George B. Dantzig (1914–2005) Richard Cottle, Ellis Johnson, and Roger Wets

The final test of a theory is its capacity to solve the methods (in particular, algorithms) are required to problems which originated it. obtain the kinds of solutions called for in practical This work is concerned with the theory and so- decision problems. lution of linear inequality systems. ... George Dantzig is best known as the father of The viewpoint of this work is constructive. It re- linear programming (LP) and the inventor of the flects the beginning of a theory sufficiently power- simplex method. The practical power of these two ful to cope with some of the challenging decision contributions is so great that, on these grounds problems upon which it was founded. alone, he was arguably one of the most influential mathematicians of the twentieth century. And yet there is much more to the breadth and significance o says George B. Dantzig in the preface of his work. Our aim in this memorial article is to his book, Linear Programming and Extensions,anowclassicworkpublishedin to document the magnitude of George Dantzig’s 1963, some sixteen years after his formu- impact on the world by weaving together many lation of the linear programming problem strands of his life and professional commitments. andS discovery of the simplex algorithm for its In so doing, we bring to light the inadequacy of the solution. The three passages quoted above repre- “father/inventor” epithet. sent essential components of Dantzig’s outlook on Theimpactwehaveinmindisofmanykinds.The linear programming and, indeed, on mathematics earliest, of course, was on military and industrial generally. The first expresses his belief in the im- planning and production. Dantzig’s work greatly portance of real world problems as an inspiration impacted economics, mathematics, operations for the development of mathematical theory, not research, computer science, and various fields of for its own sake, but as a means to solving impor- applied science and technology. In response to tant practical problems. The second statement is these developments, there emerged the concomi- based on the theoretical fact that although a linear tant growth of educational programs. Dantzig programming problem, is, prima facie, concerned himself was a professor for more than half of his with constrained optimization, it is really all about solving a linear inequality system. The third state- professional life and in that capacity had a pro- mentrevealsDantzig’sconvictionthatconstructive found impact on the lives and contributions of his more than fifty doctoral students. Richard Cottle is professor emeritus of management As already suggested, George Dantzig passion- science and engineering at Stanford University. His ately believed in the importance of real world prob- email address is [email protected]. Ellis Johnson is lems as a wellspring of mathematical opportunity. professor of industrial and systems engineering at the Whether this was a lesson learned or a conviction Georgia Institute of Technology. His email address is [email protected]. Roger Wets is professor of held since early adulthood is unclear, but it served mathematics at the University of California, Davis. His him very well throughout his long and productive email address is [email protected]. life.

344 Notices of the AMS Volume 54, Number 3 Formation he had a remarkable ex- Our knowledge of George Dantzig’s childhood is perience that was to be- largely derived from part of an interview [1] con- come a famous legend. ducted in November 1984 by Donald Albers. (Much Arriving late to one of the same article is available in [2].) George was the Neyman’sclasses,Dantzig son of mathematician Tobias Dantzig (1884–1956) saw two problems writ- and Anja Ourisson who had met while studying ten on the blackboard mathematics at the Sorbonne. There, Tobias was and mistook them for greatly impressed by Henri Poincaré (1854–1912) a homework assignment. andlaterwroteabookonhim[41],thoughheisbest He found them more chal- known for his Number, The Language of Science lenging than usual, but [40]. managed to solve them GeorgeDantzigwasborninPortland,Oregon,on and submitted them di- November 8, 1914. His parents gave him the mid- George B. Dantzig rectly to Neyman. As it dle name“Bernard”hoping that he would becomea turned out, these prob- writer like George Bernard Shaw. George’s younger lems were actually two open questions in the brother Henry (1918–1973), who was given the mid- theory of mathematical statistics. Dantzig’s 57- dle name Poincaré, became an applied mathemati- page Ph.D. thesis [12] was composed of his cian working for the Bendix Corporation. solutions to these two problems. One of these By his own admission, George Dantzig lacked was immediatelysubmitted for publication and ap- interest in schoolwork until grade seven. He then peared in 1940 as [11]. For reasons that are not became keen about science and mathematics al- altogether clear, the other appeared only in 1951 though in ninth grade he made a “poor start” in his asajointpaperwithAbrahamWald[39]. first algebra course. “To be precise,” he said, “I was By June 1941, the content of Dantzig’s disser- flunking.” Furious with himself, he buckled down tation had been settled: it was to be on the two and went on to excel in high school mathematics problems and their solutions. Although he still had and science courses. He was particularly absorbed various degree requirements to complete, he was by projective geometry and worked “thousands” of eager to contribute to the war effort and joined the such problems given to him by his father, Tobias, U.S. Air Force Office of Statistical Control. He was who was then on the mathematics department put in charge of the CombatAnalysis Branch where faculty at the University of Maryland. he developed a system through which combat units As an undergraduate, George concentrated reported data on missions. Some of this work also in mathematics and physics at the University involved planning (and hence modeling) that, in of Maryland, taking his A.B. degree in 1936. Af- light of the primitive computing machinery of the ter this he went to the University of Michigan day, was a definite challenge. The War Department and obtained the M.A. in mathematics in 1938. recognized his achievements by awarding him its In Ann Arbor, he took a statistics course from Exceptional Civilian Service Medal in 1944. Harry C. Carver (founding editor of the Annals of Dantzig returned to Berkeley in the spring of Mathematical Statistics and a founder of the Insti- 1946 to complete his Ph.D. requirements, mainly tute of Mathematical Statistics). He found the rest a minor thesis and a dissertation defense. Around of the curriculum excessively abstract and decided that time he was offered a position at Berkeley but to get a job after finishing his master’s degree turneditdowninfavorofbecomingamathematical program in 1938. advisor at the U.S. Air Force Comptroller’s Office. Dantzig’s decision to take a job as a statistical clerk at the Bureau of Labor Statistics (BLS) turned Thissecondfatefuldecisionsethimonapathtothe out to be a fateful one. In addition to acquiring a discovery of linear programming and the simplex knowledge of many practical applications, he was algorithm for its solution in 1947. assigned to review a paper written by the eminent mathematical statistician Jerzy Neyman who was Origins then at University College, London, and soon there- Dantzig’s roles in the discovery of LP and the after at the University of California at Berkeley. Ex- simplex method are intimately linked with the cited by Neyman’s paper, Dantzig saw in it a logi- historical circumstances, notably the Cold War cally based approach to statistics rather than a bag and the early days of the Computer Age. The de- oftricks.HewrotetoNeyman(atBerkeley)declaring fense efforts undertaken during World War II and his desire to complete a doctorate under Neyman’s its aftermath included very significant contribu- supervision, and this ultimately cameto pass. tions from a broad range of scientists, many of In 1939 Dantzig enrolled in the Ph.D. program whom had fled the horrors of the Nazi regime.This of the Berkeley Mathematics Department where trend heightened the recognition of the power that Neyman’s professorship was located. Dantzig took mathematical modeling and analysis could bring only two courses from Neyman, but in one of them to real-world problem solving. Beginning in 1946,

March 2007 Notices of the AMS 345 Dantzig’s responsibility at the Pentagon involved mention here. In [58], Kantorovich proposed an the “mechanization” of the Air Force’s planning approachto solving some classesof extremalprob- procedures to support time-staged deployment of lems that would include the linear programming training and supply activities. Dantzig’s approach problem. A brief discussion of the second paper to the mathematization of this practical problem [59] is given below. It is worth noting here that these usheredinanewscientificeraandledtohisfamein two articles—written in English during World War an ever-widening circle of disciplines. II—were reviewed by H. H. Goldstine [55] and Max In [19], George Dantzig gives a detached, histor- Shiffman [75], respectively. Hence they were not ical account of the origins of—and influences on— altogether unknown in the West. linear programming and the simplex method. Al- In 1940 Koopmans emigrated from the Nether- most twenty years later, as he approached the age lands to the United States. During World War II he of seventy, Dantzig began turning out numerous in- was employed by the Combined Shipping Adjust- vited articles [21]–[26] on this subject, most of them ment Board, an agency based in Washington, D.C., having an autobiographical tone. From the mathe- that coordinated the merchant fleets of the Allied matical standpoint, there is probably none better governments, chiefly the United States and Britain. than [25] which, unlike the rest, explicitly relates Koopmans’sfirstpaperofalinearprogrammingna- partofhisdoctoraldissertationtothefieldoflinear ture dates from 1942. For security reasons, the pa- programming. per was classified. It became available as part of a Dantzig was not alone in writing about the dis- Festschriftin1970[63].Basedonthiswartimeexpe- covery of linear programming. One of the most rience, Koopmans’s 1947 paper [64] develops what informative articles onthis subjectis thatofRobert later came to be called the transportation problem Dorfman, a professor of economics at Harvard [14]. These works are not algorithmic; they empha- (now deceased). In telling the story, Dorfman [46] size the modeling aspectfor the particular shipping clarifies “the roles of the principal contributors”. applicationthat Koopmans had in mind. As he goes on to say, “it is not an especially com- As Koopmans was later to discover, the special plicated story, as histories of scientific discoveries class of linear programming problems he had writ- go, butneither is itentirely straightforward.”These tenabout(namely,thetransportationproblem)had opening remarks are meant to suggest that many alreadybeenpublishedin1941byanMITalgebraist elements of linear programming had already been named Frank L. Hitchcock. The title of Hitchcock’s come upon prior to their independent discovery by paper, “The distribution of a product from several Dantzig in 1947. For the convenience of readers, sources to numerous localities” [56], describes the present article will revisit a bit of this lore and what the transportation problem is about, includ- willendeavortoestablishthepointmadeattheout- ing its important criterion of least cost. (A review set that Dantzig’s greatness rests not only on the of the paper [56] can be found in [62].) Hitchcock discovery of linear programming and the simplex makes only the slightest of suggestions on the method, but on the depth of his commitment to application of the model and method he advances their development and extensions. inhis paper. At the outset of Dantzig’s work on linear pro- Just as Hitchcock’s paper was unknown to gramming, there already existed studies by two of Koopmans (and to Dantzig), so too the work of the principal contributors to its discovery. The first Kantorovich on the transportation problem was of these was done by the mathematician Leonid unknown to Hitchcock. Moreover, it appears that V. Kantorovich. The second was by the mathemat- in writing his paper [59] on the “translocation of ical statistician/economist Tjalling C. Koopmans. masses”,Kantorovichwasnotfamiliarwiththevery Forgood reasons,these advances were unknown to much earlier paper [68] of Gustave Monge (1746– Dantzig. 1818) on “cutting and filling”, a study carried out As a professor of mathematics and head of in conjunction with the French mathematician’s the Department of Mathematics at the Institute work on the moving of soil for building military of Mathematics and Mechanics of Leningrad State fortifications. The formulation in terms of contin- University, Kantorovich was consulted by some uous mass distributions has come to be called the engineers from the Laboratory of the Veneer Trust “Monge-Kantorovich Problem”. who were concerned with the efficient use of ma- A few others contributed to the “pre-history” of chines. From that practical contact sprung his linear programming. In 1826 Jean-Baptiste Joseph report on linear programming [57], which, though Fourier announced a method for the solution of it appearedin 1939, seemsnot to have been known linear inequality systems [52]; it has elements in in the West (or the East) until the late 1950s and was common with the simplex method of LP. Charles not generally available in English translation [60] de la Vallée Poussin in 1911 gave a method for until 1960. (This historically important document finding minimum deviation solutions of systems is a 68-page booklet in the style of a preprint. Kan- of equations [81]. John von Neumann’s famous torovich [61, p. 31] describes it as a “pamphlet”.) game theory paper [70] of 1928, and his book [72] Two other publications of Kantorovich deserve written with Oscar Morgenstern and published in

