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January  2015, 11(1): 83-104. doi: 10.3934/jimo.2015.11.83

## Optimization analysis of the machine repair problem with multiple vacations and working breakdowns

Received  April 2013 Revised  December 2013 Published  May 2014

This paper investigates the M/M/1 warm-standby machine repair problem with multiple vacations and working breakdowns. We first apply a matrix-analytic method to obtain the steady-state probabilities. Next, we construct the total expected profit per unit time and formulate an optimization problem to find the maximum profit. The particle swarm optimization (PSO) algorithm is implemented to determine the optimal number of warm standbys and two variable service rates simultaneously at the optimal maximum profit. We compare the searching results of the PSO algorithm with those of Genetic algorithm (GA) and Exhaustive Search Method (ESM) to ensure the superior searching quality of the PSO algorithm. Sensitivity analysis with numerical illustrations is also provided to improve the design quality of system engineers.
Citation: Cheng-Dar Liou. Optimization analysis of the machine repair problem with multiple vacations and working breakdowns. Journal of Industrial and Management Optimization, 2015, 11 (1) : 83-104. doi: 10.3934/jimo.2015.11.83
##### References:
 [1] M. Clerc, Particle Swarm Optimization, Translated from the 2005 French original, ISTE, London, 2006. doi: 10.1002/9780470612163. [2] B. T. Doshi, Queueing systems with vacations-a Survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327. [3] R. C. Eberhart and Y. Shi, Particle swarm optimization: Developments, applications and resources, in Proceedings of IEEE International Conference on Evolutionary Computation, Coex, Seoul, 1 (2001), 81-86. doi: 10.1109/CEC.2001.934374. [4] R. C. Eberhart and Y. Shi, Computational Intelligence: Concepts to Implementations, Morgan Kaufmann, Burlington, 2007. [5] S. W. Fuhrmann and R. B. Cooper, Stochastic decompositions in the M/G/1 queue with generalize vacations, Operations Research, 33 (1995), 1117-1129. doi: 10.1287/opre.33.5.1117. [6] M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, John-Wiley & Sons, Inc., New York, 2007. doi: 10.1002/9780470172261. [7] N. Gharbi and M. Ioualalen, Numerical investigation of finite-source multi server systems with different vacation policies, Journal of Computational and Applied Mathematics, 234 (2010), 625-635. doi: 10.1016/j.cam.2009.11.040. [8] W. J. Gray, P. P. Wang and M. Scott, A vacation queueing model with service breakdowns, Applied Mathematical Modelling, 24 (2000), 391-400. doi: 10.1016/S0307-904X(99)00048-7. [9] S. M. Gupta, N-policy queueing system with finite source and warm spares, Journal of Operational Research Society of India, 36 (1999), 189-217. [10] S. M. Gupta, Machine interference problem with warm spares, sever vacations and exhaustive service, Performance Evaluation, 29 (1997), 195-211. [11] L. Haque and M. J. Armstrong, A survey of the machine interference problem, European Journal of Operational Research, 179 (2007), 469-482. doi: 10.1016/j.ejor.2006.02.036. [12] J. H. Holland, Adaptation in Natural and Artificial Systems, An introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press, Ann Arbor, Mich., 1975. [13] M. Jain and R. S. Maheshwari, N-policy for a machine repair system with spares and reneging, Applied Mathematical Modelling, 28 (2004), 513-531. doi: 10.1016/j.apm.2003.10.013. [14] J. Jia and S. Wu, A replacement policy for a repairable system with its repairman having multiple vacations, Computers and Industrial Engineering, 57 (2009), 156-160. doi: 10.1016/j.cie.2008.11.003. [15] F. Karaesmen and S. M. Gupta, The finite GI/M/1 queue with server vacations, Journal of the Operational Research Society, 47 (1996), 817-828. doi: 10.2307/3010289. [16] J. C. Ke, The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns, Computers and Industrial Engineering, 44 (2003), 567-579. doi: 10.1016/S0360-8352(02)00235-8. [17] J. C. Ke, Vacation policies for machine interference problem with an unreliable server and state-dependent service rate, Journal of the Chinese Institute of Engineers, 23 (2006), 100-114. [18] J. C. Ke and C. H. Lin, Sensitivity analysis of machine