# Robust Additive Value-Based Efficiency Analysis
## 3 A Hierarchical Structure of Inputs and Outputs
### Exact Methods
\[\sum_{l\in A_{i}^{(l)}}\hat{w}_{q}^{c_{i}^{(l)}}u_{q}(DMU_{o})-\sum_{l\in A _{i}^{(l)}}\hat{w}_{q}^{c_{i}^{(l)}}u_{q}(DMU_{k})\leq C(1-b_{k}),\text{ for }k=1,\ldots,K;k\
eq o,\] \[b_{k}\in\{0,1\},\text{ for }k=1,2,\ldots,K;k\
eq o, \tag{9}\]  
Note that when \(b_{k}=1\), \(k=1,\ldots,K\), the respective constraint ensures that \(DMU_{k}\) is ranked no worse than \(DMU_{o}\) since \(E_{o}^{c_{i}^{(l)}}\leq E_{k}^{c_{i}^{(l)}}\). This deteriorates the worst rank of \(DMU_{o}\) by one. When \(b_{k}=0\), the respective constraint is satisfied for all variable values; hence, it relaxed. The third perspective focuses on the pairwise comparisons between DMUs using two relations: necessary and possible. Given the uncertainty of selecting a specific weight vector, the necessary relation can be considered robust. Specifically, \(DMU_{o}\) is necessarily preferred to \(DMU_{k}\), given category \(c_{i}^{(l)}\) (\(DMU_{o}\succ_{E}^{N,c_{i}^{(l)}}DMU_{k}\)), when \(DMU_{o}\) is not worse at level \(c_{i}^{(l)}\) in terms of efficiency than \(DMU_{k}\) for all feasible weight vectors. Its truth for pair \((DMU_{o},DMU_{k})\) and category \(c_{i}^{(l)}\) can be verified using the following model:  
\[\text{Minimize }d_{o,k}^{c_{i}^{(l)}},\;\;s.t.\;\sum_{q\in A_{i}^{(l)}}\hat{w} _{q}^{c_{i}^{(l)}}u_{q}(DMU_{o})-\sum_{q\in A_{i}^{(l)}}\hat{w}_{q}^{c_{i}^{(l) }}u_{q}(DMU_{k})\leq d_{o,k}^{c_{i}^{(l)}}\text{ and }\mathcal{W}. \tag{10}\]  
Its optimal solution \(d_{o,k,*}^{c_{i}^{(l)}}\) is equal to the minimal difference between efficiencies of \(DMU_{o}\) and \(DMU_{k}\) observable in the set of feasible weights \(\mathcal{W}\), given category \(c_{i}^{(l)}\). If \(d_{o,k,*}^{c_{i}^{(l)}}\geq 0\), then for all feasible weights \(E_{o}^{c_{i}^{(l)}}\geq E_{k}^{c_{i}^{(l)}}\), and hence \(DMU_{o}\succ_{E}^{N,c_{i}^{(l)}}DMU_{k}\). Otherwise, \(\
eg(DMU_{o}\succ_{E}^{N,c_{i}^{(l)}}DMU_{k})\) because there is at least one feasible weight vector, such that \(E_{o}^{c_{i}^{(l)}}<E_{k}^{c_{i}^{(l)}}\). Furthermore, \(DMU_{o}\) is possibly preferred to \(DMU_{k}\), given category \(c_{i}^{(l)}\) (\(DMU_{o}\succ_{E}^{P_{c_{i}^{(l)}}}DMU_{k}\)), when \(DMU_{o}\) is not worse at level \(c_{i}^{(l)}\) in terms of efficiency than \(DMU_{k}\) for at least one feasible weight vector. Its truth for pair \((DMU_{o},DMU_{k})\) and category \(c_{i}^{(l)}\) is verified using the following model:  
\[\text{Maximize }d_{o,k}^{c_{i}^{(l)}},\;\;s.t.\;\;\sum_{q\in A_{i}^{(l)}}\hat{w} _{q}^{c_{i}^{(l)}}u_{q}(DMU_{o})-\sum_{q\in A_{i}^{(l)}}\hat{w}_{q}d_{q}^{c_{ i}^{(l)}}(DMU_{k})\geq d_{o,k}^{c_{i}^{(l)}}\text{ and }\mathcal{W}. \tag{11}\]  
Its optimal solution \(d_{o,k}^{c_{i}^{(l)},*}\) is equal to the maximal difference between efficiencies of \(DMU_{o}\) and \(DMU_{k}\) observable in the set of feasible weights \(\mathcal{W}\), given category \(c_{i}^{(l)}\). If \(d_{o,k}^{c_{i}^{(l)},*}\geq 0\), then for at least one feasible weight \(E_{o}^{c_{i}^{(l)}}\geq E_{k}^{c_{i}^{(l)}}\), and hence, \(DMU_{o}\succ_{E}^{P_{c_{i}^{(l)}}}DMU_{k}\). Otherwise, \(\
eg(DMU_{o}\succ_{E}^{P_{c_{i}^{(l)}}}DMU_{k})\) because there is no feasible weight vector, such that \(E_{o}^{c_{i}^{(l)}}\geq E_{k}^{c_{i}^{(l)}}\). The relevant properties of the exact robust results given the hierarchical structure are presented in Appendix A. The formulations of example mathematical programming models that support understanding the general formulations are given in Appendix B.