A prototype column generation strategy for the multiple container loading problem

The multiple container loading cost minimization problem (MCLCMP) is a practical and useful problem in the transportation industry, where products of various dimensions are to be loaded into containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally formulated as a set cover problem and solved using column generation techniques, which is a popular method for handling huge numbers of variables. However, the direct application of column generation is not effective because feasible solutions to the pricing subproblem is required, which for the MCLCMP is NP-hard. We show that efficiency can be greatly improved by generating prototypes that approximate feasible solutions to the pricing problem rather than actual columns. For many hard combinatorial problems, the subproblem in column generation based algorithms is NP-hard; if suitable prototypes can be quickly generated that approximate feasible solutions, then our strategy can also be applied to speed up these algorithms.     

Column generation approaches to ship scheduling with flexible cargo sizes

We present a Dantzig–Wolfe procedure for the ship scheduling problem with flexible cargo sizes. This problem is similar to the well-known pickup and delivery problem with time windows, but the cargo sizes are defined by intervals instead of by fixed values. The flexible cargo sizes have consequences for the times used in the ports because both the loading and unloading times depend on the cargo sizes. We found it computationally hard to find exact solutions to the subproblems, so our method does not guarantee to find the optimum over all solutions. To be able to say something about how good our solution is, we generate a bound on the difference between the true optimal objective and the objective in our solution. We have compared our method with an a priori column generation approach, and our computational experiments on real world cases show that our Dantzig–Wolfe approach is faster than the a priori generation of columns, and we are able to deal with larger or more loosely constrained instances. By using the techniques introduced in this paper, a more extensive set of real world cases can be solved either to optimality or within a small deviation from optimality.     

Integrating column generation in a method to compute a discrete representation of the non-dominated set of multi-objective linear programmes

In this paper we propose the integration of column generation in the revised normal boundary intersection (RNBI) approach to compute a representative set of non-dominated points for multi-objective linear programmes (MOLPs). The RNBI approach solves single objective linear programmes, the RNBI subproblems, to project a set of evenly distributed reference points to the non-dominated set of an MOLP. We solve each RNBI subproblem using column generation, which moves the current point in objective space of the MOLP towards the non-dominated set. Since RNBI subproblems may be infeasible, we attempt to detect this infeasibility early. First, a reference point bounding method is proposed to eliminate reference points that lead to infeasible RNBI subproblems. Furthermore, different initialisation approaches for column generation are implemented, including Farkas pricing. We investigate the quality of the representation obtained. To demonstrate the efficacy of the proposed approach, we apply it to an MOLP arising in radiotherapy treatment design. In contrast to conventional optimisation approaches, treatment design using column generation provides deliverable treatment plans, avoiding a segmentation step which deteriorates treatment quality. As a result total monitor units is considerably reduced. We also note that reference point bounding dramatically reduces the number of RNBI subproblems that need to be solved.     

A lexicographic pricer for the fractional bin packing problem

We propose an exact lexicographic dynamic programming pricing algorithm for solving the Fractional Bin Packing Problem with column generation. The new algorithm is designed for generating maximal columns of minimum reduced cost which maximize, lexicographically, one of the measures of maximality we investigate. Extensive computational experiments reveal that a column generation algorithm based on this pricing technique can achieve a substantial reduction in the number of columns and the computing time, also when combined with a classical smoothing technique from the literature.     

Vehicle routing via column generation

This paper explores an approximate method for solving a routing problem in a four-level distribution which has “double-ended” demand. Routes are represented as columns in a linear program and column generation is used to improve the solution by generating new routes. The generation of new routes is based on an LP sub-problem. Its solution is rounded down to integer values to insure its feasibility as a route for inclusion in the restricted master problem. Finally, an illustrative problem is solved.     

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

A method is presented for generating a well-distributed Pareto set in nonlinear multiobjective optimization. The approach shares conceptual similarity with the Physical Programming-based method, the Normal-Boundary Intersection and the Normal Constraint methods, in its systematic approach investigating the objective space in order to obtain a well-distributed Pareto set. The proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space. It is shown that the proposed modification allows the method to generate an even representation of the entire Pareto surface. The generation is performed for both convex and nonconvex Pareto frontiers. A simple algorithm has been proposed to remove local Pareto solutions. The suggested approach has been verified by several test cases, including the generation of both convex and concave Pareto frontiers.     

Large-scale optimization planning methods for the distribution of United States army munitions

Coordinating the distribution of ammunition and scheduling strategic transportation resources during military contingency operations is a complex process. This paper presents a large-scale optimization-based planning method that uses column generation to schedule the movement of ammunition and transportation resources through a time-space network representation of the distribution system. The optimization-based planner is initialized using a feasible solution generated by a heuristic planning method. Both the optimization-based planner and the heuristic planner generate plans with improved ship utilization and delivery tardiness values as compared to plans generated using current planning techniques. In addition, the heuristic planner is implemented within a closed-loop planning and control framework, and is used to generate plans on a rolling horizon basis.     

