Capacitated, finish-to-order production planning with customer ordering day assignments

We examine the production and ordering fulfillment system of a firm offering a large variety of products to time-sensitive customers. The firm wants to combine a finish-to-order strategy with an existing customer ordering system. The finish-to-order policy guarantees significant inventory savings due to the large number of end-product offerings, but it may also increase setup costs due to a short delivery commitment. The ordering system assigns periodic ordering dates to customers, encourages them to place their orders only on these dates, and promises them the same short delivery time irrespective of their geographic locations. The ordering system facilitates a smooth capacity load as well as transportation economies by allowing shipments to each location once during each ordering cycle. We propose a formulation for the combinatorial problem of choosing an ordering-day assignment and a finish-to-order production schedule that minimizes setup costs and maintains a balanced capacity load. We examine a Lagrangian relaxation of this formulation and heuristic solutions based on our relaxation. Our numerical work demonstrates that our best heuristic differs from the optimal solution by 5% on average for small problems (with up to 10 products and 10 customers). Moreover, we show that the difference between the objective value of the relaxation and that of the best heuristic decreases significantly as the problem size increases, being as low as 4% on average for problems with 30 products and 30 customers.