# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
### The Category of Financial Management
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline _PEOI_ & _ZPM_ & _POM_ & _WM_ & _PDL_ & _LBU_ & _WLKP_ & _KP_ & _MAZ_ & _LBL_ & _DSL_ & _OPO_ & _LDZ_ & _SL_ & _SW_ & _MLP_ & _PKR_ \\ \hline ZPM & 1 & 0.948 & 1 & 1 & 0.973 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ POM & 0.052 & 1 & 1 & 1 & 0.897 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ WM & 0 & 0 & 1 & 1 & 0 & 0.995 & 0.070 & 0.904 & 0.921 & 1 & 0.851 & 0 & 0.786 & 1 & 1 \\ PDL & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0.240 & 0.014 & 1 & 0.320 & 0 & 0.209 & 0.844 & 0.767 \\ LBU & 0.027 & 0.103 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0.987 & 1 & 1 & 1 \\ WLKP & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ KP & 0 & 0 & 0.005 & 1 & 0 & 0 & 1 & 0 & 0.565 & 0.395 & 1 & 0.640 & 0 & 0.524 & 1 & 0.996 \\ MAZ & 0 & 0 & 0.930 & 1 & 0 & 0 & 1 & 0.989 & 0.939 & 1 & 1 & 0 & 1 & 1 & 1 \\ LBL & 0 & 0 & 0.096 & 0.760 & 0 & 0 & 0.435 & 0.011 & 1 & 0.278 & 1 & 0.688 & 0 & 0.489 & 0.962 & 1 \\ DSL & 0 & 0 & 0.079 & 0.986 & 0 & 0 & 0.605 & 0.061 & 0.722 & 1 & 1 & 0.728 & 0 & 0.635 & 0.942 & 0.947 \\ OPO & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0.015 \\ LDZ & 0 & 0 & 0.149 & 0.680 & 0 & 0 & 0.360 & 0 & 0.312 & 0.272 & 1 & 1 & 0 & 0.011 & 1 \\ SL & 0 & 0 & 1 & 1 & 0.013 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ SW & 0 & 0 & 0.214 & 0.791 & 0 & 0 & 0.476 & 0 & 0.511 & 0.365 & 1 & 0.989 & 0 & 1 & 1 \\ MLP & 0 & 0 & 0 & 0.156 & 0 & 0 & 0 & 0 & 0.038 & 0.058 & 1 & 0 & 0 & 0 & 1 & 0.478 \\ PKR & 0 & 0 & 0 & 0.233 & 0 & 0 & 0.004 & 0 & 0 & 0.053 & 0.985 & 0 & 0 & 0 & 0.522 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Pairwise efficiency outranking indices for pairs of Polish voivodeships when considering inhabitants’ health improvement level. has the broadest range of distances (\([0.140,0.416]\)), confirming its performance's sensitivity to the selection of particular priorities. This results from an imbalanced performance profile with a highly favorable value on output \(F_{1}\) and a relatively poor value on output \(F_{3}\). Finally, when complete order is desired, it can be imposed by the expected distances. In this case, LBU (\(Ed^{F}=0.010\)) and POM (\(Ed^{F}=0.021\)) are safely ranked at the top, and DSL (\(Ed^{F}=0.509\)) and SL (\(Ed^{F}=0.735\)) are ranked at the bottom. The distance distribution at the financial management level is presented in Table 9. For both efficient provinces (POM and LBU), the distance is always within the first bucket (\([0.0,0.1]\)). Among the inefficient provinces, WLKP confirms its superiority over the remaining units, as for over 99% samples, its distance from the efficient unit is not greater than 0.2. The only two other provinces with positive \(DAII\)s for bucket \((0.1,0.2]\) are KP (14.2%) and MLP (4.3%). The greatest stability of distances among inefficient provinces is observed for ZPM with \(DAII^{F}(ZPM,(0.3,0.4])=1\). There are only two provinces, PDL and KP, for which most samples confirm no single distance bucket. Furthermore, KP is the only unit with positive \(DAII\)s for more than three buckets (\((0.1,0.5]\)). The results of the robustness analysis for efficiency ranks at the financial management level are presented in Figure 13 (extreme and expected positions) and Table 10 (efficiency  
Figure 12: Extreme and expected distances to the best unit for Polish vovodeships when considering financial management level.