346 Notices of the AMS Volume 54, Number 3 1944, treated finite two-person zero-sum games, Dantzig’s discovery of the linear programming a subject that is intimately connected with linear problem and the simplex algorithm was indepen- programming; von Neumann’s paper [71] is of dent of [52], [81], [57], [56], and [63]. Yet, as he particular interest in this regard. In addition to has often related, it was not done in isolation. On theseprecursors,thereremainsTheodorMotzkin’s the formulation side, he was in contact with Air scholarly dissertation Beiträge zur Theorie der Force colleagues, particularly Murray Geisler and Linearen Ungleichungen accepted in 1933 at the Marshall K. Wood, and with National Bureau of University of Basel and published in 1936 [69]. (For Standards (NBS) personnel. At the suggestion of a loose English translation of this work, see [7].) Albert Kahn at NBS, Dantzig consulted Koopmans Apart from studying the general question of the who was by then at the Cowles Commission for existence (or nonexistence) of solutions to linear in- Research in Economics (which until 1955 was based equality systems, Motzkingave anelimination-type at the Universityof Chicago). The visit took place in solution method resembling the technique used June 1947 and got off to a slow start; before long, earlierbyFourier(andDines[43]whoseemstohave Koopmans perceived the broad economic signif- concentrated on strict inequalities). icance of Dantzig’s general (linear programming) Initially, Dantzig knew nothing of these prece- model. This might have prompted Koopmans to dents, yet he did respond to a powerful influence: disclose information about his 1942 work on the Wassily Leontief’s work [67] on the structure of the transportation problem and Hitchcock’s paper of American economy. This was brought to Dantzig’s 1941. attention by a former BLS colleague and friend, Another key visit took place in October 1947 Duane Evans. The two apparently discussed this at the Institute for Advanced Study (IAS) where subject at length. In Dantzig’s opinion [19, p. 17] Dantzig met with John von Neumann. Dantzig’s “Leontief’s great contribution . . . was his construc- vivid account of the exchange is givenin [1, p. 309] tion of a quantitative model ...for the purpose where he recalls, “I began by explaining the for- of tracing the impact of government policy and mulation of the linear programming model ...I consumertrendsuponalargenumberofindustries describedittohimasIwouldtoanordinarymortal. which were imbedded in a highly complex series of He responded in a way which I believe was unchar- interlocking relationships.” Leontief’s use of “an acteristic of him. ‘Get to the point,’ he snapped. empirical model” as distinct from a “purely formal ...In less than a minute, I slapped the geometric model” greatly impressedDantzig as did Leontief’s and algebraic versions of my problem on the black- organizational talent in acquiring the data and his board. He stood up and said, ‘Oh that.’” Just a few “marketing” of the results. “These things,” Dantzig years earlier von Neumann had co-authored and declares, “are necessary steps for successful appli- published the landmark monograph [72]. Dantzig cations, and Leontief took them all. That is why in goes on, “for the next hour and a half [he] pro- mybookheisahero”[1, p. 303]. ceeded to give me a lecture on the mathematical Carrying out his assignment at the U.S.A.F. theory of linear programs.” Dantzig credited von Comptroller’s Office, George Dantzig applied him- Neumann with edifying him onFarkas’slemma and selftothemechanizationoftheAirForce’splanning the duality theorem (of linear programming). On procedures tosupportthe time-stageddeployment a separate visit to Princeton in June 1948, Dantzig of training andsupply activities.He createda linear met Albert Tucker who with his students Harold mathematical model representing what supplies KuhnandDavidGalegavethefirstrigorousproofof were available and what outputs were required the duality theorem that von Neumann and Dantzig over a multi-period time horizon. Such conditions had discussedattheir initial meeting. normallyleadtoanunder-determinedsystem,even During the summer of 1947, months before his when the variablesare required to assume nonneg- encounter with von Neumann at the IAS, Dantzig ative values, which they were in this case, reflecting proposed his simplex method (much as described their interpretation as physical quantitities. To elsewhere in this article); in the process of this dis- single out a “best” solution, Dantzig introduced a covery he discussed versions of the algorithm with linear objective function, that is, a linear minimand economists Leonid Hurwicz and Tjalling Koop- or maximand. This was an innovation in planning mans. It was recognized that the method amounts circles, an achievement in which Dantzig took great to traversing a path of edges on a polyhedron; for pride. As he put it in 1957, “linear programming is that reason he set it aside, expecting it to be too an anachronism”; here he was alluding to the work inefficient for practical use. These reservations of economists François Quesnay, Léon Walras, and were eventually overcome when he interpreted the Wassily Leontief as well as mathematician John method in what he called the “column geometry”, von Neumann all of whom could (and in his mind, presumably inspired by Part I of his Ph.D. thesis should) have introduced objective functions in [12], [39], [25, p. 143]. There, the analogue of a their work [42, p. 102]. so-called convexity constraint is present. With a

March 2007 Notices of the AMS 347 constraint of the form basic solution that would be obtained if it were to replace one of the basic columns. The convex hull (1) x +···+ x = 1, x ≥ 0 j = 1,...,n 1 n j of this new point and σ is an (m + 1)-simplex τ (which is common, but by no means generic)the re- in IRm+1. The requirement line meets the boundary maining constraints of τ in two points: the previous solution and one

(2) A.1x1 +···+ A.nxn = b other point with a better objective function value. In the (nondegenerate) case where the new point amount to askingfor a representationof b as a con- liesintherelativeinteriorofafacetof τ,say ρ, there vex combination of the columns A. ,...A. ∈ IRm. 1 n is a unique (currently basic) column for which the (Of course, the solutions of (1) alone constitute corresponding weight (barycentric coordinate) is an n − 1 simplex in IRn, but that is not exactly zero. This point is opposite the new facet ρ where where the name “simplex method” comes from.) the requirement line meets the boundary of τ. Note By adjoining the objective function coefficients that ρ is an m-simplex and corresponds to a new c to the columns A. , thereby forming vectors j j andimprovedfeasiblebasis.Theprocessofmaking (A. ,c ) and adjoining a variable component z to j j the basis change is called simplex pivoting. The the vector b to obtain (b, z), Dantzig viewed the name can be seen as an apt one when the facets of corresponding linear programming problem as τ are likened to the (often triangle-shaped) body one of finding a positively weighted average of the partsof ahinge. vectors (A. ,c ), . . . , (A. ,c ) that equals (b, z) and 1 1 n n Beyond the fact that not all linear program- yields the largest(orsmallest)value of z. ming problems include convexity constraints, it Itwasknownthatifalinearprogram(instandard was clear that, without significant advances in au- form) has an optimal solution, then it must have tomatic computing machines, the algorithm—in an optimal solution that is also an extreme point of whatever form it took—was no match for the size the feasible region (the set of all vectors satisfying of the planning problems for which numerical so- the constraints). Furthermore, the extreme points lutions were needed. One such need is typified by a of the feasible region correspond (albeit not neces- critical situation that came up less than a year after sarily in a bijective way) to basic feasible solutions Dantzig’s discoveries: the Berlin Airlift. Lasting of the constraints expressed as a system of linear 463 days, this program called for the scheduling of equations. Under the reasonable assumption that aircraft and supply activities, including the training thesystem(1),(2)hasfullrank,oneisledtoconsider of pilots, on a very large, dynamic scale. During nonsingular matrices of the form this crisis, Britain, France, and the United States 1 1 · · · 1 airlifted more than two million tons of food and = (3) B . . · · · . " A j1 A j2 A jm+1 # supplies tothe residents ofWestBerlinwhose road, rail, and water contacts with Western Europe had such that beenseveredbythe USSR. 1 (4) B−1 ≥ 0. " b # Initial Impacts . . . “Mathematical Techniques of Program Planning” is The columns A j1 , A j2 ,...,A jm+1 viewed as points in IRm are easily seen to be in general position. Ac- the title of the brief talk in which Dantzig publicly cordingly, their convex hull is an m-simplex. announced his discovery of the simplex Dantzig conceptualized the n points A.j as lying method. The presentation took place at a ses- in the “horizontal” space IRm and then pictured sion of the joint annual meeting of the American each (m + 1)-tuple (A.j ,cj ) as a point on the line Statistical Association (ASA) and the Institute of m orthogonal to IR and passing through A.j with MathematicalStatistics (IMS) onDecember 29, 1947 cj measuring the vertical distance of the point [48, p. 134]. As Dorfmanreports [46, p. 292], “there above or below the horizontal “plane”, according is no evidence that Dantzig’s paper attracted any to the sign of cj . The requirement line consisting particular interest or notice.” The paper was not of points (b, z) where b ∈ IRm is as above and z published.Themethod’snextappearanceoccurred is the (variable) value of the objective function is in a session (chaired by von Neumann) at a joint likewise orthogonal to the horizontal plane. For national meeting of the IMS and the Econometric any feasible basis, the requirement line meets Society on September 9, 1948. Dantzig’s talk, titled σ , the convex hull of the corresponding points “ProgramminginaLinearStructure”,arousedmore . . . (A j1 ,cj1 ), (A j2 ,cj2 ), . . . , (A jm+1 ,cjm+1 ), in the point interest than its predecessor. The abstract [13] is (b, z), the ordinate z being the value of the objec- remarkable for its visionary scope. In it we find tive functiongivenbythe associated basic solution. mention of the LP model, the notion of dynamic The m + 1 vertices of the simplex σ determine a systems, connections with the theory of games, a hyperplane in IRm+1. The vertical distance from a reference to computational procedures for “large point (A.j ,cj ) to this hyperplane indicates whether scale digital computers”, and the bold sugges- this point will improve the objective value of the tion that the solutions of such problems could be

