Dantzig Homework

Tales of Statisticians
George B Dantzig
8 Nov 1914 -13 May 2005

George Dantzig was the son of Russian-born Tobias Dantzig, author of the widely popular book Number: The Language of Science. Though named for George Bernard Shaw, he majored in mathematics at the University of Maryland, where his father then taught (it was cheaper than other schools, and the family was not well off). He began graduate work on a scholarship at the University of Michigan in 1936, and got his MA in 1937, but did not continue, due to his distaste for the abstractness of the mathematics he encountered there. After working as a statistician in Seattle, he wrote in 1939 to Neyman, whose papers had interested him, and an assistantship was arranged for him at Berkeley. This story from that period is a classic:

"During my first year at Berkeley I arrived late one day to one of Neyman's classes. On the blackboard were two problems which I assumed had been assigned for homework. I copied them down. A few days later I apologized to Neyman for taking so long to do the homework -- the problems seemed to be a little harder to do than usual. I asked him if he still wanted the work. He told me to throw it on his desk. I did so reluctantly because his desk was covered with such a heap of papers that I feared my homework would be lost there forever."

"About six weeks later, one Sunday morning about eight o'clock, Anne and I were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited: "I've just written an introduction to one of your papers. Read it so I can send it out right away for publication." For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard which I had solved thinking they were homework were in fact two famous unsolved problems in statistics. That was the first inkling I had that there was anything special about them."

[from Albers, More Mathematical People]

The point of the story, which has been quoted frequently in inspirational writings, is that if Dantzig had known the problems were unsolved, he might not have made any serious attempt to solve them, whereas the "positive" assumption that they were not only solvable, but routinely solvable, focused his attention simply on finding the solution. Ignorance can thus be a help to the discoverer. So can youth. A similar point was made by pianist Ruth Laredo, looking back on her acquisition of the fiendishly difficult Ravel literature:

 "I learned the Ravel repertoire mostly when I was so young that the extreme difficulties somehow didn't bother me. Gaspard de la Nuit came into my life at fifteen. I just didn't know how hard it was."

It seems there is a lot to be said for not knowing how hard things are.

Dantzig had all but completed his degree in 1941, but WW2 then interrupted. WW2 had its problems for many of us, but it undeniably gave enormous opportunities for the development of statistics, the practical stepsister of mathematics. Dantzig want to Washington in 1941 as Head of the Combat Analysis Branch of the Air Force's Headquarters Statistical Control.

"I also helped other divisions of the Air Staff prepare plans called "programs." Everything was planned in greatest detail: all the nuts and bolts, the procurement of airplanes, the detailed manufacture of everything. There were hundreds of thousands of different kinds of material goods and perhaps fifty thousand specialties of people."

This gigantic problem of finding the optimal intersection, the optimal "program," for all these interconnected flowlines, was one of great difficulty. For his contribution to this and other urgent problems of managerial logistics, he was awarded the War Department's Exceptional Civilian Service Medal in 1944.

He left this post in 1946, and returned to Berkeley for a semester to finish his degree. He declined an offered junior position at Berkeley due to the tiny salary. In June he was back in Washington, where he accepted a post as Mathematical Advisor to the Defense Department, to work on mechanizing the planning process. In 1947, based partly on his earlier work with aircraft supply flowlines, he discovered what is called the simplex method of linear programming. Its first large-scale test involved a problem with 9 equations in 77 unknowns, which, with the calculating machinery available at the time, took 120 man-days of labor to solve. Computers, which had been developed but not fully exploited during the war, were the obvious next step. In 1952 Dantzig went to work for the RAND Corporation, on implementing the simplex method on computers. In 1960 he accepted a teaching position at Berkeley, moving in 1966 to Stanford as Professor of Operations Research and Computer Science.

The power of the simplex method (as was also true of Wilcoxon and his nonparametric methods, which had been published two years earlier, in 1945) continued to surprise Dantzig himself. Thanks in large part to his own vigorous following up of his initial success, the simplex method is now said to underlie more computer-time use than anything else. His still classic book on the subject, Linear Programming and Extensions, appeared in 1963. In 1975 came the first of many prizes recognizing the importance of the method. Appropriately enough, given von Neumann's role in pushing for the first computer during WW2, this was the von Neumann Theory Prize in Operational Research. Other recognitions followed. Not including the Nobel (technically, the Bank of Sweden Prize in Economics), which in 1975 went to Koopmans and Kantorovich for an achievement to which Dantzig had also made a decisive contribution: the mathematical theory of the allocation of scarce resources. So upset was Koopmans at Dantzig's omission, that he suggested to Kantorovich that they refuse the prize themselves. Famous once a year is Stockholm, and famous down the years are the lapses of Stockholm.

