We consider multi-agent transport task problems where, e.g. in a factory setting, items have to be delivered from a given start to a goal pose while the delivering robots need to avoid collisions with each other on the floor. We introduce a Task Conflict-Based Search (TCBS) Algorithm to solve the combined delivery task allocation and multi-agent path planning problem optimally. The problem is known to be NP-hard and the optimal solver cannot scale. However, we introduce it as a baseline to evaluate the sub-optimality of other approaches. We show experimental results that compare our solver with different sub-optimal ones in terms of regret.
A different type of problem is studied in vehicle routing with capacities @cite_4 , which focuses on the deliveries from one central depot based on certain demands. This is a different problem in the sense that it considers only one origin and additionally considers capacity constraints.
@cite_4
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In this paper we deal with a vehicle routing problem on a tree-shaped network with a single depot. Customers are located on vertices of the tree, and each customer has a positive demand. Demands of customers are served by a fleet of identical vehicles with limited capacity. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. Each tour begins at the depot, visits a subset of the customers and returns to the depot without violating the capacity constraint. We show that the problem is NP-complete and propose a 1.5-approximation algorithm for the problem. We also give some computational results. This paper presents a new approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot. Customers are located on vertices of the tree, and each customer has a positive demand. Demands of customers are served by a fleet of identical vehicles with limited capacity. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. Each tour begins at the depot, visits a subset of the customers and returns to the depot without violating the capacity constraint We propose a 1.35078-approximation algorithm for the problem (exactly, (√41 - 1) 4), which is an improvement over the existing 1.5-approximation. The vehicle routing problem with multiple routes consists in determining the routing of a fleet of vehicles when each vehicle can perform multiple routes during its operations day. This problem is relevant in applications where the duration of each route is limited, for example when perishable goods are transported. In this work, we assume that a fixed-size fleet of vehicles is available and that it might not be possible to serve all customer requests, due to time constraints. Accordingly, the objective is first to maximize the number of served customers and then, to minimize the total distance traveled by the vehicles. An adaptive large neighborhood search, exploiting the ruin-and-recreate principle, is proposed for solving this problem. The various destruction and reconstruction operators take advantage of the hierarchical nature of the problem by working either at the customer, route or workday level. Computational results on Euclidean instances, derived from well-known benchmark instances, demonstrate the benefits of this multi-level approach. In the capacitated vehicle routing problem, introduced by Dantzig and Ramser in 1959, we are given the locations of n customers and a depot, along with a vehicle of capacity k, and wish to find a minimum length collection of tours, each starting from the depot and visiting at most k customers, whose union covers all the customers. We give a quasi-polynomial time approximation scheme for the setting where the customers and the depot are on the plane, and distances are given by the Euclidean metric.