Column Generation Scheme Research Articles

Genetic algorithm‐based meta‐heuristic for target coverage problem

In wireless sensor networks (WSNs), network lifetime and energy consumption are two important parameters which directly impacts each other. In order to enhance the global network lifetime, one should need to utilise the available sensors' energy in an optimise way. There are several approaches discussed in the literature to maximise the network lifetime for well-known target coverage problem in WSN. The target coverage problem is presented as a maximum network lifetime problem (MLP) and solved heuristically using various approaches. In this study, the authors propose a genetic algorithm (GA)-based meta-heuristic to solve the above said MLP. The GA is a non-linear optimisation solution method which is proven to be better as compared to the column generation or approximation schemes.

A column generation approach for location-routing problems with pickup and delivery

In this paper we formulate an integer programming model for the Location and Routing Problem with Pickup and Delivery. We propose a column generation scheme and implement, for the subproblem, a label-setting algorithm for the shortest path with pickup and delivery and time windows problem. We also propose a set of heuristics to speed up this process. To validate the model, we implement the column generation scheme and test it on different instances developed in this paper. We also provide an analysis of how the costs of opening depots and the fixed cost of routes affect the optimal solution.

The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation

A new and strong convexified formulation of the fixed charge transportation problem is provided. This formulation is obtained by integrating the concepts of Lagrangian decomposition and column generation. The decomposition is made by splitting the shipping variables into supply and demand side copies, while the columns correspond to extreme flow patterns for single sources or sinks. It is shown that the combination of Lagrangian decomposition and column generation provides a formulation whose strength dominates those of three other convexified formulations of the problem. Numerical results illustrate that our integrated approach has the ability to provide strong lower bounds. The Lagrangian decomposition yields a dual problem with an unbounded set of optimal solutions. We propose a regularized column generation scheme which prioritizes an optimal dual solution with a small $$l_1$$ -norm. We further demonstrate numerically that information gained from the strong formulation can also be used for constructing a small-sized core problem which yields high-quality upper bounds.

Column generation algorithms for bi-objective combinatorial optimization problems with a min–max objective

Many practical combinatorial optimization problems can be described by integer linear programs having an exponential number of variables, and they are efficiently solved by column generation algorithms. For these problems, column generation is used to compute good dual bounds that can be incorporated in branch-and-price algorithms. Recent research has concentrated on describing lower and upper bounds of bi-objective and general multi-objective problems with sets of points (bound sets). An important issue to address when computing a bound set by column generation is how to efficiently search for columns corresponding to each point of the bound set. In this work, we propose a generalized column generation scheme to compute bound sets for bi-objective combinatorial optimization problems. We present specific implementations of the generalized scheme for the case where one objective is a min–max function by using a variant of the $$\varepsilon $$ -constraint method to efficiently model these problems. The proposed strategies are applied to a bi-objective extension of the multi-vehicle covering tour problem, and their relative performances based on different criteria are compared. The results show that good bound sets can be obtained in reasonable times if columns are efficiently managed. The variant of the $$\varepsilon $$ -constraint presented is also better than a standard $$\varepsilon $$ -constraint method in terms of the quality of the bound sets.

A column generation approach for the strong network orientation problem

In this study, an aggregated flow formulation and a column generation strategy are proposed for the Strong Network Orientation Problem (SNOP) that consists in setting an orientation for each edge in a given graph, such that the resulting digraph is strongly connected and the total travel distance between all pairs of vertices is minimized. SNOP is NP-hard and finds application in urban networks.

