# Robust Additive Value-Based Efficiency Analysis
## 3 A Hierarchical Structure of Inputs and Outputs
### Simulation-Based Methods  
The results determined with mathematical programming are often insufficiently conclusive. In particular, the difference between extreme distances or ranks may be significant, the necessary relation may be poor, and the possible relation may be very rich. If so, it would be helpful to determine the distribution of results observed for the set of feasible weight vectors. Unfortunately, such distribution cannot be computed exactly. However, using Monte Carlo simulations, we can estimate the share of feasible weight space confirming a particular outcome. Specifically, we use the hit-And-run algorithm to generate a predefined number of weight vector samples [27]. We generate weights for all categories and factors while respecting that the sum of weights of categories or factors with the same parent must be equal to one. In the example problem, the sum of the weights of two categories \(C_{1}^{(1)}\) and \(C_{2}^{(1)}\) must be equal to one (\(w_{C_{1}^{(1)}}+w_{C_{2}^{(1)}}=1\)), and the sums of weights assigned to the elementary indicators in the same category also need to be one (\(w_{i_{1}}+w_{o_{1}}=1\) and \(w_{i_{2}}+w_{o_{2}}=1\)). Moreover, we obey the provided weight constraints for all hierarchy levels. After generating a predefined number of weight samples, we compute the efficiencies of all DMUs for each. This lets us calculate the relevant stochastic acceptability indices estimating the respective shares of feasible weight vectors.  
In what follows, when considering category \(c_{i}^{(l)}\) and referring to weight vectors, we mean the weights assigned to all categories and factors that are direct or indirect children of \(c_{i}^{(l)}\) in the hierarchy. The most interesting stochastic acceptabilities are defined as follows:  
* Distance Acceptability Interval Index (\(DAII^{c_{i}^{(l)}}(DMU_{o},b_{i})\)) for unit \(DMU_{o}\), interval \(b_{i}\subseteq[0,1]\), and category \(c_{i}^{(l)}\) is the share of feasible weight vectors for which \(E_{o}^{c_{i}^{(l)}}\) belongs to \(b_{i}\). Note that all intervals must be disjoint (\(b_{i}\cap b_{j}=\emptyset\), \(i\
eq j\)), and their sum must cover the space of possible distances (\(b_{1}\cup b_{2}\cup\ldots\cup b_{z}=[0,1]\); z--the number of intervals).
* Efficiency Rank Acceptability Index (\(ERAI^{c_{i}^{(l)}}(DMU_{o},r)\)) for unit \(DMU_{o}\) and rank \(r\) is the share of feasible weight vectors for which \(DMU_{o}\) attains \(r\)-th position in the efficiency ranking of all DMUs given category \(c_{i}^{(l)}\).
* Pairwise Efficiency Outranking Index (\(PEO\Gamma^{[l]}(DMU_{o},DMU_{k})\)) for pair (\(DMU_{o}\), \(DMU_{k}\)) and category \(c_{i}^{(l)}\) is the share of feasible weight vectors for which \(DMU_{o}\) is at least as efficiency as \(DMU_{k}\) at level \(c_{i}^{(l)}\), i.e., \(E_{o}^{c_{i}^{(l)}}\geq E_{k}^{c_{i}^{(l)}}\).  
Moreover, we compute the expected distance \(Ed\) to the efficient unit and expected rank \(ER\) for each DMU [28]. This is performed by averaging the distances or ranks observed for all samples. Note that by default, we use uniform distribution for weight sampling. However, the weights can be generated from any predefined distribution, but it is hard to define as it requires in-depth knowledge about the specific application domain.  
In Appendix C, we illustrate the process of computing the stochastic results on a small sample of weight vectors.