Performance evaluation using Data Envelopment Analysis (DEA)

Abstract

The study proposes the use of Data Envelopment Analysis (DEA) to analyse the performance of Decision Making Units (DMUs). DEA models are optimisation models, whereyou have an objective function to minimise or maximise based on data. Efficiency in DEA heavily relies on this concept. However, the optimal value is a single value which changes as the sample data changes. In other words, DMUs may be deemed efficient under one data set and inefficient under another. Due to the sample data nature used in statistics and DEA is a data driven method, the question is: How does one conclude whether a particular DMU is or is not efficient? This study attempts to answer this question by creating confidence intervals for such DMUs using the non-parametric bootstrap resamplingmethod. The core concept of bootstrapping involves using resampled data from an original sample to mimic the process of making inferences about a larger population. Since the actual population remains unknown, determining the accuracy of a sample statistic compared to its true population value is challenging. However, bootstrapping resolves this issue by treating the sample as a proxy for the population, allowing for measurable assessments of inference quality when applying the resampled data to the original sample. More formally, the bootstrap works by treating inference of the true probability distribution, given the original data, as being analogous to an inference of the empirical distribution, given the resampled data. The accuracy of inferences regarding the true probability distribution using the resampled data can be assessed because we know its estimate. If the estimate is a reasonable approximation to the true probability distribution, then the quality of inference on the true probability distribution can in turn be inferred. This study explores the concept of efficiency and its evolution within the framework of Data Envelopment Analysis (DEA), motivated by the need for objective performance evaluation across various domains. A comprehensive theoretical foundation is laid out, detailing key DEA models and mathematical extensions, including those that accommodate ratio data which are critical for real-world applications. The study also incorporates the non-parametric bootstrap resampling method to enhance the robustness of the efficiency analysis. The models considered include the Banker, Charnes, and Cooper (BCC) model, the Slacks-Based Measure (SBM) model, and the Additive (ADD) model. The DEA methodology is applied to two primary domains: professional football and finance. In the context of football, the study evaluates the performance of players during the 2020/2021 season across 18 top leagues and competitions worldwide. It critiques the current FIFA awards selection process, which heavily relies on subjective expert voting and often favours players from European teams or those in attacking positions. DEA is proposed as a more objective alternative, focusing on individual player performance based on 20 selected variables such as minutes played, assists, penalty goals, and clean sheets. The DEA results aligned with the FIFA rankings for 4 of the 11 nominees: Robert Lewandowski, Lionel Messi, Mohammed Salah, and Karim Benzema, whilst offering contrasting evaluations for others like Cristiano Ronaldo and Neymar. Ultimately, the models supported Lewandowski’s win but highlighted inconsistencies in the current selection system. In the financial domain, DEA models are employed to evaluate the performance of companies listed on the Johannesburg Stock Exchange (JSE) Top 40. The goal is to identify efficient companies based on fundamental performance metrics, such as dividends per share, current ratio, and quick ratio, rather than stock price behaviour. This approach is especially beneficial for investors prioritising dividend income. The results showed that both classical and bootstrap versions of the SBM and BCC models were consistent in selecting efficient companies, while the bootstrap version of the Additive model was overly conservative, labeling all companies as inefficient. Building on the financial analysis, the study applies portfolio theory to the DEA identified efficient companies. Optimal portfolios are constructed and tested using out-of-sample data, with their performance compared to that of the JSE Top 40 index. While the index outperformed most portfolios, the medium-risk portfolio derived from the classical Additive model exceeded the index in terms of profitability. This portfolio included stocks such as Amplats, Anglogold, Bats, Capitec, Richemont, Firstrand, and Pepkorh. Overall, the research demonstrates the value of DEA as a rigorous, data-driven tool for evaluating performance in both sports and financial markets, offering a more objective foundation for decision-making and recognition.