# Adjustable Pheromone Reinforcement Strategies for Problems with Efficient Heuristic Information
## 1 Introduction  
Ant colony optimization (ACO) is a nature-inspired metaheuristic that has been used as a design template for many successful algorithms. These algorithms are mostly used to solve NP-hard combinatorial optimization problems, although ACO has also been used for continuous and mixed-variable optimization problems [1; 2; 3; 4; 5; 6]. ACO algorithms are stochastic and cannot guarantee optimal solutions, but they are commonly applied to NP-hard optimization problems because, for such problems, exact algorithms generally cannot yield optimal solutions in a reasonable time. It is sometimes also possible to combine ACO techniques like pheromone trails with Monte Carlo methods to maintain some advantages from both procedures [7]. The key concept in ACO is the use of pheromone trails associated with solution components to allow the colony of artificial ants to accumulate collective knowledge and experience about the problem instance that they are trying to solve. These pheromone trailsbias future attempts of artificial ants to construct better solutions. Being metaheuristic, ACO only gives general recipes on how to devise an algorithm. For solving a specific type of problem, it is necessary to devise a specific ACO algorithm, and success depends not only on the nature of the problem but also greatly on how these recipes are applied. It is important to enable good guidance in the search space by properly balancing between exploration and exploitation. A too greedy algorithm might result in fast convergence toward moderate or bad solutions, but too little greediness can result in slow convergence or unguided roaming through search space with a very low probability of obtaining good solutions. For some NP-hard problems, ACO can get substantial help from heuristic information, but for other NP-hard problems such useful heuristic information is unknown or maybe impossible. Heuristic information is very useful for problems where some kind of optimal route is objective, such as the traveling salesman problem (TSP) [8], asymmetric traveling salesman problem (ATSP) [8], sequential ordering problem (SOP) [9; 10], various vehicle routing problems (VRPs) [11; 12], car renter salesman problem (CaRS) [13], etc. In these cases, the searching process can be faster because of heuristic information, and thus, the parameters and strategies used in ACO should be adjusted accordingly. The standard methods for reinforcing pheromone trails in modern ACO variants have diametrically opposite strategies, leaving a large gap between too much and too little greediness. In this paper, we conduct experimental research on generalized reinforcement strategies to gain better control over algorithmic behavior and, thus, achieve better performance in ant colony optimization algorithms. The research questions that we tried to answer were the following. Can adjustable pheromone reinforcement strategies improve the algorithmic performance of ACO algorithms? How can \(\kappa\)-best and max-\(\kappa\)-best be generalized or extended to provide less greedy algorithmic behavior than with \(\kappa=1\), in the case when this is desirable? How do different adjustable pheromone strategies influence the behavior of ACO algorithms (with and without local optimization) for combinatorial problems that have efficient heuristic information? For experiments performed to answer these research questions, we have chosen well-known instances of TSP and ATSP. Initial ideas for such strategies were presented at the conference [14], and here, we extend our proposal and carry out a comprehensive experimental evaluation. The results show that, by using adjustable reinforcement strategies, an ACO algorithm can obtain better solutions and allow greater control over algorithmic behavior.