Data Envelopment Analysis Problems Research Articles

The solution of super-scale DEA models based on preference character

The basic relationships and intrinsic laws in the data envelopment analysis (DEA) are the basis for designing fast solutions for big data DEA models. By delving into these relationships, it becomes feasible to address super-scale DEA problems more swiftly, aligning with the need for timeliness in real-world efficiency evaluations and reducing time costs. This paper leverages the intrinsic characteristics of DEA to formulate a DEA model scale reduction approach, offering reliable theoretical support for solving super- scale DEA problems. Initially, the partially ordered set theory to prove the relationship between the efficient decision making units (DMUs) of DEA and the maximal element of the partially ordered set. Building on this, a DEA model based on maximal element set (DEA-PS) is constructed based on the theory of partially ordered set theory, and the equivalence between DEA-PS model and the original DEA model is proved. Furthermore, it is proved that the dominant theorem exists in the solution of multiple classical DEA models, and a method of super-scale DEA models based on dominant theorem is given. The conclusions of this paper theoretically reveal the relationship between efficient DMUs and maximal elements, from this relationship, two key insights emerge: (1) Regardless of how huge the number of DMU is, the proposed method in this paper can realize the scale reduction of the DEA model quickly, (2) The algorithm has good compatibility with all current big data DEA model algorithms.

A combinatorial data envelopment analysis with uncertain interval data with application to ICT evaluation

Information and Communication Technologies (ICTs) have been extensively adopted by firms worldwide due to the significant positive effect on their performance. This fact contrasts with the uncertainty faced by decision makers when entering a country and selecting local firms with which to interact. Consider selecting Decision Making Units (DMUs) according to their relative efficiency, this efficiency being determined via Data Envelopment Analysis (DEA) based on the potential inputs consumed and outputs produced. The values of these variables are uncertain and defined through interval evaluations. Assume now that the interactions may be interrupted several times and new DMUs selected in place of previous ones. The new DMUs may require higher or lower amounts of inputs to produce variable amounts of outputs. The consequences derived from the potential realizations resolving the uncertainty should be incorporated into the DEA problem when deciding which DMUs to interact with and in which order. We study the combinatorial decision framework arising from the potential interactions with new DMUs. A numerical example is provided to complement the problem statement and outline the drawbacks of the existing approaches. It is shown that the selected DMUs and their order may differ substantially when accounting for the complementarities existing among all the DMUs. Moreover, the selection process and any subsequent decision vary with the number of modifications considered relative to the DMU initially selected. A case study analyzing the productive and environmental efficiency of a group of European countries displaying uncertain interval levels of ICT development is presented.

Sequential Malmquist-Luenberger productivity index for interval data envelopment analysis

<p style='text-indent:20px;'>Data envelopment analysis (DEA) based productivity indexes models are widely applied to evaluate the productivity of decision-making units over a period. This study proposes a productivity index for evaluating environmentally sensitive productivity growth while excluding the possibility of spurious technical regress. This innovative index has been created by combining directional distance functions, sequential DEA, undesirable data, and the concept of interval DEA. With this combination, the traditional sequential Malmquist-Luenberger productivity index (SMLPI) has been reformulated as an interval DEA problem to present a novel productivity index named interval SMLPI. We propose a decomposition of the developed index utilizing both constant returns to scale and variable returns to scale frontiers as the benchmark, which allows us to quantify scale efficiency change with interval data. To exhibit the capability of the proposed extension of SMLPI, we model a framework for Indian commercial banks and measure productivity change intervals for twenty-one banks from 2011 to 2018. The empirical findings elucidate that the scale efficiency change plays an essential role in driving productivity change. ICICI Bank had the highest average marginal productivity gain of 1.5007 between 2011 and 2018, whereas Karur Vysya bank had the highest average marginal productivity decline of 0.9411.</p>

