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A conditional-logic interpretation for Miller–Tucker–Zemlin inequalities and extensions

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Abstract

Routing problems that seek to traverse a set of cities are faced with the challenge of avoiding subtours. To address this challenge, attention has been given to devising subtour elimination constraints. A classical approach is through the use of “Miller–Tucker–Zemlin” (MTZ) inequalities. MTZ inequalities have the advantage of being few in number but have the disadvantage of yielding weak continuous relaxations. As a result, strengthenings have been computed over time in a seemingly unrelated fashion. In this note, we provide a unifying conditional-logic interpretation of MTZ inequalities for the Traveling Salesman Problem (TSP). Our emphasis is on linear inequalities but our analysis also provides a new family of tightened quadratic forms. We apply the interpretation to the more general Capacitated Vehicle Routing Problem (CVRP), both explaining existing and motivating new inequalities.

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Acknowledgements

This work of the first author and the third author was supported in part by the Office of Naval Research under Grant Numbers N00014-16-1-2168 and N00014-20-1-2154, respectively.

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Audrey Dietz

  2. School of Mathematical and Statistical Sciences (emeritus), Clemson University, Clemson, SC, 29634, USA

    Warren Adams

  3. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634, USA

    Boshi Yang

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  1. Audrey Dietz
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  2. Warren Adams
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  3. Boshi Yang
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Correspondence to Boshi Yang.

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Dietz, A., Adams, W. & Yang, B. A conditional-logic interpretation for Miller–Tucker–Zemlin inequalities and extensions. Optim Lett 17, 245–264 (2023). https://doi.org/10.1007/s11590-022-01947-w

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