Sensitivity Analysis of the Replacement Problem
Affiliation(s)
Academic Area of Engineering, Autonomous University of Hidalgo, Pachuca, México. Monterrey Institute of Technology-Mexico State Campus, Pachuca, México.
Academic Area of Engineering, Autonomous University of Hidalgo, Pachuca, México. Monterrey Institute of Technology-Mexico State Campus, Pachuca, México.
ABSTRACT
The replacement
problem can be modeled as a finite, irreducible, homogeneous Markov Chain. In
our proposal, we modeled the problem using a Markov decision process and then,
the instance is optimized using linear programming. Our goal is to analyze the
sensitivity and robustness of the optimal solution across the perturbation of
the optimal basis obtained from the simplex algorithm. The
perturbation 
can be
approximated by a given matrix H such
that 
. Some algebraic relations between the optimal
solution and the perturbed instance are obtained and discussed.
Cite this paper
Gress, E. , Lechuga, G. and Gress, N. (2014) Sensitivity Analysis of the Replacement Problem. Intelligent Control and Automation, 5, 46-59. doi: 10.4236/ica.2014.52006.
Gress, E. , Lechuga, G. and Gress, N. (2014) Sensitivity Analysis of the Replacement Problem. Intelligent Control and Automation, 5, 46-59. doi: 10.4236/ica.2014.52006.
References
[1] Childress, S. and Durango-Cohen. (2005) On Parallel Machine Replacement Problems with General Replacement Cost Functions and Stochastic Deterioration. Naval Research Logistics, 52, 409-419. http://dx.doi.org/10.1002/nav.20088 [2] Cooper, L. and Cooper, M. (1981) Introduction to Dynamic Programming. Pergamon Press, London. [3] Howard, R. (1960) Dynamic Programming and Markov Processes. The MIT Press, Cambridge. [4] Sernik, E. and Marcus, S. (1991) Optimal Cost and Policy for a Markovian Replacement Problem. Journal of Optimization Theory and Applications, 71, 105-126.
http://dx.doi.org/10.1007/BF00940042 [5] Kristensen, A. (1996) Dynamic Programming and Markov Processes. Dina Notat No. 49.
http://www.prodstyr.ihh.kvl.dk/pdf/notat49.pdf [6] Sethi, S., Sorger, G. and Zhou, X. (2000) Stability of Real-Time Lot Scheduling and Machine Replacement Policies with Quality Levels. IEEE Transactions on Automatic Control, 45, 2193-2196.
http://dx.doi.org/10.1109/9.887687 [7] Hadley G. (1964) Nonlinear and Dynamic Programming. Addison-Wesley, Massachussetts. [8] Fu, M., Hu, J. and Shi, L. (1993) An Application of Perturbation Analysis to a Replacement Problem in Maintenance Theory. Proceedings of the 1993 Winter Simulation Conference, 12-15 December 1993, 329-337. [9] Pérez, G., álvarez, M. and Garnica, J. (2006) Stochastic Linear Programming to Optimize some Stochastic Systems. WSEAS Transactions on Systems, 9, 2263-2267. [10] Sherwin, J. and Al-Najjar, B. (1999) Practical Models for Condition-Based Monitoring Inspection Intervals. Journal of Quality on Maintenance Engineering, 5, 203-221.
http://dx.doi.org/10.1108/13552519910282665 [11] Lewis, E. (1987) Introduction to Reliability Theory. Wiley, Singapore. [12] Cheng, T. (1992) Optimal Replacement of Ageing Equipment Using Geometric Programming. International Journal of Production Research, 3, 2151-2158.
http://dx.doi.org/10.1080/00207549208948142 [13] Karabakal, N., Bean, C. and Lohman, J. (2000) Solving Large Replacement Problems with Budget Constraints. Engineering Economy, 5, 290-308.
http://dx.doi.org/10.1080/00137910008967554 [14] Dohi, T., Ashioka, A., Kaio, N. and Osaki, S. (2004) A Simulation Study on the Discounted Cost Distribution under Age Replacement Policy. Journal of Management and Engineering Integration, 3, 134-139. [15] Bellman, R. (1955) Equipment Replacement Policy. Journal of the Society for Industrial and Applied Mathematics, 3, 133-136.
