Although the aim of the present study is not to pinpoint accurately the beginning and end of each phase of the literature, it was observed that the model design phase encompasses a longer period. This results primarily from the small number of publications in the first years of study emerging from the literature maturation process. Then, the first phase of the evolution of DDEA literature presumably involves the models published from to The models applied in the initial phase were not yet sufficiently adapted to extract the inefficiency results for several real processes.

Thus, the reformulation of previously developed dynamic models aimed at improving and designing new dynamic models became more common. The second phase included improvements in the introduction of other methods and more advanced DEA models aimed at accessing properties that are not separately available from their use. The period encompassing the second phase of the literature spans from to However, it is worth noting again that the transition periods between phases are malleable.

Moreover, the validation of these models in practice may demand modeling advancements that can cover a large number of real situations and particularities not previously considered. The categorization of studies enables the observation that some studies became theoretical landmarks for DDEA references because they enhanced the evolution of the literature and provided the rationale for further studies in the field. Thus, the present study aims to identify the studies with higher impact on theoretical and applied constructs in the DDEA modeling literature.

A key factor considered in this design was the relationship between one model and other published studies. Such applied studies are termed the structuring models of the DDEA literature in the present study. Those models were innovative because they introduced new DDEA approaches based on classical models and other studies of the DEA literature, while their proposals enabled authors to conduct several studies. Thus, understanding structuring studies is essential to our general understanding of DDEA efficiency evaluation and its differentiation from static efficiency.

These authors introduce the dynamic aspects of the productive environment to conventional DEA modeling through proposals, and they believe that the decision of one period affects the outputs in another period. The authors also introduced another aspect of modeling with the possibility of entering storable inputs. In this case, the system allows the use of inputs and the production of outputs in each period t , the deduction reserve of inputs from the period for storage and future use, and the use of inventories from previous periods.

The authors establish a set of production possibilities for three periods of analysis in the dynamic context, according to the input resources, and the initial intermediate,. The objective function should be adjusted to each intended analysis situation and the variables to be optimized, and the informed parameters should be changed.

## Data envelopment analysis with missing data a reliable solution method

Each technology uses exogenous inputs and intermediate inputs to produce final outputs and intermediates. The key for the model by FG indicates the possible variables that will be used depending on the chosen method of optimization. However, their objective was to optimize the intermediate input. A transition element has the same characteristic as the outputs, functioning as a quantity of a finished product that will be used only during the following period.

Constraints FG. It is important to note that the basic technology of FG also considers the inputs, intermediate and outputs as parameters. These data may be compared with the variables optimized to measure efficiency according to the optimization focus.

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In the FG model, the measures of efficiency also depend on the chosen objective function. The profits maximization function was adopted in the empirical example provided by the authors, which enabled them to calculate the dynamic efficiency by subtracting the optimized profits by the observed profits and dividing by the observed gross income of the DMU.

The model by NG seeks to minimize the costs of a company by proposing an econometric approach to evaluate its dynamic efficiency. The model by Nemoto and Goto is shown below in detail:. Subject to: NG. Key for the model by Nemoto and Goto : Variables : input i in period t ; : intensity variable of period t ; : intermediate input f in period t ; Parameters : discount factor; : weight of inputs i in period t for the DMU analyzed k ; : weight of intermediate inputs f in period t for the DMU analyzed k ; : intermediate input f of the j th DMU in the period t ; : intermediate input f established for DMUs in the initial period 0; : input i of the j th DMU in period t ; : output r of the j th DMU in period t ; : output r of the DMU analyzed k in period t ;.

Nemoto and Goto do not consider the existence of an intermediate input in the last period of analysis, as shown in the constraint NG. The representation of this specificity and other characteristics of the structure of this model are shown in Fig. Simultaneously, the researcher informs the previous values of those variables for each DMU and period to compare and measure the inefficiency.

Conversely, both elements are evaluation parameters in the structuring model by Tone and Tsutsui The total inefficiency measured by NG assessed the variation of the predicted optimal use of inputs and investments. In conclusion, the model by NG only measures the total inefficiency of a period t , with no reference to an overall dynamic measure resulting from all periods. A positive result for the total inefficiency indicates excessive use of inputs or investments, while a negative result indicates insufficient inputs or investments.

