## 1 Introduction
Each step entails a plethora of parameters that are used in either simple techniques like the degree-day's method or quite sophisticated like dynamic thermal models in order to estimate the transmission loads. After that, the cost analysis, using the Net present value analysis, determines the total cost and the optimum insulation thickness. The optimization of thermal insulation thickness has been investigated in many studies previously. For example, [PERSON] et al. [15] investigated the optimal thermal insulation thickness of archetypal exterior walls in Iran. They performed the optimization analysis using the life-cycle cost analysis method together with the numerical solution to identify the best insulation thickness. They identified that the \(X_{opt}\) in Iran is not more than 4 cm and that the \(X_{opt}\) varies depending on the climatic location. Similarly, [PERSON] et al. and [PERSON] (2016); [PERSON] (2016) used the complex finite Fourier transform to determine \(X_{opt}\), energy savings, and payback time for a wall, based on the research described above. In addition, they studied the influence of the Tunisian climate and the impact of wall orientation. In Turkey, [PERSON] (2018, 2019) employed the finite difference method to analyze optimal insulation thickness (\(X_{opt}\)), energy savings, and payback periods. His study also highlighted the influence of wall orientation on thermal performance, revealing that the north-facing wall had the lowest \(X_{opt}\), had the smallest cooling load, and was the most cost-effective, with an \(X_{opt}\) of 3.1 cm. Similarly, [PERSON] (2018) used the heating degree-hours approach to determine \(X_{opt}\) for a building in Bursa, Turkey. His findings indicated that \(X_{opt}\) ranged between 5.3 and 12.4 cm, depending on the type of fuel used, with natural gas emerging as the most economical option. Additionally, [PERSON] et al. (2018) applied the cost-optimal methodology to an office building in Italy and found that the \(X_{opt}\) based on cost considerations could be lower than the regulatory insulation requirements. A study in Palestine using life-cycle cost analysis demonstrated that polyurethane insulation required only half the \(X_{opt}\) compared to polystyrene ([PERSON], 2018). Meanwhile, in China, [PERSON] et al. investigated \(X_{opt}\) across various climatic zones, incorporating factors such as wall orientation and surface color ([PERSON], 2018). Their findings revealed that polystyrene is the most economical insulation material, providing the quickest payback period and the greatest life-cycle savings. [PERSON] (2018) conducted a comprehensive analysis of economic thermal insulation systems for building exterior walls, focusing on determining the optimal insulation thickness (\(X_{opt}\)) and its impact on energy consumption. The study compared different optimization and economic analysis approaches, presenting various insulation materials and fuel types while assessing their influence on energy efficiency and emissions in buildings. Figure 1: Optimum insulation thickness ([PERSON], 2019). Similarly, [PERSON] et al. [24] investigated the optimal insulation thickness for residential building exterior walls in Athens, Greece, considering factors such as construction techniques, wall orientation, and insulation placement. Their study found that \(X_{opt}\) ranged from 11.2 to 23.4 cm, depending on these variables. Additionally, the research highlighted that incorporating insulation could lower annual CO2 emissions per unit wall area by 63.2-72.2% compared to non-insulated walls, with the extent of emission reduction influenced by both the type of insulation material and its positioning within the wall. [PERSON] et al. [25] assessed the thermal performance of different insulation materials and fuel types across various geographical regions in Turkey. It evaluated optimal insulation thickness, annual cost savings, payback periods, and emission impacts. Additionally, new empirical models were formulated using degree-day calculations for XPS and EPS materials.