A primal–dual algorithm for the economic lot-sizing problem with multi-mode replenishment

Abstract

The classical economic lot-sizing problem assumes that a single supplier and a single transportation mode are used to replenish the inventory. This paper studies an extension of this problem where several suppliers and transportation modes are available. The decision-making process in this case involves identifying (i) the timing for an order; (ii) the choice of shipment modes; and (iii) the order size for each mode. The problem is defined as a network flow problem with multiple setups cost function and additional side constraints. This study provides an MIP formulation for the problem. We also provide an additional formulation of the problem by redefining its decision variables and show that the dual of the corresponding LP-relaxation has a special structure. We take advantage of the structure of the dual problem to develop a primal–dual algorithm that generates tight lower and upper bounds. Computational results demonstrate the effectiveness of the algorithm.

Introduction

The classical economic lot-sizing problem is well studied in the literature of inventory control, production planning and capacity planning. In 1958, Wagner and Whitin (1958) solved the problem efficiently, but a continuing interest in the problem remains because of its practical applications. For example, economic lot-sizing is the core problem in aggregate production planning in MRP systems (Nahmias, 1997). For an extensive review of literature that discusses this problem, see: Aggarwal and Park, 1993, Bahl et al., 1987, Belvaux and Wolsey, 1998, Belvaux and Wolsey, 1999, Belvaux and Wolsey, 2001, Nemhauser and Wolsey, 1988, Wolsey, 1995.

The classical economic lot-sizing problem is defined as follows: Given the demand, the unit production cost, the unit inventory holding cost for a commodity, and the set-up costs for each time period over a finite and discrete-time horizon find a production schedule that satisfies demand at minimum costs. This model assumes a fixed and a variable component of production costs as well as a deterministic time-varying demand. Wagner and Whitin (1958) developed a dynamic programming algorithm that solves the problem in $O(T^2)$, where T is the length of the planning horizon. Almost thirty years later, Wagelmans et al., 1992, Aggarwal and Park, 1993, and Federgruen and Tzur (1991) independently studied and improved the time complexity for obtaining an optimal solution to $O(T\log T)$ in the general case and to $O(T)$ when the costs have a special structure.

A number of studies have generalized the classical model with various considerations. These extensions include finite production capacity models (Florian and Klein, 1971, Bitran and Yanasse, 1982), multi-echelon models (Kaminsky and Simchi-Levi, 2003) and multi-item models (Manne, 1958, Barany et al., 1984). The general purpose of these extensions was to replicate real problems faced by manufacturing companies. However, because of today’s changing logistics practices, the new prevailing extensions of this problem need to be studied. This is one of the motivations for this research.

This paper studies the economic lot-sizing problem with multi-mode replenishment. The classical problem assumes that items can be purchased from a single supplier and/or delivered using a single transportation mode. This assumption holds true when a particular supplier or transportation mode is always superior to others or only one mode is feasible due to restrictions or requirements such as using tank trucks or using company-owned trucks. However, studies have shown that a more cost-effective practice considers the availability of several suppliers and transportation modes when making inventory replenishment decisions.

A number of studies in the area of supply chain management have discussed coordination of inventory and distribution decisions (Burns et al., 1985, Gupta, 1992, Hahm and Yano, 1992, Hahm and Yano, 1995, Ekşioğlu et al., 2006, Ekşioğlu et al., 2007, Chopra and Meindl, 2006). These studies analyze the trade-offs between inventory and transportation costs, however, they consider that a single transportation mode is available, and corresponding transportation costs are either assumed to be linear with respect to distance and volume or to have a fixed charge cost structure. Researchers have also analyzed the impact of multi-mode delivery opportunities on transportation costs. For example, Baumol and Vinod (1970) identify factors that impact the selection of a particular transportation mode. They show that faster, more dependable transportation modes reduce shippers or receivers inventories, including the safety stock and in-transit. These studies propose a mathematical model that identifies the single, best transportation mode when given the annual demand forecasts. Other studies consider transportation related problems with multi-mode delivery options; however, they do not relate to the economic lot-sizing literature.

To our knowledge, Jaruphongsa et al. (2005) are the only ones who have studied the economic lot-sizing problem with multi-mode replenishment and cargo-capacity considerations. However, they analyze and provide exact solution algorithms to the special case of the problem when two transportation modes are available. Our paper studies the multi-mode version of this problem and provides an efficient solution approach.

The model and the solution method we discuss can be used as a sub-module in MRP systems for requirements planning with multi-mode replenishment options, or as a decision tool to determine whether a particular replenishment mode, or a combination of different modes, should be employed to replenish the inventory. The main contribution of this paper is developing an algorithm that, as demonstrated by our computational results, provides good quality solutions in a reasonable amount of time. Jaruphongsa et al. (2005) describe an application of this model in a third-party logistics (TPL) setting.

Initially, the problem is defined as a network flow problem with multiple setups cost function and additional side constraints, and formulated as a mixed-integer program (MIP) (Section 2). Next, an additional formulation is proposed by redefining the decision variables (Section 3). The dual of the corresponding LP-relaxation has a special structure. A primal–dual algorithm has been developed that takes advantage of the structure of the dual problem (Section 3.2). Computational results demonstrate the effectiveness of the algorithm (Section 4). The special case of the economic lot-sizing problem with multi-mode replenishment where the cost function has a fixed charge structure has been analyzed, and a dynamic programming algorithm that solves the problem in $O({\mathit{IT}}^2)$ has been developed (Section 2.1), where I is the number of replenishment modes available.