## 5 Algorithm
### Movement Operators
#### 5.2.1 Search strategy  
VLE begins the search by randomly creating a single starting solution. To do this, VLE considers, independently, the initial domain specified for each decision variable. Once this solution is created, VLE begins to explore its environment in a parallel search space, making a predefined number of changes in the value of each decision variable, keeping the other ones constant. This parameter is called alpha (\(\alpha\)) and it can be any odd number greater or equal to 3. If this number is equal to 5, the algorithm will apply, for each decision variable, \((5-1)/2=2\) times the bubble point operator and then \((5-1)/2=2\) times the dew point operator. In other words, VLE starts creating new saturated binary mixes for each decision variable. For each created mixture, VLE evaluates its aptitude by using the equivalent values of each variable in its real domain. Next, for each decision variable, VLE looks for the most suitable mixture, that is, the one with the best aptitude, and thus the new value of the decision variable is determined. With the new values of the decision variables, the aptitude of the new solution is evaluated and compared with the best aptitude obtained from the last movement. If a better result is obtained, the procedure iterates until the change between the current aptitude and the best aptitude is less than an established tolerance or a stop criterion is reached. This criterion is usually the maximum number of movements or restarts. If the last solution found is the best one so far and it is no longer possible to continue exploring its environment, given the correspondence established between the real and parallel domains of the variables when the search starts or restarts, then VLE narrows the relationship between these two domains around the solution found. Once the relationship is narrowed, the exploitation phase begins. Nevertheless, if the result is worse, VLE either proceeds to accept the solution found or reject it to restart the search by creating a new initial solution elsewhere. The decision of acceptance or rejection depends on the probability of acceptance calculated for the solution found. If its probability of acceptance is grater than the predefined probability _beta_, then VLE accepts the solution, otherwise it is rejects it. The relationship between the real domain and the parallel domain is given by (12) or (14). Figures 7-12, show the temperature diagrams versus molar fraction, in terms of ordered pairs, versus the respective values for each decision variable. The narrowing of this relationship is autonomous and depends exclusively on the equilibrium relationship between the chemical species that make up the binary mixture. If initially the relationship between these domains is from \([-100,+100]\) to \([0,1]\), and if the values of the decision variable closest to their current value are, for example, \(+4\) and \(+12.2\), assuming a value for the decision variable equal to \(+7\), the narrowing for this variable will be from \([+4,+12.2]\) to \([0,1]\). In other words, VLE amplifies the environment closest to the solution, allowing the exploitation stage. Summarizing, the exploration is performed covering a wide area that contains a certain number of binary chemical mixtures, all of them possible solutions of the optimization problem, and the exploitation is performed covering a reduced or local area that contains the same number of binary