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Determining lower bound on number of vehicle blocks in multi-depot vehicle scheduling problem with mixed fleet covering electric buses

Duda Jerzy, Fierek Szymon, Karkula Marek, Kisielewski Piotr, Puka Radosław, Redmer Adam, Skalna Iwona

Archives of Transport

2023

Vol. 65, iss. 1

27--38

Scheduling buses in public transport systems consists in assigning trips to vehicle blocks. To minimize the cost of fuel and environmental impact of public transport, the number of vehicle blocks used should be as small as possible, but sufficient to cover all trips in a timetable. However, when solving real life transportation problems, it is difficult to decide whether the number of vehicle blocks obtained from an algorithm is minimal, unless the actual minimal number is already known, which is rare, or the theoretical lower bound on the number of vehicles has been determined. The lower bound on the number of vehicle blocks is even more important and useful since it can be used both as a parameter that controls the optimization process and as the minimum expected value of the respective optimization criterion. Therefore, methods for determining the lower bound in transportation optimization problems have been studied for decades. However, the existing methods for determining the lower bound on the number of vehicle blocks are very limited and do not take multiple depots or heterogeneous fleet of vehicles into account. In this research, we propose a new practical and effective method to assess the lower bound on the number of vehicle blocks in the Multi-Depot Vehicle Scheduling Problem (MDVSP) with a mixed fleet covering electric vehicles (MDVSP-EV). The considered MDVSP-EV reflects a problem of public transport planning encountered in medium-sized cities. The experimental results obtained for a real public transport system show the great potential of the proposed method in determining the fairly strong lower bound on the number of vehicle blocks. The method can generate an estimated distribution of the number of blocks during the day, which may be helpful, for example, in planning duties and crew scheduling. An important advantage of the proposed method is its low calculation time, which is very important when solving real life transportation problems.