A regression analysis of the impact of routing and packing dependencies on the expected runtime

Abstract

Problems with multiple interdependent components offer a better representation of the real-world situations where globally optimal solutions are preferred over optimal solutions for the individual components. One such model is the Travelling Thief Problem (TTP); while it may offer a better benchmarking alternative to the standard models, only one form of inter-component dependency is investigated. The goal of this paper is to study the impact of different models of dependency on the fitness landscape using performance prediction models (regression analysis). To conduct the analysis, we consider a generalised model of the TTP, where the dependencies between the two components of the problem are tunable through problem features. We use regression trees to predict the instance difficulty using an efficient memetic algorithm that is agnostic to the domain knowledge to avoid any bias. We report all the decision trees resulting from the regression model, which is the core in understanding the relationship between the dependencies (represented by the features) and problem difficulty (represented by the runtime). The regression model was able to predict the expected runtime of the algorithm based on the problem features. Furthermore, the results show that the contribution of the item value drop dependency is significantly higher than the velocity change dependency.

Additional results

In this appendix, we show the results for additional experiments on different sets of instances for $n=6$ and $n=7$. The same process defined in the earlier sections was used to create them, but just different seeds of the random number general were used. The goal is to show that different sets of instances generate roughly the same regression model, i.e. regression trees with a similar logic as shown in Fig. 10.

When comparing these two with the corresponding trees in Figs. 7 and 8, we can see that the conditions at the inner nodes are almost always identical (i.e. for 10 of 13 inner nodes) or very similar, and the respective errors (shown in Table 5) and sample numbers are very close matches, too.

Fig. 10

Regression tree $\mathrm{{RT}}_{3,380}^{6*}$ for a new dataset with $n=6$ (top) and $\mathrm{{RT}}_{3,380}^{7*}$ with $n=7$ (bottom)

Hence, we conclude that even though the methodology is based on randomly created instances and even though it employs a memetic algorithm as a randomised search heuristic, the achievable insights at a high level (i.e. when reasoning about the effects of dependencies) are unaffected.

Table 5 Evaluation metrics of the regression trees