J. van de Klundert

Cyclic scheduling in robotic flowshops

Fully automated production cells consisting of flexible machines and a material handling robot have become commonplace in contemporary manufacturing systems. Much research on scheduling problems arising in such cells, in particular in flowshop-like production cells, has been reported recently. Although there are many differences between the models, they all explicitly incorporate the interaction between the materials handling and the classical job processing decisions, since this interaction determines the efficiency of the cell. This paper surveys cyclic scheduling problems in robotic flowshops, models for such problems, and the complexity of solving these problems, thereby bringing together several streams of research that have by and large ignored one another, and describing and establishing links with other scheduling problems and combinatorial topics.

Exact algorithms for procurement problems under a total quantity discount structure

In this paper, we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so-called total quantity discount policy. This policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount. Moreover, the discounted prices apply to all goods bought from the supplier, not only to those goods exceeding the volume threshold. We refer to this cost-minimization problem as the total quantity discount (TQD) problem. We give a mathematical formulation for this problem and argue that not only it is NP-hard, but also that there exists no polynomial-time approximation algorithm with a constant ratio (unless P = NP). Apart from the basic form of the TQD problem, we describe four variants. In a first variant, the market share that one or more suppliers can obtain is constrained. Another variant allows the buyer to procure more goods than strictly needed, in order to reach a lower total cost. We also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought. In a third variant, the number of winning suppliers is limited, both in general and per product. Finally, we investigate a multi-period variant, where the buyer not only needs to decide what goods to buy from what supplier, but also when to do this, while considering the inventory costs. We show that the TQD problem and its variants can be solved by solving a series of min-cost flow problems. Finally, we investigate the performance of three exact algorithms (min-cost flow based branch-and-bound, linear programming based branch-and-bound, and branch-and-cut) on randomly generated instances involving 50 suppliers and 100 goods. It turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time. However, we find that different algorithms perform best in terms of computation time for different variants.

Lifting valid inequalities for the precedence constrained knapsack problem

This paper considers the precedence constrained knapsack problem. More specifically, we are interested in classes of valid inequalities which are facet-defining for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities arising from minimal induced covers, we identify a class of lifting coefficients for which this problem can be solved in polynomial time, by using a supermodular function, and for which the values of the lifting coefficients have a combinatorial interpretation. For the remaining lifting coefficients it is shown that this optimization problem is strongly NP-hard. The same lifting procedure can be applied to (1,k)-configurations, although in this case, the same combinatorial interpretation no longer applies. We also consider K-covers, to which the same procedure need not apply in general. We show that facets of the polytope can still be generated using a similar lifting technique. For tree knapsack problems, we observe that all lifting coefficients can be obtained in polynomial time. Computational experiments indicate that these facets significantly strengthen the LP-relaxation.Abstract.This paper considers the precedence constrained knapsack problem. More specifically, we are interested in classes of valid inequalities which are facet-defining for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities arising from minimal induced covers, we identify a class of lifting coefficients for which this problem can be solved in polynomial time, by using a supermodular function, and for which the values of the lifting coefficients have a combinatorial interpretation. For the remaining lifting coefficients it is shown that this optimization problem is strongly NP-hard. The same lifting procedure can be applied to (1,k)-configurations, although in this case, the same combinatorial interpretation no longer applies. We also consider K-covers, to which the same procedure need not apply in general. We show that facets of the polytope can still be generated using a similar lifting technique. For tree knapsack problems, we observe that all lifting coefficients can be obtained in polynomial time. Computational experiments indicate that these facets significantly strengthen the LP-relaxation.