## 2 Materials and Methods
### Data  
Data used in the present work for creating the efficiency scores refer to the Packages of Optimal Solutions (POS) defined in the HAPPEN project implementation and collected in the report D3.4 [2] of the project, where they are freely disposable. The project identifies 16 POS starting from four pilot case studies of deep retrofitting of residential buildings carried on in 4 different Countries. The Package of Optimal Solution (POS) is defined as a set of 12 solutions, each of which minimizes the LCC of the buildings after the renovation process with different combinations of technical renovation measures. Each POS was defined specifically for each reference building of the Country of the four pilot studies and for each climate zone (4 Countries and, for each Country, 4 climatic zones). The total 192 solutions referred to the 16 defined POS were formulated considering different combinations of renovation measures for the facades, for the renovation of the roofs, of the floors, of the thermal bridges, of the ventilation system, of the shading elements, and of different types of glazing [2]. The proposed analysis selected 2 different building typologies (i.e., 2 Single-Family Houses, SFH and 2 Multi-Family Houses, MFH) and 4 climate zones (i.e., W1S2; W2S2; W2S3; W3S2). The Single-Family Houses are homes including single- and two- family houses and terraced houses; the Multi-Family Houses refer to buildings with more than two apartments and fewer than 9 floors. Buildings that are higher than 8 floors are not considered in the present study (from the report D3.2 \"Catalogue of reference buildings classes in MED countries\"). Considering the four-pilot case-studies analyzed, they are characterized by two main measures: the thermal transmittance \(\hat{U}\), (W/m\({}^{2}\)K) and the compactness C (m). In details: Cyprus: \(\hat{U}=2.12\) and C = 1.57; Labin: \(\hat{U}=2.23\) and C = 1.33; Marseille: \(\hat{U}=1.89\) and C = 3.46; Castellon: \(\hat{U}=2.54\) and C = 2.96 (data from the report [2]). To identify the climate zones, the project considered both the Climate Severity Index (CSI) and the aim to cover the Mediterranean area. Different zones are identified with some acronyms where W means winter and S corresponds to summer (Climate zones were defined considering the Climate Severity Index equations for winter and for summer. In Appendix A, Table A1 reports details for the four climate zones considered). Croatia (HR) and Cyprus (CY) represent the pilots for SFH, while France (FR) and Spain (SP) are the front-runner pilots where the MFH typology was considered. Table 1 presents the strategy adopted for the DDF application. Indeed, considering each building typology and climate zone, 8 Directional Distance Functions were estimated. Let we consider, for instance, the MFH typology and the climate zone W1S2: 24 observations represent the sample size of DDF model (12 solutions for the corresponding POS13 from Spain and 12 solutions for the POS9 from France). The results are (holistic/environmental) efficiency scores that can be used for comparing the 24 solutions of each initial sample. Indeed, the methodology proposed allows to obtain efficiency scores based on minimization of CO2 emissions and the aim is to evaluate the efficiency of each solution for each POS considering the CO2 emissions as a bad output. With this intent, two model definitions were proposed, different for the input variables chosen. The outputs are two and they do not change in the two formulations. The final energy savings per year (The final energy savings per year (Mwh) represents the planned absolute value of the final energy savings per year) is the good output to maximize, while the CO2 emissions (The CO2 emissions represent the value of the CO2 production after the implementation of the optimal solution proposed) represent the output to minimize (i.e., the bad or undesirable output).