The nonparametric approach to the measurement of productive efficiency can be specified either through a flexible form of the production function which satisfies the efficiency hypothesis, or a set-theoretic characterization of an efficient isoquant. In the first case the production frontier can be of any general shape satisfying some very weak conditions like quasi-concavity or monotonicity, although in most empirical and applied work piecewise linear or, log-linear functions have been frequently used. Thus both Farrell and Johansen applied linear programming models in the specification of the production frontier. Farrell’s efficiency measure is based on estimating by a sequence of linear programs (LPs) a convex hull of the observed input coefficients in the input space. Two features of Farrell efficiency make it very useful in applied research. One is that it is completely data-based i.e., it uses only the observed inputs and outputs of the sample units while assuming production functions to be homogeneous of degree one. Hence it has many potential applications for the public sector units, where for most of the inputs and outputs the price data are not available. For example consider educational production functions for public schools, where outputs such as test scores in achievement tests are only proxy variables for learning; inputs such as average class size, experience of teachers or ethnic background of students do not have observed market prices. Secondly, Farrell’s method uses a set of LP models to estimate the efficiency parameters, so that the production frontier appears as piecewise linear functions. Nonnegativity conditions on the parameter estimates can therefore be easily incorporated.

The Nonparametric Approach

The pressing need of real-world problems for quick, simple, and implementable solutions coupled with a recently increased research productivity on improved and rigorously evaluated heuristic methods are rapidly increasing the need, usage, and respect for heuristics. This paper presents a framework for heuristic “optimization” by systematically examining this change in attitudes towards heuristics, their desirable features and proper usage.

Heuristic “Optimization”: Why, When, and How to Use It

An important list of topics in the physical and social sciences involves continuum concepts and modelling with infinite sets of inequalities in a finite number of variables. Topics include: engineering design, variational inequalities and saddle value problems, nonlinear parabolic and bang-bang control, experimental regression design and the theory of moments, continuous linear programming, geometric programming, sequential decision theory, and fuzzy set theory. As an optimization involving only finitely many variables, semi-infinite programming can be studied with various reductions to finiteness, such as finite subsystems of the infinite inequality system or finite probability measures.

Semi-Infinite Programming and Applications

Multiple criteria decision problems with one decision maker have been recognized and discussed in the recent literature in optimization theory, operations research and management science. The corresponding concept with n-decision makers, namely multicriteria n-person games, has not yet been extensively explored. In this paper we first demonstrate that existing solution concepts for single criterion n-person games in both normal form and characteristic function form induce domination structures (similar to those defined and studied by Yu [39] for multicriteria single decision maker problems) in various spaces, including the payoff space, the imputation space and the coalition space. This discussion provides an understanding of some underlying assumptions of the solution concepts and provides a basis for generalizing and generating new solution concepts not yet defined. Also we illustrate that domination structures may be regarded as a measure of power held by the players. We then illustrate that a multicriteria problem can naturally arise in decision situations involving (partial) conflict among n-persons. Using our discussion of solution concepts for single criterion games as a basis, various approaches for resolving both normal form and characteristic function form multicriteria n-person games are proposed. For multicriteria games in characteristic function form, we define a multicriteria core and show that there exists a single ‘game point’ whose core is equal to the multicriteria core. If we reduce a multicriteria game to a single criterion game, domination structures which are more general than ‘classical’ ones must be considered, otherwise some crucial information in the game may be lost. Finally, we discuss a parametrization process which, for a given multicriteria game, associates a single criterion game to each point in a parametric space. This parametrization provides a basis for the discussion of solution concepts in multicriteria n-person games.

Domination structures and multicriteria problems in n-person games

Some important conceptual problems concerning the application of chance constrained programming CCP to risky practical decision problems are discussed by comparing CCP to stochastic programming with recourse SPR. We expand on Garstka's distinction between mathematical equivalence and economic equivalence showing that much of practical usefulness is lost in the transition between SPR and CCP. By examining the literature on CCP applications we conclude that there is little evidence that CCP is used with the care that is necessary. Finally we conclude that CCP is seriously deficient as a modeling technique and of limited value as a computational device.

Decision Problems Under Risk and Chance Constrained Programming: Dilemmas in the Transition

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

: Several important and efficient methods of solution of specific types of linear programming problems have the feature of sometimes providing optimal solutions which are not extreme-point (or basic) solutions, so that important and useful analyses provided by knowledge of the optimal dual evaluators are not available. It is also often desirable to be able to begin with a solution suggested by knowledgeable persons with experience in the field (or other considerations) and to proceed immediately to a basic solution at least as good as the suggested one. In this paper it is shown how part of the technique of proof of the opposite sign theorem can be employed in a simple algorithm to achieve this end. This method is equally valid when maximizing a nonlinear but convex objective function. A tested ALGOL code is provided for executing the algorithm in a manner compatible (as a procedure) with other programs.

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Extreme point solutions in mathematical programming: an opposite sign algorithm

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.

An extension to the theory of linear programming over generalized networks is presented which replaces the generalized Kirchoff node conditions by chance constraints. The extension is motivated by a class of problems in sanitary and chemical engineering in which the nonzero entries in the generalized incidence matrix may be random variables. Duality relations are established for appropriate pairs of such chance-constrained programming problems by showing that their deterministic equivalents consist of a deterministic generalized network problem and its dual. It is also shown how these duality relations may be exploited in order to obtain actual solutions to chance-constrained generalized network problems.

Chance-Constrained Generalized Networks

A variety of objectives and constraints are presented for use by a regulatory agency in pricing public services and evaluating capacities under conditions where the demands are random variables with probability distributions that depend on prices in different parts of the system. The models studied include a maximization of the joint probability of achieving at least specified levels of consumers' surpluses in these interacting markets, as well as models that constrain the choices of prices and services to ensure that these surpluses are satisfied to at least specified levels of probability, while observing expected fair return constraints for the producers. Chance constrained programming formulations and reductions to deterministic equivalents are developed to use for evaluating levels of service, etc. Some new “coefficients conditions” establishing concavity for quadratic forms are stated and proved in the Appendix to this paper. This also establishes convexity for the deterministic equivalents develope...

William M Raike

Papers

On generation of test problems for linear programming codes

Extreme point solutions in mathematical programming: an opposite sign algorithm

Chance-constrained games with partially controllable strategies.

Chance-Constrained Generalized Networks

Regulatory Models for Pricing and Evaluation of Transport Services