José M. P. Paixão

Multiobjective Metaheuristics for the Bus Driver Scheduling Problem

We present new multiobjective metaheuristics for solving real-world crew scheduling problems in public bus transport companies. Since the crews of these companies are drivers, we will designate the problem as bus-driver scheduling. Crew scheduling problems are well known, and several mathematical programming-based techniques have been proposed to solve them, in particular, using the single-objective set-covering formulation. However, in practice, there exists the need to consider multiple objectives, some of them in conflict with each other; for example, the cost and service quality, implying also that alternative solution methods have to be developed. We propose multiobjective metaheuristics based on the tabu search and genetic algorithms. These metaheuristics also present some innovation features related with the structure of the crew scheduling problem that guide the search efficiently and enable them to find good solutions. Some of these new features can also be applied to the development of heuristics to other combinatorial optimization problems. A summary of computational results with real-data problems is presented. The methods have been successfully incorporated in the GIST Planning Transportation Systems and are actually used by several companies.

Models and Algorithms for Single-Depot Vehicle Scheduling

Vehicle scheduling is the process of assigning vehicles to a set of predetermined trips with fixed starting and ending times, while minimizing capital and operating costs. This paper considers modeling, algorithmic, and computational aspects of the polynomially solvable case in which there is a single depot and vehicles are identical. A quasiassignment formulation is reviewed and an alternative asymmetric assignment formulation is given. The main contributions of the paper are a new two-phase approach which is valid in the case of a special cost structure, an auction algorithm for the quasiassignment problem, a core-oriented approach, and an extensive computational study. New algorithms are compared with the most successful algorithms for the vehicle-scheduling problem, using both randomly generated and real-life data. The new algorithms show a significant performance improvement with respect to computation time. Such improvement can, for example, be very important when this particular vehicle-scheduling problem appears as a subproblem in more complex vehicle- and crew-scheduling problems.

Vehicle Scheduling for Public Mass Transit — An Overview

Vehicle scheduling always plays an important role in urban mass transit companies. Efficient schedules of a given timetable have a significant influence on the demand for vehicles and, hence, the demand for personnel. This means that the economic results of a urban mass transit company are closely linked to the planning activities in this area.

Multiple Depot Vehicle Scheduling Problem: A New Heuristic Based on Quasi-Assignment Algorithms

This work is concerned with the multiple depot vehicle scheduling problem where one has to group timetabled trips into chains and, at the same time, assign those chains to depots in order to minimize fleet size costs and dead-heading time. A new mathematical programming formulation is presented for such problem which is known to be NP-hard (Bertossi et al, 1987). Lagrangean relaxation is used, with the corresponding relaxed problem being decomposed into: a semi-assignment problem; a multiple depot vehicle scheduling problem without the requirement that each vehicle must return to the source depot. This last problem is solved by an adaptation of the quasi-assignment algorithm for the multiple depot vehicle scheduling problem (Branco,1989), which takes into account the depots capacities. Computational experience is reported for problems randomly generated according to real life data.

Exact Algorithms for the Multi-Depot Vehicle Scheduling Problem Based on Multicommodity Network Flow Type Formulations

We compare the linear programming relaxation of different mathematical formulations for the multi-depot vehicle scheduling problem. As a result of this theoretical analysis, we select for development a tree search procedure based on a multicommodity network flow formulation that involves two different types of decision variables: one type is used to describe the connections between trips, in order to obtain the vehicle blocks, while the other type is related to the assignment of trips to depots. We also develop a branch and bound algorithm based on the linear relaxation of a more compact multicommodity network flow formulation, in the sense that it contains just one type of variable and fewer constraints than the previous model. Computational experience is presented to compare the two algorithms.

Vehicle Scheduling with Time Constraint

We present methods for solving the vehicle scheduling problem with time constraint. Such problem consists of minimizing the costs related to the assignment of vehicles for performing a set of short trips. The vehicles are located at a single depot and one must consider the additional constraint that no vehicle can be away from the depot longer than a maximum time period. For two integer programming models we consider the corresponding Linear Programming and Lagrangean relaxations. The mathematical programming approach as well as a heuristic approach are tested on real-life problems.

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.

Driver scheduling problem modelling

The Drivers Scheduling Problem (DSP) consists of selecting a set of duties for vehicle drivers, such as bus, train and boat drivers or plane pilots, for the transportation of passengers or goods. This is a complex problem as it involves several constraints related to labour and company rules and may also entail different evaluation criteria and objectives. The ability to develop an adequate model for this problem, which can represent the real problem as closely as possible is an important research area.In this paper we present new mathematical models for the DSP which embody the very complexity of the drivers scheduling problem, besides demonstrating that the solutions generated by these models can easily be implemented in real situations.On the strength of extensive passenger transportation experience in bus companies in Portugal, we propose and test new alternative models to formulate the DSP. These models are based on Set Partitioning/Covering models. Moreover, they also take into account the bus operator issues and the user’s standpoint and environment.

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 New Approach for the Crew Rostering Problem

In this paper an algorithm is presented for determining a lower bound for the Crew Rostering Problem (CRP). The formulation of the CRP presented is basically that of a Set Covering Problem (SCP) with additional constraints. The method proposed to derive a lower bound consists on solving a linear relaxation based on an adaptation of the classic Column Generation approach, since the reduced costs can only be underestimated. The associated sub-problem is shown to be a Shortest Path with Additional Constraints Problem. Furthermore, a very efficient cut for the formulation is presented.