Spatial Resource Constrained Project Scheduling

In this paper, we present a new topic in connection with project scheduling. Since a limited work space can form a restriction to project scheduling problems, an efficient usage of these work spaces (or so-called spatial resources) is required to produce good project schedules. The requirement for the resource units of these spatial resources is in an adjacent manner (we cannot place a product under construction in separate parts of the work space). A spatial resource unit is not required by a single activity, but by a so-called activity group. The requirement for adjacent resource units not only appears in project scheduling, but also in several other applications. For example, the problem of scheduling check-in desks at an airport. The reserved check-in desks for each single flight have to be adjacent. Another example appears in the berthing problem, where quay space has to be assigned to ships of different length, which berth during specified time periods. Within warehouse management and the reservation of hotel rooms, resource adjacency is not required but desirable. The aim of this paper is to derive a solution method which provides an efficient spatial resource usage within project scheduling. This solution method allocates the activity groups to the spatial resources and schedules the activity groups, such that good solution possibilities for the overall project scheduling problem are provided. In this paper, we present an ILP formulation for the group scheduling problem, and we show how the input for the group scheduling problem can be extracted from the overall project scheduling problem. Furthermore, we propose a way of translating back the output of the group scheduling problem to the overall project scheduling problem. Afterwards, we show that the group scheduling problem is NP -hard. Since the size of realistic problem instances for the group scheduling problem is not very large, we test the ILP model of the group scheduling problem to see upto what input sizes the problem can be solved within reasonable time. Before we perform these tests, we present a modification of the ILP formulation in order to decrease the size of the ILP.