The two most common families of electoral systems (ESs), defining the rules used to elect assemblies and legislative institutions, are proportional representation (PR) and plurality (Bormann and Golder, 2013; Farrell, 2011). When they are evaluated, most often the arguments come from social choice theory and political science. The former overall uses an axiomatic approach that includes a list of mathematical criteria a system should fulfill (Urken et al., 1995; Sen, 1995). The latter predominantly focuses on the tradeoff between proportionality of apportionment and governability (Monroe, 1994; Carey and Hix, 2011). However, there is an ongoing discussion about which ES is the best (Bowler et al., 2005; Farrell and Gallagher, 1999) and which set of indexes and measures would be the most important in such assessment (Pennisi, 1998). Although previous research addressed various perceptions of fairness related to the proportionality of different ESs (Blau, 2004; Plescia et al., 2020), the sensitivity of ES to efforts that influence opinions has been neglected. Here, we address this research gap with a framework that can measure ESs’ susceptibility to different means of influence. Using a simulation study, we show that plurality ESs are less stable than PR. They are more susceptible to coordinated efforts to influence opinions, for example, by political agitators and media propaganda. A review of real-world ES reveals possible improvements in their design, leading to lower susceptibility. Additionally, our simulation framework allows the computation of popular indexes, such as the Gallagher index and the effective number of parties, in different scenarios. Our work provides a new tool for dealing with modern threats to democracy that could destabilize voting processes (Hyde, 2020). Furthermore, our results add an important argument to a long-standing discussion on the evaluation of ES.

When evaluating the operational costs of a public transport system, the most important factor is the number of vehicles needed for operation. In contrast to the canonical sequential approach of first fixing a timetable and then adding a vehicle schedule, we consider a sequential approach where a vehicle schedule is determined for a given line plan and only afterwards a timetable is fixed. We compare this new sequential approach to a model that integrates both steps. To represent various operational requirements, we consider multiple possibilities to restrict the vehicle circulations to be short, as this can provide operational benefits. The sequential approach can efficiently determine public transport plans with a low number of vehicles. This is evaluated theoretically and empirically demonstrated for two close-to real-world instances.

We consider the following problem: given a set of lines in a public transportation network with their round trip times and frequencies, a maximum number of vehicles and a maximum number of lines that can be combined into a vehicle circulation, does there exist a set of vehicle circulations that covers all lines given the constraints. Solving this problem provides an estimate of the costs of operating a certain line plan, without having to compute a timetable first. We show that this problem is NP-hard for any restriction on the number of lines that can be combined into a circulation which is equal to or greater than three. We pay special attention to the case where at most two lines can be combined into a circulation, which is NP-hard if a single line can be covered by multiple circulations. If this is not allowed, a matching algorithm can be used to find the optimal solutions, which we show to be a 16/15-approximation for the case where it is allowed. We also provide an exact algorithm that is able to exploit low tree-width of the so-called circulation graph and small numbers of vehicles required to cover single circulations.

Abstract A promising new delivery model involves the use of a delivery truck that collaborates with a drone to make deliveries. Effectively combining a truck and a drone gives rise to a new planning problem that is known as the traveling salesman problem with drone (TSP‐D). This paper presents exact solution approaches for the TSP‐D based on dynamic programming and provides an experimental comparison of these approaches. Our numerical experiments show that our approach can solve larger problems than the mathematical programming approaches that have been presented in the literature thus far. Moreover, we show that restrictions on the number of locations the truck can visit while the drone is away can help significantly reduce the solution times while having relatively little impact on the overall solution quality.