Jati K. Sengupta

Safety-First Rules under Chance-Constrained Linear Programming

The approach of chance-constrained linear programming is analyzed here in the context of safety-first principles based on Tchebycheff-type inequalities. The analysis attempts to define relatively distribution-free tolerance levels and the incidence of nonnormality in chance-constrained linear programming.

An application of sensitivity analysis to a linear programming problem

SummaryIn an ordinary linear programming problem with a given set of statistical data, it is not known generally how reliable is the optimal basic solution. Our object here is to indicate a general method of reliability analysis for testing the sensitivity of the optimal basic solution and other basic solutions, in terms of expectation and variance when sample observations are available. For empirical illustration the time series data on input-output coefficients of a single farm producing three crops with three resources is used. The distributions of the first, second, and third best solutions are estimated assuming the vectors of net prices and resources to be constant and the coefficient matrix to be stochastic. Our method of statistical estimation is a combination of the Pearsonian method of moments and the maximum likelihood method.In our illustrative example we observe that the skewness of the distribution of the first best solution exceeds that of the distributions of the second and third best solution. We have also analyzed the time paths for the three ordered solutions to see how far one could apply the idea of a regression model based on inequality constraints. A sensitivity index for a particular sample is suggested based on the spread of the maximum and minimum values of the solutions.ZusammenfassungIm allgemeinen ist bei Linear-Programming-Problemen mit statistischen Einflüssen die Zuverlässigkeit der optimalen Basislösung nicht bekannt. Unser Ziel ist es, eine allgemeine Methode anzugeben, um die Empfindlichkeit der optimalen Basislösung und anderer Basislösungen durch den Erwartungswert und die Varianz bei gegebener Stichprobe zu testen. Zur Illustration wird eine Zeitreihe der input-output-Koeffizienten einer einzigen Farm benutzt, die drei Getreidesorten erzeugt, wobei drei Ressourcen benützt werden. Es werden die Verteilungen der ersten drei besten Lösungen geschätzt bei vorausgesetzten konstanten Nettopreisen und Ressourcen und stochastischer Koeffizientenmatrix. Die verwendete Methode der statistischen Schätzung ist eine Kombination der Pearsonschen Momentenmethode und der Maximum-Likelihood-Methode.In unserem Beispiel stellen wir fest, daß die Schiefe der Verteilung der besten Lösung größer ist als die der Verteilung der zweit- und drittbesten Lösungen. Ferner wurden die Zeitläufe der ersten drei geordneten Lösungen analysiert, um festzustellen, wie weit sich die Idee eines Regressionsmodells, das auf Ungleichungsrestriktionen basiert, anwenden läßt. Für eine Stichprobe wird ein Empfindlichkeitsindex empfohlen, der sich aus der Spannweite der maximalen und minimalen Werte der Lösungen ableitet.

A weak duality theorem for stochastic linear programming

SummaryA linear programming problem is said to be stochastic if one or more of the coefficients in the objective function or the system of constraints or resource availabilities is known only by its probability distribution. A distinction is usually made between two related approaches to stochastic linear programming, the active and passive approach respectively. An extension of the duality theorem of non-stochastic or deterministic programming problem has been attempted in this paper in the area of stochastic linear programming in its two approaches. The method of proof is based on the idea that since the parameter space defined by a stochastic linear programme is the topological product of the real line with itself, it forms a first countable topological space. Using a set of distinct and selected points in the parameter space the concepts of feasibility, optimality and duality are extended to stochastic linear programming problems of arbitrary dimensionality. Based on the non-singular regions of the parameter space of a stochastic linear programming problem the theorem utilizes the conditions of convergence of the sequence of distinct and selected points in the parameter space to a limit point and thereby generalizes the duality theorem in the stochastic case. Furthermore it is shown that the regions of feasibility of the active and passive approaches of stochastic linear programming may be different, so that on this basis it may be possible to establish some inequality relations for the optimal solutions defined for the respective feasible regions.ZusammenfassungEin lineares Programmproblem wird stochastisch genannt, wenn ein oder mehrere Koeffizienten der Zielfunktion oder des Systems der Beschränkungen oder der verfügbaren Ressourcen nur durch ihre Wahrscheinlichkeitsverteilung bekannt sind. Gewöhnlich wird zwischen zwei verwandten Verfahren für das stochastische lineare Programmieren unterschieden, dem aktiven und dem passiven Verfahren.In der vorliegenden Arbeit wird versucht, das für nichtstochastische oder deterministische Programmprobleme gültige Dualitätstheorem unter Berücksichtigung beider Verfahrensweisen auf den Bereich des stochastischen linearen Programmierens auszudehnen. Der Beweis gründet sich auf den Gedanken, daß der Parameterraum einen abzählbaren topologischen Raum bildet, da er — durch ein stochastisches lineares Programm definiert — das topologische Produkt der reellen Achse mit sich selbst ist. Unter Benutzung einer Menge von verschiedenen und ausgewählten Punkten im Parameterraum werden die Begriffe der Zulässigkeit, der Optimalität und der Dualität auf stochastische lineare Programmprobleme beliebiger Dimension ausgedehnt. Auf der Grundlage nichtsingulärer Bereiche des Parameterraumes eines stochastischen linearen Programmproblems benutzt das Theorem die Bedingungen für die Konvergenz einer Folge verschiedener und ausgewählter Punkte des Parameterraumes nach einem Grenzpunkt und verallmeinert damit das Dualitätstheorem für den stochastischen Fall. Weiter wird gezeigt, daß die Zulässigkeitsbereiche der aktiven und passiven Verfahren des stochastischen linearen Programmierens verschieden sein können, so daß es möglich sein kann, gewisse Ungleichungen für die optimalen Lösungen aufzustellen, die für die entsprechenden zulässigen Bereiche definiert sind.

