Minimizing the makespan in job shop scheduling under conflict graph constraints

Abstract

We study the job shop scheduling problem with a conflict graph (JSC), in which adjacent jobs in the conflict graph cannot be processed simultaneously on different machines, with the objective of minimizing the makespan. The problem models settings where jobs share additional resources while retaining their individual machine routings. We first investigate its computational complexity and establish a polynomial equivalence between JSC and a variant of the resource-constrained job shop problem with unit-capacity resources. Although the general problem on two machines is NP-hard, we identify a polynomially solvable special case. For the general problem, we develop precedence-based and time-indexed mixed-integer linear formulations, along with lower bounds on the makespan. We also propose a genetic algorithm using permutation-with-repetition encoding and active, non-delay, and hybrid schedule evaluation procedures. Computational experiments on instances derived from the Lawrence and Taillard benchmarks, as well as randomly generated generalized job shop instances, are conducted to evaluate the performance of the proposed formulations, lower bounds, and genetic algorithm.