao C, Tang H (2010) Single machine scheduling with past-sequence-dependent setup times and deteriorating jobs. Comput Ind Eng 59(4):663–666. https://doi.org/10.1016/j.cie.2010.07.015
Funding
This work is supported by the Key research and development Projects in Anhui (1804b06020377), the Basic scientific research Projects in central colleges and Universities (JZ2018HGTB0232), the National Natural Science Foundation of China (Nos. 71601065, 71690235, 71690230) and the Innovative Research Groups of the National Natural Science Foundation of China (71521001).
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Communicated by Yaroslav D. Sergeyev.
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Shao, K., Fan, W., Yang, Z. et al. Patient scheduling with deteriorating treatment duration and maintenance activity. Soft Comput 24, 17649–17668 (2020). https://doi.org/10.1007/s00500-020-05156-4
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DOI: https://doi.org/10.1007/s00500-020-05156-4