# Robust Additive Value-Based Efficiency Analysis
## 3 A Hierarchical Structure of Inputs and Outputs
On the contrary, the example constraint \(w_{i_{1}}\leq w_{i_{2}}\) is not allowed, because \(i_{1}\) and \(i_{2}\) have different parents (\(p(i_{1})\
eq p(i_{2})\)). The space of weight vectors that meet these restrictions is denoted by \(S_{w}\). To introduce weight restrictions, we consider additional variables (\(\hat{w}_{q}\)), representing the aggregated weights of elementary factors in the hierarchy. They are defined as the products of all weights on the path from the analyzed category (\(c_{i}^{(l)}\)) at the hierarchy level \(l\) to the analyzed factor \(f\):  
\[\hat{w}_{q}^{c_{i}^{(l)}}=w_{q}\cdot\prod_{t=1,\ldots,l-1\wedge t\in c_{i}^{( l)}\wedge f\in A_{t}}w_{t}. \tag{3}\]  
For factor \(i_{1}\) in the considered example, when taking into account the root category (\(C_{1}^{(2)}\)), the above formula takes the following form \(\hat{w}_{i_{1}}^{C_{1}^{(2)}}=w_{i_{1}}\cdot w_{C_{1}^{(1)}}\), and when considering category \(C_{1}^{(1)}\), it is expressed as follows \(\hat{w}_{i_{1}}^{C_{1}^{(1)}}=w_{i_{1}}\). We analyze the efficiency of \(DMU_{o}\) in each node of the hierarchy. For category \(c_{i}^{(l)}\), such efficiency is defined as follows:  
\[E_{o}^{c_{i}^{(l)}}=\sum_{q\in A_{c_{i}^{(l)}}}\hat{w}_{q}^{c_{i}^{(l)}}\cdot u _{q}(DMU_{o}). \tag{4}\]  
Figure 3: An example hierarchical structure of inputs and outputs. The true weights assigned to each hierarchy category \(c_{i}^{(l)}\) from the set of the indirect children of the analyzed category \(c_{j}^{(k)}\) are defined as the ratio of the sum of weights of indicators contained in this category and the sum of weights of indicators in the parent category:  
\[w_{c_{i}^{(l)}}^{c_{i}^{(l)}}=\frac{\sum_{f\in A_{c_{i}^{(l)}}}\hat{w}_{f}^{c_{ j}^{(k)}}}{\sum_{f_{p}\in A_{p(c_{i}^{(l)})}}\hat{w}_{f_{p}}^{c_{j}^{(l)}}}. \tag{5}\]  
Note that the value of weight \(w_{c_{i}^{(l)}}^{c_{i}^{(k)}}\) is always the same, regardless of the considered category (\(c_{j}^{(k)}\)), so we replace symbol \(w_{c_{i}^{(l)}}^{c_{i}^{(l)}}\) with \(w_{c_{i}^{(l)}}\). For example, when considering the root category (\(C_{1}^{(2)}\)), the weight \(w_{i_{1}}\) of indicator \(i_{1}\) in the considered example can be calculated as \(w_{i_{1}}=\frac{\hat{w}_{i_{1}}^{C_{1}^{2}}}{\hat{w}_{i_{1}^{C_{1}^{2}}}+\hat {w}_{i_{1}}^{C_{1}^{2}}}\), whereas the weight \(w_{C_{2}^{(1)}}\) of category \(C_{2}^{(1)}\) is \(w_{C_{2}^{(1)}}=\frac{\hat{w}_{i_{1}}^{C_{2}^{(2)}}+\hat{w}_{i_{2}}^{C_{2}^{(2 )}}}{\hat{w}_{i_{1}}^{C_{1}^{(2)}}+\hat{w}_{i_{1}}^{C_{1}^{(2)}}+\hat{w}_{i_{2} }^{C_{2}^{(2)}}}\). Robustness Analysis for Additive Value-Based Efficiency Analysis with a Hierarchical Structure of Factors  
The standard value-based efficiency model verifies if each DMU is efficient. Such an analysis builds on the weight vector that is the most advantageous for a given DMU, allowing it to minimize the distance from some efficient DMU. In this section, we introduce a suite of methods that investigate the robustness of efficiency outcomes given all feasible weights. They can be divided into two groups. First, the exact approaches use mathematical programming to find the extreme outcomes for each DMU. In turn, the probabilistic methods estimate the stochastic acceptability indices based on Monte Carlo simulations, reflecting the distributions of possible results. Each group concerns three relevant viewpoints: distances to the efficient unit, ranks, and pairwise preference relations. In what follows, we present the approaches that are flexible enough to determine the relevant results in each hierarchy node.