Integrated transportation system planning with gravitational search algorithm approach based on fuzzy mutant controller

<p style="text-align: left;"><strong>Abstract:</strong> <p style="text-align: left;">Gravitational search algorithm is a newly emerging meta-heuristic algorithm. In this algorithm, the gravitational force between answers can be calculated in two ways. In the first method, a response is selected from the local neighborhood space of the current response and the gravitational force between these two responses is calculated. In the second method, the gravitational force is calculated among all the neighboring responses in the neighborhood space of the current response and is not limited to a neighboring response. This algorithm has premature convergence in some optimization problems and gets stuck in the local optimum and does not make progress to find the optimal solution, which is one of the disadvantages of this algorithm. In the article, this problem was tested in the first step by comparing the two mentioned methods, and it was found that the first method has relative superiority in terms of the speed of reaching the response, in particular, the optimal response, and then by defining a heuristic mutation function, it uses a Fuzzy controller to control the jump rate. The proposed method has been evaluated on standard benchmark functions, including single-mode and multi-mode functions, and the results of evaluating these two methods have been compared with the standard gravitational search algorithm (GSA) and the gravitational particle population algorithm (GPS), the particle population optimization algorithm (PSO) and real genetic algorithm (RGA). The observed tests show that the presented method has better results than other compared algorithms. <p style="text-align: left;"><strong>Key Words:</strong> gravitational force algorithm, integrated combined transport, routing, multimodal transport, vehicles, VRP, MTP <p style="text-align: left;">&nbsp; <p style="text-align: left;">&nbsp; <p style="text-align: left;"><strong>1.Introduction</strong> <p style="text-align: left;">The vehicle routing problem has long been a subject of extensive research. However, a review of the literature reveals that simultaneous constraints related to warehouse capacity, vehicle capacity, and route length have not been considered together. Considering these three constraints simultaneously brings the problem closer to reality. <p style="text-align: left;">In many logistics environments, the location routing problem helps managers to make decisions such as the location of facilities (distribution centers or warehouses), the allocation of customers to these facilities, and then the transportation plans to connect customers and facilities. In fact, the problems of location routing are defined in order to find the right place and number of facilities as well as distribution routes by means of vehicles. The main difference between routing positioning problem and traditional positioning problem is that in the first one, after determining the location of facilities, communication routes between customers and facilities are examined as a tour, but in the second one, it is assumed that there are direct routes between customers and facilities, which ultimately leads to an increase in distribution costs. So, unlike the traditional positioning method, in the routing positioning method, considering the tour, it simultaneously looks for the optimal places of facilities and also the optimal routes. <ol style="text-align: left;" start="2"> <li><strong>Literature Review</strong></li> </ol> <p style="text-align: left;">By reviewing the literature in this field, it was seen that until now, the limitations related to warehouse capacity, vehicle capacity, and route length have not been considered simultaneously, while considering these three limitations together brings the problem closer to reality. Rabbani, Sepehri and Zagerdi, in a research entitled &ldquo;the problem of vehicle routing connected to multimodal transportation: an integrated approach&rdquo;, designed and modeled a network consisting of vehicle routes and multimodal transportation routes for the first time. In another study, entitled "Analysis of the Role of Financial Factors in the Whiplash Effect in a Two-tier Supply Chain&rdquo;, Movahedi, Zulfaqari and Jolai, paid attention to the financial factors affecting the escalation of the whiplash effect and quantified this effect. Moreover, by entering the financial factors in the obtained model, they examined the result carefully examined in two ways: first, investigating the time value of money in estimating the financial value of the demand met during p period; second, investigating changes in the monetary value of orders due to the waiting time for order delivery (L). Dera Mirki also investigated the effectiveness of this proposed algorithm by solving 11 VRP problems in a research titled "New Innovative Algorithm for Solving Vehicle Routing Problems". <ol style="text-align: left;" start="3"> <li><strong>Methodology</strong></li> </ol> <p style="text-align: left;">In this research, the goal of developing the gravitational force algorithm is to solve the problems of integrated and multi-purpose transportation systems, and in the real world, the effect of gravity on objects in a mutual way and the application of gravitational force to all objects in space is the main idea of ​​this algorithm. Obviously, this force changes the speed of objects and pulls them towards larger objects, which in nature can be referred to the relationship between the earth, the moon, terrestrial objects and other celestial bodies. Gravitational force algorithm or GSA local search is a new algorithm which, as its name suggests, is a heuristic local search method. In addition, this algorithm belongs to the group of those methods that imitate some natural or physical processes (such as simulated cooling and genetic algorithm). <p style="text-align: left;">In the gravitational force algorithm, the search space is imagined as a galaxy, and the solutions in the search space are like the objects inside the galaxy. Each of these bodies creates a mutual force, which is obtained using Newton's law of gravity with a slight change for F=(m-n)/r^2 problems. Based on the formula, this force is equal to the difference between the value of the evaluation function of the candidate response and the current response divided by the radius of the neighborhood to the square of two. Since the high speed of the gravitational force algorithm is due to the local search, it is very important to create and select the neighborhood as well as to choose the correct conditions for the solutions to be neighbors in this algorithm. In addition to the high speed in reaching the optimal response, the algorithm avoids reaching and staying in a local optimum due to the inherent properties of gravity. <ol style="text-align: left;" start="4"> <li><strong>Result</strong></li> </ol> <p style="text-align: left;">In this section, the results of the implementation of true genetic algorithm (RGA), particle population optimization algorithm (PSO), gravity particle population algorithm (GPS), standard gravity search algorithm(SGSA) and mutated gravity search algorithm(MUGSA) on 23 functions of the standard benchmark are presented in Tables 2,3 and 4. <p style="text-align: left;">Real Genetic Algorithm (RGA) <p style="text-align: left;">The true genetic algorithm is randomly generated with an initial population of 50 members in the search space. The dimensions of the problem are the number of genes that are continuously evaluated in the problem space. In each generation, based on the fitness of the chromosomes and using the roulette wheel, parents are selected and then randomly paired with each other. Children are generated by hemming and mutation operators. The rate of hembray operator is 0.3 and the mutation probability is 0.1. <p style="text-align: left;">The continuous symmetry operator is performed on each pair of X i and X j in the form of equation 1. In this relation, a is a random value with a uniform distribution in the interval [0,1]. <p style="text-align: left;">Xi, new = Xi &minus;a (Xi &minus; X j), X j,new= X j &minus;a (Xi &minus; X j )&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;(1) <p style="text-align: left;">&nbsp; <p style="text-align: left;">In the mutation operator on each child Xi, a gene X id is randomly selected and its value is replaced with a new random value according to relations 2 and 3 in the range of the search space. <p style="text-align: left;">Sigma = Scale&times; (1&minus; Tt), Scale = X hi &minus;5 X lo&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;(2) <p style="text-align: left;">a Id, new = X id + rand &times;Sigma&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (3) <p style="text-align: left;">&nbsp; <p style="text-align: left;">In the relations above, Xhi and Xlo are the upper limit and lower limit of the search space, respectively. T is the total number of repetitions and rand is a random number with a normal distribution and zero mean. <p style="text-align: left;"><strong>&nbsp;</strong> <p style="text-align: left;"><strong>Particle Population Optimization Algorithm (PSO)</strong> <p style="text-align: left;">The particle population optimizer algorithm is randomly generated with an initial population of 50 birds in the search space, and Vid, X id is calculated according to relations 4 and 5. <p style="text-align: left;">X id (t+1)=X id (