Requiem for the Miller-Tucker-Zemlin subtour elimination constraints?

Bektas, T. and Gouveia, L. (2014) Requiem for the Miller-Tucker-Zemlin subtour elimination constraints? European Journal of Operational Research, 236 (3), 820-832. (doi:10.1016/j.ejor.2013.07.038).

Record type: Article

Abstract

The Miller-Tucker-Zemlin (MTZ) subtour elimination constraints (SECs) and the improved version by Desrochers and Laporte (DL) have been and are still in regular use to model a variety of routing problems. This paper presents a systematic way of deriving inequalities that are more complicated than the MTZ and DL inequalities and that, in a certain way, ``generalize" the underlying idea of the original inequalities. We present a polyhedral approach that studies and analyses the convex hull of feasible sets for small dimensions. This approach allows us to generate generalizations of the MTZ and DL inequalities, which are ``good" in the sense that they define facets of these small polyhedra. It is well known that DL inequalities imply a subset of Dantzig-Fulkerson-Johnson (DFJ) SECs for two-node subsets. Through the approach presented, we describe a generalization of these inequalities which imply DFJ SECs for three-node subsets and show that generalizations for larger subsets are unlikely to exist. Our study presents a similar analysis with generalizations of MTZ inequalities and their relation with the lifted circuit inequalities for three node subsets.

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