data envelopment analysis in operation research

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• Data Envelopment Analysis

Data Envelopment Analysis

Theory and techniques for economics and operations research.

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• Subhash C. Ray , University of Connecticut
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Book description

Using the neo-classical theory of production economics as the analytical framework, this book, first published in 2004, provides a unified and easily comprehensible, yet fairly rigorous, exposition of the core literature on data envelopment analysis (DEA) for readers based in different disciplines. The various DEA models are developed as nonparametric alternatives to the econometric models. Apart from the standard fare consisting of the basic input- and output-oriented DEA models formulated by Charnes, Cooper, and Rhodes, and Banker, Charnes, and Cooper, the book covers developments such as the directional distance function, free disposal hull (FDH) analysis, non-radial measures of efficiency, multiplier bounds, mergers and break-up of firms, and measurement of productivity change through the Malmquist total factor productivity index. The chapter on efficiency measurement using market prices provides the critical link between DEA and the neo-classical theory of a competitive firm. The book also covers several forms of stochastic DEA in detail.

"...none of the books on DEA that are available in the market today examines DEA from an economist's point of view. Subhash Ray's Data Envelopment Analysis is an exception. This book familiarizes the reader with the microeconomic foundations of the various DEA models that are currently available and widely used in the literature. Professor Ray explains the neo-classical production theory behind various DEA models. It is up-to-date and gives an excellent overview of DEA models from both empirical and theoretical points of view. Professor Ray is a leading authority on DEA. Every chapter of the book shows his mastery in the field. I am confident that the researchers at all levels working on DEA will immensely benefit from reading this book." Subal C. Kumbhakar, State University of New York, Binghamton

"Firmly grounded in the economic theory of production, and richly illustrated with a battery of numerical and computer-driven examples, Data Envelopment Analysis achieves both objectives. Professor Ray is a leading researcher in the field, and he has written a valuable book from which scholars and their students will learn a great deal." C.A. Knox Lovell, University of Georgia

"This lucid treatment of data envelopment analysis is the most thorough and extensive available, and unifies economics, operations research, and management science on this topic. The exposition contains plenty for both those wanting a theoretical treatment and for practitioners primarily interested in applications. The comprehensive book fills a critical gap in the literature, and will be highly valued by applied researchers in government, universities, and the private sector interested in measuring economic efficiency and productivity." Dale Squires, National Oceanic and Atmospheric Administration

"Only a deep understanding of the economics of production can provide an Ariadne's thread to find oneas way through the existing literature on efficiency analysis in general and on the DEA method in particular. In this book, Professor Ray offers the reader an enlightening guidance through this maze, thanks to his equal mastery of the economic conceptual foundations of the field and of the mathematical programming techniques that have rendered it widely applicable. The result is a choice of topics that bear on basic aspects of the efficiency issue more than on technical virtuosity. Among these, the way FDH is put in its proper nonparametric production theoretic perspective pleases me most." Henry Tulkens, CORE, Université Catholique de Louvain

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Frontmatter pp i-vi

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Contents pp vii-viii

Preface pp ix-xii, 1 - introduction and overview pp 1-11, 2 - productivity efficiency, and data envelopment analysis pp 12-45, 3 - variable returns to scale: separating technical and scale efficiencies pp 46-81, 4 - extensions to the basic dea models pp 82-110, 5 - nonradial models and pareto–koopmans measures of technical efficiency pp 111-133, 6 - efficiency measurement without convexity assumption: free disposal hull analysis pp 134-158, 7 - dealing with slacks: assurance region/cone ratio analysis, weak disposability, and congestion pp 159-186, 8 - efficiency of merger and breakup of firms pp 187-207, 9 - efficiency analysis with market prices pp 208-244, 10 - nonparametric approaches in production economics pp 245-273, 11 - measuring total productivity change over time pp 274-306, 12 - stochastic approaches to data envelopment analysis pp 307-326, 13 - looking ahead pp 327-328, references pp 329-338, index pp 339-353, full text views.

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Efficiency measurement using data envelopment analysis (dea) in public healthcare: research trends from 2017 to 2022.

1. Introduction

• What is the recent trend of efficiency research using DEA in the public healthcare sector?
• What is the popular selection and development of DEA models to analyze the efficiency of public healthcare?
• What subject has been examined in the recent DEA-based public healthcare research?

2. Search Process

3.1. research purposes & methods, 3.2. dmus & variables, 3.3. regions and analysis subjects, 4. discussion, 5. conclusions, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Jung, S.; Son, J.; Kim, C.; Chung, K. Efficiency Measurement Using Data Envelopment Analysis (DEA) in Public Healthcare: Research Trends from 2017 to 2022. Processes 2023 , 11 , 811. https://doi.org/10.3390/pr11030811

Jung S, Son J, Kim C, Chung K. Efficiency Measurement Using Data Envelopment Analysis (DEA) in Public Healthcare: Research Trends from 2017 to 2022. Processes . 2023; 11(3):811. https://doi.org/10.3390/pr11030811

Jung, Sungwook, Jiyoon Son, Changhee Kim, and Kyunghwa Chung. 2023. "Efficiency Measurement Using Data Envelopment Analysis (DEA) in Public Healthcare: Research Trends from 2017 to 2022" Processes 11, no. 3: 811. https://doi.org/10.3390/pr11030811

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Benchmarking in data envelopment analysis: balanced efforts to achieve realistic targets

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• Published: 09 September 2024

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• Hernán P. Guevel   ORCID: orcid.org/0000-0003-4434-9178 1 , 2 ,
• Nuria Ramón   ORCID: orcid.org/0000-0002-3532-3139 1 &
• Juan Aparicio   ORCID: orcid.org/0000-0002-0867-0004 1 , 3  

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The minimum distance models have undoubtedly represented a significant advance for the establishment of targets in Data Envelopment Analysis (DEA). These models may help in defining improvement plans that require the least overall effort from the inefficient Decision Making Units (DMUs). Despite the advantages that come with Closest Targets, in some cases unsatisfactory results may be given, since improvement plans, even in that context, differ considerably from the actual performances. This generally occurs because all the effort employed to reach the efficient DEA frontier is channeled into just a few variables. In certain contexts these exorbitant efforts in some inputs/outputs become unapproachable. In fact, proposals for sequential improvement plans can be found in the literature. It could happen that the sequential improvement plans continue to be so demanding in some variable that it would be difficult to achieve such targets. We propose an alternative approach where the improvement plans require similar efforts in the different variables that participate in the analysis. In the absence of information about the limitations of improvement in the different inputs/outputs, we consider that a plausible and conservative solution would be the one where an equitable redistribution of efforts would be possible. In this paper, we propose different approaches with the aim of reaching an impartial distribution of efforts to achieve optimal operating levels without neglecting the overall effort required. Therefore, we offer different alternatives for planning improvements directed towards DEA efficient targets, where the decision-maker can choose the one that best suits their circumstances. Moreover, and as something new in the benchmarking DEA context, we will study which properties satisfy the targets generated by the different models proposed. Finally, an empirical example used in the literature serves to illustrate the methodology proposed.

