A goal-driven prototype column generation strategy for the multiple container loading cost minimization problem

In the multiple container loading cost minimization problem (MCLCMP), rectangular boxes of various dimensions are loaded into rectangular containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally modeled as a set cover problem. We generalize the set cover formulation by introducing a new parameter to model the gross volume utilization of containers in a solution. The state-of-the-art algorithm tackles the MCLCMP using the prototype column generation (PCG) technique. PCG is an effective technique for speeding up the column generation technique for extremely hard optimization problems where their corresponding pricing subproblems are NP-hard. We propose a new approach to the MCLCMP that combines the PCG technique with a goal-driven search. Our goal-driven prototype column generation (GD-PCG) algorithm improves the original PCG approach in three respects. Computational experiments suggest that all three enhancements are effective. Our GD-PCG algorithm produces significantly better solutions for the 350 existing benchmark instances than all other approaches in the literature using less computation time. We also generate two new set instances based on industrial data and the classical single container loading instances.     

A tree search method for the container loading problem with shipment priority

This paper addresses a special kind of container loading problem with shipment priority. We present a tree search method, which is based on a greedy heuristic. In the greedy heuristic, blocks made up of identical items with the same orientation are selected for packing into a container. Five evaluation functions are proposed for block selection, and the different blocks selected by each evaluation function constitute the branches of the search tree. A method of space splitting and merging is also embedded in the algorithm to facilitate efficient use of the container space. In addition, the proposed algorithm covers an important constraint called shipment priority to solve practical problems. The validity of the proposed algorithm is examined by comparing the present results with those of published algorithms using the same data.     

A caving degree approach for the single container loading problem

Inspired by an old adage “Gold corner, silver side and strawy void”, and improved by a new observation “Maximum value in diamond cave”, a new heuristic approach is proposed for solving the three-dimensional single container loading problem. Differing from several previous approaches, its key issue is to pack the outside item into a corner or even a cave in the container such that the item is as compactly and closely as possible with other packed items. Experiments are on two groups of public and difficult benchmarks. For the 47 without-orientation-constraint instances from the OR-Library, experiments indicate an average packing utilization of 94.9%, which improves current best result reported in the literature by 3.9%. For the 800 strongly heterogeneous instances among 1500 representative benchmarks proposed by Bischoff et al., (100 instances in a set), experiments show an average packing utilization of 87.97%, which improves current best record reported in the literature by 0.28%. Besides, new best records are achieved on the latter five sets among the eight sets of strongly heterogeneous benchmarks.     

A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size

We consider a two-dimensional cutting stock problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as its subproblem the solution of a two-dimensional knapsack problem with two-staged guillotines cuts. A further contribution of the paper consists in the definition of a mixed integer linear programming model for the solution of this knapsack problem, as well as a heuristic procedure based on dynamic programming. Computational experiments show the effectiveness of the proposed approach, which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by Cintra et al. [3].     

The multiple container loading cost minimization problem

In the shipping and transportation industry, there are several types of standard containers with different dimensions and different associated costs. In this paper, we examine the multiple container loading cost minimization problem (MCLCMP), where the objective is to load products of various types into containers of various sizes so as to minimize the total cost. We transform the MCLCMP into an extended set cover problem that is formulated using linear integer programming and solve it with a heuristic to generate columns. Experiments on standard bin-packing instances show our approach is superior to prior approaches. Additionally, since the optimal solutions for existing test data is unknown, we propose a technique to generate test data with known optimal solutions for MCLCMP.     

A column generation approach for the unconstrained binary quadratic programming problem

This paper proposes a column generation approach based on the Lagrangean relaxation with clusters to solve the unconstrained binary quadratic programming problem that consists of maximizing a quadratic objective function by the choice of suitable values for binary decision variables. The proposed method treats a mixed binary linear model for the quadratic problem with constraints represented by a graph. This graph is partitioned in clusters of vertices forming sub-problems whose solutions use the dual variables obtained by a coordinator problem. The column generation process presents alternative ways to find upper and lower bounds for the quadratic problem. Computational experiments were performed using hard instances and the proposed method was compared against other methods presenting improved results for most of these instances.     

