Stationary analysis of an \((R,Q)\) inventory model with normal and emergency orders. (English) Zbl 1509.90004
Summary: 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.
MSC:
| 90B05 | Inventory, storage, reservoirs |
| 60K30 | Applications of queueing theory (congestion, allocation, storage, traffic, etc.) |
References:
| [1] | Axsäter, S. (2007). A heuristic for triggering emergency orders in an inventory system. Europ. J. Operat. Res.172, 880-891. · Zbl 1103.90302 |
| [2] | Axsäter, S. (2014). An improved decision rule for emergency replenishments. Internat. J. Product. Econom.157, 313-317. |
| [3] | Bar-Ilan, A. and Ben-David, N. (1996). An algorithm for evaluating the number of controls in trigger-target models. J. Econom. Dynam. Control20, 1367-1371. · Zbl 0875.90130 |
| [4] | Bar-Ilan, A. and Sulem, A. (1995). Explicit solution of inventory problems with delivery lags. Math. Operat. Res.20, 709-720. · Zbl 0846.90031 |
| [5] | Bar-Ilan, A., Marion, N. P. and Perry, D. (2007). Drift control of international reserves. J. Econom. Dynam. Control31, 3110-3137. · Zbl 1163.91493 |
| [6] | Bar-Ilan, A., Perry, D. and Stadje, W. (2004). A generalized impulse control model of cash management. J. Econom. Dynam. Control28, 1013-1033. · Zbl 1179.91251 |
| [7] | 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. · Zbl 1063.90001 |
| [8] | Brill, P. H. (2017). Level Crossing Methods in Stochastic Models. Springer. · Zbl 1380.60002 |
| [9] | Bulinskaya, E. (1964). Some results concerning optimal inventory policies. Prob. Theory Appl.9, 502-507. · Zbl 0131.18805 |
| [10] | Chiang, C. (2001). Order splitting under periodic review inventory systems. Internat. J. Product. Econom.70, 67-76. |
| [11] | 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. · Zbl 0852.90061 |
| [12] | Chiang, C. and Gutierrez, G. J. (1996). A periodic review inventory system with two supply modes. Europ. J. Operat. Res.94, 527-547. · Zbl 0947.90501 |
| [13] | Cohen, J. W. (1977). On up- and downcrossings. J. Appl. Prob.14, 405-410. · Zbl 0365.60108 |
| [14] | Federgruen, A. and Wang, M. (2013). Monotonicity policies of a class of stochastic inventory systems. Ann. Operat. Res.218, 155-186. · Zbl 1274.90020 |
| [15] | Ganeshan, R., Tyworth, J. E. and Guo, Y. (1999). Dual sourced supply chains: the discount supplier option. Transport. Res. E35, 11-23. |
| [16] | 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. · Zbl 0381.60092 |
| [17] | Hederra, F. J. (2008). Periodic-review policies for a system with emergency orders. PhD dissertation, Georgia Institute of Technology. |
| [18] | 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. · Zbl 1237.90013 |
| [19] | 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. |
| [20] | Johansen, S. G. and Thorstenson, A. (1998). An inventory model with Poisson demands and emergency orders. Internat. J. Product. Econom.56-57, 275-289. |
| [21] | Kelle, P. and Silver, E. A. (1990). Safety stock reduction by order splitting. Naval Res. Logistics37, 725-743. · Zbl 0721.90033 |
| [22] | 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. |
| [23] | 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. · Zbl 0777.90014 |
| [24] | Miller, M. and Orr, D. (1966). A model of the demand for money by firms. Quart. J. Econom.81, 413-435. |
| [25] | Moinzadeh, K. and Nahmias, S. (1988). A continuous review model for an inventory system with two supply modes. Manag. Sci.34, 761-773. · Zbl 0659.90038 |
| [26] | Moinzadeh, K. and Schmidt, C. P. (1991). An \((S-1,S)\) inventory system with emergency orders. Operat. Res.39, 308-321. · Zbl 0738.90023 |
| [27] | 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. · Zbl 0729.90035 |
| [28] | 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. |
| [29] | 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. |
| [30] | Tagaras, G. and Vlachos, D. (2001). A periodic review inventory system with emergency replenishments. Manag. Sci.47, 415-429. · Zbl 1232.90096 |
| [31] | Veinott, A. F., Jr (1966). The status of mathematical inventory theory. Manag. Sci.12, 745-777. · Zbl 0143.21801 |
| [32] | Whittmore, A. S. and Saunders, S. (1977). Optimal inventory under stochastic demand with two supply options. SIAM J. Appl. Anal.32, 293-305. · Zbl 0358.90019 |
| [33] | Wright, G. P. (1968). Optimal policies for a multi-product inventory system with negotiable lead times. Naval Res. Logistics Quart.15, 375-401. · Zbl 0169.51505 |
| [34] | Zipkin, P. H. (2000). Foundations of Inventory Management, Vol. 2. McGraw-Hill. · Zbl 1370.90005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
