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The inverse parallel machine scheduling problem with minimum total completion time
A timedependent scheduling problem to minimize the sum of the total weighted tardiness among two agents
1.  Department of Business Administration, KangNing 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 
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. 
[2] 
A. Agnetis, D. Pacciarelli and A. Pacifici, Multiagent single machine scheduling,, Annals of Operations Research, 150 (2007), 3. doi: 10.1007/s104790060164y. 
[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. 
[4] 
A. Allahverdi and F. S. AlAnzi, Using twomachine 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/S03050548(00)000976. 
[5] 
A. Bachman and A. Janiak, Scheduling Jobs with Special Type of Start Time Dependent Processing Times,, Report No 34/97, (1997). 
[6] 
K. R. Baker and J. C. Smith, A multiplecriterion model for machine scheduling,, Journal of Scheduling, 6 (2003), 7. doi: 10.1023/A:1022231419049. 
[7] 
D. BenArieh 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. 
[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. 
[9] 
S. R. Cheng, A singlemachine twoagent scheduling problem by GA approach,, AsiaPacific Journal of Operational Research, 29 (2012). doi: 10.1142/S0217595912500133. 
[10] 
T. C. E. Cheng, Q. Ding and B. M. T. Lin, A concise survey of scheduling with timedependent processing times,, European Journal of Operational Research, 152 (2004), 1. doi: 10.1016/S03772217(02)009098. 
[11] 
T. C. E. Cheng, C. T. Ng and J. J. Yuan, Multiagent 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. 
[12] 
T. C. E. Cheng, C. T. Ng and J. J. Yuan, Multiagent scheduling on a single machine with maxform criteria,, European Journal of Operational Research, 188 (2008), 603. doi: 10.1016/j.ejor.2007.04.040. 
[13] 
T. C. E. Cheng, S. R. Cheng, W. H. Wu, P. H. Hsu and C. C. Wu, A twoagent single machine scheduling problem with truncated sumofprocessingtimesbased learning considerations,, Computers & Industrial Engineering, 60 (2011), 534. doi: 10.1016/j.cie.2010.12.008. 
[14] 
T. C. E. Cheng, W. H. Wu, S. R. Cheng and C. C. Wu, Twoagent scheduling with position based deteriorating jobs and learning effects,, Applied Mathematics and Computation, 217 (2011), 8804. doi: 10.1016/j.amc.2011.04.005. 
[15] 
T. C. E. Cheng, Y. H. Chung, S. C. Liao and W. C Lee, Twoagent singemachine 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. 
[16] 
C. Chu, A branchandbound algorithm to minimize total tardiness with different release dates,, Naval Research Logistics, 39 (1992), 859. 
[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). 
[18] 
A. Colorni, M. Dorigo, V. Maniezzo and M. Trubian, Ant system for jobshop scheduling,, Belgian Journal of Operations Research, 34 (1994), 39. 
[19] 
M. Dorigo, Di Caro G and L. M. Gambardella, Ant algorithms for discrete optimization,, Artificial Life, 5 (1999), 137. doi: 10.1162/106454699568728. 
[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. 
[21] 
M. L. Fisher, A dual algorithm for the onemachine scheduling problem,, Math Programming, 11 (): 229. doi: 10.1007/BF01580393. 
[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/S01675060(08)70356X. 
[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. 
[24] 
S. Kirkpatrick, C. Gelatt and M. Vecchi, Optimization by simulated annealing,, Science, 220 (1983), 671. doi: 10.1126/science.220.4598.671. 
[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: 10.1016/03772217(90)90089T. 
[26] 
K. Lee, B. C. Choi, J. Y. T. Leung and M. L. Pinedo, Approximation algorithms for multiagent scheduling to minimize total weighted completion time,, Information Processing Letters, 109 (2009), 913. doi: 10.1016/j.ipl.2009.04.018. 
[27] 
J. Lenstra, A. H. G. Rinnooy Kan and P. Brucker, Complexity of Machine Scheduling Problems,, Annals of Discrete Mathematics, 1 (1977), 343. 
[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. 
[29] 
P. Liu and L. Tang, Twoagent scheduling with linear deteriorating jobs on a single machine,, Lecture Notes in Computer Science, 5092 (2008), 642. 
[30] 
P. Liu, X. Y. Zhou and L. X. Tang, Twoagent singlemachine scheduling with positiondependent processing times,, International Journal of Advanced Manufacturing Technology, 48 (2010), 325. doi: 10.1007/s0017000922595. 