Dantzig had technically retired in 1973 from Stanford, but continued active until 1977. As late as 2001, he was listed as Chief of Operations Research and Computer Systems at Stanford, as well as Co-Director of the Systems Optimization Lab, and Director of the PILOT Energy-Economic Model Project. His achievements are recorded in detail, along with many reminiscences by himself and others, at the web site maintained by his student Saul Gass.

is Copyright © 2001- by E Bruce Brooks 

1 Feb 2006 / Contact The Project / Exit to Statistics Page

George Bernard Dantzig

Gerald R. Ford awarded George B. Dantzig at the National Medal of Science Awards Ceremony, 1976

Born(1914-11-08)November 8, 1914
Portland, Oregon
DiedMay 13, 2005(2005-05-13) (aged 90)
Stanford, California
Alma materBachelor's degree:University of Maryland
Master's degree: University of Michigan
Doctor of Philosophy: University of California, Berkeley
Known forLinear programming
Simplex algorithm
Dantzig-Wolfe decomposition principle
Generalized linear programming
Generalized upper bounding
Max-flow min-cut theorem of networks
Quadratic programming
Complementary pivot algorithms
Linear complementarity problem
Stochastic programming
AwardsJohn von Neumann Theory Prize(1975)
National Medal of Science in Mathematical, Statistical, and Computational Sciences (1975)
Harvey Prize(1985)
Harold Pender Award(1995)
Scientific career
Operations research
Computer science
InstitutionsU.S. Air Force Office of Statistical Control
RAND Corporation
University of California, Berkeley
Stanford University
Doctoral advisorJerzy Neyman
Doctoral students
Robert Fourer
Alfredo Noel Iusem
Ellis L. Johnson
Thomas Magnanti
Roger J-B Wets
Yinyu Ye
InfluencesWassily Leontief
John von Neumann
Marshal K. Wood
InfluencedKenneth J. Arrow
Robert Dorfman
Leonid Hurwicz
Tjalling C. Koopmans
Thomas L. Saaty
Paul Samuelson

Harry M. Markowitz
Philip Wolfe

George Bernard Dantzig (; November 8, 1914 – May 13, 2005) was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics.

Dantzig is known for his development of the simplex algorithm,[1] an algorithm for solving linear programming problems, and for his other work with linear programming. In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman.[2]

Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford.


Born in Portland, Oregon, George Bernard Dantzig was named after George Bernard Shaw, the Irish writer.[3][4] His father, Tobias Dantzig, was a BalticGerman mathematician and linguist, and his mother, Anja Dantzig (née Ourisson), was a French linguist of Jewish origin. Dantzig's parents met during their study at the University of Paris, where Tobias studied mathematics under Henri Poincaré, after whom Dantzig's brother was named.[4] The Dantzigs immigrated to the United States, where they settled in Portland, Oregon.

Early in the 1920s the Dantzig family moved from Baltimore to Washington. His mother became a linguist at the Library of Congress, and his father became a math tutor at the University of Maryland, College Park. Dantzig attended Powell Junior High School and Central High School; one of his friends there was Abraham Seidenberg, who also became a professional mathematician.[4] By the time he reached high school he was already fascinated by geometry, and this interest was further nurtured by his father, challenging him with complicated problems, particularly in projective geometry.[2][4]

George Dantzig received his B.S. from University of Maryland in 1936 in mathematics and physics, which is part of the University of Maryland College of Computer, Mathematical, and Natural Sciences. He earned his master's degree in mathematics from the University of Michigan in 1938. After a two-year period at the Bureau of Labor Statistics, he enrolled in the doctoral program in mathematics at the University of California, Berkeley, where he studied statistics under Jerzy Neyman.

