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

The usual properties of a characteristic function game were derived byvon Neumann andMorgenstern from the properties of a game in normal form. In this paper we give a linear programming principle for the calculation of the characteristic function. The principle is a direct application ofCharnes' linear programming method for the calculation of the optimal strategies and the value of a two-person zero-sum game. The linear programming principle gives another method for proving the standard properties of a characteristic function when it is derived from a game in normal form. Using an idea originated byCharnes for two person games, we develop the concept of a constrainedn-person game as a simple, practical extension of ann-person game. However the characteristic function for a constrainedn-person game may not satisfy properties, such as superadditivity, usually associated with a characteristic function.

Constrainedn-person games

Two problems in linear programming associated with data envelopment analysis (DEA) namely, employing non-archimedean infinitesimals, transcendemals (‘big Ms’) and categorical variables (in a new non-archimedean formulation) are addressed. A new more sophisticated pricing procedure as part of an adjacent extreme point algorithm solves these efficiently in the base field. Employing this in Charnes's non-archimedean simplex algorithm led to a ninefold increase in computational speed on large DEA problems with about 1000 decision-making units. Additionally, computational failures due to cycling and/or conditioning instabilities were eliminated.

Non-archimedean infinitesimals, transcendentals and categorical inputs in linear programming and data envelopment analysis

Asymptotic duality over closed convex sets.

This paper is the second of a series directed at the investigation of relations of chance-constrained programming to problems in game theory. The model introduced and analyzed herein is a two-person game model with zero-sum payoff matrix in which the strategies selected by the players do not in themselves determine the payoffs, but in which random perturbations with known distributions modify the strategy of each player before actual implementation of the strategies. A major hypothesis is that, while the strategy perturbations may be random vectors with known distributions, the selection of strategies (or more accurately strategy policies) is to be made before any observations of the random variables are made so that the strategies chosen are to be “zero-order” decision functions of the random perturbations, in the customary terminology of chance-constrained programming. The strategy selected by each player is to be chosen to extremize that payoff which the player can be assured with at least some (a prio...

Chance-constrained games with partially controllable strategies.

Some theorems are given which relate to approximating and establishing the existence of solutions to systemsF(x) = y ofn equations inn unknowns, for variousy, in a region of euclideann-space E
 n
 . They generalize known theorems. Viewing complementarity problems and fixed-point problems as examples, known results or generalizations of known results are obtained. A familiar use is made of homotopies H: E
 n
 × [0, 1]→E
 n
 of the formH(x, t) = (1 −t)F
0
(x) + t[F(x) − y] where theF
0 in this paper is taken to be linear. Simplicial subdivisionsT

 k
 of E
 n
 × [0, 1] furnish piecewise linear approximatesG

 k
 toH. The basic computation is via the generation of piecewise linear curvesP

 k
 which satisfyG

 k
 
(x, t) = 0. Visualizing a sequence {T

 k
 } of such subdivisions, with mesh size going to zero, arguments are made on connected, compact limiting curvesP on whichH(x, t) = 0. This paper builds upon and continues recent work of C.B. Garcia.

Abraham Charnes

Papers

Constrainedn-person games

Non-archimedean infinitesimals, transcendentals and categorical inputs in linear programming and data envelopment analysis

Asymptotic duality over closed convex sets.

Chance-constrained games with partially controllable strategies.

Constructive proofs of theorems relating to:F(x) = y, with applications