Column Generation Technique Research Articles

Reliability-based stochastic transit assignment with capacity constraints: Formulation and solution method

This paper proposes a Linear Complementarity Problem (LCP) formulation for the reliability-based stochastic transit assignment problem with capacity constraints and non-additive link costs, where in-vehicle travel times and waiting times are uncertain. The capacity constraints are developed via the notions of effective capacity and chance constraints. An equivalent route-based linear program (LP) for the proposed problem is formulated to determine the patronage of each line section, critical links, critical service frequencies, unmet demand and the network capacity, which considers the risk-aversive behavior of travelers. A solution method is developed, utilizing the K-shortest path algorithm, the column generation technique, and the revised simplex method, to solve the proposed LP with guaranteed finite convergence. Numerical experiments are also set up to illustrate the properties of the problem and the application of the proposed model for reliability analysis.

Exact and hybrid methods for the multiperiod field service routing problem

This article deals with a particular class of routing problem, consisting of the planning and routing of technicians in the field. This problem has been identified as a multiperiod, multidepot uncapacitated vehicle routing problem with specific constraints that we call the multiperiod field service routing problem (MPFSRP). We propose a set covering formulation of the problem for the column generation technique and we develop an exact branch and price solution method for small-sized instances. We also propose several heuristic versions for larger instances. We present the results of experiments on realistic data adapted from an industrial application.

Improving a Lagrangian decomposition for the unconstrained binary quadratic programming problem

This paper presents a new alternative of Lagrangian decomposition based on column generation technique to solve the unconstrained binary quadratic programming problem. We use a mixed binary linear version of the original quadratic problem with constraints represented by a graph. This graph is partitioned into clusters of vertices forming subproblems whose solutions use the dual variables obtained by a coordinator problem. Computational experiments consider a set of difficult instances and the results are compared against other methods reported recently in the literature.

Anycast Routing for Survivable Optical Grids: Scalable Solution Methods and the Impact of Relocation

In this paper, we address the issue of resiliency against single link network failures in optical grids and show how the anycast routing principle, which is typical of grids, can be exploited in providing efficient shared path protection. We investigate two different integer linear program models for the full anycast routing problem, deciding on the primary and backup server locations as well as on the lightpaths toward them. The first model is a classical integer linear programming (ILP) model, which lacks scalability. The second model is a large-scale optimization model which can be efficiently solved using column generation techniques. We also design two new heuristics: the first one is an improvement of a previously proposed one which, although providing near optimal solutions, lacks scalability, while the second one is highly scalable, at the expense of reduced accuracy. Numerical results are presented for three mesh networks with varying node degrees. They allow an illustration of the scalability of the newly proposed approaches. Apart from highlighting the difference in performance (i.e., scalability and optimality) among the algorithms, our case studies demonstrate the bandwidth savings that can be achieved by exploiting relocation rather than using a backup path to the original (failure-free) destination site. Numerical results for varying network topologies, as well as different numbers of server sites show that relocation allows bandwidth savings in the range of 7–21%.

A column generation approach for power-aware optimization of virtualized heterogeneous server clusters

Increasingly, clusters of servers have been deployed in large data centers to support the development and implementation of many kinds of services, having distinct workload demands that vary over time, in a scalable and efficient computing environment. Emerging trends are utility/cloud computing platforms, where many network services, implemented and supported using server virtualization techniques, are hosted on a shared cluster infrastructure of physical servers. The energy consumed to maintain these large server clusters became a very important concern, which in turn, requires major investigation of optimization techniques to improve the energy efficiency of their computing infrastructure.In this work, we propose an efficient approach to solve a relevant cluster optimization problem which, in practice, can be used as an embedded module to implement an integrated power and performance management solution in a real server cluster. The optimization approach simultaneously deals with (i) CPU power-saving techniques combined with server switching on/off mechanisms, (ii) the case of server heterogeneity, (iii) virtualized server environments, (iv) an efficient optimization method, which is based on column generation techniques. The key aspects of our approach are the basis on rigorous and robust optimization techniques, given by high quality solutions in short amount of processing time, and experimental results on the cluster configuration problem for large-scale heterogeneous server clusters that can make use of virtualization techniques.

A note on the primal-dual column generation method for combinatorial optimization

Abstract In this paper, we address the primal-dual column generation technique, which relies on well-centred suboptimal solutions of the restricted master problems. We summarize new theoretical developments and present computational results for two classical combinatorial optimization problems, in which this technique has not been tested before. The results show that the primal-dual column generation technique usually leads to substantial reductions in the number of iterations and CPU time when compared to two other well-established approaches: the classical column generation technique and the analytic centre cutting plane method.

