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doi: 10.3934/jimo.2019009

## A nonhomogeneous quasi-birth-death process approach for an $(s, S)$ policy for a perishable inventory system with retrial demands

 Department of Industrial Engineering, Konkuk University, Seoul, Korea

Received  March 2018 Revised  October 2018 Published  March 2019

In this paper, an $(s, S)$ continuous inventory model with perishable items and retrial demands is proposed. In addition, replenishment lead times that are independent and identically distributed according to phase-type distribution are implemented. The proposed system is modeled as a three-dimensional Markov process using a level-dependent quasi-birth-death (QBD) process. The ergodicity of the modeled Markov system is demonstrated and the best method for efficiently approximating the steady-state distribution at the inventory level is determined. This paper also provides performance measure formulas based on the steady-state distribution of the proposed approximation method. Furthermore, in order to minimize the system cost, the optimum values of $s$ and $S$ are determined numerically and sensitivity analysis is performed on the main parameters.

Citation: Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $(s, S)$ policy for a perishable inventory system with retrial demands. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019009
##### References:
 [1] M. Alizadeh, H. Eskandari and S. M. Sajadifar, A modified $(S-1, S)$ inventory system for deteriorating items with Poisson demand and non-zero lead time, Applied Mathematical Modelling, 38 (2014), 699-711. doi: 10.1016/j.apm.2013.07.014. [2] W. J. Anderson, Continuous-time Markov Chains: An Applications-oriented Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0. [3] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 51 (2010), 1071-1081. doi: 10.1016/j.mcm.2009.12.011. [4] J. Artalejo and G. Falin, Standard and retrial queueing systems: A comparative analysis, Revista Matematica Complutense, 15 (2002), 101-129. doi: 10.5209/rev_REMA.2002.v15.n1.16950. [5] J. R. Artalejo, A. Krishnamoorthy and M. J. Lopez-Herrero, Numerical analysis of $(s, S)$ inventory systems with repeated attempts, Annals of Operations Research, 141 (2006), 67-83. doi: 10.1007/s10479-006-5294-8. [6] J. R. Artalejo and M. J. Lopez-Herrero, A simulation study of a discrete-time multiserver retrial queue with finite population, Journal of Statistical Planning and Inference, 137 (2007), 2536-2542. doi: 10.1016/j.jspi.2006.04.018. [7] O. Baron, O. Berman and D. Perry, Continuous review inventory models for perishable items ordered in batches, Mathematical Methods of Operations Research, 72 (2010), 217-247. doi: 10.1007/s00186-010-0318-1. [8] L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic Models, 11 (1995), 497-525. doi: 10.1080/15326349508807357. [9] B. D. Choi and B. Kim, Non-ergodicity criteria for denumerable continuous time Markov processes, Operations Research Letters, 32 (2004), 574-580. doi: 10.1016/j.orl.2004.03.001. [10] G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. [11] A. Gómez-Corral, A bibliographical guide to the analysis of retrial queues through matrix analytic techniques, Annals of Operations Research, 141 (2006), 163-191. doi: 10.1007/s10479-006-5298-4. [12] Ü. Gürler and B. Y. Özkaya, Analysis of the $(s, S)$ policy for perishables with a random shelf life, IIE Transactions, 40 (2008), 759-781. [13] S. Kalpakam and G. Arivarignan, A continuous review perishable inventory model, Statistics, 19 (1988), 389-398. doi: 10.1080/02331888808802112. [14] S. Kalpakam and G. Arivarignan, Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145. doi: 10.1007/BF01719709. [15] S. Kalpakam and K. P. Sapna, Continuous review $(s, S)$ inventory system with random lifetimes and positive leadtimes, Operations Research Letters, 16 (1994), 115-119. doi: 10.1016/0167-6377(94)90066-3. [16] S. Kalpakam and K. P. Sapna, $(S-1, S)$ Perishable systems with stochastic leadtimes, Mathematical and Computer Modelling, 21 (1995), 95-104. doi: 10.1016/0895-7177(95)00026-X. [17] T. Karthick, B. Sivakumar and G. Arivarignan, An inventory system with two types of customers and retrial demands, International Journal of Systems Science: Operations & Logistics, 2 (2015), 90-112. [18] C. Kouki, E. Sahin, Z. Jemai and Y. Dallery, Periodic Review Inventory Policy for Perishables with Random Lifetime, In Eighth International Conference of Modeling and Simulation, 2010. [19] A. Krishnamoorthy and P. V. Ushakumari, Reliability of a k-out-of-n system with repair and retrial of failed units, Top, 7 (1999), 293-304. doi: 10.1007/BF02564728. [20] S. Kumaraswamy and E. Sankarasubramanian, A continuous review of $(s, S)$ inventory systems in which depletion is due to demand and failure of units, Journal of the Operational Research Society, 32 (1981), 997-1001. [21] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, 1999. doi: 10.1137/1.9780898719734. [22] A. S. Lawrence, B. Sivakumar and G. Arivarignan, A perishable inventory system with service facility and finite source, Applied Mathematical Modelling, 37 (2013), 4771-4786. doi: 10.1016/j.apm.2012.09.018. [23] P. Vijaya Laxmi and M. L. Soujanya, Perishable inventory system with service interruptions, retrial demands and negative customers, Applied Mathematics and Computation, 262 (2015), 102-110. doi: 10.1016/j.amc.2015.04.013. [24] Z. Lian and L. Liu, Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822. [25] L. Liu, (s, S) Continuous Review Models for Inventory with Random Lifetimes, Operations Research Letters, 9 (1990), 161-167. doi: 10.1016/0167-6377(90)90014-V. [26] L. Liu and D. H. Shi, An $(s, S)$ model for inventory with exponential lifetimes and renewal demands, Naval Research Logistics, 46 (1999), 39-56. doi: 10.1002/(SICI)1520-6750(199902)46:1<39::AID-NAV3>3.0.CO;2-G. [27] L. Liu and T. Yang, An $(s, S)$ random lifetime inventory model with a positive lead time, European Journal of Operational Research, 113 (1999), 52-63. doi: 10.1016/0167-6377(90)90014-V. [28] E. Mohebbi and M. J. Posner, A continuous review inventory system with lost sales and variable lead time, Naval Research Logistics, 45 (1998), 259-278. doi: 10.1002/(SICI)1520-6750(199804)45:3<259::AID-NAV2>3.0.CO;2-6. [29] S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708. [30] S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. [31] M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981. [32] F. Olsson and P. Tydesjö, Inventory problems with perishable items: Fixed lifetimes and backlogging, European Journal of Operational Research, 202 (2010), 131-137. doi: 10.1016/j.ejor.2009.05.010. [33] C. Periyasamy, A finite source perishable inventory system with retrial demands and multiple server vacation, International Journal of Engineering Research and Technology, 2 (2013), 3802-3815. [34] G. P. Prastacos, Blood inventory management: An overview of theory and practice, Management Science, 30 (1984), 777-800. doi: 10.1287/mnsc.30.7.777. [35] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37. [36] N. Ravichandran, Stochastic analysis of a continuous review perishable inventory system with positive lead time and Poisson demand, European Journal of Operational Research, 84 (1995), 444-457. [37] G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2 (1961), 421–430. [38] C. P. Schmidt and S. Nahmias, $(S-1, S)$ Policies for perishable inventory, Management Science, 31 (1985), 719-728. doi: 10.1287/mnsc.31.6.719. [39] L. I. Sennott, P. A. Humblet and R. L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Operations Research, 31 (1983), 783-789. doi: 10.1287/opre.31.4.783. [40] B. Sivakumar, Two-commodity inventory system with retrial demand, European Journal of Operational Research, 187 (2008), 70-83. doi: 10.1016/j.ejor.2007.02.036. [41] B. Sivakumar, A perishable inventory system with retrial demands and a finite population, Journal of Computational and Applied Mathematics, 224 (2009), 29-38. doi: 10.1016/j.cam.2008.03.041. [42] R. L. Tweedie, Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, Journal of Applied Probability, 18 (1981), 122-130. doi: 10.2307/3213172. [43] P. V. Ushakumari, On $(s, S)$ inventory system with random lead time and repeated demands, International Journal of Stochastic Analysis, 2006 (2006), Art. ID 81508, 22 pp. doi: 10.1155/JAMSA/2006/81508. [44] H. J. Weiss, Optimal ordering policies for continuous review perishable inventory models, Operations Research, 28 (1980), 365-374. doi: 10.1287/opre.28.2.365.