Column Generation for Outbound Baggage Handling at Airports

The planning of outbound baggage handling at international airports is challenging. Outgoing flights have to be assigned and scheduled to handling facilities at which the outgoing baggage is loaded into containers. To avoid disruptions of the system the objective is to minimize workload peaks over the entire system. The resource demand of the jobs that have to be scheduled depends on the arrival process of the baggage. In this paper we present a time-indexed mathematical programming formulation for planning the outbound baggage. We propose an innovative decomposition procedure in combination with a column generation scheme to solve practical problem instances. The decomposition significantly reduces the symmetry effect in the time-indexed formulation and also speeds up the computational time of the corresponding Dantzig–Wolfe formulation. To further improve our column generation algorithm we propose state-of-the-art acceleration techniques for the primal problem and the pricing problem. Computational results based on real data from a major European airport show that the proposed procedure reduces the maximal workloads by more than 60% compared to the current assignment procedure used. The online appendix is available at https://doi.org/10.1287/trsc.2017.0739 .

Column generation strategies and decomposition approaches for the two-stage stochastic multiple knapsack problem

Many problems can be formulated by variants of knapsack problems. However, such models are deterministic, while many real-life problems include some kind of uncertainty. Therefore, it is worthwhile to develop and test knapsack models that can deal with disturbances. In this paper, we consider a two-stage stochastic multiple knapsack problem. Here, we have a multiple knapsack problem together with a set of possible disturbances. For each disturbance, or scenario, we know its probability of occurrence and the resulting reduction in the sizes of the knapsacks. For each knapsack we decide in the first stage which items we take with us, and when a disturbance occurs we are allowed to remove items from the corresponding knapsack. Our goal is to find a solution where the expected revenue is maximized. We use branch-and-price to solve this problem. We present and compare two solution approaches: the separate recovery decomposition (SRD) and the combined recovery decomposition (CRD). We prove that the LP-relaxation of the CRD is stronger than the LP-relaxation of the SRD. Furthermore, we investigate numerous column generation strategies and methods to create additional columns outside the pricing problem. These strategies reduce the solution time significantly. To the best of our knowledge, there is no other paper that investigates such strategies so thoroughly.

Application of a homogenous patch mean kernel with within-class collaborative representation for hyperspectral imagery classification

ABSTRACTSpectral–spatial-based classification methods demonstrate satisfying performance for hyperspectral imagery (HSI) classification. In this letter, in order to make full use of spectral and contexture information with simultaneously considering within-class information, we propose a new algorithm for HSI classification based on within-class collaborative representation and column generation (CG) strategy. The proposed accelerated homogeneous patch mean kernel (HPMK) can automatically assign a homogeneous patch for the target sample and represent the similarities between training set and assigned homogeneous patch in kernel feature space based on CG strategy. Further, for including intra-class information and improve classification efficiency, within-class collaborative representation classification (WCRC) is incorporated into new feature space to enhance the classification performance. Experiments on two real HSI data sets demonstrate that the proposed algorithm presents satisfying results in terms of classification accuracy and efficiency.

Heuristic decomposition approaches for an integrated task scheduling and personnel rostering problem

Real world combinatorial optimisation problems do not often reduce to neatly delineated theoretical problems. Rather, they combine characteristics of various subproblems which then appear to be strongly intertwined. The present contribution introduces a challenging integration of task and personnel scheduling in which both tasks and shifts must be assigned to a set of multi-skilled employees. Three constructive heuristics, based on column generation and other decomposition schemes, are presented, as well as a very large-scale neighbourhood search algorithm to further decrease the schedule's cost. The performance of these algorithms is evaluated on a large set of diverse instances. Computational results illustrate the effectiveness of the proposed approaches, and provide insight into their behaviour. The initial benchmarks are published so as to encourage further research.

Model-hierarchical column generation and heuristic for the routing and wavelength assignment problem

The routing and wavelength assignment (RWA) problem typically occurs in wavelength division multiplexing optical networks. Given a number of available wavelengths, we consider here the problem of maximising the number of accepted connections with respect to the clash and continuity constraints. We first propose a new strategy which combines two existing models. This leads to an improved column generation scheme. We also present two heuristics to compute feasible solutions: a hybrid heuristic and the integer solution at the root node of the column generation. Our approaches are compared with the best existing results on a set of classic RWA instances.

