Data envelopment analysis cross-efficiency model in fuzzy environments

<abstract><p>Data Envelopment Analysis (DEA) is a prominent technique for evaluating the performance and ranking of a set of decision-making units (DMUs) that transform multiple inputs into multiple outputs. However, one of the challenges of the primary DEA models is facing imprecise data in real practical problems. To address this issue, fuzzy DEA have been proposed, which have been successfully applied in many real fields. On the other hand, in some real-world DEA applications, the primary objective of performance evaluation is the ranking of a group that consists of several DMUs that are typically under the control of a centralized management. In this paper, we try to use the theory of cooperative games and Shapley value method as a fair method to solve such games in order to rank groups in DEA. In this way, the resulting rank for groups is based on the average marginal shares of groups in different coalitions that are formed based on the theory of cooperative games. We applied the proposed method to rank groups of airlines considering fuzzy data. To the best of authors' knowledge, so far, no method has been presented in DEA literature for ranking groups in fuzzy environment and using game theory techniques.</p></abstract>

Data Envelopment Analysis (DEA) method is used to evaluate the efficiency of the bank branches. The present paper prioritized the indicators, using fuzzy questionnaires and by gathering the managers’ ideas, in order to obtain the most important evaluating indicators. After obtaining these indicators, Data Envelopment Analysis method was used to determine the performance bound interval, regarding the fact that the data are related to a certain period of time and their nature is imprecise and fuzzy. Then, Fuzzy Theory method and Fuzzy Data Envelopment Analysis method were employed to rank the branches, and finally the results were presented.

DEA cross-efficiency evaluation based on Pareto improvement

In this paper, we examine the cross-efficiency concept in data envelopment analysis (DEA). Cross efficiency links one decision-making unit's (DMU) performance with others and has the appeal that scores arise from peer evaluation. However, a number of the current cross-efficiency approaches are flawed because they use scores that are arbitrary in that they depend on a particular set of optimal DEA weights generated by the computer code in use at the time. One set of optimal DEA weights (possibly out of many alternate optima) may improve the cross efficiency of some DMUs, but at the expense of others. While models have been developed that incorporate secondary goals aimed at being more selective in the choice of optimal multipliers, the alternate optima issue remains. In cases where there is competition among DMUs, this situation may be seen as undesirable and unfair. To address this issue, this paper generalizes the original DEA cross-efficiency concept to game cross efficiency. Specifically, each DMU is viewed as a player that seeks to maximize its own efficiency, under the condition that the cross efficiency of each of the other DMUs does not deteriorate. The average game cross-efficiency score is obtained when the DMU's own maximized efficiency scores are averaged. To implement the DEA game cross-efficiency model, an algorithm for deriving the best (game cross-efficiency) scores is presented. We show that the optimal game cross-efficiency scores constitute a Nash equilibrium point.

Data envelopment analysis (DEA) provides a relative efficiency measure for peer decision making units (DMUs) with multiple inputs and outputs. While DEA has been proven an effective approach in identifying the best practice frontiers, its flexibility in weighting multiple inputs and outputs and its nature of self-evaluation have been criticized. The cross efficiency method was developed as a DEA extension to rank DMUs with the main idea being to use DEA to do peer evaluation, rather than in pure self-evaluation mode. However, cross efficiency scores obtained from the original DEA model are generally not unique, and depend on which of the alternate optimal solutions to the DEA linear programs is used. The current chapter discusses various cross efficiency approaches in dealing with non-unique solutions from DEA

Group decision making based on the modified probability calculation method and DEA cross-efficiency with probabilistic hesitant fuzzy preference relations