[31] 
D. C. Li and P. H. Hsu, Solving a twoagent singlemachine scheduling problem considering learning effect,, Computers & Operations Research, 39 (2012), 1644. doi: 10.1016/j.cor.2011.09.018. 
[32] 
W. Luo, L. Chen and G. Zhang, Approximation schemes for twomachine flow shop scheduling with two agents,, Journal of Combinatorial Optimization, 24 (2012), 229. doi: 10.1007/s1087801193782. 
[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. 
[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. 
[35] 
G. Mosheiov, $V$shaped policies for scheduling deteriorating jobs,, Operations Research, 39 (1991), 979. doi: 10.1287/opre.39.6.979. 
[36] 
G. Mosheiov, Scheduling jobs under simple linear deterioration,, Computers & Operations Research, 21 (1994), 653. doi: 10.1016/03050548(94)900809. 
[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. 
[38] 
C. T. Ng, T. C. E. Cheng and J. J. Yuan, A note on the complexity of the problem of twoagent scheduling on a single machine,, Journal of Combinatorial Optimization, 12 (2006), 387. doi: 10.1007/s1087800690010. 
[39] 
Q. Q. Nong, T. C. E. Cheng and C. T. Ng, Twoagent scheduling to minimize the total cost,, European Journal of Operational Research, 215 (2011), 39. doi: 10.1016/j.ejor.2011.05.041. 
[40] 
J. B. Wang and T. C. E. Cheng, Scheduling problems with the effects of deterioration and learning,, AsiaPacific Journal of Operational Research, 24 (2007), 245. doi: 10.1142/S021759590700122X. 
[41] 
J. B. Wang and Q. Guo, A duedate assignment problem with learning effect and deteriorating jobs,, Applied Mathematical Modelling, 34 (2010), 309. doi: 10.1016/j.apm.2009.04.020. 
[42] 
J. B. Wang, L. H. Sun and L. Y. Sun, Singlemachine total completion time scheduling with a timedependent deterioration,, Applied Mathematical Modelling, 35 (2011), 1506. doi: 10.1016/j.apm.2010.09.028. 
[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. 
[44] 
W. H. Wu, S. R. Cheng, C. C. Wu and Y. Yin, Ant colony algorithms for a twoagent scheduling with sumof processing timesbased learning and deteriorating considerations,, Journal of Intelligent Manufacturing, 23 (2012), 1985. doi: 10.1007/s1084501105255. 
[45] 
C. C. Wu, S. K. Huang and W. C. Lee, Twoagent scheduling with learning consideration,, Computers & Industrial Engineering, 61 (2011), 1324. doi: 10.1016/j.cie.2011.08.007. 
[46] 
D. L. Yang and W. H. Kuo, Singlemachine scheduling with both deterioration and learning effects,, Annals of Operations Research, 172 (2009), 315. doi: 10.1007/s1047900906153. 
[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. 
[48] 
S. H. Yang and J. B. Wang, Minimizing total weighted completion time in a twomachine flow shop scheduling under simple linear deterioration,, Applied Mathematics and Computation, 217 (2011), 4819. doi: 10.1016/j.amc.2010.11.037. 
[49] 
S. H. Yang and D. L. Yang, Minimizing the total completion time in singlemachine 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. 
[50] 
Y. Yin and D. Xu, Some singlemachine 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. 
[51] 
Y. Yin, S. R. Cheng and C. C. Wu, Scheduling problems with two agents and a linear nonincreasing deterioration to minimize earliness penalties,, Information Sciences, 189 (2012), 282. doi: 10.1016/j.ins.2011.11.035. 
[52] 
Y. Yin, S. R. Cheng, T. C. E. Cheng, W. H. Wu and C. C. Wu, Twoagent singlemachine scheduling with release times and deadlines,, International Journal of Shipping and Transport Logistics, 5 (2013), 75. doi: 10.1504/IJSTL.2013.050590. 
[53] 
Y. Yin, T. C. E. Cheng, J. Xu, S. R. Cheng and C. C. Wu, Singlemachine scheduling with pastsequencedependent delivery times and a linear deterioration,, Journal of Industrial and Management Optimization, 9 (2013), 323. doi: 10.3934/jimo.2013.9.323. 
