## 1 Introduction  
Global energy consumption is rapidly increasing, leading to a rise in Greenhouse Gas (GHG) emissions. This trend is expected to worsen in the future unless substantial efforts are made to reduce fossil fuel consumption. Climate change, primarily caused by human activities and GHG emissions from fossil fuels, is considered one of the most significant threats to humanity [1]. To address climate change, it is crucial to transition to low-carbonenergy systems. The building sector, with its substantial energy and environmental impact, must play a key role in this transition toward a sustainable energy future [2]. Buildings consume nearly 40% of global energy, with approximately 60% of this used for heating and cooling, representing the highest percentage of energy use [3]. To limit the Earth's temperature rise to 2 \({}^{\circ}\)C, it is recommended to reduce overall Carbon Dioxide (CO\({}_{2}\)) emissions by 32 Giga Tons in the global building sector by 2050 [4]. Saudi Arabia is one of the leading countries globally regarding per capita energy consumption and CO\({}_{2}\) emissions. The country's per capita annual electricity consumption is among the highest in the world, equivalent to 9401.5 kWh/capita, while the world average is 3131.7 kWh/capita [5]. Its building sector is experiencing fast growth in energy demand as the building stock continues to expand. Research suggests that energy consumption in buildings is influenced by multiple factors, including rapid population growth, the extreme desert climate, inefficient building designs, subsidized energy prices, and the absence of strict energy policies [6]. In Saudi Arabia, buildings face considerable energy challenges, with air conditioning (AC) systems being the primary contributors to electricity consumption. AC units account for nearly 70% of residential electricity use, largely due to the widespread lack of thermal insulation in building envelopes--approximately 70% of existing structures remain uninsulated [7; 8; 9]. The region's hot-arid climate necessitates continuous cooling to maintain indoor comfort, making it a major driver of energy demand. To mitigate economic burdens and reduce environmental pollution, enhancing building energy efficiency is essential [2]. Earlier research has highlighted that applying thermal insulation to exterior walls is one of the most efficient methods for lowering energy consumption [9; 10; 11]. Envelope insulation thickness and effectiveness are influenced by several weather-related parameters [12]. The thickness of the insulation added to the building envelope is one of the most critical factors to consider when designing such energy-efficient structures [13; 14]. Economically, thermal insulation thickness considers the initial investment of the insulation systems as well as the continuing benefit of energy savings over the planned service life of the insulation. The thickness that offers the lowest overall lifetime cost is the economically optimum insulation thickness (\(X_{opt}\)). As illustrated in Figure 1, increasing the insulation thickness leads to higher installation costs, whereas heating and cooling energy expenses decline. The optimal insulation thickness is the point at which the combined cost of energy consumption and insulation is minimized. Beyond the optimal thickness, energy savings continue to increase; however, the overall cost rises due to the disproportionate growth in insulation expenses. Thus, while additional insulation beyond the optimal point may result in lower energy consumption, it does not provide economic benefits. The objective of optimizing insulation thickness is to achieve the most cost-effective balance between the energy cost savings (from sources such as electricity, natural gas, fuel oil, etc.) and the expenses of thermal insulation materials. Generally, there are two main steps for optimizing insulation thickness in buildings. First, the annual heating and cooling transmission loads should be determined. Then, an economic analysis takes place.