Solving large scale crew scheduling problems in practice

This paper deals with large-scale crew scheduling problems arising at the main Dutch railway operator, Netherlands Railways (NS). NS operates about 30000 trains a week. All these trains need a driver and a certain number of guards. Some labor rules restrict the duties of a certain crew base over the complete week. Therefore, splitting the problem in several subproblems per day leads to suboptimal solutions.In this paper, we present an algorithm, called LUCIA, which can solve such huge instances without splitting. This algorithm combines Lagrangian heuristics, column generation and fixing techniques. We compare the results with existing practice. The results show that the new method significantly improves the solution.

A Freight Service Design Problem for a Railway Corridor

We study the problem of designing a set of highly profitable freight routes in a railway corridor, taking into account the level of service requested by different goods; in particular, the profit achieved by transporting a good is a nonlinear function of the associated travel time. We propose an ILP model that is solved heuristically by column generation and fixing techniques. Computational results on a real corridor crossing 11 European countries are reported, showing that we can find solutions that are provably close to optimal. Given the large size of our instances, a key issue of our approach is to avoid finding an optimal solution of the continuous relaxation of our model, stopping as soon as near-optimal primal and dual solutions are available.

An efficient solution algorithm for solving multi-class reliability-based traffic assignment problem

The multi-class reliability-based user equilibrium (RUE) problem has been intensively studied in recent years, as it can capture the route choice behaviors of users with heterogeneous risk-aversion under demand and supply uncertainties. Few solution algorithms, however, are available for solving the RUE problems in large-scale road networks. This is mainly due to the non-additive property of the path finding sub-problem in the RUE model. An efficient traffic assignment solution algorithm for solving the multi-class RUE problems in large-scale road networks is proposed in this study. First, an effective shortest path algorithm is developed to explicitly overcome the non-additive difficulty. The algorithm is capable of finding optimal paths for all user classes in one search process and hence the repeated search process for each user class is avoided. This property can save not only computational time but also memory requirement. The proposed shortest path algorithm is then, further incorporated into a path-based traffic assignment algorithm using a column generation technique. Such traffic assignment algorithms can solve the multi-class RUE problem without the requirement of path enumeration. Experimental results show that the proposed solution algorithms can, even for large-scale networks with multi-user classes, efficiently achieve highly accurate RUE solutions within satisfactory computational time.

Design of optimum component test plans in the demonstration of diverse system performance measures

While component-level tests have many advantages over system-level tests, the actual protection offered in making inferences about system reliability is not the same as what is expected. Thus, a significant proportion of research has concentrated on the design of system-based component test plans that also have minimum cost. This article extends those previous studies by considering two additional system performance measures: expected system lifetime and system availability. After explicitly expressing these performance measures as a function of failure rates for various system types, the component testing problem is formulated as a semi-infinite linear programming problem and solved with a column generation technique incorporating signomial geometric programming. Several numerical examples are presented that provide insight on the model parameters.

Ship routing problem with berthing time clash avoidance constraints

We consider a ship routing problem in which multiple vessels have to perform pickups and deliveries of cargoes at various locations. The loading and unloading time of cargoes at pickup and delivery locations is significant, and at each of these locations we need to assign a time slot to each vessel to perform the loading/unloading task so as to avoid time clashes. This problem is motivated by the operations of feeder vessels and company-owned cargo terminals, where the shipping company wishes to coordinate the routing and the berthing time of its vessels. We develop a heuristic algorithm for the problem using set partitioning formulation and column generation techniques. The effectiveness of the heuristic is tested via extensive computational experiments.

An Inventory-Distribution System with LTL Deliveries – Mixed Integer Approach

The paper concerns a class of supply chain management problems called Joint Transportation-and-Inventory Problems (JTIPs). These problems are characterized by the presence of both transportation and inventory considerations, either as policy-variables or constraints. The research presented in this paper aims at determining an optimal joint inventory/transportation policy. Mathematical modeling is applied to find the optimal solution to JTIPs thus defined. The set partitioning formulation for solving the vehicle routing problem is incorporated into a discrete time, finite-horizon inventory planning problem. Two models will be discussed: time-discretized integer programming and the new approach with predefined quantities of delivery. As is shown in this paper, the approach allows us to use a column generation technique.

Freight Locomotive Rescheduling Algorithm under Disordered Train Operation

Railway operators adjust timetables, and reschedule rolling stock allocation and crew duties when train operations are disrupted due to accidents. This paper discusses the problem of rescheduling locomotive assignment to freight trains after timetable adjustment has been completed. The problem is modeled as an integer programming problem with set-partitioning constraints, and is solved by column generation technique. Numerical experiments using real data have revealed that our method provides a locomotive-rescheduling plan of satisfactory quality in acceptable computing time.

