Single‐warehouse multi‐retailer inventory systems with full truckload shipments

We consider a multistage inventory system composed of a single warehouse that receives a single product from a single supplier and replenishes the inventory of n retailers through direct shipments. Fixed costs are incurred for each truck dispatched and all trucks have the same capacity limit. Costs are stationary, or more generally monotone as in Lippman (1969). Demands for the n retailers over a planning horizon of T periods are given. The objective is to find the shipment quantities over the planning horizon to satisfy all demands at minimum system-wide inventory and transportation costs without backlogging. Using the structural properties of optimal solutions, we develop (1) an O(T 2 ) algorithm for the single-stage dynamic lot sizing problem; (2) an O(T 3 ) algorithm for the case of a one-warehouse single-retailer system; and (3) a nested shortest-path algorithm for the one-warehouse multi-retailer problem that runs in polynomial time for a given number of retailers. To overcome the computational burden when the number of retailers is large, we propose aggregated and disaggregated Lagrangian decomposition methods that make use of the structural properties and the ecient single-stage algorithm. Computational experiments show the eectiveness of these algorithms and the gains associated with coordinated versus decentralized systems. Finally, we show that the decentralized solution is asymptotically optimal.

[1]  E. Arkin,et al.  Computational complexity of uncapacitated multi-echelon production planning problems , 1989 .

[2]  Teodor Gabriel Crainic,et al.  Multicommodity Capacitated Network Design , 1999 .

[3]  R. Roundy.  98%-Effective Integer-Ratio Lot-Sizing for One-Warehouse Multi-Retailer Systems , 1985 .

[4]  Brunilde Sansò,et al.  Telecommunications Network Planning , 1998 .

[5]  Michal Tzur,et al.  Algorithms for the multi-item multi-vehicles dynamic lot sizing problem , 2006 .

[6]  David Simchi-Levi,et al.  The logic of logistics , 1997 .

[7]  H. Pirkul,et al.  New formulation and relaxation to solve a concave-cost network flow problem , 1997 .

[8]  Osman Alp,et al.  Optimal Lot-Sizing/Vehicle-Dispatching Policies Under Stochastic Lead Times and Stepwise Fixed Costs , 2003, Oper. Res..

[9]  Guillermo Gallego,et al.  On the effectiveness of direct shipping strategy for the one-warehouse multi-retailer R -systems , 1990 .

[10]  Alain Martel,et al.  Dynamic Lot Sizing for Multi-Echelon Distribution Systems with Purchasing and Transportation Price Discounts , 1993, Oper. Res..

[11]  Laurence A. Wolsey,et al.  Lot-Sizing with Constant Batches: Formulation and Valid Inequalities , 1993, Math. Oper. Res..

[12]  Dev Joneja,et al.  The Joint Replenishment Problem: New Heuristics and Worst Case Performance Bounds , 1990, Oper. Res..

[13]  Richard W. Cuthbertson,et al.  The Logic of Logistics: Theory, Algorithms and Applications for Logistics Management , 1998, J. Oper. Res. Soc..

[14]  Yue Jin.  Essays on supply chain management , 2007 .

[15]  Michal Tzur,et al.  Shipping Multiple Items by Capacitated Vehicles: An Optimal Dynamic Programming Approach , 2005, Transp. Sci..

[16]  Chung-Yee Lee,et al.  A Dynamic Model for Inventory Lot Sizing and Outbound Shipment Scheduling at a Third-Party Warehouse , 2003, Oper. Res..

[17]  M. Posner,et al.  Approximation Procedures for the One-Warehouse Multi-Retailer System , 1994 .

[18]  Chung-Yee Lee,et al.  A dynamic lot-sizing model with multi-mode replenishments: polynomial algorithms for special cases with dual and multiple modes , 2005 .

[19]  Leroy B. Schwarz,et al.  A Simple Continuous Review Deterministic One-Warehouse N-Retailer Inventory Problem , 1973 .

[20]  Lap Mui Ann Chan,et al.  On the Effectiveness of Zero-Inventory-Ordering Policies for the Economic Lot-Sizing Model with a Class of Piecewise Linear Cost Structures , 2002, Oper. Res..

[21]  Chung-Lun Li,et al.  Dynamic Lot Sizing with Batch Ordering and Truckload Discounts , 2004, Oper. Res..

[22]  S. Lippman.  Optimal inventory policy with multiple set-up costs , 1968 .

[23]  David Simchi-Levi,et al.  Supply Chain Design and Planning - Applications of Optimization Techniques for Strategic and Tactical Models , 2003, Supply Chain Management.

[24]  K. Holmberg,et al.  A Lagrangean heuristic for the facility location problem with staircase costs , 1997 .

[25]  Stephen C. Graves,et al.  A composite algorithm for a concave-cost network flow problem , 1989, Networks.

[26]  K. Holmberg.  Solving the staircase cost facility location problem with decomposition and piecewise linearization , 1994 .

[27]  Thomas L. Magnanti,et al.  Models and Methods for Merge - in - Transit Operations , 2003, Transp. Sci..

[28]  David B. Shmoys,et al.  A constant approximation algorithm for the one-warehouse multi-retailer problem , 2005, SODA '05.

[29]  S. Graves,et al.  Supply chain management : design, coordination and operation , 2003 .

[30]  Michal Tzur,et al.  Algorithms for the Multi-Item Capacitated Dynamic Lot Sizing Problem , 2006 .

[31]  Lap Mui Ann Chan,et al.  Effective Zero-Inventory-Ordering Policies for the Single-Warehouse Multiretailer Problem with Piecewise Linear Cost Structures , 2002, Manag. Sci..