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1 Introduction

Benchmarking consists of evaluating and analyzing the processes, products, services, or other aspects of companies in order to draw a comparison and take this information as a reference point to guide future strategic decision-making. The intention is to learn from the experience of other units in the sector to improve the performance of the evaluated unit. Camp ( 1989 ) defines benchmarking as the search for industry best practices that lead to superior performance. Coldling ( 2000 ) describes benchmarking as a way to develop best practices, learn from other entities, organizations and cultures while there is no universal definition of what the term Benchmarking would mean. Different definitions have been given throughout the literature, always taking as a reference the experience of the Xerox Company in 1979 when it began to question its management model (De Cárdenas, 2006 ). Benchmarking allows the strengths and weaknesses of competitors to be identified and thereby could improve the position of companies in the market. It can therefore be said that it is an evidence-based management technique. Benchmarking has its origin in the Total Quality Management framework and, in recent years, it is leading to a revolution in the way that organizations approach and manage operational structure and behavior. In general, the evaluation of management and decision-making in organizations are currently becoming a burgeoning area of scientific attention, mainly motivated by resource limitations and the need to choose. In this sense, the methods related to benchmarking are especially useful for this purpose. Recently, Sutia et al. ( 2020 ) have identified the benefits of benchmarking methods in the industry.

Several support tools for developing management and decision-making in organizations are used, including statistical and econometric approaches and operational research methods. Data Envelopment Analysis (DEA) (Charnes et al., 1978 ) has proven to be a useful tool for the benchmarking of Decision Making Units (DMUs) involved in production process. In fact, benchmarking in DEA has been applied in multiple fields such as transportation, the service sector, product planning, maintenance, the hospitality industry, education, distribution, environmental factors, among others. See Rostamzadeh et al. ( 2021 ) for a comprehensive review of the application of DEA in benchmarking.

The identification of best practices allows the establishment of targets and, with them, the design of improvement plans in susceptible areas. In benchmarking process, it is important to consider the following considerations: 1) the identification of best practices must be done taking into account the circumstances and characteristics of the organizations being evaluated, 2) these best practices must reflect efficient behaviors, and 3) the established targets must be achievable and, as far as possible, require control over the effort necessary for their achievement. Our work revolves around this third consideration. On the one hand, we will seek to control the global efforts necessary to achieve the proposed targets and, on the other, we will focus our interest on obtaining achievable improvement plans.

DEA is a methodology designed for the evaluation of the efficiency of DMUs in production processes, as well as for benchmarking and ranking purposes. DEA classifies DMUs as efficient and inefficient, so that the latter are evaluated in relation to an efficient frontier determined by the former, assuming certain postulates (enveloping the data from above, convexity, constant or variable returns to scale, free disposability and minimal extrapolation). In the context of benchmarking, the efficient DMUs, as defined by DEA, may not necessarily form a “production frontier”, but rather lead to a “best-practice frontier”. Specifically, the points on the best-practice frontier allow the identification of benchmarks for the inefficient units, and the coordinates represent levels of the inputs and the outputs of the inefficient DMUs that would lead them to operate efficiently. In other words, the benchmarks provided by DEA allow the establishment of objectives (targets) for inefficient units, thus making it possible to design improvement plans for their management.

It is worth highlighting the following aspects that make DEA an appealing tool for benchmarking: 1) The relative nature of the efficiency evaluation that is carried out, in which the participating DMUs are compared with each other, makes the DEA a methodology that can naturally be applied for benchmarking, 2) DEA, in particular the non-radial models, allow the identification of benchmarks that are efficient in the Pareto sense; that is, those for which it is not possible to find a possible production plan that is better or equal in all the inputs and all the outputs, being strictly better in some of these variables, and 3) the DEA technique enables the individual circumstances of each DMU to be taken into account in the evaluation performed. Regarding benchmarking, this means that each unit may have its own references, which will be selected precisely according to those circumstances.

Despite these positive features for benchmarking, the literature has not ceased to provide new formulations of the basic DEA models, which have been modified to introduce improvements in many different aspects, particularly in those that refer to benchmarking. One of the aspects of benchmarking that has attracted increasing interest in recent years is the attempt to identify the Closest Targets (CT). In the context of non-radial models, the basic additive DEA model provides a selection of benchmarks that results from maximizing the distance from the DMU that is evaluated to the efficient frontier. It seems logical, however, that such distance should be minimized instead of maximized. The identification of closer references leads to benchmarks that are more similar to the units being evaluated and, with this, the design improvement plans that require less effort from the DMUs to achieve efficiency. However, finding the shortest distance to a point within the technology is equivalent to calculating the distance between a point belonging to a convex set (a polyhedron) and the complement of that convex set, which is computationally challenging. In this regard, Aparicio et al. ( 2007 ) theoretically solve the problem of minimizing the distance to the efficient frontier by implementing a Mixed Integer Linear Problem (MILP) that is used to set the Closest Targets. Several authors have worked since then with the same idea: Ramón et al. ( 2016 ) by incorporating weight restrictions in the model, Ruiz and Sirvent ( 2016 ) by developing a common benchmarking framework, Aparicio et al. ( 2017 ) by working with oriented models, Zhu et al. ( 2018 ) by considering non-oriented DEA models with a simple MILP, and Cook et al. ( 2019 ) by dealing with a benchmarking approach within the context of pay-for-performance incentive plans, are just some examples. See also the survey by Aparicio ( 2016 ).

Despite the significant progress made with Closest Targets which require the least overall effort, in practice we often find situations for which the Closest Targets remain unattainable. This situation would occur when the improvement plan that results from applying Closet Targets requires a very ambitious effort in only one variable for which it is technically impossible to improve. To be more concise, let’s simply imagine as an example the evaluation of university professors where inputs such as public economic founds or contracted hours, on one hand, and outputs such as research stays, supervised theses or conferences attended are involved. We solve the Closest Targets problem and we could find the situation that the improvement plan for a professor that globally requires the least effort only imposes an improvement of 80% of its supervised doctoral theses which is impossible due to not having doctoral students in the department. These would be the situations where the literature sometimes describes closet targets as unattainable or unapproachable. In any case, our critique is not directed at DEA as a whole, but rather at specific applications where the assumptions of certain models may not align with the practical realities of the DMUs under evaluation. Our approach aims to complement existing DEA models by providing alternative models that ensure a more balanced effort distribution.

Moreover, the literature has already shown that Closest Targets sometimes lead to unsatisfactory results and raises the need to design new alternatives to the benchmarking process. In this regard, see the recent contributions by An et al. ( 2021 ), who classify Closest Targets on certain occasions as impractical. See also Ramón et al. ( 2018 ), who propose a two-step benchmarking approach for setting more realistically achievable targets, and by Lozano and Villa ( 2010 ), who introduce an approach to deal with gradual technical and scale efficiency improvements. The minimum distance models, such as the Closest Target (CT) model proposed by Aparicio et al. ( 2007 ), represent a significant advancement in DEA for establishing targets with minimal overall effort. However, these models may sometimes produce improvement plans that are impractical due to excessive focus on a few variables. This critique is not against the DEA methodology as a whole but highlights specific scenarios where certain models may not be suitable. Our work proposes alternative approaches that balance efforts across different variables, providing more practical and achievable improvement plans for decision-makers.