A column generation approach for determining optimal fleet mix,schedules, and transshipment facility locations for a vessel transportation problem

This paper presents a column generation approach for a storage replenishment transportation-scheduling problem. The problem is concerned with determining an optimal combination of multiple-vessel schedules to transport a product from multiple sources to different destinations based on demand and storage information at the destinations, along with cost-effective optimal strategic locations for temporary transshipment storage facilities. Such problems are faced by oil/trucking companies that own a fleet of vessels (oil tankers or trucks) and have the option of chartering additional vessels to transport a product (crude oil or gasoline) to customers (storage facilities or gas stations) based on agreed upon contracts. An integer-programing model that determines a minimum-cost operation of vessels based on implicitly representing feasible shipping schedules is developed in this paper. Due to the moderate number of constraints but an overwhelming number of columns in the model, a column generation approach is devised to solve the continuous relaxation of the model, which is then coordinated with a sequential fixing heuristic in order to solve the discrete problem. Computational results are presented for a range of test problems to demonstrate the efficacy of the proposed approach.     

Integrated airline crew scheduling: A bi-dynamic constraint aggregation method using neighborhoods

The integrated crew scheduling (ICS) problem consists of determining, for a set of available crew members, least-cost schedules that cover all flights and respect various safety and collective agreement rules. A schedule is a sequence of pairings interspersed by rest periods that may contain days off. A pairing is a sequence of flights, connections, and rests starting and ending at the same crew base. Given its high complexity, the ICS problem has been traditionally tackled using a sequential two-stage approach, where a crew pairing problem is solved in the first stage and a crew assignment problem in the second stage. Recently, Saddoune et al. (2010b) developed a model and a column generation/dynamic constraint aggregation method for solving the ICS problem in one stage. Their computational results showed that the integrated approach can yield significant savings in total cost and number of schedules, but requires much higher computational times than the sequential approach. In this paper, we enhance this method to obtain lower computational times. In fact, we develop a bi-dynamic constraint aggregation method that exploits a neighborhood structure when generating columns (schedules) in the column generation method. On a set of seven instances derived from real-world flight schedules, this method allows to reduce the computational times by an average factor of 2.3, while improving the quality of the computed solutions.     

A column generation method for inverse shortest path problems

In this paper we formulate an inverse shortest path problem as a special linear programming problem. A column generation scheme is developed that is able to keep most columns of the LP model implicit and to generate necessary columns by shortest path algorithms. This method can get an optimal solution in finitely many steps. Some numerical results are reported to show that the algorithm has a good performance.The authors gratefully acknowledge the partial support of Croucher Foundation.     

Deriving compact extended formulations via LP-based separation techniques

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.     

New developments in the primal–dual column generation technique

The optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and, consequently, originates an unnecessarily large number of iterations of the method. To overcome this drawback, variations of the standard approach use interior points of the dual feasible set instead of optimal solutions. In this paper, we focus on a variation known as the primal–dual column generation technique which uses a primal–dual interior point method to obtain well-centered non-optimal solutions of the restricted master problems. We show that the method converges to an optimal solution of the master problem even though non-optimal solutions are used in the course of the procedure. Also, computational experiments are presented using linear-relaxed reformulations of three classical integer programming problems: the cutting stock problem, the vehicle routing problem with time windows, and the capacitated lot sizing problem with setup times. The numerical results indicate that the appropriate use of a primal–dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times, on average. Furthermore, the results show that the larger the instance, the better the relative performance of the primal–dual column generation technique.     

The Optimization of Slender Reinforced Concrete Columns

Slenderness is an important issue in design of reinforced concrete (RC) columns. Especially for long columns, second order effects may be not so small to neglect, but the calculation of second order effects may take too much time. For that reason, ACI 318 design code includes a simple approach in order to increase the flexural moment of columns according to their slenderness. Thus, second order effects are considered. In optimization, the effect of slenderness can be considered by using the factored design flexural moments. In this paper, harmony search (HS) algorithm is employed to find the optimum design variables of slender RC columns. These design variables are web width, height, diameter and number of reinforcements. The optimization objective is total cost of materials including concrete and steel. The developed method is effective to find the optimal design for axial force, flexural moment and shear force values. As numerical examples, optimum design of columns with different lengths, but with the same loadings and material properties were investigated. Thus, the effect of slenderness was seen on the optimum costs. By the increase of column length, increase of total material cost is more than a linear increase. This situation shows us the effect of slenderness on optimum RC columns (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vectorization of conjugate-gradient methods for large-scale minimization in meteorology

