# Multi-Agent Logistics Optimization

This is a simplified version of problem faced by logistics company.

Problem: Given there are # dispatchers and # of locations for delivery, optimize the paths for each dispatcher.

\begin{aligned} & \min & & \sum_{i,j,k} W_{ij}X_{ijk} \\ & \text{where} & & W := \text{Euclidean distance matrix} \\ &&& X := \text{Adjacency matrix} \\ & \text{s.t.} & & i, j \in \{1, \dots, \#Locations \} \\ &&& k \in \{1, \dots, \#Dispatchers \} \\ \end{aligned}

Solution:

Linear Programming. Computing adjacency matrices for all dispatcher in a single run is very expensive and slow. One could adopt divide and conquer strategy for scalability. For example, during the first iteration, set number of dispatcher to 1 to get overall shortest path, then cut it into 2 and rerun the optimizer for each of the parts. More constraints can be introduced to set the minimum and maximum work to be carried out by each of the dispatcher.