빅데이터 DEA를 위한 기계학습의 적용

Data envelopment analysis (DEA) is a tool for identifying best-practices when multiple performance metrics or measures are present for decision-making units (DMUs). As big data issue becomes an important area of supply chain and operations management, DEA is evolving into a data-oriented data science tool for benchmarking, performance evaluation, composite index construction and others. As the number of DMUs increases, the running-time to solve the standard DEA model sharply rises. Such situations are appearing more frequently in the era of big data. This issue could be an important challenge particularly when real-time data stream in at extremely high rates and the DEA analysis needs to be performed very quickly. Therefore, there exist practical needs for developing an efficient way of solving large-scale DEA problems. In this paper, we propose a practical approach for speeding up the DEA efficiency estimation process based on machine learning. In this approach, a sample of DMUs is selected from the population as a training data set, based on which a machine is trained to predict the efficiency scores of unselected or newly streamed-in DMUs. We also suggest a data augmentation technique to enhance the learning process under severe data class imbalance. The superior performance of the proposed approach over the conventional one in terms of efficiency prediction power as well as model computation time is shown through a series of computational experiments using randomly generated data.

The Environmental Efficiency Analysis Based on the Three-Step Method for Two-Stage Data Envelopment Analysis

This paper suggests that the efficiency of a system (decision-making unit) and its subsystem cannot be properly measured using a two-stage data envelopment analysis (DEA) model either in cooperative or non-cooperative evaluation. Indeed, the existing methods subjectively determine the status of the subsystems in the whole system. The two-stage DEA models, either cooperative game or non-cooperative game, are used to analyze the environmental efficiency. However, when the actual relationship between the two subsystems is inconsistent with the subjective relationship assumptions, the overall efficiency of the system and the efficiency of each subsystem will be biased. The conventional two-stage DEA models require predetermining the relationship between the subsystems within the system based on the subjective judgment of the decision-maker. Based on this, this paper proposes a three-step method to solve the two-stage DEA. First, the position relation among subsystems is determined according to the optimal weights through the model. According to the status relationship among subsystems, the decision units are grouped, and the two-stage DEA model of cooperative game or non-cooperative game is used to analyze the efficiency in each group. This method reduces the subjectivity of decision making and analyzes the efficiency of each decision unit applying the most appropriate two-stage DEA model to find the source of inefficiency. Finally, this paper verifies the rationality and validity of the method by analyzing the water use efficiency of industrial systems in China. It is found that most regions in China value economic development more than environmental protection (as evidenced by the DEA weights). What is more, the method proposed by the paper can be generalized for any two-stage DEA problem.

A Fuzzy Approach to Support Evaluation of Fuzzy Cross Efficiency

Cross-efficiency evaluation effectively distinguishes a set of decision-making units (DMUs) via self- and peer-evaluations. In constant returns to scale, this evaluation technique is usually applied for data envelopment analysis (DEA) models because negative efficiencies will not occur in this case. For situations of variable returns to scale, the negative cross-efficiencies may occur in this evaluation method. In the real world, the observations could be uncertain and difficult to measure precisely. The existing fuzzy cross-evaluation methods are restricted to production technologies with constant returns to scale. Generally, symmetry is a fundamental characteristic of binary relations used when modeling optimization problems. Additionally, the notion of symmetry appeared in many studies about uncertain theories employed in DEA problems, and this approach can be considered an engineering tool for supporting decision-making. This paper proposes a fuzzy cross-efficiency evaluation model with fuzzy observations under variable returns to scale. Since all possible weights of all DMUs are considered, a choice of weights is not required. Most importantly, negative cross-efficiencies are not produced. An example shows that this paper’s fuzzy cross-efficiency evaluation method has discriminative power in ranking the DMUs when observations are fuzzy numbers.

A new non-radial directional distance model for data envelopment analysis problems with negative and flexible measures

Data envelopment analysis (DEA) is a mathematical approach for evaluating the efficiency of decision-making units that convert multiple inputs into multiple outputs. Traditional DEA models measure technical (radial) efficiencies by assuming the input and output status of each performance measure is known, and the data associated with the performance measures are non-negative. These assumptions are restrictive and limit the applications of DEA to real-world problems. We propose a new extended non-radial directional distance model, which is a variant of the weighted additive model, to cope with negative data. We then extend our model and use flexible measures, which play the role of both inputs and outputs, to cope with the unknown status of the performance measures. We also present a case study in the automotive industry to exhibit the efficacy of the models proposed in this study.

