## 2 Basic Theory
### Multi Objective Decision Making on Product Option Selection Based on NSGA-II  
Different kinds of decision-making requirements are needed to meet in multi-objective optimization problems, and the objective functions converging in different directions should be balanced through the selection mechanism of an optimization scheme. Parts of multi-objective optimization algorithms use weight indexes to decompose the losses of each objective function, and its operation speed and convergence accuracy are limited [32; 33; 34]. In fact, its essence is still a single objective optimization problem. If the dimension of the two objective functions cannot be unified, one scheme will even swallow up another one; however, the Improved Non-dominated Sorting Genetic Algorithm attaches importance to the convergence of each objective function. It can search the optimal scheme globally through the basic operation of the genetic algorithm and the advantages of biological evolution law, with a strong optimization ability and high operation accuracy. The convergence of the objective function is expressed in multi-dimensions, and the elite individuals are searched in the global space. At the same time, the introduction of an elite retention mechanism can also speed up the convergence speed.  
The method provides an objective means of assessing the product option selection for aircraft systems in the stages of aircraft design or in motion, leading to the minimization of subsequent maintenance actions and thus reducing the operating cost. The steps of multi-objective cyclic optimization (Figure 2) based on the algorithm to generate a decision are as follows:  
STEP 1: The initial population with a scale of 'n' is generated randomly, and the population is classified by non-dominated sorting. The non-dominated meaning is that any decision variable \(\rm X_{a}\) corresponding to the objective \(\rm f_{i}(x)\) in the population satisfies Equation (7), and the decision variable is called a non-dominated solution. This step finds all non-dominated solution sets \(\rm f_{i}\) from the solution set f and marks the non-dominated sorting in number one. Then, it finds all non-dominated solutions in f-\(\rm f_{i}\), and its non-dominated sorting number is two. This repeats until f-\(\rm F_{I}\) is an empty set, and the non-dominated sorting is completed.  
\[\begin{array}{ll}\forall\rm i=1,2,\ldots n&\rm f_{i}\ \ (X_{a})\leq f_{i}(X_{b})\\ \exists\rm i=1,2,\ldots n&\rm f_{i}\ \ (X_{a})<f_{i}(X_{b})\end{array} \tag{7}\]  
STEP 2: The basic operations, such as selection, crossover and mutation, are used to generate a new population. The purpose of selection is to select crossover individuals. The choice is based on the crowding degree. The calculation of the crowding degree is expressed in Equation (8). The higher the crowding degree, the higher the density in the space around the point, the more obvious is the advantage. Cross operations simulate binary crossover, crossover points are randomly selected from two parents, and the offspring are gradually convergent with evolution. The offspring produced after crossing enter variation, and the offspring produced by the mutation are compared with the individuals with higher crowding degrees, preferentially entering the new population.  
\[\rm n_{d}=n_{d}+(f_{m}(i+1)-f_{m}(i-1))/\left(f_{m}^{max}-f_{m}^{min}\right) \tag{8}\]  
STEP 3: In order to keep the excellent individuals in the cycle or iteration as much as possible, the algorithm adopts the elite retention strategy to expand the population size for basic operations to '2N'. Through the non-dominated sorting and crowding degree of individuals, the individual's number of 'N' are selected to enter the sub-population. This can reduce the amount of calculation and accelerate the convergence speed of the algorithm.  
STEP 4: Repeat steps two and three until the end condition of the algorithm is met, that is, the number of iterations exceeds the initial setting. The best population and the last generation are output.