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

binary knapsack

Lagrangian relaxation


unrelated parallel machine problem


Edvin Åblad

Fraunhofer-Chalmers Centre

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Ann-Brith Strömberg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Domenico Spensieri

Chalmers, Industrial and Materials Science, Product Development

Open Journal of Mathematical Optimization

2777-5860 (eISSN)

Vol. 2 1-15 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

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

Swedish Foundation for Strategic Research (SSF) (RIT15-0025), 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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