The author of the article describes the theoretical assumptions of the DEA method used for measuring the productivity of airports described in the article ‘Research into the methods of analysing the productivity indicators of transport terminals’ (Jaržemskienė 2009). The essential insights presented in the above mentioned paper reveal that Data Envelopment Analysis (DEA) is a relatively new ‘data oriented’ approach to evaluating the performance of the so-called Decision Making Units (DMU) that convert multiple inputs into multiple outputs. The article focuses on the findings of the study carried out by the author in accordance with those assumptions. Research represents the Data Envelopment Analysis (DEA) method testing 15 selected airports situated in the Baltic Region, including Vilnius, Kaunas, Palanga, Riga, Tallinn, St. Petersburg, Helsinki, Turku, Stockholm, Malme, Copenhagen, Hamburg, Gdansk, Warsaw and Minsk. Airport productivity indicators are ranked considering importance and using the method of Delphi expert survey made of two rounds. The author presented the following indicators (expressed as ‘ratio’) as the major ones estimated by PAX/LAND, AIR/LAND, PAX/AIR, PAX/RW, PAX/RWA, GA and INTER experts. The succeeding indicators were introduced by PAX/TERMAREA, PAX/GATES, AIR/RW, AIR/RWA, AIR/TERMAREA, AIR/GATES, FR/LAND and FR/RW. 10 indicators were accepted as the most important and selected from the current set in the following sequence: AIR/LAND, AIR/RW, PAX/RW, PAX/LAND, AIR/RWA, PAX/AIR, PAX/RWA, AIR/TERMAREA, PAX/GATES and PAX/TERMAREA. AIR/LAND and AIR/RW were submitted as two main indicators. The acronyms are explained as follows: LAND – airport area, RWA – runway length, PAX – the number of passengers, AIR – the number of aircraft take-offs and landings, RW – the number of runways, GATES – the number of gates, FR – the amount of freight served, TERMAREA – the area of passenger terminal, GA – a general aviation market share of airport served aircraft by percentage, INTER – the percentage of international passengers considering all passengers served by airports. After two key productivity indicators were chosen conducting the expert survey, airport productivity was compared applying the DEA method.

In this paper, we focus on measuring the performance efficiencies of decision making units (DMUs) using dual slack based measure (SBM) model with fuzzy data in data envelopment analysis (DEA). In the conventional dual SBM model, the data and the weights of input and output are found as crisp quantities. However, in real world applications, input-output data and input-output weights may have vague/uncertainty due to various factors such as quality of treatment and medicines, number of medical and non-medical staffs, number of patients, etc. in health sector. To deal with such uncertainty, we can apply fuzzy set theory. In this paper, we propose a SBM model with fuzzy weights in Fuzzy DEA (FDEA) for fuzzy input and fuzzy output. This model is then reduced to a crisp LPP by using expected values of a fuzzy number (FN). Finally, we present an application of the proposed model to the health sector, consisting of two input variables as (i) sum of number of doctors and staff nurses (ii) number of pharmacists and two output variables as (i) number of inpatients (ii) number of outpatients. Both the input variables and output variables are considered as TFNs.

The ranking of the decision making units (DMUs) is an essential problem in data envelopment analysis (DEA). Numerous approaches have been proposed for fully ranking of units. Majority of these methods consider DMUs with optimistic approach, whereas their weaknesses are ignored. In this study, for fully ranking of the units, a modified optimistic–pessimistic approach, which is based on game cross efficiency idea is proposed. The proposed game like iterative optimistic-pessimistic DEA procedure calculates the efficiency scores according to weaknesses and strengths of units and is based on non-cooperative game. This study extends the optimistic-pessimistic DEA approach to obtain robust rank values for DMUs. The proposed approach yields Nash equilibrium solution, thus overcomes the problem of non-uniqueness of the DEA optimal weights that can possibly reduce the usefulness of cross efficiency. Finally, in order to verify the validity of the proposed model and to show the practicability of algorithm, we apply a real-world example for selection of industrial R&D projects. The proposed model can increase the discriminating power of DMUs and can fully rank the DMUs.

The article describes the Data Envelopment Analysis (DEA) method and the main features of its application. The main problems of heat and power systems are described, which are addressed by the DEA method of efficiency assessment presented in the article. The approbation of this method is presented at the objects of the centralized municipal heat supply system of the fuel and energy complex: boiler houses and heat and power plants. 9 objects were analyzed according to four input indicators: available heat capacity, installed heat capacity, heat consumption for own needs, fuel consumption. Also, the efficiency of the system was evaluated according to two output indicators: the release of thermal energy to the grid and the mass of the emission. As a result of the analysis and calculations made, it was revealed that 5 objects have the maximum possible efficiency indicator equal to 1, that is, they function as efficiently as possible. 4 objects of the centralized municipal heat supply system have an efficiency indicator less than 1. Accordingly, improvements are required for the operation of the above Decision-Making Units (DMU)s. These objects have deviations in terms of the inputs and outputs of the actual data and those obtained using the DEA method. Based on the calculations obtained for these 4 objects, the article provides recommendations for changing the quantitative values of their input and output indicators. For example, for object number 2, it is recommended to reduce the installed heat capacity in the grid by 72.57%, without changing the available heat capacity and fuel consumption. Reduce the heat consumption for your own needs by 69.383%. In addition, it is recommended to increase the supply of thermal energy to the grid by 6,034%, and reduce the mass of emission by 11.5%. Specific measures have also been developed to modernize the studied objects in order to achieve the recommended indicators of inputs and outputs. The research results presented in the article are of scientific and practical interest and can be used to improve the efficiency of heat and power systems facilities

