# Robust Additive Value-Based Efficiency Analysis
## 3 A Hierarchical Structure of Inputs and Outputs  
The DMUs consume \(m\) inputs \(x\) and produce \(n\) outputs \(y\). To simplify the notation, we aggregate all inputs and outputs into a single set of factors \(\mathbf{f}=\mathbf{x}\cup\mathbf{y}=\{f_{1},f_{2},\ldots,f_{Q_{0}}\}\). Set \(\mathbf{f}\) forms level 0 of the hierarchy. These factors are grouped into \(Q_{1}\) categories of the first level, named \(\mathbf{C^{(1)}}=\{c_{1}^{(1)},c_{2}^{(1)},\ldots,c_{Q_{1}}^{(1)}\}\). Analogously, the first-level categories can be grouped into second-level categories, forming a set \(\mathbf{C^{(2)}}=\{c_{1}^{(2)},c_{2}^{(2)},\ldots,c_{Q_{2}}^{(2)}\}\), etc. The entire structure contains \(L\) levels. In the last (\(L\)-th) level, there is only a single category (\(c_{1}^{(L)}\)), called a _root_. From the mathematical viewpoint, the factors and categories form a tree (see Figure 2). The set of all nodes in the tree (factors and categories) is denoted by \(\mathbf{N}=f\cup C^{(1)}\cup C^{(2)}\cup\ldots\cup C^{(L)}\). For each node \(t\in\mathbf{N}\setminus\{root\}\), we define its parent \(p(t)\) as a category in which it is directly contained. The set of direct children of category \(c_{i}^{(l)}\) is marked as \(ch(c_{i}^{(l)})=\{t\in\mathbf{N}\text{:}p(t)=c_{i}^{(l)}\}\). The set of indirect children of category \(c_{i}^{(l)}\) contains all direct children of \(c_{i}^{(l)}\) (\(ch(c_{i}^{(l)})\)) and their direct and indirect children until reaching the tree's leaves. For each category at hierarchy level \(c_{i}^{(l)}\), we define set \(A_{c_{i}^{(l)}}\) as a subset of \(\mathbf{f}\) (inputs and outputs), which are the indirect children of \(c_{i}^{(l)}\). In particular, all factors are indirect children of the root category, i.e., \(A_{c_{1}^{(l)}}=\mathbf{f}\). On the contrary, for an elementary factor \(f\), \(A_{f}\) is a singleton, i.e., \(A_{f}=\{f\},f\in\mathbf{f}\). To maintain the spirit of DEA, for each category \(c_{i}^{(l)}\), the respective set of factors (\(A_{c_{i}^{(l)}}\)) needs to contain at least one input and one output, i.e., \(A_{c_{i}^{(l)}}\cap\mathbf{x}\
eq\emptyset\) and \(A_{c_{i}^{(l)}}\cap\mathbf{y}\
eq\emptyset\), for \(l=1,2,\ldots,L,\ i=1,2,\ldots,Q_{l}\). Figure 2: A hierarchical structure of inputs and outputs. To illustrate the notation used in the paper, we will describe it using a simple hierarchy of factors in Figure 3. This example involves two inputs (\(\mathbf{x}=\{i_{1},i_{2}\}\)) and two outputs (\(\mathbf{y}=\{o_{1},o_{2}\}\)). The set of factors \(\mathbf{f}\) containing all elements from \(\mathbf{x}\) and \(\mathbf{y}\) is \(\mathbf{f}=\{i_{1},o_{1},i_{2},o_{2}\}\). Overall, there are four factors (\(Q_{0}=4\)), two first-level categories (\(\mathbf{C_{1}}=\{C_{1}^{(1)},C_{2}^{(1)}\}\); \(Q_{1}=2\)), and one root corresponding to a second-level category (\(\mathbf{C_{2}}=\{C_{1}^{(2)}\}\); \(Q_{2}=1\)). The hierarchy contains two levels of categories (\(L=2\)). The parent category for input \(i_{1}\) is \(C_{1}^{(1)}\) (\(p(i_{1})=C_{1}^{(1)}\)). For \(C_{1}^{(1)}\) and \(C_{2}^{(1)}\), the parent category is \(C_{1}^{(2)}\), i.e., \(p(C_{1}^{(1)})=p(C_{2}^{(1)})=C_{1}^{(2)}\). The considered sets of factors \(A_{C_{i}^{(k)}}\) for example categories are the following: \(A_{i_{1}}=\{i_{1}\}\), \(A_{C_{1}^{(1)}}=\{i_{1},o_{1}\}\), \(A_{C_{2}^{(1)}}=\{i_{2},o_{2}\}\), and \(A_{C_{1}^{(2)}}=\mathbf{f}=\{i_{1},i_{2},o_{1},o_{2}\}\). Given a hierarchy of relevant factors and categories, we assign weight \(w_{t}\) to each node \(t\) except the root. Moreover, we admit specifying the linear constraints for these weights at each hierarchy level. Factors or categories involved in a single constraint must have a common parent. For example, for the considered hierarchy, the constraint can take the form \(w_{i_{1}}\leq 2\cdot w_{o_{1}}\) or \(w_{c_{1}^{(1)}}\geq w_{c_{2}^{(1)}}\), as \(i_{1}\) and \(o_{1}\) or \(C_{1}^{(1)}\) and \(C_{2}^{(1)}\) have the same parent.