In microeconomics, a production function is a mathematical function that transforms all combinations of inputs of an entity, firm or organization into the output. Given the set of all technically feasible combinations of outputs and inputs, only the combinations encompassing a maximum output for a specified set of inputs would constitute the production function. Data Envelopment Analysis (DEA), which has initially been originated by Charnes, Cooper and Rhodes in 1978, is a well-known non-parametric mathematical method with the aim of estimating the production function. In fact, DEA evaluates the relative performance of a set of homogeneous decision making units with multiple inputs and multiple outputs.

This book covers some basic DEA models and disregards more complicated ones, such as network DEA, and mainly stresses the importance of weights in DEA and some of their applications. As a result, this book mainly considers the multiplier form of DEA models to extend some new approaches, however, the envelopment forms are introduced in some possible approaches. This book also aims at dealing with some innovative uses of binary variables in extended DEA model formulations. The auxiliary variables enable us to formulate Mixed Integer Programming (MIP) DEA models for addressing the problem of finding a single efficient and ranking efficient DMUs. In some cases, the status of input(s) or output(s) measure is unknown and binary variables are utilized to accommodate these flexible measures. Furthermore, the binary variables approach tackles the problem of selecting input or output measures.

The book also stresses the mathematical aspects of selected DEA models and their extensions so as to illustrate their potential uses with applications to different contexts, such as banking industry in the Czech Republic, financing decision problem, technology selection problem, facility layout design problem, and selecting the best tennis player. In addition, the majority of the extended models in this book can be extended to some other DEA models, such as slacks-based measures, hybrids, non-discretionary, and fuzzy DEA which are applicable in some other contexts.

This research-based book contains six chapters as follows:

The first chapter (General Discussion) starts with a simple numerical example to explain the concept of relative efficiency and to clarify the importance of input and output weights in measuring the efficiency score. Then these basic concepts are extended to some more complex cases. Efficient frontiers and projection points are illustrated by means of some constructive and insightful graphs.

The second chapter (Basic DEA Models) presents both envelopment and multiplier forms of the DEA models in the presence of multiple inputs and multiple outputs. However, this book mainly focuses on the multiplier form of DEA models. In addition, this chapter illustrates the role of each axiom to construct the production possibility set (PPS). It is also concerned with some DEA models to deal with pure input data as long as with pure output data sets. Apart from basic input- and output-oriented DEA models with different returns to scale, the chapter includes a model that combines both orientations. Three various case studies involving banking industry, technology selection, and asset financing are provided in this section.

In chapter 3 (GAMS Software), we briefly introduce General Algebraic Modeling System (GAMS) software, a modelling system for linear, nonlinear and mixed integer optimization problems for solving DEA models.

Chapter 4 (Weights in DEA) treats the weights in DEA and their importance along with various weight restrictions and common set of weights (CSW) approaches. The chapter includes Assurance Region (AR) and Assurance Region Global (ARG) methods to restrict weight flexibility in DEA. Two DEA models with different types of efficiency, i.e. minsum and minimax, with their integrated versions are introduced in this chapter. The evaluation of facility layout design problem is addressed as a numerical example.

Chapter 5 (Best Efficient Unit) considers CSW and binary variable approaches as the main tool for developing models that have the capability to find the most efficient DMU and also rank DMUs. We cover WEI/WEO data sets along with multiple input and multiple output data sets. Some epsilon-free DEA models are introduced to overcome the problem of finding a set of positive weights. The problem of finding the most cost-efficient under certain and uncertain input prices is also discussed. Two real data sets involving professional tennis players and a Turkish automotive company are rendered to validate the approaches in this chapter.

 Chapter 6 (Data Selection in DEA) closes the book by considering the data selection problem in DEA and presenting some modifications of the standard DEA models to accommodate flexible and selective measures. To deal with these problems, two multiplier and envelopment DEA models are developed where each model contains two alternative approaches: individual and integrated models. The individual approach classifies flexible measures and identifies selective measures for each DMU, and the aggregate approach accommodates these measures using integrated DEA models. We present three case studies to examine and validate the approaches in this chapter.

Evidently, my deepest gratitude and love go to my family, Laleh and Arad, for supporting me in writing this book. Ronak Azizi saved me a lot of trouble by tackling all formatting issues in Microsoft. Last, but certainly not least, I would like to extend my thanks to my friend, Dr Adel Hatami-Marbini, for helping me with editing the book and for invaluable ideas and comments.

This publication has been elaborated in the framework of the project “Support research and development in the Moravian-Silesian Region 2013 DT 1 - International research teams“ (02613/2013/RRC). Financed from the budget of the Moravian-Silesian Region.