In fact, in applications frequently point to substantial differences in the effort required in the different variables in inputs and / or outputs to reach the marked levels. Thus, in practice, and following with the same example we could find the case where the Closest Target for a professor supposes an improvement of 200% over its current levels in supervised doctoral theses and 0% in attendance to congresses See also the application on airlines in Aparicio et al. ( 2007 ), where this problem was empirically highlighted. Perhaps for that particular researcher, this alternative is totally infeasible despite assuming the least overall effort to reach efficient levels. In terms of benchmarking, this means that the references used in the evaluation of each unit can lead us to establish objectives where the required efforts are clearly unbalanced in terms of improvement directions. Although the use of a DEA-based approach (which carries out an evaluation of each unit taking into account its circumstances), and in particular minimum distance models (which look for the most similar references), can mitigate this problem, in many cases this is not enough since the models used often give rise to unrealistic objectives as described above.

The proposed models in this paper do not assume the use of additional information beyond the data itself and the standard DEA assumptions (inputs, outputs, returns to scale, and convexity). Our mention of additional information pertains to its potential availability for conducting sensitivity analyses, which can help assess the robustness of the results. We recognize that the production possibility set constructed by DEA inherently includes all feasible and applicable situations, and thus, the targets generated by an appropriate DEA model are valid within this context. Our critique is not aimed at the DEA methodology as a whole but at specific models and their practical implementation. For instance, while the Closest Target (CT) model effectively minimizes overall effort, it may result in impractical improvement plans by focusing too much effort on a single variable. Our alternative models aim to provide more balanced improvement plans by distributing the required efforts more evenly across different dimensions, thus enhancing their practical applicability.

The idea that in the absence of additional information, the most reasonable option is an equitable distribution has already been put forward in the literature in other DEA contexts. In that regard, Alcaraz et al. ( 2022 ) state: “ Any fair evaluation should start from a neutral situation ”. Thus, in efficiency measures or cross-efficiency evaluation, many authors have aimed not only to avoid zero weights but also to pursue weights that are as balanced as possible (see Ramón et al., 2010a , 2010b , and Cooper et al., 2011 ). The problem of avoiding unattainability targets, which are mathematically feasible solutions but may not be appropriate from an administrative or economic point of view, has already been addressed in the benchmarking context. Lozano and Villa ( 2010 ), Fang ( 2015 ) and Ramón et al. ( 2018 ), Aparicio et al. ( 2021 ) and Aparicio and Monge ( 2022 ) have already highlighted the need for more balanced efforts in improvement plans within the benchmarking context. In fact, Aparicio et al. ( 2021 ) propose a new measure, the Range-Adjusted Measure (RAM) in its multiplicative version (MRAM), which although it was originally proposed as an alternative measure to RAM with more desirable properties, it also ended up providing improvement plans with balanced efforts.

In this paper, we want to draw attention to the use of the DEA for establishing targets and, in general terms, the use of benchmarking within this context. The literature has multiple efficiency measures and, consequently, multiple alternatives for reaching the efficient frontier by searching benchmarks and improvement plans. In this article, our objective is to draw a comparison with the two most desired features in the benchmarking literature, Closest Targets; because they represent the least global effort, and the benchmarks that result from applying MRAM; due to the nature of that model since it allows strategies to be established for reaching optimal levels of operation with balanced efforts. In the total absence of additional information, we considered that the most realistic and conservative solution would be the one where the effort necessary to achieve optimal operating levels is as balanced as possible in all dimensions without neglecting the overall necessary effort. Therefore, we would like to highlight the advantages of our approach since we will combine the benefits of both methods (CT and MRAM). In this regard, we introduce three alternative models to state benchmarks in the context of DEA. On the other hand, and no less important, we study which properties satisfy the targets generated by the different models proposed. To date, interest in this matter has only focused on the properties that the value of the efficiency score fulfilled. However, as far as we are aware, no one before has ever established what list of properties the targets satisfy. We consider that it is important to study certain properties of the targets to decide on the goodness or suitability of the different targets offered by the DEA literature. Finally, using a real-data example, we show that the new approaches achieve improvement plans with a global effort very close to that of the Closest Targets and much smaller than those obtained with MRAM. Our goal is to complement existing DEA models by introducing approaches that ensure balanced efforts across different dimensions. This addresses a practical need for improvement plans that are both achievable and equitable. We acknowledge the foundational purposes of DEA and aim to build on this robust framework to offer additional tools for decision-makers. While DEA provides achievable targets, our contribution lies in proposing methods that ensure these targets are more evenly distributed across different variables. Our approach does not contradict the principles of DEA but rather complements them by addressing practical implementation concerns.

The paper is organized as follows: In Sect.  2 , we develop different approaches to avoid unbalanced and unacceptable efforts in the different dimensions (inputs and outputs). We then propose different alternatives for improvement plans so that the decision-maker can choose the one that best suits their circumstances. This section includes a study of the main properties met by the targets obtained in the different models proposed, an example for illustrative purposes which has already been used in the benchmarking DEA literature and will allow us to highlight the advantages of the approach. Section  3 concludes.

2 Balanced efforts with alternative approaches

Within the standard DEA framework, we define n decision making units (DMUs) which use m inputs to produce s outputs. These are denoted by (X j , Y j ), j = 1,..., n. It is assumed that X j  = (x 1j ,…, x mj ) ≥ 0  m , j = 1,..., n and Y j  = (y 1j ,…, y sj ) ≥ 0  s , j = 1,..., n. The Production Possibility Set is denoted by T = {(X,Y) ≥ 0  m+s : X produce Y}, which can be empirically constructed from the n observations by assuming several postulates (see Banker et al., 1984 ). If, in particular, Variable Returns to Scale (VRS) is assumed, then T can be characterized as follows:

\({\text{T = }}\left\{ {{\text{(X,Y) }} \ge {0}_{{\text{m + s}}} {: }\sum\limits_{{\text{j = 1}}}^{{\text{n}}} {{\text{X}}_{{\text{j}}} {\uplambda }_{{\text{j}}} \le {\text{X, }}\sum\limits_{{\text{j = 1}}}^{{\text{n}}} {{\text{Y}}_{{\text{j}}} {\uplambda }_{{\text{j}}} \ge {\text{Y}}} } {,}\sum\limits_{{\text{j = 1}}}^{{\text{n}}} {{\uplambda }_{{\text{j}}} } { = 1, }\forall j} \right\}\) .

The measure of efficiency for DMU 0 is obtained as the result of its comparison with a dominating projection point on the efficient frontier of the Production Possibility Set. The coordinates of this projection will be the targets for DMU 0 . Regarding the determination of Closest Targets, our approach fundamentally builds on the characterization of the set of Pareto-efficient points of T that dominate DMU 0 .

Next, we propose three approaches to avoid unbalanced efforts reaching the efficient frontier. As Aparicio et al., ( 2021 , p. 265) state “… measures like RAM or SBM can set unrealistic targets generating zero value slacks, involving unbalanced efforts in some dimensions to reach the efficient frontier, which in the benchmarking context can be seen as a drawback since unreachable directions of improvements can be proposed in some variables while others barely improve” . The assertion that measures like RAM or SBM can set unrealistic targets generating zero value slacks and unbalanced efforts is derived from specific applications and observations. When SBM is applied appropriately, it indeed finds the farthest feasible point in any direction and is designed to handle non-radial, non-oriented situations effectively. However, our critique is not of the SBM model itself but of the practical challenges that can arise in certain contexts where the resulting targets may require highly unbalanced efforts. We do not employ an additive model because any inefficient DMU always projects to the farthest efficient point, and this benchmark may or may not be achievable in practice depending on the considered context. In our research, we seek to explore alternative models that balance the required efforts across different variables, thereby providing more practical and achievable improvement plans. Our main goal is to build on this existing knowledge and propose new models that offer decision-makers additional tools for achieving balanced and feasible improvement plans.

