## 2 Related Studies  
There are numerous algorithms reported in the literature for solving the COPs of knapsack, quadratic assignment, and maximum cut. Previous studies have agglutinated different algorithms within the ILS for performance improvement. The authors in [45] employed meta-learning to improve the performance of ILS on the Google machine reassignment problem (GMRP). The idea was to learn from some instances of a problem and recommend suitable ILS components based on the instance being solved. [PERSON] et al. [46] combined a genetic algorithm (GA) with ILS to solve the instances of a dominating tree problem (DTP). In their approach, multiple solutions were kept and each solution in the population of solutions was subjected to possible improvement through the ILS procedure. The GA presented in the study utilized a specialized mutation procedure with high diversification strength and ignored crossover operators. The specialized mutation operator was applied after the ILS phase to each solution in the population during the second stage of the search process. The second stage helps to improve diversity in the solutions returned by the ILS in the first stage according to the study. The study reported in [47] combined the ILS with other techniques to effectively solve numerical optimization problems. The perturbation strategy was based on the success-history-based parameter adaptation for differential evolution (SHADE) while the local search phase was based on a mathematical model. The ILS keeps multiple solutions while it relies primarily on the perturbation mechanism of the SHADE to control the strength of the ILS perturbation. The partition crossover operator was incorporated into an ILS framework for the computational design of proteins [39]. The crossover operator was utilized as an additional perturbation operator to combine two solutions generated by the ILS framework to produce a new solution that is further enhanced by the steepest descent algorithm. The algorithm contested favorably with a classical ILS and the Rosetta fixbb method. The APW employed in the present paper is not a new acceptance mechanism because it has been used in the literature for ILS [21; 48; 49]. However, a strategy is proposed in this study to oscillate its important parameter called temperature during the search process. In [50], ant colony optimization (ACO) was employed to generate rules for the selection of heuristics for the knapsack problem. Candidate hyper-heuristics were constructed by the virtual ants over time using the current problem state until all items had been packed. The best hyper-heuristic constructed during the simulation can now be applied to unseen instances. The studies in [51; 52] designed a feature-independent hyper-heuristic through an evolutionary algorithm to solve the 0/1 knapsack problem. Similar to the approach in the previous study [50], a set of training instances were used to construct viable hyper-heuristics that were superior to the individual LLHs of the problem investigated. [PERSON] et al. [15; 53] incorporated fuzzy logic into the inference process of the selection of heuristics for the knapsack problem. The authors considered seven features of the problem as inputs to the fuzzy inference engine and four LLHs as outputs. The fuzzy-based hyper-heuristic was optimized by a genetic algorithm that was benchmarked against traditional hyper-heuristics optimized by a particle swarm optimization (PSO) algorithm. The other optimization methods not based on hyper-heuristics that have been applied to solve the knapsack problem include the binary monarch optimization algorithm [54], binary bat algorithm [55], whale optimization [56], list-based simulated annealing [57], and artificial bee colony optimization [58]. A recent study [59] compared the genetic algorithm (GA), simulated annealing (SA), dynamic programming (DP), branch-and-bound (BB), greedy search (GS), and a hybrid of GA-SA to solve the knapsack problem. [PERSON] et al.