Computational study of a column generation algorithm for bin packing and cutting stock problems

In this paper, we propose a generalised 4-level integrated lot-sizing and cutting stock problem that addresses key decisions within the supply chain, including the availability of multiple suppliers for purchasing raw materials that will be cut into small pieces, and the transportation of final products to a warehouse. To solve this integrated problem, we propose hybrid heuristics designed to overcome its inherent complexities. The hybrid methods incorporate two decomposition approaches within each iteration: a column generation procedure and a relax-and-fix procedure. Given the specific characteristics of the problem, we propose an innovative column generation approach to tackle cutting patterns in the cutting stock problem and cargo configurations in the transportation problem. First, the performance of the integrated approach is compared to a sequential approach which is commonly used in practice, showing the benefits of integrating processes in an industrial setting. Next, the proposed methods are employed to assess the impact of integrating these decisions across the supply chain, and the performance is compared against a benchmark heuristic from the literature to assess their effectiveness in solving this integrated problem.

We are given a set of items, each characterized by a weight and a fragility, and a large number of uncapacitated bins. Our aim is to find the minimum number of bins needed to pack all items, in such a way that in each bin the sum of the item weights is less than or equal to the smallest fragility of an item in the bin. The problem is known in the literature as the Bin Packing Problem with Fragile Objects, and appears in the telecommunication field, when one has to assign cellular calls to available channels by ensuring that the total noise in a channel does not exceed the noise acceptance limit of a call.We propose several techniques to compute lower and upper bounds for this problem. For what concerns lower bounds, we present combinatorial techniques with guaranteed worst case and a more complex bound based on a column generation algorithm. We also present a technique to compute, in a fast heuristic way, dual information that is used to strengthen the convergence of the column generation. For what concerns upper bounds, we present a large set of constructive heuristics followed by a Variable Neighborhood Search algorithm. Our heuristic techniques are aimed at both computing upper bounds and strengthening the behavior of the lower bounds in a matheuristic fashion. Extensive computational tests show the effectiveness of the proposed algorithms.

Column generation is a fundamental technique for solving large-scale combinatorial optimization problems such as unit commitment and vehicle routing, yet its performance is often limited by dual oscillation. This study explores the intrinsic cause of this phenomenon from the perspective of shadow price theory and demonstrates that dual oscillation arises from the lack of marginal interpretability of Lagrange multipliers when multiple dual solutions coexist. To address this issue, an improved column generation framework is proposed in which traditional multipliers are replaced with minimum-norm multipliers that possess clear economic meaning and act as directional shadow prices. A generalized pricing subproblem is formulated, and partial minimum-norm multipliers are obtained through convex quadratic optimization to guide column generation. Numerical experiments on a simplified single-period unit commitment case and large-scale cutting stock problems showed that the proposed approach eliminated invalid column generation and achieved speedy convergence to the optimal solution within only two iterations for the unit commitment case, and the classical column generation exhibited slow convergence with dual oscillation in large-scale scenarios while the improved algorithm achieved fast and stable convergence. The results indicate that the stabilization method enhances the consistency of dual variables and provides a more robust foundation for the theoretical and practical development of column generation algorithms.

Algorithms for the one-dimensional two-stage cutting stock problem

Approximation and online algorithms for multidimensional bin packing: A survey

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.

In difference to one-dimensional Bin Packing Problems (1BPP) where each item is considered to be a unique one, in a one-dimensional Cutting Stock Problem (1CSP), the number of different piece types is rather small, but their order demands (or availability in case of packing problems) are mostly large. Another differentiator of bin packing and cutting stock problems could be the magnitude of the respective optimal value. If it is small in comparison to the total number of items, then the instance is of BPP type, otherwise the problem type depends on the number of different patterns in a solution.