The set of DMUs can be classified into the classes E, E′, F, NE, NE′, and NF following Charnes et al. ( 1991 ) classification. E and E´ DMUs are Pareto-efficient units, while E are the extreme efficient units, E′ can be described as a linear combination of DMUs in E. F are the weakly efficient units, that means, efficient units but not in the Pareto sense and are located onto the weak efficient frontier. Finally, units NE, NE′, and NF are inefficient units that are projected onto points that are in E, E′, and F, respectively (Ramón et al., 2010b ).

Starting from the previous classification, we focus on the non-efficient units in the Pareto sense and from there we explore, through different proposals, their projection towards the Pareto-efficient frontier. The following subsections describe three different alternatives to determine targets satisfying two requirements at the same time: closer and more balanced targets.

2.1 Minimum range model

The absence of additional information leads us to a neutral situation where the difficulty of making improvements should be the same in the different dimensions. Therefore, extreme imbalances in the efforts required in the different variables to reach optimal levels should be avoided. The Minimum Range Model (MRM) has been designed to ensure that the greatest and least efforts to reach the optimum levels in the different variables would be as similar as possible. The ideal situation would be when the improvement plans indicate the same efforts throughout the different dimensions.

The underlying assumption is as follows: Firstly, with (1.1)–(1.3) we consider all the points (X, Y) in the Production Possibility Set that are linear combinations of units in E and dominate DMU 0 . Secondly, (1.4) along with the conditions v i  ≥ 1; i = 1,..., m; and u r  ≥ 1, r = 1,..., s, are the constraints corresponding to the multiplier formulation of the additive DEA model (see Ali & Seiford, 1993 ), but only considering extreme efficient units ( \({\text{h}}_{0}\) is the offset of the hyperplane in (1.4), which is related to the assumed VRS, under CRS, \({\text{h}}_{0}\)  = 0). With these constraints, we allow for all the hyperplanes such that all the points of T lie on or below these hyperplanes. Finally, we remark on the conditions that connect the previous groups of constraints: (1.5)–(1.6). Note that if \({\uplambda }_{{\text{j}}} > 0\) then (1.6) implies b j  = 0 and, consequently, k j  = 0 by virtue of (1.5). Thus, if DMU j participates actively as a peer then it necessarily belongs to the hyperplane \({ - }\sum\nolimits_{{{\text{j}} \in {\text{E}}}}^{{}} {{\text{v}}_{{\text{i}}} {\text{x}}_{{{\text{ij}}}} } { + }\sum\nolimits_{{{\text{j}} \in {\text{E}}}}^{{}} {{\text{u}}_{{\text{i}}} {\text{y}}_{{{\text{rj}}}} } {\text{ + h}}_{0} { = 0}\) . In case of \({\uplambda }{}_{j}\)  = 0, then k j  ≥ 0, so that nothing can be stated about whether or not DMU j is located on this hyperplane, Nevertheless, this is not relevant since, in that case, it is not a peer for DMU 0 . (Aparicio et al., 2007 ).

The constraints (1.9)-(1.10) oblige the effort to take values between LE (lower effort) and HE (higher effort). As the objective function is to maximize precisely the ratio LE/HE, in reality want the smallest and largest effort to be as close as possible; the ideal situation being when the optimum is 1, which will occur when the variable-specific efforts, measured as \({{{\text{s}}_{{{\text{io}}}}^{ - } } \mathord{\left/ {\vphantom {{{\text{s}}_{{{\text{io}}}}^{ - } } {{\text{x}}_{{{\text{io}}}} }}} \right. \kern-0pt} {{\text{x}}_{{{\text{io}}}} }}\) for inputs and \({{{\text{s}}_{{{\text{ro}}}}^{ + } } \mathord{\left/ {\vphantom {{{\text{s}}_{{{\text{ro}}}}^{ + } } {{\text{y}}_{{{\text{ro}}}} }}} \right. \kern-0pt} {{\text{y}}_{{{\text{ro}}}} }}\) for outputs, to avoid problems with units of measure, are identical throughout the variables.

Constraints (1.5)–(1.6) in model (1) include the classical big M and binary variables, restrictions that allow that the DMUs in E that participate actively as a referent in the evaluation of DMU 0 necessarily belong to a supporting hyperplane containing a facet of T. Nevertheless, (1) can be solved in practice by reformulating these constraints using Special Ordered Sets (SOS) (Beale & Tomlin, 1970 ), which avoid the need to specify a value for M. SOS Type 1 is a set of variables where at most one variable may be nonzero. Therefore, if we remove these two groups of constraints from the formulation and instead define an SOS Type 1 for each pair of variables \({\text{\{b}}_{{\text{j}}} {,}\lambda_{{\text{j}}} {\text{\} }}{\kern 1pt} {\kern 1pt} {\text{j}} \in {\text{E}}\) , then it is ensured that \({\text{b}}_{{\text{j}}}\) and \({\uplambda }_{{\text{j}}}\) cannot be simultaneously positive for DMU j , \({\text{j}} \in {\text{E}}\) . CPLEX Optimizer (and also LINGO) can solve LP problems with SOS. SOS variables have already been used for solving different variations of DEA models in Ruiz et al. ( 2016 ), Aparicio et al. ( 2016 ) and Cook et al. ( 2017 ), Ramón et al. ( 2020 ), among others.

Lastly, model (1) presents a fractional objective function and linear restrictions. Its optimal value can easily be obtained using the transformation given by Charnes and Cooper ( 1962 ) (for more details see the Appendix).

2.2 Distance to the minimum squared model

Another alternative that allows to explore the efficient frontier by controlling unbalanced efforts is the Distance to the Minimum Squared Model (DMISM), whose main characteristic is to minimize the sum of efforts to their minimum. The philosophy of this proposal again seeks to shorten distances between the requirements necessary for achieving optimal levels of efficiency. Note that in the objective function we use the square of the difference to the minimum and not the absolute value, which would lead to a linear model after a simple change of variable, to penalize large efforts. In this way, not only do we look for efforts that are as similar as possible, but we also penalize moving away from the minimum effort with the square to also control the overall effort.

In model (2) the restrictions (1.1) to (1.8) from (1) are repeated. Restrictions (2.1) and (2.2) represent the distance to the efficient frontier of each variable (efforts). Restrictions (2.3) and (2.4) demand that no distance must be less than the minimum. Restrictions (2.5)-(2.7) establish that at least one input or output must reach the minimum.

2.3 Distance to the mean squared model

The last model is called the Distance to the Mean Squared Model (DMESM) where we seek to minimize the deviation from the average efforts. Here again, and for the same reason explained in model (2), we opted for the square and not for the absolute values.

As for the constraints, the mathematical formulation of (3) is made up with the set of constraints (1.1) to (1.8) of model (1) and constraints (2.1)-(2.2) of model (2). Additionally, constraint (3.1) defines \({\overline{\text{d}}}\) as the average of the relative distances.