348 Notices of the AMS Volume 54, Number 3 implemented and not merely discussed. A lively gasolines”. This—and, more generally, blending discussion followed Dantzig’s talk; afterwards in the petrochemical and other industries—was he participated in a panel discussion beside such to become an important early application area for distinguished figures as Harold Hotelling, Irving linear programming. From this groupofpapers one Kaplansky, Samuel Karlin, Lloyd Shapley, and John cansense the transitionof the subjectfrom military Tukey. The ball was now rolling. applications to its many fruitful applications in In an autobiographical piece [66] written around the civilian domain. Before long, the writings of the time he (and Kantorovich) received the 1975 No- Dantzig and others attracted the attention of a wide bel Prize in Economics, Tjalling Koopmans relates circle of applied scientists, including many from how in 1944 his “work at the Merchant Shipping industry. See [3]. Mission fizzled” due to a “reshuffling of respon- Besides the transportation problem, which con- sibilities”. Renewing his contact with economist tinues to have utility in shipping and distribution Jacob Marschak, Koopmans secured a position at enterprises, an early direction of applied linear the Cowles Commission in Chicago. He goes on to programming is to be found in agriculture, a ven- say “my work on the transportation model broad- erable topic of interest within economics. Our ened out into the study of activity analysis at the illustration is drawn from a historically important CowlesCommissionasaresultofabriefbutimpor- article that provided a natural setting for the LP. In tant conversation with George Dantzig, probably 1945, George J. Stigler [79] published a paper that in early 1947. It was followed by regular contacts developsa model for finding a least-cost“diet” that anddiscussionsextendingoverseveralyearsthere- would provide at least a prescribed set of nutri- after. Some of these discussions included Albert tional requirements. [See the companion piece by W. Tucker of Princeton who added greatly to my Gale in this issue for a discussion of the formula- understanding of the mathematical structure of tion.] Dantzig reports [19, p. 551] how in 1947 the duality.” simplex method was tried out on Stigler’s nutrition Oddly, Koopmans’s autobiographical note model (or “diet problem” as it is now called). The makes no mention of his instrumental role in constraints involved nine equations in seventy- staging what must be considered one of the most seven unknowns. At the time, this was considered influential events in the development of linear (and a large problem. In solving the LP by the sim- nonlinear) programming: the “Conference onActiv- plex method, the computations were performed ity Analysis of Production and Allocation” held in on hand-operated desk calulators. According to Chicago under the auspicesof the Cowles Commis- Dantzig, this process took “approximately 120 sionforResearchinEconomicsinJune1949.Edited man-days to obtain a solution.” Such was the state by Koopmans, the proceedings volume [65] of this of computing in those days. In 1953, the problem conference comprises twenty-five papers, four of wassolvedandprintedoutbyanIBM701computer which were authored—and one co-authored—by in twelve minutes. Today, the problem would be Dantzig. The speakers and other participants, ap- considered small and solved in less than a second proximately fifty in all, constitute an impressive on a personal computer [54]. One may question the set of individuals representing academia, govern- palatabilityofa dietfor humanconsumptionbased ment agencies, and the military establishment. It on LP principles. Actually, the main application of is a matter of some interest that the proceedings this methodology lies in the animal feed industry volume mentions no participants from industry. where itisof greatinterest. Exactlythesametypesofindividualswerespeakers The Air Force’s desire to mechanize planning at the “Symposium on Linear Inequalities and Pro- procedures and George Dantzig’s contributions gramming” held in Washington, D.C., June 14–16, to that effort through his creation of linear pro- 1951. Of the nineteen papers presented, twelve gramming and the simplex method had a very were concerned with mathematical theory and significant impact on the development of comput- computational methods while the remaining seven ing machines. The encouraging results obtained dealt with applications. In addition to a paper by by Dantzig and his co-workers convinced the Air George Dantzig and Alex Orden offering “A duality Force Comptroller, Lt. Gen. Edwin W. Rawlings, that theorem based on the simplex method”, the first much could be accomplished with more powerful part of the proceedings contains Merrill Flood’s computers. Accordingly, he transferred the (then paper “On the Hitchcock distribution problem”, large) sum of US$400,000 to the National Bureau that is, the transportation problem. Flood rounds of Standards which in turn funded mathematical up the pre-existing literature on this subject in- and electronic computer research (both in-house) cluding the 1942 paper of Kantorovich though not as well as the development of several computers the 1781 paper of Monge. Among the papers in the such as UNIVAC, IBM, SEAC, and SWAC. Speaking portion of the proceedings on applications, there to a military audience in 1956, General Rawlings appears the abstract (though not the paper) by notes, “I believe it can be fairly said that the Air Abraham Charnes, William Cooper and Bob Mellon Force interest in actual financial investments in (an employee of Gulf Oil Co.) on “Blending aviation the development of electronic computers has been

March 2007 Notices of the AMS 349 one of the important factors in their rapid devel- Including an Analysis of Monopolistic Firms by opment in this country” [74]. Dantzig took pride Non-linear Programming appeared in book form in the part that linear programming played in the [45] in 1951. It might well be the first book that developmentof computers. ever used “linear programming” in its title. The The list of other industrial applications of linear “non-linear programming” mentioned therein is programming—eveninits earlyhistory—is impres- what he calls “quadratic programming”, the opti- sive.A quick sense of this canbe gleanedfrom Saul mization of a quadratic function subject to linear Gass’s1958 textbook[53],one ofthe firstto appear inequality constraints. The first book on linear on the subject. In addition to a substantial chapter programming per se was [9], An Introduction to on applications of linear programming, it contains Linear Programming by A. Charnes, W. W. Cooper, a bibliography of LP applications of all sorts or- and A. Henderson, published in 1953. Organized in ganized under twelve major headings (selected two parts (applications and theory), the material of from the broader work [54]). Under the heading this slender volume is based on seminar lectures “Industrial Applications”, Gass lists publications given at Carnegie Institute of Technology (now pertaining to the following industries: chemical, Carnegie-Mellon University). coal, commerical airlines, communications, iron and steel, paper,petroleum, and railroad. Organizations and Journals Also published in 1958 was the book Linear Linear and nonlinear programming (collectively Programming and Economic Analysis by Dorfman, subsumed under the name “mathematical pro- Samuelson, and Solow [47]. “Intended not as a text gramming”) played a significant role in the forma- butasageneralexpositionoftherelationshipoflin- tionofprofessionalorganizations.AsearlyasApril ear programming to standard economic analysis” 1948, the Operational Research Club was founded it had “been successfully used for graduate classes in London; five years later, it became the Opera- in economics.” Its preface proclaims that “linear tional Research Society (of the UK). The Operations programming has been one of the most important Research Society of America (ORSA) was founded postwar developments in economic theory.” The in 1952, followed the next year, by the Institute of authors highlight LP’s interrelations with von Neu- ManagementSciences (TIMS). These parallelorgani- mann’s theory of games, with welfare economics, zations, each with its own journals, merged in 1995 and with Walrasian equilibrium. Arrow [4] gives a to form the Institute for Operations Research and perspective on Dantzig’s role in the development the Management Sciences (INFORMS) with a mem- of economic analysis. bership today of about 12,000. Worldwide, there Early Extensions are now some forty-eight national OR societies with a combined membership in the vicinity of 25,000. From its inception, the practical significance of lin- Mathematical programming (or “optimization” as earprogrammingwasplaintosee.Equallyvisibleto it now tends to be called) was only one of many Dantzigandotherswasafamilyofrelatedproblems subjects appearing in the pages of these journals. that came to be known as extensions of linear pro- Indeed, the first volumes of these journals devoted gramming. We turn to these now to convey another a small portion of their space to mathematical senseof theimpactof LPandits applications. programming. But that would change dramatically In this same early period, important advances with time. were made in the realm of nonlinear optimization. Over the years, many scientific journals have H. W. Kuhn and A. W. Tucker presented their work “Nonlinear programming” at the Second Berke- been established to keep pace with this active field ley Symposium on Mathematical Statistics and ofresearch.Inadditiontothejournal Mathematical Probability, the proceedings of which appeared Programming, there are today about a dozen more in 1951. This paper gives necessary conditions of whose name includes the word “optimization”. optimality for the (possibly) nonlinear inequality These, of course, augment the OR journals of OR constrained minimization of a (possibly) nonlinear societies and other such publications. objective function. The result is a descendant of the so-called “Method of Lagrange multipliers”. A Educational Programs similar theorem had been obtained by F. John and Delivering a summarizing talk at “The Second Sym- published in the “Courant Anniversary Volume” posium in Linear Programming” (Washington, D.C., of 1948. For some time now, these optimality con- January 27–29, 1955) Dantzig said “The great body ditions have been called the Karush-Kuhn-Tucker of my talk has been devoted to technical aspects conditions, in recognition of the virtually identical of linear programming.I have discussedsimple de- result presented in Wm. Karush’s (unpublished) vices that can make possible the efficient solution master’sthesisattheUniversityofChicagoin1939. ofavarietyofproblemsencounteredinpractice.In- Dorfman’s doctoral dissertation Application of terest in this subject has been steadily growing in Linear Programming to the Theory of the Firm, industrial establishments and in government and

