# American Institute of Mathematical Sciences

April  2014, 10(2): 591-611. doi: 10.3934/jimo.2014.10.591

## A time-dependent scheduling problem to minimize the sum of the total weighted tardiness among two agents

 1 Department of Business Administration, Kang-Ning Junior College of Medical Care and Management, Taipei, Taiwan, Taiwan 2 College of Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China 3 Department of Healthcare Management, Yuanpei University, Hsinchu, Taiwan 4 Department of Statistics, Feng Chia University, Taichung, Taiwan

Received  September 2012 Revised  April 2013 Published  October 2013

The problem of scheduling with time-dependent processing times has been studied for more than two decades and significant advances have been made over the years. However, most work has paid more attention to the single-criterion models. Furthermore, most heuristics are constructed for the time-dependent scheduling problems in a step-by-step way. Motivated by the observations, this paper studies a two-agent scheduling model with increasing linear deterioration jobs in which the processing time of a job is modeled as an increasing linear function of its starting time. The objective function is to minimize the sum of the maximum weighted tardiness of the jobs of the first agent and the total weighted tardiness of the jobs of the second agent. This problem is known to be strongly NP-hard. Thus, as an alternative, the branch-and-bound, ant colony algorithm and simulated annealing algorithms are developed for the problem. Computational results are also presented to determine the performance of the proposed algorithms.
Citation: Wen-Hung Wu, Yunqiang Yin, Wen-Hsiang Wu, Chin-Chia Wu, Peng-Hsiang Hsu. A time-dependent scheduling problem to minimize the sum of the total weighted tardiness among two agents. Journal of Industrial & Management Optimization, 2014, 10 (2) : 591-611. doi: 10.3934/jimo.2014.10.591
##### References:
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Gupta, Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem,, European Journal of Operation Research, 47 (1990), 56.  doi: 10.1016/0377-2217(90)90089-T.  Google Scholar [26] K. Lee, B. C. Choi, J. Y. T. Leung and M. L. Pinedo, Approximation algorithms for multi-agent scheduling to minimize total weighted completion time,, Information Processing Letters, 109 (2009), 913.  doi: 10.1016/j.ipl.2009.04.018.  Google Scholar [27] J. Lenstra, A. H. G. Rinnooy Kan and P. Brucker, Complexity of Machine Scheduling Problems,, Annals of Discrete Mathematics, 1 (1977), 343.   Google Scholar [28] M. Lai and X. Tong, A metaheuristic method for vehicle routing problem based on improved ant colony optimization and Tabu search,, Journal of Industrial and Management Optimization, 8 (2012), 469.  doi: 10.3934/jimo.2012.8.469.  Google Scholar [29] P. Liu and L. Tang, Two-agent scheduling with linear deteriorating jobs on a single machine,, Lecture Notes in Computer Science, 5092 (2008), 642.   Google Scholar [30] P. Liu, X. Y. Zhou and L. X. Tang, Two-agent single-machine scheduling with position-dependent processing times,, International Journal of Advanced Manufacturing Technology, 48 (2010), 325.  doi: 10.1007/s00170-009-2259-5.  Google Scholar [31] D. C. Li and P. H. Hsu, Solving a two-agent single-machine scheduling problem considering learning effect,, Computers & Operations Research, 39 (2012), 1644.  doi: 10.1016/j.cor.2011.09.018.  Google Scholar [32] W. Luo, L. Chen and G. Zhang, Approximation schemes for two-machine flow shop scheduling with two agents,, Journal of Combinatorial Optimization, 24 (2012), 229.  doi: 10.1007/s10878-011-9378-2.  Google Scholar [33] W. Luo and L. Chen, Approximation schemes for scheduling a maintenance and linear deteriorating jobs,, Journal of Industrial and Management Optimization, 8 (2012), 271.  doi: 10.3934/jimo.2012.8.271.  Google Scholar [34] E. Mokotoff, Algorithms for bicriteria minimization in the permutation flow shop scheduling problem,, Journal of Industrial and Management Optimization, 7 (2011), 253.  