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May  2020, 16(3): 1415-1433. 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, 2020, 16 (3) : 1415-1433. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. Google Scholar [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.  Google Scholar [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.   Google Scholar [13] S. Kalpakam and G. Arivarignan, A continuous review perishable inventory model, Statistics, 19 (1988), 389-398.  doi: 10.1080/02331888808802112.  Google Scholar [14] S. Kalpakam and G. Arivarignan, Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145.  doi: 10.1007/BF01719709.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] Z. Lian and L. Liu, Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708.   Google Scholar [30] S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. Google Scholar [31] M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [35] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37.   Google Scholar [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.   Google Scholar [37] G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2 (1961), 421–430.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] G. Falin and J. G. Templeton, Retrial Queues (Vol. 75). CRC Press, 1997. Google Scholar [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.  Google Scholar [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.   Google Scholar [13] S. Kalpakam and G. Arivarignan, A continuous review perishable inventory model, Statistics, 19 (1988), 389-398.  doi: 10.1080/02331888808802112.  Google Scholar [14] S. Kalpakam and G. Arivarignan, Inventory system with random supply quantity, Operations Research Spektrum, 12 (1990), 139-145.  doi: 10.1007/BF01719709.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] Z. Lian and L. Liu, Continuous review perishable inventory systems: Models and heuristics, IIE Transactions, 33 (2001), 809-822.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] S. Nahmias, Perishable inventory theory: A review, Operations Research, 30 (1982), 680-708.   Google Scholar [30] S. Nahmias, Perishable Inventory Systems, Springer Science & Business Media, 2011. Google Scholar [31] M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, Courier Corporation, 1981.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [35] F. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of the Operational Research society, 42 (1991), 27-37.   Google Scholar [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.   Google Scholar [37] G. E. H. Reuter, Competition processes, In Proc. 4th Berkeley Symp. Math. Statist. Prob, 2 (1961), 421–430.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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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