# The Pheromone-Based Harmony Search Algorithm for the Asymmetric Traveling Salesman Problem
## 1 Introduction  
Due to the considerable complexity of many problems occurring in business practice, the use of classical deterministic algorithms that enable determination of exact solutions becomes impossible in an acceptable time. For this reason, approximate methods are increasingly used, among which one can distinguish metaheuristics. These algorithms have found application in the optimization of logistic processes, e.g., Nondominated Sorting Genetic Algorithm II and Multiobjective Particle Swarm Optimization were used for sustainable ship routing and scheduling with draft restrictions [1], Non-Dominated Sorting Genetic Algorithm II was applied for the design of the Shuttle-Based Storage and Retrieval System [2] and Particle Swarm Optimization-Composite Particle was used for sustainable integrated dynamic ship routing and scheduling optimization [3], as well as sustainable maritime inventory routing problem with time window constraints [4]. The Harmony Search method (HS) is an interesting metaheuristic, which despite causing a great many controversies (it is falsely [5] accused of a lack of innovativeness and similarity to the Evolution Strategies (ES) [6]), found application in the process of solving many utilitarian problems (it was applied, among others, in the process of flood control management [7], in the fourth-party logistics routing problems [8] and in the optimization of allocation of containers around a harbor [9]). The character of the technique, implying its preferential usage to the continuous space, imposes the application of sophisticated approaches aimed at adjusting the algorithm to combinatorial optimization problems, at the same time influencing the appearance of difficulties connected with an effective designing of a method of the HS adjusted to solving a significant number of practical problems. One of the most tested concepts of combinatorial optimization is the Traveling Salesman Problem (TSP). It is characterized by unquestionable utilitarian meaning--its asymmetric version (ATSP) models the linear infrastructure located in urbanized areas, thus becoming groundwork for the planning of routes of vehicles collecting municipal waste [10] and waste electrical and electronic equipment [11]. The use of ATSP goes beyond the area of reverse logistics (that can be defined as 'theprocess of planning, implementing, and controlling the efficient, effective inbound flow, inspection and disposition of returned products and related information for the purpose of recovering value' [12]) and can be used to solve various problems occurring in distribution logistics, e.g., the distribution process of perishable products [13], post delivery service, fast food delivery or school bus operation [14]. Utilitarian importance of ATSP and HS features influenced the onset of research on the adjustment of the HS to the ATSP. Paper [15] presents suggestions of modification of the HS, owing to which the method enables defining relatively good solutions for the issue in question. In the quoted publication, however, a noticeable decline of effectiveness of the technique for instances of the problem described by the means of more than 71 vertices was emphasized. In the process of solving the TSP substantial effectiveness is ascribed to techniques that base on analogy to the process of searching for food by ants, using an information exchange pheromone [16; 17]. Due to the lack of long-term memory in the suggested approach to designing the HS, the key element of Ant Systems (AS) was adjusted to the needs of the tested algorithm. The need to use long-term memory can be argued with the results presented by [18], according to which the elimination of the memory reset mechanism made it possible to improve the results for larger ATSP instances (worsening them at the same time for smaller ones, pointing to the need to support HS memory).