This paper addresses the problem of enumerating all supported efficient solutions for a linear multi-objective integer minimum cost flow problem (MOIMCF). It derives an output-polynomial time algorithm to determine all supported efficient solutions for MOIMCF problems. This is the first approach to solve this general problem in output-polynomial time. Moreover, we prove that the existence of an output-polynomial time algorithm to determine all weakly supported nondominated vectors (or all weakly supported efficient solutions) for a MOIMCF problem with a fixed number of d ≥ 3 objectives can be excluded unless P = NP .

On-demand ridepooling systems offer flexible services pooling multiple passengers into one vehicle, complementing traditional bus services. We propose a transportation system combining the spatial aspects of a fixed sequence of bus stops with the temporal flexibility of ridepooling. In the line-based Dial-a-Ride problem (liDARP), vehicles adhere to a fixed, ordered sequence of stops in their routes, with the possibility of taking shortcuts and turning if they are empty. We propose three MILP formulations for the liDARP with a multi-objective function balancing environmental aspects with customer satisfaction, comparing them on a real-world bus line. Our experiments show that the formulation based on an Event-Based graph is the fastest, solving instances with up to 50 requests in under one second. Compared to the classical DARP, the liDARP is computationally faster, with minimal increases in total distance driven and average ride times.

Abstract We propose a multi-swarm approach to approximate the Pareto front of general multi-objective optimization problems that is based on the consensus-based optimization method (CBO). The algorithm is motivated step by step beginning with a simple extension of CBO based on fixed scalarization weights. To overcome the issue of choosing the weights we propose an adaptive weight strategy in the second modeling step. The modeling process is concluded with the incorporation of a penalty strategy that avoids clusters along the Pareto front and a diffusion term that prevents collapsing swarms. Altogether the proposed K-swarm CBO algorithm is tailored for a diverse approximation of the Pareto front and, simultaneously, the efficient set of general non-convex multi-objective problems. The feasibility of the approach is justified by analytic results, including convergence proofs, and a performance comparison to the well-known non-dominated sorting genetic algorithms NSGA2 and NSGA3 as well as the recently proposed one-swarm approach for multi-objective problems involving consensus-based optimization.

In many ridepooling applications transportation requests arrive throughout the day and have to be answered and integrated into the existing (and operated) vehicle routing. To solve this dynamic dial-a-ride problem we present a rolling-horizon algorithm that dynamically updates the current solution by solving an MILP formulation. The MILP model is based on an event-based graph with nodes representing pick-up and drop-off events associated with feasible user allocations in the vehicles. The proposed solution approach is validated on a set of real-word instances with more than 500 requests. In 99.5% of all iterations the rolling-horizon algorithm returned optimal insertion positions w.r.t. the current schedule in a time-limit of 30 seconds. On average, incoming requests are answered within 2.8 seconds.

A problem of image registration is considered in the context of optimal mass transportation. The properties and limitations of an optimal image transportation are analyzed. A modified formulation of this approach is proposed in order to overcome the morphing effect. Finally, a fast and simple scale-space approach for the new formulation is introduced, and numerical examples are presented.