A column generation approach for the split delivery vehicle routing problem

A column generation approach is presented for the split delivery vehicle routing problem with large demand. Columns include route and delivery amount information. Pricing sub-problems are solved by a limited-search-with-bound algorithm. Feasible solutions are obtained iteratively by fixing one route once. Numerical experiments show better solutions than in the literature.     

A column generation approach for the rail crew re-scheduling problem

When tracks are out of service for maintenance during a certain period, trains cannot be operated on those tracks. This leads to a modified timetable, and results in infeasible rolling stock and crew schedules. Therefore, these schedules need to be repaired. The topic of this paper is the re-scheduling of crew.     

A column generation approach to the discrete barycenter problem

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation.In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig–Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.     

A two-stage intelligent search algorithm for the two-dimensional strip packing problem

This paper presents a two-stage intelligent search algorithm for a two-dimensional strip packing problem without guillotine constraint. In the first stage, a heuristic algorithm is proposed, which is based on a simple scoring rule that selects one rectangle from all rectangles to be packed, for a given space. In the second stage, a local search and a simulated annealing algorithm are combined to improve solutions of the problem. In particular, a multi-start strategy is designed to enhance the search capability of the simulated annealing algorithm. Extensive computational experiments on a wide range of benchmark problems from zero-waste to non-zero-waste instances are implemented. Computational results obtained in less than 60 seconds of computation time show that the proposed algorithm outperforms the supposedly excellent algorithms reported recently, on average. It performs particularly better for large instances.     

A heuristic for the container loading problem: A tertiary-tree-based dynamic space decomposition approach

Increasing fuel costs, post-911 security concerns, and economic globalization provide a strong incentive for container carriers to use available container space more efficiently, thereby minimizing the number of container trips and reducing socio-economic vulnerability. A heuristic algorithm based on a tertiary tree model is proposed to handle the container loading problem (CLP) with weakly heterogeneous boxes. A dynamic space decomposition method based on the tertiary tree structure is developed to partition the remaining container space after a block of homogeneous rectangular boxes is loaded into a container. This decomposition approach, together with an optimal-fitting sequencing and an inner-right-corner-occupying placement rule, permits a holistic loading strategy to pack a container. Comparative studies with existing algorithms and an illustrative example demonstrate the efficiency of this algorithm.     

A heuristic for the three-dimensional strip packing problem

The contribution presents a heuristic for the three-dimensional strip packing problem (3D-SPP) with rectangular pieces (boxes). The considered 3D-SPP can be formulated as follows: for a given set of boxes and a given longitudinal open container, determine an arrangement of all boxes within the container so that the required container length is minimized.     

Solving the pallet loading problem

This paper presents new bounds, heuristics, and an exact algorithm for the Pallet Loading Problem (PLP). PLP maximizes the number of boxes placed on a rectangular pallet. All boxes have identical rectangular dimensions and, when placed, must be located completely within the pallet. Boxes may be rotated 90° so long as they are placed with edges parallel to the pallet’s edges. The set of all PLP instances with an area ratio (pallet area divided by box area) less than 101 boxes can be represented by 3,080,730 equivalent classes. Our G5-heuristic finds optimal solutions to 3,073,724 of these 3,080,730 classes and in the remaining 7006 classes only differs from the best known bound by one box. We develop three other heuristics that solve another 54 instances. Finally, we solve the 6952 remaining classes with our exact HVZ algorithm. Only a subset of these classes has been solved previously.     

A column generation approach for solving the patient admission scheduling problem

This paper addresses the Patient Admission Scheduling (PAS) problem. The PAS problem entails assigning elective patients to beds, while satisfying a number of hard constraints and as many soft constraints as is possible, and arises at all planning levels for hospital management. There exist a few, different variants of this problem. In this paper we consider one such variant and propose an optimization-based heuristic building on branch-and-bound, column generation, and dynamic constraint aggregation to solve it. We achieve tighter lower bounds than previously reported in the literature and, in addition, we are able to produce new best known solutions for five out of twelve instances from a publicly available repository.     

