The bilevel optimisation of a multi-agent project scheduling and staffing problem

In this paper, we study a multi-agent project staffing problem involving a single project, which has to be scheduled under resource constraints. We consider a functional organisational structure where a team leader and a project manager are together responsible for the operational execution of a project. The team leader, which has the formal authority over the resources, is standing at the top of the hierarchy and determines the number and mix of (additional) employees and tries to level the workload over the planning period in order to avoid idle resource times. The project manager is responsible for the scheduling of the project activities and his/her objective is to minimise the project duration. The interaction between both agents in the decision-making process is, on the one hand, hierarchical, i.e. the team leader imposes his/her decision on the project manager. On the other hand, the decision taken by the team leader should comply to the objective of the project manager such that the staffing plan and project schedule is agreed by both parties. We propose a bilevel optimisation model that embeds a nested inner optimisation problem, i.e. the project manager decision problem, as a constraint in the outer optimisation problem, i.e. the decision problem of the team leader. The algorithm is a mathematical programming method thriving on the generation of additional lazy constraints via feasibility callbacks so that the team leader problem has to respect the requirements formulated in the project manager problem. In the computational experiments, we compare this solution approach to alternative classical optimisation approaches and we validate the design choices related to the proposed speed-up mechanisms and parameter settings.