doi: 10.3934/jimo.2011.7.253.  Google Scholar [35] G. Mosheiov, $V$-shaped policies for scheduling deteriorating jobs,, Operations Research, 39 (1991), 979.  doi: 10.1287/opre.39.6.979.  Google Scholar [36] G. Mosheiov, Scheduling jobs under simple linear deterioration,, Computers & Operations Research, 21 (1994), 653.  doi: 10.1016/0305-0548(94)90080-9.  Google Scholar [37] B. Mor and G. Mosheiov, Scheduling problems with two competing agents to minimize minmax and minsum earliness measures,, European Journal of Operational Research, 206 (2010), 540.  doi: 10.1016/j.ejor.2010.03.003.  Google Scholar [38] C. T. Ng, T. C. E. Cheng and J. J. Yuan, A note on the complexity of the problem of two-agent scheduling on a single machine,, Journal of Combinatorial Optimization, 12 (2006), 387.  doi: 10.1007/s10878-006-9001-0.  Google Scholar [39] Q. Q. Nong, T. C. E. Cheng and C. T. Ng, Two-agent scheduling to minimize the total cost,, European Journal of Operational Research, 215 (2011), 39.  doi: 10.1016/j.ejor.2011.05.041.  Google Scholar [40] J. B. Wang and T. C. E. Cheng, Scheduling problems with the effects of deterioration and learning,, Asia-Pacific Journal of Operational Research, 24 (2007), 245.  doi: 10.1142/S021759590700122X.  Google Scholar [41] J. B. Wang and Q. Guo, A due-date assignment problem with learning effect and deteriorating jobs,, Applied Mathematical Modelling, 34 (2010), 309.  doi: 10.1016/j.apm.2009.04.020.  Google Scholar [42] J. B. Wang, L. H. Sun and L. Y. Sun, Single-machine total completion time scheduling with a time-dependent deterioration,, Applied Mathematical Modelling, 35 (2011), 1506.  doi: 10.1016/j.apm.2010.09.028.  Google Scholar [43] G. Wan, R. S. Vakati, J. Y. T. Leung and M. Pinedo, Scheduling two agents with controllable processing times,, European Journal of Operational Research, 205 (2010), 528.  doi: 10.1016/j.ejor.2010.01.005.  Google Scholar [44] W. H. Wu, S. R. Cheng, C. C. Wu and Y. Yin, Ant colony algorithms for a two-agent scheduling with sum-of processing times-based learning and deteriorating considerations,, Journal of Intelligent Manufacturing, 23 (2012), 1985.  doi: 10.1007/s10845-011-0525-5.  Google Scholar [45] C. C. Wu, S. K. Huang and W. C. Lee, Two-agent scheduling with learning consideration,, Computers & Industrial Engineering, 61 (2011), 1324.  doi: 10.1016/j.cie.2011.08.007.  Google Scholar [46] D. L. Yang and W. H. Kuo, Single-machine scheduling with both deterioration and learning effects,, Annals of Operations Research, 172 (2009), 315.  doi: 10.1007/s10479-009-0615-3.  Google Scholar [47] D. L. Yang and W. H. Kuo, Scheduling with deteriorating jobs and learning effects,, Applied Mathematics and Computation, 218 (2011), 2069.  doi: 10.1016/j.amc.2011.07.023.  Google Scholar [48] S. H. Yang and J. B. Wang, Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration,, Applied Mathematics and Computation, 217 (2011), 4819.  doi: 10.1016/j.amc.2010.11.037.  Google Scholar [49] S. H. Yang and D. L. Yang, Minimizing the total completion time in single-machine scheduling with aging/deteriorating effects and deteriorating maintenance activities,, Computers and Mathematics with Applications, 60 (2010), 2161.  doi: 10.1016/j.camwa.2010.08.003.  Google Scholar [50] Y. Yin and D. Xu, Some single-machine scheduling problems with general effects of learning and deterioration,, Computers and Mathematics with Applications, 61 (2011), 100.  doi: 10.1016/j.camwa.2010.10.036.  Google Scholar [51] Y. Yin, S. R. Cheng and C. C. Wu, Scheduling problems with two agents and a linear non-increasing deterioration to minimize earliness penalties,, Information Sciences, 189 (2012), 282.  doi: 10.1016/j.ins.2011.11.035.  Google Scholar [52] Y. Yin, S. R. Cheng, T. C. E. Cheng, W. H. Wu and C. C. Wu, Two-agent single-machine scheduling with release times and deadlines,, International Journal of Shipping and Transport Logistics, 5 (2013), 75.  doi: 10.1504/IJSTL.2013.050590.  Google Scholar [53] Y. Yin, T. C. E. Cheng, J. Xu, S. R. Cheng and C. C. Wu, Single-machine scheduling with past-sequence-dependent delivery times and a linear deterioration,, Journal of Industrial and Management Optimization, 9 (2013), 323.  doi: 10.3934/jimo.2013.9.323.  Google Scholar [54] C. L. Zhao, Q. L. Zhang and H. Y. Tang, Scheduling problems under linear deterioration,, Acta Automatica Sinica, 29 (2003), 531.   Google Scholar

