# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
### The Category of Inhabitants' Health Improvement
Table 6 presents the distribution of distances for all voivodeships when considering the inhabitants' health improvement level. The only two provinces for which this distance is lower than 0.1 are WLKP (100%) and ZPM (33.2%). Such a favorable result was not attained by ZPM at the comprehensive level for any feasible weight vector. Then, its distance could drop even above 0.4. For the two units mentioned above, as well as POM and LBU, all samples confirm distances not higher than 0.2. Furthermore, for all provinces, we can indicate a single bucket in which the unit's distance falls for most samples. For example, it is \((0.2,0.3]\) for LBU, \((0.3,0.4]\) for MAZ, and \((0.7,0.8]\) for OPO. For these three provinces, the predominating distance buckets at the comprehensive level were better. However, for other voivodeships, including SW, MLP, and PKR, the most often repeated distance range worsened when limiting the analysis to the inhabitants' health improvement level. The extreme and expected efficiency ranks at the inhabitants' health improvement level are provided in Figure 10. WLKP, as the only efficient province, is always ranked first. Both ZPM and POM are ranked between second and fourth, while for LBU, this range is slightly wider (\([2,5]\)). Conversely, only two voivodeships--OPO and PKR--fall to the bottom ranking at any point, and three others (WM, DSL, and MLP) are ranked fifteenth in the least favorable scenario. Finally, for DSL and MAZ, the difference between their extreme ranks is the greatest. For example, DSL is ranked between sixth and fifteenth. According to the expected ranks, WLKP (\(ER=1\)), ZPM (2.110), POM (3.034), and LBU (3.860) are the best, and PKR (14.205), MLP (14.339), and OPO (15.990) are the worst. Figure 9: Extreme and expected distances to the best unit for Polish voivodeships when considering inhabitants’ health improvement level. When compared to the comprehensive level, the greatest improvement in the attained ranks can be observed for ZPM (\([2,4]\) rather than \([4,15]\)) and SL (\([4,5]\) rather than \([7,16]\)). On the contrary, the greatest deterioration of possible positions is noted for OPO (\([15,16]\) rather than \([8,16]\)) and MLP (\([11,16]\) rather than \([4,15]\)). For many provinces, including WM, POL, MAZ, LBL, LDZ, and SW, the ranking intervals got significantly narrower, confirming the lower diversity of results when limiting the scope of the analysis to health improvement. The distribution of efficiency ranks given the inhabitants' health improvement level is presented in Table 7. The most stable individual positions were observed for WLKP (1-100%), SL (5-99.6%), OPO (16-99.0%), and ZPM (2-91.6%). Such high acceptabilities were not observed at the comprehensive level for any position and unit. In fact, the greatest share of weights (72.4%) supported the third position of LBU. Returning to the health improvement category, for some other voivodeships, the vast majority of weights indicate a pair of ranks (e.g., MLP and PKR are ranked 14 th or 15 th for 92.3% or 78.3% samples, respectively). Finally, the ranks of some units are more dependent on the chosen weight vector. For example, KP attained positions between 8 th and 12 th, with no \(ERAI\) exceeding 28%. Similarly, SW is ranked within the range \([7,12]\) with \(ERAI\)s not less than