Lecture Notes in Economics and Mathematical Systems
Kurt Marti · Ching-Lai Hwang · Diethard Pallaschke · Andrzej P. Wierzbicki · Klaus Neumann · Wei-Bin Zhang · Yuri Ermoliev
Latest release: June 21, 2023
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579
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About this ebook series
This study has grown out of a part of the author's thesis "Some Simple and Bulk Queueing Systems: A Study of Their Transient Behavior" submitted to the University of Western Australia (1964) and a course on Queueing Theory given to graduate students in the Operations Research Group of Case Institute of Technology, Cleveland, Ohio. The one semester course (approximately 35 hours) consisted of the following topics. (i) Some of the important special queues such as M/M/s, M/D/s, M/Ek/l etc., with emphasis on the different methods employed in the transient as well as steady state solution. (ii) Imbedded Markov chain analysis of M/G/l and GI/M/l as given in the joint paper of the author and N. U. Prabhu as well as the papers of D. G. Kendall. [All notations and papers are referred to later in the notes]. (iii) The contents of this memorandum. The author feels that such a course prepares the students adequately for an advanced course in Queueing Theory involving topics on Waiting Times, the General Queue GI/G/l and other ramifications such as Priorities, etc. A few words regarding the approach adopted in this study may not be out of place. So far, the time dependent behavior of queueing systems has not found a place in courses given outside the Department of Mathematics.
A Study of the Queueing Systems M/G/1 and GI/M/1
Book 2 · Dec 2012 ·
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This study has grown out of a part of the author's thesis "Some Simple and Bulk Queueing Systems: A Study of Their Transient Behavior" submitted to the University of Western Australia (1964) and a course on Queueing Theory given to graduate students in the Operations Research Group of Case Institute of Technology, Cleveland, Ohio. The one semester course (approximately 35 hours) consisted of the following topics. (i) Some of the important special queues such as M/M/s, M/D/s, M/Ek/l etc., with emphasis on the different methods employed in the transient as well as steady state solution. (ii) Imbedded Markov chain analysis of M/G/l and GI/M/l as given in the joint paper of the author and N. U. Prabhu as well as the papers of D. G. Kendall. [All notations and papers are referred to later in the notes]. (iii) The contents of this memorandum. The author feels that such a course prepares the students adequately for an advanced course in Queueing Theory involving topics on Waiting Times, the General Queue GI/G/l and other ramifications such as Priorities, etc. A few words regarding the approach adopted in this study may not be out of place. So far, the time dependent behavior of queueing systems has not found a place in courses given outside the Department of Mathematics.
An Introduction to Optimal Control Theory
Book 3 · Dec 2012 ·
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This paper is intended for the beginner. It is not a state of-the-art paper for research workers in the field of control theory. Its purpose is to introduce the reader to some of the problems and results in control theory, to illustrate the application of these re sults, and to provide a guide for his further reading on this subject. I have tried to motivate the results with examples, especial ly with one canonical, simple example described in §3. Many results, such as the maximum principle, have long and difficult proofs. I have omitted these proofs. In general I have included only the proofs which are either (1) not too difficult or (2) fairly enlightening as to the nature of the result. I have, however, usually attempted to draw the strongest conclusion from a given proof. For example, many existing proofs in control theory for compact targets and uniqueness of solutions also hold for closed targets and non-uniqueness. Finally, at the end of each section I have given references to generalizations and origins of the results discussed in that section. I make no claim of completeness in the references, however, as I have often been content merely to refer the reader either to an exposition or to a paper which has an extensive bibliography. IV These 1ecture notes are revisions of notes I used for aseries of nine 1ectures on contro1 theory at the International Summer Schoo1 on Mathematica1 Systems and Economics held in Varenna, Ita1y, June 1967.
