On the \(\varvec{(Q,r)}\) policy for perishables with positive lead times and multiple outstanding orders

  • Emre BerkEmail author
  • Ülkü Gürler
  • Saeed Poormoaied
Original Research


We consider an inventory system for perishables with fixed lifetimes, positive replenishment lead times and lost sales in the presence of non-negligible fixed ordering costs. The system is studied under the lotsize reorder level (Qr) policy. An exact analysis of this system based on the stationary distribution of the remaining lifetime process is provided by Berk and Gürler (Oper Res 56(5):1238–1246, 2008) under the restriction that there is at most one outstanding order at any time (\(r<Q\)). In this work, we generalize their results to allow for more than one outstanding orders \((r\ge Q)\). We provide the operating characteristics of the inventory system and construct the exact expected cost rate expression using a renewal theoretic approach. An illustrative numerical study indicates that allowing for multiple outstanding orders \((r\ge Q)\) may result in significant savings in the expected cost rate, compared to the case with \(r<Q\). In particular, when the fixed lifetimes are short and the ordering costs are low, expected costs can be reduced by more than half.


Perishable inventory Lot size-reorder point policy Lost sales Effective lifetime Multiple outstanding orders 


  1. Al Hamadi, H. M., Sangeetha, N., & Sivakumar, B. (2015). Optimal control of service parameter for a perishable inventory system maintained at service facility with impatient customers. Annals of Operations Research, 233(1), 3–23.CrossRefGoogle Scholar
  2. Amirthakodi, M., Radhamani, V., & Sivakumar, B. (2015). A perishable inventory system with service facility and feedback customers. Annals of Operations Research, 233(1), 25–55.CrossRefGoogle Scholar
  3. Avinadav, T., Chernonog, T., Lahav, Y., & Spiegel, U. (2017). Dynamic pricing and promotion expenditures in an EOQ model of perishable products. Annals of Operations Research, 248(1–2), 75–91.CrossRefGoogle Scholar
  4. Berk, E., & Gürler, Ü. (2008). Analysis of the \((Q, r)\) inventory model for perishables with positive lead times and lost sales. Operations Research, 56(5), 1238–1246.CrossRefGoogle Scholar
  5. Chintapalli, P. (2015). Simultaneous pricing and inventory management of deteriorating perishable products. Annals of Operations Research, 229(1), 287–301.CrossRefGoogle Scholar
  6. Chiu, H. N. (1995). An approximation to the continuous review inventory model with perishable items and lead times. European Journal of Operational Research, 87(1), 93–108.CrossRefGoogle Scholar
  7. Cinlar, E. (2013). Introduction to stochastic processes. Chelmsford, MA: Courier Corporation.Google Scholar
  8. Gürler, Ü., & Özkaya, B. Y. (2008). Analysis of the \((s, S)\) policy for perishables with a random shelf life. IIE Transactions, 40(8), 759–781.CrossRefGoogle Scholar
  9. Hill, R. M. (1992). Numerical analysis of a continuous-review lost-sales inventory model where two orders may be outstanding. European Journal of Operational Research, 62(1), 11–26.CrossRefGoogle Scholar
  10. Hill, R. M. (1994). Continuous review lost sales inventory models where two orders may be outstanding. International Journal of Production Economics, 35(1–3), 313–319.CrossRefGoogle Scholar
  11. Ioannidis, S., Jouini, O., Economopoulos, A. A., & Kouikoglou, V. S. (2013). Control policies for single-stage production systems with perishable inventory and customer impatience. Annals of Operations Research, 209(1), 115–138.CrossRefGoogle Scholar
  12. Kalpakam, S., & Shanthi, S. (2001). A perishable inventory system with modified \((S-1, S)\) policy and arbitrary processing times. Computers and Operations Research, 28(5), 453–471.CrossRefGoogle Scholar
  13. Kalpakam, S., & Shanthi, S. (2006). A continuous review perishable system with renewal demands. Annals of Operations Research, 143(1), 211–225.CrossRefGoogle Scholar
  14. Kouki, C., Jemaï, Z., & Minner, S. (2015). A lost sales \((r, Q)\) inventory control model for perishables with fixed lifetime and lead time. International Journal of Production Economics, 168, 143–157.CrossRefGoogle Scholar
  15. Laslett, G., Pollard, D., & Tweedie, R. (1978). Techniques for establishing ergodic and recurrence properties of continuous-valued markov chains. Naval Research Logistics (NRL), 25(3), 455–472.CrossRefGoogle Scholar
  16. Liu, G., Zhang, J., & Tang, W. (2015). Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand. Annals of Operations Research, 226(1), 397–416.CrossRefGoogle Scholar
  17. Liu, L. (1990). \((s, S)\) continuous review models for inventory with random lifetimes. Operations Research Letters, 9(3), 161–167.CrossRefGoogle Scholar
  18. Nahmias, S. (1975). Optimal ordering policies for perishable inventory-II. Operations Research, 23(4), 735–749.CrossRefGoogle Scholar
  19. Nahmias, S., & Wang, S. S. (1979). A heuristic lot size reorder point model for decaying inventories. Management Science, 25(1), 90–97.CrossRefGoogle Scholar
  20. Olsson, F. (2014). Analysis of inventory policies for perishable items with fixed leadtimes and lifetimes. Annals of Operations Research, 217(1), 399–423.CrossRefGoogle Scholar
  21. Perry, D., & Posner, M. J. (1998). An \((S-1, S)\) inventory system with fixed shelf life and constant lead times. Operations Research, 46(3–Suppl–3), S65–S71.CrossRefGoogle Scholar
  22. Ross, S. M. (1970). Average cost semi-markov decision processes. Journal of Applied Probability, 7(3), 649–656.CrossRefGoogle Scholar
  23. Schmidt, C. P., & Nahmias, S. (1985). \((S-1, S)\) policies for perishable inventory. Management Science, 31(6), 719–728.CrossRefGoogle Scholar
  24. Tekin, E., Gürler, Ü., & Berk, E. (2001). Age-based vs. stock level control policies for a perishable inventory system. European Journal of Operational Research, 134(2), 309–329.CrossRefGoogle Scholar
  25. Tijms, H. C. (1994). Stochastic models: An algorithmic approach (Vol. 303). London: Wiley.Google Scholar
  26. Weiss, H. J. (1980). Optimal ordering policies for continuous review perishable inventory models. Operations Research, 28(2), 365–374.CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Faculty of Business AdministrationBilkent UniversityAnkaraTurkey
  2. 2.Department of Industrial EngineeringBilkent UniversityAnkaraTurkey

Personalised recommendations