http://dx.doi.org/10.1137/0103011 [16] White, R. (1969) Dynamic Programming. Holden Hay, San Francisco. [17] Davidson, D. (1970) An Overhaul Policy for Deteriorating Equipment. In: Jardine, A.K.S., Ed., Operational Research in Maintenance, Manchester University Press, Manchester, 72-99. [18] Walker, J. (1992) Graphical Analysis for Machine Replacement: A Case Study. International Journal of Operations and Production Management, 14, 54-63.
http://dx.doi.org/10.1108/01443579410067252 [19] Bertsekas, D. (2000) Dynamic Programming and Optimal Control. Athena Scientific, Belmont. [20] Plá, L., Pomar, C. and Pomar, J.A (2004) Markov Sow Model Representing the Productive Lifespan of Herd Sows. Agricultural Systems, 76, 253-272.
http://dx.doi.org/10.1016/S0308-521X(02)00102-6 [21] Nielsen, L. and Kristensen, A. (2006) Finding the K Best Policies in a Finite-Horizon Markov Decision Process. European Journal of Operational Research, 175, 1164-1179.
http://dx.doi.org/10.1016/j.ejor.2005.06.011 [22] Nielsen, L., Jorgensen, E., Kristensen, A. and Ostergaard, S. (2009) Optimal Replacement Policies for Dairy Cows Based on Daily Yield Measurements. Journal of Dairy Science, 93, 75-92.
http://dx.doi.org/10.3168/jds.2009-2209 [23] Hillier, F. and Lieberman, J. (2002) Introduction to Operations Research. Mc Graw Hill, New York. [24] Meyer, C. (1994) Sensitivity of the Stationary Distribution of a Markov Chain. SIAM Journal on Matrix Analysis and Applications, 15, 715-728.
http://dx.doi.org/10.1137/S0895479892228900 [25] Abbad, M. and Filar, J. (1995) Algorithms for Singularly Perturbed Markov Control Problems: A Survey. Control and Dynamic Systems, 73, 257-288. [26] Feinberg, E. (2000) Constrained Discounted MarkovDecision Processes and Hamiltonian Cycles. Mathematics of Operations Research, 25, 130-140.
http://dx.doi.org/10.1137/S0895479892228900 [27] Schrijner, P. and van Doorn, E. (2001) The Deviation Matrix of a Continuous-Time Markov Chain. Probability in the Engineering and Informational Sciences, 16, 351-366. [28] Ross, S. (1992) Applied Probability Models with Optimization Applications. Holden Day, San Francisco.
[1] Childress, S. and Durango-Cohen. (2005) On Parallel Machine Replacement Problems with General Replacement Cost Functions and Stochastic Deterioration. Naval Research Logistics, 52, 409-419. http://dx.doi.org/10.1002/nav.20088 [2] Cooper, L. and Cooper, M. (1981) Introduction to Dynamic Programming. Pergamon Press, London. [3] Howard, R. (1960) Dynamic Programming and Markov Processes. The MIT Press, Cambridge. [4] Sernik, E. and Marcus, S. (1991) Optimal Cost and Policy for a Markovian Replacement Problem. Journal of Optimization Theory and Applications, 71, 105-126.
http://dx.doi.org/10.1007/BF00940042 [5] Kristensen, A. (1996) Dynamic Programming and Markov Processes. Dina Notat No. 49.
http://www.prodstyr.ihh.kvl.dk/pdf/notat49.pdf [6] Sethi, S., Sorger, G. and Zhou, X. (2000) Stability of Real-Time Lot Scheduling and Machine Replacement Policies with Quality Levels. IEEE Transactions on Automatic Control, 45, 2193-2196.
