Column Generation Technique Research Articles

Vehicle Scheduling in Public Transit and Lagrangean Pricing

This paper investigates the solution of the linear programming (LP) relaxation of the multi-commodity flow formulation of the multiple-depot vehicle scheduling problems arising in public mass transit. We develop a column generation technique that makes it possible to solve the huge linear programs that come up there. The technique, which we call Lagrangean pricing, is based on two different Lagrangean relaxations. We describe in detail the basic ingredients of our approach and give computational results for large-scale test data (with up to 70 million variables) from three German public transportation companies. Because of these results, we propose Lagrangean pricing as one of the basic ingredients of an effective method to solve multiple-depot vehicle scheduling problems to proven optimality.

A Note on Optimal Component Test Plans for Series System Reliability with Exponential Failure Times

In a recent article, Rajgopal and Mazumdar (1995) considered a series system consisting of n components and studied the problem of designing optimum component test plans for series system reliability. Specifically, one of the problems considered by them was described as follows. The n components have different component testing costs of cj per unit time, j = 1, 2,..., n. The component lifetimes are assumed to be exponentially distributed with Aj the failure rate of component j, and one may further assume that Aj RI and definitely unacceptable if Rs R1) may be erroneously rejected by the test plan. 3 is the maximum acceptable consumer risk, which is the probability that an unacceptable system (Rs R1 is less than or equal to a and that the probability of accepting the system when Rs < Ro is less than or equal to 3. It is required to find optimal values of m and tj, j = 1, 2,..., n. Rajgopal and Mazumdar (1995) suggested solving the problem first for a given value of m and then determining the best value of rn. They proposed two algorithms to obtain the optimal tj values for a given m. The algorithms appear to be complicated and essentially involve formulating an abstract dual and solving it by the column-generation technique. In this note, it is shown that the original problem is equivalent to a linear programming problem that can be solved directly by any of the well-known computer routines. 1. OPTIMIZATION PROBLEM AND SOLUTION

Routing and scheduling temporal and heterogeneous freight car traffic on rail networks

Abstract Freight car scheduling is taking on a more important role in rail operating plans as more shippers demand trip plan information for their procurement, production and distribution plans, and as railroads pursue operations that are better scheduled and planned. This paper presents several ways to improve current freight car scheduling practices and describes a dynamic freight car routing and scheduling model that can produce more achievable and market-sensitive car schedules. A time–space network representation technique was used to represent car moves on possible sequences of car-to-block and block-to-train assignments on a general-merchandise rail service network. The problem was formulated as a linear multicommodity flow problem; the column generation technique was used as a solution approach. The model was tested on a hypothetical rail network based on the sub-network of a major U.S. railroad.

Routing helicopters for crew exchanges on off-shore locations

This paper deals with a vehicle routing problem with split demands, namely the problem of determining a flight schedule for helicopters to off-shore platform locations for exchanging crew people employed on these platforms. The problem is formulated as an LP model and solved by means of a column-generation technique including solving TSP problems. Since the final solution needs to be integral, we have chosen a rounding procedure to obtain an integer solution. Since the LP approach needs a considerable amount of computer time, it is only suitable for long-term planning practices. For the usual short-term planning, we have designed the so-called Cluster-and-Route Heuristic together with a number of improvement heuristics. The Cluster-and-Route procedure constructs a suitable clustering of the platforms and simultaneously forms the routes of the helicopter flights associated with the clusters. This approach is different from the usual heuristics, in which the clusters are constructed first, and the routes for each cluster are made afterwards. Simulations with various data sets show that the new heuristic outperforms the usual heuristics for vehicle routing problems. Even better results are obtained when improvement heuristics are applied. We use four improvement heuristics, including, so-called 1-opt and 2-opt procedures.

Daily Aircraft Routing and Scheduling

In this paper we consider the daily aircraft routing and scheduling problem (DARSP). It consists of determining daily schedules which maximize the anticipated profits derived from the aircraft of a heterogeneous fleet. This fleet must cover a set of operational flight legs with known departure time windows, durations and profits according to the aircraft type. We present two models for this problem: a Set Partitioning type formulation and a time constrained multicommodity network flow formulation. We describe the network structure of the subproblem when a column generation technique is applied to solve the linear relaxation of the first model and when a Dantzig-Wolfe decomposition approach is used to solve the linear relaxation of the second model. The linear relaxation of the first model provides upper bounds. Integer solutions to the overall problem are derived through branch-and-bound. By exploiting the equivalence between the two formulations, we propose various optimal branching strategies compatible with the column generation technique. Finally we report computational results obtained on data provided by two different airlines. These results show that significant profit improvement can be generated by solving the DARSP using our approach and that this can be obtained in a reasonable amount of CPU time.

