History of Operations Research
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An Analog of the Minimax Theorem for Vector Payoffs
Pacific Journal of Mathematics AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL Vol. 6, No. 1 November 1956 AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL 1. Introduction* The von Neumann minimax theorem [2] for finite games asserts that for every rxs matrix M=\\m(i, j)\\ with real elements there exist a number v and vectors P=(Pi, •••, Pr)f Q={QU •••> Qs)f Pi, Qj>β, such that i> 3) for all i, j. Thus in the (two-person, zero-sum) game with matrix Λf, player I has a strategy insuring an expected gain of at least v, and player II has a strategy insuring an expected loss of at most v. An alternative statement, which follows from the von Neumann theorem and an appropriate law of large numbers is that, for any ε>0, I can, in a long series of plays of the game with matrix M, guarantee, with probability approaching 1 as the number of plays becomes infinite, that his average actual gain per play exceeds v — ε and that II can similarly restrict his average actual loss to v-he. These facts are assertions about the extent to which each player can control the center of gravity of the actual payoffs in a long series of plays. In this paper we investigate the extent to which this center of gravity can be controlled by the players for the case of matrices M whose elements m(i9 j) are points of ΛΓ-space. Roughly, we seek to answer the following question. -
Blackwell, David; Diaconis, Persi a Non-Measurable Tail Set
Blackwell, David; Diaconis, Persi A non-measurable tail set. Statistics, probability and game theory, 1--5, IMS Lecture Notes Monogr. Ser., 30, Inst. Math. Statist., Hayward, CA, 1996. Blackwell, David Operator solution of infinite $G\sb \delta$ games of imperfect information. Probability, statistics, and mathematics, 83--87, Academic Press, Boston, MA, 1989. 60040 Blackwell, David; Mauldin, R. Daniel Ulam's redistribution of energy problem: collision transformations. Lett. Math. Phys. 10 (1985), no. 2-3, 149--153. Blackwell, David; Dubins, Lester E. An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 (1983), no. 4, 691--692. 60B10 Blackwell, David; Maitra, Ashok Factorization of probability measures and absolutely measurable sets. Proc. Amer. Math. Soc. 92 (1984), no. 2, 251--254. Blackwell, David; Girshick, M. A. Theory of games and statistical decisions. Reprint of the 1954 edition. Dover Publications, Inc., New York, 1979. xi+355 pp. ISBN: 0-486-63831-6 90D35 (62Cxx) Blackwell, David On stationary policies. With discussion. J. Roy. Statist. Soc. Ser. 133 (1970), no. 1, 33--37. 90C40 Blackwell, David The stochastic processes of Borel gambling and dynamic programming. Ann. Statist. 4 (1976), no. 2, 370--374. Blackwell, David; Dubins, Lester E. On existence and non-existence of proper, regular, conditional distributions. Ann. Probability 3 (1975), no. 5, 741--752. Blackwell, David; MacQueen, James B. Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 (1973), 353--355. Blackwell, David; Freedman, David On the amount of variance needed to escape from a strip. Ann. Probability 1 (1973), 772--787. Blackwell, David Discreteness of Ferguson selections. -
Positional Notation Or Trigonometry [2, 13]
The Greatest Mathematical Discovery? David H. Bailey∗ Jonathan M. Borweiny April 24, 2011 1 Introduction Question: What mathematical discovery more than 1500 years ago: • Is one of the greatest, if not the greatest, single discovery in the field of mathematics? • Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? • Was fiercely resisted in Europe for hundreds of years after its discovery? • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, to- gether with the basic arithmetic computational schemes, which were discov- ered in India prior to 500 CE. ∗Bailey: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Email: [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. yCentre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. 1 2 Why? As the 19th century mathematician Pierre-Simon Laplace explained: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very sim- plicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appre- ciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. -
Print Prt489138397779225351.Tif (16 Pages)
