The lot–sizing polytope is a fundamental structure contained in many practical production planning problems. Here we study this polytope and identify facet–defining inequalities that cut off all fractional extreme points of its linear programming relaxation, as well as liftings from those facets. We give a polynomial– time combinatorial separation algorithm for the inequalities when capacities ...

We study the classical multistage lot sizing problem that arises in distribution and inventory systems. A celebrated result in this area is the 94% and 98% approximation guarantee provided by power-of-two policies. In this paper, we propose a simple randomized rounding algorithm to establish these performance bounds. We use this new technique to extend several results for the capacitated lot si...

Jobs can be batched if they share the same setup times on a machine in many practical situations. Another reason for batching arises when a machine can process several jobs at the same time. On the other hand, splitting a job or lot which consists of many identical items often increases customer service and decreases throughput time. Especially in CIM (Computer Integrated Manufacturing) these d...

Companies are trying to improve their ability to deliver and to reduce their production costs simultaneously. Nevertheless ongoing orders of customers are causing difficulties to deliver because of bottlenecks in the production. Furthermore constantly incoming customer orders effect that every actual optimal production plan with its already realized and planned guidelines has induced states tha...

A dynamic knapsack set is a natural generalization of the 0-1 knapsack set with a continuous variable studied recently. For dynamic knapsack sets a large family of facet-defining inequalities, called dynamic knapsack inequalities, are derived by fixing variables to one and then lifting. Surprisingly such inequalities have the simultaneous lifting property, and for small instances provide a sign...

We study a multiechelon supply chain model that consists of a production level and several transportation levels, where the demands can exist in the production echelon as well as any transportation echelons. With the presence of stationary production capacity and general cost functions, our model integrates production, inventory and transportation decisions and generalizes existing literature o...

We study the lot-sizing problem with piecewise concave production costs and concave holding costs. This problem is a generalization of the lot-sizing problem with quantity discounts, minimum order quantities, capacities, overloading, subcontracting or a combination of these. We develop a dynamic programming (DP) algorithm to solve this problem and answer an open question in the literature: we s...