Exact makespan minimization of unrelated parallel machines
Journal article, 2021

We study methods for the exact solution of the unrelated parallel machine problem with makespan minimization, generally denoted as R||Cmax. Our original application arises from the automotive assembly process where tasks needs to be distributed among several robots. This involves the solutions of several R||Cmax instances, which proved hard for a MILP solver since the makespan objective induces weak LP relaxation bounds. To improve these bounds and to enable the solution of larger instances, we propose a branch–and–bound method based on a Lagrangian relaxation of the assignment constraints. For this relaxation we derive a criterion for variable fixing and prove the zero duality gap property for the case of two parallel machines. Our computational studies indicate that the proposed algorithm is competitive with state-of-the-art methods on different types of instances. Moreover, the impact of each proposed feature is analysed.

variable fixing


Lagrangian relaxation

unrelated parallel machine problem

binary knapsack


Edvin Åblad

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Fraunhofer-Chalmers Centre

Ann-Brith Strömberg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Domenico Spensieri

Fraunhofer-Chalmers Centre

Chalmers, Industrial and Materials Science, Product Development

Open Journal of Mathematical Optimization

2777-5860 (eISSN)

Vol. 2 2

Smart Assembly 4.0

Swedish Foundation for Strategic Research (SSF) (RIT15-0025), 2016-05-01 -- 2021-06-30.

Interlinked combinatorial and geometrical optimization problems in an autonomous automotive manufacturing industry

Swedish Foundation for Strategic Research (SSF) (RIT15-0025), 2017-08-15 -- 2022-09-05.

Fraunhofer-Chalmers Centre, 2017-08-15 -- 2022-09-05.

Areas of Advance

Information and Communication Technology


Subject Categories

Computational Mathematics

Other Mathematics


Discrete Mathematics


Basic sciences

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