# Article

Full entry | PDF   (0.6 MB)
Keywords:
inventory control; Markovian analysis; stochastic lead-times; distribution inventory systems
Summary:
This paper considers a distribution inventory system that consists of a single warehouse and several retailers. Customer demand arrives at the retailers according to a continuous-time renewal process. Material flow between echelons is driven by reorder point/order quantity inventory control policies. Our objective in this setting is to calculate the long-run inventory, backorder and customer service levels. The challenge in this system is to characterize the demand arrival process at the warehouse. We present a Markovian methodology to elucidate and approximate this process. We illustrate the use of this methodology in the distribution inventory system under stochastic transportation times with identical and non-identical retailers.
References:
[1] Albin, S. L.: Approximating a point process by a renewal process, II: Superposition arrival processes to queues. Oper. Res. 32 (1984), 5, 1133-1162. DOI 10.1287/opre.32.5.1133 | MR 0770513
[2] Altiok, T.: Performance Analysis of Manufacturing Systems. Springer Series in Operations Research and Financial Engineering. Springer, New York 1997. DOI 10.1007/978-1-4612-1924-8
[3] Axsäter, S.: Simple solution procedures for a class of two-echelon inventory problems. Oper. Res. 38 (1990), 1, 64-69. DOI 10.1287/opre.38.1.64
[4] Axsäter, S.: Exact and approximate evaluation of batch-ordering policies for two-level inventory systems. Oper. Res. 41 (1993), 4, 777-785. DOI 10.1287/opre.41.4.777
[5] Axsäter, S.: Exact analysis of continuous review $(R,Q)$ policies in two-echelon inventory systems with compound Poisson demand. Oper. Res. 48 (2000), 5, 686-696. DOI 10.1287/opre.48.5.686.12403 | MR 1792775
[6] Axsäter, S.: Approximate optimization of a two-level distribution inventory system. Int. J. Product. Econom. 81 (2003), 545-553. DOI 10.1016/s0925-5273(02)00270-0
[7] Axsäter, S., Marklund, J.: Optimal position-based warehouse ordering in divergent two-echelon inventory systems. Oper. Res. 56 (2008), 4, 976-991. DOI 10.1287/opre.1080.0560 | MR 2454108
[8] Balcıoğlu, B., Jagerman, D. L., Altiok, T.: Approximate mean waiting time in a $GI/D/1$ queue with autocorrelated times to failures. IIE Trans. 39 (2007), 10, 985-996. DOI 10.1080/07408170701275343
[9] Beamon, B. M.: Supply chain design and analysis: Models and methods. Int. J. Product. Econom. 55 (1998), 3, 281-294. DOI 10.1016/s0925-5273(98)00079-6
[10] Benjaafar, S., Cooper, W. L., Kim, J.-S.: On the benefits of pooling in production-inventory systems. Management Sci. 54 (2005), 548-565. DOI 10.1287/mnsc.1040.0303
[11] Bitran, G. R., Dasu, S.: Approximating nonrenewal processes by Markov chains: Use of Super-Erlang (SE) chains. Oper. Res. 41 (1993), 5, 903-923. DOI 10.1287/opre.41.5.903
[12] Bitran, G. R., Dasu, S.: Analysis of the $\sum Ph_i/Ph/1$ queue. Oper. Res. 42 (1994), 1, 158-174. DOI 10.1287/opre.42.1.158 | MR 1261375
[13] Buzacott, J. A., Kostelski, D.: Matrix-geometric and recursive algorithm solution of a two-stage unreliable how line. IIE Trans. 19 (1987), 4, 429-438. DOI 10.1080/07408178708975416
[14] Cachon, G. P.: Exact evaluation of batch-ordering inventory policies in two-echelon supply chains with periodic review. Oper. Res. 49 (2001), 1, 79-98. DOI 10.1287/opre.49.1.79.11188 | MR 1814611
[15] Chen, F., Zheng, Y.-S.: One-warehouse multiretailer systems with centralized stock information. Oper. Res. 45 (1997), 2, 275-287. DOI 10.1287/opre.45.2.275
[16] Chew, E. P., Tang, L. C.: Warehouse-retailer system with stochastic demands - Non-identical retailer case. Europ. J. Oper. Res. 82 (1995), 1, 98-110. DOI 10.1016/0377-2217(93)e0279-7
[17] Deuermeyer, B. L, Schwarz, L. B.: A model for the analysis of system service level in warehouse-retailer distribution systems: The identical retailer case. Institute for Research in the Behavioral, Economic, and Management Sciences, Krannert Graduate School of Management, Purdue University, 1979.
