Arc routing problems with time-dependent service costs

This paper studies an arc routing problem with capacity constraints and time-dependent service costs. This problem is motivated by winter gritting applications where the “timing” of each intervention is crucial. The exact problem-solving approach reported here first transforms the arc routing problem into an equivalent node routing problem. Then, a column generation scheme is used to solve the latter. The master problem is a classical set covering problem, while the subproblems are time-dependent shortest path problems with resource constraints. These subproblems are solved using an extension of a previously developed algorithm. Computational results are reported on problems derived from a set of classical instances of the vehicle routing problem with time windows.     

Branching in branch-and-price: a generic scheme

Developing a branching scheme that is compatible with the column generation procedure can be challenging. Application specific and generic schemes have been proposed in the literature, but they have their drawbacks. One generic scheme is to implement standard branching in the space of the compact formulation to which the Dantzig-Wolfe reformulation was applied. However, in the presence of multiple identical subsystems, the mapping to the original variable space typically induces symmetries. An alternative, in an application specific context, can be to expand the compact formulation to offer a wider choice of branching variables. Other existing generic schemes for use in branch-and-price imply modifications to the pricing problem. This is a concern because the pricing oracle on which the method relies might become obsolete beyond the root node. This paper presents a generic branching scheme in which the pricing oracle of the root node remains of use after branching (assuming that the pricing oracle can handle bounds on the subproblem variables). The scheme does not require the use of an extended formulation of the original problem. It proceeds by recursively partitioning the subproblem solution set. Branching constraints are enforced in the pricing problem instead of being dualized via Lagrangian relaxation, and the pricing problem is solved by a limited number of calls to the pricing oracle. This generic scheme builds on previously proposed approaches and unifies them. We illustrate its use on the cutting stock and bin packing problems. This is the first branch-and-price algorithm capable of solving such problems to integrality without modifying the subproblem or expanding its variable space.     

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 constraint generation scheme to probabilistic linear problems with an application to power system expansion planning

In this paper, we first describe a constraint generation scheme for probabilistic mixed integer programming problems. Next, we present a decomposition approach to the peak capacity expansion planning of interconnected hydrothermal generating systems, with bounds on the transmission capacity between the regions. The objective is to minimize investments in generating units and interconnection links, subject to constraints on supply reliability. The problem is formulated as a stochastic integer program. The constraint generation scheme, which is similar to Benders decomposition, is applied in the solution of the peak capacity expansion problem. The master problem in this decomposition scheme is an integer program, solved by implicit enumeration. The operating subproblem corresponds to a stochastic network flow problem, and is solved by a maximum flow algorithm and Monte Carlo simulation. The approach is illustrated through a case study involving the expansion of the system of the Brazilian Southeastern region.     

A reactive variable neighborhood tabu search for the heterogeneous fleet vehicle routing problem with time windows

This paper presents a solution methodology for the heterogeneous fleet vehicle routing problem with time windows. The objective is to minimize the total distribution costs, or similarly to determine the optimal fleet size and mix that minimizes both the total distance travelled by vehicles and the fixed vehicle costs, such that all problem’s constraints are satisfied. The problem is solved using a two-phase solution framework based upon a hybridized Tabu Search, within a new Reactive Variable Neighborhood Search metaheuristic algorithm. Computational experiments on benchmark data sets yield high quality solutions, illustrating the effectiveness of the approach and its applicability to realistic routing problems. This work is supported by the General Secretariat for Research and Technology of the Hellenic Ministry of Development under contract GSRT NM-67.     

Improved branch-cut-and-price for capacitated vehicle routing

The best performing exact algorithms for the capacitated vehicle routing problem developed in the last 10 years are based in the combination of cut and column generation. Some authors only used cuts expressed over the variables of the original formulation, in order to keep the pricing subproblem relatively easy. Other authors could reduce the duality gaps by also using a restricted number of cuts over the master LP variables, stopping when the pricing becomes prohibitively hard. A particularly effective family of such cuts are the subset row cuts. This work introduces a technique for greatly reducing the impact on the pricing of these cuts, thus allowing much more cuts to be added. The newly proposed branch-cut-and-price algorithm also incorporates and combines for the first time (often in an improved way) several elements found in previous works, like route enumeration and strong branching. All the instances used for benchmarking exact algorithms, with up to 199 customers, were solved to optimality. Moreover, some larger instances with up to 360 customers, only considered before by heuristic methods, were solved too.     

Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints, such as capacity or time windows, are solved via column generation and branch-and-price, it is common that the pricing subproblem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the vertices. A common solution technique for this problem is dynamic programming. In this paper we illustrate how the basic dynamic programming algorithm can be improved by bounded bi-directional search and we experimentally evaluate the effectiveness of the enhancement proposed. We consider as benchmark problems the elementary shortest path problems arising as pricing subproblems in branch-and-price algorithms for the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the capacitated vehicle routing problem with time windows.     

A branch-and-price algorithm for the capacitated vehicle routing problem with stochastic demands

This article introduces a new exact algorithm for the capacitated vehicle routing problem with stochastic demands (CVRPSD). The CVRPSD can be formulated as a set partitioning problem and it is shown that the associated column generation subproblem can be solved using a dynamic programming scheme. Computational experiments show promising results.     

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.     

On Compact Formulations for Integer Programs Solved by Column Generation

Column generation has become a powerful tool in solving large scale integer programs. It is well known that most of the often reported compatibility issues between pricing subproblem and branching rule disappear when branching decisions are based on imposing constraints on the subproblem's variables. This can be generalized to branching on variables of a so-called compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems, each of which contributes only one column in an integer solution. This construction has an interpretation as reversing a Dantzig-Wolfe decomposition. Our proposal opens the way for the development of branching rules adapted to the subproblem's structure and to the linking constraints.     

Vehicle routing with soft time windows and stochastic travel times: A column generation and branch-and-price solution approach

We study a vehicle routing problem with soft time windows and stochastic travel times. In this problem, we consider stochastic travel times to obtain routes which are both efficient and reliable. In our problem setting, soft time windows allow early and late servicing at customers by incurring some penalty costs. The objective is to minimize the sum of transportation costs and service costs. Transportation costs result from three elements which are the total distance traveled, the number of vehicles used and the total expected overtime of the drivers. Service costs are incurred for early and late arrivals; these correspond to time-window violations at the customers. We apply a column generation procedure to solve this problem. The master problem can be modeled as a classical set partitioning problem. The pricing subproblem, for each vehicle, corresponds to an elementary shortest path problem with resource constraints. To generate an integer solution, we embed our column generation procedure within a branch-and-price method. Computational results obtained by experimenting with well-known problem instances are reported.     

The fleet assignment problem: Solving a large-scale integer program

Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.This work was supported by NSF and AFORS grant DDM-9115768 and NSF grant SES-9122674.Corresponding author.     

Constraint Programming Based Column Generation for Crew Assignment

Airline crew assignment problems are large-scale optimization problems which can be adequately solved by column generation. The subproblem is typically a so-called constrained shortest path problem and solved by dynamic programming. However, complex airline regulations arising frequently in European airlines cannot be expressed entirely in this framework and limit the use of pure column generation. In this paper, we formulate the subproblem as a constraint satisfaction problem, thus gaining high expressiveness. Each airline regulation is encoded by one or several constraints. An additional constraint which encapsulates a shortest path algorithm for generating columns with negative reduced costs is introduced. This constraint reduces the search space of the subproblem significantly. Resulting domain reductions are propagated to the other constraints which additionally reduces the search space. Numerical results based on data of a large European airline are presented and demonstrate the potential of our approach.     

Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problem

We compare two branch-and-price approaches for the cutting stock problem. Each algorithm is based on a different integer programming formulation of the column generation master problem. One formulation results in a master problem with 0–1 integer variables while the other has general integer variables. Both algorithms employ column generation for solving LP relaxations at each node of a branch-and-bound tree to obtain optimal integer solutions. These different formulations yield the same column generation subproblem, but require different branch-and-bound approaches. Computational results for both real and randomly generated test problems are presented.     

