# American Institute of Mathematical Sciences

January  2014, 10(1): 243-258. doi: 10.3934/jimo.2014.10.243

## A difference of convex functions algorithm for optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors

 1 Laboratory of Theoretical and Applied Computer Science (LITA), University of Lorraine, Ile du Saulcy, 57045, Metz, France, France 2 LGIPM, INRIA Costeam, Ecole National d’Ing´enieurs de Metz, France

Received  January 2012 Revised  July 2013 Published  October 2013

In this paper, we introduce a new approach based on DC (Difference of Convex functions) Programming and DCA (DC Algorithm) for minimizing the maintenance cost involving flow-time and tardiness penalties by optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors. The main idea is to divide the horizon considered into $H$ intervals. The problem is first formulated as a mixed integer linear problem (MILP) and then reformulated as a DC program. A solution method based on DCA is used to solve the resulting problem. The efficiency of DCA is compared with the algorithm based on the new flow-time and tardiness rule (FTR) given in [1]. The computational results on several test problems show that the solutions provided by DCA are better.
Citation: Le Thi Hoai An, Tran Duc Quynh, Kondo Hloindo Adjallah. A difference of convex functions algorithm for optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors. Journal of Industrial and Management Optimization, 2014, 10 (1) : 243-258. doi: 10.3934/jimo.2014.10.243
##### References:
 [1] K. H. Adjallah and K. P. Adzakpa, Minimizing maintenance cost involving flow-time and tardiness penalty with unequal release dates, In Press of Journal of Risk and Reliability, Part O, 221 (2007), 57-66. [2] K. P. Adzakpa, "Maintenance of Distributed Systems: Methods for Real-Time Decision-Making," PhD Thesis, University of Technology of Troyes, France, October 2004. [3] M. Berg, The marginal cost analysis and its application to repair and replacement policies, European Journal of Operational Research, 82 (1995), 214-224. [4] E. K. Burke and A. J. Smith, Hybrid evolutionary techniques for the maintenance scheduling problem, Ieee2000 Trans. on Power Systems, 15 (2000), 122-128. [5] C. Derman, G. J. Lieberman and S. M. Ross, On the optimal assignment of servers and a repairman, Journal of Applied Probability, 17 (1980), 577-581. doi: 10.2307/3213050. [6] C. Duron, M. A. Ould Louly and J.-M. Proth, The one machine scheduling problem: Insertion of a job under the real-time constraint, European Journal of Operational Research, 199 (2009), 695-701. doi: 10.1016/j.ejor.2007.09.048. [7] E. Frostig, Jointly optimal allocation of a repairman and optimal control of service rate for machine repairman problem, European Journal of Operational Research, 116 (1999), 274-280. [8] E. Frostig, Optimal allocation of machines to distinguishable repairmen in order to maximize some reward functions, Probability in the Engineering and Informational Sciences, 9 (1995), 633-645. doi: 10.1017/S0269964800004113. [9] E. Frostig, Optimal policies for machine repairmen problems, Journal of Applied Probability, 30 (1993), 703-715. doi: 10.2307/3214776. [10] M. Gopalakrishnan, S. Mohan and Z. He, A tabu search heuristic for preventive maintenance scheduling, Computers & Industrial Engineering, 40 (2001), 149-160. [11] G. Graves and C. Lee, Scheduling maintenance and semiresumable jobs on a single machine, Naval Research Logistics, 