repair problems in manufacturing systems with service interruptions, Applied Mathematical Modelling, 32 (2008), 2087-2105. doi: 10.1016/j.apm.2007.07.004. [19] J. C. Ke and C. H. Lin, A markov repairable system involving an imperfect service station with multiple vacations, Asia Pacific Journal of Operational Research, 22 (2005), 555-582. doi: 10.1142/S021759590500073X. [20] J. C. Ke, C. H. Lin, H. I. Huang and Z. G. Zhang, An algorithm analysis of multi-server vacation model with service interruptions, Computers and Industrial Engineering, 61 (2011), 1302-1308. doi: 10.1016/j.cie.2011.08.003. [21] J. C. Ke and K. H. Wang, Cost analysis of the M/M/R machine repair problem with balking, reneging, and server breakdowns, Journal of the Operational Research Society, 50 (1999), 275-282. doi: 10.2307/3010691. [22] J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares, Applied Mathematical Modelling, 31 (2007), 880-894. doi: 10.1016/j.apm.2006.02.009. [23] J. Kennedy and R. C. Eberhart, Particle swarm optimization, in Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ, (1995), 1942-1948. [24] J. Kennedy, R. C. Eberhart and Y. Shi, Swarm Intelligence, Morgan Kaufmann, CA, 2001. [25] B. K. Kumar and S. P. Madheswari, An M/M/2 queueing system with heterogeneous servers and multiple vacations, Mathematical and Computer Modelling, 41 (2005), 1415-1429. doi: 10.1016/j.mcm.2005.02.002. [26] Y. Li and J. Xu, A deteriorating system with its repairman having multiple vacations, Applied Mathematics and Computation, 217 (2011), 4980-4989. doi: 10.1016/j.amc.2010.11.048. [27] C. J. Lin and C. Y. Lee, Non-linear system control using a recurrent fuzzy neural network based on improved particle swarm optimization, International Journal of Systems and Science, 41 (2010), 381-395. [28] C. D. Liou, Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method'', Journal of Industrial and Management Optimization, 8 (2012), 727-732. doi: 10.3934/jimo.2012.8.727. [29] L. D. Servi and S. G. Finn, M/M/1 queue with working vacation (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3. [30] Y. Shi and R. C. Eberhart, Parameter selection in particle swarm optimization, Lecture Notes in Computer Science, 1447 (1998), 591-600. doi: 10.1007/BFb0040810. [31] K. E. Stecke and J. E. Aronson, Review of operator/machine interference models, International Journal of Production Research, 23 (1985), 129-151. doi: 10.1080/00207548508904696. [32] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research and Management Science, Springer-Verlag, New York, LLC 2006. [33] K. H. Wang, Profit analysis of the MRP with a single service station subject to breakdowns, Journal of the Operational Research Society, 41 (1990), 1153-1160. [34] K. H. Wang, W. L. Chen and D. Y. Yang, Optimal management of the machine repair problem with working vacation: Newton's method, Journal of Computational and Applied Mathematics, 233 (2009), 449-458. doi: 10.1016/j.cam.2009.07.043. [35] K. H. Wang and M. Y. Kuo, Profit analysis of the M/Ek/1 machine repair problem with a non-reliable service station, Computers and Industrial Engineering, 32 (1997), 587-594. doi: 10.1016/S0360-8352(96)00313-0. [36] K. H. Wang, C. D. Liou and Y. H. Lin, Comparative analysis of the machine repair problem with imperfect coverage and service pressure condition, Applied Mathematical Modelling, 37 (2013), 2870-2880. doi: 10.1016/j.apm.2012.06.024. [37] K. H. Wang, C. D. Liou and Y. L. Wang, Profit Optimization of the Multiple-Vacation Machine Repair Problem Using Particle Swarm Optimization, International Journal of Systems and Science, 2014, (Accepted DOI:10.1080/00207721.2012.757378). [38] L. Yuan, Reliability analysis for a k-out-of-n: G system with redundant dependency and repairmen having multiple vacations, Applied Mathematics and Computation, 218 (2012), 11959-11969. doi: 10.1016/j.amc.2012.06.006. [39] D. Yue, J. Yu and W. Yue, A Markovian queue with two heterogeneous servers and multiple vacations, Journal of Industrial and Management Optimization, 5 (2009), 453-465. doi: 10.3934/jimo.2009.5.453. [40] D. Yue, W. Yue, Z. Saffer and X. Chen, Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, Journal of Industrial and Management Optimization, 10 (2014), 89-112. doi: 10.3934/jimo.2014.10.89.

