# Robust Additive Value-Based Efficiency Analysis
## 3 A Hierarchical Structure of Inputs and Outputs
### Exact Methods  
In this section, we present the mathematical programming models that determine the exact robust results. These include extreme (the most and the least advantageous), necessary (observable for all feasible weight vectors), and possible (holding for at least one feasible weight vector) conclusions. Let us first focus on verifying the stability of distances to the efficient unit. The best (minimal) distance \(\hat{d}_{s,\theta}^{c_{i}^{(l)}}\) for \(DMU_{o}\) considering category \(c_{i}^{(l)}\) can be computed by solving the following model:  
\[\text{Minimize }\hat{d}_{o}^{c_{i}^{(l)}}\]  
s.t. \[\left. \begin{array}{l}\sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q}^{c_{i}^{(l)}}u_{q}( DMU_{k})-\sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q}^{c_{i}^{(l)}}u_{q}(DMU_{o})\leq d _{o}^{c_{i}^{(l)}}\text{, for }k=1,\ldots,K,\\ d_{o}^{c_{i}^{(l)}}\geq 0,\\ \sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q}^{c_{i}^{(l)}}=1,\\ \hat{w}_{q}\geq 0,\ q\in A_{c_{i}^{(l)}},\\ w_{t}=\frac{\sum_{f\in A_{q}}\hat{w}_{f}^{c_{i}^{(l)}}}{\sum_{f\in A_{p(t)}} \hat{w}_{f}^{c_{i}^{(l)}}}\in S_{w}\text{, for }t\in N\setminus\{root\}.\end{array}  \right\}\mathcal{W} \tag{6}\]  
Similarly to the standard efficiency analysis, \(DMU_{o}\) with \(d_{*,o}^{c_{i}^{(l)}}=0\) is deemed efficient, given category \(c_{i}^{(l)}\), while \(d_{*,o}^{c_{i}^{(l)}}>0\) implies inefficiency. To compute the worst (maximal) distance \(d_{o}^{*,c_{i}^{(l)}}\) for \(DMU_{o}\), given category \(c_{i}^{(l)}\), we solve the following Mixed-Integer Linear Programming (MILP) model:  
\[\text{Maximize }d_{o}^{c_{i}^{(l)}}\] s.t. \[\left. \begin{array}{l}\sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q}^{c_{i}^{(l)} }\;u_{q}(DMU_{k})-d_{o}^{c_{i}^{(l)}}\geq\sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q }^{c_{i}^{(l)}}u_{q}(DMU_{o})-C(1-b_{k})\text{, for }k=1,\ldots,K,\\ \sum_{k=1,\ldots,K}b_{k}=1,\\ b_{k}\in\{0,1\}\text{, for }k=1,2,\ldots,K,\\ d_{o}^{c_{i}^{(l)}}\geq 0,\\ \mathcal{W},\end{array} \right. \tag{7}\]  
where \(C\) is a large positive constant. The above model uses binary variables \(b_{k}\), \(k=1,\ldots,K\), to ensure that \(d_{o}^{c_{i}^{(l)}}\) is equal to the efficiency difference between \(DMU_{o}\) and some \(DMU_{k}\), \(k=1,\ldots,K\), for which \(b_{k}=1\). Maximizing \(d_{o}^{c_{i}^{(l)}}\) guarantees that we obtain the greatest possible difference observable in the set of feasible weights \(\mathcal{W}\). Note that when \(b_{k}=0\), the respective constraint is satisfied for all possible variable values; hence, it is relaxed. The second perspective concerns the bounds of efficiency ranks attained by \(DMU_{o}\). To find the best (minimal) rank \(R_{*,o}^{c_{i}^{(l)}}\) of \(DMU_{o}\), given category \(c_{i}^{(l)}\), we minimize the number of other DMUs with greater efficiencies than \(E_{o}^{c_{i}^{(l)}}\):  
\[\text{Minimize }1+\sum_{k=1,\ldots,K;\ k\
eq o}b_{k}\] s.t. \[\left. \begin{array}{l}\sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q}^{c_{i}^{(l )}}\;u_{q}(DMU_{k})-\sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q}^{c_{i}^{(l)}}u_{q}( DMU_{o})\leq Cb_{k}\text{, for }k=1,\ldots,K;k\
eq o,\\ b_{k}\in\{0,1\}\text{, for }k=1,2,\ldots,K;\ k\
eq o,\\ \mathcal{W}.\end{array} \right\} \tag{8}\]  
Note that when \(b_{k}=0\), \(k=1,\ldots,K\), the respective constraint ensures that \(DMU_{o}\) is ranked not worse than \(DMU_{k}\) since \(E_{k}^{c_{i}^{(l)}}\leq E_{o}^{c_{i}^{(l)}}\). When \(b_{k}=1\), \(DMU_{k}\) is ranked better than \(DMU_{o}\), deteriorating its best rank by one. To obtain the worst (maximal) rank \(R_{o}^{*,c_{i}^{(l)}}\) for \(DMU_{o}\), given category \(c_{i}^{(l)}\), we maximize the number of DMUs with the efficiencies not worse than \(E_{o}^{c_{i}^{(l)}}\) :  
\[\text{Maximize }1+\sum_{k=1,\ldots,K;\;k\
eq o}b_{k}\] s.t.