Information Theory for Systems Engineers
Book 5 · Dec 2012 ·
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This book is based on lectures given by the author at the IBM European Systems Research Institute (ESRI) in Geneva. Information Theory on the syntactic level, as introduced by Claude Shannon in 1949, has many limitations when applied to information processing by computers. But in spite of some obvious shortcomings, the underlyin~ principles are of fundamental importance for systems engineers in understanding the nature of the problems of handling information, its acquisition, storage, processing, and interpretation. The lectures, as presented in this book, attempt to give an exposition of the lovical foundation and basic principles, and to provide at the same time a basis for further study in more specific areas of this expan1in~ theory, such as coding, detection, pattern recognition, and filtering. Most of the problems in Appendix C are intended as extensions of the text, while calling for actjve participation by the stu1ent. Some other problems are direct applications of the theory to specific situations. Some problems require extensive numerical calculations. It is assumed in those cases that the student has access to a computer and that he is capable of writing the necessary programs. The stu1ent is assumed to have a good command of the calculus, and of the theory of probability as well as statistics. Therefore no basic mathematical concepts are discussed in this IV book. The Fourier transform and some related mathematical concepts are introduced in Appendix A.
Computing Methods in Optimization Problems: Papers presented at the 2nd International Conference on Computing Methods in Optimization Problems, San Remo, Italy, September 9–13, 1968
Book 14 · Mar 2013 ·
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Economic Models, Estimation and Risk Programming: Essays in Honor of Gerhard Tintner: Essays in Honor of Gerhard Tintner
Book 15 · Dec 2012 ·
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These essays in honor of Professor Gerhard Tintner are substantive contributions to three areas of econometrics, (1) economic models and applications,. (2) estimation, and (3) stochastic programming, in each of which he has labored with outstanding success. His own work has extended into multivariate analysis, the pure theory of decision-making under un certainty, and other fields which are not touched upon here for reasons of space and focus. Thus, this collection is appropriate to his interests but covers much less than their full range. Professor Tintner's contributions to econometrics through teaching, writing, editing, lecturing and consulting have been varied and inter national. We have tried to highlight them in "The Econometric Work of Gerhard Tintner" and to place them in historical perspective in "The Invisible Revolution in Economics: Emergence of a Mathematical Science. " Professor Tintner's career to date has spanned the organizational life of the Econometric Society and his contributions have been nearly coextensive with its scope. His principal books and articles up to 1968 are listed in the "Selected Bibliography. " Professor Tintner's current research involves the intricate problems of specification and application of stochastic processes to economic systems, particularly to growth, diffusion of technology, and optimal control. As always, he is moving with the econometric frontier and a portion of the frontier is moving with him. IV Two of the editors wrote dissertations under Professor Tintner's sup- vision; the third knew him as a colleague and friend.
Mathematical Modeling for Industrial Processes
Book 19 · Dec 2012 ·
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These notes are based on the material presented in a series of lec tures in the IBM Systems Research Institute (ESRI) in Geneva durJng 1967-1969 to systems engineers working in the design and programming of computer systems for control and monitoring of i~nustrial proc esses. The purpose of the lectures and this book is to give a survey of dif ferent approaches in developing models to describe the behavior of the process in terms of controllable variables. It does not cover the theory of control, stability of control systems, nor techniques in data acquisition or problems in instrumentation and sampling. But certain aspects in the organization of data collection and design of experiments are obtained as side products, notably the concept of orthogonality. The reader is assumed to have a working knowledge of elementary prob ability theory and mathematical statistics. Therefore, the text con tains no introduction to these concepts. The author is aware of some inaccuracies in not making proper dis tinction between population parameters and their sample estimates in the text, but this should alw~s be evident from the context. The same applies to the occasional replacement of number of degrees of freedom by the number of samples in the data. In practice, computer collected sets of data consist of a high number of samples and the difference between the two is inSignificant.
Some Aspects of Queueing and Storage Systems
Book 23 · Dec 2012 ·
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In these lecture notes which grew out of my lectures at the Honours classes of Honash University, I have attempted to develop queueing and storage problems from a unified viewpoint. It has been recognized a decade ago that many of the queueing and storage problems belong to the same family of stochastic processes; so a problem in storage theory may bave an analogue in a queueing situation and vice versa. In my notes I have highlighted this aspect and tried to develop a broad perspective in a student rather than to work out in detail the various e~rcises in queueing and inventory problems which are mostly available in the literature produced so far. I have also pointed out some of the practical applications of some theoretical results which appear to be important for an Operational Research worker. Hany of the results given in these notes have cropped out of the author's own research over the last decade. Some new ideas which have scope for further exploitation have been given in most of the chapters (e. g. concept of cybernetic systems in Chapter 3, optimality problems in Chapter 4 and some problems in Chapter 2, etc. ): these may benefit graduate students. I thank my students for various discussions inside and outside the classrooms. I am also grateful to Mrs A. Darby for painfully and accurately typing out my manuscript. Ami tava Ghosal September 1969 Index Chapter 1.