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##### References:
 [1] M. Clerc, Particle Swarm Optimization, Translated from the 2005 French original, ISTE, London, 2006. doi: 10.1002/9780470612163. [2] B. T. Doshi, Queueing systems with vacations-a Survey, Queueing Systems, 1 (1986), 29-66. doi: 10.1007/BF01149327. [3] R. C. Eberhart and Y. Shi, Particle swarm optimization: Developments, applications and resources, in Proceedings of IEEE International Conference on Evolutionary Computation, Coex, Seoul, 1 (2001), 81-86. doi: 10.1109/CEC.2001.934374. [4] R. C. Eberhart and Y. Shi, Computational Intelligence: Concepts to Implementations, Morgan Kaufmann, Burlington, 2007. [5] S. W. Fuhrmann and R. B. Cooper, Stochastic decompositions in the M/G/1 queue with generalize vacations, Operations Research, 33 (1995), 1117-1129. doi: 10.1287/opre.33.5.1117. [6] M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, John-Wiley & Sons, Inc., New York, 2007. doi: 10.1002/9780470172261. [7] N. Gharbi and M. Ioualalen, Numerical investigation of finite-source multi server systems with different vacation policies, Journal of Computational and Applied Mathematics, 234 (2010), 625-635. doi: 10.1016/j.cam.2009.11.040. [8] W. J. Gray, P. P. Wang and M. Scott, A vacation queueing model with service breakdowns, Applied Mathematical Modelling, 24 (2000), 391-400. doi: 10.1016/S0307-904X(99)00048-7. [9] S. M. Gupta, N-policy queueing system with finite source and warm spares, Journal of Operational Research Society of India, 36 (1999), 189-217. [10] S. M. Gupta, Machine interference problem with warm spares, sever vacations and exhaustive service, Performance Evaluation, 29 (1997), 195-211. [11] L. Haque and M. J. Armstrong, A survey of the machine interference problem, European Journal of Operational Research, 179 (2007), 469-482. doi: 10.1016/j.ejor.2006.02.036. [12] J. H. Holland, Adaptation in Natural and Artificial Systems, An introductory analysis with applications to biology, control, and artificial intelligence. University of Michigan Press, Ann Arbor, Mich., 1975. [13] M. Jain and R. S. Maheshwari, N-policy for a machine repair system with spares and reneging, Applied Mathematical Modelling, 28 (2004), 513-531. doi: 10.1016/j.apm.2003.10.013. [14] J. Jia and S. Wu, A replacement policy for a repairable system with its repairman having multiple vacations, Computers and Industrial Engineering, 57 (2009), 156-160. doi: 10.1016/j.cie.2008.11.003. [15] F. Karaesmen and S. M. Gupta, The finite GI/M/1 queue with server vacations, Journal of the Operational Research Society, 47 (1996), 817-828. doi: 10.2307/3010289. [16] J. C. Ke, The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns, Computers and Industrial Engineering, 44 (2003), 567-579. doi: 10.1016/S0360-8352(02)00235-8. [17] J. C. Ke, Vacation policies for machine interference problem with an unreliable server and state-dependent service rate, Journal of the Chinese Institute of Engineers, 23 (2006), 100-114. [18] J. C. Ke and C. H. Lin, Sensitivity analysis of machine repair problems in manufacturing systems with service interruptions, Applied Mathematical Modelling, 32 (2008), 2087-2105. doi: 10.1016/j.apm.2007.07.004. [19] J. C. Ke and C. H. Lin, A markov repairable system involving an imperfect service station with multiple vacations, Asia Pacific Journal of Operational Research, 22 (2005), 555-582. doi: 10.1142/S021759590500073X. [20] J. C. Ke, C. H. Lin, H. I. Huang and Z. G. Zhang, An algorithm analysis of multi-server vacation model with service interruptions, Computers and Industrial Engineering, 61 (2011), 1302-1308. doi: 10.1016/j.cie.2011.08.003. [21] J. C. Ke and K. H. Wang, Cost analysis of the