# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
### The Category of Consumer Satisfaction  
In this section, we present the results of provinces' performance analysis at the level of consumer satisfaction with the healthcare system. To save space, we refer only to the exact and expected results without reporting the complete tables with stochastic acceptabilities.  
Figure 15 presents the extreme and expected distances when considering the satisfaction of consumers. There are five efficient provinces with \(d_{*}^{S}=0\): POM, KP, LBL, SW, and PKR. This is the greatest number for any node in the considered hierarchy. Among them, two provinces, LBL and PKR, have particularly narrow distance intervals, being close to the efficient units regardless of the weight vector (\(d^{S,*}=0.081\) for LBL and \(d^{S,*}=0.057\) for PKR). They are also the best regarding the expected distance (for PKR--\(Ed^{S}=0.005\) and for LBL--\(Ed^{S}=0.029\)). In turn, the maximal (worst) possible distances for SW (0.285) and POM (0.307) are vastly greater, emphasizing their sensitivity to the selection of a particular weight vector. Among the inefficient units, the best minimal (optimistic) distance is achieved by OPO (0.049), and the best maximal (pessimistic) distance is attained by WLKP (0.312). The three provinces that are the worst considering both minimal and maximal distance are PDL (\(d_{*}^{S}=0.454\), \(d^{S,*}=0.556\)), DSL (\(d_{*}^{S}=0.449\), \(d^{S,*}=0.623\)), and ZPM (\(d_{*}^{S}=0.513\  
The extreme and expected efficiency ranks in consumer satisfaction are presented in Figure 16. Among the five efficient provinces, PKR has the narrowest rank interval; in the least favorable scenario, it is ranked third. On the contrary, even if POM is efficient, it can drop even to the tenth position, making its performance the least stable among all vovodeships. LBU is the best inefficient province when it comes to the best rank (\(R_{s}^{s}=3\)), and the best-ranked inefficient units in the pessimistic settings are WM, LBU, and OPO (\(R^{S,*}=9\)). The worst provinces considering both the minimal and maximal ranks are ZPM (\([15,16]\)), PDL (\([14,15]\)), and DSL (\([14,16]\)). Notably, their rank intervals are very narrow. As far as the expected ranks are concerned, the best province is PKR (1.426), followed by LBL (2.249) and KP (3.441). On the other extreme, PDL (14.192), DSL (14.988), and ZPM (15.820) are the least favorable.  
The necessary efficiency preference relation is presented as the Hasse diagram in Figure 17. The provinces form the multiple-level structure, with efficient ones (POM, PKR, LBL, KP, and SW) at the top. Among them, PKR and LBL are necessarily preferred to the most significant number of eleven other provinces, while POM proves robustly better than only six other provinces. Five other vovodeships (WM, LBU, OPO, WLKP, and MLP) are placed in the second level. All of them are necessarily preferred to four or five other provinces. Finally, DSL and ZPM are confirmed as the worst provinces, as they are not necessarily preferred to any other province. In fact, DSL is possibly preferred to only two other units (ZPM and PDL), while ZPM is at least as efficient for at least one weight vector only when compared to DSL.