An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

The vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem. It arises when each vehicle performs several routes during the workday due to strict time limits on route duration (e.g., when perishable goods are transported). The routes are defined over customers with a revenue, a demand and a time window. Given a fixed-size fleet of vehicles, it might not be possible to serve all customers. Thus, the customers must be chosen based on their associated revenue minus the traveling cost to reach them. We introduce a branch-and-price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation, using column generation. The pricing subproblems are elementary shortest path problems with resource constraints. Computational results are reported on euclidean problems derived from well-known benchmark instances for the vehicle routing problem with time windows.     

Scheduling a triple round robin tournament for the best Danish soccer league

In this paper, we present a solution method for the highly constrained problem of finding a seasonal schedule for the best Danish soccer league. The league differs from most sports leagues, since it plays a triple round robin tournament which leads to an uneven distribution of home and away games. The solution method presented here uses a logic-based Benders decomposition in which the master problem finds home-away pattern sets while the subproblem finds timetables. Furthermore, column generation techniques are used to enhance the speed of the master problem. The computational results show that the solution method is capable of solving the problem within reasonable time and the Danish Football Association has used it for scheduling the 2006/2007 season.     

A new model and hybrid approach for large scale inventory routing problems

This paper studies an inventory routing problem (IRP) with split delivery and vehicle fleet size constraint. Due to the complexity of the IRP, it is very difficult to develop an exact algorithm that can solve large scale problems in a reasonable computation time. As an alternative, an approximate approach that can quickly and near-optimally solve the problem is developed based on an approximate model of the problem and Lagrangian relaxation. In the approach, the model is solved by using a Lagrangian relaxation method in which the relaxed problem is decomposed into an inventory problem and a routing problem that are solved by a linear programming algorithm and a minimum cost flow algorithm, respectively, and the dual problem is solved by using the surrogate subgradient method. The solution of the model obtained by the Lagrangian relaxation method is used to construct a near-optimal solution of the IRP by solving a series of assignment problems. Numerical experiments show that the proposed hybrid approach can find a high quality near-optimal solution for the IRP with up to 200 customers in a reasonable computation time.     

An improved local branching approach for train formation planning

The train formation plan (TFP) determines routing and frequency of trains, and assigns the demands to trains. In this paper, an improved local branching algorithm is proposed for the TFP model in Iranian railway. This solution strategy is exact in nature, although it is designed to improve the heuristic behavior of the mixed integer programming (MIP) solver at hand. In the local branching algorithm, additional constraints are built in the model for the binary variables, but in the improved local branching algorithm, the additional constraints are built in the model for integer variables. A state-of-the-art method is applied for parameter tuning using design of experiments approach. To evaluate the proposed solution method, we have simulated and solved twenty test problems. The results show the efficiency and effectiveness of the proposed approach. The proposed algorithm is implemented for Iranian Railway network as a case study.     

Exact approaches for lifetime maximization in connectivity constrained wireless multi-role sensor networks

In this paper, we consider the duty scheduling of sensor activities in wireless sensor networks to maximize the lifetime. We address full target coverage problems contemplating sensors used for sensing data and transmit it to the base station through multi-hop communication as well as sensors used only for communication purposes. Subsets of sensors (also called covers) are generated. Those covers are able to satisfy the coverage requirements as well as the connection to the base station. Thus, maximum lifetime can be obtained by identifying the optimal covers and allocate them an operation time. The problem is solved through a column generation approach decomposed in a master problem used to allocate the optimal time interval during which covers are used and in a pricing subproblem used to identify the covers leading to maximum lifetime. Additionally, Branch-and-Cut based on Benders’ decomposition and constraint programming approaches are used to solve the pricing subproblem. The approach is tested on randomly generated instances. The computational results demonstrate the efficiency of the proposed approach to solve the maximum network lifetime problem in wireless sensor networks with up to 500 sensors.