show all references

##### References:
 [1] M. Alizadeh, H. Eskandari and S. M. Sajadifar, A modified $(S-1, S)$ inventory system for deteriorating items with Poisson demand and non-zero lead time, Applied Mathematical Modelling, 38 (2014), 699-711. doi: 10.1016/j.apm.2013.07.014. [2] W. J. Anderson, Continuous-time Markov Chains: An Applications-oriented Approach, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-3038-0. [3] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 51 (2010), 1071-1081. doi: 10.1016/j.mcm.2009.12.011. [4] J. Artalejo and G. Falin, Standard and retrial queueing systems: A comparative analysis, Revista Matematica Complutense, 15 (2002), 101-129. doi: 10.5209/rev_REMA.2002.v15.n1.16950. [5] J. R. Artalejo, A. Krishnamoorthy and M. J. Lopez-Herrero, Numerical analysis of $(s, S)$ inventory systems with repeated attempts, Annals of Operations Research, 141 (2006), 67-83. doi: 10.1007/s10479-006-5294-8. [6] J. R. Artalejo and M. J. Lopez-Herrero, A simulation study of a discrete-time multiserver retrial queue with finite population, Journal of Statistical Planning and Inference, 137 (2007), 2536-2542. doi: 10.1016/j.jspi.2006.04.018. [7] O. Baron, O. Berman and D. Perry, Continuous review inventory models for perishable items ordered in batches, Mathematical Methods of Operations Research, 72 (2010), 217-247. doi: 10.1007/s00186-010-0318-1. [8] L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic Models, 11 (1995), 497-525. doi: 10.1080/15326349508807357. [9] B. D. Choi and B. Kim, Non-ergodicity criteria for denumerable continuous time Markov processes, Operations Research Letters, 32 (2004), 574-580. doi: 10.1016/j.orl.2004.03.001. [10] G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. [11] A. Gómez-Corral, A bibliographical guide to the analysis of retrial queues through matrix analytic techniques, Annals of Operations Research, 141 (2006), 163-191. doi: 10.1007/s10479-006-5298-4. [12] Ü. Gürler and B. Y. Özkaya, Analysis of the $(s, S)$ policy for perishables with a random shelf life, IIE Transactions, 40 (2008), 759-781. [13] S. Kalpakam and G. Arivarignan, A continuous review perishable inventory model, Statistics, 19 (1988), 389-398. doi: 10.1080/02331888808802112. [14] S. Kalpakam and G. Arivarignan, Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145. doi: 10.1007/BF01719709. [15] S. Kalpakam and K. P. Sapna, Continuous review $(s, S)$ inventory system with random lifetimes and positive leadtimes, Operations Research Letters, 16 (1994), 115-119. doi: 10.1016/0167-6377(94)90066-3. [16] S. Kalpakam and K. P. Sapna, $(S-1, S)$ Perishable systems with stochastic leadtimes, Mathematical and Computer Modelling, 21 (1995), 95-104. doi: 10.1016/0895-7177(95)00026-X. [17] T. Karthick, B. Sivakumar and G. Arivarignan, An inventory system with two types of customers and retrial demands, International Journal of Systems Science: Operations & Logistics, 2 (2015), 90-112. [18] C. Kouki, E. Sahin, Z. Jemai and Y. Dallery, Periodic Review Inventory Policy for Perishables with Random Lifetime, In Eighth International Conference of Modeling and Simulation, 2010. [19] A. Krishnamoorthy and P. V. Ushakumari, Reliability of a k-out-of-n system with repair and retrial of failed units, Top, 7 (1999), 293-304. doi: 10.1007/BF02564728. [20] S. Kumaraswamy and E. Sankarasubramanian, A continuous review of $(s, S)$ inventory systems in which depletion is due to demand and failure of units, Journal of the Operational Research Society, 32 (1981), 997-1001. [21] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, 1999. doi: 