## 1 Introduction
Based on the findings, climatic regions that are extremely hot and extremely cold are much more suitable for applying thermal insulation for energy saving. [PERSON] et al. [16] determined the carbon payback time through a life cycle assessment method, taking into account the global warming potential of nine different insulation materials at their ideal thicknesses. The findings showed that among the four cardinal orientations (north, south, east, and west), aerogel exhibited the longest carbon payback time of 2.34 years, while glass wool had the shortest carbon payback time of only 0.09 years. [PERSON] et al. [17] evaluated the impact of different insulation materials like perlite, expanded polystyrene, foamed polyvinyl chloride, and rock wool on ideal thermal insulation thickness for a greenhouse, which is a unique construction unlike a standard building. Although foamed polyvinyl chloride has lower thermal conductivity than others, they did not recommend it as an energy-saving material because of its high price. [PERSON] [18] conducted research on determining the ideal thermal insulation thickness for all cities in Turkiye through life cycle cost analysis. Polyurethane foam, extruded polystyrene, expanded polystyrene, and rock wool as insulation materials were considered due to their common usage. Natural gas was chosen for heating the buildings, while electricity was chosen for cooling purposes. In this research, a regression model was developed utilizing MATLAB v.2024a to determine the ideal insulation thickness, yielding results closely aligned with theoretical findings. [PERSON] et al. [19] investigated the condensation effect on insulation applications, considering the water vapor diffusion resistance factor. In addition to insulation thickness minimization, they performed heat and mass transfer calculations for different relative humidity values and indoor-outdoor temperatures. They concluded that when employing insulation materials with low vapor diffusion resistance factors, the thermal conductivity of the insulation material does not significantly impact the minimum required insulation thickness to prevent condensation. [PERSON] [20] performed a multiple optimization study considering the total cost and CO2 emissions for four climatic regions, three heating sources, four insulation materials, and three scenarios of seasonal usage. He determined ideal insulation thicknesses, energy cost savings, energy saving rates, payback periods, carbon savings, and carbon saving rates for numerous cases. It was found that when the primary aim of applying insulation is to reduce CO2 emissions, the determined ideal thicknesses of insulation tend to be greater than those determined when the focus is primarily on minimizing overall costs. [PERSON] et al. [21] performed a transient simulation to determine the ideal insulation thickness, life cycle energy savings, and payback period considering the orientation of the outer walls and solar radiation effect for a building in Miskolc City, Hungary. They found that for the north-oriented wall,the ideal insulation thicknesses were 17 cm for EPS, 22 cm for glass wool, and 12 cm for rock wool. Correspondingly, the life cycle energy savings were 142.5 kWh/m\({}^{2}\) for EPS, 153 kWh/m\({}^{2}\) for glass wool, and 132.12 kWh/m\({}^{2}\) for rock wool. As seen from the literature review, many studies have been conducted lately, focusing on different factors and using different approaches to the optimum insulation thickness. Correlatively, due to the energy crisis we have been through, energy efficiency in buildings will continue to be a priority in the foreseeable future and will remain relevant. In the existing literature, studies on insulation thickness optimization have often been limited to considering only the cardinal directions (south, north, east, and west), without examining the impact of intermediate directions, which can significantly affect energy performance due to variations in solar irradiance.