## 3 Methodology: Development of the Evaluation Framework
### Definition of the Optimization Problem  
Multi-objective optimization, also called multi-criteria optimization, can be defined as \"the problem of finding a solution vector of decision variables which satisfies the project constraints introduced by the end users and optimizes a vector function whose elements represent individual objective functions\" [28]. These functions form a mathematical description of performance criteria (DPIs). Hence, the term \"optimise\" means finding such a solution vector (refurbishment scenario) which would give the values of selected DPIs acceptable according to the predefined targets and boundaries.  
Formally, a multi-objective optimization problem is defined so as to find the solution vector \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{T}\) which optimizes the district performance vector defined as:  
\[D(x)=[f_{1}(x),f_{2}(x),\ldots f_{n}(x)]^{T} \tag{1}\]  
In other words, the goal is to determine among the set of all feasible refurbishment scenarios that satisfies the constraints, the particular set \(\mathbf{x}\) which yields the optimum values of the selected DPIs defined in equation f(\(\mathbf{x}\)).  
Note that in general there is not a single point optimizing the set of objective functions, whereas a set of optimal solutions considered being non-dominated solutions or Pareto set approximation. Therefore, the notion of \"optimum\" varies with respect to mono-objective approaches, which aims at achieving a unique solution that simultaneously meets the constraints and provides the best value for the objective function. In multi-objective approaches, a solution vector \(\mathbf{x}\) is Pareto optimal if there does not exist another \(\mathbf{x}\), such that \(\mathbf{f}_{i}(\mathbf{x})\leq\mathbf{f}_{i}(\mathbf{x})\) for all i \(>\{1,\ldots,n\}\) and \(\mathbf{f}_{j}(\mathbf{x})<\mathbf{f}_{j}(\mathbf{x})\) for at least one \(j\).  
This definition means that x is Pareto optimal if no feasible vector of decision variables exist which would decrease some criterion without causing a simultaneous increase in at least another criterion. The plot of the objective functions whose non-dominated vectors are in the Pareto optimal set is called the Pareto front [29]. In the present approach and as depicted before, the evaluation criteria has been divided into three groups representing the costs the first, and the benefits the second and third (being this last one a subcategory of the second). Therefore, the optimization problem here is the one that aims at finding the set \(\mathbf{x}\) that minimizes the costs (considering not only economic costs but also environmental or social costs) and maximizes the benefits, which are plotted in order to find those optimal contained in the Pareto front.  
However, in order to find the cost and benefit values for each candidate retrofitting scenario, a weighted sum of the indicators that represent these two indexes has to be carried out, where the weights have to represent the prioritization criteria established by the stakeholders.  
\begin{table}
\begin{tabular}{c c c} \hline
**Group 1—Cost** & **Group 2—Benefit 1** & **Group 3—Benefit 2** \\ \hline ENV01.Global Warming Potential - & & ENE 01.Energy demand & ENE 14.Energy use from Biomass \\ GWP (kg CO\({}_{2}\)) & & & \\ \hline ENV04.Primary energy consumption & & & \\ \hline ENV06.Energy payback time & & ENE 06.Net fossil energy & ENE 16.Energy use from Solar \\  & consumed & Thermal \\ \hline ECO02.2 Investments (in \(\mathbf{\upin}\)) & & ENE 09.Energy demand covered & ENE 17.Energy use from \\  & by renewable sources & Hydraulic \\ \hline ECO03.Life cycle cost & & ENE 13.Energy use from \\  & Heating & Mini-Eolic \\ \hline ECO05.Payback Period & & ENE 19.Energy use from \\  & & Geothermal \\ \hline \end{tabular}
\end{table}
Table 1: List of District Performance Indicators (DPIs) per evaluation group.