2.4 Properties of targets

DEA is a nonparametric methodology for determining technical efficiency. Due to its nature, the researcher does not need to assume a particular functional form for the underlying production frontier of the corresponding sector under performance evaluation. This fact contrasts to other well-known alternatives in the literature for measuring technical efficiency, such as Stochastic Frontier Analysis (SFA), which does require specifying a particular mathematical expression for the efficient frontier to start with the analysis. Regarding SFA, a very relevant characteristic of this approach is that a measure of goodness of fit of the model can be computed and tested by standard statistical tools (for example, R 2 ). In contrast, Data Envelopment Analysis lacks a goodness of fit measure.

In this regard, an alternative way of testing the goodness of DEA technical efficiency measures that has been utilized in the literature is stating a list of properties that the measure should verify, which represents an axiomatic approach of the problem. Among them, we mention the following properties. First, the value of the efficiency measure is always between zero and one. Second, monotonicity, i.e., if unit A dominates, in Pareto sense, unit B, then the value of the technical efficiency measure associated with A is greater than the value of the technical efficiency measure related to B. Third, units invariance. Fourth, translation invariance. And fifth, the evaluated observation is Pareto–Koopmans efficient if and only if the value of the efficiency measure is one. The fulfillment of the biggest possible number of these properties has been one of the explanations why many different technical efficiency measures have been defined over the last decades in the Data Envelopment Analysis literature (see, for example, Cooper et al., 1999, Pastor et al., 1999 or, more recently, Aparicio & Monge, 2022 ).

As we have already mentioned, DEA provides not only an efficiency score but also benchmarking information, as the input and output targets that would make the assessed inefficient unit perform efficiently. However, as far as we know, no one before has ever established what list of properties the targets generated by an efficiency measure should satisfy. The spotlight has always been focused on the properties that the value of the efficiency score (i.e., the optimal value of the objective function of the corresponding optimization model) must meet, despite the relevance of providing suitable benchmarking knowledge for company managers. The only exception that we could find in the specialized literature is the property of “unique projection for efficiency comparison” defined by Sueyoshi and Sekitani ( 2009 ), which was a property related to targets introduced alongside the typical tests associated with the value of the efficiency measures. This property ensures that the reference point on the frontier generated by the DEA efficiency model is unique for each assessed DMU. If not, it should be necessary to invoke some (subjective) secondary criterion as a rule to select a benchmark among the existing ones to deal with this problem in empirical applications. Also, Sueyoshi and Sekitani ( 2009 ) highlighted that this property is very hard to meet. Indeed, in their Proposition 9, they proved that no standard DEA model satisfies the property of unique projection.

Consequently, so far, DEA has lacked a list of properties defined for the input–output targets yielded by the existing alternative models associated with different technical efficiency measures. Regarding this issue, we go on to fill this gap in the literature by proposing an initial list, which could be extended with additional tests in the future. These properties are defined as follows:

(P1) [Dominance] The evaluated DMU is Pareto-dominated by the projection point generated by the DEA model.

(P2) [Pareto-efficiency] The projection point is Pareto–Koopmans efficient.

(P3) [Consistency to units of measurement changes] If some variable (input or output) is multiplied by a strictly positive constant and the DEA model is applied on the transformed database, then the projection point must be altered in the same way.

(P4) [Consistency to data translation] If some variable (input or output) is translated by a constant and the DEA model is applied on the transformed database, then the projection point must be altered in the same way.

(P5) [Unique projection for efficiency comparison] The DEA model can only generate a projection point for any assessed DMU.

Property P1 says that the projection point is unquestionably performing better than the evaluated unit, producing at least the same quantity of outputs by using at most the same quantity of inputs. Property P2 ensures that we cannot find a better benchmark in the sense that, with the current knowledge (technology), it is not possible to improve any dimension (input or output) without making it worse with respect to another dimension. Properties P3 and P4 say that if the data are transformed and the same DEA model is applied over the new database, then the original benchmark transformed in the same way as the original data is also a valid benchmark for the transformed model. Finally, P5 is the property established by Sueyoshi and Sekitani ( 2009 ) regarding the uniqueness of the benchmark.

Next, we show that the Minimum Range Model satisfies the first four properties of the above list, but it does not fulfill the ‘unique projection for efficiency comparison’ property of Sueyoshi and Sekitani ( 2009 ), as expected.

The MRM meets P1-P4 and it does not satisfy P5 .

Before starting with the proofs, let us introduce some necessary notation. Let \(\pi^{*} = \left( {LE^{*} ,HE^{*} ,\lambda_{1}^{*} ,...,\lambda_{\left| E \right|}^{*} ,s_{10}^{ - *} ,...,s_{m0}^{ - *} ,s_{10}^{ + *} ,...,s_{s0}^{ + *} ,v_{1}^{*} ,...,v_{m}^{*} ,u_{1}^{*} ,...,u_{s}^{*} ,h_{0}^{*} ,k_{1}^{*} ,}\right.\) \(\left.{...,k_{\left| E \right|}^{*} ,b_{1}^{*} ,...,b_{\left| E \right|}^{*} } \right)\) be an optimal solution of model (1). And let \(x_{i0}^{*} = x_{i0}^{{}} - s_{i0}^{ - *}\) , \(i = 1,...,m\) , and \(y_{r0}^{*} = y_{r0}^{{}} - s_{r0}^{ + *}\) , \(r = 1,...,s\) , the input and output targets corresponding to the optimal solution π * , which define the benchmark \(\left( {X_{0}^{*} ,Y_{0}^{*} } \right)\) for the assessed unit \(\left( {X_{0}^{{}} ,Y_{0}^{{}} } \right)\) . Next, we prove Theorem 1 property by property. P1 is evident by the non-negativity constraints in model (1) for the slacks. This means that \(s_{i0}^{ - *} \ge 0\) , \(i = 1,...,m\) , and \(s_{r0}^{ + *} \ge 0\) , \(r = 1,...,s\) , and, consequently, \(x_{i0}^{*} \le x_{i0}^{{}}\) , \(i = 1,...,m\) , and \(y_{r0}^{*} \ge y_{r0}^{{}}\) , \(r = 1,...,s\) . Regarding P2, model (1) resorts to the set of constraints introduced by Aparicio et al. ( 2007 ) to ensure that the benchmark is Pareto-efficient. Therefore, this property is also verified by the MRM. As for the satisfaction of P3, let us assume that input \(i^{\prime}\) is multiplied by the strictly positive scalar \(\delta\) . Then, it is not difficult to prove that \(\hat{\pi } = \left( {LE^{*} ,HE^{*} ,\lambda_{1}^{*} ,...,\lambda_{\left| E \right|}^{*} ,s_{10}^{ - *} ,...,s_{{i^{\prime} - 10}}^{ - *} ,\hat{s}_{{i^{\prime}0}}^{ - } ,s_{{i^{\prime} + 10}}^{ - *} ,...,s_{m0}^{ - *} ,s_{10}^{ + *} ,...,s_{s0}^{ + *} ,v_{1}^{*} ,...,}\right.\) \(\left.{v_{{i^{\prime} - 1}}^{*} ,\hat{v}_{{i^{\prime}}}^{{}} ,} \right.v_{{i^{\prime} + 1}}^{*} ,\) \(\left. {...,v_{m}^{*} ,u_{1}^{*} ,...,u_{s}^{*} ,h_{0}^{*} ,k_{1}^{*} ,...,k_{\left| E \right|}^{*} ,b_{1}^{*} ,...,b_{\left| E \right|}^{*} } \right)\) , with \(\hat{s}_{{i^{\prime}0}}^{ - } = \delta s_{{i^{\prime}0}}^{ - *}\) and \(\hat{v}_{{i^{\prime}}}^{{}} = {{v_{{i^{\prime}}}^{*} } \mathord{\left/ {\vphantom {{v_{{i^{\prime}}}^{*} } \delta }} \right. \kern-0pt} \delta }\) , is a feasible and optimal solution of the model (1) when it is transformed. Then, the benchmark associated with the optimal solution \(\hat{\pi }\) when \(\left( {x_{10} ,...,x_{{i^{\prime} - 10}} ,\delta x_{{i^{\prime}0}} ,x_{{i^{\prime} + 10}} ,...,x_{m0} ,Y_{0}^{{}} } \right)\) is evaluated would have the input and output targets as follows: \(\hat{x}_{i0}^{*} = x_{i0}^{{}} - s_{i0}^{ - *} ,{\kern 1pt} \forall i \ne i^{\prime}\) , \(\hat{x}_{{i^{\prime}0}}^{*} = \delta x_{{i^{\prime}0}}^{{}} - \hat{s}_{{i^{\prime}0}}^{ - } = \delta x_{{i^{\prime}0}}^{{}} - \delta s_{{i^{\prime}0}}^{ - *} = \delta \left( {x_{{i^{\prime}0}}^{{}} - s_{{i^{\prime}0}}^{ - *} } \right) = \delta x_{{i^{\prime}0}}^{*}\) and \(\hat{y}_{r0}^{*} = y_{r0}^{{}} + s_{r0}^{ + *} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} r = 1,...,s\) . Consequently, P3 holds. Property P4 can be analogously proved.

