## 2 Framework
### Mathematical Model  
Although several research papers suggest decision-making models to select the optimum energy retrofitting strategy, the literature shows many limitations in terms of ignoring some cost components or constraints. For instance, some models ignored the maintenance cost [30], which is a significant operational cost and a component of the LCC. In another study [8], the thermal comfort requirement was ignored. Moreover, many studies consider building envelope measures and neglect electrical and mechanical equipment measures or they neglect the interdependence between them [31], for instance, the effect of the building envelope measures on the adequate HVAC system. As a result, and in order to achieve a successful energy retrofitting plan, the suggested model should:  
Consider all possible measures that are available and within the reach of the decision maker;  
Include the comfort constraints such as thermal comfort and visual comfort;  
Cover the interdependence between the measures. While there are multiple options for the objective function, minimizing the life cycle cost is a widely used objective function in many papers as it calculates the overall costs of the retrofitting project or equipment over its lifetime, does not ignore any incurred costs in the future, and ensures the sustainability of the retrofitting project. For the mathematical model, the energy retrofitting measures are collected and then the life cycle cost is determined for each retrofitting measure. Then, each retrofitting measure will be represented by a binary 0-1 decision variable as follows:  
\[\text{X}_{\text{ij}}=\{1,\text{ implement energy retrofitting measure ij; 0, otherwise}\}\]  
where i represents the retrofitting measure type and j represents the different available options within that type i. For example,  
\[\text{X}_{\text{ij}}=\{1,\text{ Replace the electrical equipment j = (1, 2, 3,..., J); 0, otherwise}\}\]  
where i = 1 represents the \"replace electrical equipment\" retrofitting measure type and j denotes the different options of electrical equipment available. Using the binary decision variables, the resulting objective function is as follows:  
\[\text{Minimize z}=\sum_{\text{i}=1,\text{j}=1}^{\text{IJ}}\text{LCC}_{\text{ ij}}*\text{X}_{\text{ij}} \tag{6}\]  
Examples of similar objective functions and decision variables can be found in [14; 30; 31]. Constraints  
According to ASHRAE Standard 55, Thermal Environmental Conditions for Human Occupancy, \"Thermal comfort is the condition of mind that expresses satisfaction with the thermal environment\" [32]. As the downsizing or change of the HVAC system is one of the retrofitting measures, the optimal one should satisfy the building's cooling and heating loads to maintain the required thermal comfort. * Cooling Load Cooling load is defined as \"the amount of heat that would need to be removed from the building\" while the capacity of the HVAC system is defined as \"the ability of the equipment to remove the heat.\" In the HVAC industry, the British thermal unit (Btu) and ton refrigeration (TR) are the units used to measure the quantity of heat that should be removed from the space [22]. The cooling load should be calculated in peak summer time when the heat reaches its highest levels in order to select an adequate HVAC system. The cooling load is the sum of the total sensible loads and latent loads. * Sensible load is the solar heat gain from the area of exposure, transmission gain (solar radiation through transparent surfaces, such as windows; heat conduction through exterior walls and roofs; and heat conduction through interior partitions, ceilings, and floors), outside air heat gain (load as a result of ventilation and infiltration of outdoor air), and internal load. * Latent load includes people, appliances, and outside air load. Formulas on how to calculate the cooling load can be found in [14].