Effective Zero-Inventory-Ordering Policies for the Single-Warehouse Multiretailer Problem with Piecewise Linear Cost Structures

Abstract

W e analyze the problem faced by companies that rely on TL (Truckload) and LTL (Less than Truckload) carriers for the distribution of products across their supply chain. Our goal is to design simple inventory policies and transportation strategies to satisfy timevarying demands over a finite horizon, while minimizing systemwide cost by taking advantage of quantity discounts in the transportation cost structures. For this purpose, we study the cost effectiveness of restricting the inventory policies to the class of zero-inventory-ordering (ZIO) policies in a single-warehouse multiretailer scenario in which the warehouse serves as a cross-dock facility. In particular, we demonstrate that there exists a ZIO inventory policy whose total inventory and transportation cost is no more than 4/3 (5.6/4.6 if transportation costs are stationary) times the optimal cost. However, finding the best ZIO policy is an NPhard problem as well. Thus, we propose two algorithms to find an effective ZIO policy: An exact algorithm whose running time is polynomial for any fixed number of retailers, and a linear-programming-based heuristic whose effectiveness is demonstrated in a series of computational experiments. Finally, we extend the worst-case results developed in this paper to systems in which the warehouse does hold inventory.

A Constant Approximation Algorithm for the One-Warehouse Multiretailer Problem

“…Chan et al (2000) have considered a variant of OWMR, in which the ordering costs are piecewise-linear functions, and the holding cost is linear and additive. They considered the class of zero-inventory ordering (ZIO) policies, in which the warehouse and retailers order if and only if their current on hand inventory is 0.…”

“…They established the effectiveness of these policies, showing that the cost of the optimal ZIO policy is at most 4 3 times the cost of the optimal policy. In Chan et al (2000) and in a subsequent paper by Shen et al (2002), they have proposed an integer program to find the optimal ZIO policy, which is NP-hard. Next they have developed heuristics to round the optimal solution of the LP relaxation to get an approximation algorithm for finding the best ZIO policy.…”

The submodular joint replenishment problem

“…Becchetti et al (2006) and Khanna et al (2002) have considered special cases of the tree JRP with assumptions on the holding cost structure. Chan et al (2002) show that zero-inventory policies are near-optimal for joint replenishment problems with piecewise linear costs, however their conditions do not imply submodularity.…”

Transportation pricing of a truckload carrier

“…While many of the papers in this area consider the problem in a single echelon setting (e.g., Aucamp, 1982;Lee, 1986;Tersine and Barman, 1994;Shinn et al, 1996;Burwell et al, 1997;Hwang et al, 1990;Jaruphongsa et al, 2005;Mendoza and Ventura, 2008;Rieksts and Ventura, 2008;Toptal, 2009), relatively recent papers model and solve the replenishment problems of multiple echelons under transportation considerations (see Hoque and Goyal, 2000;Çetinkaya and Lee, 2002;Chan et al, 2002;Toptal et al, 2003;Toptal and Çetinkaya, 2006;Rieksts and Ventura, 2010).…”