Ana Paias

Set partitioning/covering-based approaches for the integrated vehicle and crew scheduling problem

In the integrated vehicle and crew scheduling problem (VCSP) one has to simultaneously assign vehicles to timetabled trips and drivers to vehicles. In this paper, the VCSP is described by an integer linear programming formulation combining a multicommodity network flow model with a set partitioning/covering model. We propose an algorithm that starts with a pre-processing phase to define the set of tasks and to obtain an initial set of duties. In a second phase, we solve the linear programming relaxation of the models using a column generation scheme. Whenever the resulting solution is not integer, branch-and-bound techniques are used over the set of feasible crew duties generated while solving the linear programming relaxation in order to obtain a feasible solution for the VCSP. We show that, under some conditions, just one subset of the decision variables is required to be integer. Computational experience, with randomly generated data publicly available for benchmarking on the WEB is reported, and the results show the effectiveness of our approach concerning both, the quality of the solutions and the CPU time needed to obtain them. Moreover, the approach can be applied to large instances.

A new model for the integrated vehicle-crew-rostering problem and a computational study on rosters

Operational planning within public transit companies has been extensively tackled but still remains a challenging area for operations research models and techniques. This phase of the planning process comprises vehicle-scheduling, crew-scheduling and rostering problems. In this paper, a new integer mathematical formulation to describe the integrated vehicle-crew-rostering problem is presented. The method proposed to obtain feasible solutions for this binary non-linear multi-objective optimization problem is a sequential algorithm considered within a preemptive goal programming framework that gives a higher priority to the integrated vehicle-crew-scheduling goal and a lower priority to the driver rostering goals. A heuristic approach is developed where the decision maker can choose from different vehicle-crew schedules and rosters, while respecting as much as possible management’s interests and drivers’ preferences. An application to real data of a Portuguese bus company shows the influence of vehicle-crew-scheduling optimization on rostering solutions.

Models for a traveling purchaser problem with additional side-constraints

The traveling purchaser problem (TPP) is the problem of determining a tour of a purchaser that needs to buy several items in different shops such that the total amount of travel and purchase costs is minimized. Motivated by an application in machine scheduling, we study a variant of the problem with additional constraints, namely, a limit on the maximum number of markets to be visited, a limit on the number of items bought per market and where only one copy per item needs to be bought. We present an integer linear programming (ILP) model which is adequate for obtaining optimal integer solutions for instances with up to 100 markets. We also present and test several variations of a Lagrangian relaxation combined with a subgradient optimization procedure. The relaxed problem can be solved by dynamic programming and can also be viewed as resulting from applying a state space relaxation technique to a dynamic programming formulation. The Lagrangian based method is combined with a heuristic that attempts to transform relaxed solutions into feasible solutions. Computational results for instances with up to 300 markets show that with the exception of a few cases, the reported differences between best upper bound and lower bound values on the optimal solutions are reasonably small.

State space relaxation for set covering problems related to bus driver scheduling

Abstract This paper reports on a lower bound technique based on state space relaxation for a dynamic program associated with a particular class of covering problems related with crew scheduling. Both dynamic programming (DP) and state space relaxation (SSR) techniques may be applied to any type of set covering problem (SCP), but, in particular, the SSR described in this paper revealed itself as an interesting approach for the bus driver scheduling problem. SSR provides a lower bound on the optimal value for the SCP and some reduction tests are derived in order to reduce the number of columns and rows for the instances. Also, feasible solutions may be build upon the SSR solutions yielding an upper bound on the optimum. Computational experience with real life test problems shows that the technique described in this paper is worthwhile trying when dealing with such applications. In fact, for most of the cases, we were able to improve on the quality of the feasible solutions obtained through the using of the greedy heuristics described in the literature for the set covering problem.

A decomposition approach for the integrated vehicle-crew-roster problem with days-off pattern

The integrated vehicle-crew-roster problem with days-off pattern aims to simultaneously determine minimum cost vehicle and daily crew schedules that cover all timetabled trips and a minimum cost roster covering all daily crew duties according to a pre-defined days-off pattern. This problem is formulated as a new integer linear programming model and is solved by a heuristic approach based on Benders decomposition that iterates between the solution of an integrated vehicle-crew scheduling problem and the solution of a rostering problem. Computational experience with data from two bus companies in Portugal and data from benchmark vehicle scheduling instances shows the ability of the approach for producing a variety of solutions within reasonable computing times as well as the advantages of integrating the three problems.

Restricted dynamic programming based neighborhoods for the hop-constrained minimum spanning tree problem

In this paper we develop, study and test new neighborhood structures for the Hop-constrained Minimum Spanning Tree Problem (HMSTP). These neighborhoods are defined by restricted versions of a new dynamic programming formulation for the problem and provide a systematic way of searching neighborhood structures based on node-level exchanges. We have also developed several local search methods that are based on the new neighborhoods. Computational experiments for a set of benchmark instances with up to 80 nodes show that the more elaborate methods produce in a quite fast way, heuristic solutions that are, for all cases, within 2% of the optimum.

A decompose-and-fix heuristic based on multi-commodity flow models for driver rostering with days-off pattern

Facing severe budgetary constraints, public transport companies are forced to efficiently manage staff, one of the most expensive resources.

Solving the family traveling salesman problem

In this paper we address the family traveling salesman problem (FTSP), an NP-hard problem in which the set of nodes of a graph is partitioned into several subsets, which are called families. The objective is to visit a predefined number of nodes in each family at a minimum cost. We present several compact and non-compact models for the FTSP. Computational experiments with benchmark instances show that the non-compact models outperform the compact ones. One of the non-compact models is able to solve instances with 127 nodes, in less than 70 seconds, and one of the instances with 280 nodes in 3615 seconds. The optimal values of these instances were not known. For the higher dimensioned instances, the ones whose optimal value remains unknown, we propose an iterated local search algorithm that is able to improve the best known upper bounds from the literature.

Metaheuristics based on decision hierarchies for the traveling purchaser problem

In this paper we address the traveling purchaser problem, an NP-hard problem that generalizes the traveling salesman problem. We present several metaheuristics that combine genetic algorithms and local search. The genetic algorithms are induced by different hierarchic orderings of the decision making regarding the route and the acquisition of the items. Computational experiments were carried out with benchmark instances and the results show that the proposed metaheuristics are a suitable tool to solve high-dimensioned instances for which the exact methods do not provide solutions within a reasonable CPU time. For several instances, best new upper bounds for the optimum value of the objective function were obtained.

An Integrated Vehicle-Crew-Roster Problem with Days-Off Pattern

The integrated vehicle-crew-roster problem with days-off pattern aims to simultaneously determine minimum cost sets of vehicle and daily crew schedules that cover all timetabled trips and a minimum cost roster covering all daily crew duties according to a pre-defined days-off pattern. This problem is modeled as a mixed binary linear programming problem. A heuristic approach with embedded column generation and branch-and-bound techniques within a Benders decomposition is proposed. The new methodology was tested on real instances and the computational results are promising.