350 Notices of the AMS Volume 54, Number 3 some of these ideas may make the difference be- Dantzig at RAND tween interest and use.” The notion of “making the In 1952 George Dantzig joined the RAND Corpo- difference between interest and use” was a deeply ration in Santa Monica, California, as a research held conviction of Dantzig’s. He knew how much mathematician. The RAND Corporation had come could be accomplished through the combination of into being in 1948 as an independent, private non- modeling, mathematical analysis, and algorithms profit organization after having been established like the simplex method. And then he added a pre- three years earlier as the Air Force’s Project RAND, established in 1945 through a special contract diction.“Duringthenexttenyearswewillseeagreat with the Douglas Aircraft Company. Despite its body of important applications; indeed, so rich in separation from Douglas—under which Project content and value to industry and government that RAND reported to Air Force Major General Curtis mathematics ofprogramming willmove toa princi- LeMay, the Deputy Chief of Air Staff for Research pal positionin collegecurriculums” [15, p. 685]. and Development—the newly formed RAND Cor- By the early 1950s, operations research (OR) porationmaintained a close connectionwiththe Air courses had begun appearing in university curric- Force and was often said to be an Air Force “think ula,andalongwiththemcamelinearprogramming. tank”. We have already mentioned the lectures of Charnes The reason for Dantzig’s job change (from and Cooper at Carnegie Tech in 1953. In that year mathematical advisor at the Pentagon to research both OR and LP were offered as graduate courses mathematician at RAND) is a subject on which the at Case Institute of Technology (the predecessor interviews and memoirs are silent. In [1, p. 311] of Case Western Reserve University). At Stanford he warmly describes the environment that existed OR was first taught in the Industrial Engineering in the Mathematics Division under its head, John D. Williams. The colleagues were outstanding and Program in the 1954–55 academic year, and LP was the organizational chart was flat. These were rea- one of the topics covered. At Cornell instruction in sons enough to like it, but there must have been these subjects began in 1955, while at Northwest- another motivation: the freedom to conduct re- ern it began in 1957. In the decades that followed, search that would advance the subject and to write the worldwide teaching of OR at post-secondary the book [19]. institutions grew dramatically. In some cases, The 1950s were exciting times for research in the separate departments of operations research and Mathematics Division at RAND. There, Dantzig en- corresponding degree programs were established; joyed the stimulation of excellent mathematicians in other cases these words (or alternatives like and the receptiveness of its sponsors. During his “management science” or “decision science”) were eight years (1952–1960) at RAND, Dantzig wrote added to the name of an existing department, and many seminal papers and internal research memo- there were other arrangements as well. The inter- randa focusing on game theory, LP and variants of disciplinary nature of the field made for a wide the simplex method, large-scale LP, linear program- ming under uncertainty, network optimization range of names and academic homes for the sub- including the traveling salesman problem, integer ject, sometimes even within the same institution, linear programming, and a variety of applications. such as Stanford, about which George Dantzig was Much of this work appeared first in the periodical fond of saying “it’s wall-to-wall OR.” Wherever op- literature and then later in [19]. erations research gained a strong footing, subjects The topic of methods for solving large-scale lin- like linear programming outgrew their place within earprograms,whichhaddrawnDantzig’sattention introductory survey courses; they became many at the Pentagon, persisted at RAND—and for years individual courses intheir own right. to come. Although the diet problem described Alongwiththedevelopmentofacademiccourses above was considered “large” at the time, it was came a burst of publishing activity. Textbooks for puny by comparison with the kinds of problems the classroom and treatises for seminars and re- the Air Force had in mind. Some inkling of this searchers on a wide range of OR topics came forth. can be obtained from the opening of a talk given Books onmathematicalprogramming were a major by M. K. Wood at the 1951 “Symposium on Linear part of this trend. One of the most important of Inequalities and Programming”. He says in its pro- ceedings[73, p. 3],“Justtoindicatethegeneralsize these was George Dantzig’s Linear Programming of the programming problem with which we are and Extensions [19]. Published in1963, itsucceeded faced,Iwouldliketogiveyouafewstatistics.Weare bytwoyearsthemonographofCharnesandCooper discussing an organization of over a million people [8]. Dantzig’s book was so rich in ideas that it soon who are classified in about a thousand different came to be called “The Bible of Linear Program- occupational skills. These people are organized ming”. Two of the sections to follow treat Dantzig’s into about ten thousand distince [sic] organiza- direct role in academic programs. For now, we tional units, each with its own staff and functions, return to the chronology ofhis professionalcareer. located at something over three hundred major

March 2007 Notices of the AMS 351 operating locations. The organizational units use upperbound,say αij .Inlinearprogrammingterms, approximately one million different kinds of items the problem was of the following type, of supplies and equipment, at a total annual costof min c · x subject to Ax = b, lb ≤ x ≤ ub, something overfifteen billion dollars.” The challenge of solving problems on such a with m linear constraints (Ax = b) and, rather than grand scale was one that Dantzig took seriously. just nonnegativity restrictions on the x-variables, Speaking at the “First International Conference on the vectors lb and ub imposed lower and upper Operational Research” in 1957, he acknowledged, bounds on x. Here after an appropriate change of “While the initial proposal was to use a linear pro- variables and inclusion of the slack variables, the gramming model to develop Air Force programs, problem would pass from one with m linear con- + it was recognized at an early date that even the straints to one with at least m n such constraints. most optimistic estimates of the efficiency of fu- Solving the scheduling problem might have been the source of further delays! That is when Dantzig ture computing procedures and facilities would introduced a new pivoting scheme that would be not be powerful enough to prepare detailed Air integrated in any further implementation of the Force programs” [17]. Nevertheless, he steadfastly simplex method. It would essentially deal with the devotedhimselftothecauseofimprovingthecapa- problem as one with only m linear constraints re- bilities of mathematical programming, especially quiring just a modicum of additional bookkeeping the simplex method of linear programming. to keep track of variables at their lower bounds, If one had to characterize Dantzig’s work on the basicvariables,and those at their upper bound, large-scale systems, more specifically, large-scale rather than just with basic and nonbasic variables linear programs, it could be viewed as the design asin the originalversionof the simplexmethod. of variants of the simplex method that require Dantzig’s work on the transportation problem only the use of a “compact” basis, i.e., that simplex andcertainnetwork problems hadmade him aware iterationswill becarriedout byrelying on a basisof that the simplex method becomes extremely ef- significantly reduced size. Early on, it was observed ficient when the basis has either a triangular or that given a linear program in standard form, i.e., a nearly triangular structure. Moreover, as the min c · x subject to Ax = b, x ≥ 0, the efficiency of simplex method was being used to solve more the simplex method is closely tied to the number of sophisticated models, in particular involving dy- linear constraints (Ax = b) that determine the size namical systems, the need for variants of the of the basis rather than to the number of variables. simplex method to handle such larger-scale prob- It is part of the linear programming folklore that lems became more acute. Although the simplex “in practice” the number of steps required to solve basesofsuchproblemsarenotpreciselytriangular, a linear program is of the order of 3m/2 where they have a block triangular structure,e.g., m is the number of linear constraints. Moreover, A working with a compact basis (i.e., one of small 11 A A  21 22  order), which was essentially going be inverted, , .. would make the method significantly more reliable  .    from a numerical viewpoint. A A ... A   T 1 T 2 T T  The first challenging test of the simplex method   that can be exploited to “compactify” the numerical ona“large-scale”problemhadcomeindealingwith operations required by the simplex method. All Stigler’s diet problem; the introduction of upper present-day, commercial-level, efficient implemen- bounds on the amounts of various foods greatly tations of the simplex method make use of the enlarged the number of constraints, thereby ex- shortcuts proposed byDantzig in[16], inparticular acerbating an already difficult problem. The need in the choice of heuristics designed to identify a to have a special version of the simplex method to good starting basis known among professionals as deal with upper bounds came up again at RAND. a “crash start”. Researchers were dissatisfied with the turn-around In the late 1950s, Dantzig and Wolfe proposed time for the jobs submitted to their computer cen- the Decomposition Principle for linear programs on ter, mostly because certain top-priority projects which they lectured at the RAND Symposium on would absorb all available computing resources Mathematical Programming in March 1959. Their for weeks. That is when a more flexible priority approach was inspired by that of Ford and Fulker- scheduling method was devised in which the value son on multistage commodity network problems assigned to a job decreased as its completion date [51]. Actually, the methodology is based on funda- was delayed. The formulation of this scheduling mental results dating back to the pioneering work problem turned out to be a linear program having of Minkowski and Weyl on convex polyhedral sets. special (assignment-like) structure and the restric- By the mid-1950s Dantzig had already written a tion that the number of hours xij to be assigned to couple of papers on stochastic programming; he each project i in week j could not exceed a certain realized that a method was needed to handle really