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##### References:
 [1] A. Agnetis, P. B. Mirchandani, D. Pacciarelli and A. Pacifici, Scheduling problems with two competing agents,, Operations Research, 52 (2004), 229.  doi: 10.1287/opre.1030.0092.  Google Scholar [2] A. Agnetis, D. Pacciarelli and A. Pacifici, Multi-agent single machine scheduling,, Annals of Operations Research, 150 (2007), 3.  doi: 10.1007/s10479-006-0164-y.  Google Scholar [3] B. Alidaee and N. K. Womer, Scheduling with time dependent processing times: Review and extensions,, Journal of the Operational Research Society, 50 (1999), 711.   Google Scholar [4] A. Allahverdi and F. S. Al-Anzi, Using two-machine flowshop with maximum lateness objective to model multimedia data objects scheduling problem for WWW applications,, Computers and Operations Research, 29 (2002), 971.  doi: 10.1016/S0305-0548(00)00097-6.  Google Scholar [5] A. Bachman and A. Janiak, Scheduling Jobs with Special Type of Start Time Dependent Processing Times,, Report No 34/97, (1997).   Google Scholar [6] K. R. Baker and J. C. Smith, A multiple-criterion model for machine scheduling,, Journal of Scheduling, 6 (2003), 7.  doi: 10.1023/A:1022231419049.  Google Scholar [7] D. Ben-Arieh and O. Maimon, Annealing method for PCB assembly scheduling on two sequential machines,, International Journal of Computer Integrated Manufacturing, 5 (1992), 361.  doi: 10.1080/09511929208944543.  Google Scholar [8] S. Browne and U. Yechiali, Scheduling deteriorating jobs on a single processor,, Operations Research, 38 (1990), 495.  doi: 10.1287/opre.38.3.495.  Google Scholar [9] S. R. Cheng, A single-machine two-agent scheduling problem by GA approach,, Asia-Pacific Journal of Operational Research, 29 (2012).  doi: 10.1142/S0217595912500133.  Google Scholar [10] T. C. E. Cheng, Q. Ding and B. M. T. Lin, A concise survey of scheduling with time-dependent processing times,, European Journal of Operational Research, 152 (2004), 1.  doi: 10.1016/S0377-2217(02)00909-8.  Google Scholar [11] T. C. E. Cheng, C. T. Ng and J. J. Yuan, Multi-agent scheduling on a single machine to minimize total weighted number of tardy jobs,, Theoretical Computer Science, 362 (2006), 273.  doi: 10.1016/j.tcs.2006.07.011.  Google Scholar [12] T. C. E. Cheng, C. T. Ng and J. J. Yuan, Multi-agent scheduling on a single machine with max-form criteria,, European Journal of Operational Research, 188 (2008), 603.  doi: 10.1016/j.ejor.2007.04.040.  Google Scholar [13] T. C. E. Cheng, S. R. Cheng, W. H. Wu, P. H. Hsu and C. C. Wu, A two-agent single- machine scheduling problem with truncated sum-of-processing-times-based learning considerations,, Computers & Industrial Engineering, 60 (2011), 534.  doi: 10.1016/j.cie.2010.12.008.  Google Scholar [14] T. C. E. Cheng, W. H. Wu, S. R. Cheng and C. C. Wu, Two-agent scheduling with position- based deteriorating jobs and learning effects,, Applied Mathematics and Computation, 217 (2011), 8804.  doi: 10.1016/j.amc.2011.04.005.  Google Scholar [15] T. C. E. Cheng, Y. H. Chung, S. C. Liao and W. C Lee, Two-agent singe-machine scheduling with release times to minimize the total weighted completion time,, Computers & Operations Research, 40 (2013), 353.  doi: 10.1016/j.cor.2012.07.013.  Google Scholar [16] C. Chu, A branch-and-bound algorithm to minimize total tardiness with different release dates,, Naval Research Logistics, 39 (1992), 859.   