# Robust Additive Value-Based Efficiency Analysis
## 6 Conclusions  
This paper introduce a novel framework for robustness analysis in the context of additive value-based efficiency analysis with a hierarchical structure. It admits a multiple-layer organization of relevant factors from the most general to the most detailed ones while tolerating both inputs and outputs in the same node. We accept the linear weight restrictions concerning the importance and trade-offs between various factors or subcategories with the common predecessor in the hierarchy. The results can be considered in each hierarchy node, letting the analyst view the comprehensive outcomes and draw conclusions for the subproblems where the relevant factors are limited to concise subsets of inputs and outputs reflecting a particular perspective. The proposed framework can be used in the standard efficiency analysis and the decision contexts requiring the consideration of composite indicators. We derived the results by considering three perspectives: score-based distances to the efficient unit, ranks, and pairwise preference relations. For each of them, we proposed a pair of methods. The first group was based on mathematical programming, offering the exact, extreme outcomes that can be attained in the set of feasible input/output weights. The other group was based on Monte Carlo simulations, providing the distribution of efficiency outcomes through stochastic acceptabilities that estimate the share of the weight subspaces confirming a given result. These approaches are complementary because the exact outcomes often need more conclusiveness, whereas the stochastic indices--even if approximated with high accuracy--may fail to capture some extreme results. We illustrated the framework's applicability by assessing the quality of the healthcare system of sixteen Polish voyodeships. The analysis included nine indicators of differentnatures and concerned four levels. The comprehensive results were based on all relevant characteristics considered jointly, while the three subproblems captured the perspectives of inhabitants' health improvement, financial management, and consumer satisfaction. We reported three provinces--Pomorskie (POM), Wielkopolskie (WLKP), and Lubuskie (LBU)--as the most efficient ones. We presented their strong and weak points by referring to the results in all hierarchy nodes. Moreover, we discussed the practical usefulness of the robustness analysis in terms of managerial implications. A.L.-K.: conceptualization, methodology, software, investigation, data curation, writing--original draft, visualization; M.K. : conceptualization, methodology, validation, formal analysis, investigation, writing--original draft, supervision, project administration, funding acquisition; W.M. : conceptualization, software, visualization, writing--revision. All authors have read and agreed to the published version of the manuscript. [PERSON] is grateful for the support from the Polish Ministry of Education and Science (grant no. 0311/SBAD/0735). [PERSON] acknowledges support from the Polish National Science Center under the SONATA BIS project (grant no. DEC-2019/34/E/HS4/00045). Not applicable. The complete research data produced within the study is contained within this article. The authors declare no conflict of interest. The following abbreviations are used in this manuscript:  
AHP Analytical Hierarchy Process  
CE Cross-efficiency  
DEA Data Envelopment Analysis  
DMU Decision Making Unit  
LP Linear Programming  
MAVT Multi-Attribute Value Theory  
MCDA Multiple Criteria Decision Analysis  
MCHP Multiple Criteria Hierarchy Process  
MILP Mixed-Integer Linear Programming  
SE Super-efficiency  
This section presents the relevant properties of the exact robust results given the hierarchical structure. In particular, when the minimal distances in all children nodes of some more general category are equal to zero, the minimal distance in this category must also be zero.