# Robust Additive Value-Based Efficiency Analysis
## 1 Introduction
These preferences form the linear weight restrictions between factor categories at the same level. The benefits of using a hierarchical structure have been explored in MCDA. The example methods that handle such a decomposition include the Analytical Hierarchy Process (AHP) [13], the Multiple Criteria Hierarchy Process (MCHP) [14; 15], and ELECTRES-III-H [16]. In the DEA context, the first attempt was made with a two-layer nonlinear model [17] and its linear counterpart [18]. Then, [PERSON]. [19] proposed a multiple-layer DEA model (MLDEA) handling an arbitrary number of levels of inputs and outputs. Further, MLDEA was combined with AHP to consider relative priorities of various factors, mainly in the scenarios where DEA is used as a mathematical tool for constructing so-called composite indicators [20; 21]. Finally, the latter approach was generalized to the setting of Network and Fuzzy DEA [22]. The above-mentioned DEA models require inputs and outputs to be considered in separate hierarchies. We fill this research gap by admitting a single multiple-layer hierarchical structure containing inputs and outputs. In this way, the properly defined efficiency can be analyzed in each hierarchy node. Second, in the proposed framework, we go beyond classifying the DMUs only into efficient and inefficient, as in the original value-based efficiency analysis [8]. This is attained by verifying the robustness of efficiency results observable for the entire space of feasible input and output weights. We focus on three perspectives: distances to the efficient DMU, ranks, and pairwise preference relations. For each of them, we compute the exact (necessary, possible, and extreme) outcomes by solving dedicated mathematical programming models. Moreover, we estimate the distribution of results using Monte Carlo simulations. The proposed framework is inspired by [23], being adapted to the multiple-level hierarchy value-based efficiency analysis. The formulations of dedicated procedures and types of considered results are similar to those considered in [24]. However, we admit the stakeholders studying the stability of efficiency outcomes in each hierarchy node instead of forcing them to consider all inputs and outputs simultaneously. In this regard, the main challenge is adequately handling the indicator weights considered at different hierarchy levels. We also formulate the properties of the exact efficiency outcomes observed along the hierarchy tree. Typically, they relate the results observed in all children nodes of some more general category to the outcomes obtained in the parent node. Hence, they help understand the evolution of necessary, impossible, and extreme conclusions. Third, we apply the proposed framework to a case study concerning healthcare. As noted in [25], the efficiency of using resources to ensure a decent level of healthcare has become one of the most critical public policy issues in recent decades. Its assessment can be conducted from the perspective of the entire system or individual organization (e.g., hospitals). A detailed review of the applications of efficiency analysis in healthcare can be found in [26], and a detailed description of the healthcare system in Poland is given in [25]. We analyzed nine indicators capturing the quality of healthcare systems in sixteen Polish ovivodeships. The indicators were grouped under three main categories: inhabitants' health improvement, financial management, and consumer satisfaction. We elicited the preferences in the form of marginal value functions for all inputs and outputs and weight constraints. We report the results given a comprehensive efficiency index encompassing all relevant viewpoints and the three subproblems that allow for an understanding of each ovivodeship's strong and weak points. The paper's three significant contributions, along with their essential aspects, are summarized in Figure 1.  
The paper's remainder is organized as follows. Section 2 describes an additive value-based efficiency model.