This paper considers a project scheduling environment in which the activities of the project network are partitioned among a set of agents. Activity durations are controllable, i.e., every agent is allowed to shorten the duration of its activities, incurring a crashing cost. If the project makespan is reduced with respect to its normal value, a reward is offered to the agents and each agent receives a given ratio of the total reward. Agents want to maximize their profit. Assuming a complete knowledge of the agents' parameters and of the activity network, this problem is modeled as a non-cooperative game and Nash equilibria are analyzed. We characterize Nash equilibria in terms of the existence of certain types of cuts on the project network. We show that finding one Nash equilibrium is easy, while finding a Nash strategy that minimizes the project makespan is NP-hard in the strong sense. The particular case where each activity belongs to a different agent 1 is also studied and some polynomial-time algorithms are proposed for this case.
For example, in REF Nash equilibrium is characterized in terms of the existence of certain types of cuts on the project network, and in [9] a solver specifically designed to find an equilibrium in concave games is introduced; it is based on the primaldual interior-point method for nonlinear programming.
Nash equilibria for the multi-agent project scheduling problem with controllable processing times
Journal of Scheduling
Computer Science