http://dx.doi.org/10.1109/9.887687 [7] Hadley G. (1964) Nonlinear and Dynamic Programming. Addison-Wesley, Massachussetts. [8] Fu, M., Hu, J. and Shi, L. (1993) An Application of Perturbation Analysis to a Replacement Problem in Maintenance Theory. Proceedings of the 1993 Winter Simulation Conference, 12-15 December 1993, 329-337. [9] Pérez, G., álvarez, M. and Garnica, J. (2006) Stochastic Linear Programming to Optimize some Stochastic Systems. WSEAS Transactions on Systems, 9, 2263-2267. [10] Sherwin, J. and Al-Najjar, B. (1999) Practical Models for Condition-Based Monitoring Inspection Intervals. Journal of Quality on Maintenance Engineering, 5, 203-221.
http://dx.doi.org/10.1108/13552519910282665 [11] Lewis, E. (1987) Introduction to Reliability Theory. Wiley, Singapore. [12] Cheng, T. (1992) Optimal Replacement of Ageing Equipment Using Geometric Programming. International Journal of Production Research, 3, 2151-2158.
http://dx.doi.org/10.1080/00207549208948142 [13] Karabakal, N., Bean, C. and Lohman, J. (2000) Solving Large Replacement Problems with Budget Constraints. Engineering Economy, 5, 290-308.
http://dx.doi.org/10.1080/00137910008967554 [14] Dohi, T., Ashioka, A., Kaio, N. and Osaki, S. (2004) A Simulation Study on the Discounted Cost Distribution under Age Replacement Policy. Journal of Management and Engineering Integration, 3, 134-139. [15] Bellman, R. (1955) Equipment Replacement Policy. Journal of the Society for Industrial and Applied Mathematics, 3, 133-136.
http://dx.doi.org/10.1137/0103011 [16] White, R. (1969) Dynamic Programming. Holden Hay, San Francisco. [17] Davidson, D. (1970) An Overhaul Policy for Deteriorating Equipment. In: Jardine, A.K.S., Ed., Operational Research in Maintenance, Manchester University Press, Manchester, 72-99. [18] Walker, J. (1992) Graphical Analysis for Machine Replacement: A Case Study. International Journal of Operations and Production Management, 14, 54-63.
http://dx.doi.org/10.1108/01443579410067252 [19] Bertsekas, D. (2000) Dynamic Programming and Optimal Control. Athena Scientific, Belmont. [20] Plá, L., Pomar, C. and Pomar, J.A (2004) Markov Sow Model Representing the Productive Lifespan of Herd Sows. Agricultural Systems, 76, 253-272.
http://dx.doi.org/10.1016/S0308-521X(02)00102-6 [21] Nielsen, L. and Kristensen, A. (2006) Finding the K Best Policies in a Finite-Horizon Markov Decision Process. European Journal of Operational Research, 175, 1164-1179.
http://dx.doi.org/10.1016/j.ejor.2005.06.011 [22] Nielsen, L., Jorgensen, E., Kristensen, A. and Ostergaard, S. (2009) Optimal Replacement Policies for Dairy Cows Based on Daily Yield Measurements. Journal of Dairy Science, 93, 75-92.
http://dx.doi.org/10.3168/jds.2009-2209 [23] Hillier, F. and Lieberman, J. (2002) Introduction to Operations Research. Mc Graw Hill, New York. [24] Meyer, C. (1994) Sensitivity of the Stationary Distribution of a Markov Chain. SIAM Journal on Matrix Analysis and Applications, 15, 715-728.
http://dx.doi.org/10.1137/S0895479892228900 [25] Abbad, M. and Filar, J. (1995) Algorithms for Singularly Perturbed Markov Control Problems: A Survey. Control and Dynamic Systems, 73, 257-288. [26] Feinberg, E. (2000) Constrained Discounted MarkovDecision Processes and Hamiltonian Cycles. Mathematics of Operations Research, 25, 130-140.
http://dx.doi.org/10.1137/S0895479892228900 [27] Schrijner, P. and van Doorn, E. (2001) The Deviation Matrix of a Continuous-Time Markov Chain. Probability in the Engineering and Informational Sciences, 16, 351-366. [28] Ross, S. (1992) Applied Probability Models with Optimization Applications. Holden Day, San Francisco.