[54] 
C. L. Zhao, Q. L. Zhang and H. Y. Tang, Scheduling problems under linear deterioration,, Acta Automatica Sinica, 29 (2003), 531. 
show all references
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. 
[2] 
A. Agnetis, D. Pacciarelli and A. Pacifici, Multiagent single machine scheduling,, Annals of Operations Research, 150 (2007), 3. doi: 10.1007/s104790060164y. 
[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. 
[4] 
A. Allahverdi and F. S. AlAnzi, Using twomachine 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/S03050548(00)000976. 
[5] 
A. Bachman and A. Janiak, Scheduling Jobs with Special Type of Start Time Dependent Processing Times,, Report No 34/97, (1997). 
[6] 
K. R. Baker and J. C. Smith, A multiplecriterion model for machine scheduling,, Journal of Scheduling, 6 (2003), 7. doi: 10.1023/A:1022231419049. 
[7] 
D. BenArieh 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. 
[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. 
[9] 
S. R. Cheng, A singlemachine twoagent scheduling problem by GA approach,, AsiaPacific Journal of Operational Research, 29 (2012). doi: 10.1142/S0217595912500133. 
[10] 
T. C. E. Cheng, Q. Ding and B. M. T. Lin, A concise survey of scheduling with timedependent processing times,, European Journal of Operational Research, 152 (2004), 1. doi: 10.1016/S03772217(02)009098. 
[11] 
T. C. E. Cheng, C. T. Ng and J. J. Yuan, Multiagent 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. 
[12] 
T. C. E. Cheng, C. T. Ng and J. J. Yuan, Multiagent scheduling on a single machine with maxform criteria,, European Journal of Operational Research, 188 (2008), 603. doi: 10.1016/j.ejor.2007.04.040. 
[13] 
T. C. E. Cheng, S. R. Cheng, W. H. Wu, P. H. Hsu and C. C. Wu, A twoagent single machine scheduling problem with truncated sumofprocessingtimesbased learning considerations,, Computers & Industrial Engineering, 60 (2011), 534. doi: 10.1016/j.cie.2010.12.008. 
[14] 
T. C. E. Cheng, W. H. Wu, S. R. Cheng and C. C. Wu, Twoagent scheduling with position based deteriorating jobs and learning effects,, Applied Mathematics and Computation, 217 (2011), 8804. doi: 10.1016/j.amc.2011.04.005. 
[15] 
T. C. E. Cheng, Y. H. Chung, S. C. Liao and W. C Lee, Twoagent singemachine 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. 
[16] 
C. Chu, A branchandbound algorithm to minimize total tardiness with different release dates,, Naval Research Logistics, 39 (1992), 859. 
[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). 
[18] 
A. Colorni, M. Dorigo, V. Maniezzo and M. Trubian, Ant system for jobshop scheduling,, Belgian Journal of Operations Research, 34 (1994), 39. 
[19] 
M. Dorigo, Di Caro G and L. M. Gambardella, Ant algorithms for discrete optimization,, Artificial Life, 5 (1999), 137. doi: 10.1162/106454699568728. 
[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. 
[21] 
M. L. Fisher, A dual algorithm for the onemachine scheduling problem,, Math Programming, 11 (): 229. doi: 10.1007/BF01580393. 
[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/S01675060(08)70356X. 
[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. 
[24] 
S. Kirkpatrick, C. Gelatt and M. Vecchi, Optimization by simulated annealing,, Science, 220 (1983), 671. doi: 10.1126/science.220.4598.671. 
[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: 10.1016/03772217(90)90089T. 
[26] 
K. Lee, B. C. Choi, J. Y. T. Leung and M. L. Pinedo, Approximation algorithms for multiagent scheduling to minimize total weighted completion time,, Information Processing Letters, 109 (2009), 913. doi: 10.1016/j.ipl.2009.04.018. 
[27] 
J. Lenstra, A. H. G. Rinnooy Kan and P. Brucker, Complexity of Machine Scheduling Problems,, Annals of Discrete Mathematics, 1 (1977), 343. 
[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. 
[29] 
P. Liu and L. Tang, Twoagent scheduling with linear deteriorating jobs on a single machine,, Lecture Notes in Computer Science, 5092 (2008), 642. 
[30] 
P. Liu, X. Y. Zhou and L. X. Tang, Twoagent singlemachine scheduling with positiondependent processing times,, International Journal of Advanced Manufacturing Technology, 48 (2010), 325. doi: 10.1007/s0017000922595. 
