An Evolutionary Approach to the Resource-Constrained Project Scheduling Problem

The resource-constrained project-scheduling problem (RCPSP ) may be stated as follows. A project consists of a set of n activities numbered 1 to n, where each activity has to be processed without interruption to complete the project. The dummy activities 1 and n represent the beginning and end of the project. The duration of an activity j is denoted by dj where d1 = dn = 0. There are R renewable resource types. The availability of each resource type k in each time period is Rk units, k = 1, ..., R. Each activity j requires rjk units of resource k during each period of its duration where r1k = rnk = 0, k = 1, ..., R. All parameters are assumed to be non-negative integer valued. There are precedence relations of the finish-start type with a zero parameter value (i.e., FS = 0) defined between the activities. In other words, activity i precedes activity j if j cannot start until i has been completed. The structure of a project can be represented by an activity-on-node network G = (V,A), where V is the set of activities and A is the set of precedence relationships. Sj(Pj) is the set of successors (predecessors) of activity j. It is assumed that 1 ∈ Pj , j = 2, ..., n, and n ∈ Sj , j = 1, ..., n − 1. The objective of the RCPSP is to find a schedule S of the activities, i.e., a set of starting times (s1, s2, ..., sn) where s1 = 0 and the precedence and resource constraints are satisfied, such that the schedule duration T (S) = sn is minimised. Let T ∗ be the minimum schedule duration or minimum makespan.