A new ranking method to fuzzy data envelopment analysis

This study aims to analyze the technical efficiency of herbal product sales at PT X, Sleman branch, with a focus on determining the most efficient input and output variables. The research adopts a descriptive quantitative approach, utilizing the Data Envelopment Analysis (DEA) method to measure efficiency. Several input variables are assessed, including the number of active employees, basic salary, allowances, and bonuses, while the output variable considered is the total amount of herbal product sales. The results, obtained through efficiency measurement using the WinDEAP software, reveal that 50% of the Decision-Making Units (DMUs) are inefficient. Specifically, DMU 1, DMU 3, DMU 4, DMU 5, and DMU 6 fail to achieve optimal efficiency levels. To address this, recommendations for improvement are provided by referring to the efficient DMUs. These recommendations involve tailored input and output adjustments aimed at optimizing sales. Efficiency can be realized if each employee earns a salary of Rp 4,109,733.00 while achieving a minimum sales target of 146 herbal products. Following these recommendations, an assessment of employee performance is conducted. The findings indicate that 30% of marketing employees demonstrate efficient sales performance, whereas 70% still struggle to reach efficiency targets, necessitating further adjustments.

Self-reported behavioural data, being often linguistic variables that represent a qualitative measure of respondents’ opinions/attitudes, are vague, uncertain, and fuzzy in nature. A road safety performance index, based on these fuzzy data, should consider this uncertainty. In this study, fuzzy numbers were used to describe self-reported behaviour on Montenegrin roads, which was further integrated into the data envelopment analysis (DEA), a technique for measuring the relative performance of decision-making units (DMUs). The vagueness of the performance scores obtained in this way was treated with grey relational analysis (GRA). GRA was applied to the cross-efficiency (CE) matrix constructed by the DEA to distinguish Montenegrin municipalities’ performance, with the main goal of describing road safety in the observed territories in the environment of uncertain/grey data. It is concluded that the proposed DEA–GRA model, based on fuzzy data, provides a more reasonable and encompassing measure of performance, and with which the overall ranking position of municipalities can be obtained.

Data Envelopment Analysis (DEA) is the most popular mathematical approach to assess efficiency of decision-making units (DMUs). In complex organizations, DMUs face a heterogeneous condition regarding environmental factors which affect their efficiencies. When there are a large number of objects, non-homogeneity of DMUs significantly influences their efficiency scores that leads to unfair ranking of DMUs. The aim of this study is to deal with non-homogeneous DMUs by implementing a clustering technique for further efficiency analysis. This paper proposes a common set of weights (CSW) model with ideal point method to develop an identical weight vector for all DMUs. This study proposes a framework to measuring efficiency of complex organizations, such as banks, that have several operational styles or various objectives. The proposed framework helps managers and decision makers (1) to identify environmental components influencing the efficiency of DMUs, (2) to use a fuzzy equivalence relation approach proposed here to cluster the DMUs to homogenized groups, (3) to produce a common set of weights (CSWs) for all DMUs with the model developed here that considers fuzzy data within each cluster, and finally (4) to calculate the efficiency score and overall ranking of DMUs within each cluster.

ABSTRACTThis paper proposes a new method to evaluate decision-making units (DMUs) under uncertainty using fuzzy data envelopment analysis (DEA). In the proposed multi-objective nonlinear programming methodology, both the objective functions and the constraints are considered fuzzy. This model is comprehensive in dealing with uncertainty, in the sense that coefficients of the decision variables in the objective functions and in the constraints, as well as the DMUs under assessment, are assumed to be fuzzy numbers with triangular membership functions. A comparison between the current fuzzy DEA models and the proposed method is illustrated by a numerical example.