A novel method for solving data envelopment analysis problems with weak ordinal data using robust measures

The efficiency measurement with traditional data envelopment analysis (DEA) requires precise input and output data. However, the input and output data is imprecise in many real-world problems. Various imprecise methods are developed to estimate the lower and upper bound efficiency scores in the presence of interval and weak ordinal data. We show the existing methods provide an estimation of the lower or upper bound efficiency score and cannot find their exact values in the presence of weak ordinal data. As a result, the ranking of decision making units (DMUs) may not be reliable since they are based on estimated efficiency scores. We propose a pair of DEA models to find the lower and upper bound efficiency scores. We prove the proposed models can find the exact values of the lower and upper bound efficiency scores in the presence of weak ordinal data. However, these exact values depend on a set of parameters in the presence of ordinal data. Therefore, we further propose a novel measure, fathoming the robustness of the efficiency function for the parameter space, to select the best practice DMUs in the presence of imprecise data. We present a real-world application in the space industry to demonstrate the applicability and superiority of the proposed method over the existing methods.

Parallel processing of the Build Hull algorithm to address the large-scale DEA problem

Build Hull is currently one of the most efficient methods to address the large-scale data envelopment analysis problem. In this paper, the computational complexity of Build Hull is established, implying that the dimension of the data set has a strong influence on computational efficiency. Based on the complexity result, we find that the “worst density” of Build Hull monotonically increases with respect to dimension. In addition, a two-phase parallel Build Hull algorithm is proposed to enhance the computational efficiency of Build Hull. The parallel procedure is based on the minimum volume enclosing ellipsoid technique, which enables the exclusion of a large number of DMUs in the first phase. A sensitivity analysis-based technique is also proposed to substantially reduce computational time in the second phase. Numerical tests support our theoretical results.

Metaheuristics for data envelopment analysis problems

In conventional data envelopment analysis (DEA) models, some inputs and outputs may lose their weights in the process of measuring the relative efficiency of DMUs and it leads to ignore corresponding inputs and outputs. Also, using the DEA for a large dataset needs to have a long time with a high-speed computer. In order to solve the problem, genetic algorithm (GA) and differential evolution (DE) algorithm are adopted and proposed in this paper. The GA is used in previous papers but here some operators are used for the first time. Besides, DE algorithm is adopted and used to solve the problem via new operators which are first introduced in this paper. Moreover, a plan is utilised to generate test problems in different sizes in this work. Since the crossover and mutation operators have significant impacts on the algorithms’ performance, all the operators and parameters are calibrated by means of the Taguchi experimental design in order to improve their performances. The performances of the proposed algorithms are then evaluated by comparing their solutions based on the presented instances. Finally, the impacts of increasing the problem size on the performance of our proposed algorithms are investigated.

Modelling Efficiency in Regional Innovation Systems: A Two-Stage Data Envelopment Analysis Problem with Shared Outputs within Groups of Decision-Making Units

Regional Innovation Systems (RISs) literature usually focuses on the comparative performance of different regions and analyzes how each region is utilizing its own dedicated resources. The available resources can be shared by many firms which are grouped by industry, just as universities collaborate with many firms in many industries. This paper studies a region in Mexico and the firms within that region with the aim being to identify which of those firms are using the available resources in the best way. We use Data Envelopment Analysis (DEA) as a methodology for evaluating the relative efficiencies of the firms, based on their multiple inputs and outputs, and considering their processes as being divided into two stages. An important problem in this setting is that the two-stage process exhibits the characteristic of having outputs being shared among the firms in each industry; this makes it more challenging to determine independent efficiency scores for each firm in each industry, where we need to cater for this phenomenon. To address this, the current article presents a methodology for measuring efficiency in situations where Decision Making Units (DMUs) share outputs with other units within the same group. By solving this problem, we can identify the best-performers and their strategies regarding how they use the available resources in the region.

Hybrid Data Envelopment Analysis for Large-Scale Smartphone Data Modeling

This paper deals with the problem of improving the existing optimization techniques for Data Envelopment Analysis (DEA). The algorithm proposed herein is a combination of the “quickhull algorithm” and a DEA algorithm written in Python programming language. To the best of the authors’ knowledge no prior effort has been made to date to propose a methodology for reducing the running time of a DEA problem that incorporates multiple inputs and outputs. The algorithmic implementation is applied on the existing problem of driving efficiency evaluation by exploiting a driving data sample of 10,088 trips collected from smartphone devices. Results indicate that the proposed algorithm is performing relatively well for Big Data compared to other existing DEA algorithmic methodologies that yield the same optimal solution such as Standard DEA and RBE DEA methodologies. The results obtained are calculated for the test sets of 100, 500, 1000, 5,000 and 10,088 Decision-Making-Units (DMUs) and compared in terms of running time of each of the algorithms applied. The results of per trip analysis can be exploited in order to classify trips into different efficiency categories (such as efficient, less efficient, non-efficient) and present their main characteristics.