[31] 
D. C. Li and P. H. Hsu, Solving a twoagent singlemachine scheduling problem considering learning effect,, Computers & Operations Research, 39 (2012), 1644. doi: 10.1016/j.cor.2011.09.018. 
[32] 
W. Luo, L. Chen and G. Zhang, Approximation schemes for twomachine flow shop scheduling with two agents,, Journal of Combinatorial Optimization, 24 (2012), 229. doi: 10.1007/s1087801193782. 
[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. 
[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. 
[35] 
G. Mosheiov, $V$shaped policies for scheduling deteriorating jobs,, Operations Research, 39 (1991), 979. doi: 10.1287/opre.39.6.979. 
[36] 
G. Mosheiov, Scheduling jobs under simple linear deterioration,, Computers & Operations Research, 21 (1994), 653. doi: 10.1016/03050548(94)900809. 
[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. 
[38] 
C. T. Ng, T. C. E. Cheng and J. J. Yuan, A note on the complexity of the problem of twoagent scheduling on a single machine,, Journal of Combinatorial Optimization, 12 (2006), 387. doi: 10.1007/s1087800690010. 
[39] 
Q. Q. Nong, T. C. E. Cheng and C. T. Ng, Twoagent scheduling to minimize the total cost,, European Journal of Operational Research, 215 (2011), 39. doi: 10.1016/j.ejor.2011.05.041. 
[40] 
J. B. Wang and T. C. E. Cheng, Scheduling problems with the effects of deterioration and learning,, AsiaPacific Journal of Operational Research, 24 (2007), 245. doi: 10.1142/S021759590700122X. 
[41] 
J. B. Wang and Q. Guo, A duedate assignment problem with learning effect and deteriorating jobs,, Applied Mathematical Modelling, 34 (2010), 309. doi: 10.1016/j.apm.2009.04.020. 
[42] 
J. B. Wang, L. H. Sun and L. Y. Sun, Singlemachine total completion time scheduling with a timedependent deterioration,, Applied Mathematical Modelling, 35 (2011), 1506. doi: 10.1016/j.apm.2010.09.028. 
[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. 
[44] 
W. H. Wu, S. R. Cheng, C. C. Wu and Y. Yin, Ant colony algorithms for a twoagent scheduling with sumof processing timesbased learning and deteriorating considerations,, Journal of Intelligent Manufacturing, 23 (2012), 1985. doi: 10.1007/s1084501105255. 
[45] 
C. C. Wu, S. K. Huang and W. C. Lee, Twoagent scheduling with learning consideration,, Computers & Industrial Engineering, 61 (2011), 1324. doi: 10.1016/j.cie.2011.08.007. 
[46] 
D. L. Yang and W. H. Kuo, Singlemachine scheduling with both deterioration and learning effects,, Annals of Operations Research, 172 (2009), 315. doi: 10.1007/s1047900906153. 
[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. 
[48] 
S. H. Yang and J. B. Wang, Minimizing total weighted completion time in a twomachine flow shop scheduling under simple linear deterioration,, Applied Mathematics and Computation, 217 (2011), 4819. doi: 10.1016/j.amc.2010.11.037. 
[49] 
S. H. Yang and D. L. Yang, Minimizing the total completion time in singlemachine 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. 
[50] 
Y. Yin and D. Xu, Some singlemachine 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. 
[51] 
Y. Yin, S. R. Cheng and C. C. Wu, Scheduling problems with two agents and a linear nonincreasing deterioration to minimize earliness penalties,, Information Sciences, 189 (2012), 282. doi: 10.1016/j.ins.2011.11.035. 
[52] 
Y. Yin, S. R. Cheng, T. C. E. Cheng, W. H. Wu and C. C. Wu, Twoagent singlemachine scheduling with release times and deadlines,, International Journal of Shipping and Transport Logistics, 5 (2013), 75. doi: 10.1504/IJSTL.2013.050590. 
[53] 
Y. Yin, T. C. E. Cheng, J. Xu, S. R. Cheng and C. C. Wu, Singlemachine scheduling with pastsequencedependent delivery times and a linear deterioration,, Journal of Industrial and Management Optimization, 9 (2013), 323. doi: 10.3934/jimo.2013.9.323. 
[54] 
C. L. Zhao, Q. L. Zhang and H. Y. Tang, Scheduling problems under linear deterioration,, Acta Automatica Sinica, 29 (2003), 531. 
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