A Local Search Algorithm for Solving Large Scale Conflict Graphs in Freight Railway Timetabling

We consider the problem of planning the annual timetable for all freight trains in Germany simultaneously. That is, for each train, construct a slot through the network such that no two slots of different trains have a conflict. We denote this task by the Train Path Assignment Problem (TPAP) and consider a column generation approach where iteratively, we are given for each train request a growing set of possible slots. In each iteration, we look for a maximum subset without any conflicts. We model this problem as the Maximum Independent Set problem (MIS). Due to the many slots that are constructed, hence variables that are generated, we deal with large scale MIS instances. Therefore, we solve the MIS heuristically and develop a local search algorithm called Conflict Resolving (CR) that is tailored to the specially structured instances from the application. To solve the MIS, CR iteratively perturbs the current solution in order to leave local optima and then repeatedly improves the solution by replacing k-1 solution vertices by k non-solution vertices. These steps are embedded in a simulated annealing framework. In this paper, we present the column generation approach that is solved as an MIS. Furthermore, we introduce the CR algorithm and numerically compare it to both, a MIP solver and Iterated Local Search (ILS), a state-of-the-art heuristics. It turns out that CR performs best for the instances from real-world timetabling, and is also comparable to the ILS on selected MIS benchmark instances

Local Search Approximation Algorithms for Clustering Problems

In this research we study the use of local search in the design of approximation algorithms for NP-hard optimization problems. For our study we have selected several well-known clustering problems: k-facility location problem, minimum mutliway cut problem, and constrained maximum k-cut problem. We show that by careful use of the local optimality property of the solutions produced by local search algorithms it is possible to bound the ratio between solutions produced by local search approximation algorithms and optimum solutions. This ratio is known as the locality gap. The locality gap of our algorithm for the k-uncapacitated facility location problem is 2+sqrt(3) +epsilon for any constant epsilon \u3e0. This matches the approximation ratio of the best known algorithm for the problem, proposed by Zhang but our algorithm is simpler. For the minimum multiway cut problem our algorithm has locality gap 2-2/k, which matches the approximation ratio of the isolation heuristic of Dahlhaus et al; however, our experimental results show that in practice our local search algorithm greatly outperforms the isolation heuristic, and furthermore it has comparable performance as that of the three currently best algorithms for the minimum multiway cut problem. For the constrained maximum k-cut problem on hypergraphs we proposed a local search based approximation algorithm with locality gap 1-1/k for a variety of constraints imposed on the k-cuts. The locality gap of our algorithm matches the approximation ratio of the best known algorithm for the max k-cut problem on graphs designed by Vazirani, but our algorithm is more general

Comparison of heuristic approaches for the multiple depot vehicle scheduling problem

Given a set of timetabled tasks, the multi-depot vehicle scheduling problem is a well-known problem that consists of determining least-cost schedules for vehicles assigned to several depots such that each task is accomplished exactly once by a vehicle. In this paper, we propose to compare the performance of five different heuristic approaches for this problem, namely, a heuristic \\mip solver, a Lagrangian heuristic, a column generation heuristic, a large neighborhood search heuristic using column generation for neighborhood evaluation, and a tabu search heuristic. The first three methods are adaptations of existing methods, while the last two are novel approaches for this problem. Computational results on randomly generated instances show that the column generation heuristic performs the best when enough computational time is available and stability is required, while the large neighborhood search method is the best alternative when looking for a compromise between computational time and solution quality

The congested multicommodity network design problem

This paper studies a version of the fixed-charge multicommodity network design problem where in addition to the traditional costs of flow and design, congestion at nodes is explicitly considered. The problem is initially modeled as a nonlinear integer programming formulation and two solution approaches are proposed: (i) a reformulation of the problem as a mixed integer second order cone program to optimally solve the problem for small to medium scale problem instances, and (ii) an evolutionary algorithm using elements of iterated local search and scatter search to provide upper bounds. Extensive computational results on new benchmark problem instances and on real case data are presented

The Steiner Multi Cycle Problem with Applications to a Collaborative Truckload Problem

