# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
### The Category of Inhabitants' Health Improvement  
In this section, we focus on the results attained when considering only inputs and outputs from the inhabitants' health improvement category. In addition, we emphasize the differences with respect to the comprehensive level. Figure 9 presents the extreme and expected distances to the best unit for all considered vovodeships. WLKP is the only efficient province given this category, so its distance to  
Figure 8: The Hasse diagram of the necessary efficiency preference relation in the comprehensive analysis of the healthcare system. \begin{table}
\begin{tabular}{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 & 0.835 & 1 & 0 & 0 & 0.690 & 0.962 & 0.852 & 1 & 1 & 0.989 & 1 & 0.791 & 1 & 0.994 \\ POM & 1 & 1 & 1 & 1 & 0.870 & 0.625 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ WM & 0.165 & 0 & 1 & 1 & 0 & 0 & 0.286 & 0.609 & 0.630 & 1 & 1 & 0.890 & 0.918 & 0.960 & 0.967 \\ PDL & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0.760 & 0.748 & 0.006 & 0.095 & 0 & 0.039 & 0.146 \\ LBU & 1 & 0.130 & 1 & 1 & 1 & 0.181 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ WLKP & 1 & 0.375 & 1 & 1 & 0.819 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ KP & 0.310 & 0 & 0.714 & 1 & 0 & 0 & 1 & 0.758 & 0.805 & 1 & 1 & 0.953 & 0.974 & 0.651 & 0.996 & 0.990 \\ MAZ & 0.038 & 0 & 0.391 & 1 & 0 & 0.242 & 1 & 0.534 & 1 & 1 & 0.850 & 0.981 & 0.378 & 0.993 & 0.958 \\ LBL & 0.148 & 0 & 0.370 & 1 & 0 & 0 & 0.195 & 0.466 & 1 & 1 & 1 & 0.879 & 0.909 & 0.295 & 0.988 & 0.988 \\ DSL & 0 & 0 & 0 & 0.240 & 0 & 0 & 0 & 0 & 0 & 1.0487 & 0.001 & 0.016 & 0 & 0.052 & 0.061 \\ OPO & 0 & 0 & 0 & 0.252 & 0 & 0 & 0 & 0 & 0.513 & 1 & 0.002 & 0.023 & 0 & 0 & 0.020 \\ LDZ & 0.011 & 0 & 0.110 & 0.994 & 0 & 0 & 0.047 & 0.150 & 0.121 & 0.999 & 0.998 & 1 & 0.745 & 0.051 & 0.757 & 0.849 \\ SL & 0 & 0 & 0.082 & 0.905 & 0 & 0 & 0.026 & 0.019 & 0.984 & 0.977 & 0.255 & 1 & 0.049 & 0.489 & 0.524 \\ SW & 0.209 & 0 & 0.519 & 1 & 0 & 0 & 0.349 & 0.622 & 0.705 & 1 & 1 & 0.949 & 0.951 & 1 & 0.997 & 0.992 \\ MLP & 0 & 0 & 0.040 & 0.961 & 0 & 0 & 0.004 & 0.007 & 0.102 & 0.948 & 1 & 0.243 & 0.511 & 0.003 & 1 & 0.568 \\ PKR & 0.006 & 0 & 0.033 & 0.854 & 0 & 0 & 0.010 & 0.042 & 0.012 & 0.939 & 0.980 & 0.151 & 0.476 & 0.008 & 0.432 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Pairwise efficiency outranking indices for pairs of Polish vovodeships in the comprehensive analysis of the healthcare system. the best unit always equals zero. Hence, POM and LBU lose the status of efficient units. However, these two voivodeships and ZPM have relatively low distances to the best one: ZPM (\(d^{H}\in[0.065,0.146]\)), POM (\(d^{H}\in[0.130,0.180]\)), and LBU (\(d^{H}\in[0.114,0.246]\)). In general, the widths of distance intervals are notably more precise than when considering all relevant factors jointly. They are also more diverse, encompassing a greater range. In particular, seven provinces have maximal distances greater than 0.6, with three even exceeding the threshold of 0.7. At the comprehensive level, this was not observed for any voivodeship. Among the inefficient provinces, ZPM can be considered the best, as it has the lowest distances to the best unit in both optimistic and pessimistic settings. Moreover, its best expected distance (0.109) is twice lower than at the comprehensive level, letting it overtake POM and LBU, which were judged efficient in the hierarchy's root. The other twelve inefficient provinces are significantly worse. The least favorable among them is OPO, with the distance to the best unit in its optimistic scenario equal to 0.73 and an expected distance of 0.754. In the average case, OPO is directly preceded by MLP (0.659) and PKR (0.652). Note that at the comprehensive level, the worst maximal distances were attained by SL, DSL, and PDL. They all prove slightly better regarding inhabitants' health improvement results.