# American Institute of Mathematical Sciences

April  2016, 12(2): 609-624. doi: 10.3934/jimo.2016.12.609

## An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure

 1 Department of Industrial Engineering, Konkuk University, 120 Neungdong-ro, Gwangjin-gu, Seoul, 143-701, South Korea 2 Department of Industrial Engineering, Chosun University, 309 Pilmun-daero, Dong-gu, Gwangju, 501-759, South Korea, South Korea

Received  July 2014 Revised  March 2015 Published  June 2015

Inventory models are widely used in a variety of real-world applications. In particular, inventory systems with perishable items have received a significant amount of attention. We consider an $(s,S)$ continuous inventory model with perishable items, impatient customers, and random lead times. Two characteristic behaviors of impatient customers are balking and reneging. Balking is when a customer departs the system if the item they desire is unavailable. Reneging occurs when a waiting customer leaves the system if their demand is not met within a set period of time. The proposed system is modeled as a two-dimensional Markov process with level-dependent $G/M/1$-type structure. We also consider independent and identically distributed replenishment lead times that follow a phase-type distribution. We find an efficient approximation method for the joint stationary distribution of the number of items in the system, and provide formulas for several performance measures. Moreover, to minimize system costs, we find the optimal values of $s$ and $S$ numerically and perform a sensitivity analysis on key parameters.
Citation: Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial and Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609
##### References:
 [1] E. Altman and A. A. Borovkov, On the stability of retrial queues, Queueing Syst., 26 (1997), 343-363. doi: 10.1023/A:1019193527040. [2] S. Asmussen, Applied Probability and Queues, John Wiley & Sons, 1987. [3] A. Brandt and M. Brandt, On the M(n)/M(n)/s queue with impatient calls, Perform. Eval., 35 (1999), 1-18. [4] A. Brandt and M. Brandt, Asymptotic results and a markovian approximation for the M(n)/M(n)/s+GI system, Queueing Syst., 41 (2002), 73-94. doi: 10.1023/A:1015781818360. [5] L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Commun. Statist. - Stochastic Models, 11 (1995), 497-525. doi: 10.1080/15326349508807357. [6] S. Charkravarthy and J. Daniel, A markovian inventory system with random shelf time and back orders, Computers and Industrial Engineering, 47 (2004), 315-337. [7] G. I. Falin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls, Adv. Appl. Prob., 16 (1984), 447-448. doi: 10.2307/1427079. [8] Qi-Ming He, E. M. Jewkes and J. Buzacott, The value of information used in inventory control of a make-to-order inventory-production system, IIE Transactions, 34 (2002), 999-1013. doi: 10.1080/07408170208928929. [9] S. Ioannidis, O. Jouini, A. A. Economopoulos and V. S. Kouikoglou, Control policies for single-stage production systems with perishable inventory and customer impatience, Annals of Operations Research, (2012), 1-24. doi: 10.1007/s10479-012-1058-9. [10] 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. [11] S. Kalpakam and K. P. Sapna, $(S-1,S)$ perishable systems with stochastic lead times, Mathematical and Computer Modelling, 21 (1995), 95-104. doi: 10.1016/0895-7177(95)00026-X. [12] I. Karaesmen, A. Scheller-Wolf and B. Deniz, Managing perishable and aging invetories: Review and future research directions, In Planning Production and Inventories in the Extended Enterprise, Springer, (2011), 393-438. [13] A. Krishnamoorthy, K. P. Jose and V. C. Narayanan, Numerical investigation of a PH/PH/1 inventory system with positive service time and shortage, Neural Parallel & Scientific Comp., 16 (2008), 579-591. [14] 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 Operational Research Society, 32 (1981), 997-1001. [15] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM series on statistics and applied probability, 1999. doi: 10.1137/1.9780898719734. [16] L. Liu, $(s,S)$ continous review models for inventory with random lifetimes, Operations Research Letters, 9 (1990), 161-167. doi: 10.1016/0167-6377(90)90014-V. [17] L. Liu and T. Yang, An $(s,S)$ random lifetime inventory model with a positive lead time, European Journal of Operational Research, 112 (1999), 52-63. doi: 10.1016/S0377-2217(97)00426-8. [18] S. Nahmias, Perishable inventory theory: A review, Operational Research, 30 (1982), 680-708. doi: 10.1287/opre.30.4.680. [19] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, MD., 1981. [20] M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, Inc., 1989. [21] D. Perry and W. Stadje, Perishable inventory systems with impatient demands, Math. Meth. of OR, 50, (1999), 77-90. [22] G. P. Prestacos, Blood inventory management, Management Science, 30 (1984), 777-801. [23] M. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of Operational Research Society, 42 (1991), 27-37. [24] N. Ravichandran, Stochastic analysis of a continous review perishable inventory system with positive lead time and Poisson demand, European Journal of Operational Research, 84 (1995), 444-457. [25] 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. [26] A. R. Ward and P. W. Glynn, A diffusion approximation for a markovian queue with reneging, Queueing Syst., 43 (2003), 103-128. doi: 10.1023/A:1021804515162. [27] S. Zeltyn and A. Mandelbaum, Call centers with impatient customers: Many-server asymptotics of the M/M/n + G queue, Queueing Syst., 51 (2005), 361-402. doi: 10.1007/s11134-005-3699-8.

