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Probabilistic Analysis of Renewal Cycles: An Application to a Non-Markovian Inventory Problem with Multiple Objectives

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  • Mahmut Parlar

    (DeGroote School of Business, McMaster University, Hamilton, Ontario L8S 4M4 Canada)

Abstract

Many stochastic optimization problems are solved using the renewal reward theorem (RRT). Once a regenerative cycle is identified, the objective function is formed as the ratio of the expected cycle cost to the expected cycle time and optimized using the standard techniques. Application of the RRT requires only the first moments of the cycle-related random variables. However, if the start of a cycle corresponds to an important event, e.g., end of a period of shortages in an inventory problem, knowing only the expected time---and the cost---of the cycle may not give enough information on the functioning of the stochastic system. For example, it may be useful to know the probability that the cycle cost, or more importantly, the average cost per unit time will exceed predetermined levels. In this paper we provide a complete description of the cycle-related random variables for a stochastic inventory problem with supply interruptions. We assume a general phase-type distribution for the supplier's availability (ON) periods and an exponential distribution for the OFF periods. The first passage time of an embedded Markov chain of the ON/OFF process is used to develop the expressions for the exact distribution and the moments of the cycle time and cycle cost random variables. We then describe a method for computing the probability that the average cost per unit time will exceed a predetermined level. This method is used to construct an “efficient frontier” for the two criteria of (i) average cost and (ii) the probability of exceeding it. The efficient frontier is used to find a solution to the multiple-criteria optimization problem.

Suggested Citation

  • Mahmut Parlar, 2000. "Probabilistic Analysis of Renewal Cycles: An Application to a Non-Markovian Inventory Problem with Multiple Objectives," Operations Research, INFORMS, vol. 48(2), pages 243-255, April.
    • Handle: RePEc:inm:oropre:v:48:y:2000:i:2:p:243-255
    DOI: 10.1287/opre.48.2.243.12377
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    References listed on IDEAS

    as
    1. Mahmut Parlar & Defne Berkin, 1991. "Future supply uncertainty in EOQ models," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(1), pages 107-121, February.
    2. Yao, David D. W. & Buzacott, J. A., 1985. "Queueing models for a flexible machining station Part II: The method of Coxian phases," European Journal of Operational Research, Elsevier, vol. 19(2), pages 241-252, February.
    3. Mahmut Parlar & David Perry, 1996. "Inventory models of future supply uncertainty with single and multiple suppliers," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(2), pages 191-210, March.
    4. Edward A. Silver, 1981. "Operations Research in Inventory Management: A Review and Critique," Operations Research, INFORMS, vol. 29(4), pages 628-645, August.
    5. Parlar, Mahmut, 1997. "Continuous-review inventory problem with random supply interruptions," European Journal of Operational Research, Elsevier, vol. 99(2), pages 366-385, June.
    6. Ülkü Gürler & Mahmut Parlar, 1997. "An Inventory Problem with Two Randomly Available Suppliers," Operations Research, INFORMS, vol. 45(6), pages 904-918, December.
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    Cited by:

    1. Mohebbi, Esmail & Hao, Daipeng, 2006. "When supplier's availability affects the replenishment lead time--An extension of the supply-interruption problem," European Journal of Operational Research, Elsevier, vol. 175(2), pages 992-1008, December.
    2. Mohebbi, Esmail, 2006. "A production-inventory model with randomly changing environmental conditions," European Journal of Operational Research, Elsevier, vol. 174(1), pages 539-552, October.
    3. Francisco Arcelus & T. Pakkala & Gopalan Srinivasan, 2008. "Retailer’s inventory policies for a one time only manufacturer trade deal of uncertain duration," Annals of Operations Research, Springer, vol. 164(1), pages 3-15, November.
    4. Mohebbi, Esmail & Hao, Daipeng, 2008. "An inventory model with non-resuming randomly interruptible lead time," International Journal of Production Economics, Elsevier, vol. 114(2), pages 755-768, August.
    5. Ahiska, S. Sebnem & Appaji, Samyuktha R. & King, Russell E. & Warsing, Donald P., 2013. "A Markov decision process-based policy characterization approach for a stochastic inventory control problem with unreliable sourcing," International Journal of Production Economics, Elsevier, vol. 144(2), pages 485-496.
    6. F. Arcelus & T. Pakkala & G. Srinivasan, 2006. "On the interaction between retailers inventory policies and manufacturer trade deals in response to supply-uncertainty occurrences," Annals of Operations Research, Springer, vol. 143(1), pages 45-58, March.
    7. Xiaobo, Zhao & Xu, Deju & Zhang, Hanqin & He, Qi-Ming, 2007. "Modeling and analysis of a supply-assembly-store chain," European Journal of Operational Research, Elsevier, vol. 176(1), pages 275-294, January.

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