Production-Inventory Systems with Lost Sales and Compound Poisson Demands

Jim (Junmin) Shi (), Michael N. Katehakis (), Benjamin Melamed () and Yusen Xia ()
Additional contact information
Jim (Junmin) Shi: School of Management, New Jersey Institute of Technology, Newark, New Jersey 07102
Michael N. Katehakis: Department of Management Science and Information Systems, Rutgers Business School -- Newark and New Brunswick, Piscataway, New Jersey 08854
Benjamin Melamed: Department of Supply Chain Management and Marketing Sciences, Rutgers Business School -- Newark and New Brunswick, Piscataway, New Jersey 08854
Yusen Xia: Department of Managerial Sciences, Georgia State University, Atlanta, Georgia 30303

Operations Research, 2014, vol. 62, issue 5, 1048-1063

Abstract: This paper considers a continuous-review, single-product, production-inventory system with a constant replenishment rate, compound Poisson demands, and lost sales. Two objective functions that represent metrics of operational costs are considered: (1) the sum of the expected discounted inventory holding costs and lost-sales penalties, both over an infinite time horizon, given an initial inventory level; and (2) the long-run time average of the same costs. The goal is to minimize these cost metrics with respect to the replenishment rate. It is, however, not possible to obtain closed-form expressions for the aforementioned cost functions directly in terms of positive replenishment rate ( PRR ). To overcome this difficulty, we construct a bijection from the PRR space to the space of positive roots of Lundberg's fundamental equation , to be referred to as the Lundberg positive root ( LPR ) space. This transformation allows us to derive closed-form expressions for the aforementioned cost metrics with respect to the LPR variable, in lieu of the PRR variable. We then proceed to solve the optimization problem in the LPR space and, finally, recover the optimal replenishment rate from the optimal LPR variable via the inverse bijection. For the special cases of constant or loss-proportional penalty and exponentially distributed demand sizes, we obtain simpler explicit formulas for the optimal replenishment rate.

Keywords: compound Poisson arrivals; integro-differential equation; Laplace transform; Lundberg's fundamental equation; lost sales; production-inventory system; constant replenishment rate (search for similar items in EconPapers)
Date: 2014
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (14)

Downloads: (external link)
http://dx.doi.org/10.1287/opre.2014.1299 (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:inm:oropre:v:62:y:2014:i:5:p:1048-1063

Access Statistics for this article

More articles in Operations Research from INFORMS Contact information at EDIRC.
Bibliographic data for series maintained by Chris Asher ().