Mathematical Models and Decomposition Algorithms for Cutting and Packing Problems

In this thesis, we provide (or review) new and effective algorithms based on Mixed-Integer Linear Programming (MILP) models and/or decomposition approaches to solve exactly various cutting and packing problems. The first three contributions deal with the classical bin packing and cutting stock problems. First, we propose a survey on the problems, in which we review more than 150 references, implement and computationally test the most common methods used to solve the problems (including branch-and-price, constraint programming (CP) and MILP), and we successfully propose new instances that are difficult to solve in practice. Then, we introduce the BPPLIB, a collection of codes, benchmarks, and links for the two problems. Finally, we study in details the main MILP formulations that have been proposed for the problems, we provide a clear picture of the dominance and equivalence relations that exist among them, and we introduce reflect, a new pseudo-polynomial formulation that achieves state of the art results for both problems and some variants. The following three contributions deal with two-dimensional packing problems. First, we propose a method using Logic based Benders’ decomposition for the orthogonal stock cutting problem and some extensions. We solve the master problem through an MILP model while CP is used to solve the slave problem. Computational experiments on classical benchmarks from the literature show the effectiveness of the proposed approach. Then, we introduce TwoBinGame, a visual application we developed for students to interactively solve two-dimensional packing problems, and analyze the results obtained by 200 students. Finally, we study a complex optimization problem that originates from the packaging industry, which combines cutting and scheduling decisions. For its solution, we propose mathematical models and heuristic algorithms that involve a non-trivial decomposition method. In the last contribution, we study and strengthen various MILP and CP approaches for three project scheduling problems

Hospital-wide therapist scheduling and routing: exact and heuristic methods

In this paper, we address the problem of scheduling and routing physical therapists hospital-wide. At the beginning of a day, therapy jobs are known to a hospital's physical therapy scheduler who decides for each therapy job when, where and by which therapist a job is performed. If a therapist is assigned to a sequence which contains two consecutive jobs that must take place in different treatment rooms, then transfer times must be considered. We propose three approaches to solve the problem. First, an Integer Program (IP) simultaneously schedules therapies and routes therapists. Second, a cutting plane algorithm iteratively solves the therapy scheduling problem without routing constraints and adds cuts to exclude schedules which have no feasible routes. Since hospitals are interested in obtaining quick solutions, we also propose a heuristic algorithm, which schedules therapies sequentially by simultaneously checking routing and resource constraints. Using real-world data from a hospital, we compare the performance of the three approaches. Our computational analysis reveals that our IP formulation fails to solve test, which have more than~30 jobs, to optimality in an acceptable solution time. In contrast, the cutting plane algorithm can solve instances with more than 100 jobs optimally. The heuristic approach obtains good solutions for large real-world instances within fractions of a second