Multi-mode resource-constrained project scheduling problem with time-dependent resource costs and capacities: A bi-objective approach

This research explores a realistic variant of the Multi-mode Resource-Constrained Project Scheduling Problem (MRCPSP), a well-known optimization problem in project planning where each activity can be executed in one of several modes under limited resources. Each execution mode represents an alternative combination of resource requirements and activity duration. We extend the classical setting by introducing time-dependent resource costs and capacities, meaning that the availability and cost of a given resource depend not only on the resource itself, but also on the specific time periods during which it is employed—a situation that often arises in real-world scenarios. Furthermore, we adopt a multi-objective framework, with the objectives of minimizing the makespan and the total cost of resource usage. A mathematical formulation is presented for this problem, and an attempt is made to solve it using an exact method to find the Pareto front (PF), i.e., the set of non-dominated solutions where no objective can be improved without deteriorating another. Nonetheless, the inability of exact techniques to handle large-sized instances of this hard problem motivates the development of a metaheuristic to approximate the PF. We design an innovative multi-objective genetic algorithm that effectively integrates the new problem elements to build high-quality solutions. A comprehensive computational study is conducted on a benchmark set of instances for this novel bi-criteria, multi-mode problem. The results demonstrate the effectiveness of the proposed metaheuristic.