One Benders cut to rule all schedules in the neighbourhood

TL;DR

This paper addresses the challenge of solving scheduling on unrelated parallel machines with sequence-dependent setups and renewable resource constraints, which is NP-hard. It combines Logic-Based Benders Decomposition (LBBD) with Local Branching by introducing a single neighbourhood cut that eliminates an entire $k$-OPT neighbourhood and a CP-based subproblem to select the best schedule within that neighbourhood. The method is instantiated on two objectives, total completion time and total tardiness, and strengthened with domination rules and strong valid inequalities to tighten lower bounds, yielding faster convergence and smaller optimality gaps. Computational experiments on randomly generated instances with up to 100 jobs show meaningful improvements over standard Branch-and-Check, especially when resource constraints exist, and demonstrate good transferability to related neighbourhoods and sequencing problems. The approach is straightforward to port to related problems and opens avenues for exploring larger neighbourhoods and different sequencing contexts.

Abstract

Logic-Based Benders Decomposition (LBBD) and its Branch-and-Cut variant, namely Branch-and-Check, enjoy an extensive applicability on a broad variety of problems, including scheduling. Although LBBD offers problem-specific cuts to impose tighter dual bounds, its application to resource-constrained scheduling remains less explored. Given a position-based Mixed-Integer Linear Programming (MILP) formulation for scheduling on unrelated parallel machines, we notice that certain $k-$OPT neighbourhoods could implicitly be explored by regular local search operators, thus allowing us to integrate Local Branching into Branch-and-Check schemes. After enumerating such neighbourhoods and obtaining their local optima - hence, proving that they are suboptimal - a local branching cut (applied as a Benders cut) eliminates all their solutions at once, thus avoiding an overload of the master problem with thousands of Benders cuts. However, to guarantee convergence to optimality, the constructed neighbourhood should be exhaustively explored, hence this time-consuming procedure must be accelerated by domination rules or selectively implemented on nodes which are more likely to reduce the optimality gap. In this study, the realisation of this idea is limited on the common 'internal (job) swaps' to construct formulation-specific $4$-OPT neighbourhoods. Nonetheless, the experimentation on two challenging scheduling problems (i.e., the minimisation of total completion times and the minimisation of total tardiness on unrelated machines with sequence-dependent and resource-constrained setups) shows that the proposed methodology offers considerable reductions of optimality gaps or faster convergence to optimality. The simplicity of our approach allows its transferability to other neighbourhoods and different sequencing optimisation problems, hence providing a promising prospect to improve Branch-and-Check methods.

Figures (3)

• Figure 1: An indicative solution for an instance of $5$ jobs on $2$ machines
• Figure 2: An indicative example of moving the starting job $j_{1}$ of machine $m_{1}$ to the starting slot of machine $m_{2}$
• Figure 3: A slot change by three jobs

Theorems & Definitions (5)

• Lemma 1
• Definition 2
• Proposition 3
• Proposition 4
• Proposition 5