# Study on Multi-Objective Optimization-Based Climate Responsive Design of Residential Building
## 2 Background and Literature
### Algorithm Used in Optimization  
Currently in multi-objective optimization research, representative algorithms include MOGA (Multi-Objective Genetic Algorithm), NSGA (Non dominated Sorting Genetic Algorithm), NSGA-II (Non dominated Sorting Genetic Algorithm-II), PESA (Pareto Envelope-Based Selection Algorithm), and SPEA-II (Strength Pareto Evolutionary Algorithm-II). The performance of multi-objective algorithms mainly depends on three aspects, namely convergence, the distribution of solutions set and robustness. NSGA-II, SPEA-II, and PESA all have good convergence and stability, but compared with SPEA-II and PESA, NSGA-II has worse convergence. Because of the truncation characteristics in NSGA-II algorithm, its distribution performance is obviously not as good as SPEA-II. Furthermore, when the number of targets is more than one, PESA will have poor distribution. It can be seen that the SPEA-II algorithm is superior to other algorithms in terms of convergence and solution set distribution. Therefore, the SEPA-II algorithm is used in this study. SEPA2 is an improved version of SPEA (Strength Pareto Evolutionary Algorithm) proposed by [PERSON] and [PERSON] in 2001 [22]. It is a Pareto algorithm for solving multi-objective problems. In this algorithm, the fitness of an individual is also called Pareto strength. The fitness of individuals in a non-dominated set is defined as the proportion of the total number of individuals dominating in the group. The fitness of other individuals is defined as the total number of individuals dominating it plus one, and individuals with low fitness are corresponding to a higher probability of selection. In addition to the evolutionary population, an external population is also set up to save the current non-dominated individuals. When the number of individuals in the external population exceeds the predefined value, clustering techniques are used to delete them. Tournament is used to select individuals from evolutionary groups and external populations to enter the mating pool for crossover and mutation operations. The SPEA-II algorithm flow chart is shown in Figures 8 and 9. Suppose the size of the group P is N, the archive set Q is M, and the number of iterations is T, then the work flow of the SPEA-II algorithm [23] is:  
1. Randomly generate an archive set and initial population Q0, P0, and iterator t = 0. 2. Calculate fitness for individuals in PT and QT. 3. Keep all non-dominated individuals in PT and QT into the next generation archive set QT+1. If \(|\)QT+1\(|\) > M, then remove excess individuals. If \(|\)QT+1\(|\) < M, then select some individuals in PT and QT, to join QT+1 to make \(|\)QT+1\(|\) = M.
4. If t = T or other termination conditions are met, the non-dominated solution in QT+1 is output as the algorithm result. 5. If not satisfied, perform tournament selection, crossover and mutation on QT+1, keep the result in PT+1, t = t + 1, and back to (2). Figure 7: Parametric building optimization process. The logic of fitness allocation and environment selection is as follows:  
(1) Fitness allocation  
In order to make each individual have a different fitness value, while considering the situation of all individuals in the iterative group and the external group. The crowding situation is determined by calculating the distance between an individual and its neighbors, that is, calculating the fitness of each individual in the population \(\mathrm{P_{T}}\) (initial set) and \(\mathrm{Q_{T}}\) (archive set). The total fitness F(i) is determined by R(i) and D(i) (as in Equation (5)). \[\mathrm{F(i)=R(i)+D(i)} \tag{5}\]  
where R(i) is the integer part, and R(i) is calculated as Equation (6). \[\mathrm{R(i)=\sum_{j\in P_{i}\cup\mathrm{Q_{j}},j-i}S(j)} \tag{6}\]  
Figure 8: SPEA-II work flow. Figure 9: Diagram of SPEA-II algorithm work flow. where S(j) is the number of individuals dominated by j in the population \(\mathrm{P_{t}}\) and \(\mathrm{Q_{T}}\). The lower the R(i), the better the quality of the solution.