[18] Diks, E. B., Kok, A. G. de, Lagodimos, A. G.: Multi-echelon systems: A service measure perspective. Europ. J. Oper. Res. 95 (1996), 2, 241-263. DOI 10.1016/s0377-2217(96)00120-8
[19] Erenguc, S. S., Simpson, N. C., Vakharia, A. J.: Integrated production/distribution planning in supply chains: An invited review. Europ. J. Oper. Res. 115 (1999), 2, 219-236. DOI 10.1016/s0377-2217(98)90299-5
[20] Eruguz, A. S., Sahin, E., Jemai, Z., Dallery, Y.: A comprehensive survey of guaranteed-service models for multi-echelon inventory optimization. Int. J. Product. Econom. 172 (2016), 110-125. DOI 10.1016/j.ijpe.2015.11.017
[21] Forsberg, R.: Exact evaluation of $(R,Q)$-policies for two-level inventory systems with Poisson demand. Europ. J. Oper. Res. 96 (1997), 1, 130-138. DOI 10.1016/s0377-2217(96)00137-3
[22] Girish, M. K., Hu, J.-Q.: Higher order approximations for the single server queue with splitting, merging and feedback. Europ. J. Oper. Res. 124 (2000), 3, 447-467. DOI 10.1016/s0377-2217(99)00174-5 | MR 1782301
[23] Gurgur, C. Z., Altiok, T.: Approximate analysis of decentralized, multi-stage, pull-type production/inventory systems. Ann. Oper. Res. 125 (2004), 1-4, 95-116. DOI 10.1023/b:anor.0000011187.52502.37 | MR 2052322
[24] Karaman, A., Altiok, T.: Approximate Analysis of Batch Ordering Policies in Distribution Inventory Systems. Technical Report TR-2007-050, Rutgers University, Department of Industrial and Systems Engineering, 2007.
[25] Karaman, A., Altiok, T.: Approximate analysis and optimization of batch ordering policies in capacitated supply chains. Europ. J. Oper. Res. 193 (2009), 1, 222-237. DOI 10.1016/j.ejor.2007.10.018
[26] Karaman, A. S.: Performance Analysis and Design of Batch Ordering Policies in Supply Chains. PhD Thesis, Rutgers, The State University of New Jersey, 2007.
[27] Lucantoni, D. M.: New results on the single server queue with a batch Markovian arrival process. Commun. Statist. Stoch. Models 7 (1991), 1, 1-46. DOI 10.1080/15326349108807174 | MR 1102528
[28] Moinzadeh, K., Lee, H. L.: Batch size and stocking levels in multi-echelon repairable systems. Management Sci. 32 (1986), 12, 1567-1581. DOI 10.1287/mnsc.32.12.1567
[29] Neuts, M. F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover Publications, 1994. MR 1313503
[30] Norros, I., Roberts, J. W., Simonian, A., Virtamo, J. T.: The superposition of variable bit rate sources in an ATM multiplexer. IEEE J. Selected Areas Commun. 9 (1991), 3, 378-387. DOI 10.1109/49.76636
[31] Osogami, T., Harchol-Balter, M.: A closed-form solution for mapping general distributions to minimal PH distributions. In: Computer Performance Evaluation. Modelling Techniques and Tools (P. Kemper and W. H. Sanders, eds.), Lecture Notes in Computer Science 2794, pp. 200-217. Springer Berlin Heidelberg, 2003. DOI 10.1007/978-3-540-45232-4_13
[32] Schwarz, L. B., Deuermeyer, B. L., Badinelli, R. D.: Fill-rate optimization in a one-warehouse $N$-identical retailer distribution system. Management Sci. 31 (1985), 5, 488-498. DOI 10.1287/mnsc.31.4.488 | MR 0788081
[33] Sherbrooke, C. C.: Metric: A multi-echelon technique for recoverable item control. Operr. Res. 16 (1968), 1, 122-141. DOI 10.1287/opre.16.1.122
[34] Svoronos, A., Zipkin, P.: Estimating the performance of multi-level inventory systems. Oper. Res. 36(1988), 57-72. DOI 10.1287/opre.36.1.57
[35] Svoronos, A., Zipkin, P.: Evaluation of one-for-one replenishment policies for multiechelon inventory systems. Management Sci. 37 (1991), 1, 68-83. DOI 10.1287/mnsc.37.1.68
[36] Tempelmeier, H.: A multi-level inventory system with a make-to-order supplier. Int. J. Product. Res. 51 (2013), 23-24, 6880-6890. DOI 10.1080/00207543.2013.776190
[37] Vuuren, M. van, Adan, I. J. B. F.: Approximating multiple arrival streams by using aggregation. Stoch. Models 22 (2006), 3, 423-440. DOI 10.1080/15326340600820398 | MR 2247591
[38] Whitt, W.: Approximating a point process by a renewal process, I: Two basic methods. Oper. Res. 30 (1982), 1, 125-147. DOI 10.1287/opre.30.1.125 | MR 0647405
[39] Zipkin, P.: The use of phase-type distributions in inventory-control models. Naval Res. Logistics 35 (1988), 2, 247-257. DOI 10.1002/1520-6750(198804)35:2<247::aid-nav3220350209>3.0.co;2-l | MR 0930967

Partner of