In could not lead to the improvement of all

In order to survey the optimal performance of flat tubes equipped with porous layer, the GMDH models obtained in Section 4 are now employed in a multi-objective optimization method using NSGA II algorithms. This algorithm was presented by Deb et al. 42 as an improvement to NSGA. For maximizing the amount of heat transfer, it is essential to choose proper values for design parameters. However, variation of the values of these parameters for the purpose of improving the heat transfer normally increases the fluid pressure drop, which is not desirable. Therefore, a point at which heat transfer is  at most and pressure drop is minimum cannot be found.  This two conflicting objectives h and  should be optimized simultaneously with respect to the design variables: H, Hp, , Q and .

To solve this problem, instead of finding one special state as the optimal state, a set of optimal states are obtained. These groups of points that are known as the Pareto optimal points or Pareto front are obtained via multi objective optimization. The concept of Pareto optimal points in multi-objective optimization problems is based on a set of solutions that are non-dominated to each other but are superior to the rest of solutions. This means that there is not a single solution to be superior to all other solutions due to the all objectives. Therefore, variation of the vector of design variables in such a Pareto front including these non-dominated solutions could not lead to the improvement of all objectives simultaneously. Consequently, such a change will lead to decline of at least one objective. Thus, each solution of the Pareto set includes at least one objective inferior to that of another solution in that Pareto set, although both are superior to others in the rest of search space. In all runs a population size of 60 has been chosen with crossover probability (Pc) and mutation probability (Pm) as 0.7 and 0.07 respectively.

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