## 2 Materials and Methods
### Data
Concerning the inputs, the present paper proposes two formulations of the DDF estimations: the first one (Model#1) considers as input the LCC estimation for each solution; while Model#2 represents a more standard definition of the DDF, considering the total costs of solution. The purpose to define two different input-output space strategies allows to consider different meanings of efficiency. In the case of Model#2, solutions are compared considering their ability to minimize the CO2 emissions and simultaneously to maximize the final energy savings, using the same amount of costs. These results represent the environmental efficiency scores, well known in literature. However, thanks to the data collected in the HAPPEN project, scores with a more informative meaning were calculated considering the LCC as input. This happens in the Model#1, which considers not only the negative externalities of production process as bad output (i.e., emissions), but also costs and investments as input (LCC estimation). For this reason, the obtained results catch the holistic impact of the retrofitting interventions and allow to compare, in a simple way, different solutions. All data used are available on the EU website of the project, and they were extracted from the [2]. The LCC is defined in the HAPPEN project as the initial investment plus the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Building** & **Climate** & **Pilot Study** & **Pilot Study** & **Sample Size** & **DDF** \\
**Typology** & **Zone** & **(Country/POS n\({}^{\circ}\))** & **(Country/POS n\({}^{\circ}\))** & **(Number of Solutions)** & **Frontier** \\ \hline \multirow{4}{*}{SFH} & W1S2 & CY (POS1) & HR (POS5) & 24 & 1 \\  & W2S2 & CY (POS2) & HR (POS6) & 24 & 2 \\ \cline{1-1}  & W2S3 & CY (POS3) & HR (POS7) & 24 & 3 \\ \cline{1-1}  & W3S2 & CY (POS4) & HR (POS8) & 24 & 4 \\ \hline \multirow{4}{*}{MFH} & W1S2 & SP (POS13) & FR (POS9) & 24 & 5 \\ \cline{1-1}  & W2S2 & SP (POS14) & FR (POS10) & 24 & 6 \\ \cline{1-1}  & W2S3 & SP (POS15) & FR (POS11) & 24 & 7 \\ \cline{1-1}  & W3S2 & SP (POS16) & FR (POS12) & 24 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Strategy design for Directional Distance Function estimation. operational costs over 30 years after implementing the optimal solutions; and a deeper explanation of the computation can be found in [36]. In details, the Life-Cycle Costs are estimated as follows:  
\[LCC(\epsilon)=Investment(\epsilon)+\sum_{i=1}^{30}(OC_{i})\cdot Discount\ rate\]  
where OC represent the operational costs for each year expressed in \(\epsilon\)/year, and in this particular case, they refer to heating, cooling, energy consumptions, and maintenance costs. The Discount Rate adopted is an annuity calculated as follows:  
\[Discount\ rate=\frac{1-\left(1+\frac{k}{100}\right)^{-30}}{k/100}\]  
where the current interest rate k is 1.47%, calculated considering an interest rate of 3.5% and an inflation rate of 2% as suggested by the delegated regulation [37]. In addition, following [38], the size of input-output space (3 variables: 1 input + 2 outputs) allows to obtain robust results as an OLS with a sample equal to 69 units and the robustness of results is improved by the bootstrap resampling procedure. Finally, we can sum up that the two input-output space strategies (Model#1: holistic efficiency, and Model#2: environmental efficiency) were used for defining the Directional Distance Functions considering the two building typologies (i.e., SFH and MFH), and the four climate zones (i.e., W1S2, W2S2, W2S3, W3S2). This means that 8 frontiers were built considering as input the Life-Cycle Cost estimations (Model#1), and 8 with total costs (Model#2). Notice that costs, as well as the final energy consumptions, vary significantly among climate zones (W1S2, W2S2, W2S3, W3S2), and different solutions of retrofitting interventions were analyzed separately. Descriptive statistics of input and outputs are presented in Table 2.