Computational Methods in Optimal Control Problems
Book 27 · Dec 2012 ·
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The purpose of this modest report is to present in a simplified manner some of the computational methods that have been developed in the last ten years for the solution of optimal control problems. Only those methods that are based on the minimum (maximum) principle of Pontriagin are discussed here. The autline of the report is as follows: In the first two sections a control problem of Bolza is formulated and the necessary conditions in the form of the minimum principle are given. The method of steepest descent and a conjugate gradient-method are dis cussed in Section 3. In the remaining sections, the successive sweep method, the Newton-Raphson method and the generalized Newton-Raphson method (also called quasilinearization method) ar~ presented from a unified approach which is based on the application of Newton Raphson approximation to the necessary conditions of optimality. The second-variation method and other shooting methods based on minimizing an error function are also considered. TABLE OF CONTENTS 1. 0 INTRODUCTION 1 2. 0 NECESSARY CONDITIONS FOR OPTIMALITY •••••••• 2 3. 0 THE GRADIENT METHOD 4 3. 1 Min H Method and Conjugate Gradient Method •. •••••••••. . . . ••••••. ••••••••. • 8 3. 2 Boundary Constraints •••••••••••. ••••. • 9 3. 3 Problems with Control Constraints ••. •• 15 4. 0 SUCCESSIVE SWEEP METHOD •••••••••••••••••••• 18 4. 1 Final Time Given Implicitly ••••. •••••• 22 5. 0 SECOND-VARIATION METHOD •••••••••••••••••••• 23 6. 0 SHOOTING METHODS ••••••••••••••••••••••••••• 27 6. 1 Newton-Raphson Method ••••••••••••••••• 27 6.
Theoretical Approaches to Non-Numerical Problem Solving: Proceedings of the IV Systems Symposium at Case Western Reserve University
Book 28 · Dec 2012 ·
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Advances in computer technology have pointed out the next important area of computer applications: solution of non-numerical problems. It is hardly necessary to emphasize the importance of these kind of problems. First of all most of the decisions one has to make in real-life situations are non-numerical in the first instance and can be represented as numerical problems only as approximations which are often only partially valid. Second, to use the computer to its full potential it should be employed as a logical machine, capable of deduction, and not just as a numerical calculating machine. Thus the computer would extend man's capability for logical reasoning and not just for his capability to do fast and accurate calculation. It is not a new area; indeed non-numerical problems are central in fields such as artificial intelligence, heuristic programming, pattern recognition, classification and information-processing (and retrival) etc. However, it is fair to assess that progress in the area has not been quite as expected. One of the reasons was a lack of conceptual and theoretical framework in which to investigate different classes of non-numerical problems to improve understanding of various types of problems and methods for their solutions and furthermore to enable the methods which have been proven as effective in one situation to be used in another situation with appropriately similar structure.
Some Network Models in Management Science
Book 29 · Dec 2012 ·
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Foundations of Non-stationary Dynamic Programming with Discrete Time Parameter
Book 33 · Dec 2012 ·
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The present work is an extended version of a manuscript of a course which the author taught at the University of Hamburg during summer 1969. The main purpose has been to give a rigorous foundation of stochastic dynamic programming in a manner which makes the theory easily applicable to many different practical problems. We mention the following features which should serve our purpose. a) The theory is built up for non-stationary models, thus making it possible to treat e.g. dynamic programming under risk, dynamic programming under uncertainty, Markovian models, stationary models, and models with finite horizon from a unified point of view. b) We use that notion of optimality (p-optimality) which seems to be most appropriate for practical purposes. c) Since we restrict ourselves to the foundations, we did not include practical problems and ways to their numerical solution, but we give (cf.section 8) a number of problems which show the diversity of structures accessible to non stationary dynamic programming. The main sources were the papers of Blackwell (65), Strauch (66) and Maitra (68) on stationary models with general state and action spaces and the papers of Dynkin (65), Hinderer (67) and Sirjaev (67) on non-stationary models. A number of results should be new, whereas most theorems constitute extensions (usually from stationary models to non-stationary models) or analogues to known results.
