Supplemental Material for “ New Insights Into Diversification of Hyper - Heuristics ”

In this document, we present the supplemental materials for the paper “New Insights Into Diversification of HyperHeuristics”. More specifically, the document consists of 4 sections. In Section I, we present the complete results of the sensitivity analysis, for the Low Level Heuristic (LLH) parameters. In Section II, the numerical results for both HIP-HOP (recursive acronym of “HIP-HOP is an Instance Perturbation based Hyper-heuristic Optimization Procedure”) and SOPHY (SOlution Perturbation based HYper-heuristics). Then in Section III, we give the numerical results of SOPHY-LONG (SOPHY with longer cut off time). Finally in Section IV, we compare HIP-HOP with an ant based hyper-heuristic AHSAR (Ant based Hyper-heuristic with SpAce Reduction), to evaluate the performance of HIP-HOP from more aspects. I. SENSITIVITY ANALYSIS OF ALL THE LLH PARAMETERS In this section, we present the complete results of the parameter sensitivity analysis. The experiment is conducted as follows. First, the LLH parameters are assigned with the output of irace [1]. Then, for each LLH parameter, we enumerate several values in its feasible range, with step size 0.1, meanwhile keep the other LLH parameters assigned with respect to irace. For each set of modified LLH parameter configuration, HIP-HOP is executed for 10 independent trials over the representative instances. Through this experiment scheme, we are able to evaluate the quality, as well as the robustness of the offline tuning process. For example, in Figs. 1–2, we present the influences of the LLH parameter configurations over the framework. In the figures, each line indicates the algorithm’s behavior (captured by the %err) when changing the value of each LLH parameter, over typical Ising Spin Glass instances (19×19-3 and 20×20-8). Note that the experimental results for both IPLLHs and SPLLHs are presented, with different types of legends. Similarly, the same experiment is conducted over typical p-Median instances (fl1400 with p = 500 and fl1577 with p = 500). From Figs. 1–2, the following observations could be drawn. First, HIP-HOP is not very sensitive to the LLH parameter configurations. For example, over the Ising Spin Glass instances, the average percentage error always range within [0.5%, 1.52%]. Meanwhile, over the p-Median instances, the average percentage error lies within [0.1%, 0.8%]. One possible reason might be that, in hyper-heuristics, there are multiple LLHs that are able to provide the diversification mechanism. In this sensitivity analysis, the LLH parameters are examined in a sequential paradigm, thus changing one LLH parameter configuration may not have a great impact on the whole framework. Second, for each LLH parameter, the %err is relatively small within the 0 0.4 0.8 1.2 1.6 2 2.4 2.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 % e rr parameter configuration i-mutation-portion