46 (1999), 845-863. [12] G. Koole, Optimal repairman assignment in two symmetric maintenance models, European Journal of Operational Research, 82 (1995), 295-301. [13] H. A. Le Thi and T. Pham Dinh, Solving a class of linearly constrained indefinite quadratic problems by DC algorithms, Journal of Global Optimization, 11 (1997), 253-285. doi: 10.1023/A:1008288411710. [14] H. A. Le Thi and T. Pham Dinh, A Continuous approach for globally solving linearly constrained quadratic zero-one programming problem, Optimization, 50 (2001), 93-120. doi: 10.1080/02331930108844555. [15] H. A. Le Thi and T. Pham Dinh, The DC(difference of convex functions) Programming and DCA revisited with DC models of real world non convex optimization problems, Annals of Operations Research, 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1. [16] H. A. Le Thi, T. Pham Dinh and M. Le Dung, Exact penalty in d.c. programming, Vietnam Journal of Mathematics, 27 (1999), 169-178. [17] H. A. Le Thi and T. Pham Dinh, A continuous approach for the concave cost supply problem via DC programming and DCA, Discrete Applied Mathematics, 156 (2008), 325-338. doi: 10.1016/j.dam.2007.03.024. [18] C. Lee and Z. Chen, Scheduling jobs and maintenance activities on parallel machines, Naval Research Logistics, 47 (2000), 145-165. doi: 10.1002/(SICI)1520-6750(200003)47:2<145::AID-NAV5>3.0.CO;2-3. [19] J. Lenstra, A. R. Kan and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 1 (1977), 343-362. [20] Z. Li, H. Liao, L-Y. Chan and A. Elsayed, Maintenance of continuously degrading systems considering random service delays, Proceedings of the 2008 Industrial Research Conference (IERC), Vancouver, British Columbia, Canada, May 17-21, 2008. [21] S-T. Lo, R-M. Chen, Y-M. Huang and C-L. Wu, Multiprocessor system scheduling with precedence and resource constraints using an enhanced ant colony system, Expert Systems with Applications, 34 (2008), 2071-2081. [22] T. Pham Dinh and H. A. Le Thi, Convex analysis approach to d.c programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, dedicated to Professor Hoang Tuy on the occasion of his 70th birthday, 22 (1997), 289-355. [23] T. Pham Dinh and H. A. Le Thi, DC optimization algorihms for solving the trust region subproblem, SIAM J.Optimization, 8 (1997), 476-505. [24] K. Park, Optimal number of minimal repairs before replacement, IEEE Transactions on Reliability, R-28 (1979), 137-140. [25] X. Qi, T. Chen and F. Tu, Scheduling the maintenance on a single machine, Journal of the Operation Research Society, 50 (1999), 1071-1078. [26] R.T. Rockafellar, "Convex analysis Princeton University Press," Princeton University Press, Princeton, N. J. 1970 xviii+451 pp. [27] S. Seshadri, Determination of aggregate preventive maintenance programs using production schedules, Computers and Industrial Engineering, 14 (1988), 193-200. [28] W. Stadje and D. Zuckerman, Optimal repair policies with general degree of repair in two maintenance models, Operations Research Letters, 11 (1992), 77-80. doi: 10.1016/0167-6377(92)90036-3. [29] H. Tajima, J. Sugimoto, T. Funabashi and R. Yokoyama, Auction price-based thermal unit maintenance scheduling by reactive tabu search, Proc. 41st IEEE Int. UPEC'06 (Universities Power Engineering Conference), 2006. [30] L. Weinstein and C. Chung, Integrating maintenance and production decisions in a hierarchical production planning environment, Computer and Operations Research, 26 (1999), 1059-1074.

