# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
### The Category of Inhabitants' Health Improvement
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(DAII\) & [0.0, 0.1] & [0.1, 0.2] & [0.2, 0.3] & [0.3, 0.4] & [0.4, 0.5] & [0.5, 0.6] & [0.6, 0.7] & [0.7, 0.8] & [0.8, 0.9] & [0.9, 1.0] \\ \hline ZPM & 0.332 & 0.668 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ POM & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ WM & 0 & 0 & 0 & 0.055 & 0.727 & 0.218 & 0 & 0 & 0 \\ PDL & 0 & 0 & 0 & 0 & 0.658 & 0.342 & 0 & 0 & 0 \\ LBU & 0 & 0.713 & 0.287 & 0 & 0 & 0 & 0 & 0 & 0 \\ WLKP & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ KP & 0 & 0 & 0 & 0.341 & 0.659 & 0 & 0 & 0 & 0 \\ MAZ & 0 & 0 & 0 & 0.817 & 0.183 & 0 & 0 & 0 & 0 \\ LBL & 0 & 0 & 0 & 0 & 0.342 & 0.618 & 0.040 & 0 & 0 \\ DSL & 0 & 0 & 0 & 0.033 & 0.510 & 0.433 & 0.024 & 0 & 0 \\ OPO & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ LDZ & 0 & 0 & 0 & 0.150 & 0.710 & 0.140 & 0 & 0 & 0 \\ SL & 0 & 0 & 0.433 & 0.567 & 0 & 0 & 0 & 0 & 0 \\ SW & 0 & 0 & 0 & 0 & 0.256 & 0.744 & 0 & 0 & 0 \\ MLP & 0 & 0 & 0 & 0 & 0 & 0.088 & 0.715 & 0.197 & 0 \\ PKR & 0 & 0 & 0 & 0 & 0 & 0.125 & 0.708 & 0.167 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Distribution of distances to the best unit for Polish voivodeships when considering inhabitants’ health improvement level. Figure 10: Extreme and expected ranks for Polish voivodeships when considering inhabitants’ health improvement level. 6.9% and not greater than 31.1%. Still, these outcomes exhibit less diversity than at the comprehensive level. The graph of the necessary relation at the inhabitants' health improvement level is shown in Figure 11. The robust conclusions are richer than at the comprehensive level. For example, the number of pairs of different provinces that are related by the necessary preference increased from 47 to 85. Moreover, the number of levels in the respective Hasse diagram increased from 3 to 7. In particular, WLKP is necessarily preferred to all other provinces, and four other voivodeships--POM, ZMP, LBU, and SL--proves to be necessarily better than the remaining eleven units. The worst unit is OPO, which is not necessarily preferred to any other province while being possibly preferred only to PKR. Further, MLP and PKR are necessarily worse than 10 and 9 other provinces, respectively. Finally, SL benefited the most from limiting the scope to the health improvement level, because in the hierarchy's root, it was not robustly better than any other voivodeship. The respective \(PEO\)Is are given in Table 8. Similarly, as at the comprehensive level, for some pairs, one unit is significantly better than the other (see, e.g., ZPM and POM with \(PEO\)\(I(ZPM,POM)=94.8\%\) or LBU and SL with \(PEO\)\(I^{H}(LBU,SL)=98.7\%\)). In turn, other pairs are characterized by more balanced stochastic accepabilities, indicating an advantage of either vivodeship (see, e.g., PKR and MLP with \(PEO\)\(I^{H}(PKR,MLP)=52.2\%\) or KP and SW with \(PEO\)\(I^{H}(KP,SW)=52.4\%\)). However, the absolute values of \(POE\)Is for some pairs differ vastly compared to the hierarchy's root. For example, POM is necessarily better than ZPM at the comprehensive level, but when considering only health improvement, ZPM attains no worse efficiency for 94.8% of feasible weights.