Formulations and branch-and-cut algorithms for multi-product multi-vehicle production routing problems with startup cost

Highlights

• MILP models for multi-product multi-vehicle production routing problems with startup.

• Branch-and-cut algorithm with relax-and-fix heuristic.

• Big-bucket versus small-bucket formulation.

• Comparison with multi-product multi-vehicle inventory routing problems.

• Case study for multi-product multi-vehicle production routing problems with startup.

Abstract

In multi-product multi-vehicle production routing problems (MMPRPs), it is necessary to accurately model the capacity utilization of multiple products to obtain feasible production plans. Thus, start-up variables are required to model the capacity consumed or cost incurred when a machine starts a production batch, or when a machine switches from one product to another. In this work, we present a mixed integer linear programming model of the MMPRP with the startup cost (MMPRPSC), which is an extension of the single-item multi-vehicle production routing problem. It also generalizes the multi-item multi-vehicle inventory routing problems by incorporating production decisions. This formulation is tightened with three families of valid inequalities in which the generalized (l, S) inequalities are used in these problem settings for the first time. Using the formulation and valid inequalities, we implement a branch-and-cut algorithm for the solution of the MMPRPSC. Computational experiments also confirm the effectiveness of the valid inequalities. The computational results of a case study show a 15% percent decrease in the total cost.

Introduction

Vehicle routing problems (VRPs) lie at the heart of distribution management (Cinar, Gakis, Pardalos, 2016, Li, Pardalos, Sun, Pei, Zhang, 2015, Shi, Boudouh, Grunder, 2017). Production routing problems (PRPs) are variants of VRPs and involve the joint optimization of decisions regarding production,inventory, distribution and routing (Adulyasak, Cordeau, & Jans, 2015b). Such integrated optimization problems are of practical relevance to success in business competition, especially in relation to vendor managed inventory (VMI) and just-in-time (JIT) practices, which prevail in the use of information technology in modern production practice.

Single-product PRPs have been investigated using exact algorithms and heuristics over the last decade, e.g., with multiple plants (Lei, Liu, Ruszczynski, & Park, 2006), under demand uncertainty (Adulyasak, Cordeau, & Jans, 2015a), with uncapacitated production (Absi, Archetti, Dauzere-Peres, & Feillet, 2015), and with multiple homogeneous capacitated vehicles (Adulyasak, Cordeau, & Jans, 2014a), which enhanced our understanding of the PRP.

An obvious gap exists between these studies and business practices, which are often characterized by multiple products. In fact, in the seminal papers (Chandra, Fisher, 1994, Fumero, Vercellis, 1999), PRPs were presented as multi-product multi-vehicle production routing problems (MMPRPs). In these problems, multi-product lot-sizing problems (LSPs) are important parts that include many variants. Each variant is suitable for a given production environment. For instance, setup times or startup times are usually included in production LSPs when setting up a production line with the specifications of a particular product requires the loss of several days of production (Vanderbeck, 1998). There has been little progress in the research on the MMPRP in this direction. The most recent development involved an MMPRP with backordering (Brahimi & Aouam, 2016), which was solved with the relax-and-fix heuristic. Our aim is to bridge this gap by incorporating the still-missing startup decisions in MMPRPs.

Many practical applications of lot-sizing and scheduling problems involve the startup times/costs (Karmarkar, Kekre, & Kekre, 1987), e.g., the manufacturing of food products or chemicals where significant clean-up must occur between different production batches. In some production scheduling scenarios where the duration of a production run is long, and a single product can occupy a machine for one or more periods continuously, the assumption of a fixed cost for every period with positive production is inappropriate. Then, it may be possible to carry over a setup from one period to the next without incurring an additional setup cost.

A startup occurs when a machine is set up for a product for which it was not set up in the previous period. When the startup cost is explicitly considered, the fixed startup cost represents the explicit or implicit time costs of setting up the machine for the product. In the multi-item context, this cost can be interpreted as a changeover expense and take the role of the usual setup cost, which can be considered to be a maintenance expense. In a PRP with this type of scenario, the MMPRP with the startup cost (MMPRPSC) is an important extension of the original PRP.

Although huge efforts have been made in studying PRPs with heuristics, e.g., the approximation algorithm (Chandra & Fisher, 1994), decoupled heuristic (Fumero & Vercellis, 1999), greedy randomized adaptive search procedure (Boudia, Louly, & Prins, 2007), memetic algorithm (Boudia & Prins, 2009), tabu search (Armentano, Shiguemoto, Lokketangen, 2011, Bard, Nananukul, 2009b), branch-and-price-based heuristic (Bard, Nananukul, 2009a, Bard, Nananukul, 2010), adaptive large neighborhood search (Adulyasak, Cordeau, & Jans, 2014b), iterative mixed integer programming (Absi et al., 2015), and particle-swarm optimization (Kumar et al., 2016), the research on the exact algorithms has only just begun. Branch-and-cut algorithms for the single-product PRP were investigated only recently (Adulyasak, Cordeau, Jans, 2014a, Archetti, Bertazzi, Paletta, Speranza, 2011). To the best of our knowledge, no exact algorithms have been developed for the MMPRP.

Successful branch-and-cut algorithms for the single-product multi-vehicle PRP can solve problems with up to 50 customers, 3 periods, and 3 vehicles in 2 h using parallel computing (Adulyasak et al., 2014a). A limitation of these algorithms is the lack of LSP-related valid inequalities. As indicated by Adulyasak et al. (2015b) and Díaz-Madroñero, Peidro, and Mula (2015), the LSP is a major component of the PRP. LSP-related valid inequalities are thus expected to enhance the performance of exact methods to solve the PRP. In this work, we plan to incorporate these valid inequalities from the LSP literature and show that they can contribute to exact algorithms for the MMPRPSC. Using these constraints together with valid inequalities from the VRP, we find that the resulting exact algorithms can actually solve larger problems with up to 60 customers, 2 or 3 products, 4 vehicles, and 3 periods in 1 h using a weak big bucket formulation. The successful implementation of the proposed branch-and-cut algorithms might shed light on the future research for the MMPRP with time windows, startup times and other extensions.

From an “expert systems” point of view, to solve the PRP, a knowledge base in VMI is not enough to accomplish feats of apparent intelligence. A priori routing rules must be incorporated into the expert system for the PRP to help construct a working inference engine. The incorporation of artificial-intelligence methods into clustering heuristics and other fast vehicle routing heuristics would help build a solid expert system for the PRP. Solving the PRP in turn can help build a knowledge base when confronted with combinations of NP-hard problems.

The remainder of this work is organized as follows. In Section 2, we provide a formal description of the problem. Three families of valid inequalities are then introduced in Section 3. Based on these valid inequalities and the MILP formulation, we devise a branch-and-cut algorithm in Section 4. We present the computational results of using the algorithm in Sections 5 and 6, followed by conclusions in Section 7.