352 Notices of the AMS Volume 54, Number 3 large linear programs, with “large” now taking on master program is feasible. Clearly, if this master its present-day meaning. program is unbounded, so is the originally formu- Because it illustrates so well how theory can lated problem. So, let us assume that this problem be exploited in a computational environment, we is bounded, and let (λν ,µν ) be an optimal solution ν = ν ν = ν ν present an abbreviated description of the method. where λ (λ1 ,...,λrν ) and µν (µ1 ,...,µsν ) ν Suppose we have split our system of linear con- with optimal value ζν . Let π be the vector of mul- straints in two groups so that the linear program tipliers attached to the first m linear constraints takeson theform, and θν be the multiplier attached to the (m + 1)st r = min c · x subject to Ax = b, Tx = d, x ≥ 0 constraint k=1 λk 1. In sync with the (simplex method) criterionP to find an improved solution to where the constraints have been divided up so the full master problem, one could search for a that Ax = b consists of a relatively small num- k l new column of type (γk,P , 1),oroftype (δl, Q , 0), ber, say m, of linking constraints, and the system ν k with the property that γk − π · P − θν < 0 T x = d is large. As shall be seen later on, sto- ν l or δl − π · Q < 0. Equivalently, if the linear chastic programs will provide good examples of subprogram problems that naturally fall into this pattern. The ⊤ ν n min (c − A π ) · x subject to T x = d, x ≥ 0, set K = x ∈ IR+ T x = d is a polyhedral set is unbounded, in which case the simplex method that in view of the Weyl-Minkowski Theorem ad- mits a “dual” representation as a finitely generated would provide a direction of unboundedness ql , set obtained as the sum of a polytope (bounded that would, in turn, give us a new column of type l polyhedron) (δl , Q , 0) tobeincludedinourmasterprogram.Or, r r if this linear sub-program is bounded, the optimal k k x = p λk λk = 1,λk ≥ 0, k = 1,...,r solution p generatedbythesimplexmethodwould n = = bea vertexofthe polyhedralset x ∈ IR T x = d .  kX1 kX1 + − ⊤ ν · k and a polyhedral cone As long as θν < (c A π ) p , a new column k s of type (γk,P , 1) would be included in the master l problem. On the other hand, if this last inequality x = q µl µ ≥ 0, l = 1, . . . , s was not satisfied, it would indicate that no column lX=1  can be found that would enable us to improve the with both r and s finite. Our given problem is thus optimal value ofthe problem, i.e., equivalent to the following linear program, which rν sν we shall refer to as the full master program, ∗ k ν k ν x = p λk + q µl min r γ λ + s δ µ k=1 k k l=1 l l kX=1 lX=1 r k s l isthentheoptimalsolutionofouroriginalproblem. subject to Pk=1 P λk + Pl=1 Q µl = b, Clearly,themethodwillterminateinafinitenumber Pr = P k=1 λk 1, of steps; the number of columns generated could

λPk ≥ 0, k = 1,...,r, neverexceedthose in the full masterprogram. An interesting and particular application of this µ ≥ 0, l = 1, . . . , s l methodologyistolinearprogramswithvariableco- k k k l where γk = c · p ,P = Ap , δl = c · q , and efficients. The problem is one of the following form: Ql = Aql . Because this linear program involves n n j only m + 1 linear constraints, the “efficiency and min cj xj subject to P xj = Q, x ≥ 0, numerical reliability” of the simplex method would jX=1 jX=1 be preserved if, rather than dealing with the large- j where the vectors (cj ,P ) may be chosen from a scale system, we could simply work with this latter closed convex set Cj ⊂ IRm+1. If the sets Cj are problem. That is, if we do not take into account the polyhedral, we essentially recover the method de- work of converting the constraints T x = d, x ≥ 0 scribed earlier, but in general, it leads to a method to their dual representation, and this could be a for nonlinear convex programs that generates “in- horrendous task that, depending on the structure ternal” approximations, i.e., the feasible region, of these constraints, could greatly exceed that say C, of our convex program is approximated by a of solving the given problem. What makes the succession of polyhedral sets contained in C. The Dantzig-Wolfe decomposition method a viable and strategy is much the same as that describedearlier, attractive approach is that, rather than generat- viz., one works with a master (problem) whose ing the full master program, it generates only a columns are generated from the subproblems to small, carefully selected subset of its columns, improve the succession of master solutions. For all namely, those that potentially could be included in j, these subproblems are of the following type: an optimal basis. Let us assume that, via a Phase I ν j j j type-procedure, a few columns of the (full) master min cj − π · P subject to (cj ,P ) ∈ C . programhavebeengenerated,say k = 1,...,rν

March 2007 Notices of the AMS 353 on the convex sets Cj ; the π ν correspond to the but Ray, who was a good poker player, said the bet optimal multipliers associated with the linear con- should be the closest to the actual number and he straints of the master problem. Dantzig outlined an went lower, saying eleven. In actuality, only seven elegant application of this method in “Linear con- were needed, plus two more that were presented trol processes and mathematical programming” as being somewhat ad hoc but needed to make the [20]. linear programmingoptimum be integer-valued. In this period, Dantzig co-authored another The two additional inequalities are briefly jus- major contribution: a solution of a large (by the tified and are acknowledged in a footnote, “We are standards of the day) Traveling Salesman Problem indebtedtoI.GlicksbergofRandforpointingoutre- (TSP). However, this work was considerably more lations of this kind to us.” Muchof the later research than just a way to find an optimum solution to a in combinatorial optimization is directed at find- particular problem. It pointed the way to several ing such inequalities that can be quickly identified, approaches to combinatorial optimization and when violated. For example, finding a violated sub- integer programming problems. tour elimination constraint is equivalent to finding First, the traveling salesman problem is to find a “minimum cut” in the graph where the weights on the shortest-distance way to go through a set of the arcs are the primal variables xij , which may be cities so as to visit each city once and return to the fractional. The min cut is the one that minimizes starting point. This problem was put on the com- that sum of xij overalledgesin thecut;thatvalueis putational mathematical map by the now-famous then comparedto 2. paper of Dantzig, Fulkerson, and Johnson [28]. It The method used is to start with a tour, which has a footnote on the history of the problem that can be found by any one of several heuristics. Dual includes Merrill Flood stimulating interest in the variables on the nodes can then be found, and the problem, Hassler Whitney apparently lecturing on paper explains how. Then, non-basic arcs can be it, and Harold Kuhn exploring the relation between priced out and a basis change found. If the solution the TSP and linear programming. The abstract is moves to another tour, then the step is repeated. If one sentence: “It is shown that a certain tour of it moves to a sub-tour, then a sub-tour elimination forty-nine cities, one in each of the forty-eight constraint is added. If it moves to a fractional solu- states and Washington, D.C., has the shortest road tion, then they look for a constraint to add that goes distance.” The distances were taken from a road through the current solution vector and cuts off atlas. The authors reduce the problem to forty-two the fractional solution. Thus, it is an integer-primal cities by pointing out that their solution with seven simplex method, a primal simplex method that Northeastern cities deleted goes through those staysat integerpointsby addingcuts asneeded. sevencities. Thus the solution methodis appliedto A device the paper employs is the use of a 42-city problem. “reduced-cost fixing” of variables. Once a good The solution method involves solving the linear tour is found and an optimum linear program em- program with a 0-1 variable for each link (the ter- ploying valid cuts, such as sub-tour elimination minology they use for undirected arcs) of the com- cuts, the difference between the tour cost and the plete graphonforty-two nodes. The firstsetof con- linear program optimum provides a value that can straints says the sum of the variables on links inci- be used to fix non-basic variables at their current denttoagivennodemustequaltwo.Thereareforty- non-basic values. Although the number of vari- two of these constraints, for the 42-city problem. ables is not enormous, the 42-city problem has 861 The second set of constraints is made up of what variables, which is a lot if one is solving the linear are called sub-tour elimination constraints. These program by hand, as they did. They were using the require that the sum of variables over links, with revised simplex method so only needed to price oneendinasetS ofnodesandtheotherendnotinS, out all these columns and enter the best one. Thus, must be greater-than-or-equal to two. There are of even though the revised simplex method only has order 242 of these constraints. However, they only to price out over the columns, the process can be had to add seven of them to the linear program for onerous whenthere are 861variables and computa- the 42-city problem at which point all the rest were tionisbeingdone by hand.For that reason,actually satisfied. So despite the fact that there are a huge droppingmanyofthesevariablesfromtheproblem number of such constraints, only a few were used hada realadvantage. when they were added on to the linear program as The paper also refers to a “combinatorial ap- needed. proach” which depends critically on reducing the On several occasions, George Dantzig spoke of number of variables. This approach seems to be a bet he had with Ray Fulkerson. George was con- related to the reduced-costfixing referred to above. vincedthatnotverymanysub-toureliminationcon- Although the paper is a bit sketchy about this straints would be needed, despite their large num- approach, it seems to be that as the variables are ber. He was so convinced that he proposed a bet reduced to a small number, some enumeration that there would be no more than twelve (the ex- without re-solving the linear program (but perhaps act number is probably lost forever at this time), using the reduced-cost fixing) is required. In this