STOCHASTIC LINEAR PROGRAMMING AND ITS APPLICATION TO ECONOMIC PLANNING

This chapter discusses stochastic linear programming and its application to economic planning. An extension of the duality theorem of nonstochastic programming in the area of stochastic linear programming in its two approaches has been considered using the properties of a first countable topological space and the convergence of the sequence of distinct and selected points in the parameter space to a limit point. When the cost of sampling is introduced in the parameter space defined by a stochastic linear program, this line of discussion of the active approach would lead to the sequential probability ratio tests in some modified manner. A third class of decision rules, much more general than the first two classes, may be defined by alternative rules of mapping from the uncertainty space to the certainty space.

A system reliability approach to linear programming

SummaryA system reliability approach to linear programming is developed here for the case when the restrictions are chance-constrained. Methods of characterizing a system reliability measure for a linear programming system, its implications under alternative probability distribution assumptions and its uses for specifying policies with an improved system reliability routine are analytically discussed.ZusammenfassungIn der vorliegenden Arbeit wird ein Verfahren zur Untersuchung der Zuverlässigkeit eines Systems für den Fall entwickelt, daß die Beschränkungen eines L. P.-Problems zufallsabhängig sind. Es werden Methoden zur Charakterisierung eines Maßes für die Zuverlässigkeit eines L. P.-Systems, die sich daraus ergebenden Implikationen bei verschiedenen Annahmen über die Wahrscheinlichkeitsverteilung und die Möglichkeit zur Bestimmung von Politik-Arten mittels eines verbesserten Verfahrens zur Untersuchung der System-Zuverlässigkeit diskutiert.

Distribution Problems in Stochastic and Chance-Constrained Programming

The problems of statistical distribution of the maximand are here analyzed under stochastic and chance-constrained linear programming. Uses of non-central Chi-square, truncated normal, non-central F and other non-negative distributions of statistical reliability theory are indicated. This analysis would be useful for economic models involving input-output coefficients which are usually required to be non-negative.

Economic Policy Simulation in Dynamic Control Models Under Econometric Estimation

The use of modern control theory in various dynamic economic models has raised a number of interesting issues in the theory of economic policy and its operational applications to problems of economic growth, stabilisation and development planning Two of these seem to be of great importance: econometric estimation viewed as a part of the decision-making process by a policymaker and the operational linkages between a consistency model without any explicit optimisation criterion and an optimisation model with an explicit objective function defined in a programming framework. In order to compare and evaluate alternative economic policies defined within a dynamic econometric model, these two problems become most relevant and they have to be resolved in some manner. As an example of the first type of problem one may refer to the use of the Brookings quarterly econometric model of the U.S. economy by Fromm and Taubman [1] for evaluation of the relative desirability of a set of monetary and fiscal policy actions. They noted that the method of optimum growth defined in a Ramsay-type framework of maximisation of a utility functional over a horizon is not applicable to cyclical paths; moreover it ignores the disutility of the time path of variances of the arguments (e.g. consumption, etc.) in the utility function.

On the active approach of stochastic linear programming

SummaryThe problem of statistical distribution of the optimal objective function under the so-called active approach of stochastic linear programming is investigated here from two interrelated aspects. First, the active approach is viewed as a method of decomposition. Second, some results on the asymptotic form of distribution of extreme values are utilized to derive the asymptotic form of the distribution of the maximand under the active approach.

STOCHASTIC MODELS OF ECONOMIC DEVELOPMENT

This chapter discusses the problems of specification, estimation, and empirical application of selected stochastic processes, which are believed to be useful for economic systems. In recent applications of the statistical theory of time series to economic models, two methods have gained some prominence, namely, the spectral analysis for identification of cyclical characteristics and the theory of stochastic control viewed as a modified method of forecasting. The general theory of stochastic processes offers a more satisfactory and generalized approach, particularly when models of economic growth are concerned. From an applied viewpoint, the two most important aspects of stochastic models are the specification of the stochastic process in relation to the economic trend and the statistical estimation of its parameters. The chapter highlights two aspects with particular reference to models of economic growth.

The Econometric Work of Gerhard Tintner

Gerhard Tintner was born September 29, 1907 of Austrian parents in Nurnberg, Germany. He studied economics, statistics and law at the University of Vienna, where he received his Doctor’s degree in 1929. In 1930 he spent some time in research work at the London School of Economics. Then, under a fellowship extended to him by the Rockefeller Foundation, he took postdoctoral work at Harvard University, Columbia University, the University of California (Berkeley) and Stanford University in the United States, at the Institut Henri Poincare in Paris, France and at Cambridge University in England. By training and by temperament, Tintner has been able to collate ideas from many sources; his professional work bears the mark of his own originality and has been singularly free from the stamp of a particular school or tradition.