During the last few years, conjugate-gradient methods have been found to be the best available tool for large-scale minimization of nonlinear functions occurring in geophysical applications. While vectorization techniques have been applied to linear conjugate-gradient methods designed to solve symmetric linear systems of algebraic equations, arising mainly from discretization of elliptic partial differential equations, due to their suitability for vector or parallel processing, no such effort was undertaken for the nonlinear conjugate-gradient method for large-scale unconstrained minimization.Computational results are presented here using a robust memoryless quasi-Newton-like conjugate-gradient algorithm by Shanno and Phua applied to a set of large-scale meteorological problems. These results point to the vectorization of the conjugate-gradient code inducing a significant speed-up in the function and gradient evaluation for the nonlinear conjugate-gradient method, resulting in a sizable reduction in the CPU time for minimizing nonlinear functions of 10⁴ to 10⁵ variables. This is particularly true for many real-life problems where the gradient and function evaluation take the bulk of the computational effort.It is concluded that vector computers are advantageous for largescale numerical optimization problems where local minima of nonlinear functions are to be found using the nonlinear conjugate-gradient method.This research was supported by the Florida State University Supercomputer Computations Research Institute, which is partially funded by the US Department of Energy through Contract No. DE-FC05-85ER250000.     

Polynomial bases on the numerical solution of the multivariate discrete moment problem

The multivariate discrete moment problem (MDMP) has been introduced by Prékopa. The objective of the MDMP is to find the minimum and/or maximum of the expected value of a function of a random vector with a discrete finite support where the probability distribution is unknown, but some of the moments are given. The MDMP can be formulated as a linear programming problem, however, the coefficient matrix is very ill-conditioned. Hence, the LP problem usually cannot be solved in a regular way. In the univariate case Prékopa developed a numerically stable dual method for the solution. It is based on the knowledge of all dual feasible bases under some conditions on the objective function. In the multidimensional case the recent results are also about the dual feasible basis structures. Unfortunately, at higher dimensions, the whole structure has not been found under any circumstances. This means that a dual method, similar to Prékopa??s, cannot be developed. Only bounds on the objective function value are given, which can be far from the optimum. This paper introduces a different approach to treat the numerical difficulties. The method is based on multivariate polynomial bases. Our algorithm, in most cases, yields the optimum of the MDMP without any assumption on the objective function. The efficiency of the method is tested on several numerical examples.     

Preference scheduling for nurses using column generation

The purpose of this paper is to present a new methodology for scheduling nurses in which several conflicting factors guide the decision process. Unlike manufacturing facilities where standard shifts and days off are the rule, hospitals operate 24 hours a day, 7 days a week and face widely fluctuating demand. A more flexible arrangement for working hours and days off is needed, especially in light of the growing nursing shortage. To improve retention, management must now take into account individual preferences and requests for days off in a way that is perceived as fair, while ensuring sufficient coverage at all times. This multi-objective problem is solved with a column generation approach that combines integer programming and heuristics. The integer program is formulated as a set covering-type problem whose columns correspond to alternative schedules that a nurse can work over the planning horizon. A double swapping heuristic is used to generate the columns. The objective coefficients are determined by the degree to which the individual preferences of a nurse are violated. As part of the computational scheme, feasible solutions are refined to minimize the use of outside nurses, but when gaps in coverage exist, the outside nurses are distributed as evenly as possible over the shifts. The methodology was tested on a series of problems with up to 100 nurses using data provided by a large hospital in the US. The results indicate that high-quality solutions can be obtained within a few minutes in the majority of cases.     

Efficient solution generation for multiple objective linear programming based on extreme ray generation method

In this paper we consider solution generation method for multiple objective linear programming problems. The set of efficient or Pareto optimal solutions for the problems can be regarded as global information in multiple objective decision making situation. In the past three decades as solution generation techniques various conventional algorithms based on simplex-like approach with heavy computational burden were developed. Therefore, the development of novel and useful directions in efficient solution generation method have been desired. The purpose of this paper is to develop theoretical results and computational techniques of the efficient solution generation method based on extreme ray generation method that sequentially generates efficient points and rays by adding inequality constraints of the polyhedral feasible region.     

基于非线性尺度不变性的CCG法

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A branch-and-price approach for operational aircraft maintenance routing

In recent years, considerable effort in the field of operations research has been paid to optimizing airline operations, including the logistics of an airline’s fleet of aircraft. We focus on the problem of aircraft routing, which involves generating and selecting a particular route for each aircraft of a sub-fleet that is already assigned to a set of feasible sequences of flight legs. Similar studies typically focus on long-term route planning. However, stochastic events such as severe weather changes, equipment failures, variable maintenance times, or even new regulations mandated by the Federal Aviation Administration (FAA) play havoc on these long-term plans. In addition, these long-term plans ignore detailed maintenance requirements by considering only one or two of the primary maintenance checks that must be performed on a regular, long-term basis. As a result, these plans are often ignored by personnel in airline operations who are forced on a daily basis to develop quick, ad hoc methods to address these maintenance requirements and other irregular events. To address this problem, we develop an operational aircraft maintenance routing problem formulation that includes maintenance resource availability constraints. We propose a branch-and-price algorithm for solving this problem, which, due to the resource constraints, entails a modification of the branch-on, follow-on branching rule typically used for solving similar problems. Through computational testing, we explore the efficiency of this solution approach under a combination of heuristic choices for column (route) generation and selection.