With the outbreak of World War II, Dantzig took a leave of absence from the doctoral program at Berkeley to join the U.S. Air Force Office of Statistical Control. In 1946, he returned to Berkeley to complete the requirements of his program and received his Ph.D. that year.[3] Although he had a faculty offer from Berkeley, he returned to the Air Force as mathematical advisor to the comptroller.[4]

In 1952 Dantzig joined the mathematics division of the RAND Corporation. By 1960 he became a professor in the Department of Industrial Engineering at UC Berkeley, where he founded and directed the Operations Research Center. In 1966 he joined the Stanford faculty as Professor of Operations Research and of Computer Science. A year later, the Program in Operations Research became a full-fledged department. In 1973 he founded the Systems Optimization Laboratory (SOL) there. On a sabbatical leave that year, he headed the Methodology Group at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria. Later he became the C. A. Criley Professor of Transportation Sciences at Stanford, and kept going, well beyond his mandatory retirement in 1985.[3]

He was a member of the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences. Dantzig was the recipient of many honors, including the first John von Neumann Theory Prize in 1974, the National Medal of Science in 1975,[5] an honorary doctorate from the University of Maryland, College Park in 1976. The Mathematical Programming Society honored Dantzig by creating the George B. Dantzig Prize, bestowed every three years since 1982 on one or two people who have made a significant impact in the field of mathematical programming.

Dantzig died on May 13, 2005, in his home in Stanford, California, of complications from diabetes and cardiovascular disease. He was 90 years old.[2]


Freund wrote further that "through his research in mathematical theory, computation, economic analysis, and applications to industrial problems, Dantzig has contributed more than any other researcher to the remarkable development of linear programming".[6]

Dantzig's seminal work allows the airline industry, for example, to schedule crews and make fleet assignments. Based on his work tools are developed "that shipping companies use to determine how many planes they need and where their delivery trucks should be deployed. The oil industry long has used linear programming in refinery planning, as it determines how much of its raw product should become different grades of gasoline and how much should be used for petroleum-based byproducts. It is used in manufacturing, revenue management, telecommunications, advertising, architecture, circuit design and countless other areas".[2]

Mathematical statistics[edit]

An event in Dantzig's life became the origin of a famous story in 1939, while he was a graduate student at UC Berkeley. Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.[4][7]

Six weeks later, Dantzig received a visit from an excited professor Neyman, who was eager to tell him that the homework problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[8] As Dantzig told it in a 1986 interview in the College Mathematics Journal:[9]

A year later, when I began to worry about a thesis topic, Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis.

Years later another researcher, Abraham Wald, was preparing to publish an article that arrived at a conclusion for the second problem, and included Dantzig as its co-author when he learned of the earlier solution.[4][10]

This story began to spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time Dantzig's name was removed, and facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the movie Good Will Hunting.[7]

Linear programming[edit]

Linear programming is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming arose as a mathematical model developed during World War II to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Postwar, many industries found its use in their daily planning.

The founders of this subject are Leonid Kantorovich, a Russian mathematician who developed linear programming problems in 1939, Dantzig, who published the simplex method in 1947, and John von Neumann, who developed the theory of the duality in the same year.

Dantzig's original example of finding the best assignment of 70 people to 70 jobs exemplifies the usefulness of linear programming. The computing power required to test all the permutations to select the best assignment is vast; the number of possible configurations exceeds the number of particles in the universe. However, it takes only a moment to find the optimum solution by posing the problem as a linear program and applying the Simplex algorithm. The theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked.

In 1963, Dantzig’s Linear Programming and Extensions was published by Princeton University Press. Rich in insight and coverage of significant topics, the book quickly became “the bible” of linear programming.


Books by George Dantzig:

  • 1953. Notes on linear programming. RAND Corporation.
  • 1956. Linear inequalities and related systems. With others. Edited by H.W. Kuhn and A.W. Tucker. Princeton University Press.
  • 1963. Linear programming and extensions. Princeton University Press and the RAND Corporation. pdf from RAND
  • 1966. On the continuity of the minimum set of a continuous function. With Jon H. Folkman and Norman Shapiro.
  • 1968. Mathematics of the decision sciences. With Arthur F. Veinott, Jr. Summer Seminar on Applied Mathematics 5th : 1967 : Stanford University. American Mathematical Society.
  • 1969. Lectures in differential equations. A. K. Aziz, general editor. Contributors: George B. Dantzig and others.
  • 1970. Natural gas transmission system optimization. With others.
  • 1973. Compact city; a plan for a liveable urban environment. With Thomas L. Saaty.
  • 1974. Studies in optimization. Edited with B.C. Eaves. Mathematical Association of America.
  • 1985. Mathematical programming : essays in honor of George B. Dantzig. Edited by R.W. Cottle. Mathematical Programming Society.
  • 1997. Linear programming 1: Introduction. G.B.D. and Mukund N. Thapa. Springer-Verlag.
  • 2003. Linear programming 2: Theory and Extensions. G.B.D. and Mukund N. Thapa. Springer-Verlag.
  • 2003. The Basic George B. Dantzig. Edited by Richard W. Cottle. Stanford Business Books, Stanford University Press, Stanford, California.[11]

Book chapters:

  • Dantzig, George B. (1960), "General convex objective forms", in Arrow, Kenneth J.; Karlin, Samuel; Suppes, Patrick, Mathematical models in the social sciences, 1959: Proceedings of the first Stanford symposium, Stanford mathematical studies in the social sciences, IV, Stanford, California: Stanford University Press, pp. 151–158, ISBN 9780804700214. 