10.1137/1.9780898719734. [22] A. S. Lawrence, B. Sivakumar and G. Arivarignan, A perishable inventory system with service facility and finite source, Applied Mathematical Modelling, 37 (2013), 4771-4786. doi: 10.1016/j.apm.2012.09.018. [23] P. Vijaya Laxmi and M. L. Soujanya, Perishable inventory system with service interruptions, retrial demands and negative customers, Applied Mathematics and Computation, 262 (2015), 102-110. doi: 10.1016/j.amc.2015.04.013. [24] Z. Lian and L. Liu, Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822. [25] L. Liu, (s, S) Continuous Review Models for Inventory with Random Lifetimes, Operations Research Letters, 9 (1990), 161-167. doi: 10.1016/0167-6377(90)90014-V. [26] L. Liu and D. H. Shi, An $(s, S)$ model for inventory with exponential lifetimes and renewal demands, Naval Research Logistics, 46 (1999), 39-56. doi: 10.1002/(SICI)1520-6750(199902)46:1<39::AID-NAV3>3.0.CO;2-G. [27] L. Liu and T. Yang, An $(s, S)$ random lifetime inventory model with a positive lead time, European Journal of Operational Research, 113 (1999), 52-63. doi: 10.1016/0167-6377(90)90014-V. [28] E. Mohebbi and M. J. Posner, A continuous review inventory system with lost sales and variable lead time, Naval Research Logistics, 45 (1998), 259-278. doi: 10.1002/(SICI)1520-6750(199804)45:3<259::AID-NAV2>3.0.CO;2-6. [29] S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708. [30] S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. [31] M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981. [32] F. Olsson and P. Tydesjö, Inventory problems with perishable items: Fixed lifetimes and backlogging, European Journal of Operational Research, 202 (2010), 131-137. doi: 10.1016/j.ejor.2009.05.010. [33] C. Periyasamy, A finite source perishable inventory system with retrial demands and multiple server vacation, International Journal of Engineering Research and Technology, 2 (2013), 3802-3815. [34] G. P. Prastacos, Blood inventory management: An overview of theory and practice, Management Science, 30 (1984), 777-800. doi: 10.1287/mnsc.30.7.777. [35] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37. [36] N. Ravichandran, Stochastic analysis of a continuous review perishable inventory system with positive lead time and Poisson demand, European Journal of Operational Research, 84 (1995), 444-457. [37] G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2 (1961), 421–430. [38] C. P. Schmidt and S. Nahmias, $(S-1, S)$ Policies for perishable inventory, Management Science, 31 (1985), 719-728. doi: 10.1287/mnsc.31.6.719. [39] L. I. Sennott, P. A. Humblet and R. L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Operations Research, 31 (1983), 783-789. doi: 10.1287/opre.31.4.783. [40] B. Sivakumar, Two-commodity inventory system with retrial demand, European Journal of Operational Research, 187 (2008), 70-83. doi: 10.1016/j.ejor.2007.02.036. [41] B. Sivakumar, A perishable inventory system with retrial demands and a finite population, Journal of Computational and Applied Mathematics, 224 (2009), 29-38. doi: 10.1016/j.cam.2008.03.041. [42] R. L. Tweedie, Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, Journal of Applied Probability, 18 (1981), 122-130. doi: 10.2307/3213172. [43] P. V. Ushakumari, On $(s, S)$ inventory system with random lead time and repeated demands, International Journal of Stochastic Analysis, 2006 (2006), Art. ID 81508, 22 pp. doi: 10.1155/JAMSA/2006/81508. [44] H. J. Weiss, Optimal ordering policies for continuous review perishable inventory models, Operations Research, 28 (1980), 365-374. doi: 10.1287/opre.28.2.365.