A branch-and-price algorithm for solving the cutting strips problem

After giving a suitable model for the cutting strips problem, we present a branch-and-price algorithm for it by combining the column generation technique and the branch-and-bound method with LP relaxations. Some theoretical issues and implementation details about the algorithm are discussed, including the solution of the pricing subproblem, the quality of LP relaxations, the branching scheme as well as the column management. Finally, preliminary computational experience is reported.

A dynamic model for component testing

We consider the component testing problem of a system where the main feature is that the component failure rates are not constant parameters, but they change in a dynamic fashion with respect to time. More precisely, each component has a piecewise-constant failure-rate function such that the lifetime distribution is exponential with a constant rate over local intervals of time within the overall mission time. There are several such intervals, and the rates change dynamically from one interval to another. We note that these lifetime distributions can also be used in a more general setting to approximate arbitrary lifetime distributions. 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 a numerical example. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 187–197, 1997

Statistical tolerance analysis based on beta distributions

In an earlier paper, Bozer and Srinivasan introduced the tandem concept for automated guided vehicle (AGV) systems and presented an analytical model to evaluate the throughput performance of a basic component of the system; namely, a single vehicle serving a set of workstations under the First-Encountered-First-Served rule. In this study, using the above analytical model and certain column generation techniques, we present a heuristic partitioning scheme to configure tandem AGV systems. The partitioning scheme is based on a variation of the well-known set partitioning problem. It is aimed at evenly distributing the workload among all the AGVs in the system. We demonstrate the procedure with two numerical examples. Using simulation, the performance of the tandem configuration obtained for each example is compared to that of the corresponding conventional AGV system.

Flexible transmission bandwidth management with effective channel reservation techniques for NGSO MSS networks

This paper describes efficient transmission bandwidth management based on flexible channel assignment and effective grade of service (GOS) control methods for nongeostationary orbit mobile satellite service (NGSO MSS) networks. Our basic approach for the flexible channel assignment is to use linear programming (FLEXCA-LP), and to apply the column generation technique for further reduction of the total bandwidth requirement. These methods efficiently assign channels, taking account of moving spot beam coverage and dynamic traffic load variation for large-scale networks. The GOS is investigated for the channel assignment by FLEXCA-LP to make the GOS of ongoing calls better than that of fresh calls as stipulated in ITU-T Recommendations. We propose a few effective means to improve the results of FLEXCA-LP by reducing the disruption possibility of ongoing calls. One is to add channels utilizing a hysteresis concept. The others are effective use of GOS control techniques for NGSO MSS networks. These techniques are evaluated by a simulation study to identify promising techniques. Under time-variant conditions, we show that a reliable channel assignment plan is generated first by FLEXCA-LP considering the traffic load of fresh calls only and then by GOS control techniques to improve handover failure and call disruption rates.

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra's d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities).

An Integer Programming Approach to the Bandwidth Packing Problem

We consider the bandwidth packing problem arising from telecommunication networks. The problem is to determine the set of calls and an assignment of them to the paths in an arc-capacitated network to maximize profit. We propose an algorithm to solve the integer programming formulation of the problem. An efficient column generation technique to solve the linear programming relaxation is proposed, and a modified cover inequality is used to strengthen the IP formulation. The algorithm incorporates the column generation technique and the strong cutting plane approach into a branch-and-bound scheme. We test the proposed algorithm on some random problems. The results show that the algorithm can be used to solve the problems within reasonably small time limits.

Planning the routing mix in FASs to minimize total transportation time

The routing mix problem in flexible assembly systems is considered. The problem consists of assigning the operations for each part to the machines, with the two objectives of balancing the machine workloads and minimizing the burden of the transportation system. These two objectives are sometimes conflicting, since the latter tends to support assigning operations to the same machine(s) as much as possible, and this may be bad for workload balancing. A linear programming problem is presented that, given a constraint on the workload of each machine, finds one solution that minimizes the overall time spent moving the parts from one machine to another. Since such a linear program may have an exponential number of variables, an efficient column generation technique to solve the problem is devised. The efficiency of the method is validated by experiments on a large number of random problems.

An LP-based approach to a two-stage cutting stock problem

This paper describes a linear programming model for a two-stage cutting stock problem that arises in a make-to-order steel company. For technological reasons, there are two different cutting operations. Between the cutting operations, the intermediate rolls are processed to obtain the required characteristics. The minimization of the trim loss and the reduction of the number of set-ups are primary concerns in cutting stock problems. In this case, it is also important to minimize the number of intermediate rolls, in order to reduce the processing work. These objectives may be conflicting and a balance may be necessary. We present a simplified LP model, which is solved by a column generation technique. The subproblems are relatively large knapsack problems. Computational results for a microcomputer are presented.