Articles, a selection:

  • Dantzig, George B. (June 1940). "On the Non-Existence of Tests of 'Student's' Hypothesis Having Power Functions Independent of σ". The Annals of Mathematical Statistics. 11 (2): 186–92. doi:10.1214/aoms/1177731912. JSTOR 2235875. 
  • Wood, Marshall K.; Dantzig, George B. (1949). "Programming of Interdependent Activities: I General Discussion". Econometrica. 17 (3/4): 193–9. doi:10.2307/1905522. JSTOR 1905522. 
  • Dantzig, George B. (1949). "Programming of Interdependent Activities: II Mathematical Model". Econometrica. 17 (3): 200–211. doi:10.2307/1905523. JSTOR 1905523. 
  • Dantzig, George B. (1955). "Optimal Solution of a Dynamic Leontief Model with Substitution". Econometrica. 23 (3): 295–302. doi:10.2307/1910385. JSTOR 1910385. 

See also[edit]


  1. ^Gass, Saul I. (2011). "George B. Dantzig". Profiles in Operations Research. International Series in Operations Research & Management Science. 147. pp. 217–240. doi:10.1007/978-1-4419-6281-2_13. ISBN 978-1-4419-6280-5. 
  2. ^ abcdeJoe Holley (2005). "Obituaries of George Dantzig". In: Washington Post, May 19, 2005; B06
  3. ^ abcRichard W. Cottle, B. Curtis Eaves and Michael A. Saunders (2006). "Memorial Resolution: George Bernard Dantzig". Stanford Report, June 7, 2006.
  4. ^ abcdefghAlbers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990). "George B. Dantzig". More Mathematical People. Harcourt Brace Jovanovich. pp. 60–79. ISBN 978-0-15-158175-7. 
  5. ^National Science Foundation – The President's National Medal of Science
  6. ^Robert Freund (1994). "Professor George Dantzig: Linear Programming Founder Turns 80". In: SIAM News, November 1994.
  7. ^ ab"The Unsolvable Math Problem". Snopes. June 28, 2011. 
  8. ^Dantzig, George (1940). "On the non-existence of tests of "Student's" hypothesis having power functions independent of σ". The Annals of Mathematical Statistics. 11 (2): 186–192. doi:10.1214/aoms/1177731912. 
  9. ^Allende, Sira M.; Bouza, Carlos N. (2005). "Professor George Bernard Dantzig, Life & Legend"(PDF). Revista Investigación Operacional. 26 (3): 205–11. 
  10. ^Dantzig, George; Wald, Abraham (1951). "On the Fundamental Lemma of Neyman and Pearson". The Annals of Mathematical Statistics. 22: 87–93. doi:10.1214/aoms/1177729695. Retrieved 14 October 2014. 
  11. ^Todd, Michael J. (2011). "Review: The Basic George B. Dantzig, by Richard W. Cottle". Bull. Amer. Math. Soc. (N.S.). 48 (1): 123–129. doi:10.1090/S0273-0979-2010-01303-3. 

Further reading[edit]

  • Cottle, Richard; Johnson, Ellis; Wets, Roger (March 2007). "George B. Dantzig (1914–2005)"(PDF). Notices of the American Mathematical Society. 54 (3): 344–62. 
  • "Professor George Dantzig: Linear Programming Founder Turns 80", SIAM News, November 1994
  • O'Connor, John J.; Robertson, Edmund F., "George Dantzig", MacTutor History of Mathematics archive, University of St Andrews .
  • Dantzig, George B. (1990). "The Diet Problem". Interfaces. 20 (4): 43–7. doi:10.1287/inte.20.4.43. JSTOR 25061369. 
  • Cottle, Richard W. (2005). "George B. Dantzig: a legendary life in mathematical programming". Mathematical Programming. 105 (1): 1–8. doi:10.1007/s10107-005-0674-4. ISSN 0025-5610. 

External links[edit]

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