Finally, regarding P5, this property does not hold. To demonstrate it we rely on the data in Table  1 , where five DMUs with two inputs and two outputs are presented. DMUs A, B, C and D are efficient in the Pareto sense while DMU E is inefficient.

Solving model (1) for DMU E, we find the optimum at 0.375 which can be achieved in two different Pareto-efficient projections with, therefore, different input–output targets. One takes the efficient DMUs A and B ( \(\lambda_{A}^{*}\)  = 0.5, \(\lambda_{B}^{*}\)  = 0.5) as peers and another, whose referents are DMUs C and D ( \(\lambda_{C}^{*}\)  = 0.5, \(\lambda_{D}^{*}\)  = 0.5).

Our proposed models introduce a novel approach to achieving balanced efforts across different variables, addressing a gap in existing methodologies. Traditional models like the Closest Target (CT) model and SBM focus on minimizing overall effort or finding the farthest feasible point, but they may lead to unbalanced improvement plans in certain contexts. Our models ensure that the efforts required are distributed more equitably, providing practical and achievable improvement plans for decision-makers. Regarding Theorem 1 , we acknowledge that unit invariance and translation invariance are well-known properties of additive DEA models. However, our models are not additive. They may resemble additive models due to the incorporation of constraints that combine elements from both the primal and dual forms of the additive model. Despite this resemblance, our models differ significantly in their objective functions and additional constraints. Therefore, it is necessary to demonstrate Theorem 1 to confirm that our models retain the essential properties of unit invariance and translation invariance for the yielded targets while introducing the novel feature of balanced efforts.

Regarding DMISM and DMESM, it is possible to establish similar results as Theorem 1 .

The DMISM satisfies P1-P4 but it does not satisfy P5 .

The DMESM satisfies P1-P4 but it does not satisfy P5 .

Proofs of properties

P1-P4 in Theorem 2 and Theorem 3 are analogous to those of Theorem 1 . In case of P5, we continue to use the data in Table  1 to demonstrate that it is not fulfilled. Specifically, the optimum for model DMISM is 3.781 and it is possible to obtain it through two different projections (on one hand, \(\lambda_{A}^{*} = \lambda_{B}^{*} = 0.5\) and, on the other, \(\lambda_{C}^{*} = \lambda_{D}^{*} = 0.5\) ). Something similar occurs with DMESM, whose optimal value 1.891 can be reached in two different projections (on one hand, \(\lambda_{A}^{*} = \lambda_{B}^{*} = 0.5\) and, on the other, \(\lambda_{C}^{*} = \lambda_{D}^{*} = 0.5\) ).

Consequently, establishing a new model that satisfies P5, together with the other four properties, could be an interesting future line of research.

2.5 Comparative models: a toy example

In order to graphically compare the approaches, two descriptors have been considered. The first one represents the effort measured by the percentage of change through the observed value. The second one is the global effort, measured as the sum of the individual efforts in each dimension.

Consider the DMUs in Table  2 that use two inputs to produce two outputs. The DEA-VRS analysis reveals that A, B, C and D are the extremely efficient DMUs and are the key points of the Pareto-efficient frontier, as well as the DMU E being an inefficient DMU.

As mentioned in the introduction, the three approaches developed in this paper will be compared with the Closest Targets (CT) approach Footnote 1 by Aparicio et al. ( 2007 ) and the MRAM measure by Aparicio et al. ( 2021 ). Table 3 illustrate the results as well as the advantages of the different methods.

The Closest Targets (CT) approach minimizes the distance to the frontier. Following this philosophy, it is recommended to focus all efforts on input 2, output 1, and input 1. However, despite assuming the least global effort, there may be situations where the improvement in output 2, or in those specific requirements, is not achievable. In contrast, MRAM proposes changes in both inputs and outputs, with a greater overall effort providing a better balance between the efforts required of the variables.

The MRM, DMISM, and DMESM propose changes across all variables. As expected, the total effort for these models is greater than that of the Closest Targets approach. The MRM suggests equal efforts in input 1 and both outputs (50%), while the effort in output 2 is higher (83.33%). The DMISM proposes very similar changes, ranging between 38.88% and 58.32%. Lastly, the DMESM suggests more imbalanced changes compared to the previous models, with efforts ranging from 35.66% to 53.49%. It is also noteworthy that the reference DMUs vary in each scenario, offering different decision alternatives.

These observations are significant for decision-makers as they highlight the potential for increasing decision scenarios. For example, if the maximum change a DMU can achieve is 50%, then the CT approach becomes infeasible. In another scenario, if the DMU cannot make changes in input 2 exceeding 30%, only the MRM would be feasible. The benefit of having multiple targets is that DMUs can choose different paths that align more closely with their specific expectations and constraints.

2.6 Empirical example

This section includes an empirical illustration of the use of the methodology proposed in this paper. We are particularly interested in comparing in practice the results of the new models with the methods based on Closest Targets and MRAM, in an attempt to show that in practical applications we can find substantial differences related to improvement plans within the benchmarking context. We are using a database that includes a set of 28 airlines from different regions (Coelli et al., 2002 , Ray, 2004 and Aparicio et al., 2007 ). This database was used by the authors of Closest Targets and MRAM. This will allow us to compare ourselves with them and highlight the main advantages. In the database, each airline has been represented through 4 inputs and 2 outputs. The inputs considered were: the number of employees (LAB), fuel consumed (FUEL), millions of dollars of overheads applied, excluding labor and fuel costs (MATL) and millions of dollars invested (CAP). The outputs were passenger-kilometers covered (PASS) and tonne-kilometers covered (CARGO).

