## 2 Mathematical Model
### Multi-Objective Optimization and Knee Point  
Due to the increase in carbon emissions, it has become essential to create low-carbon emission designs in building insulation works, as in every field. As with every design, cost is an important element of building insulation. Considering the above information, a multi-objective optimization is carried out in this study, aiming to both reduce carbon emissions and reduce the total cost. In multi-objective optimizations, instead of a single optimum, values that optimize both objectives are obtained. At the same time, when one objective is optimized in the Pareto curve, it can be seen at what levels the other objective can be best. In this way, the optimum range in which the system can operate can be seen with a single optimization. While the objectives of the multi-objective optimization study carried out in this study are to reduce carbon emissions and total cost, the design variables are four: insulation thickness, insulation material, wall material, and heating type. In these design variables, insulation thickness is defined as a continuous design variable, while other design variables are defined as discrete design variables. Therefore, the multi-objective optimization is given as follows:  
\[\left\{ \begin{aligned} &\text{Min}\big{(}C_{T},\;M_{CO_{2}} \big{)}\\ &\text{s.t.}\;\left\{\begin{aligned} & 0<Insulation\; Thickness\;(m)<0.5\\ & Insulation\;Material\;\in(EPS,\;XPS,\;MW,\;PUR)\\ & Wall\;Material\in(\text{Light\;Concrete,\;Reinforced\;Concrete,\;Brick} )\\ & Heating\;Source\in(Natural\;Gas,\;Electricity,\;Fuel\; Oil)\\ \end{aligned}\right.\end{aligned} \right. \tag{21}\]  
In this study, four different multi-objective optimization methods are examined to determine the optimum points and compare different optimization methods. Multi-objective optimizations are performed in the MATLAB R2023a program via PlatEMO [51]. The first optimization algorithm compared is the fast and elitist multi-objective genetic algorithm (NSGA-II) [52] method, which is one of the most preferred methods. The general flow chart of the NSGA-II method is given in Figure 2. The second method is the third-generation cooperative coevolutionary differential evolution algorithm (CCGDE3) method [30]. Other methods examined in the study are MOEA/D with an update when required (MOEA/D-UR) [31] and a decomposition-based multi-objective evolutionary algorithm guided by growing neural gas (DEA-GNG) [32], which have been introduced to the literature in recent years. While summary information about the NSGA-II method is given below, detailed information about the CCGDE3, MOEA/D-UR, and DEA-GNG methods can be found in the related method reference. The method first starts with determining the method parameters and then creating the first population. The fitness functions of each alternative are calculated. The fitness values of each individual are calculated, and the individuals are sorted in order of priority (non-dominated sorting). By calculating the crowding distance, diversity among individuals with the same priority is preserved. Parents are selected using selection methods. In this phase, fitness values and crowding distance are considered. New individuals are created by performing crossover and mutation operations on the selected parents. These new individuals create the next generation population. A large population is created by combining the old population and the newly created population. Non-dominated sorting is applied again on the large population. After the large population is sorted, the best individuals are selected when creating a new population. In this selection, diversity is preserved by using crowding distance. Depending on the stopping criterion, the algorithm is terminated when a certain number of generations is reached, or the stopping criterion is met. The final set of population solutions is presented as a Pareto front. For details of the NSGA-II method, the article presented by [PERSON] et al. [52] can be reviewed.