A Column Generation Model for a Scheduling Problem with Maintenance Constraints

In this paper we study an allocation and scheduling problem with maintenance constraints. It consists of two difficult sub-problems: a resource allocation sub-problem and a maintenance scheduling sub-problem. A Column Generation model is proposed to solve this problem. In order to be able to find optimal solutions for the relaxed (non integer) problem a model based on a graph with supernodes has been designed. The integer solution is obtained by a Branch and Price heuristic. Results obtained on a set of benchmark instances given by Bouygues show that this algorithm is very satisfactory. It gives the optimal solution for 8 out of 13 instances, and the best known results for the 5 others. 1 Problem description The problem solved in this paper is an allocation and scheduling problem taking into account maintenance constraints. It has been stated by Caseau and Kökény (1998). They present it as an inventory management problem, where resources must be assigned to orders. Each order requires a given type of resource (typically one or two items of equipment). For some problem instances, it is possible to substitute a type of resource by another (i.e. a bigger equipment than the one required for an order). When there are not enough resources, it is possible to rent additional items, and even to buy new ones. In this paper we study the problem without the purchasing possibility. The maintenance constraints are of two types. First, each item of resource cannot be used more than a given amount of time. It has to be sent in the maintenance workshop before being assigned to a new order. Second the workshop has a capacity constraint (i.e., cannot maintain more than two items at the same time). So the scheduling part of the problem deals with the work in the maintenance workshop and consists in determining at which time the workers maintain the items. 2 A Column Generation model 2.1 Column Generation as a hybrid method The Column Generation technique is a pure genuine hybrid algorithm with two co-operating sub-problems using their respective model and algorithm (see Minoux 1983, chapter 10 and Jacquet-Lagrèze 1998-a). It is based on Dantzig-Wolfe decomposition principle which is a generalised linear programming technique.