# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
### Comprehensive Evaluation of the Quality of Healthcare Systems  
In this section, we discuss the results of the comprehensive assessment of Polish vovodeships, taking into account all nine indicators. Figure 6 presents the extreme and expected distances to the best unit for each analyzed province. Three vovodeships are efficient: POM, LBU, and WLKP. POM also attains the lowest maximal distance (0.105), which confirms its most favorable evaluation of the healthcare system for all feasible weights. Moreover, the distances for POM are the most stable, as it is characterized by the narrowest range (\(d^{l,*}-d^{l}_{*}=0.105\)). Among the efficient provinces, WLKP has the worst pessimistic distance to the best province. However, its expected distance is better than that of LBU, meaning that for some weight vectors, WLKP performs worse than LBU, but its efficiency score is closer to the best province on average. The worst provinces in the most and the least favorable scenarios are DSL (\(d^{l}_{*}=0.306\), \(d^{l,*}=0.522\)) and SL (\(d^{l}_{*}=0.251\), \(d^{l,*}=0.526\)). The greatest sensitivity of the distances depending on the selected weight vector is observed for SW, as the width of its distance interval equals 0.355. The analysis of extreme distances can be enriched with the distribution of distances over all feasible weight vectors (see Table 2). The efficient provinces (POM, LBU, and WLKP) are the only ones whose distances were not greater than 0.1 for some samples. When considering these three vovodeships, only for LBU, the distance was greater than 0.1 for some marginal share of weights (0.8%). Hence, these provinces are robustly better than the remaining ones. Among the inefficient units, the most favorable results were attained by ZPM and KP. Figure 4: A hierarchy of indicators considered in the quality evaluation of healthcare systems. Some provinces are characterized by rather stable distance values. For example, for MAZ and WM, for over 95% of weight vectors, the distance is the interval \((0.2,0.3]\). On the contrary, the distance for PKR varies more depending on the chosen weight vector. In this case, positive \(DAII\)s were observed for all buckets between 0.1 and 0.5 with the greatest values for the intervals \((0.3,0.4]\) (\(DAII(PKR,(0.3,0.4])=65.9\%\)) and \([0.2,0.3]\) (\(DAII(PKR,(0.2,0.3])=29.5\%\)). The complete ranking determined by the expected distances (see Figure 6) indicates POM (0.009) and WLKP (0.017) as the best units and DSL (0.408) and OPO (0.409) as the worst. Figure 5: The marginal value functions associated with the inputs and outputs considered in the case study. Figure 6: Extreme and expected distances to the best unit for Polish vivodeships in the comprehensive analysis of the healthcare system. The results of robustness analysis for efficiency ranks are presented in Figure 7 and Table 3. The three efficient ovivodeships attain the first rank in the most favorable scenario. POM and LBU are ranked third in their worst scenario, while WLKP falls fourth in the pessimistic case. These units are also the best, given their expected ranks. In this regard, POM (1.543) is followed by WLKP (1.783) and LBU (2.674). Among the inefficient units, KP is the most advantageous when considering the most favorable ranks (\(R_{*}=2\)). Other inefficient units ranked relatively high in their best scenario are ZPM, WM, LBL, LDZ, SW, MLP, and PKR (\(R_{*}=4\)). However, all inefficient provinces are ranked low in the least favorable scenario. The best maximal rank among them is observed for KP and LBL (\(R^{*}=12\)), while five provinces can be ranked at the bottom (PDL, DSL, OPO, SL, and PKR). Finally, the best expected ranks among inefficient units are attained by ZPM (4.962), KP (5.878), and SW (6.611), while the worst expected positions are associated with DSL (15.166) and OPO (15.195). The analysis of efficiency rank acceptability indices (see Table 3) confirms the superiority of POM over other provinces.