On a General Economic Theory of Motion
Book 36 · Dec 2012 ·
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On Round-Off Errors in Linear Programming
Book 37 · Dec 2012 ·
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Due to the limited number of digits or bits per storage location in electronic computers, round-off errors arise during arithmetic operations. Depending upon the kind of operation, the structure of the data, and the skillfulness of the program, these errors increase and spread out more or less quickly during a continued computation process in which the computed data affected by errors are themselves used for generating new data. The purpose of this investigation was to learn about the increase of round-off errors in linear programming procedures. Less attention was paid to the theory of round-off errors or to the effectiveness of error elimination procedures. In regard to these questions the results of in vestigations which have been made on round-off errors in a more general context dealing with matrix inversion and eigenvalue problems could be used for the purposes of this paper. The emphasis of this investigation lay rather on studying the behavior of typical linear programming problems from the pOint of view of error cumulation.
The Coordinate-Free Approach to Gauss-Markov Estimation
Book 40 · Dec 2012 ·
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These notes originate from a couple of lectures which were given in the Econometric Workshop of the Center for Operations Research and Econometrics (CORE) at the Catholic University of Louvain. The participants of the seminars were recommended to read the first four chapters of Seber's book [40], but the exposition of the material went beyond Seber's exposition, if it seemed necessary. Coordinate-free methods are not new in Gauss-Markov estimation, besides Seber the work of Kolmogorov [11], SCheffe [36], Kruskal [21], [22] and Malinvaud [25], [26] should be mentioned. Malinvaud's approach however is a little different from that of the other authors, because his optimality criterion is based on the ellipsoid of c- centration. This criterion is however equivalent to the usual c- cept of minimal covariance-matrix and therefore the result must be the same in both cases. While the usual theory gives no indication how small the covariance-matrix can be made before the optimal es timator is computed, Malinvaud can show how small the ellipsoid of concentration can be made: it is at most equal to the intersection of the ellipssoid of concentration of the observed random vector and the linear space in which the (unknown) expectation value of the observed random vector is lying. This exposition is based on the observation, that in regression ~nalysis and related fields two conclusions are or should preferably be applied repeatedly.
Introduction to Optimization Theory in a Hilbert Space
Book 42 · Dec 2012 ·
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This book is based on lectures given in a one-quarter course at UCLA. The aim. is to present som.e of the basic concepts and techniques of Functional Analys.is of relevance to optim.ization problem.s in Control. Com.m.unication and other areas in System. Science. The students are expected to have had an introductory course in Hilbert Space theory. Som.e effort has been m.ade to be self-contained m.ainly in order that the vocabularly used can be clarified. A m.inim.al bibliography is appended. The author is indebted to Jiri Ruzicka and Jerom.e Mersky for help with proof-reading. Profes sor L. Berkovitz looked over and m.ade m.any helpful corn.rn.ents on parts of an early version. Thanks are also due to Trudy Cook for typing the m.anuscript. Grateful acknowledgem.ent is also m.ade of partial support under AFOSR Grant No. 68-1408, Applied Mathem.atics Division, United Stat s Air Force.