M/M/R machine repair problem with balking, reneging, and server breakdowns, Journal of the Operational Research Society, 50 (1999), 275-282. doi: 10.2307/3010691. [22] J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares, Applied Mathematical Modelling, 31 (2007), 880-894. doi: 10.1016/j.apm.2006.02.009. [23] J. Kennedy and R. C. Eberhart, Particle swarm optimization, in Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ, (1995), 1942-1948. [24] J. Kennedy, R. C. Eberhart and Y. Shi, Swarm Intelligence, Morgan Kaufmann, CA, 2001. [25] B. K. Kumar and S. P. Madheswari, An M/M/2 queueing system with heterogeneous servers and multiple vacations, Mathematical and Computer Modelling, 41 (2005), 1415-1429. doi: 10.1016/j.mcm.2005.02.002. [26] Y. Li and J. Xu, A deteriorating system with its repairman having multiple vacations, Applied Mathematics and Computation, 217 (2011), 4980-4989. doi: 10.1016/j.amc.2010.11.048. [27] C. J. Lin and C. Y. Lee, Non-linear system control using a recurrent fuzzy neural network based on improved particle swarm optimization, International Journal of Systems and Science, 41 (2010), 381-395. [28] C. D. Liou, Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method'', Journal of Industrial and Management Optimization, 8 (2012), 727-732. doi: 10.3934/jimo.2012.8.727. [29] L. D. Servi and S. G. Finn, M/M/1 queue with working vacation (M/M/1/WV), Performance Evaluation, 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3. [30] Y. Shi and R. C. Eberhart, Parameter selection in particle swarm optimization, Lecture Notes in Computer Science, 1447 (1998), 591-600. doi: 10.1007/BFb0040810. [31] K. E. Stecke and J. E. Aronson, Review of operator/machine interference models, International Journal of Production Research, 23 (1985), 129-151. doi: 10.1080/00207548508904696. [32] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research and Management Science, Springer-Verlag, New York, LLC 2006. [33] K. H. Wang, Profit analysis of the MRP with a single service station subject to breakdowns, Journal of the Operational Research Society, 41 (1990), 1153-1160. [34] K. H. Wang, W. L. Chen and D. Y. Yang, Optimal management of the machine repair problem with working vacation: Newton's method, Journal of Computational and Applied Mathematics, 233 (2009), 449-458. doi: 10.1016/j.cam.2009.07.043. [35] K. H. Wang and M. Y. Kuo, Profit analysis of the M/Ek/1 machine repair problem with a non-reliable service station, Computers and Industrial Engineering, 32 (1997), 587-594. doi: 10.1016/S0360-8352(96)00313-0. [36] K. H. Wang, C. D. Liou and Y. H. Lin, Comparative analysis of the machine repair problem with imperfect coverage and service pressure condition, Applied Mathematical Modelling, 37 (2013), 2870-2880. doi: 10.1016/j.apm.2012.06.024. [37] K. H. Wang, C. D. Liou and Y. L. Wang, Profit Optimization of the Multiple-Vacation Machine Repair Problem Using Particle Swarm Optimization, International Journal of Systems and Science, 2014, (Accepted DOI:10.1080/00207721.2012.757378). [38] L. Yuan, Reliability analysis for a k-out-of-n: G system with redundant dependency and repairmen having multiple vacations, Applied Mathematics and Computation, 218 (2012), 11959-11969. doi: 10.1016/j.amc.2012.06.006. [39] D. Yue, J. Yu and W. Yue, A Markovian queue with two heterogeneous servers and multiple vacations, Journal of Industrial and Management Optimization, 5 (2009), 453-465. doi: 10.3934/jimo.2009.5.453. [40] D. Yue, W. Yue, Z. Saffer and X. Chen, Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, Journal of Industrial and Management Optimization, 10 (2014), 89-112. doi: 10.3934/jimo.2014.10.89.
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