Novel criterion models in the inverse DEA problem

This article deals with 'inverse' data envelopment analysis (DEA) problem which is a mathematical programming-based technique. The process of checking perturbed DMUs is simplified by proposing a new criterion model. This yields to a reduction of computational complexity for the criterion model. In addition, more realistic criterion model is also proposed and the relationship between existing criterion model and proposed models are discussed. Moreover, it is shown that proposed models solve some problematic failures of the existing inverse DEA models in the literature. Two numerical examples are provided to illustrate the idea. The proposed model are illustrated by a real life data and a comparison between existing criterion model in the literature and proposed criterion models is also provided.

A New Method for Solving Dual DEA Problems with Fuzzy Stochastic Data

Data envelopment analysis (DEA) is a widely used mathematical programming technique for measuring the relative efficiency of decision-making units which consume multiple inputs to produce multiple outputs. Although precise input and output data are fundamentally used in classical DEA models, real-life problems often involve uncertainties characterized by fuzzy and/or random input and output data. We present a new input-oriented dual DEA model with fuzzy and random input and output data and propose a deterministic equivalent model with linear constraints to solve the model. The main contributions of this paper are fourfold: (1) we extend the concept of a normal distribution for fuzzy stochastic variables and propose a DEA model for problems characterized by fuzzy stochastic variables; (2) we transform the proposed DEA model with fuzzy stochastic variables into a deterministic equivalent linear form; (3) the proposed model which is linear and always feasible can overcome the nonlinearity and infeasibility in the existing fuzzy stochastic DEA models; (4) we present a case study in the banking industry to exhibit the applicability of the proposed method and feasibility of the obtained solutions.

Robust optimization with nonnegative decision variables: A DEA approach

Abstract Robust optimization has become the state-of-the-art approach for solving linear optimization problems with uncertain data. Though relatively young, the robust approach has proven to be essential in many real-world applications. Under this approach, robust counterparts to prescribed uncertainty sets are constructed for general solutions to corresponding uncertain linear programming problems. It is remarkable that in most practical problems, the variables represent physical quantities and must be nonnegative. In this paper, we propose alternative robust counterparts with nonnegative decision variables – a reduced robust approach which attempts to minimize model complexity. The new framework is extended to the robust Data Envelopment Analysis (DEA) with the aim of reducing the computational burden. In the DEA methodology, first we deal with the equality in the normalization constraint and then a robust DEA based on the reduced robust counterpart is proposed. The proposed model is examined with numerical data from 250 European banks operating across the globe. The results indicate that the proposed approach (i) reduces almost 50% of the computational burden required to solve DEA problems with nonnegative decision variables; (ii) retains only essential (non-redundant) constraints and decision variables without alerting the optimal value.

Modeling fuzzy data envelopment analysis under robust input and output data

This paper offers a fuzzy optimization framework for data envelopment analysis (DEA) to evaluate the relative efficiency of decision making units (DMUs) with parametric interval-valued fuzzy variable-based inputs and outputs. The parametric interval-valued fuzzy variable-based inputs and outputs is employed to capture the uncertainty of data on the basis of professional judgements or empirical estimations. The DEA problem is formulated as fuzzy expectation model with credibility constraints. When the inputs and outputs are mutually independent parametric interval-valued triangular fuzzy variables, we investigate the parametric equivalent representations of expectation objective function and chance constraints. In order to find the optimal solution of our DEA model, a domain decomposition method is proposed. Finally, the numerical example on the sustainable supplier evaluation and selection problem is provided to demonstrate the efficiency of the proposed DEA model and domain decomposition method.