Google Scholar [17] A. Colorni, M. Dorigo, M. Maniezzo, I. F. J. Varela and P. Bourgine, Distributed Optimization by Ant Colonies,, Proceedings of the first European Conference on Artificial Life, (1991).   Google Scholar [18] A. Colorni, M. Dorigo, V. Maniezzo and M. Trubian, Ant system for job-shop scheduling,, Belgian Journal of Operations Research, 34 (1994), 39.   Google Scholar [19] M. Dorigo, Di Caro G and L. M. Gambardella, Ant algorithms for discrete optimization,, Artificial Life, 5 (1999), 137.  doi: 10.1162/106454699568728.  Google Scholar [20] M. Dorigo and L. M. Gambardella, Ant colony system: A cooperative learning approach to travel salesman problem,, IEEE Trans Evol Computing, 1 (1997), 53.  doi: 10.1109/4235.585892.  Google Scholar [21] M. L. Fisher, A dual algorithm for the one-machine scheduling problem,, Math Programming, 11 (): 229.  doi: 10.1007/BF01580393.  Google Scholar [22] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and heuristic in deterministic sequencing and scheduling: A survey,, Annals of Discrete Mathematics, 5 (1979), 287.  doi: 10.1016/S0167-5060(08)70356-X.  Google Scholar [23] A. Janiak, T. Krysiak and R. Trela, Scheduling problems with learning and aging effects: A survey,, Decision Making in Manufacturing and Services, 5 (2011), 19.   Google Scholar [24] S. Kirkpatrick, C. Gelatt and M. Vecchi, Optimization by simulated annealing,, Science, 220 (1983), 671.  doi: 10.1126/science.220.4598.671.  Google Scholar [25] A. S. Kunnathur and S. K. Gupta, Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem,, European Journal of Operation Research, 47 (1990), 56.  doi: 10.1016/0377-2217(90)90089-T.  Google Scholar [26] K. Lee, B. C. Choi, J. Y. T. Leung and M. L. Pinedo, Approximation algorithms for multi-agent scheduling to minimize total weighted completion time,, Information Processing Letters, 109 (2009), 913.  doi: 10.1016/j.ipl.2009.04.018.  Google Scholar [27] J. Lenstra, A. H. G. Rinnooy Kan and P. Brucker, Complexity of Machine Scheduling Problems,, Annals of Discrete Mathematics, 1 (1977), 343.   Google Scholar [28] M. Lai and X. Tong, A metaheuristic method for vehicle routing problem based on improved ant colony optimization and Tabu search,, Journal of Industrial and Management Optimization, 8 (2012), 469.  doi: 10.3934/jimo.2012.8.469.  Google Scholar [29] P. Liu and L. Tang, Two-agent scheduling with linear deteriorating jobs on a single machine,, Lecture Notes in Computer Science, 5092 (2008), 642.   Google Scholar [30] P. Liu, X. Y. Zhou and L. X. Tang, Two-agent single-machine scheduling with position-dependent processing times,, International Journal of Advanced Manufacturing Technology, 48 (2010), 325.  doi: 10.1007/s00170-009-2259-5.  Google Scholar [31] D. C. Li and P. H. Hsu, Solving a two-agent single-machine scheduling problem considering learning effect,, Computers & Operations Research, 39 (2012), 1644.  doi: 10.1016/j.cor.2011.09.018.  Google Scholar [32] W. Luo, L. Chen and G. Zhang, Approximation schemes for two-machine flow shop scheduling with two agents,, Journal of Combinatorial Optimization, 24 (2012), 229.  doi: 10.1007/s10878-011-9378-2.  Google Scholar [33] W. Luo and L. Chen, Approximation schemes for scheduling a maintenance and linear deteriorating jobs,, Journal of Industrial and Management Optimization, 8 (2012), 271.  doi: 10.3934/jimo.2012.8.271.  Google Scholar [34] E. Mokotoff, Algorithms for bicriteria minimization in the permutation flow shop scheduling problem,, Journal of Industrial and Management Optimization, 7 (2011), 253.  doi: 10.3934/jimo.2011.7.253.  Google Scholar [35] G. Mosheiov, $V$-shaped policies for scheduling deteriorating jobs,, Operations Research, 39 (1991), 979.  doi: 10.1287/opre.39.6.979.  Google Scholar [36] G. Mosheiov, Scheduling jobs under simple linear deterioration,, Computers & Operations Research, 21 (1994), 653.  doi: 10.1016/0305-0548(94)90080-9.  Google Scholar [37] B. Mor and G. Mosheiov, Scheduling problems with two competing agents to minimize minmax and minsum earliness measures,, European Journal of Operational Research, 206 (2010), 540.  