Production planning problems and its variants are widely studied in operations management and optimization literature. One variation that has not garnered much attention is the presence of multiple production families in a coordinated and capacitated lot-sizing setting. While its single-family counterpart has been the subject of many advances in formulations and solution techniques, the latest published research on multiple family problems was over 25 years ago (Erenguc and Mercan, 1990; Mercan and Erenguc, 1993).
Chapter 2 begins with a new formulation for this coordinated capacitated lot-sizing problem for multiple product families where demand is deterministic and time-varying. The problem considers setup and holding costs, where capacity constraints limit the number of individual item and family setup times and the amount of production in each period. We use a facility location reformulation to strengthen the lower bound of our demand-relaxed model. In addition, we combine Benders decomposition with an evolutionary algorithm to improve upper bounds on optimal solutions. To assess the performance of our approach, single-family problems are solved and results are compared to those produced by state-of-the-art heuristics by de Araujo et al. (2015) and Süral et al. (2009). For the multi-family setting, we first create a standard test bed of problems, then measure the performance of our heuristic against the SDW heuristic of Süral et al. (2009), as well as an exact cutting plane approach we developed based on combinatorial Benders decomposition. We show that our Benders approach combined with an evolutionary algorithm consistently achieves better bounds, reducing the duality gap compared to other single-family methods studied in the literature.
Lot-sizing problems also exist within a vendor-managed-inventory setting, with production planning, distribution and vehicle routing problems all solved simultaneously. By considering these decisions together, companies achieve reduced inventory and transportation costs compared to when these decisions are made sequentially. We present in Chapter 3 a branch-and-cut algorithm to tackle a production-routing problem (PRP) consisting of multiple products and customers served by a heterogeneous fleet of vehicles. To accelerate the performance of this algorithm, we also construct an upper bounding heuristic that quickly solves production-distribution and routing subproblems, providing a warm-start for the branch-and-cut procedure. In four scenarios, we vary the degree of flexibility in demand and transportation by considering split deliveries and backorders, two settings that are not commonly studied in the literature. We confirm that our upper bounding procedure generates high quality solutions at the root node for reasonably-sized problem instances; as time horizons grow longer, solution quality degrades slightly. Overall costs are roughly the same in these scenarios, though cost proportions vary. When backorders are not allowed (Scenarios 1 and 3), inventory holding costs account for over 90% of total costs and transportation costs contribute less than 0.01%. When backorders are allowed (Scenarios 2 and 4), most of the cost burden is shouldered by production, with transportation inching closer to 0.1% of total costs.
In our fifth scenario for the PRP with multiple product families, we employ a decomposition heuristic for determining dedicated routes for distribution. Customers are clustered through k-means++ and a location-allocation subproblem based on their contribution to overall demand, and these clusters remain fixed over the entire planning horizon. A routing subproblem dictates the order in which to visit customers in each period, and we allow backorders in the production-distribution routine. While the branch-and-cut algorithm for Scenarios 1 through 4 quickly finds high quality solutions at the root node, Scenario 5’s dedicated routes heuristic boasts high vehicle utilization and comparable overall costs with minimal computational effort.
Prof. James Bookbinder
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