Mathematical models for the two-dimensional variable-sized cutting stock problem in the home textile industry

Solving real-world cutting stock-problems in the paper industry: Mathematical approaches, experience and challenges

Multi-capacity bin-packing is a generalization of the classical one-dimensional bin-packing problem in which the bin capacity and the item sizes are represented by d-dimensional vectors. Previous work in d-capacity bin-packing algorithms analyzed variants of single capacity bin-packing, extended to deal with the d capacities. These extended algorithms are oblivious to the additional capacity information, however, and do not scale well with increasing d. We provide new packing algorithms which use the additional capacity information to provide better packing and show how these algorithms might lead to better multi-resource allocation and scheduling solutions.

We investigate a two-dimensional two-stage cutting stock problem (2D-2CP) with multiple stock sizes when fixed-size usable leftovers are considered. A fixed-size usable leftover is an object with the predefined size that could be used later. Fixed-size usable leftovers can reduce the waste area and therefore can help construct better cutting patterns. A mathematical model is proposed in a set-partitioning form where the sub-problems corresponding to two-dimensional knapsack problem (2DKP) with fixed-size usable leftovers are generated for optimality testing. A column generation technique is applied in an attempt to find a solution that minimises a total waste. Moreover, two different practical integer solution finding strategies are proposed. The algorithms are tested on a real-world dataset using various initial solution patterns and analysed. Additionally, the effect of the usable leftovers is investigated when compared with the prior literature that used the same dataset on the 2D-2CP problem without considering usable leftovers.

This paper presents a hybrid multi-chromosome genetic algorithm (HMCGA) to solve an in integer linear programming formulation of the Cutting Stock Problem (CSP). The CSP is an important class combinatorial problem. It is appropriate to minimize the raw material used by industries for fulfilling customer’s demands. In such cases, classic models for solving the cutting stock problem are useless. HMCGA differ from previous application of genetic algorithm to CSP in that our coding contain tow chromosome, one represents the cutting pattern and another represents the frequencies of the cutting patterns in the first chromosome, rather than relying on a traditional genetic algorithm to decode each individual solution. Results obtained from computational experiments for ten benchmarks demonstrate that the performance of HMCGA is compared to that obtained using existing mete-heuristic algorithm.

In this paper, we introduce the Electric Vehicle Routing Problem with Flexible Time Windows (EVRPFTW) in which vehicles are allowed to serve customers before and after the earliest and latest time window bounds, respectively. The objective of this problem is to assign electric vehicles to feasible routes and make schedules with minimum total cost that includes the traveling costs, the costs of using electric vehicles and the penalty costs incurred for earliness and lateness. The proposed mathematical model is solved by a column generation procedure. To generate an integer solution, we solve an integer programming problem using the routes constructed by the column generation algorithm. We further develop a linear programming model to compute the optimal times to start service at each customer for the selected routes. A number of well-known benchmark instances is solved by our solution procedure to evaluate the operational gains obtained by employing flexible time windows.

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm.

In this paper, we address the electric vehicle routing problem with time windows and propose two branch-and-price-and-cut methods based on a column generation algorithm. One is an exact algorithm whereas the other is a heuristic method. The pricing sub-problem of the column generation method is solved using a label correcting algorithm. The algorithms are strengthened with the state-of-the-art acceleration techniques and a set of valid inequalities. The acceleration techniques include: (i) an intermediate column pool to prevent solving the pricing sub-problem at each iteration, (ii) a label correcting method employing the ng-route algorithm adopted to our problem, (iii) a bidirectional search mechanism in which both forward and backward labels are created, (iv) a procedure for dynamically eliminating arcs that connect customers to remote stations from the network during the path generation, (v) a bounding procedure providing early elimination of sub-optimal routes, and (vi) an integer programming model that generates upper bounds. Numerical experiments are conducted using a benchmark data set to compare the performances of the algorithms. The results favour the heuristic algorithm in terms of both the computational time and the number of instances solved. Moreover, the heuristic algorithm is shown to be specifically effective for larger instances. Both algorithms introduce a number of new solutions to the literature.