A branch-and-price algorithm for an integrated production and inventory routing problem
Introduction
Integrating production and distribution decisions is a challenging problem for manufacturers trying to optimize their supply chain. At the planning level, the immediate goal is to coordinate production, inventory, and delivery to meet customer demand so that the corresponding costs are minimized. Achieving this goal provides the foundations for streamlining the logistics network and for integrating other operational and financial components of the organization. In this paper we analyze a single production facility that serves a set of customers with time varying demand over a finite and discrete planning horizon. The capacity of the facility is assumed to be limited and each day production is scheduled, a setup cost is incurred. The focus is on the case where a routing problem must be solved daily to either restock inventory, meet that day's demand or both. When the associated decisions are made in a coordinated fashion, we have what is referred to as the production, inventory, distribution, routing problem (PIDRP).
Lei et al. [1] were the first to formulate the PIDRP as a mixed-integer program (MIP) and proposed a two-phase solution approach that avoided the need to address lot-sizing and routing simultaneously. Boudia et al. [2], [3] developed a similar MIP and proposed both a memetic algorithm with population management (MAPA) and a reactive greedy randomized adaptive search procedure (GRASP) with path-relinking [4] as solution methodologies. Following their lead, Bard and Nananukul [5] developed a tabu search procedure that provided slightly better results. Their model was based on the MIP in Boudia et al. [3] but is more efficient in terms of variable definitions and number of constraints. The same model is used in this paper.
Manufacturers who resupply a large number of retailers on a periodic basis continually struggle with the question of how to formulate a replenishment strategy. One popular approach is a balanced strategy in which an equal proportion of retailers is replenished each day of the work week. This has the advantage of evening the workload at the warehouse. A second strategy advocated by many practitioners is synchronized replacement in which all retailers are replenished concurrently and goods are moved into the manufacturer's warehouse immediately prior to distribution. This creates unbalanced workloads at the warehouse but allows for cross-docking of a significant portion of the goods [6]. For just-in-time suppliers, it is common to partition the customers into compact groups and follow the same delivery sequence daily, skipping those locations with absent demand [7].
Rather than treating the replenishment strategy in partial isolation, we take a more integrated view and develop a branch-and-price-based (B&P) scheme for solving the PIDRP as a complement to existing metaheuristics. This is our major contribution. Secondary contributions include the design of methods for dealing with symmetry during branching and the use of heuristics to achieve integrality. As part of the solution process, four critical decisions have to be made: how many items to manufacture each day, when to visit each customer, how much to deliver to a customer during a visit, and which delivery routes to use. Because the last decision requires the solution of a vehicle routing problem (VRP) each day, the PIDRP is evidently NP-hard in the strong sense, so it is not likely that large instances will be tractable for more than a few time periods. With this limitation in mind, we propose several compromises to the exact algorithm that involve the solution of a lot-sizing problem to estimate delivery quantities in each period and the use of a VRP heuristic.
In the next section, the literature related to the primary components of the PIDRP is reviewed with an emphasis on recent research. In Section 3, a formal definition of our version of the problem is given along with a new MIP formulation. The B&P algorithm is described in Section 4. In Section 5, we discuss our approach to dealing with symmetry along with the details of an enhanced branching strategy. Heuristic ideas are outlined in Section 6, and in Section 7, computational results are presented for a wide range of problem instances. Section 8 offers several ideas for extending the work.
Section snippets
Literature review
There is a vast quantity of literature on each component of the PIDRP so we will only highlight the most relevant work. A vendor managed inventory replenishment (VMI) system is a good example of the type of integration that we wish to address (e.g., see [8], [9]). In the VMI model, the manufacturer observes and controls the inventory levels of its customers, as opposed to conventional approaches where customers monitor their own inventory and decide the time and amount of product to reorder.
Model formulation
We are given a set of n customers geographically dispersed on a grid and a single facility for producing a unique item. Distances are calculated with the Euclidean metric and accordingly satisfy the triangle inequality. Over a planning horizon of τ periods, each customer i has a fixed nonnegative integral demand dit in period t that must be fulfilled; i.e., shortages are not permitted (for a discussion of backordering, see, e.g. [26]). If production takes place at the facility in period t, then
Solution methodology
This experience led first to development of a tabu search algorithm [5] and then to an exact method based on branch and price (B&P), which we discuss in this section. In simple terms, B&P combines (i) Dantzig–Wolfe (D–W) decomposition extended to accommodate integer variables, and (ii) standard branch and bound (B&B) [29], [30]. Below we outline how initial feasible solutions are obtained and then describe the principal components of our B&P algorithm.
Algorithmic issues
Two issues arise when trying to solve the PIDRP with B&P that strongly affect algorithmic efficiency. The first concerns symmetry, where columns with the same delivery quantities but with reverse or permuted tours are generated at various nodes in the search tree. This is a common difficulty with routing and scheduling problems. The second involves the generation of null columns that specify a VRP solution without delivery quantities. In the following subsections, we describe how these issues
Enhanced algorithmic features
Based on preliminary test results, the size of PIDRP instances that can be solved with the B&P algorithm within 1 h are limited to roughly 10 customers and 6 time periods. In this section, we outline two enhancements that were adopted to improve the performance of the algorithm. The first is a column generation heuristic for solving ; the second is a rounding heuristic that is used to update the upper bound at selected nodes in the B&B tree.
Computational results
Extensive testing was done on all components of the B&P algorithm. Results from the experiment designed to gauge CPLEX's performance on the full PIDRP are reported by Nananukul [36]. The next section gives results for the basic B&P algorithm while Section 7.2 discusses the performance of the B&P heuristic for a range of configurations and data sets.
All computations were performed on a 2.53 GHz processor with 512 MB of RAM. The optimization models were implemented in Java Netbean 4.1 and linked to
Discussion and future directions
Finding good solutions to large-scale production planning problems can provide measurable benefits to manufacturers as they move towards greater coordination of their supply chain. Although zero inventory is an ideal that many manufacturers strive for, there is a growing realization that holding some amount of finished goods inventory is necessary to hedge against production and distribution disruptions, especially when the distance between the plant and the customer sites is large. In this
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