Optimal size of a rental inventory with items available from a secondary source: a model with non-stationary probabilities

  • Leonardo D. EpsteinEmail author
  • Eduardo González
  • Abdón Sepúlveda
S.I.: CLAIO 2016


This article concerns operations of businesses that own inventories of rental items, and can hire additional items from secondary sources whenever they face a temporary exhaustion of their inventories. This set-up is relevant to many operations: the items may be tools, trucks, containers, communication channels, or individuals who provide services such as repairmen. A fundamental problem that emerges in the design of these operations is to determine the optimal size of the inventory of items the business should own. To solve this problem, this article takes the view of a finite horizon project and proposes an approach that chooses the inventory size that minimizes the expected present cost of the project. This approach models random times between consecutive item requests and random rental durations with corresponding expectations that may vary along the day. The expected present cost uses non-stationary transition probabilities that recent articles have computed resorting to stationary approximations. This article, in contrast, computes these probabilities faster solving a differential equation without resorting to such approximations. If the present cost is of interest, an analysis that plugs-in the optimal size into the present cost ignores the sampling variability that transfers from the traffic data to the optimal size. This article complements the analysis with simulations that provide the sampling distribution of the present cost.


Rental items Inventory models Optimal inventory level Optimal mix Call center 



The authors thank two anonymous reviewers whose comments helped improve the paper substantially.


  1. Conte, S. D., de Boor, C., Napier, C., & Amar, J. S. (Eds.). (1980). Elementary numerical analysis. New York: Mc-Graw Hill.Google Scholar
  2. Gans, N., Koole, G., & Mandelbaum, A. (2003). Telephone call centers: Tutorial, review, and research prospects. Manufacturing & Service Operations Management, 5, 79–141.CrossRefGoogle Scholar
  3. George, D. K., & Xia, C. H. (2011). Fleet-sizing and service availability for a vehicle rental system via closed queueing networks. European Journal of Operational Research, 211, 198–207.CrossRefGoogle Scholar
  4. González, E., & Epstein, L. D. (2015). Minimum cost in a mix of new and old reusable items: an application to sizing a fleet of delivery trucks. Annals of Operations Research, 232, 135–149.Google Scholar
  5. González, E., Epstein, L. D., & Godoy, V. (2012). Optimal number of bypasses: Minimizing cost of calls to wireless phones under calling party pays. Annals of Operations Research, 199, 179–191.CrossRefGoogle Scholar
  6. Gross, D., & Harris, C. M. (1985). Fundamentals of queueing theory (2nd ed.). New York: Wiley.Google Scholar
  7. Iversen, V. B. (2015). Teletraffic engineering and network planning. Technical University of Denmark, Technical University of Denmark. Accessed on September 26, 2017.
  8. Loxton, R., Lin, Q., & Teo, K. L. (2012). A stochastic fleet composition problem. Computers & Operations Research, 39, 3177–3184.CrossRefGoogle Scholar
  9. Nightingale, J., & Teohary, T. (2003). Calling party pays system bypass. US patent 6, 546, 238. 2003 Apr 8. Bell Atlantic mobile Inc, Bedminster, NJ.Google Scholar
  10. Papier, F., & Thonemann, U. W. (2008). Queuing models for sizing and structuring rental fleets. Transportation Science, 42(3), 302–317.CrossRefGoogle Scholar
  11. Tainiter, M. (1964). Some stochastic inventory models for rental situations. Management Science, 11(2), 316–326.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Leonardo D. Epstein
    • 3
    Email author
  • Eduardo González
    • 1
    • 2
  • Abdón Sepúlveda
    • 4
  1. 1.School of EngineeringUniversidad Finis TerraeSantiagoChile
  2. 2.Synopsys Inc.SantiagoChile
  3. 3.School of BusinessUniversidad de los AndesSatiagoChile
  4. 4.Mechanical and Aerospace Engineering DepartmentSchool of Engineering and Applied Science, UCLALos AngelesUSA

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