# Robust Additive Value-Based Efficiency Analysis
## 5 Case Study concerning Evaluation of Healthcare System in Poland
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Voivodeship** & **Short Name** & \(\mathbf{H_{1}}\) & \(\mathbf{H_{2}}\) & \(\mathbf{H_{3}}\) & \(\mathbf{F_{1}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{S_{1}}\) & \(\mathbf{S_{2}}\) & \(\mathbf{S_{3}}\) \\ \hline Zachodniopomorskie & ZPM & 44.44 & 4.05 & 17.6 & \(-1.5\) & 46.1 & 46.9 & 19.23 & 3.55 & 9.30 \\ Pomorskie & POM & 45.31 & 5.36 & 21.3 & 0.08 & 38.9 & 51.3 & 27.04 & 3.82 & 33.33 \\ Warmirsk\(-\)Mazurskie & WM & 43.34 & 7.11 & 15.7 & \(-1.09\) & 44.8 & 47.8 & 22.19 & 3.72 & 22.73 \\ Podlaskie & PDL & 37.54 & 6.93 & 15.8 & 2.3 & 49.4 & 43.4 & 19.47 & 3.61 & 11.11 \\ Lubuskie & LBU & 50.21 & 5.83 & 18.0 & 1.9 & 42.0 & 51.3 & 18.32 & 3.84 & 13.04 \\ Wikelkopolskie & WLKP & 47.88 & 4.36 & 25.7 & \(-0.3\) & 43.4 & 50.5 & 15.16 & 3.69 & 17.24 \\ Kujawsk\(-\)Pomorskie & KP & 39.90 & 6.80 & 15.7 & 3.8 & 46.5 & 41.8 & 22.36 & 3.77 & 33.33 \\ Mazowieckie & MAZ & 38.59 & 5.32 & 16.7 & \(-1.57\) & 47.4 & 47.8 & 20.26 & 3.62 & 20.56 \\ Lubelskie & LBL & 45.17 & 6.74 & 8.1 & 0.41 & 51.8 & 43.0 & 13.40 & 3.77 & 20.37 \\ Dolnoslaskie & DSL & 45.04 & 7.66 & 13.8 & \(-1.77\) & 50.8 & 43.8 & 24.55 & 3.57 & 17.28 \\ Opolskie & OPO & 33.71 & 6.88 & 10.0 & \(-0.36\) & 44.1 & 40.9 & 23.03 & 3.67 & 35.71 \\ Lodzkie & LDZ & 42.00 & 5.88 & 6.0 & 1.02 & 50.6 & 47.3 & 19.90 & 3.78 & 15.15 \\ Slaskie & SL & 40.70 & 4.54 & 13.9 & \(-1.63\) & 55.5 & 39.1 & 22.02 & 3.75 & 16.88 \\ Swietokrzyskie & SW & 41.14 & 5.79 & 7.8 & 0.17 & 49.5 & 47.2 & 12.44 & 3.57 & 29.17 \\ Malopolskie & MLP & 32.22 & 5.61 & 9.1 & 0.79 & 43.7 & 44.3 & 18.27 & 3.64 & 25.84 \\ Podkarpackie & PKR & 38.73 & 5.76 & 1.9 & \(-4.31\) & 48.0 & 45.8 & 14.39 & 3.84 & 20.51 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of inputs and outputs related to the healthcare systems in sixteen Polish voivodeships [29] (data publicly available at [[https://www.pwc.pl/pl/publikacje/2019/indexs-sprawnosci-ochrony-zdrowia-2018.html](https://www.pwc.pl/pl/publikacje/2019/indexs-sprawnosci-ochrony-zdrowia-2018.html)]([https://www.pwc.pl/pl/publikacje/2019/indexs-sprawnosci-ochrony-zdrowia-2018.html](https://www.pwc.pl/pl/publikacje/2019/indexs-sprawnosci-ochrony-zdrowia-2018.html)), accessed on 22 May 2023). For all factors, we elicited marginal value functions from experts in the healthcare system in Poland. They are provided in Figure 5. They are decreasing for inputs and increasing for outputs. Moreover, they differ in shape. The function is, e.g., close to linear for \(H_{2}\), convex for \(F_{3}\), concave for \(F_{2}\), and S-shaped for \(S_{2}\). Moreover, we incorporated the relative and absolute weight constraints. The category of _inhabitants' health improvement_ is more important than the two other categories. Hence, we introduced the following constraints: \(w_{H}\geq w_{F}\) and \(w_{H}\geq w_{S}\), where \(w_{H}\), \(w_{F}\), and \(w_{S}\) are the weights of the three categories. Finally, we wanted to avoid both the minor and dominating roles of any individual factor or category in the analysis. Hence, we restricted the weight of each category to be not less than 0.2 and the weights of second-level elementary factors to the interval \([0.2,0.5]\). In what follows, we discuss the results attained in the root hierarchy level and for each of the three categories separately.