354 Notices of the AMS Volume 54, Number 3 way, the options left open can be evaluated and the A relatedproblem was posedas finding the min- optimum solution identified. imum number of tankers to cover a schedule. This According to Bland and Orlin [5], “Newsweek is used today for example by charter airline com- magazine ran a story on this ‘ingenious application panies to cover required flights by the minimum of linear programming’ in the 26 July 1954 issue.” number ofaircraft. The schedule ofrequired flights However, it is highly unlikely that even Dantzig, in a charter operation is typically unbalanced re- Fulkerson, and Johnson could have anticipated quiringferryflights,ordeadheadflights,inorderto then the practical impact that this work would ul- cover the required flights. Dantzig and Fulkerson timately have.” This impact has been felt in integer [27] modeled this as a network flow problem. With programming generally where strong linear pro- more thanthe minimum number of planes, the total gramming formulations are much used. One could deadheading distance may be reduced by allowing say that the general method of finding violated, more than the minimumnumberofplanes. combinatorially derived cutting planes was intro- A notable applicationpaper [34] by G. B. Dantzig duced in this paper. Dantzig was personally firmly and J. H. Ramser introduced what is now called the convinced that linear programming was a valuable Vehicle Routing Problem (VRP), an area that has its tool in solving integer programming problems, ownextensiveliteratureandsolutionmethods.The even hard problems such as the TSP. Practically all problem is a generalizationof the TSP having many of the successful, recent work that has accepted variations. As put by Toth and Vigo [80, p. xvii], the challenge of solving larger and larger TSPs has the VRP “calls for the determination of the optimal been based on the Dantzig, Fulkerson, Johnson set of routes to be performed by a fleet of vehicles approach. to serve a given set of customers, and it is one of Another contribution of Dantzig in combina- the most important, and studied, combinatorial torial optimization is the work based on network optimization problems. More than forty years have flow. Ray Fulkerson and Alan Hoffman were two of elapsed since Dantzig and Ramser introduced the his collaborators and co-authors. The crucial ob- problem in 1959.” servation is that this class of linear programs gives In addition to these contributions to network integer basic solutions when the right-hand side optimization problems, we must also mention and bounds are all integers, so the integer program Dantzig’s article [18] pointing out the rich set is solved by the simplex method. In addition, the of problems that can be modeled as integer pro- dual is integer when the costs are integer. In partic- grams.This papermotivatedcomputationalefforts ularthe dualitytheorem provides a proofofseveral such as cutting plane methods, enumeration, and combinatorial optimization results, sometimes branch-and-bound to effectively solve these di- referred to as “min-max theorems”. An example of verse problems.These efforts have continued up to a rather complex application of this min-max theo- today. rem is the proof of Dilworth’s theorem by Dantzig Dantzig often referred to stochastic program- and Hoffman. This theorem says that for a partial ming as the “real problem”. He was very well aware order, the minimum number of chains covering that almost all important decision problems are all elements is equal to the maximum number of decision-making problems under uncertainty, i.e., pair-wise unrelated elements. where at least some of the parameters are at best Of course, George Dantzig was always interested known in a statistical sense, and sometimes not in applications, and developed a rich set of integer even reliable statistical information is available. programming applications. One of these is what is His serious commitment to this class of problems called today the “fleet assignment model”. Current dates from the middle 1950s. As in many other models for this problem are much used in airline instances, the “spark” probably came from an ap- planning. As the problems’ sizes grow and more plication, in this case allocating aircraft to routes details are included in the models, computational under uncertain demand. That problem was the challenges abound; nevertheless this model has concern of Alan Ferguson, one of his colleagues at generally provento be quite tractable despitebeing RAND[49].Aftertheydevisedanelegantprocedure a large, mixed-integer program. In fact, the second to solve this particular problem, Dantzig returned of the two papers [50] on this subject was an early to his desk and wrote a fundamental paper that exampleofastochasticintegerprogrammingprob- not only introduced the fundamental stochastic lem. The problem is to allocate aircraft to routes programmingmodel,knowntodayasthe stochastic for a given numberof aircraft and routes.The main program with recourse,butalsostartedderivingits purpose of the problem is to maximize net revenue basic properties. This certainly reaffirmed the need by routing the larger aircraft over flights that have to deal efficiently with large-scale mathematical more demand or by using more aircraft on such programs. routes. The stochastic version of the problem has With only a slight reformulation, Dantzig’s simple recourse, i.e., leave empty seats or leave model was demand unsatisfied depending on the realized demand. min c · x + E{Q(ξ, x)} subject to Ax = b, x ≥ 0,

March 2007 Notices of the AMS 355 where E{·} denotes the taking of expectation, and reliability of the solutions so obtained. They also Q(ξ, x) = inf q · y W y = ξ − T x, y ≥ 0 ; proposed a scheme based on importance sampling that reducessamplevariance.  Ξ d here ξ isarandomvectorwithvalues ξ in ⊂ IR ; an It would be misleading to measure the role extensionofthemodelwouldallowforrandomness Dantzig played in this field in terms of just his in the parameters (q,W,T), in addition to just the publications, even taking into account that he was right-hand sides, defining the function Q(·, x). He responsible for the seminal work. All along, he en- proved that in fact this is a well defined convex op- couraged students, associates, colleagues, as well timization problem, placing no restrictions on the as anybody who would listen, to enter the field, and distribution of the random elements, but requiring gave them his full support. He saw the need to deal that the problem defining Q besolvableforall x and with the computational challenges, but his major Ξ ξ ∈ , the complete recourse condition as it is now interestseemedtolieinbuildingmodelsthatwould called. This means that stochastic programs with impact policy-making in areas that would signifi- recourse, although generally not linear programs, cantly benefit society on a global scale. The “PILOT” fallintothenext“nice”class,viz.,convexprograms. project [33] provides an example such an effort. However,if the random vector ξ has finite support, Another is his continued active involvement with say Ξ = ξl , l = 1,...,L with prob{ξ = ξl} = p , l the InternationalInstitute of Applied Systems Anal- or the discretization comes about as an approxi-  ysis (IIASA) in Laxenburg (Austria) whose mission mation to a problem whose random components is to conduct inter-disciplinary scientific studies are continuously distributed, the solution x∗ canbe on environmental, economic, technological, and found by solving the large-scalelinear program: socialissuesinthecontextofhumandimensionsof L l min c · x + l=1 pl q · y global change. subject to Ax = bP Dantzig at Berkeley l l T x + W y = ξ , l = 1,...,L George Dantzig left RAND in 1960 to join the fac- x ≥ 0, y l ≥ 0, l = 1,...,L ulty of the University of California in Berkeley; he accepted a post as professor in the Department of “large”,ofcourse,dependingonthesizeof L.Justto Industrial Engineering. Established just four years render this a little bit more tangible, assume that ξ earlier as a full-fledged department, Industrial En- consistsoftenindependentrandomvariables,each gineering was chaired from 1960 to 1964 by Ronald taking on ten possible values; then this linear pro- W. Shephard who, like Dantzig, was born in Port- gram comes with at least 1011 linear constraints— land, Oregon, and had done his doctoral studies the 1010 possible realizations of the random vector under Jerzy Neyman at Berkeley. Asked why he left ξ generating 1010 systems of 10 linear equations— RAND to return to the academic world, Dantzig told certainly, a daunting task! To the rescue came (i) Albers andReidintheir1984interview,“Myleaving the Dantzig-Wolfe decomposition method and (ii) had to do with the way we teamed up to do our re- Dantzig’s statistical apprenticeship. It is pretty straightforward that, up to a sign search. ...Each of us got busy doing his own thing. change in the objective, the dual of the preceding ...There were no new people being hired to work problemcan beexpressedas with us as disciples. ...My stimulus comes from students and working closely withresearchers else- L l l min b · σ + l=1 pl ξ · π where” [1, p. 311]. As it turned out, he had disciples ⊤ PL ⊤ l aplentyat Berkeley(and later atStanford). subject to A σ + pl T π ≤ c, l=1 By 1960 Operations Research was well on its way ⊤ W π l ≤Pq, l = 1,... ; L, to becoming a separate (albeit interdisciplinary) i.e., a problem with a few linking constraints and field of study having natural ties to mathematics, a subproblem that can, itself, be decomposed statistics, computer science, economics, business into L relatively small (separable) subproblems. administration, and some of the more traditional The Dantzig-Wolfe decomposition method was branches of engineering. Dantzig (and Shephard) perfectly adapted to a structured problem of this founded the Operations Research Center (ORC) type, and this was exploited and explained in [31]. whichcoordinated teaching and researchina range Still, this elegant approach required, at each major of OR topics. For several years, the ORC was incon- iteration, solving (repeatedly) a huge number of veniently situated in a small, shabby wood-frame “standard” linear programs. And this is how the house at the university’s Richmond Field Station, idea of solving just a sampled number of these somesixmilesfromthemaincampus.Nevertheless subproblems came to the fore. Of course, one could the ORC managed to attract an enthusiastic group no longer be certain that the absolute optimal so- of faculty and students who investigated a range lution would be attained, but statisticians know of OR themes. Dantzig’s research program was how to deal with this. Dantzig, Glynn, and Infanger certainly the largest at the ORC—and possibly the [29, 30] relied on the Student-t test to evaluate the largestin the IE Departmentforthat matter.