Railway crew rescheduling with retiming

Railway operations are disrupted frequently. For example, the Dutch railway network experiences about three large disruptions per day on average. In a disrupted situation, a railway operator needs to quickly adjust the timetable and the resource schedules. Usually the timetable, the rolling stock and the crew schedule are recovered in a sequential way. In this paper, we model and solve the crew rescheduling problem with retiming. This problem extends the crew rescheduling problem by the possibility to slightly delay the departure of some trains, so that some more flexibility in the crew scheduling process is obtained. Our algorithm focuses on rescheduling the duties of the train drivers. It is based on column generation techniques combined with Lagrangian heuristics. In order to prevent a large increase in computation time, retiming is allowed only for a limited number of trains for which it seems promising. Computational experiments with train driver duties and real-life disruption data show that, compared to the classical approach, it is possible to find better solutions by using crew rescheduling with retiming.

Mission-Based Component Testing for Series Systems

We consider the component testing problem of a device that is designed to perform a mission consisting of a random sequence of phases with random durations. Testing is done at the component level to attain desired levels of mission reliability at minimum cost. The components fail exponentially where the failure rate depends on the phase of the mission. The reliability structure of the device involves a series connection of nonidentical components with different failure characteristics. The optimal component testing problem is formulated as a semi-infinite linear program. We present an algorithmic procedure to compute optimal test times based on the column generation technique, and illustrate it with numerical examples.

Column Generation with Dynamic Duty Selection for Railway Crew Rescheduling

The Dutch railway network experiences about three large disruptions per day on average. In this paper, we present an algorithm to reschedule the crews when such a disruption occurs. The algorithm is based on column generation techniques combined with Lagrangian heuristics. Since the number of duties is very large in practical instances, we first define a core problem of tractable size. If some tasks remain uncovered in the solution of the core problem, we perform a neighborhood exploration to improve the solution. Computational experiments with real-life instances show that our method is capable of producing good solutions within a couple of minutes of computation time.

The Column Generation Technique for Public Transport Line Planning by IP-Solver

The paper deals with an application of the column generation method to the public transport line planning making use of a common optimization environment. The authors focus on the opportunities offered by the optimization software for the column optimization and for manmachine approach to the process of line planning. The paper resumes the former approach to the transport line planning based on line selection from a large set of all sensible lines and, on the contrary to the former approach, introduces the column generation method for a new route design. A case study from practice is used to compare both approaches and to point out their advantages and disadvantages.

Pricing structure optimization in mixed restricted/unrestricted fare environments

In recent years, many traditional practitioners of revenue management such as airlines or hotels were confronted with aggressive low-cost competition. In order to stay competitive, these firms responded by cutting down fare restrictions that were originally meant to fence off customer segments, that is to prevent highyield customers from buying down. For the corresponding markets, unrestricted fares were introduced whose essentially only distinctive feature is its price. Some markets, however, were unaffected and here restrictions could be maintained as it was the case for long-haul flights, for example. We develop choice-based network revenue management approaches for such a mixed fare environment that can handle both the traditional opening or closing of restricted fare classes as well as handling pricing of the unrestricted fares simultaneously. For any such unrestricted fare, we assume a fixed number of price points to choose from. It is natural to ask then how these price points shall be chosen. To that end, we formulate the problem as a dynamic program and approximate it with an efficient mixed integer linear program (MIP) that selects the best n, say, price points out of a potentially large set of price candidates for each unrestricted fare. We show both theoretically and practically that it is advantageous to recompute price points later in the booking horizon using our approach. Furthermore, additional insight is gained from the fact that the dual values associated with MIP provide an upper bound on the value of having an additional price point. Numerical experiments illustrate the quality of the obtained price structure and that computational effort is relatively low given that we need to tackle the large-scale MIP with column generation techniques where each column pricing problem itself is NP-hard.

Optimization of oil tanker schedules by decomposition, column generation, and time-space network techniques

Ship scheduling problem is an important operational level planning problem in maritime logistics. In this paper, we show how we designed and developed a mathematical model for real-world tanker scheduling problem in Japan. Tanker operators own their fleet of tankers and make their schedules for the next several weeks for meeting customers’ demands. However, due to the high uncertainty of the ship operations and unexpected changes of demands, the schedules has to be revised frequently. Our methodology allows the operators to determine schedules that minimize the operational cost in a few minutes. These solutions provides cost improvements for tanker operators, as measured by reduction of 5–16%.

Solving One-Dimensional Cutting Stock Problem with Discrete Demands and Capacitated Planning Objective

Problem statement: One-dimensional cutting stock problem with discret e demands and capacitated planning objective is an NP hard problem. Approach: The mathematical model with column-generation technique by a branch-and-bound procedure and the heuristic b ased on the first fit decreasing method are propose d. Then, both approaches were compared and some characteristics were investigated such as upper-bound value, percent age above lower-bound value, computation time, and number of patterns. Results: The 24 instances were examined. The proposed heuristic provides the upper-bound value a bove the lower-bound around 0-16.78%. All upper-bound values from column-generation and integer programming are better than the proposed heuristic but all c omputation times are higher. Conclusion: The proposed heuristic has consistently high perfo rmance in computation times.