Branch-and-cut methods are among the most important techniques for solving integer programming problems. They can also be used to prove that all solutions of an integer program satisfy a given linear inequality. We examine the complexity of branch-and-cut proofs in the context of 0-1 integer programs. We prove an exponential lower bound on the length of branch-and-cut proofs which use 0-1 branc...

Different types of formulations are proposed in the literature to model vehicle routing problems. Currently, the most used ones can be fitted into two classes, namely vehicle flow formulations and set partitioning formulations. These types of formulations differ from each other not only due to their variables and constraints but also due to their main features. Vehicle flow formulations have th...

We are given a rectangle R in the plane and a nite set P of n points in the interior of R. A rectangular partition of R is a partition of the surface of R into smaller rectangles. A rectangular partition is said to be feasible with respect to P if no points in P lie in the strict interior of any rectangle of the partition. The length of a rectangular partition is computed as the sum of the leng...

This is a report on how mixed integer programming works. It starts by showing the form of a mixed integer program with n variables and m constraints. They explain the branch and bound method which is the most widely used method for solving integer programs. It goes into some detail about how the variables have only two possible restrictions, one and zero. There’s also some mention of the branch...

In this work we study a suppliers selection and routing problem where a fleet of homogeneous vehicles with a predefined capacity is available for procuring different products from different suppliers with the aim to minimize both the traveling and the purchasing costs. Decisions are further complicated by the presence of pairwise incompatibility constraints among products, implying the impossib...

In this article, we provide an overview on how the maximum weighted stable set problem can be solved exactly with Branch & Cut techniques. In addition, we provide selected references to other exact methods. We start with a brief introduction of the stable set problem and a few basic definitions but assuming that the reader is already familiar with the basic concepts. The main stress of this art...

The production routing problem (PRP) is a generalization of the inventory routing problem and concerns the production and distribution of a single product from a production plant to multiple customers using capacitated vehicles in a discrete and finite time horizon. In this study, we consider the stochastic PRP with demand uncertainty in two-stage and multi-stage decision processes. The decisio...

The Quadratic Knapsack Problem (QKP) is a well-known NP-hard combinatorial optimisation problem, with many practical applications. We present ‘cut-and-branch’ algorithm for the QKP, in which cutting-plane phase followed by branch-and-bound phase. more sophisticated than existing ones literature, incorporating several classes of cutting planes, two primal heuristics, and rules eliminating variab...