Compatibility graphs in scheduling on batch processing machines

TL;DR

This paper studies scheduling on identical batch processing machines with compatibility constraints modeled by a graph, providing polynomial algorithms for subproblems, MILP formulation, heuristics, and computational experiments.

Contribution

It introduces polynomial algorithms for certain subproblems, a MILP model for the general problem, and heuristic methods, advancing scheduling with compatibility constraints.

Findings

Several subproblems are polynomially solvable.

The MILP formulation effectively models the general problem.

Heuristics provide practical solutions with good performance.

Abstract

We consider the problem of minimizing the makespan on batch processing identical machines, subject to compatibility constraints, where two jobs are compatible if they can be processed simultaneously in a same batch. These constraints are modeled by an undirected graph $G$, in which compatible jobs are represented by adjacent vertices. We show that several subproblems are polynomial. We propose some exact polynomial algorithms to solve these subproblems. To solve the general case, we propose a mixed-integer linear programming (MILP) formulation alongside with heuristic approaches. Furthermore, computational experiments are carried out to measure the performance of the proposed methods.