Inventory Model
Contour Plot of TCR
The effect of $\lambda$
The effect of $\mu$
Total Cost Rate(TCR)
 $S \diagdown s$ 1 2 3 4 5 6 7 8 9 10 15 367.4 363.25 361.55 362.46 366.06 372.45 381.92 394.31 409.75 429.18 16 366.82 362.49 360.49 360.94 363.82 369.16 377.05 387.83 400.29 415.71 17 366.83 362.4 360.21 360.34 362.72 367.3 374.09 383.24 394.32 407.17 18 367.32 362.86 360.56 360.46 362.49 366.53 372.52 380.51 390.68 401.73 19 368.23 363.77 361.42 361.18 362.95 366.59 372 379.14 388.1 398.4 20 369.49 365.07 362.7 362.38 363.97 367.32 372.29 378.8 386.87 396.6 21 371.06 366.69 364.35 363.98 365.45 368.58 373.22 379.25 386.65 395.44 22 372.88 368.6 366.29 365.91 367.3 370.27 374.65 380.32 387.2 395.29 23 374.93 370.74 368.49 368.12 369.45 372.31 376.5 381.89 388.37 395.91 24 377.18 373.09 370.91 370.56 371.87 374.65 378.69 383.85 390.02 397.14 25 379.6 375.62 373.52 373.21 374.51 377.23 381.16 386.14 392.06 398.84 26 382.17 378.31 376.29 376.03 377.34 380.02 383.86 388.71 394.42 400.93 27 384.88 381.13 379.2 379 380.33 382.99 386.76 391.5 397.05 403.34 28 387.71 384.07 382.24 382.1 383.46 386.1 389.83 394.48 399.91 406.01 29 390.64 387.12 385.38 385.31 386.7 389.35 393.05 397.63 402.94 408.9 30 393.66 390.27 388.62 388.62 390.05 392.71 396.39 400.91 406.14 411.98
 $S \diagdown s$ 1 2 3 4 5 6 7 8 9 10 15 367.4 363.25 361.55 362.46 366.06 372.45 381.92 394.31 409.75 429.18 16 366.82 362.49 360.49 360.94 363.82 369.16 377.05 387.83 400.29 415.71 17 366.83 362.4 360.21 360.34 362.72 367.3 374.09 383.24 394.32 407.17 18 367.32 362.86 360.56 360.46 362.49 366.53 372.52 380.51 390.68 401.73 19 368.23 363.77 361.42 361.18 362.95 366.59 372 379.14 388.1 398.4 20 369.49 365.07 362.7 362.38 363.97 367.32 372.29 378.8 386.87 396.6 21 371.06 366.69 364.35 363.98 365.45 368.58 373.22 379.25 386.65 395.44 22 372.88 368.6 366.29 365.91 367.3 370.27 374.65 380.32 387.2 395.29 23 374.93 370.74 368.49 368.12 369.45 372.31 376.5 381.89 388.37 395.91 24 377.18 373.09 370.91 370.56 371.87 374.65 378.69 383.85 390.02 397.14 25 379.6 375.62 373.52 373.21 374.51 377.23 381.16 386.14 392.06 398.84 26 382.17 378.31 376.29 376.03 377.34 380.02 383.86 388.71 394.42 400.93 27 384.88 381.13 379.2 379 380.33 382.99 386.76 391.5 397.05 403.34 28 387.71 384.07 382.24 382.1 383.46 386.1 389.83 394.48 399.91 406.01 29 390.64 387.12 385.38 385.31 386.7 389.35 393.05 397.63 402.94 408.9 30 393.66 390.27 388.62 388.62 390.05 392.71 396.39 400.91 406.14 411.98
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