Currently the collection of waste in Bogotá, is carried out by five different operators divided into localities, Proambiental is responsible for the localities of Usaquén, Chapinero, Santa Fe, San Cristóbal, Usme and Sumapaz; Lime-Limpieza Metropolitana is in charge of the localities of Teusaquillo, Los Mártires, Puente Aranda, Antonio Nariño, Rafael Uribe Uribe, Tunjuelito, Ciudad Bolívar and Bosa; was awarded to Ciudad Limpia the localities of Kennedy and Fontibón; the operator Bogotá Limpia works in the localities of Engativá and Barrios Unidos; Area Limpia D.C operates in the locality of Suba . This division generates that the collection of waste is more efficient in certain areas of the city. It should be noted that the waste collection system does not takes into account the type of waste, however, a modification to the resolution 668 of 2016, states that the waste must be separated using bags of different color .
The assignment of tasks is a key issue of the mechanism of cooperation between agents in multi-agent systems. Systems consisting of collaborating mobile agents have been extensively studied and used in a broad spectrum of applications such as environmental sampling, surveillance, coverage, persistent monitoring, homework assignment, and data collection and information gathering. The general topic includes a network of cooperating mobile agents who need to frequently visit points of interest and transfer data, products or people to a base . There are several applications to solve dynamic tasks, such as linear programming, dynamic programming, game theory, the genetic algorithm or the algorithm of auction, where the goal is not only time optimization, also other restrictions are considered, such as fuel consumption and the problem of assigning weapons targets .
To solve the problems of tasks distribution in multi-agent systems, different optimization methods such as Ant Colony , Insect swarm intelligence , insects social , negotiation protocols , and among other techniques. All focus on optimizing decision-making algorithms in each of the agents that make up the network. Optimization in displacements is a feature that is sought in various fields, this is why the research about trajectory planning has increased considerably and takes into account not only the time of displacement but also external factors, such as weather conditions . Open programs like Google Maps™ or applications such as Waze have allowed users to find a quick solution and simple to move from one point to another within a city.
Path planning techniques have been linked to multi-agent systems to perform task assignments cooperatives, where various points are available through which the system must pass, and for each agent must be planned a trajectory to be the most optimal in distance, the optimization of tasks is implemented through swarms of particles . The auction algorithm refers to the routes of the D* lite based agents. D* lite is a type of modification of the A* algorithm and it can find the shortest route in a dynamic situation . These methods are being used in smart cities to improve the quality of life of the inhabitants and reduce the time in various tasks . The formulation of a single objective is extended to reflect the nature of multi-objective problems when there is no objective function to optimize, but many. Therefore, there is no single solution, but a set of solutions. This set of solutions is found through the use of Pareto Optimality Theory . A common simplified assumption is that an agent's preferences are captured by a utility function. This function provides a map of the states of the world or the result of the utility function from the game to a real number . A common problem in multi-agent systems is deciding how to reallocate a set of tasks between a set of agents. This is known as the task assignment problem.
The egalitarian social welfare solution is especially useful in scenarios where there is no agreement that provides all agents the same utility, since it is guaranteed that each problem has an egalitarian social welfare solution. However, the solution itself may in some cases seem very little egalitarian. The egalitarian solution is the agreement that maximizes the sum of the utilities that are shown in Eq. (1).
The utilitarian agreement is, by definition, an optimal Pareto agreement. There may be more than one utilitarian deal in in case of tie. The utilitarian agreement violates the independence of the assumption of cost equity, the utilitarian and egalitarian solutions are reasonable solutions. Nash proposed a solution that does not violate this assumption. The Nash negotiation solution is the treatment that maximizes the product of costs , that is shown in Eq. (2).
The Nash solution is also efficient on the Pareto front, independent of the units of the utility, symmetric and independent of irrelevant alternatives. In fact, it is the only solution that satisfies these four requirements. This means that if a solution that satisfies these four requirements is wanted, then the only option is the Nash negotiation solution .
NSGA-II is a rapid elitist non-mastered genetic classification algorithm for multi-objective optimization which has been proposed by Deb et al. in 2001 . Progress towards the Pareto set is here due to the Pareto classification that divides the population into non-dominated subsets. First: all individuals not dominated from population are labeled as rank 1; then they are temporarily removed from the population and the process is repeated: not dominated individuals from the rest of the population are given rank 2, and so successively, until the entire population is classified. The diversity conservation technique is based on the agglomeration distance, one of the possible estimates of the density of the solutions belonging to the same subset not dominated.
The selection of NSGA-II is based on tournaments, that is, to choose an individual for reproduction; the T individuals are randomly chosen from the population and compared to each other using the operator of comparison defined above. The winner then becomes a genitor. NSGA-II replacement is deterministic: it consists of merging all parents and children and choosing the best individuals of N in that global population using the same comparison operator .