doi: 10.1016/j.ejor.2010.03.003.  Google Scholar [38] C. T. Ng, T. C. E. Cheng and J. J. Yuan, A note on the complexity of the problem of two-agent scheduling on a single machine,, Journal of Combinatorial Optimization, 12 (2006), 387.  doi: 10.1007/s10878-006-9001-0.  Google Scholar [39] Q. Q. Nong, T. C. E. Cheng and C. T. Ng, Two-agent scheduling to minimize the total cost,, European Journal of Operational Research, 215 (2011), 39.  doi: 10.1016/j.ejor.2011.05.041.  Google Scholar [40] J. B. Wang and T. C. E. Cheng, Scheduling problems with the effects of deterioration and learning,, Asia-Pacific Journal of Operational Research, 24 (2007), 245.  doi: 10.1142/S021759590700122X.  Google Scholar [41] J. B. Wang and Q. Guo, A due-date assignment problem with learning effect and deteriorating jobs,, Applied Mathematical Modelling, 34 (2010), 309.  doi: 10.1016/j.apm.2009.04.020.  Google Scholar [42] J. B. Wang, L. H. Sun and L. Y. Sun, Single-machine total completion time scheduling with a time-dependent deterioration,, Applied Mathematical Modelling, 35 (2011), 1506.  doi: 10.1016/j.apm.2010.09.028.  Google Scholar [43] G. Wan, R. S. Vakati, J. Y. T. Leung and M. Pinedo, Scheduling two agents with controllable processing times,, European Journal of Operational Research, 205 (2010), 528.  doi: 10.1016/j.ejor.2010.01.005.  Google Scholar [44] W. H. Wu, S. R. Cheng, C. C. Wu and Y. Yin, Ant colony algorithms for a two-agent scheduling with sum-of processing times-based learning and deteriorating considerations,, Journal of Intelligent Manufacturing, 23 (2012), 1985.  doi: 10.1007/s10845-011-0525-5.  Google Scholar [45] C. C. Wu, S. K. Huang and W. C. Lee, Two-agent scheduling with learning consideration,, Computers & Industrial Engineering, 61 (2011), 1324.  doi: 10.1016/j.cie.2011.08.007.  Google Scholar [46] D. L. Yang and W. H. Kuo, Single-machine scheduling with both deterioration and learning effects,, Annals of Operations Research, 172 (2009), 315.  doi: 10.1007/s10479-009-0615-3.  Google Scholar [47] D. L. Yang and W. H. Kuo, Scheduling with deteriorating jobs and learning effects,, Applied Mathematics and Computation, 218 (2011), 2069.  doi: 10.1016/j.amc.2011.07.023.  Google Scholar [48] S. H. Yang and J. B. Wang, Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration,, Applied Mathematics and Computation, 217 (2011), 4819.  doi: 10.1016/j.amc.2010.11.037.  Google Scholar [49] S. H. Yang and D. L. Yang, Minimizing the total completion time in single-machine scheduling with aging/deteriorating effects and deteriorating maintenance activities,, Computers and Mathematics with Applications, 60 (2010), 2161.  doi: 10.1016/j.camwa.2010.08.003.  Google Scholar [50] Y. Yin and D. Xu, Some single-machine scheduling problems with general effects of learning and deterioration,, Computers and Mathematics with Applications, 61 (2011), 100.  doi: 10.1016/j.camwa.2010.10.036.  Google Scholar [51] Y. Yin, S. R. Cheng and C. C. Wu, Scheduling problems with two agents and a linear non-increasing deterioration to minimize earliness penalties,, Information Sciences, 189 (2012), 282.  doi: 10.1016/j.ins.2011.11.035.  Google Scholar [52] Y. Yin, S. R. Cheng, T. C. E. Cheng, W. H. Wu and C. C. Wu, Two-agent single-machine scheduling with release times and deadlines,, International Journal of Shipping and Transport Logistics, 5 (2013), 75.  doi: 10.1504/IJSTL.2013.050590.  Google Scholar [53] Y. Yin, T. C. E. Cheng, J. Xu, S. R. Cheng and C. C. Wu, Single-machine scheduling with past-sequence-dependent delivery times and a linear deterioration,, Journal of Industrial and Management Optimization, 9 (2013), 323.  doi: 10.3934/jimo.2013.9.323.  Google Scholar [54] C. L. Zhao, Q. L. Zhang and H. Y. Tang, Scheduling problems under linear deterioration,, Acta Automatica Sinica, 29 (2003), 531.   Google Scholar
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