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

Sung-Seok Ko

doi:10.3934/jimo.2019009

Article Contents

2020, Volume 16, Issue 3: 1415-1433. Doi: 10.3934/jimo.2019009

This issue Previous Article Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns Next Article Forecast horizon of dynamic lot size model for perishable inventory with minimum order quantities

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

Sung-Seok Ko

Department of Industrial Engineering, Konkuk University, Seoul, Korea

Received: February 28, 2018

Revised: September 30, 2018

Early access: March 2019

Published: March 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Inventory Model

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Figure 2. Contour Plot of TCR

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Figure 3. The effect of $ \lambda $

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Figure 4. The effect of $ \mu $

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Table 1. Total Cost Rate(TCR)

$S \diagdown s$	1	2	3	4	5	6	7	8	9	10
15	367.40	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	362.40	360.21	360.34	362.72	367.30	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	372.00	379.14	388.10	398.40
20	369.49	365.07	362.70	362.38	363.97	367.32	372.29	378.80	386.87	396.60
21	371.06	366.69	364.35	363.98	365.45	368.58	373.22	379.25	386.65	395.44
22	372.88	368.60	366.29	365.91	367.30	370.27	374.65	380.32	387.20	395.29
23	374.93	370.74	368.49	368.12	369.45	372.31	376.50	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	379.60	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	379.20	379.00	380.33	382.99	386.76	391.50	397.05	403.34
28	387.71	384.07	382.24	382.10	383.46	386.10	389.83	394.48	399.91	406.01
29	390.64	387.12	385.38	385.31	386.70	389.35	393.05	397.63	402.94	408.90
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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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.