State-Expanded Network Models for Multi-Activity Shift Scheduling

Abstract

This paper presents new formulations for multi-activity shift scheduling problems (MASSP). In these formulations, the rules governing the feasibility of shifts are encoded in state-expanded networks in which nodes are associated with states and arcs represent assignments inducing state transitions. These networks are embedded into a mixed-integer linear programming model as network flow components. A key advantage of the formulation is that for the anonymous MASSP in which all employees are considered as equal only a single network with integer flow variables is needed as long as the network encodes all shift composition rules. A challenging aspect of this type of formulation is that the networks can become very large, yielding huge model instances that are hard to solve for large problem instances. In order to address this issue, this paper proposes two exact techniques: First, it introduces a set of aggregate side constraints implicitly enforcing that an integer flow solution can be decomposed into feasible shifts. Second, it proposes to decouple the shift composition from the assignment of concrete activities to blocks of work periods. This results in two model components, both based on state-expanded networks, that are coupled to each other using linking constraints. In a computational study with two sets of MASSP instances from the literature, we demonstrate the effectiveness of these techniques for reducing the both size of the model instances and the computation time: We are able to solve all instances, including more that 70 previously open problem instances, to optimality -- the vast majority of them in less than 30 minutes on a notebook computer.