Analysis of Baseline Evolutionary Algorithms for the Packing While Travelling Problem

Although the performance of base-line Evolutionary Algorithms (EAs) on linear functions has been studied rigorously, the same theoretical analyses on non-linear objectives are still far behind. In this paper, variations of the Packing While Travelling (PWT), also known as a non-linear knapsack problem, is considered to address this gap. We investigate PWT for two cities with correlated weights and profits using single-objective and multi-objective algorithms. Our results show that RLS finds the optimal solution in $O(n^3)$ expected time while the GSEMO enhanced with a specific selection operator to deal with exponential population size, calculates all the Pareto front solutions in the same expected time. In the case of uniform weights, (1+1)~EA is able to find the optimal solution in expected time $O(n^2\log{(\max\{n,p_{max}\})})$, where $p_{max}$ is the largest profit of the given items. We also validate the theoretical results using practical experiments and present estimation for expected running time according to the experiments.