The distance constrained multiple vehicle traveling purchaser problem

In the Distance Constrained Multiple Vehicle Traveling Purchaser Problem (DC-MVTPP) a fleet of vehicles is available to visit suppliers offering products at different prices and with different quantity availabilities. The DC-MVTPP consists in selecting a subset of suppliers so to satisfy products demand at the minimum traveling and purchasing costs, while ensuring that the distance traveled by each vehicle does not exceed a predefined upper bound. The problem generalizes the classical Traveling Purchaser Problem (TPP) and adds new realistic features to the decision problem. In this paper we present different mathematical programming formulations for the problem. A branch-and-price algorithm is also proposed to solve a set partitioning formulation where columns represent feasible routes for the vehicles. At each node of the branch-and-bound tree, the linear relaxation of the set partitioning formulation, augmented by the branching constraints, is solved through column generation. The pricing problem is solved using dynamic programming. A set of instances has been derived from benchmark instances for the asymmetric TPP. Instances with up to 100 suppliers and 200 products have been solved to optimality.     

Interior point stabilization for column generation

Interior point stabilization is an acceleration method for column generation algorithms. It addresses degeneracy and convergence difficulties by selecting a dual solution inside the optimal space rather than retrieving an extreme point. The method is applied to the case of the vehicle routing problem with time windows.     

Accelerating column generation for variable sized bin-packing problems

In this paper, we study different strategies to stabilize and accelerate the column generation method, when it is applied specifically to the variable sized bin-packing problem, or to its cutting stock counterpart, the multiple length cutting stock problem. Many of the algorithms for these problems discussed in the literature rely on column generation, processes that are known to converge slowly due to primal degeneracy and the excessive oscillations of the dual variables. In the sequel, we introduce new dual-optimal inequalities, and explore the principle of model aggregation as an alternative way of controlling the progress of the dual variables. Two algorithms based on aggregation are proposed. The first one relies on a row aggregated LP, while the second one solves iteratively sequences of doubly aggregated models. Working with these approximations, in the various stages of an iterative solution process, has proven to be an effective way of achieving faster convergence.     

Solving a continuous local access network design problem with a stabilized central column generation approach

In this paper, we focus on a variant of the multi-source Weber problem. In the multi-source Weber problem, the location of a fixed number of concentrators, and the allocation of terminals to them, must be chosen to minimize the total cost of links between terminals and concentrators. In our variant, we have a third hierarchical level, two categories of link costs, and the number of concentrators is unknown. To solve this difficult problem, we propose several heuristics, and use a new stabilized column generation approach, based on a central cutting plane method, to provide lower bounds.     

Row-reduced column generation for degenerate master problems

Column generation for solving linear programs with a huge number of variables alternates between solving a master problem and a pricing subproblem to add variables to the master problem as needed. The method is known to often suffer from degeneracy in the master problem. Inspired by recent advances in coping with degeneracy in the primal simplex method, we propose a row-reduced column generation method that may take advantage of degenerate solutions. The idea is to reduce the number of constraints to the number of strictly positive basic variables in the current master problem solution. The advantage of this row-reduction is a smaller working basis, and thus a faster re-optimization of the master problem. This comes at the expense of a more involved pricing subproblem, itself eventually solved by column generation, that needs to generate weighted subsets of variables that are said compatible with the row-reduction, if possible. Such a subset of variables gives rise to a strict improvement in the objective function value if the weighted combination of the reduced costs is negative. We thus state, as a by-product, a necessary and sufficient optimality condition for linear programming.     

A partitioning algorithm for the network loading problem

This paper proposes a Benders-like partitioning algorithm to solve the network loading problem. The approach is an iterative method in which the integer programming solver is not used to produce the best integer point in the polyhedral relaxation of the set of feasible capacities. Rather, it selects an integer solution that is closest to the best known integer solution. Contrary to previous approaches, the method does not exploit the original mixed integer programming formulation of the problem. The effort of computing integer solutions is entirely left to a pure integer programming solver while valid inequalities are generated by solving standard nonlinear multicommodity flow problems. The method is compared to alternative approaches proposed in the literature and appears to be efficient for computing good upper bounds.