Measuring the efficiency of decision making units

In management contexts, mathematical programming is usually used to evaluate a collection of possible alternative courses of action en route to selecting one which is best. In this capacity, mathematical programming serves as a planning aid to management. Data Envelopment Analysis reverses this role and employs mathematical programming to obtain ex post facto evaluations of the relative efficiency of management accomplishments, however they may have been planned or executed. Mathematical programming is thereby extended for use as a tool for control and evaluation of past accomplishments as well as a tool to aid in planning future activities. The CCR ratio form introduced by Charnes, Cooper and Rhodes, as part of their Data Envelopment Analysis approach, comprehends both technical and scale inefficiencies via the optimal value of the ratio form, as obtained directly from the data without requiring a priori specification of weights and/or explicit delineation of assumed functional forms of relations between inputs and outputs. A separation into technical and scale efficiencies is accomplished by the methods developed in this paper without altering the latter conditions for use of DEA directly on observational data. Technical inefficiencies are identified with failures to achieve best possible output levels and/or usage of excessive amounts of inputs. Methods for identifying and correcting the magnitudes of these inefficiencies, as supplied in prior work, are illustrated. In the present paper, a new separate variable is introduced which makes it possible to determine whether operations were conducted in regions of increasing, constant or decreasing returns to scale in multiple input and multiple output situations. The results are discussed and related not only to classical single output economics but also to more modern versions of economics which are identified with "contestable market theories."

http://www.utdallas.edu/~ryoung/phdseminar/BCC1984.pdf

Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis

Fuzzy Set Theory - And Its Applications, Third Edition is a textbook for courses in fuzzy set theory. It can also be used as an introduction to the subject. The character of a textbook is balanced with the dynamic nature of the research in the field by including many useful references to develop a deeper understanding among interested readers. The book updates the research agenda (which has witnessed profound and startling advances since its inception some 30 years ago) with chapters on possibility theory, fuzzy logic and approximate reasoning, expert systems, fuzzy control, fuzzy data analysis, decision making and fuzzy set models in operations research. All chapters have been updated. Exercises are included.

https://www.springer.com/productFlyer_978-0-7923-7435-0.pdf?SGWID=0-0-1297-33300503-0

Fuzzy Set Theory - and Its Applications

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

List of Tables. List of Figures. Preface. 1. General Discussion. 2. The Basic CCR Model. 3. The CCR Model and Production Correspondence. 4. Alternative DEA Models. 5. Returns to Scale. 6. Models with Restricted Multipliers. 7. Discretionary, Non-Discretionary and Categorical Variables. 8. Allocation Models. 9. Data Variations. Appendices. Index.

Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software

Users of linear programming computer codes have realized the necessity of evaluating the capacity, effectiveness, and accuracy of the solutions provided by such codes. Large scale linear programming codes at most installations are assumed to be generating correct solutions without ever having been “bench-marked” by test problems with known solutions. The reason for this failure to adequately test the codes is that rarely are there large problems with known solutions readily available. This paper presents a theoretical justification and an Illustrative implementation of a method for generating linear programming test problems with known solutions. The method permits the generation of test problems that are of arbitrary size and have a wide range of numerical characteristics.

On generation of test problems for linear programming codes

: L. S. Shapley has found a necessary and sufficient condition for the non-emptiness of the core of a characteristic function n-person game stating that the core is non-empty if and only if a certain system of linear inequalities on minimal balanced collection of finite sets is consistent. Using some well known constructs of linear programming, the authors associate to any n-person game two dual linear programming problems in which the constraint set of the primal includes the core of the game, and characterize the non-emptiness of the core in terms of properties of dual optimal solutions of these problems. They then prove the Shapley conjecture on sharpness of the set of proper minimal balanced inequalities with respect to core feasibility of proper n- person games. Using the Farkas-Minkowski Theorem, they obtain a characterization of redundant inequalities with respect to core feasibility and express the rate of growth of the game as a sequence of lower bounds for successive game values corresponding to increasing subsets of the collection of N players, which vitiates the possibility of constraint redundancy. If all game values are non-negative, the characteristic growth rate induces a partial ordering on game values corresponding to subsets of N.

/pdf/on-balanced-sets-cores-and-linear-programming-51x1oprbbv.pdf

On balanced sets, cores, and linear programming

: Process analysis and related approaches to the study of energy economics have made extensive use of Shephard's lemma as well as other aspects of the Shephard-Samuelson transformation theories. A major problem is shown to be present in the use of these transforms to go from cost functions to production possibility sets in that the latter will always be unbounded above. Capacity conditions, which are especially important in energy policy studies, are therefore not adequately addressed. Troubles also occur in the use of translog approximations because of the functional forms which can result when the Shephard-Samuelson transformations are employed. Nondiffferentiability is not the primary difficulty with the translog approximations as is shown with an infinitely differentiable function. Relations between other parts of mathematical transform theory, e.g. as exhibited in Laplace transforms, are also indicated along with possible extensions that might be made in the Shephard-Samuelson 'duality' theories. (Author)

Transforms and approximations in cost and production function relations

Efficiency evaluations in data envelopment analysis are shown to be stable for arbitrary perturbations in the convex hulls of input and output data. Also, the corresponding restricted Lagrange multiplier functions are shown to be continuous. The results are proved using point-to-set mappings and a particular region of stability from input optimization.

Stability of efficiency evaluations in data envelopment analysis

Measurement and evaluation of sales response, in a multiattribute sense, for a product in the usual marketing environment of competing brands has been and continues to be an exceedingly complex and difficult task. It is made more so by the inability to obtain either comprehensive data or sample data that are free from noise factors, not all of which are recognized a priori or a posteriori. For example, even a casual review of the marketing literature would lead one to conclude that even in a heavily researched area such as the advertising—sales response curve there is little conclusive evidence as to the shape of these curves, and that all such investigations are limited by vitrue of ignoring interactions of marketing mix variables. These studies also, of course, treat only one response variable at a time.

Abraham Charnes

Papers

On generation of test problems for linear programming codes

On balanced sets, cores, and linear programming

Transforms and approximations in cost and production function relations

Stability of efficiency evaluations in data envelopment analysis

A Multiperiod Analysis of Market Segments and Brand Efficiency in the Competitive Carbonated Beverage Industry