Enhanced pseudo-polynomial formulations for bin packing and cutting stock problems

We study pseudopolynomial 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 pseudopolynomial 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

Simple and practical optimization approach based to solve a truck load and delivery problem at long haul distances with heterogenous products

This paper proposes an optimization based approach for solving the logistic processes of deliveries scheduling and product accommodation during loading with a heterogeneous fleet of vehicles. The approach focuses on the case of products with “low density values” and high heterogeneous volume and weight, and with traveling large distances to different zones, in which transportation costs constitute a important proportion of total logistic costs. The proposed approach consists of a two-phase strategy: The first uses a “Cutting Stock Problem” formulation to define utilization areas inside trucks assigned to each product family. This task is achieved by minimizing the long-haul transportation costs as a function of the vehicle size, considering a set of predefined solutions for feasible and efficient loading obtained as a result of the accumulated experience. The second phase consists of Bin Packing Problem version with a known number of bins, which were previously determined in the first phase of the approach. In this phase, different orders from a set of customers are assigned to each truck by obeying the predefined utilization areas per product category obtained in the first phase while minimizing the number of visits of each truck. The results show that the model addresses the analyzed problem in an efficient manner, which is reflected in reasonable resolution times and costs from a practical implementation perspective. Additionally, it is observed that long-haul delivery costs and vehicle utilization tend to improve with the increase of the utilized number of patterns even when the execution time is incremented.MaestríaMagister en Ingeniería Civi

Algorithms for the Bin Packing Problem with Scenarios

This paper presents theoretical and practical results for the bin packing problem with scenarios, a generalization of the classical bin packing problem which considers the presence of uncertain scenarios, of which only one is realized. For this problem, we propose an absolute approximation algorithm whose ratio is bounded by the square root of the number of scenarios times the approximation ratio for an algorithm for the vector bin packing problem. We also show how an asymptotic polynomial-time approximation scheme is derived when the number of scenarios is constant. As a practical study of the problem, we present a branch-and-price algorithm to solve an exponential model and a variable neighborhood search heuristic. To speed up the convergence of the exact algorithm, we also consider lower bounds based on dual feasible functions. Results of these algorithms show the competence of the branch-and-price in obtaining optimal solutions for about 59% of the instances considered, while the combined heuristic and branch-and-price optimally solved 62% of the instances considered

ALGORITHMS FOR OPTIMIZATION PROBLEMS WITH FRACTIONAL RESOURCES

We consider a class of optimization problems having a distinctive feature: both discrete and continuous decisions need to be taken simultaneously. These problems arise in many practical applications, for example broadband telecommunications and green transportation problems, where resources are available, that can be fractionally consumed or assigned. These problems are proven of being harder than their purely discrete counterpart. We propose effective methodologies to tackle them. Our approach is to consider variants of classical combinatorial optimization problems belonging to three domains: packing, routing and integrated routing/packing. Our results suggest that indeed effective approaches exist, reducing the computational effort required for solving the problem. Mostly, they are based on exploiting the structure of optimal solutions to reduce the search space

Arc flow formulations based on dynamic programming: Theoretical foundations and applications

Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow formulations is the arc flow, where variables represent flows on individual arcs of the network. For NP-hard problems, polynomial-sized arc flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of pseudo-polynomial arc flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation. In this survey, we present theoretical foundations of pseudo-polynomial arc flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc flow models. We also present a dual point of view to contrast the linear relaxation of arc flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc flow models based on DP in several domains such as cutting, packing, scheduling, and routing

An adaptive jellyfish search algorithm for packing items with conflict

The bin packing problem (BPP) is a classic combinatorial optimization problem with several variations. The BPP with conflicts (BPPCs) is not a well-investigated variation. In the BPPC, there are conditions that prevent packing some items together in the same bin. There are very limited efforts utilizing metaheuristic methods to address the BPPC. The current methods only pack the conflict items only and then start a new normal BPP for the non-conflict items; thus, there are two stages to address the BPPC. In this work, an adaption of the jellyfish metaheuristic has been proposed to solve the BPPC in one stage (i.e., packing the conflict and non-conflict items together) by defining the jellyfish operations in the context of the BPPC by proposing two solution representations. These representations frame the BPPC problem on two different levels: item-wise and bin-wise. In the item-wise solution representation, the adapted jellyfish metaheuristic updates the solutions through a set of item swaps without any preference for the bins. In the bin-wise solution representation, the metaheuristic method selects a set of bins, and then it performs the item swaps from these selected bins only. The proposed method was thoroughly benchmarked on a standard dataset and compared against the well-known PSO, Jaya, and heuristics. The obtained results revealed that the proposed methods outperformed the other comparison methods in terms of the number of bins and the average bin utilization. In addition, the proposed method achieved the lowest deviation rate from the lowest bound of the standard dataset relative to the other methods of comparison

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