356 Notices of the AMS Volume 54, Number 3 The research Dantzig began at RAND on com- inequality systems of the form pact basis techniques for large-scale linear pro- (5) F(x) ≥ 0, x ≥ 0, x · F(x) = 0. grams led, at Berkeley, to the development of the Generalized Upper Bound Technique (GUB). The Besides thesis supervision and book writing, Dant- motivation came from a problem that Dantzig en- zig busied himself with applied research in the life sciences,large-scaleLP,andabitofworkonoptimal countered while consulting for the Zellerbach Co., controlproblems.Withallthis,headded“educator” now the Crown Zellerbach Co. Rather than simple to his large reputation. lower/upper bound constraints, the m linking con- straints Ax = b and nonnegativity restrictions are Dantzig at Stanford augmented by a collection of L linear constraints Dantzig left UC Berkeley in 1966 to join the Com- withpositive right-hand sides and the property that puter Science Department and the Program in every variable appears at most in one of these con- Operations Research at Stanford University. Al- straints and then, witha positive coefficient. Intheir ways reluctant to discuss his motivation for the paper, Dantzig and Van Slyke [38] show that the switch, DantzigusedtoquipthatORprogramchair- optimal solution of a problem of this type has the man, Gerald J. Lieberman, promised him a parking followingproperties:(i)fromanygroupofvariables place adjacent to the office. This arrangement did associated with one of these L constraints, at least not survive the relocation of the department to one of these variables must be basic and (ii) among other quarters, but Dantzig remained at Stanford these L equations, the number of those ending up anyway. with two or more basic variables is at most m − 1. Since its creation in 1962, the OR program had These properties were exploited to show that with been authorized to grant the Ph.D. and was an inter- an appropriate modification of the pivoting rules school academic unit reporting to three deans and of the standard simplex method and some creative six departments. Fortunately, this cumbersome ar- bookkeeping, one can essentially proceed to solve rangement changed in 1967, at which time Opera- the problem as if it involved only m linear con- tions Research became a regular department in the straints. Again, a situation when a compact basis SchoolofEngineering.Dantzigretainedhisjointap- will suffice, and this resulting in much improved pointment the CS Department, but his principal ac- efficiency and numerical reliability of the method. tivity was based in OR. Berkeley followed a slightly Dantzig offered graduate courses in linear and different route; in 1966, the IE Department became nonlinear programming in addition to a course in theDepartmentofIndustrialEngineeringandOper- network flows. These were given at a time when ations Research. But with Dantzig now at Stanford, his Linear Programming and Extensions and the the West Coast center of gravity for mathematical classic work Flows in Networks by Ford and Fulk- programming shifted southward. erson were still in one stage of incompleteness or One of those recruited to the OR faculty was R. another. At best, some galleyproofs were available, W. Cottle who had been Dantzig’s student at Berke- but photocopy equipment was then in its infancy ley.Together,theyproducedafewpapersontheso- (and was a wet process at that). A while later, a called linear complementarity problem, that being corps of Dantzig’s students were called upon to the case where the mapping F in (5) is affine. One of proofread and learn from LP&E. The richness of these,“ComplementaryPivotTheoryofMathemati- the subject augmented by Dantzig’s vast store of cal Programming”had a considerablereadershipin the OR field and became something of a classic. interesting research problems gave his doctoral George Dantzig had a knack for planning but students tremendous opportunities for disserta- harbored little interest in organizational politics or tiontopics and the resources to carry them out. This administrative detail. Even so, he began his tenure included exposure to stacks of technical reports, at Stanford by serving as president of The Institute introductions to distinguished visitors at the ORC, of Management Sciences (TIMS). In 1968 he and A. and support for travel to important professional F. Veinott Jr. co-directed (and co-edited the pro- meetings. ceedings of) a very successful five-week “Summer Over his six-year period at Berkeley, Dantzig Seminar on the Mathematics of the Decision Sci- supervised eleven doctoral theses including those ences” sponsored by the AMS. He followed this by written by the authors of the present article. The directing in 1971 the “Conference on Applications types of mathematical programming in the disser- of Optimization Methods for Large-Scale Resource tations supervised by Dantzig include large-scale Allocation Problems” held in Elsinore, Denmark, linear programming, linear programming under under the sponsorship of NATO’s Scientific Affairs uncertainty, integer programming, and nonlinear Division. Still later he was program chairman of programming. A development that evolved from the “8th International Symposium on Mathematical thelattersubject(afewyearslater)cametobecalled Programming”, which was held at Stanford in 1973. complementarity theory, that is, the study of com- That year he became the first chairman of the newly plementarity problems. These are fundamental established Mathematical Programming Society.

March 2007 Notices of the AMS 357 On top of these “distractions”, Dantzig concen- in so doing established a long association with this tratedonmathematicalprogramming research,the institute. supervision of doctoral students, and the orches- Another memorable feature of 1973 was the tration of proposal writing so essential to securing well-known mideast oil crisis. This event may have its external sponsorship. been what triggered Dantzig’s interest in energy- Dantzig’s leadership and the addition of other economic modeling. By 1975 this interest evolved faculty and staff contributed greatly to the inter- into something he called the PILOT Model, a passion national stature of the Stanford OR Department. that was to occupy him and a small group of SOL Among Dantzig’s many ventures during this pe- workers untilthe late 1980s. (“PILOT” is anacronym riod, the Systems Optimization Laboratory (SOL) for “Planning Investment Levels Over Time”.) As stands out as one of the most influential and endur- Dantzig, McAllister, and Stone explain [32], PILOT ing. At the 1971 Elsinore Conference mentioned aims“toassesstheimpactofoldandproposednew above, Dantzig discoursed “On the Need for a Sys- technologies on the growth of the U.S. economy, tems Optimization Laboratory”. A year later, he and how the state of the economy and economic (and a host of co-authors) put forward another policy may affect the pace at which innovation version of this concept at an Advanced Seminar on and modernization proceeds.” The PILOT project Mathematical Programming at Madison, Wiscon- provided a context combining three streams of sin. As Dantzig put it, the purpose of an SOL was to research that greatly interested Dantzig: modeling develop “computational methods and associated of a highly relevant economic issue, large-scale computer routines for numerical analysis and opti- programming methodology, and the computation mization of large-scale systems”. By 1973 Stanford of optimal solutions or the solutions of economic had its SOL with Dantzig equilibrium (complementarity) problems. as director. For many The year 1975 brought a mix of tidings to years, the SOL was Dantzig and the mathematical programming com- blessed with an outstand- munity. The 1975 Nobel Prize in Economics went ing resident research to Kantorovich and Koopmans “for their contri- staff consisting of Philip butions to the theory of the optimum allocation of Gill, Walter Murray, resources.”Totheshockanddismayofaworldwide Michael Saunders, John body of well-wishers (including the recipients), Tomlin, and Margaret George Dantzig was not selected for this distin- Wright. The research and guishedaward.Intheirprizespeeches,Kantorovich software output of the and Koopmans recognized the independent work latter group is world fa- of Dantzig; Koopmans felt so strongly about the mous. Numerous faculty omissionthathedonatedathirdofhisprizemoney and students rounded to IIASA in Dantzig’s honor. out the laboratory’s But on a happier note, that same year George personnel. Over time, Dantzig received the prestigious National Medal Dantzig’s concept of an of Science from President Gerald Ford, the John SOL would be emulated von Neumann Theory Prize from ORSA and TIMS, Left to right: at about twenty other and membership in the American Academy of Arts T. C. Koopmans, G. B. Dantzig, institutions. and Sciences. He had, in 1971, already become a L. V. Kantorovitch, 1975 While bringing the SOL member of the National Academy of Sciences and, to fruition, Dantzig col- indeed, would become, in 1985, member of the Na- laborated with T. L. Saaty on another altogether tional Academy of Engineering. Over his lifetime, differentsortofproject:abookcalled Compact City many other awards and eight honorary doctorates [35],whichproposesaplanforaliveableurbanenvi- were conferred on Dantzig. Beginining in 1970, he ronment. The multi-story city was to be cylindrical was listed on the editorial boards of twenty-two in shape (thereby making use of the vertical dimen- different journals. sion)andwasintendedtooperateona24-hourbasis As a colleague and as a mentor, George Dantzig (thereby making better use of facilities). Although was a remarkable asset. That he had broad knowl- themainideaof thispublicationdoesnot appearto edge of the mathematical programming field and have been implemented anywhere, the book itself a wealth of experience with the uses of its method- has beentranslated into Japanese. ology on practical problems goes without saying. During the academic year of 1973–74, Dantzig What also made him so valuable is that he was spent his sabbatical leave at the International Insti- a very patient and attentive listener and would tute for Applied Systems Analysis (IIASA). Located always have a response to whatever was told him. in Laxenburg, Austria, IIASA was then about one The response was invariably a cogent observation year old. IIASA scientists worked on problems of or a valuable suggestion that would advance the energy, ecology, water resources, and methodol- discussion.AsaprofessoratStanford,Dantzigpro- ogy. Dantzig headed the Methodology Group and duced forty-one doctoral students. The subjects of

358 Notices of the AMS Volume 54, Number 3 their theses illustrates the range of his interests. the domain of an exclusive, albeit highly com- The distribution goes about like this: large-scale petent, club of mathematicians and associated linear programming (8), stochastic programming economists, with little or no impact in the “practi- (6), combinatorial optimization (4), nonlinear pro- cal” world. Dantzig’s development of the simplex gramming (4), continuous linear programming (3), method changed all that. Perhaps his greatest networks and graphs (3), complementarity and contribution was to demonstrate that one could computation of economic equilibria (2), dynamic deal (effectively) with problems involving inequal- linear and nonlinear programming (2), probabil- ity constraints. His focus and his determination ity (2), other (7). The skillful supervision Dantzig inspired a breakthrough that created a new mathe- brought to the dissertation work of these students matical paradigm which by now has flourished in a and the energy they subsequently conveyed to myriad of directions. the operations research community shows how effective his earlier plan to have disciples really turned out to be. A nearly accurate version of Dantzig’s “academic family tree” can be found at http://www.genealogy.ams.org/.

Retirement George Dantzig became an emeritus professor in 1985, though not willingly. He was “recalled to duty” and continued his teaching and research for another thirteen years. It was during this period that Dantzig’s long-standing interest in stochastic optimization got even stronger. In 1989 a young Austrian scholar named Gerd Infanger came to the OR Department as a visitor under the aegis of George Dantzig. Infanger’s doctorate at the Vienna Left to right: George Dantzig, Anne Dantzig, Technical University was concerned with energy, President Gerald Ford (National Medal of Honor economics, and operations research. He expected ceremony, 1971). to continue in that vein working toward his Habili- tation. Dantzig lured him into the ongoing research Quickly, practitioners ranging from engineers, program on stochastic optimization which, as he managers,manufacturers,agriculturaleconomists, believed,is“where the realproblemsare.”So began ecologists, schedulers, and so on, saw the potential a new collaboration. During the 1990s, Dantzig and of using linearprogramming models now that such Infanger co-authored seven papers reporting on models came with an efficient solution procedure. powerfulmethods forsolving stochasticprograms. In mathematical circles, it acted as a watershed. It Moreover, Infanger obtained his Habilitation in encouragedandinspiredmathematiciansandafew 1993. computer scientists to study a so-far-untouched Dantzig also established another collaboration class of problems, not just from a computational around this time. Having decided that much prog- but also from a theoreticalviewpoint. ress had been made since the publication of LP&E On the computational front, it suffices to visit a in 1963, he teamed up with Mukund Thapa to write site like the NEOS Server for Optimization a new book that would bring his LP&E more up to http://www-neos.mcs.anl.gov/ to get a date. They completed two volumes [36] and [37] be- glimpse at the richness of the methods, now widely fore Dantzig’s health went into steep decline (early available, to solve optimizations problems and re- 2005), leaving two more projected volumes in an lated variational problems: equilibrium problems, incomplete state. variational inequalities, cooperative and non- cooperative games, complementarity problems, Dantzig’s Mathematical Impact optimal control problems,etc. It will not be possible to give a full account of the On the theoretical front, instead of the classical mathematical impact of Dantzig’s work, some of framework for analysis that essentially restricted which is still ongoing. Instead, we focus on a few functions and mappings to those defined on open key points. sets or differentiable manifolds, a new paradigm Before the 1950s, dealing withsystems involving emerged. It liberated mathematical objects from linear—and a fortiori nonlinear—inequalities was this classical framework. A comprehensive theory of limited scope. This is not to make light of the was developed that could deal with functions that pioneering work of Fourier, Monge, Minkowski, are not necessarily differentiable, not even con- the 1930s-Vienna School, and others, but the fact tinuous, and whose domains could be closed sets remains that up to the 1950s, this area was in or manifolds that, at best, had some Lipschitzian