U.S. Department ofHo1Deland.se¢urUy U.S. Citizenship and Immigration Services Administrative Appeals Office (AAO) 20 Massachusetts Ave., N.W., MS 2090 Washington, DC 20529-2090 (b)(6) U.S. Citizenship and Immigration Services DATE : APR 1 7 2015 OFFICE: CALIFORNIA SERVICE CENTER FILE: INR E: Petitioner: Benef icia ry: PETITION: Petition for a Nonim migrant Worker Pursuant to Section 101(a)(15)(H)(i)(b) of the Immigration and Nationality Act, 8 U.S.C. § 1101(a)(15)(H)(i)(b) ON BEHALF OF PETITIONER : INSTRUCTIONS: Enclosed please find the decision of the Administrative Appeals Office (AAO) in y our case. This is a n on-pre cedent decision. The AAO does not announce new constructions of law nor establish agency policy through non-precedent decisions. I f you believe the AAO incorrectly applied current law or policy to your case or if you seek to present new facts for consideration, you may file a motion to reconsider or a motion to reopen, respectively. Any motion must be filed on a Notice of Appeal or Motion (Form I-290B) within 33 days of the date of this decision. Please review the Form I-290B instructions at http://www.uscis.gov/forms for the latest information on fee, filing location, and other requirements. See also 8 C.F.R. § 103.5. Do not file a motion directly with the AAO. Ron Rose rg Chief, Administrative Appeals Office www.uscis.gov (b)(6) NON-PRECEDENTDECISION Page2 DISCUSSION: The service center director (hereinafter "director") denied the nonimmigrant visa petition, and the matter is now before the Administrative Appeals Office on appeal. -
The Artificial Cold War Content of Game Theory and Operations Research Richard Vahrenkamp [University of Kassel, Germany]
real-world economics review, issue no. 88 subscribe for free Nominal science without data – the artificial Cold War content of Game Theory and Operations Research Richard Vahrenkamp [University of Kassel, Germany] Copyright: Richard Vahrenkamp 2019 You may post comments on this paper at https://rwer.wordpress.com/comments-on-rwer-issue-no-88/ Abstract Expanding upon literature on early digital computers, this paper shows the role mathematicians have undertaken in founding the academic fields of Game Theory and Operations Research, and details how they were supported by the mathematics departments of military agencies in branches of the US Armed Services. This paper claims that application is only decoration. Other than astronomy, physics and engineering, where experiments generate data analysed with the aid of models and appropriate software on computers, Game Theory and Operations Research are not data driven but method driven and remain a branch of applied mathematics. They use the method of “abstractification” in economy and society to derive their models but lack a layer of empirical research needed to generate data and to apply their methods in economics and society. Therefore, their models were only nominal mathematics without application. Introduction Since 1945, the United States had experienced a unique innovation push with the computer, the nuclear weapon, new air combat weapons and the transistor within just a few years. These innovations were accompanied by Game Theory and Operations Research in the academic field. Widely–held is the view that computers supplemented the mathematical concepts of Game Theory and Operations Research and gave these fields a fresh impulse. Together, they established the view of the world as a space of numbers and introduced quantitative methods in economics, political science and in sociology. -
Strength in Numbers: the Rising of Academic Statistics Departments In
Agresti · Meng Agresti Eds. Alan Agresti · Xiao-Li Meng Editors Strength in Numbers: The Rising of Academic Statistics DepartmentsStatistics in the U.S. Rising of Academic The in Numbers: Strength Statistics Departments in the U.S. Strength in Numbers: The Rising of Academic Statistics Departments in the U.S. Alan Agresti • Xiao-Li Meng Editors Strength in Numbers: The Rising of Academic Statistics Departments in the U.S. 123 Editors Alan Agresti Xiao-Li Meng Department of Statistics Department of Statistics University of Florida Harvard University Gainesville, FL Cambridge, MA USA USA ISBN 978-1-4614-3648-5 ISBN 978-1-4614-3649-2 (eBook) DOI 10.1007/978-1-4614-3649-2 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012942702 Ó Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. -
Interview of Albert Tucker
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange About Harlan D. Mills Science Alliance 9-1975 Interview of Albert Tucker Terry Speed Evar Nering Follow this and additional works at: https://trace.tennessee.edu/utk_harlanabout Part of the Mathematics Commons Recommended Citation Speed, Terry and Nering, Evar, "Interview of Albert Tucker" (1975). About Harlan D. Mills. https://trace.tennessee.edu/utk_harlanabout/13 This Report is brought to you for free and open access by the Science Alliance at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in About Harlan D. Mills by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. The Princeton Mathematics Community in the 1930s (PMC39)The Princeton Mathematics Community in the 1930s Transcript Number 39 (PMC39) © The Trustees of Princeton University, 1985 ALBERT TUCKER CAREER, PART 2 This is a continuation of the account of the career of Albert Tucker that was begun in the interview conducted by Terry Speed in September 1975. This recording was made in March 1977 by Evar Nering at his apartment in Scottsdale, Arizona. Tucker: I have recently received the tapes that Speed made and find that these tapes carried my history up to approximately 1938. So the plan is to continue the history. In the late '30s I was working in combinatorial topology with not a great deal of results to show. I guess I was really more interested in my teaching. I had an opportunity to teach an undergraduate course in topology, combinatorial topology that is, classification of 2-dimensional surfaces and that sort of thing. -