The application has been solved assuming a DEA technology under Variable Returns to Scale, according to the models presented in this paper. In this context, 16 airlines were considered efficient and the remaining 12, inefficient. Table 4 shows the application of different methods for exclusively inefficient DMUs (MRAM, CT, MRM, DMISM and DMESM). For each DMU, the target value and the individual effort to be made for each input and output are reported (columns LAB, FUEL, MATL, CAP, PASS, CARGO). Finally, in the last column, we provide the overall effort.

The first issue that should be highlighted from the table is that, for all DMUs, the CT approach, as expected, is the one that supposes a lower overall effort, but this also implies zero efforts in many dimensions, which implicitly leads to some overexertion in some sense. The second point that should be stressed is that MRAM provides balanced efforts throughout the different dimensions, but it is true that the effort required to reach the optimum levels under this proposal are far from those provided by Closest Targets. With the three alternatives that have been proposed in this paper, we provide different halfway solutions between the proposals that are closest to the desired objectives that are pursued in the context of benchmarking; minimal and balanced global efforts.

We want to draw attention to the Malaysia airline. First, observe the great difference found in the global efforts necessary to reach efficient levels with the MRAM (158.6%) and Closest Targets (58.7%) methods. Although it is true that with CT the global effort is considerably reduced, imbalance is noticeable, even assigning null efforts in three variables in exchange for overloading other dimensions. Against, we have the MRAM proposal, where the efforts are very balanced but globally an excessive effort is made to reach the levels of efficiency. In contrast, we propose three alternatives where the global effort is quite conservative while the efforts in the different dimensions are quite balanced. In particular, in this case the solution provided by MRM suggests efforts of 19%, 18%, 19%, 12.4%, 12.4%, and 19% for the four inputs and two outputs, respectively. Being DMISM and DMESM very attractive alternatives in terms of global efforts and balance between them, leaving the decision maker to choose the strategy that best suits their circumstances.

CATHAY, BRITISH, IBERIA and CANADIAN show similar overall efforts with the proposed alternatives and, in general, lower than MRAM. In the case of CANADIAN, the options presented propose an overall effort that ranges from 42% to 52.9% (Table  4 ), which is significantly lower than the effort measured through MRAM (93.6%). We also note that, except in MRM, changes have to be made in every variable to become efficient. Hence, the changes of the MRM option would be 17%, 15.2%, 6.6% in the case of LAB, FUEL MATL, respectively, and 4.7%, for the rest of the variables. Another target could be that suggested by the DMESM, where the individual efforts are 13.1% for LAB, 11.4% for FUEL, 3.3% for MATL, 1% for PASS,8.6% for CAP and 4.5% for CARGO.

The group composed of SAS, AIR CANADA, EASTERN and USAIR is also strikingly significant. In the first place because of the excessive efforts required with MRAM, which undoubtedly will be impossible to put into practice. In the case of AIR CANADA, for example, we establish improvement plans (DMISM and DMESM) with half the global effort required by MRAM and almost identical to that proposed by CT where, in addition, the efforts are much more balanced throughout the different variables. The case of EASTER is also noteworthy, which characterizes significant differences in the overall effort between the different improvement models. The possible projection paths to the frontier are very different, where MRAM proposes an effort far above the rest, mainly in the variable CARGO (471.3%), compared to 58.1%, 44.3%, 44.3%, 32.0% of the options CT, MRM, DMISM and DMESM, respectively.

Note that there are situations in which the improvement plans for certain inefficient units leave little room for any of the methods to shine. This is the case of Nippon where the distribution of efforts and the global effort hardly changes with any of the proposals. On the other hand, when there are alternative improvement plans, our models will look for that solution with the advantages of MRAM since we achieve similar efforts and the advantages of CT since we also take into account global efforts.

Table 5 shows a summary of the global effort required for each airline regarding the MRAM, CT, as well as our 3 proposals MRM, DMISM and DMESM. In general, we can sustain that the global effort necessary to achieve efficiency is significantly reduced with any of our proposals regarding the MRAM method, drawing attention to EASTERN and USAIR airlines, falling from 552.3% and 441.3% to 104.5% and 181%, respectively, regarding the MRM method (values being very similar for DMISM and DMESM). In addition, as an added value, we find that the global efforts are very close to those obtained with CT. We provide, therefore, different alternatives where the global efforts are considerably reduced and the efforts are distributed more equally among the different dimensions, thus representing improvement plans that are more in line with the reality of the units being evaluated.

In addition, Table  6 summarizes (mean and standard deviation) the results regarding the global effort determined through the approaches analyzed. It is observed that CT is the method that proposes the least average effort, a situation exposed in Ruiz et al. (2016), followed by DMESM with DMISM with 106.93%, 109.60% of average effort, MRM with 113.12% and MRAM with 197.52%. It is also observed that the standard deviation is around 72% in the methods except for MRAM which is 150.31%.

3 Conclusions

Benchmarking involves looking outside a particular organization to examine others’ performance in their industry or sector. The main objective is to identify best practices from which to learn and set improvement plans for reaching efficient levels of operation. The DEA literature contains many advances in this regard, proposing different ways to reach the efficient frontier. Although it is true that the minimum distance (Closest Targets) proposed by Aparicio et al. ( 2007 ) represented a significant advance given that it supposes the least global effort necessary to reach optimal levels; since then, there have been many studies that have striven to provide alternative improvement plans in the benchmarking context.

Despite the enormous advance that the minimum distance represented in the context of benchmarking, almost since the beginning, some voices pointed out the inconvenience of achieving efficient operating levels implying efforts only in some variables, thus showing an obvious imbalance effort in the different dimensions which in most practical situations was unattainable and/or infeasible. The MRAM measure (Aparicio et al., 2021 ), although originally proposed for other purposes, the truth is that it achieves very balanced improvement plans in terms of the necessary efforts in the different dimensions to reach the efficient frontier, which in the absence of information can be considered the most conservative option.

In this paper, we develop different approaches that take into consideration two of the basic premises that are required in any plan of improvement in the benchmarking context: control both the global efforts and the imbalances required in the necessary efforts in the different dimensions. In particular, we developed three approaches in the line described, which gives the decision maker the possibility of choosing between several improvement plans, choosing the one that best suits their particular circumstances to achieve optimal levels of efficiency.

The results in the empirical application illustrate significant improvements regarding the minimum distance and the MRAM measure. On the one hand, the global efforts are very close to those shown by the minimum distance and on the other, the notorious imbalances between the different dimensions are eliminated, reaching improvement plans with very equitable efforts.

With this article we achieve the following advances in the context of benchmarking: 1) we propose improvement plans in which the global efforts seem to be close to those that are achieved with the minimum distance model (Closest Targets), 2) disproportionate efforts are eliminated in the improvement plans that made their implementation impracticable, and 3) we propose several strategies that give the decision maker flexibility in choosing the guidelines to follow in accordance with their particular circumstances and, finally, 4) we study which properties satisfy the targets generated by the different models proposed; a topic that is completely new in the DEA literature.

