# Robust Additive Value-Based Efficiency Analysis
## 6 Conclusions
For DMU\({}_{o}\) and category \(c_{i}^{(l)}\in\mathbf{N}\setminus\mathbf{f}\), if \(\forall t\in ch(c_{i}^{(l)}):\ d_{*,o}^{t}=0\), then \(\ d_{s,o}^{c_{i}^{(l)}}=0\). Similarly, if the minimal rank of DMU\({}_{o}\) in all children nodes of some more general category is 1 (\(K\)), then it needs to be ranked first (last) in this category in the best case. For DMU\({}_{o}\) and category \(c_{i}^{(l)}\in\mathbf{N}\setminus\mathbf{f}\), if \(\forall t\in ch(c_{i}^{(l)}):R_{*,o}^{t}=1\) then \(R_{*,o}^{c_{i}^{(l)}}=1\). **Proposition A3**. : _For \(DMU_{o}\) and category \(c_{i}^{(l)}in\mathbf{N}\setminus\mathbf{f}\), if \(\forall t\in ch(c_{i}^{(l)}):R_{*,o}^{t}=K\), then \(R_{*,o}^{c_{i}^{(l)}}=K\)._  
Moreover, if the maximal rank of \(DMU_{o}\) in all children nodes of some more general category is 1, then it needs to be ranked first in this category, even in the worst case. **Proposition A4**. : _For \(DMU_{o}\) and category \(c_{i}^{(l)}\in\mathbf{N}\setminus\mathbf{f}\), if \(\forall t\in ch(c_{i}^{(l)}):R_{o}^{*,t}=1\), then \(R_{o}^{*,c_{i}^{(l)}}=1\)._  
When \(DMU_{o}\) is necessarily preferred to \(DMU_{k}\) in all children nodes of some more general category, then \(DMU_{o}\) must be necessarily preferred to \(DMU_{k}\), given this category. **Proposition A5**. : _For pair \((DMU_{o},DMU_{k})\) and category \(c_{i}^{(l)}\in\mathbf{N}\setminus\mathbf{f}\), if \(\forall t\in ch(c_{i}^{(l)}):DMU_{o}\succ_{E}^{N,t}DMU_{k}\), then \(DMU_{o}\succ_{E}^{N,c_{i}^{(l)}}DMU_{k}\)._  
Note that \(\succ_{E}^{N,c_{i}^{(l)}}\) is a partial preorder (i.e., transitive and reflexive). When \(DMU_{o}\) is not possibly preferred to \(DMU_{k}\) in all children nodes of some more general category, then \(DMU_{o}\) is not possibly preferred to \(DMU_{k}\), given this category. **Proposition A6**. : _For pair \((DMU_{o},DMU_{k})\) and category \(c_{i}^{(l)}\in\mathbf{N}\setminus\mathbf{f}\), if \(\forall t\in ch(c_{i}^{(l)}):\
eg(DMU_{o}\succ_{E}^{P,t}DMU_{k})\), then \(\
eg(DMU_{o}\succ_{E}^{P,c_{i}^{(l)}}DMU_{k})\)._  
Moreover, when \(DMU_{o}\) is necessarily preferred to \(DMU_{k}\) in all children nodes of some more general category except one node for which it is possibly preferred, then \(DMU_{o}\) needs to be possibly preferred to \(DMU_{k}\), given this category. **Proposition A7**. : _For pair \((DMU_{o},DMU_{k})\) and category \(c_{i}^{(l)}\in\mathbf{N}\setminus\mathbf{f}\), if \(\forall t\in ch(c_{i}^{(l)})\setminus a:DMU_{o}\succ_{E}^{N,t}DMU_{k}\)\(\land\)\(DMU_{o}\succ_{E}^{P,a}DMU_{k}\), then \(DMU_{o}\succ_{E}^{P,c_{i}^{(l)}}DMU_{k}\)._  
Note that \(\succ_{E}^{P,c_{i}^{(l)}}\) is negatively transitive and strongly complete. Moreover, the truth of the necessary preference implies the truth of the possible preference (\(\succ_{E}^{N,c_{i}^{(l)}}\subsetneq\succ_{E}^{P,c_{i}^{(l)}}\)). Appendix B Formulations of Example Mathematical Programming Models for Computing the Exact Robust Results  
This section illustrates mathematical programming models for computing the exact robust results. They support understanding the general formulations presented in Section 4. We consider a hierarchical structure involving four indicators as presented in Figure 3. The input and output values of four DMUs and the indicator bounds are shown in Table 1. For simplicity, we assume that the marginal value functions are linear to make them easily computable. For example, for \(DMU_{1}\) and input \(i_{1}\), the marginal value is calculated as \(u_{i_{1}}(DMU_{1})=u_{i_{1}}(9)=\frac{10-9}{10-0}=0.1\). Analogously, for \(DMU_{3}\) and output \(o_{2}\), the value of marginal function is equal to \(u_{o_{2}}(DMU_{3})=u_{o_{2}}(6)=\frac{6-5}{10-5}=0.2\). Moreover, the following weight constraints are considered: \(w_{c_{1}^{(1)}}\geq w_{c_{1}^{(1)}}\), \(w_{c_{2}^{(1)}}\geq 0.1\), \(w_{i_{1}}\geq 0.2\), and \(w_{o_{2}}\geq 0.6\). \[\left.