A Century of Mathematics in America, Peter Duren Et Ai., (Eds.), Vol
Garrett Birkhoff has had a lifelong connection with Harvard mathematics. He was an infant when his father, the famous mathematician G. D. Birkhoff, joined the Harvard faculty. He has had a long academic career at Harvard: A.B. in 1932, Society of Fellows in 1933-1936, and a faculty appointmentfrom 1936 until his retirement in 1981. His research has ranged widely through alge bra, lattice theory, hydrodynamics, differential equations, scientific computing, and history of mathematics. Among his many publications are books on lattice theory and hydrodynamics, and the pioneering textbook A Survey of Modern Algebra, written jointly with S. Mac Lane. He has served as president ofSIAM and is a member of the National Academy of Sciences. Mathematics at Harvard, 1836-1944 GARRETT BIRKHOFF O. OUTLINE As my contribution to the history of mathematics in America, I decided to write a connected account of mathematical activity at Harvard from 1836 (Harvard's bicentennial) to the present day. During that time, many mathe maticians at Harvard have tried to respond constructively to the challenges and opportunities confronting them in a rapidly changing world. This essay reviews what might be called the indigenous period, lasting through World War II, during which most members of the Harvard mathe matical faculty had also studied there. Indeed, as will be explained in §§ 1-3 below, mathematical activity at Harvard was dominated by Benjamin Peirce and his students in the first half of this period. Then, from 1890 until around 1920, while our country was becoming a great power economically, basic mathematical research of high quality, mostly in traditional areas of analysis and theoretical celestial mechanics, was carried on by several faculty members. -
ORMS 1020: Operations Research with GNU Octave
ORMS 1020 Operations Research with GNU Octave Tommi Sottinen [email protected] www.uwasa.fi/ tsottine/or_with_octave/ ∼ October 19, 2011 Contents I Introduction and Preliminaries6 1 Selection of Optimization Problems7 1.1 Product Selection Problem.......................7 1.2 Knapsack Problem........................... 10 1.3 Portfolio Selection Problem*...................... 12 1.4 Exercises and Projects......................... 13 2 Short Introduction to Octave 14 2.1 Installing Octave............................ 14 2.2 Octave as Calculator.......................... 15 2.3 Linear Algebra with Octave...................... 18 2.4 Function and Script Files....................... 28 2.5 Octave Programming: glpk Wrapper................. 32 2.6 Exercises and Projects......................... 37 II Linear Programming 39 3 Linear Programs and Their Optima 40 3.1 Form of Linear Program........................ 40 3.2 Location of Linear Programs’ Optima................ 43 3.3 Solution Possibilities of Linear Programs............... 48 3.4 Karush–Kuhn–Tucker Conditions*.................. 53 3.5 Proofs*................................. 54 3.6 Exercises and Projects......................... 56 0.0 CONTENTS 2 4 Simplex Algorithm 58 4.1 Simplex tableaux and General Idea.................. 59 4.2 Top-Level Algorithm.......................... 62 4.3 Initialization Algorithm........................ 66 4.4 Optimality-Checking Algorithm.................... 68 4.5 Tableau Improvement Algorithm................... 71 4.6 Exercises and Projects........................ -
Curriculum Vitæ
Aleksandr M. Kazachkov May 30, 2021 Contact Department of Industrial and Systems Engineering Website akazachk.github.io Information Herbert Wertheim College of Engineering Email akazachkov@ufl.edu 303 Weil Hall, P.O. Box 116595 Office 401B Weil Hall Gainesville, FL 32611-6595, USA Phone +1.352.273.4902 Research Discrete optimization, including theoretical, methodological, and applied aspects, with an emphasis on devel- Interests oping better cutting plane techniques and open-source computational contributions Computational economics, particularly on the fair allocation of indivisible resources and fair mechanism design Social good applications, such as humanitarian logistics and donation distribution Academic University of Florida, Department of Industrial and Systems Engineering, Gainesville, FL, USA Positions Assistant Professor January 2021 – Present Courtesy Assistant