A novel column generation strategy for large scale airline crew pairing problems

A crew pairing is a sequence of flight legs beginning and ending at the same crew domicile. Crew pairing planning is the primary cost-determining phase in airline crew scheduling. Optimizing crew pairings of an airline timetable is an extremely important process which helps to minimize operational crew costs and to maximize crew utilization.There are various restrictions imposed by regulations or company policies that must be considered and satisfied in crew pairing generation process. Keeping these restrictions and regulations in mind, the main goal of the optimization is the generation of low cost sets of valid crew pairings which cover all flights in airline's timetable.For this research study, already existing works related to crew pairing optimization are examined and a new column generation strategy, a pricing network design and a pairing elimination heuristic are developed as a contribution to the previous studies. In the proposed strategy, the main problem is modeled and solved as a set-covering problem and the pricing sub problem is modeled as a shortest-path problem which is efficiently solved over a duty-flight overnight connection graph by the combined usage of heuristic and exact algorithms. The proposed strategy has been tested with real world data obtained from Turkish Airlines and it is seen that it is capable of generating very competitive solutions compared to current practices in Turkish Airlines. It is also observed that there are various advantages of proposed solution approach such as sensitivity to penalty coefficients, generating less deadheads, very close solution times with a single threaded software and light weight hardware.

Integrating Timetabling and Crew Scheduling at a Freight Railway Operator

We investigate to what degree we can integrate a train timetabling/engine scheduling problem with a crew scheduling problem. In the timetabling/engine scheduling problem, we determine for each demand a specific time within its time window when the demand should be serviced. Furthermore, we generate engine duties for the demands. In our solution approach for the overall problem, we first obtain an optimal solution for the timetabling/engine scheduling problem. When solving the crew scheduling problem, we then exploit the fact that numerous optimal and near optimal solutions exist for the previous problem. We consider all these solutions that can be obtained from the optimal engine schedule by shifting the demands in time, while keeping the order of demands in the engine duties intact. In particular, in the crew scheduling stage it is allowed to retime the service of demands if the additional cost is outweighed by the crew savings. This information is implemented in a mathematical model for the crew scheduling problem. The model is solved using a column generation scheme. We perform computational experiments based on a case at a freight railway operator, DB Schenker Rail Scandinavia, and show that significant cost savings can be achieved.

Special issue: Smart transportation: Theory and practice

Special issue: Smart transportation: Theory and practice

Solving the stochastic multi‐class traffic assignment problem with asymmetric interactions, route overlapping, and vehicle restrictions

SummaryIn this paper, we develop a customized path‐based algorithm for solving the stochastic multi‐class traffic assignment problem with asymmetric interactions, route overlapping, and vehicle restrictions. The algorithm consists of an iterative balancing scheme to find the search direction, a self‐regulated averaging line search scheme to determine a suitable stepsize, and a column generation scheme to generate a universal path set for multiple vehicle classes. These three schemes work together in the customized path‐based algorithm to solve the stochastic multi‐class traffic assignment problem. The solution algorithm simultaneously considers the asymmetric interactions among different vehicle types through the link travel time functions, various vehicle restrictions in a transportation network, and route overlapping using the path‐size logit model for accounting random perceptions of network conditions in a stochastic user equilibrium framework. A real network in the city of Winnipeg, Canada, is used to examine the computational performance of the customized path‐based algorithm. In addition, sensitivity analyses are conducted to test the algorithmic effectiveness with respect to several model parameters and percentages of trucks in the transportation network. Numerical results reveal that the path‐based algorithm with the self‐regulated averaging line search scheme is computationally effective in solving the stochastic multi‐class traffic assignment problem with different modeling considerations. The algorithm is also computationally robust against various model parameters in the sensitivity analyses. Copyright © 2015 John Wiley & Sons, Ltd.

