Computing strong lower and upper bounds for the integrated multiple-depot vehicle and crew scheduling problem with branch-and-price

In the problem of the title, vehicle and crew schedules are to be determined simultaneously in order to satisfy a given set of trips over time. The vehicles and the crew are assigned to depots, and a number of rules have to be observed in the course of constructing feasible schedules. The main contribution of the paper is a novel mathematical programming formulation which combines ideas from known models, and an exact solution procedure based on branch-and-price. The method is tested on benchmark instances from the literature and it provides suboptimal schedules using limited computational resources

Combining Column Generation and Lagrangian Relaxation

Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems.column generation;Lagrangean relaxation;cutting stock problem;lotsizing;vehicle and crew scheduling

Multiple-Depot Integrated Vehicle and Crew Scheduling

This paper presents two different models and algorithms for integrated vehicle and crew scheduling in the multiple-depot case. The algorithms are both based on a combination of column generation and Lagrangian relaxation. Furthermore, we compare those integrated approaches with each other and with the traditional sequential one on random generated as well as real-world data instances for a suburban/extra-urban mass transit system. To simulate such a transit system, we propose a new way of generating randomly data instances such that their properties are the same as for our real-world instances

Combining Column Generation and Lagrangian Relaxation

Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems

On a Vehicle Routing Problem with Customer Costs and Multi Depots

The Vehicle Routing Problem with Customer Costs (short VRPCC) was developed for railway maintenance scheduling. In detail, corrective maintenance jobs for unexpected occurring failures are planned to a short time horizon. These jobs are geographically distributed in the railway net. Furthermore, dependent on the severity of the failure, it can be necessary to reduce the top speed on the track section in order to avoid safety risks or a too fast deterioration. For fatal failures, it can even be necessary to close the track section. The resulting limitations on railway service lead to penalty costs for the maintenance operator. These must be paid until the track is repaired and the restrictions are removed. By scheduling the maintenance tasks, these penalty costs can be reduced by proceeding corresponding maintenance tasks earlier. However, this may in return lead to increased costs for moving the maintenance machines and crews. For this scheduling problem, the VRPCC was developed. With it, for each maintenance vehicle and crew, a route is defined that describes the order to proceed maintenance tasks. Two kinds of costs are considered: Firstly, travel costs for machinery and crew; and secondly, penalty costs for an unsafe track condition that have to be paid for each day from failure detection to maintenance completion. To model the penalties, the novel customer costs are defined. In detail, for each maintenance activity a customer cost coefficient is given which incur for each day between failure detection and failure repair. The objective function of this problem is defined by the sum of travel costs and time-dependent customer costs. With it, the priority of customers can be taken into account without losing the sight on travel costs. This new vehicle routing problem was introduced in this thesis by a non-linear partition and permutation model. In this model, a feasible solution is defined by a partition of the job set into subsets that represent the allocation of jobs to vehicles and a permutation for each subset that represent the order of processing the jobs. Then, the start times of the jobs were calculated based on the order given by the permutations. It was taken into account that work can only be done in eight hour shifts during the night. Based on the start times, the customer cost value of each job is computed which equals to the paid penalty costs. Then, the costs of a schedule are calculated via the sum of travel costs and customer costs. To solve the VRPCC by a commercial linear programming solver, different formulations of the VRPCC as mixed-integer linear program were developed. In doing so, the start times became decision variables. It turned out that including customer costs led to problems harder to solve than vehicle routing problems where only travel costs are minimized. Further, in the thesis several construction heuristics for the VRPCC were designed and investigated. Also two local search algorithms, first and best improvement, were applied. The computational experiments showed that the solutions generated by the local search algorithm were much better than the solutions of the construction heuristics. The main part of this thesis was to design a Branch-and-Bound algorithm for the VRPCC. For this purpose, new lower bounds for the customer cost part of the objective function were formulated. The computational experiments showed that a lower bound computed from the LP relaxation of a specific bin packing problem had the best trade-off between computational effort and bound quality. For the travel cost part of the objective function, several known lower bounds from the TSP were compared. To design a Branch-and-Bound algorithm, beside efficient lower bound, also suitable branching strategies are necessary to split the problem space into smaller subspaces. In this thesis two branching strategies were developed which are based on the non-linear partition and permutation model to take advantage from the problem structure. To be more precise, new branches are generated by appending or including a job to an uncompleted schedule. Consequently, the start times can be computed directly from the so far planned jobs and more tight lower bounds can be computed for the so far unplanned jobs. By means of computational experiments, the developed Branch-and-Bound algorithms were compared with the classical approach, which means solving a mixed-integer linear program of the VRPCC by a commercial solver. The results showed that both Branch-and-Bound algorithms solved the small instances faster than the classical approach