# Mixed integer formulations using natural variables for single machine scheduling around a common due date

## TL;DR

This paper introduces new mixed integer programming formulations for single machine scheduling around a common due date, using natural variables, and demonstrates their theoretical validity and practical effectiveness through algorithms and experiments.

## Contribution

It proposes novel compact and non-compact mixed integer formulations with polynomial separation algorithms, advancing optimization methods for scheduling problems.

## Key findings

- The non-overlapping inequalities formulation is valid and solvable in polynomial time.
- The proposed Branch-and-Cut algorithm effectively solves the scheduling problem.
- Experimental results show practical relevance of the formulations and algorithms.

## Abstract

While almost all existing works which optimally solve just-in-time scheduling problems propose dedicated algorithmic approaches, we propose in this work mixed integer formulations. We consider a single machine scheduling problem that aims at minimizing the weighted sum of earliness tardiness penalties around a common due-date. Using natural variables, we provide one compact formulation for the unrestrictive case and, for the general case, a non-compact formulation based on non-overlapping inequalities. We show that the separation problem related to the latter formulation is solved polynomially. In this formulation, solutions are only encoded by extreme points. We establish a theoretical framework to show the validity of such a formulation using non-overlapping inequalities, which could be used for other scheduling problems. A Branch-and-Cut algorithm together with an experimental analysis are proposed to assess the practical relevance of this mixed integer programming based methods.