A generic column generation principle: derivation and convergence analysis

Given a non-empty, compact and convex set, and an a priori defined condition which each element either satisfies or not, we want to find an element belonging to the former category. This is a fundamental problem of mathematical programming which encompasses nonlinear programs, variational inequalities, and saddle-point problems. We present a conceptual column generation scheme, which alternates between solving a restriction of the original problem and a column generation phase which is used to augment the restricted problems. We establish the general applicability of the conceptual method, as well as to the three problem classes mentioned. We also establish a version of the conceptual method in which the restricted and column generation problems are allowed to be solved approximately, and of a version allowing for the dropping of columns. We show that some solution methods (e.g., Dantzig–Wolfe decomposition and simplicial decomposition) are special instances, and present new convergent column generation methods in nonlinear programming, such as a sequential linear programming type method. Along the way, we also relate our quite general scheme in nonlinear programming presented in this paper with several other classic, and more recent, iterative methods in nonlinear optimization.

Hybrid Algorithm for Route Design on Bus Rapid Transit Systems

In recent years, well-designed bus rapid transit (BRT) systems have become a real alternative to more expensive rail-based public transportation systems around the world. However, once the BRT system is operational, its success often depends on the routes offered to passengers. Thus, the bus rapid transit route design problem (BRTRDP) is the problem of finding a set of routes and frequencies that minimizes the operational and passenger costs (travel time) while simultaneously satisfying the system’s technical constraints, such as meeting the demands for trips, bus frequencies, and lane capacities. To address this problem, we propose a mathematical formulation of the BRTRDP as a mixed-integer program (MIP) with an underlying network structure. However, because of the vast number of routes, solving the MIP via branch and bound is out of reach for most practical instances. Hence, we propose a decomposition strategy that, given a certain set of routes, decouples the route selection decisions from the BRT system performance evaluation. The latter evaluation is done by solving a linear optimization problem using a column generation scheme. We embedded this decomposition strategy in a hybrid genetic algorithm (HGA) and tested it in 14 instances ranging from 5 to 40 stations with different BRT system topologies. The results show that in 8 of 14 problems, the HGA was able to obtain a solution that is provably optimal within 0.20%. Additionally, in 4 of 14 instances, HGA obtained the optimal solution.

Freight railway operator timetabling and engine scheduling

In this paper we consider timetable design at a European freight railway operator. The timetable is designed by choosing the time of service for customer unit train demands among a set of discrete points. These discrete points are all found within the a time-window. The objective of the model is to minimize cost while adhering to constraints regarding infrastructure usage, demand coverage, and engine availability. The model is solved by a column generation scheme where feasible engine schedules are designed in a label setting algorithm with time-dependent cost and service times.

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.

Optimal interval scheduling with a resource constraint

We consider a scheduling problem where n jobs have to be carried out by m parallel identical machines. The attributes of a job j are a fixed start time sj, a fixed finish time fj, a resource requirement rj, and a value vj. Every machine owns R units of a renewable resource necessary to carry out jobs. A machine can process more than one job at a time, provided the resource consumption does not exceed R. The jobs must be processed in a non-preemptive way. Within this setting, we ask for a subset of jobs that can be feasibly scheduled with the maximum total value. For this strongly NP-hard problem, we first discuss an approximation result. Then, we propose a column generation scheme for the exact solution. Finally, we suggest some greedy heuristics and a restricted enumeration heuristic. All proposed algorithms are implemented and tested on a large set of randomly generated instances. It turns out that the column generation technique clearly outperforms the direct resolution of a natural compact formulation; the greedy algorithms produce good quality solutions in negligible time, whereas the restricted enumeration averages the performance of the greedy methods and the exact technique.

Delivering Guaranteed Display Ads under Reach and Frequency Requirements

We propose a novel idea in the allocation and serving of online advertising. We show that by using predetermined fixed-length streams of ads (which we call patterns) to serve advertising, we can incorporate a variety of interesting features into the ad allocation optimization problem. In particular, our formulation optimizes for representativeness as well as user-level diversity and pacing of ads, under reach and frequency requirements. We show how the problem can be solved efficiently using a column generation scheme in which only a small set of best patterns are kept in the optimization problem. Our numerical tests suggest that with parallelization of the pattern generation process, the algorithm has a promising run time and memory usage.