# Robust Additive Value-Based Efficiency Analysis
## 1 Introduction  
Data Envelopment Analysis (DEA) measures the relative efficiency of Decision Making Units (DMUs) that convert inputs to outputs. It was originally proposed by [PERSON] et al. [1] as a nonparametric approach, making no assumptions about the production frontier or the weights assigned to the various factors relevant to the analysis. To assess the efficiency of DMUs, they are compared to the best-practice frontier determined by the group of units with the most favorable input-output performance. The traditional methods divide the units into efficient ones, i.e., those on the efficient frontier, and inefficient ones, i.e., those below the frontier. Due to its versatility, DEA has been widely used in various areas such as management, economics, agriculture, education, healthcare, and logistics [2]. The recent example applications concerned the assessment of public administration [3] and the urban rail transit network [4]. Since its first formulation, DEA has been extended in multiple ways [5, 6]. For example, various efficiency models have been introduced to admit static or dynamic analysis or handle constant or variable returns to scale. In particular, an additive model was formulated to guarantee that the units it indicates as efficient satisfy this property in [PERSON]'s sense [7]. However, this model has also been criticized for assuming equal weights of all factors, vulnerability to the factors' scale differences, and nonintuitive interpretation of the efficiency scores. These drawbacks have motivated the development of an additive value-based efficiency analysis [8, 9], inspired by Multi-Attribute Value Theory (MAVT) [10]. This model transforms the input and output values using the marginal functions. Such per-criterion components are aggregated into a comprehensive efficiency measure with an additive value model incorporating weights assigned to various factors. The units thatattain the greatest comprehensive value for at least one feasible weight vector are deemed efficient. Such an analysis is insensitive to scale problems due to applying value functions with a common scale. Moreover, the efficiency scores have an intuitive interpretation built on the notion of \"min-max regret\". Note that the hybrid methods combining ideas from DEA and Multiple Criteria Decision Analysis (MCDA) have become more and more popular in recent years (see, e.g., [11; 12]). This paper contributes to the literature concerning an additive value-based efficiency analysis in a three-fold way. This methodology handles only flat structures of inputs and outputs considered at the same level, without subcategories [8]. Hence, our first contribution consists of adjusting it to handle hierarchical structures of factors used to assess the performance of DMUs. This is useful in real-world decision analysis for a few reasons. First, it helps to structure inputs and outputs logically and systematically. The higher-level factors are more general, whereas those at lower hierarchy levels are more specific. Moreover, when new information becomes available, the hierarchy can be easily modified or updated, allowing it to handle evolving decision problems. Second, a hierarchical decomposition of factors allows for the breaking down of complex problems into manageable, coherent pieces representing different levels of abstraction. By analyzing the efficiency at various levels of the hierarchy, it is possible to understand the strengths and weaknesses of DMUs and explain the comprehensive results taking into account their evolution along the hierarchy. Third, a hierarchical structure of factors makes efficiency analysis more transparent, flexible, and adaptable. In particular, we support the trade-off analysis, where weights can be associated with lower and higher-level categories of factors, and hence, their relative and absolute impact can be controlled more easily. In this regard, we incorporate the preferences elicited at each hierarchy level into the analysis.