Three possibilities were presented regarding direction: 1 input, 2 output and 3 nonoriented. Tone and Tsutsui have a nonradial structure and consider the nonradial slacks when assessing inefficiency using a different approach from the previous models. The model by TT oriented toward the output and assuming variable returns is described below:.

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Subject to: TT. Fixed inputs and outputs have no radial slacks. The model by TT has one constraint for each performance in period t , given the various possibilities of performance of variables and parameters. The model by FG and the structuring study by NG fail to predict fixed parameters or even the different possibilities of intermediates. Thus, this ability to predict may be indicated as an advantage of the model by TT, which allows different analyses according to the characteristics of the elements through specific constraints.

The model by TT has a number of variables to be optimized that is proportional to the total number of elements considered for the DMUs because it is based in slacks. The other structuring models have no such correlation. Furthermore, it is noteworthy that no optimization of inputs or intermediates occurs in the model by TT as occurs in one of the different possibilities of the model by FG and in the model by NG.

This condition is already shown in the structure of the model by FG in constraint FG. In this case, the following constraints TT. This approach allows for a more accurate evaluation because it considers that the first period of analysis is also affected by the decisions made in the previous period and the performance results from the intertemporal dependence between these periods.

The second possible extension of the model by TT concerns the sharing of products, namely, in situations in which the profit of the period is transformed into outputs and intermediate inputs. According to the authors, this artifice is represented as a desirable link in Fig. The constraints TT. The study by TT is one of the few showing the calculation of the overall efficiency and periods simultaneously. Both studies Tone and Tsutsui, , ; Kao, claim that period efficiencies may have multiple optimal solutions.

Unlike the overall dynamic efficiency, period efficiencies are not compatible between DMUs Kao, In the present study, the three models identified as structuring models have different characteristics and approaches to the evaluation of dynamic efficiency. DEA may be widely applied in business management. Thus, there is a clear opportunity to apply the models designed toward solving real problems and enriching the DEA literature. The applications also allow for comparison of the results generated among the different approaches proposed, showing the advantages and disadvantages of the models Kao, Indeed, DDEA models can also be modeled and solved by different optimization tools, requiring the user to write down the model in the appropriated languages required by the solvers.

In this case, we compiled the map of the evolution of the DDEA literature. We observe that 72 studies with applications of DDEA modeling are distributed in various areas, which may be related to the recent expansion of publications, as shown in the evolution map. Other representative sectors are common to both models, including banking and agriculture and farms. Thus, the authors used the artifice of classifying the industrial sectors with few applications as miscellaneous, whereas disciplines indicates studies addressing management issues without observing a specific segment of companies.

Therefore, such studies were categorized as miscellaneous.

Conversely, the study by Chen and van Dalen , observing the impact of marketing campaigns on the automotive and pharmaceutical industries, was classified as disciplines. DMUs may cover the analysis of various sets because of the flexibility of DEA studies: industrial sectors; service companies, including banks, hotels, hospitals, employment agencies; transport infrastructure, including railways and harbors; countries, among others. In regard to the geographical distribution of the evaluated DMUs and, consequently, the studies, such evaluation was observed at a higher concentration in the United States, followed by China and Taiwan and Japan.

Some studies analyzing the global performance perspective were found, while the dissemination of studies in other countries with small ratios, including Australia, Brazil, Chile and Croatia, among others, was also observed. This distribution indicates the dissemination of modeling concepts resulting from the maturation of the literature, allowing the use of DDEA in areas and places previously not covered. The numbers of DMUs and elements inputs, outputs and intermediates are also different for each study. It was observed that some studies treated the data only to test models and, therefore, did not use the proportionality criterion Kao, Such studies included the intermediates between input variables aimed at complying with the guidelines.

Progress was made in detailing the information on DDEA applications when specifically observing the intermediates indicated in the dynamic models as responsible for the interdependence between periods. Accordingly, the following transition elements among facilities were considered: generation capacity of electric utilities evaluated by Tone and Tsutsui , ; the land, buildings, machinery and equipment of the farms observed by Silva and Stefanou , , among others.

