Grammar-Based Integer Programming Models for Multiactivity Shift Scheduling

Abstract

This paper presents a new implicit formulation for shift scheduling problems, using context-free grammars to model the rules for the composition of shifts. From the grammar, we generate an integer programming (IP) model having a linear programming relaxation equivalent to that of the classical set covering model. When solved by a state-of-the-art IP solver on problem instances with a small number of shifts, our model, the set covering formulation, and a typical implicit model from the literature yield comparable solution times. On instances with a large number of shifts, our formulation shows superior performance and can model a wider variety of constraints. In particular, multiactivity cases, which cannot be modeled by existing implicit formulations, can easily be handled with grammars. We present comparative experimental results on a large set of instances involving one work activity, as well as on problems dealing with up to 10 work activities. This paper was accepted by Dimitris Bertsimas, optimization.shift scheduling, implicit models, mixed integer programming, context-free grammars

Orbital shrinking: Theory and applications

“…Consider for example the regular polytope in its extended form: the optimal solution is always a flow of integral value, say k, and basic network flow theory guarantees that it can be decomposed into k paths of unitary flow (and since each path in the expanded graph corresponds to a word in the language, this is a feasible solution for the original explicit problem). Similar reasoning applies to the grammar polytope (although it is not a flow model), as successfully shown in Côté et al (2011b). It is interesting to note that this gives the current state-of-the-art for solving multi-activity shift scheduling problems.…”

“…This means that we disable dual reductions (otherwise the decomposition would not be correct) and use CPLEX callbacks framework to implement the decomposition. Table 3 reports a comparison between the proposed method and others in the literature, for a number of activities from 1 to 10. cpx-reg refers to the explicit model based on the regular constraint in Côté et al (2011a), while grammar refers to the implicit model based on the grammar constraint in Côté et al (2011b). Note that for grammar we are reporting the results from Côté et al (2011b), which were obtained on a different machine and, more importantly, with an older version of CPLEX, so the numbers are meant to give just a reference.…”

“…First of all, it has been proven for both the regular and context-free languages that the derived polyhedron is integral Pesant et al (2009), and thus, if the are no other constraints, it is possible to optimize a linear function over the set of feasible shifts by solving just a linear program. Even more importantly, these results have been extended also to polyhedra describing sets of feasible shifts Côté et al (2011b). It is then possible to consider an aggregated (implicit) model and reconstruct an optimal solution of the original one with a polynomial post-processing phase.…”

Nurse rostering: models and algorithms for theory, practice and integration with other problems

“…Côté et al [36] and Musliu et al [96] represent pre-emptive tasks as varying staffing requirements in intervals. Côté et al [36] use implicit models with context-free grammars to model complex rules regarding shift design.…”

“…Côté et al [36] use implicit models with context-free grammars to model complex rules regarding shift design. Musliu et al [96] study the minimum shift design problem in which the goal is to decide on an efficient shift structure and a minimal workforce that can carry out all the work without, however, explicitly assigning tasks within the shifts.…”