March 2007 Notices of the AMS 359 properties. This has brought about new notions Epilogue of (sub)derivatives that can be be used effectively GeorgeDantzig hada fertilemind andwaspassion- to characterize critical points in situations where ately dedicated to his work throughout his adult classical analysis could not contribute any in- life. Although his career began with an interest sight. It created a brand new approximation theory in mathematical statistics, circumstances guided where the classical workhorse of pointwise limits him to becoming a progenitor of mathematical is replaced by that of set limits; they enter the pic- programming (or optimization). In creating the ture because of the intrinsic one-sided (unilateral) simplex method, he gave the world what was later nature of the mathematical objects of interest. to be hailed as one of “the top 10 algorithms” of The study of integral functionals was also given the twentieth century [44]. A selection of Dantzig’s some solid mathematical foundations, and much researchoutput appearsin the anthology [10]. progresswasachievedinsolvingproblemsthathad Dantzig built his life around mathematical puzzled us for a very long time. programming, but not to the exclusion of col- Although we could go much further in this di- leagues around the world. Through his activities he rection, we conclude with one shining example. touched the lives of a vast number of mathemati- The study of (convex) polyhedral sets was imme- cians,computerscientists,statisticians,operations diately revived once Dantzig’s simplex method researchers, engineers, and applied scientists of gained some foothold in the mathematical com- all sorts. Many of these individuals benefited from munity. At the outset, in the 1950s, it was led their bonds of friendship with him. Dantzig’s nat- by a Princeton team under the leadership of A. ural warmth engendered a strong sense of loyalty, W. Tucker. But quickly, it took on a wider scope and he returned it in full. spurred by the Hirsch conjecture: Given a linear Many of Dantzig’s major birthdays were cele- program instandard form, i.e., withthe description brated by conferences, banquets, Festschrifts, and n the like. Even the simplex method had a fiftieth of the feasible polyhedral set as S = IR+ ∩ M where th the affine set M is determined by m (not redun- birthday party in 1997 at the 16 International dant) linear equations, the conjecture was that it is Symposium on Mathematical Programming (ISMP) possible to pass from any vertex of S to any other held in Lausanne, Switzerland. In the year 2000, one via a feasible path consisting of no more than George Dantzig was honored as a founder of the th m − 1 edges,or in simplex method parlance, a path field at the 17 ISMP in Atlanta, Georgia. Perhaps requiring no more than m (feasible) pivot steps. the most touching of all the festivities and tributes This question went to the core of the efficiency of in Dantzig’s honor was the conference and banquet the simplex method. The conjecture turned out to held in November 2004. He had turned ninety just be incorrect, at least as formulated, when m > 4, a few days earlier. In attendance were colleagues but it led to intensive and extensive research asso- from the past and present, former students of ciated with the names of Victor Klee, David Walkup, Dantzig, and current students of the department. Branko Grünbaum, David Gale, Micha Perlis, Peter To everyone’s delight, he was in rare form for a McMullen,GilKalai,DanielKleitman,amongothers, man of his age; moreover, he seemed to relish the entire event. Sadly, though, within two months, his that explored the geometry and the combinatorial health took a serious turn for the worse, and on the properties of polyhedral sets. Notwithstanding its following May 13th (a Friday), he passed away. actual performance in the field, it was eventually George B. Dantzig earned an enduring place shown that examples could be created, so that the in the history of mathematics. He will be warmly simplex method would visit every vertex of such remembered for years to come by those who were a polyhedral set S having a maximal number of privileged to know him. By some he will be known vertexes. The efficiency question then turned to only through his work and its impact, far into the researchonthe “expected”number ofsteps; partial future. In part, this will be ensured by the creation answers to this question being provided, in the of the Dantzig Prize jointly by the Mathematical early 1980s, by K. H. Borgwardt [6] and S. Smale Programming Society and the Society for Indus- [77]. An estimate of the importance attached to this trial and Applied Mathematics (1982), the Dantzig subject area, can be gleaned from Smale’s 1998 ar- Dissertation Award by INFORMS (1994), and an ticle [78] in which he states eighteen “mathematical endowed Dantzig Operations Research fellowship problems for the next century”. Problem 9 in this in the Department of Management Science and group asks the question: Is there a polynomial-time Engineering at Stanford University (2006). algorithm over the real numbers which decides the feasibility of the linear system of inequalities References Ax ≥ b? (Thisdecisionversionof the problemrests [1] D. J. Albers and C. Reid, An interview with George ondualitytheoryforlinearprogramming.)Problem B. Dantzig: The father of linear programming, 9 asks for an algorithm given by a real number College Math. J. 17 (1986) 292–314. machine with time being measured by the number [2] D. J. Albers et al. (Eds.), More Mathematical Peo- of arithmetic operations. ple: Contemporary Conversations, Harcourt Brace

360 Notices of the AMS Volume 54, Number 3 Jovanovich, Boston, 1990. (Based on Albers and B. Korte (Eds.), Mathematical Programming: The Reid [1].) State of the Art, Springer-Verlag, Berlin, 1983. [3] H. A. Antosiewicz (Ed.), Proceedings of the Sec- [23] , Reminiscences about the origins of linear ond Symposium in Linear Programming, National programming, in R. W. Cottle, M. L. Kelmanson, Bureau of Standards, Washington, D.C., 1955. and B. Korte (Eds.), Mathematical Programming [4] K. J. Arrow, George Dantzig in the development [Proceedings of the International Congress on of economic analysis, manuscript January 2006, to Mathematical Programming, Rio de Janeiro, Brazil, appear. 6–8 April, 1981], North-Holland, Amsterdam, 1984. [5] R. G. Bland and J. B. Orlin, IFORS’ operational [24] , Impact of linear programming on computer research hall of fame: Delbert Ray Fulkerson, Inter- development, ORMS Today 14, August (1988), 12– national Transactions in Operational Research 12 17. Also in D. V. Chudnovsky and R. D. Jenks (2005), 367. (Eds.), Computers in Mathematics, Lecture Notes in [6] K. H. Borgwardt The average number of pivot Pure and Applied Mathematics, v. 125, New York, steps required by the simplex method is polyno- Marcel Dekker, Inc., 1990, 233–240. mial, Zeitschrift für Operations Research 7 (1982), [25] , Origins of the simplex method, in S. 157–177. G. Nash (Ed.), A History of Scientific Computing, [7] D. Cantor et al. (Eds.), Theodore S. Motzkin: ACM Press, Reading, MA, 1990. Selected Papers, Birkhäuser, Boston, 1983. [26] , Linear programming, in J. K. Lenstra et al. [8] A. Charnes and W. W. Cooper, Management (Eds.), History of Mathematical Programming, CWI Models and Industrial Applications of Linear and North-Holland, Amsterdam, 1991. Programming, John Wiley & Sons, New York, 1961. [27] G. B. Dantzig and D. R. Fulkerson, Minimizing the [9] A. Charnes, W. W. Cooper and A. Henderson, An number of tankers to meet a fixed schedule, Naval Introduction to Linear Programming, John Wiley & Research Logistics Quarterly 1 (1954), 217–222. Sons, New York, 1953. [28] G. B. Dantzig, D. R. Fulkerson and S. M. Johnson, [10] R. W. Cottle (Ed.), The Basic George B. Dantzig, Solution of a large-scale Traveling-Salesman Prob- Stanford Univ. Press, Stanford, Calif., 2003. lem, Journal of the Operations Research Society of [11] G. B. Dantzig, On the non-existence of tests of America 2 (1954), 393–410. “Student’s” hypothesis having power functions in- [29] G. B. Dantzig and P. W. Glynn, Parallel processors dependent of sigma, Ann. Math. Stat. 11 (1940), for planing under uncertainty, Annals of Operations 186–192. Research 22 (1990), 1–21. [12] , I Complete Form of the Neyman-Pearson [30] G. B. Dantzig and G. Infanger, Large-scale Lemma; II On the Non-Existence of Tests of stochastic linear programs: Importance sampling “Student’s” Hypothesis Having Power Functions and Benders decomposition, in C. Brezinski Independent of Sigma. Doctoral dissertation, De- and U. Kulsich (Eds.), Computation and Applied partment of Mathematics, University of California Mathematics—Algorithms and Theory, Lecture at Berkeley, 1946. [QA276.D3 at Math-Stat Library, Notes in Mathematics, Proceedings of the 13th University of California, Berkeley.] IMACS World Congress on Computational and [13] , Programming in a linear structure, Econo- Applied Mathematics, Dublin, 1992. metrica 17 (1949), 73–74. [31] G. B. Dantzig and A. Madansky, On the solution [14] , Application of the simplex method to a of two-staged linear programs under uncertainty, transportation problem, in [65]. in J. 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362 Notices of the AMS Volume 54, Number 3