We introduce a new problem called Steiner Multi Cycle Problem that extends the Steiner Cycle problem in the same way the Steiner Forest extends the Steiner Tree problem. In this problem we are given a complete weighted graph G=(V,E), which respects the triangle inequality, a collection of terminal sets {T_1,..., T_k}, where for each a in [k] we have a subset T_a of V and these terminal sets are pairwise disjoint. The problem is to find a set of disjoint cycles of minimum cost such that for each a in [k], all vertices of T_a belong to a same cycle. Our main interest is in a restricted case where |T_a| = 2, for each a in [k], which models a collaborative less-than-truckload problem with pickup and delivery. In this problem, we have a set of agents where each agent is associated with a set T_a containing a pair of pickup and delivery vertices. This problem arises in the scenario where a company has to periodically exchange goods between two different locations, and different companies can collaborate to create a route that visits all its pairs of locations sharing the total cost of the route. We show that even the restricted problem is NP-Hard, and present some heuristics to solve it. In particular, a constructive heuristic called Refinement Search, which uses geometric properties to determine if agents are close to each other. We performed computational experiments to compare this heuristic to a GRASP based heuristic. The Refinement Search obtained the best solutions in little computational time

Heuristically Driven Search Methods for Topology Control in Directional Wireless Hybrid Network

Information and Networked Communications play a vital role in the everyday operations of the United States Armed Forces. This research establishes a comparative analysis of the unique network characteristics and requirements introduced by the Topology Control Problem (also known as the Network Design Problem). Previous research has focused on the development of Mixed-Integer Linear Program (MILP) formulations, simple heuristics, and Genetic Algorithm (GA) strategies for solving this problem. Principal concerns with these techniques include runtime and solution quality. To reduce runtime, new strategies have been developed based on the concept of flow networks using the novel combination of three well-known algorithms; knapsack, greedy commodity filtering, and maximum flow. The performance of this approach and variants are compared with previous research using several network metrics including computation time, cost, network diameter, dropped commodities, and average number of hops per commodity. The results conclude that maximum flow algorithms alone are not quite as effective as previous findings, but are at least comparable and show potential for larger networks

Hybrid Statistical Data Mining Framework for Multi-Commodity Fixed Charge Network Flow Problem

This paper presents a new approach to analyze the network structure in multi-commodity fixed charge network flow problems (MCFCNF). This methodology uses historical data produced from repeatedly solving the traditional MCFCNF mathematical model as input for the machine-learning framework. Further, we reshape the problem as a binary classification problem and employ machine-learning algorithms to predict network structure. This predicted network structure is further used as an initial solution for our mathematical model. The quality of the initial solution generated is judged on the basis of predictive accuracy, feasibility and reduction in solving time

Optimization models and methods for transportation services

Managing transportation services efficiently is essential to both public and private sectors. This dissertation addresses three scheduling problems in modern transportation systems: the network design problem, the train dispatching problem, and the service route design problem. The transportation network design problem with service requirements designs arcs on a directed network and route commodities on the designed arcs so that i) commodities satisfy service requirements and ii) the total cost is minimized. We develop three mathematical programming models: a compact but weak arc-flow formulation, a large but strong path-flow formulation, and a hybrid formulation that uses both the arc-flow and the path-flow representations. We show that the hybrid formulation can significantly strengthen the LP formulation without introducing many variables. To find a good hybrid formulation, we develop columnization and decolumnization algorithms that uses the LP relaxation information to identify commodities that should use the path-flow representation. We also develop valid inequalities for commodities using the path-flow representation. The train dispatching problem schedules the movements of trains on scarce railroad tracks so as to improve the average velocity of trains. We develop a mathematical programming model and strengthen the model using valid inequalities. Besides, we present a heuristic to find a feasible solution quickly, which can serve as the warm-start solution to the MIP solver. For the third problem, we seek to design vehicle routes to deliver and pickup orders for a major grocery chain. We design a GRASP that can incorporate various operational requirements, including warehouse loading capacity, loading sequence, time window requirements, truck volume and weight capacities, and driver time limits. Our GRASP procedure consists of two phases: the solution construction (Phase I) and the Tabu search (Phase II). We show that the neighborhood structure of solutions is highly degenerate, which limits the solution space explored by the Tabu search. We apply the Tabu search with random variable neighborhood to increase the solution space explored.Operations Research and Industrial Engineerin