# 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} \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 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ POM & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ WM & 0 & 0 & 0.757 & 0.243 & 0 & 0 & 0 & 0 & 0 \\ PDL & 0 & 0 & 0.475 & 0.483 & 0.042 & 0 & 0 & 0 & 0 \\ LBU & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ WLKP & 0 & 0.993 & 0.007 & 0 & 0 & 0 & 0 & 0 & 0 \\ KP & 0 & 0.142 & 0.453 & 0.379 & 0.026 & 0 & 0 & 0 & 0 \\ MAZ & 0 & 0 & 0 & 0.857 & 0.143 & 0 & 0 & 0 & 0 & 0 \\ LBL & 0 & 0 & 0 & 0.139 & 0.775 & 0.086 & 0 & 0 & 0 \\ DSL & 0 & 0 & 0 & 0 & 0.312 & 0.688 & 0 & 0 & 0 & 0 \\ OPO & 0 & 0 & 0 & 0.240 & 0.708 & 0.052 & 0 & 0 & 0 \\ LDZ & 0 & 0 & 0.705 & 0.295 & 0 & 0 & 0 & 0 & 0 \\ SL & 0 & 0 & 0 & 0 & 0 & 0 & 0.220 & 0.743 & 0.037 & 0 \\ SW & 0 & 0 & 0.366 & 0.628 & 0.006 & 0 & 0 & 0 & 0 \\ MLP & 0 & 0.043 & 0.957 & 0 & 0 & 0 & 0 & 0 & 0 \\ PKR & 0 & 0 & 0 & 0 & 0.545 & 0.455 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Distribution of distances to the best unit for Polish vovodeships when considering financial management level. rank acceptability indices). They provide additional insights into the comparisons of efficient provinces. Specifically, even if LUB and POM can be ranked first for some weight vectors, the former is ranked first almost twice as often as the latter. This makes it the most favorable province regarding financial management, as additionally confirmed by the expected ranks (\(ER^{F}(LUB)=1.377\) vs. \(ER^{F}(POM)=1.623\)). The remaining provinces are at most third in the best case. Again, WLKP proves to be the most advantageous among them, with \(ER^{F}(WLKP)=3.107\). Further, KP has the broadest range of efficiency positions, being ranked between third and eleventh. \(ERAI\)s confirm that its ranks are rather equally distributed between these extreme positions, with the maximal value for the fifth rank (\(ERAI^{F}(KP,5)=24.8\%\)) and acceptabilities greater than 6% for all ranks within the range \([3,11]\). Such great diversity is a consequence of its extreme performances. In fact, it is the best among all provinces on \(F_{1}\) while being in the bottom three on \(F_{3}\). Thus, KP is focused on the profit attained by healthcare institutions rather than on the number of treated patients. This aspect needs to be improved when aiming for higher ranks. There are three other provinces with relatively wide possible efficiency rank intervals: ZPM (\([6,12]\)), PDL (\([5,11]\)), and OPO (\([9,15]\)). Among them, only ZPM attains a single rank for most samples (\(ERAI^{F}(ZPM,10)=62.4\%\)). Finally, SL is ranked at the bottom regardless of the weight vector. This is related to its greatest value on input \(F_{2}\), the lowest value on output \(F_{3}\), and a relatively low value on output \(F_{1}\). Thus, even if the financial input of SL is the greatest, its outputs are less favorable than for provinces with lesser financial resources. The complete ranking established with the expected efficiency ranks aligns with the one based on expected distances, with LBU, POM, WLKP, and MLP being ranked among the best provinces and SL, DSL, and PKR placed at the bottom. Figure 14 presents the necessary ranking at the financial management level. POM and LBU are necessarily preferred to all other provinces, confirming their superiority given the financial category. The second level includes KP, WLKP, and MLP, and the lowest one contains PKR, DSL, and SL. They are possibly preferred to 4, 3, and 0 other units, respectively. Moreover, SL is necessarily worse than all other voivodeships. The pairwise comparisons are useful for analysts particularly familiar with some provinces. For example, the authorities of OPO can compare it to other provinces searching for possible improvements. They can note that OPO is robustly worse than PDL or LBU and robustly better than SL.