A branch-and-price algorithm for an integrated production and inventory routing problem

Abstract

With globalization, the need to better integrate production and distribution decisions has become ever more pressing for manufacturers trying to streamline their supply chain. This paper investigates a previously developed mixed-integer programming (MIP) model aimed at minimizing production, inventory, and delivery costs across the various stages of the system. The problem being modeled includes a single production facility, a set of customers with time varying demand, a finite planning horizon, and a fleet of homogeneous vehicles. Demand can be satisfied from either inventory held at a customer site or from daily product distribution. Whether a customer is visited on a particular day is determined by an implicit tradeoff between inventory and distribution costs. Once the decision is made, a vehicle routing problem must be solved for those customers who are scheduled for a delivery.

A hybrid methodology that combines exact and heuristic procedures within a branch-and-price framework is developed to solve the underlying MIP. The approach takes advantage of the efficiency of heuristics and the precision of branch and price. Implementation required devising a new branching strategy to accommodate the unique degeneracy characteristics of the master problem, and a new procedure for handling symmetry. A novel column generation heuristic and a rounding heuristic were also implemented to improve algorithmic efficiency. Computational testing on standard data sets showed that the hybrid scheme can solve instances with up to 50 customers and 8 time periods within 1 h. This level of performance could not be matched by either CPLEX or standard branch and price alone.

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.