Bayesian Full Information Structrual Analysis: with an Application to the Study of the Belgian Beef Market
Book 43 · Dec 2012 ·
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Invariant Imbedding: Proceedings of the Summer Workshop on Invariant Imbedding held at the University of Southern California, June – August 1970
Book 52 · Dec 2012 ·
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Imbedding is a powerful and versatile tool for problem solving. Rather than treat a question in isolation, we view it as a member of a family of related problems. Each member then becomes a stepping stone in a path to a simultaneous solution of the entire set of problems. As might be expected, there are many ways of accomplishing this imbedding. Time and space variables have been widely employed in the past, while modern approaches combine these structural features with others less immediate. Why should one search for alternate imbeddings when elegant classical formalisms already exist? There are many reasons. To begin with, different imbeddings are useful for different purposes. Some are well suited to the derivation of existence and uniqueness theorems, some to the derivation of conservation relations, some to perturbation techniques and sensitivity analysis, some to computa tional studies. The digital computer is designed for initial value problems; the analog computer for boundary-value problems. It is essential then to be flexible and possess the ability to use one device or the other, or both. In economics, engineering, biology and physics, some pro cesses lend themselves more easily to one type of imbedding rather than another. Thus, for example, stochastic decision processes are well adapted to dynamic programming. In any case, to go hunting in the wilds of the scientific world armed with only one arrow in one's quiver is quite foolhardy.
Steady State Capital Theory
Book 54 · Dec 2012 ·
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The following lecture notes were written shortly after I gave a course on capital theory in the winter-semester 1970/71 at the Univer sity of Heidelberg. While the general line of the argument is similar to the one in the course, I have modified and added a large number' of specific points in the process of writing the English version. I should like to emphasize the narrow limitations of the material covered in these notes. I have completely concentrated on steady states of stationary and exponentially growing economies, even up to the point where there is the danger of misleading the reader1 I have done this for several reasons. Other activities have not left me with a sufficient amount of time to be able to find the unifying principle of analysis and mode of presentation for the dynamic aspects of capi tal theory which would have made it worthwhile to add a sizeable book to the large body of literature in this field. On the other hand over the last couple of years I have become increasingly aware that some of the results in steady state capital theory (which could be derived without too much mathematical effort) are of relevance in present day dis cussions about the political role of economic theory and the relative merits of orthodox and radical economics. Also these results seemed not to be known by' mO$ of the participants in these discussions.
Statistical Inference in Random Coefficient Regression Models
Book 55 · Dec 2012 ·
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This short monograph which presents a unified treatment of the theory of estimating an economic relationship from a time series of cross-sections, is based on my Ph. D. dissertation submitted to the University of Wisconsin, Madison. To the material developed for that purpose, I have added the substance of two subsequent papers: "Efficient methods of estimating a regression equation with equi-correlated disturbances", and "The exact finite sample properties of estimators of coefficients in error components regression models" (with Arora) which form the basis for Chapters 11 and III respectively. One way of increasing the amount of statistical information is to assemble the cross-sections of successive years. To analyze such a body of data the traditional linear regression model is not appropriate and we have to introduce some additional complications and assumptions due to the hetero geneity of behavior among individuals. These complications have been discussed in this monograph. Limitations of economic data, particularly their non-experimental nature, do not permit us to know a priori the correct specification of a model. I have considered several different sets of assumptionR about the stability of coeffi cients and error variances across individuals and developed appropriate inference procedures. I have considered only those sets of assumptions which lead to opera tional procedures. Following the suggestions of Kuh, Klein and Zellner, I have adopted the linear regression models with some or all of their coefficients varying randomly across individuals.
Constrained Extrema Introduction to the Differentiable Case with Economic Applications
Book 56 · Dec 2012 ·
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These notes are the result of an interrupted sequence of seminars on optimiza tion theory with economic applications starting in 1964-1965. This is mentioned by way of explaining the uneven style that pervades them. Lately I have been using the notes for a two semester course on the subject for graduate students in economics. Except for the introductory survey, the notes are intended to provide an appetizer to more sophisticated aspects of optimization theory and economic theory. The notes are divided into three parts. Part I collects most of the results on constrained extremf! of differentiable functionals on finite and not so finite dimensional spaces. It is to be used as a reference and as a place to find credits to various authors whose ideas we report. Part II is concerned with finite dimensional problems and is written in detail. Needless to say, my contributions are marginal. The economic examples are well known and are presented by way of illustrating the theory. Part III is devoted to variational problems leading to a discussion of some optimal control problems. There is a vast amount of literature on these problems and I tried to limit my intrusions to explaining some of the obvious steps that are usually left out. I have borrowed heavily from Akhiezer [ 1], Berkovitz [ 7], Bliss [lOJ and Pars [40J. The economic applications represent some of my work and are presented in the spirit of illustration.