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##### References:
 [1] K. H. Adjallah and K. P. Adzakpa, Minimizing maintenance cost involving flow-time and tardiness penalty with unequal release dates, In Press of Journal of Risk and Reliability, Part O, 221 (2007), 57-66. [2] K. P. Adzakpa, "Maintenance of Distributed Systems: Methods for Real-Time Decision-Making," PhD Thesis, University of Technology of Troyes, France, October 2004. [3] M. Berg, The marginal cost analysis and its application to repair and replacement policies, European Journal of Operational Research, 82 (1995), 214-224. [4] E. K. Burke and A. J. Smith, Hybrid evolutionary techniques for the maintenance scheduling problem, Ieee2000 Trans. on Power Systems, 15 (2000), 122-128. [5] C. Derman, G. J. Lieberman and S. M. Ross, On the optimal assignment of servers and a repairman, Journal of Applied Probability, 17 (1980), 577-581. doi: 10.2307/3213050. [6] C. Duron, M. A. Ould Louly and J.-M. Proth, The one machine scheduling problem: Insertion of a job under the real-time constraint, European Journal of Operational Research, 199 (2009), 695-701. doi: 10.1016/j.ejor.2007.09.048. [7] E. Frostig, Jointly optimal allocation of a repairman and optimal control of service rate for machine repairman problem, European Journal of Operational Research, 116 (1999), 274-280. [8] E. Frostig, Optimal allocation of machines to distinguishable repairmen in order to maximize some reward functions, Probability in the Engineering and Informational Sciences, 9 (1995), 633-645. doi: 10.1017/S0269964800004113. [9] E. Frostig, Optimal policies for machine repairmen problems, Journal of Applied Probability, 30 (1993), 703-715. doi: 10.2307/3214776. [10] M. Gopalakrishnan, S. Mohan and Z. He, A tabu search heuristic for preventive maintenance scheduling, Computers & Industrial Engineering, 40 (2001), 149-160. [11] G. Graves and C. Lee, Scheduling maintenance and semiresumable jobs on a single machine, Naval Research Logistics, 46 (1999), 845-863. [12] G. Koole, Optimal repairman assignment in two symmetric maintenance models, European Journal of Operational Research, 82 (1995), 295-301. [13] H. A. Le Thi and T. Pham Dinh, Solving a class of linearly constrained indefinite quadratic problems by DC algorithms, Journal of Global Optimization, 11 (1997), 253-285. doi: 10.1023/A:1008288411710. [14] H. A. Le Thi and T. Pham Dinh, A Continuous approach for globally solving linearly constrained quadratic zero-one programming problem, Optimization, 50 (2001), 93-120. doi: 10.1080/02331930108844555. [15] H. A. Le Thi and T. Pham Dinh, The DC(difference of convex functions) Programming and DCA revisited with DC models of real world non convex optimization problems, Annals of Operations Research, 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1. [16] H. A. Le Thi, T. Pham Dinh and M. Le Dung, Exact penalty in d.c. programming, Vietnam Journal of Mathematics, 27 (1999), 169-178. [17] H. A. Le Thi and T. Pham Dinh, A continuous approach for the concave cost supply problem via DC programming and DCA, Discrete Applied Mathematics, 156 (2008), 325-338. doi: 10.1016/j.dam.2007.03.024. [18] C. Lee and Z. Chen, Scheduling jobs and maintenance activities on parallel machines, Naval Research Logistics, 47 (2000), 145-165. doi: 10.1002/(SICI)1520-6750(200003)47:2<145::AID-NAV5>3.0.CO;2-3. [19] J. Lenstra, A. R. Kan and P. Brucker, Complexity of machine scheduling problems, Annals of Discrete Mathematics, 1 (1977), 343-362. [20] Z. Li, H. Liao, L-Y. Chan and A. Elsayed, Maintenance of continuously degrading systems considering random service delays, Proceedings of the 2008 Industrial Research Conference (IERC), Vancouver, British Columbia, Canada, May 17-21, 2008. [21] S-T. Lo, R-M. Chen, Y-M. Huang and C-L. Wu, Multiprocessor system scheduling with precedence and resource constraints using an enhanced ant colony system, Expert Systems with Applications, 34 (2008), 2071-2081. [22] T. Pham Dinh and H. A. Le Thi, Convex analysis approach to d.c programming: Theory, algorithms and applications, Acta Mathematica Vietnamica, dedicated to Professor Hoang Tuy on the occasion of his 70th birthday, 22 (1997), 289-355. [23] T. Pham Dinh and H. A. Le Thi, DC optimization algorihms for solving the trust region subproblem, SIAM J.Optimization, 8 (1997), 476-505. [24] K. Park, Optimal number of minimal repairs before replacement, IEEE Transactions on Reliability, R-28 (1979), 137-140. [25] X. Qi, T. Chen and F. Tu, Scheduling the maintenance on a single machine, Journal of the Operation Research Society, 50 (1999), 1071-1078. [26] R.T. Rockafellar, "Convex analysis Princeton University Press," Princeton University Press, Princeton, N. J. 1970 xviii+451 pp. [27] S. Seshadri, Determination of aggregate preventive maintenance programs using production schedules, Computers and Industrial Engineering, 14 (1988), 193-200. [28] W. Stadje and D. Zuckerman, Optimal repair policies with general degree of repair in two maintenance models, Operations Research Letters, 11 (1992), 77-80. doi: 10.1016/0167-6377(92)90036-3. [29] H. Tajima, J. Sugimoto, T. Funabashi and R. Yokoyama, Auction price-based thermal unit maintenance scheduling by reactive tabu search, Proc. 41st IEEE Int. UPEC'06 (Universities Power Engineering Conference), 2006. [30] L. Weinstein and C. Chung, Integrating maintenance and production decisions in a hierarchical production planning environment, Computer and Operations Research, 26 (1999), 1059-1074.
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