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We can state that future research directions in DDEA have three main axes. The first axis concerns the evolution in the variable selection procedures. When too many variables exist, the researcher may apply a filtering procedure in order to identify which variables are the most representative for the application at hand.

This can be performed by techniques such as neural networks, Analytic Hierarchy Process and Principal Component Analysis. We believe that more studies are necessary in this research direction to evaluate the impact of the proportionality between the number of variables, parameters, DMUs and, particularly, the number of time periods of the model, in order to produce meaningful results for decision analysis. The second axis is related to reformulation of DEA models to a dynamic context as well as the hybridization of the proposed DDEA models with mathematical techniques.

A third axis concerns the proposition of new robustness tests for validating DDEA results. Skevas and Oude Lansink used ordinary least squares regression and bootstrap techniques to explain factors that influence DMU's performance. However, this is very limiting, especially if the secondary data is employed, since the use of the tool is reduced to only one picture of the current situation. We believe that the assessment of dynamic performance is an open research venue. We expect that in some cases, researchers may visit in loco their DMUs in order to have a better portrait of their reality and understand the reasons that lead to inefficiency.

Finally, we would like to emphasize the importance of software developments in order to popularize the DDEA models among practitioners. Many dynamic models have been proposed in the literature, though the vast majority of them are not available as software modules or packages. The evolution of the dynamic models literature in the context of DEA could be mapped in the present study by structuring the 80 articles published from to Previously, much had already been researched and published on intertemporal evaluations of performance.

However, these DDEA models initially included transition elements among their variables to represent the dependency of DMUs between periods of time. Following such developments, several studies on DDEA modeling were conducted.

The categorization showed an increase in the number of studies in recent years, accompanied by a trend of even further growth in publications. Structuring the analyzed studies enabled identifying that the evolution of the DDEA literature consists of three phases: 1 design; 2 refinement; and 3 application. The design phase characterizes the introduction of DDEA models driven by the limitations of existing DEA models in the evaluation of intertemporal performance.

The refining phase covers the period of improvement of the initial DDEA models, the inclusion of concepts related to other methods and the design of new models combining more robust DEA models to evaluate more complex systems. The third phase is the period of the literature and reflects the dissemination of models through various applications of models already established. The performance of the evaluated literature also suggests that the refinement phase of the dynamic models shall be repeated in the coming years, continuing the inclusion and advancement of several DEA approaches.

The trend to continue those advances conducting studies on models even more adapted to real situations is predicted. When analyzing the areas approached thus far, energy and transportation have the highest number of applied studies. The applications of DDEA modeling also proved interested in introducing elements related to the DMU facilities in the evaluation, including equipment, work areas and production capacities, among others. This inclusion demonstrates the importance of considering these factors in intertemporal evaluations. Although assuming few variations over time, the published studies show that such elements affect the performance of DMUs, and dynamic efficiency measures are usually more accurate than static measures.

Two other review papers on DDEA models were recently published in the literature. Our review is specially addressed to new researchers in the domain, showing the starting point for DDEA studies. The performed comparison of the structuring DDEA models contributes to the complete comprehension of these models as well as to understanding the evolution of the models and computational methods over time, as shown in Figs. Bayesian Data Analysis.

## Data envelopment analysis (DEA) – Thirty years on - Dimensions

Data analysis briefbook. Applied Life Data Analysis. Dyadic Data Analysis. Longitudinal Data Analysis. Microarray Data Analysis. Recommend Documents. Hillier Stanf Goos, J. Hartmanis, and J. Your name. Close Send. This study proposes a fuzzy set approach to deal with missing values.

The value of a DMU in an input or output which is missing is represented by a triangular fuzzy number constructed from the values of other DMUs in that input or output. A fuzzy DEA model is then used to calculate the efficiencies, which are usually also fuzzy numbers.

While the conventional DMU-deletion method will overestimate the efficiencies of the remaining DMUs, the fuzzy set approach produces results which are very close to those calculated from complete data. The average error in estimating the true efficiency is less than 0. Most importantly, the fuzzy set approach is able to calculate the efficiencies of all DMUs, including those with some values missing. Data envelopment analysis with missing data a reliable solution method.