Professor March 2020 – December 2020 Polytechnique Montréal, Department of Mathematics and Industrial Engineering, Montréal, QC, Canada Postdoctoral Researcher under Andrea Lodi May 2018 – December 2020 Education Carnegie Mellon University, Tepper School of Business, Pittsburgh, PA, USA Ph.D. in Algorithms, Combinatorics, and Optimization under Egon Balas May 2018 Dissertation: Non-Recursive Cut Generation M.S. in Algorithms, Combinatorics, and Optimization May 2013 Cornell University, College of Engineering, Ithaca, NY, USA B.S. in Operations Research and Engineering with Honors, Magna Cum Laude May 2011 Publications [1] P.I. Frazier and A.M. Kazachkov. “Guessing Preferences: A New Approach to Multi-Attribute Ranking and Selection”, Winter Simulation Conference, 2011. [2] J. Karp, A.M. Kazachkov, and A.D. Procaccia. “Envy-Free Division of Sellable Goods”, AAAI Conference on Artificial Intelligence, 2014. [3] J.P. Dickerson, A.M. Kazachkov, A.D. Procaccia, and T. -
Operations Research for Resource Planning and -Use in Radiotherapy: a Literature Review Bruno Vieira1,2,4*, Erwin W
Vieira et al. BMC Medical Informatics and Decision Making (2016) 16:149 DOI 10.1186/s12911-016-0390-4 RESEARCH ARTICLE Open Access Operations research for resource planning and -use in radiotherapy: a literature review Bruno Vieira1,2,4*, Erwin W. Hans2,3, Corine van Vliet-Vroegindeweij1, Jeroen van de Kamer1 and Wim van Harten1,4,5 Abstract Background: The delivery of radiotherapy (RT) involves the use of rather expensive resources and multi-disciplinary staff. As the number of cancer patients receiving RT increases, timely delivery becomes increasingly difficult due to the complexities related to, among others, variable patient inflow, complex patient routing, and the joint planning of multiple resources. Operations research (OR) methods have been successfully applied to solve many logistics problems through the development of advanced analytical models for improved decision making. This paper presents the state of the art in the application of OR methods for logistics optimization in RT, at various managerial levels. Methods: A literature search was performed in six databases covering several disciplines, from the medical to the technical field. Papers included in the review were published in peer-reviewed journals from 2000 to 2015. Data extraction includes the subject of research, the OR methods used in the study, the extent of implementation according to a six-stage model and the (potential) impact of the results in practice. Results: From the 33 papers included in the review, 18 addressed problems related to patient scheduling (of which 12 focus on scheduling patients on linear accelerators), 8 focus on strategic decision making, 5 on resource capacity planning, and 2 on patient prioritization. -
Rational Inattention with Sequential Information Sampling∗
Rational Inattention with Sequential Information Sampling∗ Benjamin Hébert y Michael Woodford z Stanford University Columbia University July 1, 2016 Preliminary and Incomplete. WARNING: Some results in this draft are not corrected as stated. Please contact the authors before citing. Comments welcome. Abstract We generalize the rationalize inattention framework proposed by Sims [2010] to allow for cost functions other than Shannon’s mutual information. We study a prob- lem in which a large number of successive samples of information about the decision situation, each only minimally informative by itself, can be taken before a decision must be made. We assume that the cost required for each individual increment of in- formation satisfies standard assumptions about continuity, differentiability, convexity, and monotonicity with respect to the Blackwell ordering of experiments, but need not correspond to mutual information. We show that any information cost function sat- isfying these axioms exhibits an invariance property which allows us to approximate it by a particular form of Taylor expansion, which is accurate whenever the signals acquired by an agent are not very informative. We give particular attention to the case in which the cost function for an individual increment to information satisfies the ad- ditional condition of “prior invariance,” so that it depends only on the way in which the probabilities of different observations depend on the state of the world, and not on the prior probabilities of different states. This case leads to a cost function for the overall quantity of information gathered that is not prior-invariant, but “sequentially prior-invariant.” We offer a complete characterization of the family of sequentially prior-invariant cost functions in the continuous-time limit of our dynamic model of in- formation sampling, and show how the resulting theory of rationally inattentive choice differs from both static and dynamic versions of a theory based on mutual information.