Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T09:20:46.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary analysis of an (R, Q) inventory model with normal and emergency orders

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  04 October 2022

Onno Boxma ,
David Perry  and
Wolfgang Stadje
Onno Boxma*
Affiliation:
Eindhoven University of Technology
David Perry*
Affiliation:
Holon Institute of Technology
Wolfgang Stadje*
Affiliation:
University of Osnabrück
*
*Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: o.j.boxma@tue.nl
**Postal address: Holon Institute of Technology, PO Box 305, Holon 5810201, Israel. Email address: dperry@stat.haifa.ac.il
***Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wstadje@uos.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an (R, Q) inventory model with two types of orders, normal orders and emergency orders, which are issued at different inventory levels. These orders are delivered after exponentially distributed lead times. In between deliveries, the inventory level decreases in a state-dependent way, according to a release rate function $\alpha({\cdot})$ . This function represents the fluid demand rate; it could be controlled by a system manager via price adaptations. We determine the mean number of downcrossings $\theta(x)$ of any level x in one regenerative cycle, and use it to obtain the steady-state density f (x) of the inventory level. We also derive the rates of occurrence of normal deliveries and of emergency deliveries, and the steady-state probability of having zero inventory.

Keywords

(R, Q) inventorystochastic lead timesteady-state analysislevel crossings

MSC classification

Primary: 90B05: Inventory, storage, reservoirs
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 106 - 126
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Axsäter, S. (2007). A heuristic for triggering emergency orders in an inventory system. Europ. J. Operat. Res. 172, 880–891.CrossRefGoogle Scholar
Axsäter, S. (2014). An improved decision rule for emergency replenishments. Internat. J. Product. Econom. 157, 313–317.CrossRefGoogle Scholar
Bar-Ilan, A. and Ben-David, N. (1996). An algorithm for evaluating the number of controls in trigger–target models. J. Econom. Dynam. Control 20, 1367–1371.CrossRefGoogle Scholar
Bar-Ilan, A. and Sulem, A. (1995). Explicit solution of inventory problems with delivery lags. Math. Operat. Res. 20, 709–720.CrossRefGoogle Scholar
Bar-Ilan, A., Marion, N. P. and Perry, D. (2007). Drift control of international reserves. J. Econom. Dynam. Control 31, 3110–3137.10.1016/j.jedc.2006.10.005CrossRefGoogle Scholar
Bar-Ilan, A., Perry, D. and Stadje, W. (2004). A generalized impulse control model of cash management. J. Econom. Dynam. Control 28, 1013–1033.CrossRefGoogle Scholar
Boxma, O. J., Kaspi, H., Kella, O. and Perry, D. (2005). On/off storage systems with state-dependent input, output and switching rates. Prob. Eng. Inf. Sci. 19, 1–14.CrossRefGoogle Scholar
Brill, P. H. (2017). Level Crossing Methods in Stochastic Models. Springer.CrossRefGoogle Scholar
Bulinskaya, E. (1964). Some results concerning optimal inventory policies. Prob. Theory Appl. 9, 502–507.CrossRefGoogle Scholar
Chiang, C. (2001). Order splitting under periodic review inventory systems. Internat. J. Product. Econom. 70, 67–76.CrossRefGoogle Scholar
Chiang, C. and Chiang, W. C. (1996). Reducing inventory costs by order splitting in the sole sourcing environment. J. Operat. Res. Soc. 47, 446–456.CrossRefGoogle Scholar
Chiang, C. and Gutierrez, G. J. (1996). A periodic review inventory system with two supply modes. Europ. J. Operat. Res. 94, 527–547.CrossRefGoogle Scholar
Cohen, J. W. (1977). On up- and downcrossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
Federgruen, A. and Wang, M. (2013). Monotonicity policies of a class of stochastic inventory systems. Ann. Operat. Res. 218, 155–186.CrossRefGoogle Scholar
Ganeshan, R., Tyworth, J. E. and Guo, Y. (1999). Dual sourced supply chains: the discount supplier option. Transport. Res. E 35, 11–23.CrossRefGoogle Scholar
Harrison, J. M. and Resnick, S. (1976). The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Operat. Res. 4, 347–358.CrossRefGoogle Scholar
Hederra, F. J. (2008). Periodic-review policies for a system with emergency orders. PhD dissertation, Georgia Institute of Technology.Google Scholar
Huang, S., Axsäter, S., Don, Y. and Chen, J. (2011). A real-time decision rule for an inventory system with committed service time and emergency orders. Europ. J. Operat. Res. 215, 70–79.CrossRefGoogle Scholar
Jain, A., Groenevelt, H. and Rudi, N. (2010). Continuous review inventory model with dynamic choice of two freight modes with fixed costs. Manufact. Serv. Operat. Manag. 12, 120–139.CrossRefGoogle Scholar
Johansen, S. G. and Thorstenson, A. (1998). An inventory model with Poisson demands and emergency orders. Internat. J. Product. Econom. 56–57, 275–289.CrossRefGoogle Scholar
Kelle, P. and Silver, E. A. (1990). Safety stock reduction by order splitting. Naval Res. Logistics 37, 725–743.3.0.CO;2-O>CrossRefGoogle Scholar
Lau, H.-S. and Lau, A. H.-L. (1994). Coordinating two suppliers with offsetting lead time and price performance. J. Operat. Manag. 11, 327–337.CrossRefGoogle Scholar
Lau, H.-S. and Zhao, L. G. (1993). Optimal ordering policies with two suppliers when lead-times and demands are all stochastic. Europ. J. Operat. Res. 68, 120–133.CrossRefGoogle Scholar
Miller, M. and Orr, D. (1966). A model of the demand for money by firms. Quart. J. Econom. 81, 413–435.CrossRefGoogle Scholar
Moinzadeh, K. and Nahmias, S. (1988). A continuous review model for an inventory system with two supply modes. Manag. Sci. 34, 761–773.CrossRefGoogle Scholar
Moinzadeh, K. and Schmidt, C. P. (1991). An $(S-1,S)$ inventory system with emergency orders. Operat. Res. 39, 308–321.CrossRefGoogle Scholar
Pan, A. C., Ramasesh, R. V., Hayya, J. C. and Keith Ord, J. (1991). Multiple sourcing: the determination of lead times. Operat. Res. Lett. 10, 1–7.CrossRefGoogle Scholar
Poormoaied, S., Atan, Z., de Kok, T. and van Woensel, T. (2020). Optimal inventory and timing decisions for emergency shipments. IISE Trans. 52, 904–925.CrossRefGoogle Scholar
Ramasesh, R. V., Keith Ord, J., Hayya, J. C. and Pan, A. (1991). Sole versus dual sourcing in stochastic lead-time (s, Q) inventory models. Manag. Sci. 37, 428–443.Google Scholar
Tagaras, G. and Vlachos, D. (2001). A periodic review inventory system with emergency replenishments. Manag. Sci. 47, 415–429.CrossRefGoogle Scholar
Veinott, A. F., Jr (1966). The status of mathematical inventory theory. Manag. Sci. 12, 745–777.CrossRefGoogle Scholar
Whittmore, A. S. and Saunders, S. (1977). Optimal inventory under stochastic demand with two supply options. SIAM J. Appl. Anal. 32, 293–305.Google Scholar
Wright, G. P. (1968). Optimal policies for a multi-product inventory system with negotiable lead times. Naval Res. Logistics Quart. 15, 375–401.CrossRefGoogle Scholar
Zipkin, P. H. (2000). Foundations of Inventory Management, Vol. 2. McGraw-Hill.Google Scholar