Relation Between Imprecise DESA and MOLP Methods

It is generally accepted that Data Envelopment Analysis (DEA) is a method for indicating efficiency. The DEA method has many applications in the field of calculating the relative efficiency of Decision Making Units (DMU) in explicit input-output environments. Regarding imprecise data, several definitions of efficiency can be found. The aim of our work is showing an equivalence relation between one of the models of DEA with imprecise data and Multiple Objective Linear Programming (MOLP). The relation between DEA and MOLP was studied to use interactive multiple objective models for solving the DEA problem in exact situation and find the most preferred solution. The aim of this study is to analyze an equivalent relation between imprecise DEA (IDEA) and MOLP models. In this context, we tried to solve IDEA models with interactive project procedure. The Project method is the responsible method, because it can estimate any efficient solution, and it indicates Most Preferred Solution (MPS). In addition, we will use the Data Envelopment Scenario Analysis (DESA) model. The main characteristic of DESA model is to decrease all inputs and increase all outputs and estimate one problem instead of n problems.

Fuzzy efficiency measures in data envelopment analysis using lexicographic multiobjective approach

There is an extensive literature in data envelopment analysis (DEA) aimed at evaluating the relative efficiency of a set of decision-making units (DMUs). Conventional DEA models use definite and precise data while real-life problems often consist of some ambiguous and vague information, such as linguistic terms. Fuzzy sets theory can be effectively used to handle data ambiguity and vagueness in DEA problems. This paper proposes a novel fully fuzzified DEA (FFDEA) approach where, in addition to input and output data, all the variables are considered fuzzy, including the resulting efficiency scores. A lexicographic multi-objective linear programming (MOLP) approach is suggested to solve the fuzzy models proposed in this study. The contribution of this paper is fivefold: (1) both fuzzy Constant and Variable Returns to Scale models are considered to measure fuzzy efficiencies; (2) a classification scheme for DMUs, based on their fuzzy efficiencies, is defined with three categories; (3) fuzzy input and output targets are computed for improving the inefficient DMUs; (4) a super-efficiency FFDEA model is also formulated to rank the fuzzy efficient DMUs; and (5) the proposed approach is illustrated, and compared with existing methods, using a dataset from the literature.

Solving generalized fuzzy data envelopment analysis model: a parametric approach

Data envelopment analysis (DEA) is a non-parametric technique to assess the performance of a set of homogeneous decision making units (DMUs) with common crisp inputs and outputs. Regarding the problems that are modelled out of the real world, the data cannot constantly be precise and sometimes they are vague or fluctuating. So in the modelling of such data, one of the best approaches is using the fuzzy numbers. Substituting the fuzzy numbers for the crisp numbers in DEA, the traditional DEA problem transforms into a fuzzy data envelopment analysis (FDEA) problem. Different methods have been suggested to compute the efficiency of DMUs in FDEA models so far but the most of them have limitations such as complexity in calculation, non-contribution of decision maker in decision making process, utilizable for a specific model of FDEA and using specific group of fuzzy numbers. In the present paper, to overcome the mentioned limitations, a new approach is proposed. In this approach, the generalized FDEA problem is transformed into a parametric programming, in which, parameter selection depends on the decision maker’s ideas. Two numerical examples are used to illustrate the approach and to compare it with some other approaches.

In Search of Efficient Networks Using Bilevel Evolutionary Optimization

This study introduces a method for finding input and output values that result in efficiency for all decision making units (DMUs) in all stages of a production network. This method contrasts with the usual application of DEA for measurement of efficiency and comparison of decision making units. Two-stage data envelopment analysis (DEA) networks represent internal production processes, including inputs to each stage, intermediate variables that serve as outputs from one stage and inputs to another, and feedback among stages. A directed acyclic network is usually used for the measure of efficiency, but the presence of feedback requires an undirected network. The static network with feedback is important for analysis of recycling and re-use processes, but the measures in these cases are non-linear with multiple solutions. Here, a two-stage DEA network was defined as a bilevel optimization where the upper level maximized ParetoKoopmans efficiency while efficiency of each DMU in each stage was calculated in the lower level. In an application to published data from a two-stage network with feedback, hypervolumes were observed where every interior point was Pareto-Koopmans efficient. The Pareto-Koopmans hypervolume associated with the feedback network was modeled with a Bayesian network, and simulations from the model used to visualize efficient space. The use of Bayesian networks to generate virtual DMUs provided a computationally inexpensive method to explore the trade-offs among efficient networks to meet stakeholder preferences. Overall, this application of bilevel evolutionary optimization provides a novel method for solving nonlinear DEA problems with feedback.