In this paper we present the Group Seat Reservation Knapsack Problem with Profit on Seat. This is an extension of the the Offline Group Seat Reservation Knapsack Problem. In this extension we introduce a profit evaluation dependant on not only the space occupied, but also on the individual profit brought by each reserved seat. An application of the new features introduced in the proposed extension is to influence the distribution of passengers, such as assigning seats near the carriage centre for long journeys, and close to the door for short journeys. Such distribution helps to reduce the excess of dwelling time on platform. We introduce a new GRASP based algorithm that solves the original problem and the newly proposed one. In the experimental section we show that such algorithm can be useful to provide a good feasible solution very rapidly, a desirable condition in many real world systems. Another application could be to use the algorithm solution as a startup for a successive branch and bound procedure when optimality is desired. We also add a new class of problem with five test instances that represent some challenging real-world scenarios that have not been considered before. Finally, we evaluate both the existing model, the newly proposed model, and analyse the pros and cons of the proposed algorithm.

Introduction

In this paper we extend the Offline Group Seat Reservation Knapsack Problem (GSR-KP) presented in [6]. In the original formulation, a train with W seats stops in H stations. It is required to allocate n reservations. Each reservation i occupy a set of contiguous seats for wi people from one initial station yi to a final one hi . The profit is identified as to maximise the space occupied during the journey. In our extension the value of the profit of the reservation is dependent also on the profits assigned to seats in which the reservation is eventually allocated. Our extension makes the problem more realistic, allowing the modelling of scenarios that were not possible to model with the original formulation. The new scenarios cover the cases where the ideal position of an item is affected by how long the item must be kept in its position. We exploit the original naming style and call the new extension Group Seat Reservation Knapsack Problem with Profit on Seat (GSR-KPPS). Moreover, solving realistically sized instances is challenging for a general solver and often having a good solution rapidly may be better than having an optimal solution later, e.g. when there are fixed time constraints.