The congested facility location problem

Abstract Consider a network facility location problem where congestion arises at facilities, and is represented by delay functions that approximate the queueing process. We strive to minimize the sum of customers' transportation and waiting times, and facilities' fixed and variable costs. The problem is solved using a column generation technique within a branch-and-bound scheme. Numerical results are reported and a bilevel (user-optimized) formulation considered, among other extensions.

Stochastic user equilibrium assignment in networks with queues

A stochastic user equilibrium assignment algorithm is presented for steady state store-and-forward networks. The links of the network have constant travel times and the links or nodes have finite capacities. When capacity is reached, delay sufficient to match demand to the available capacity is generated. It has been shown by others that the equilibrium assignment in networks of this kind is the solution to a particular linear programming problem. By adding an entropy term to the objective function, a convex nonlinear programming problem is formed which yields a stochastic user equilibrium assignment. For the case of link constraints, it is proven that the Lagrange multipliers of both the linear and the non-linear programming problems give the equilibrium delays in the network. The requirements for uniqueness are investigated. Iterative algorithms are formulated for solving the nonlinear programming problem with either link or node constraints and convergence is proven. For networks where path enumeration is likely to be a problem, a column generation technique is proposed. An illustrative example is presented.

A Multiple Route Conditional Risk Model For Transporting Hazardous Materials

AbstractThis paper considers the problem of developing effective routes for transporting hazardous materials for the case where the routing ends after the occurrence of the first accident. A multiple route situation is permitted, the objective being to minimize the expected risk of the first accident, with constraints on the accident probabilities, the expected a prioririsk, the transportation cost, and the equity of risk. A column generation technique is used to generate heuristic solutions. A brief summary of our results from extensive computational testing, based on a “real-life” hazardous materials routing scenario, is presented. Also presented is an example problem that illustrates several features of the model.

The design of optimum component test plans in the demonstration of system reliability

System Testing is often not feasible, and it is a common practice in these situations to attempt to draw inferences about the reliability of the system from the results of component tests. An acceptance procedure based on the sum of failures during component tests has been previously suggested in the literature for a series system, a parallel system, and a serial connection of redundant subsystems. Methods for computing the individual component test times which minimize total test cost and guarantee the probability requirements for this acceptance procedure and for such system structures have also been given; none of them considers a priori information on component reliabilities. In this paper we consider the question of whether these optimum solutions are modified if a priori deterministic information on individual component failure rates is available. Results have shown that it is profitable to use the given information. We also provide a procedure to compute optimum component test times for general systems. This procedure is based on the well known cutting plane idea and column generation technique, and eventually solves a semi-infinite linear program. We also discuss the conditions for the existence of optimum component test times.

A column generation and partitioning approach for multi-commodity flow problems

Multi-commodity network flow problems, prevalent in transportation, production and communication systems, can be characterized by a set of commodities and an underlying network. The objective is to flow the commodities through the network at minimum cost without exceeding arc capacities. In this paper, we present a partitioning solution procedure for large-scale multi-commodity flow problems with many commodities, such as those encountered in the telecommunications industry. Using a cycle-based multi-commodity formulation and column generation techniques, we solve a series of reduced-size linear programs in which a large number of constraints are relaxed. Each solution to a reduced-size problem is an improved basic dual feasible solution to the original problem and, after a finite number of steps, an optimal multi-commodity flow solution is determined. Computational experience is gained in solving randomly generated test problems and message routing problems in the communications industry. The tests show that the procedure solves large-scale multi-commodity flow problems significantly faster than existing linear programming or column generation solution procedures.

A set partitioning heuristic for the generalized assignment problem

This paper discusses a heuristic for the generalized assignment problem (GAP). The objective of GAP is to minimize the costs of assigning J jobs to M capacity constrained machines, such that each job is assigned to exactly one machine. The problem is known to be NP-Hard, and it is hard from a computational point of view as well. The heuristic proposed here is based on column generation techniques, and yields both upper and lower bounds. On a set of relatively hard test problems the heuristic is able to find solutions that are on average within 0.13% from optimality.

Tandem AGV systems: A partitioning algorithm and performance comparison with conventional AGV systems

In an earlier paper, Bozer and Srinivasan introduced the tandem concept for automated guided vehicle (AGV) systems and presented an analytical model to evaluate the throughput performance of a basic component of the system; namely, a single vehicle serving a set of workstations under the First-Encountered-First-Served rule. In this study, using the above analytical model and certain column generation techniques, we present a heuristic partitioning scheme to configure tandem AGV systems. The partitioning scheme is based on a variation of the well-known set partitioning problem. It is aimed at evenly distributing the workload among all the AGVs in the system. We demonstrate the procedure with two numerical examples. Using simulation, the performance of the tandem configuration obtained for each example is compared to that of the corresponding conventional AGV system.