Lower bounds for a multi-skill project scheduling problem

This paper deals with an extension of the classical Resource Constrained Project Scheduling Problem (RCPSP). We present a new type of resource constraints in which staff members are involved. On the one hand, we focus on staff members, each of them having several skills, i.e, is able to perform more than one kind of activity. On the other hand, an activity has specific skill requirements that must be satisfied. To solve this problem, we propose two lower bounds. The first one uses a linear programming scheme proposed for the RCPSP and the second one is based on energetic reasoning. 1. Presentation of the Multi-Skill Project Scheduling Problem The Multi-Skill Project Scheduling Problem (MSPSP) mixes both the classical RCPSP, and the Multi-Purpose Machine model [3], [4]. From the RCPSP, we use the project description, and we add new resource constraints inspired by the Multi-Purpose Machine model. For instance let us consider that resources are staff members having more than one skill, and that each activity needs a "given amount" of skills to be performed. Thus scheduling an activity at time t, requires matching its skill requirements with the skills of the staff members that are available at t. Our goal is to minimize the overall project duration, i.e, min(Cmax). Figure 1 presents a 4-activities and 4-members example with a feasible solution. Table 1.a gives the processing times of actives along with their skill requirements. Table 1.b, describes staff members in terms of skills. Figure 1.a, presents the precedence constraints between activities. Figure 1.b, shows a feasible solution. Figure 1 : Example of Multi-Skill Project Scheduling Problem (MSPSP) In the example above, one can see that activity A3 cannot start at time 2, because it requires 2 programmers, while the available staff members at this time cannot meet this skill requirement (P1 is not a programmer). A1