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##### References:
 [1] E. Altman and A. A. Borovkov, On the stability of retrial queues, Queueing Syst., 26 (1997), 343-363. doi: 10.1023/A:1019193527040. [2] S. Asmussen, Applied Probability and Queues, John Wiley & Sons, 1987. [3] A. Brandt and M. Brandt, On the M(n)/M(n)/s queue with impatient calls, Perform. Eval., 35 (1999), 1-18. [4] A. Brandt and M. Brandt, Asymptotic results and a markovian approximation for the M(n)/M(n)/s+GI system, Queueing Syst., 41 (2002), 73-94. doi: 10.1023/A:1015781818360. [5] L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Commun. Statist. - Stochastic Models, 11 (1995), 497-525. doi: 10.1080/15326349508807357. [6] S. Charkravarthy and J. Daniel, A markovian inventory system with random shelf time and back orders, Computers and Industrial Engineering, 47 (2004), 315-337. [7] G. I. Falin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls, Adv. Appl. Prob., 16 (1984), 447-448. doi: 10.2307/1427079. [8] Qi-Ming He, E. M. Jewkes and J. Buzacott, The value of information used in inventory control of a make-to-order inventory-production system, IIE Transactions, 34 (2002), 999-1013. doi: 10.1080/07408170208928929. [9] S. Ioannidis, O. Jouini, A. A. Economopoulos and V. S. Kouikoglou, Control policies for single-stage production systems with perishable inventory and customer impatience, Annals of Operations Research, (2012), 1-24. doi: 10.1007/s10479-012-1058-9. [10] 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. [11] S. Kalpakam and K. P. Sapna, $(S-1,S)$ perishable systems with stochastic lead times, Mathematical and Computer Modelling, 21 (1995), 95-104. doi: 10.1016/0895-7177(95)00026-X. [12] I. Karaesmen, A. Scheller-Wolf and B. Deniz, Managing perishable and aging invetories: Review and future research directions, In Planning Production and Inventories in the Extended Enterprise, Springer, (2011), 393-438. [13] A. Krishnamoorthy, K. P. Jose and V. C. Narayanan, Numerical investigation of a PH/PH/1 inventory system with positive service time and shortage, Neural Parallel & Scientific Comp., 16 (2008), 579-591. [14] 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 Operational Research Society, 32 (1981), 997-1001. [15] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM series on statistics and applied probability, 1999. doi: 10.1137/1.9780898719734. [16] L. Liu, $(s,S)$ continous review models for inventory with random lifetimes, Operations Research Letters, 9 (1990), 161-167. doi: 10.1016/0167-6377(90)90014-V. [17] L. Liu and T. Yang, An $(s,S)$ random lifetime inventory model with a positive lead time, European Journal of Operational Research, 112 (1999), 52-63. doi: 10.1016/S0377-2217(97)00426-8. [18] S. Nahmias, Perishable inventory theory: A review, Operational Research, 30 (1982), 680-708. doi: 10.1287/opre.30.4.680. [19] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, MD., 1981. [20] M. F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, Inc., 1989. [21] D. Perry and W. Stadje, Perishable inventory systems with impatient demands, Math. Meth. of OR, 50, (1999), 77-90. [22] G. P. Prestacos, Blood inventory management, Management Science, 30 (1984), 777-801. [23] M. Raafat, Survey of literature on continuously deteriorating inventory models, Journal of Operational Research Society, 42 (1991), 27-37. [24] N. Ravichandran, Stochastic analysis of a continous review perishable inventory system with positive lead time and Poisson demand, European Journal of Operational Research, 84 (1995), 444-457. [25] 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. [26] A. R. Ward and P. W. Glynn, A diffusion approximation for a markovian queue with reneging, Queueing Syst., 43 (2003), 103-128. doi: 10.1023/A:1021804515162. [27] S. Zeltyn and A. Mandelbaum, Call centers with impatient customers: Many-server asymptotics of the M/M/n + G queue, Queueing Syst., 51 (2005), 361-402. doi: 10.1007/s11134-005-3699-8.
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