All in all, in this paper, we proposed three new DEA models focused on achieving balanced efforts in the improvement plans for inefficient Decision Making Units (DMUs). While our models primarily consider Variable Returns to Scale (VRS), it is essential to explore how these models could be adapted and influenced by other technological assumptions such as Constant Returns to Scale (CRS), Non-Increasing Returns to Scale (NIRS), Non-Decreasing Returns to Scale (NDRS), and Free Disposal Hull (FDH). In particular, under CRS, the proportional scaling of inputs and outputs is assumed to be constant. Integrating CRS into our models would imply that any increase in inputs would result in a proportional increase in outputs. This could simplify the effort balancing process since the linear relationship might lead to more straightforward calculations of LE and HE. However, CRS might not capture the complexities of many real-world scenarios where returns to scale vary. NIRS and NDRS consider scenarios where scaling up inputs does not lead to proportional increases in outputs (NIRS) or scaling down inputs does not lead to proportional decreases in outputs (NDRS). These assumptions would introduce asymmetry into the effort balancing, potentially leading to different sets of constraints and optimization strategies for LE and HE. Future research could explore the incorporation of these conditions to provide more nuanced improvement plans tailored to specific DMU characteristics. Regarding FDH, this technology does not assume convexity of the production possibility set, allowing for more flexible and potentially more realistic representations of production processes. Applying FDH could lead to more accurate and context-specific benchmarking targets but might also increase computational complexity. The integration of FDH into our models would necessitate developing new algorithms to handle the non-convex nature of the technology set.

Given the potential benefits and challenges of incorporating these alternative technologies, future research should focus on the following areas:

Model Adaptation: Developing adaptations of our proposed models to integrate CRS, NIRS, NDRS, and FDH technologies, analyzing how these changes affect the balanced effort targets.

Algorithm Development: Creating efficient computational methods to handle the increased complexity, particularly for non-convex technologies like FDH.

Empirical Validation: Conducting extensive empirical studies across various industries to validate the performance and applicability of the adapted models under different technological assumptions.

Comparative Analysis: Performing comparative analyses to understand the trade-offs between different technologies, providing decision-makers with insights into the most appropriate models for their specific contexts.

By considering these future research directions, we aim to enhance the robustness and applicability of our DEA models, ensuring they remain valuable tools for benchmarking and performance improvement across diverse operational environments.

Finally, just as in most cases, we do not have any type of additional information about the efforts that can be assumed in each dimension, other times, due to the nature of the problem, we are aware of the existing limitations to some or all of the variables that participate in the analysis. As a future line of research, we propose the development of models that allow information to be incorporated into the analysis, eliminating those improvement plans that would be unattainable. We will analyze how the efficient frontier changes and, consequently, the targets. In addition, the existence of real or fictitious “model” DMUs can be a starting point for the implementation of strategies in the competitive world of benchmarking within different public and private organizations. Moreover, much has been written in the DEA literature about the different properties fulfilled by the different measures that have been developed in order to establish a ranking of the goodness of the different measures. Something similar should be studied for benchmarks, extending the initial list of properties included in this paper.

We assume the Variable Returns to Scale version of the Slacks-Based Measure (SBM) adaptation introduced by Aparicio et al. ( 2007 ).

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Acknowledgements

J. Aparicio and N. Ramon thank the grant PID2022-136383NB-I00 funded by MICIU/AEI/ 10.13039/501100011033 and by ERDF/EU. Moreover, N. Ramón thanks the grant PID2021-122344NB-I00 funded by MCIN/AEI/ https://doi.org/10.13039/501100011033 and by “ERDF A way of making Europe”, and the Generalitat Valenciana, project CIGE/2021/161. Additionally, J. Aparicio thanks the grant PROMETEO/2021/063 funded by the Valencian Community (Spain).

Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Ministerio de Ciencia e Innovación, GB-I00,Juan Aparicio, GB-I00,Nuria Ramón,PID2022-136383NB-I00/ AEI / https://doi.org/10.13039/501100011033,Juan Aparicio,PID2022-136383NB-I00/ AEI / https://doi.org/10.13039/501100011033,Nuria Ramón,Generalitat Valenciana,PROMETEO/2021/063,Juan Aparicio,CIGE/2021/161,Nuria Ramón,MCIN/AEI/ https://doi.org/10.13039/501100011033,PID2021-122344NB-I00,Nuria Ramón

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Juan Aparicio

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Correspondence to Nuria Ramón .

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Formulation (1) is a program with a fractional objective function and linear constraints. Its optimal value can be easily obtained by using the following change of variables (see Charnes & Cooper, 1962 ).

\(\beta { = }{{1} \mathord{\left/ {\vphantom {{1} {{\text{HE}}}}} \right. \kern-0pt} {{\text{HE}}}}\) , \(\widehat{{\text{v}}}_{{\text{i}}} { = }{{{\text{v}}_{{\text{i}}} } \mathord{\left/ {\vphantom {{{\text{v}}_{{\text{i}}} } \beta }} \right. \kern-0pt} \beta }\) , \(\widehat{{\text{u}}}_{{\text{r}}} = {{{\text{u}}_{r} } \mathord{\left/ {\vphantom {{{\text{u}}_{r} } \beta }} \right. \kern-0pt} \beta }\) , \(\widehat{{\text{L}}}{\text{E}} = {{{\text{LE}}} \mathord{\left/ {\vphantom {{{\text{LE}}} \beta }} \right. \kern-0pt} \beta }\) , \(\widehat{\lambda }_{{\text{j}}} { = }{{{\uplambda }_{{\text{j}}} } \mathord{\left/ {\vphantom {{{\uplambda }_{{\text{j}}} } \beta }} \right. \kern-0pt} \beta }\) , \(\widehat{{\text{s}}}_{{{\text{io}}}}^{ - } { = }{{{\text{s}}_{{{\text{io}}}}^{ - } } \mathord{\left/ {\vphantom {{{\text{s}}_{{{\text{io}}}}^{ - } } \beta }} \right. \kern-0pt} \beta }\) , \(\widehat{{\text{s}}}_{{{\text{ro}}}}^{ + } { = }{{{\text{s}}_{{{\text{ro}}}}^{ + } } \mathord{\left/ {\vphantom {{{\text{s}}_{{{\text{ro}}}}^{ + } } \beta }} \right. \kern-0pt} \beta }\) , \(\widehat{{\text{h}}}_{0} { = }{{{\text{h}}_{0} } \mathord{\left/ {\vphantom {{{\text{h}}_{0} } \beta }} \right. \kern-0pt} \beta }\) , \(\widehat{{\text{k}}}_{{\text{j}}} { = }{{{\text{k}}_{{\text{j}}} } \mathord{\left/ {\vphantom {{{\text{k}}_{{\text{j}}} } \beta }} \right. \kern-0pt} \beta }\) which leads to the following LP problem whose optimal value coincides with that of (1)

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Guevel, H.P., Ramón, N. & Aparicio, J. Benchmarking in data envelopment analysis: balanced efforts to achieve realistic targets. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-06216-w

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Received : 12 October 2023

Accepted : 06 August 2024

Published : 09 September 2024

DOI : https://doi.org/10.1007/s10479-024-06216-w

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