Weighted-Scenario Optimisation for the Chance Constrained Travelling Thief Problem

TL;DR

This work tackles uncertainty in the Travelling Thief Problem by introducing a weighted-scenario, chance-constrained formulation. The authors define a weighted-scenario model with $k$ scenarios, each carrying a probability, to maximize the expected TTP score while ensuring the knapsack constraint is satisfied with probability at least $\alpha$. They adapt three algorithmic families—(1+1)EA, Pack, and local-search heuristics (S5 and C5 variants)—to the scenario-based setting and evaluate them on benchmark instances under two confidence levels ($\alpha=0.8,0.9$). Experimental results show that heuristic methods, especially S5$_{ws}$ and C5$_{ws}$, typically outperform the basic evolutionary approach, with performance nuances depending on instance size and scenario set, highlighting the value of scenario-based robustness in stochastic multi-component optimization. Overall, the study demonstrates the viability of a weighted-scenario relaxation for handling chance constraints in complex, interdependent routing and packing problems, offering practical guidance for robust decision-making under uncertainty.

Abstract

The chance constrained travelling thief problem (chance constrained TTP) has been introduced as a stochastic variation of the classical travelling thief problem (TTP) in an attempt to embody the effect of uncertainty in the problem definition. In this work, we characterise the chance constrained TTP using a limited number of weighted scenarios. Each scenario represents a similar TTP instance, differing slightly in the weight profile of the items and associated with a certain probability of occurrence. Collectively, the weighted scenarios represent a relaxed form of a stochastic TTP instance where the objective is to maximise the expected benefit while satisfying the knapsack constraint with a larger probability. We incorporate a set of evolutionary algorithms and heuristic procedures developed for the classical TTP, and formulate adaptations that apply to the weighted scenario-based representation of the problem. The analysis focuses on the performance of the algorithms on different settings and examines the impact of uncertainty on the quality of the solutions.