Decision SupportUncertain Data Envelopment Analysis
Section snippets
Introduction and Motivation
Data envelopment analysis (DEA) is a well established optimization framework to conduct relative performance measurements among a group of decision making units (DMUs). There are numerous reviews of DEA, see, e.g., Cooper, Seiford, and Tone (2007); Emrouznejad, Parker, and Tavares (2008); Liu, Lu, Lu, and Lin (2013); Zhu (2014), and Hwang, Lee, and Zhu (2016); and the concept has found a wide audience in both research and application. The principal idea is to solve an optimization problem for
General Data Envelopment Analysis
We assume the (standard) input oriented model with variable returns-to-scale from among the many DEA formulations, where the efficiency score of DMU is defined by solving the linear program, where e is the vector of ones. In this model, there are M outputs, indexed by m; N inputs, indexed by n; and D DMUs, indexed by i. The matrices Y and X are the nonnegative output and input matrices so that
Uncertain Data Envelopment Analysis
The reliability of a DMU’s efficiency score is jeopardized if the data is erroneous, which points to a desire to accommodate suspect data within a DEA application. Uncertain data fits seamlessly into the paradigm of robust linear optimization, and our overarching model adapts this robust perspective. Each constraint is replaced with a set of constraints which reduces to the original constraint if the uncertainty set is restricted to the singleton, .
Our goal
Configurations of uncertainty
Both the analytical outcomes and the computational tractability of a uDEA problem rely on the type of uncertainty that is being considered and on how the amount of uncertainty is evaluated. Hence an analysis depends on the pair (Ω, m), which defines a configuration.
Definition 6 A configuration of uncertainty, or more simply a configuration, is the pair (Ω, m), where Ω is a universe of possible collections of uncertainty satisfying for all and m is an amount of uncertainty.
A configuration defines
Examples
Consider the three DMUs pictured in Fig. 1 and whose nominal data are listed in Table 1. DMU C is inefficient, and model (1) would scale C’s input of 2 by the efficiency score of 1/2 to identify A as C’s efficient target. The inefficiency of DMU C means that it has an interest in knowing if it is capable under a configuration of uncertainty. We divide the discussion into three examples with different configurations to help explore possible outcomes. This collection
- •
demonstrates a capable,
Traditional DEA as a special case of uncertain DEA
A uDEA problem obviously reduces to its certain DEA progenitor if in which case the optimal solution satisfies With the outer supremum over γ and the inner minimization over are meaningless in the uncertain model (6), and the overhead of the uncertain paradigm is unwarranted with regard to solving the DEA problem. However, the traditional DEA model in (1) is essentially a parametric query that asks, how much do the inputs of the th DMU need to
Solving uncertain DEA problems
Solving a uDEA problem is generally more difficult than is calculating the efficiency score of a DMU. Indeed, even if the configuration is designed to reasonably accommodate efficient calculations, computing γ* necessitates the layering of three optimization problems, which complicates algorithm design. We restrict ourselves here to the case in which the robust DEA problem defining can be efficiently solved as a second-order cone problem, i.e. we assume in our algorithmic development
A case study in radiotherapy
External radiation therapy is one of the major cancer treatments along with surgery and chemotherapy, and about two thirds of all cancer patients undergo a course of radiotherapy. Radiotherapy exploits a therapeutic advantage in which cancerous cells are unable to recover as well as healthy cells from radiation damage. Moreover, radiotherapy has the advantage of delivering near conformal dose distributions to tumors with complex geometries. While radiotherapy is generally regarded as a
Conclusion
We investigated how DEA is affected by uncertain data. We first presented a robust DEA model that defines a robust efficiency score for known uncertainty sets. We then formally showed that an increase in the uncertainty harbored by a collection of uncertainty increases the efficiency score of a DMU. This led to the question of how much uncertainty is needed to classify a DMU as efficient. We introduced the definition of an amount of uncertainty, which allowed us to formulate an optimization
Acknowledgment
The authors thank Emma Stubington for her thoughtful conversations on several of the topics herein. They also thank three anonymous referees for their thorough and suggestive reviews. O. N. acknowledges the support of the NSF grant CMMI-1463489.
References (30)
- et al.
Robust solutions of uncertain linear programs
Operations Research Letters
(1999) - et al.
Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA
Socio-Economic Planning Sciences
(2008) - et al.
A novel method for the evaluation of uncertainty in dose volume histogram computation
International Journal of Radiation Oncology, Biology, Physics
(2008) - et al.
A survey of DEA applications
Omega
(2013) - et al.
Variation in external beam treatment plan quality: An inter-institutional study of planners and planning systems
Practical Radiation Oncology
(2012) - et al.
Stochastic data envelopment analysis
European Journal of Operational Research
(2016) - et al.
Commissioning and quality assurance of computerized planning systems for radiation treatment of cancer
International Atomic Energy Agency Technical Report Series
(2004) - et al.
A new robust DEA model and super-efficiency measure
Optimization
(2017) - et al.
Robust optimization
(2009) - et al.
Robust optimization – methodology and applications
Mathematical Programming
(2002)
Theory and applications of robust optimization
SIAM Review
Nonconvex robust optimization for problems with constraints
INFORMS Journal on Computing
A robust optimization approach to inventory theory
Operations Research
Review IMRT: A review and preview
Physics in Medicine and Biology
Planning study to compare dynamic and rapid arc techniques for postprostatectomy radiotherapy of prostate cancer
Strahlentherapie und Onkologie
Cited by (45)
A novel robust data envelopment analysis with asymmetric uncertainty and an application to National Basketball Association
2024, Expert Systems with ApplicationsData Envelopment Analysis models with imperfect knowledge of input and output values: An application to Portuguese public hospitals
2023, Expert Systems with ApplicationsRobust optimization and its duality in data envelopment analysis
2022, Omega (United Kingdom)A box-uncertainty in DEA: A robust performance measurement framework
2022, Expert Systems with ApplicationsCitation Excerpt :Their method can be applied to evaluate the absolute efficiency score of units for different values of robustness levels in order to rank them. Ehrgott et al. (2018) used the framework of RO to propose a DEA model in case of data uncertainty. They provided a first-order algorithm to solve their model and showed that the optimal solution of it determined the maximum possible efficiency score of a unit.
Robustness of Farrell cost efficiency measurement under data perturbations: Evidence from a US manufacturing application
2021, European Journal of Operational ResearchCitation Excerpt :It was accordingly discussed that the decision maker can catch the maximum possible efficiency of a DMU over all permissible uncertainties, and the minimal amount of uncertainty is required to achieve this efficiency. In this respect, Theorem 2 is in line with Definition 2 of Ehrgott et al. (2018, p. 233) showing that the maximal robust TE measure remains 1 under the polyhedral uncertainty set, and conforming to Proposition 1 of Ehrgott et al. (2018, p. 233), Theorem 3 demonstrates that the robust TE measure is non-decreasing when uncertainties increase. Despite the theoretical aspect, the applicability of these essential properties will be illustrated in an empirical application in Section 6.
New patterns in China's regional green development: An interval Malmquist–Luenberger productivity analysis
2021, Structural Change and Economic Dynamics