We study the bin packing problem with minimum color fragmentation (BPPMCF), an extension of the well-known bin packing problem (BPP) in which a given set of weighted colored items has to be packed into a set of identical capacitated bins. Differently from the BPP, in this problem, the number of available bins is fixed and the objective is to minimize the total number of times that colors appear in the bins. After reviewing the integer linear programming models proposed in the literature, we show that one of these models, a flow formulation, shares several features with existing BPP flow formulations. We then exploit these ideas to develop three new flow formulations for the BPPMCF and demonstrate their effectiveness on a set of benchmark instances. We also outline theoretical and empirical dominance relations between the studied flow models. Finally, we empirically show how the number of color fragmentations varies when the number of available bins changes. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the Dutch Ministry of Education and the Air Force Office of Scientific Research [Grant FA8655-25-1-7013]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0972 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0972 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We consider the two bar charts packing problem (2-BCPP), a recent combinatorial optimization problem whose aim is to pack a set of one-dimensional items into the minimum number of bins. As opposed to the well-known bin packing problem, pairs of items are grouped to form bar charts, and a solution is only feasible if the first and second items of every bar chart are packed in consecutive bins. After providing a complete picture of the connections between the 2-BCPP and other relevant packing problems, we show how we can use these connections to derive valid lower and upper bounds for the problem. We then introduce two new integer linear programming (ILP) models to solve the 2-BCPP based on a nontrivial extension of the arcflow formulation. Even though both models involve an exponential number of constraints, we show that they can be solved within a constraint generation framework. We then empirically evaluate the performance of our bounds and exact approaches against an ILP model from the literature and demonstrate the effectiveness of our techniques on both benchmarks inspired by the literature and new classes of instances that are specifically designed to be hard to solve. The outcomes of our experiments are important for the packing community because they indicate that arcflow formulations can be used to solve targeted packing problems with precedence constraints and also that some of these formulations can be solved with constraint generation. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.1256 .

In the Tutor Allocation Problem, the objective is to assign a set of tutors to a set of workshops in order to maximize tutors’ preferences. The problem is solved every year by many universities, each having its own specific set of constraints. In this work, we study the tutor allocation in the School of Mathematics at the University of Edinburgh, and solve it with an integer linear programming model. We tested the model on the 2019/2020 case, obtaining a significant improvement with respect to the manual assignment in use and we showed that such improvement could be maintained while optimizing other key metrics such as load balance among groups of tutors and total number of courses assigned. Further tests on randomly created instances show that the model can be used to address cases of broad interest. We also provide meaningful insights on how input parameters, such as the number of workshop locations and the length of the tutors’ preference list, might affect the performance of the model and the average number of preferences satisfied.

We study pseudo-polynomial formulations for the classical bin packing and cutting stock problems. We first propose an overview of dominance and equivalence relations among the main pattern-based and pseudo-polynomial formulations from the literature. We then introduce reflect, a new formulation that uses just half of the bin capacity to model an instance and needs significantly fewer constraints and variables than the classical models. We propose upper- and lower-bounding techniques that make use of column generation and dual information to compensate reflect weaknesses when bin capacity is too high. We also present nontrivial adaptations of our techniques that solve two interesting problem variants, namely the variable-sized bin packing problem and the bin packing problem with item fragmentation. Extensive computational tests on benchmark instances show that our algorithms achieve state of the art results on all problems, improving on previous algorithms and finding several new proven optimal solutions.