Linear Programming

LINEAR PROGRAMMING GEORGE B. DANTZIG Department of Management Science and Engineering, Stanford University, Stanford, California 94305-4023 The Story About How It Began: Some legends, a little the transportation problem was also overlooked until after about its historical significance, and comments about where others in the late 1940’s and early 1950’s had independently its many mathematical programming extensions may be rediscovered its properties. headed. What seems to characterize the pre-1947 era was lack of any interest in trying to optimize. T. Motzkin in his schol- inear programming can be viewed as part of a great arly thesis written in 1936 cites only 42 papers on linear Lrevolutionary development which has given mankind inequality systems, none of which mentioned an objective the ability to state general goals and to lay out a path of function. detailed decisions to take in order to “best” achieve its goals The major influences of the pre-1947 era were Leon- when faced with practical situations of great complexity. tief’s work on the Input-Output Model of the Economy Our tools for doing this are ways to formulate real-world (1933), an important paper by von Neumann on Game The- problems in detailed mathematical terms (models), tech- ory (1928), and another by him on steady economic growth niques for solving the models (algorithms), and engines for (1937). executing the steps of algorithms (computers and software). My own contributions grew out of my World War II This ability began in 1947, shortly after World War II, experience in the Pentagon. During the war period (1941– and has been keeping pace ever since with the extraordinary 45), I had become an expert on programming-planning growth of computing power. So rapid has been the advance methods using desk calculators. In 1946 I was Mathemat- in decision science that few remember the contributions of ical Advisor to the US Air Force Comptroller in the Pen- the great pioneers that started it all. Some of their names tagon. I had just received my PhD (for research I had are von Neumann, Kantorovich, Leontief, and Koopmans. done mostly before the war) and was looking for an aca- The first two were famous mathematicians. The last three demic position that would pay better than a low offer I received the Nobel Prize in economics. had received from Berkeley. In order to entice me to not In the years from the time when it was first proposed take another job, my Pentagon colleagues, D. Hitchcock in 1947 by the author (in connection with the planning and M. Wood, challenged me to see what I could do to activities of the military), linear programming and its many mechanize the planning process. I was asked to find a way extensions have come into wide use. In academic circles to more rapidly compute a time-staged deployment, train- decision scientists (operations researchers and management ing and logistical supply program. In those days “mecha- scientists), as well as numerical analysts, mathematicians, nizing” planning meant using analog devices or punch-card and economists have written hundreds of books and an equipment. There were no electronic computers. uncountable number of articles on the subject. Consistent with my training as a mathematician, I set Curiously, in spite of its wide applicability today to out to formulate a model. I was fascinated by the work of everyday problems, it was unknown prior to 1947. This Wassily Leontief who proposed in 1932 a large but simple is not quite correct; there were some isolated exceptions. matrix structure which he called the Interindustry Input- Fourier (of Fourier series fame) in 1823 and the well- Output Model of the American Economy. It was simple in known Belgian mathematician de la Vallée Poussin in 1911 concept and could be implemented in sufficient detail to each wrote a paper about it, but that was about it. Their be useful for practical planning. I greatly admired Leontief work had as much influence on Post-1947 developments as for having taken the three steps necessary to achieve a suc- would finding in an Egyptian tomb an electronic computer cessful application: built in 3000 BC. Leonid Kantorovich’s remarkable 1939 1. Formulating the inter-industry model. monograph on the subject was also neglected for ideolog- 2. Collecting the input data during the Great Depression. ical reasons in the USSR. It was resurrected two decades 3. Convincing policy makers to use the output. later after the major developments had already taken place Leontief received the Nobel Prize in 1976 for developing in the West. An excellent paper by Hitchcock in 1941 on the input-output model. Subject classification: Professional: comments on. Area of review: Anniversary Issue (Special). Operations Research © 2002 INFORMS 0030-364X/02/5001-0042 $05.00 Vol. 50, No. 1, January–February 2002, pp. 42–47 42 1526-5463 electronic ISSN Dantzig / 43 For the purpose I had in mind, however, I saw that Leon- disadvantages, that it would be impossible to compare all tief’s model had to be generalized. His was a steady-state the cases and choose which among them would be the best. model and what the Air Force wanted was a highly dynamic Invariably, man in the past has turned to a leader whose model, one that could change over time. In Leontief’s ‘experience’ and ‘mature judgment’ would guide the way. model there was a one-to-one correspondence between the Those in charge like to do this by issuing a series of ground production processes and the items being produced by these rules (edicts) to be executed by those developing the plan. processes. What was needed was a model with alternative This was the situation prior to 1946. In place of an activities. Finally it had to be computable. Once the model explicit goal or objective function, there were a large num- was formulated, there had to be a practical way to com- ber of ad hoc ground rules issued by those in authority to pute what quantities of these activities to engage in that guide the selection. Without such rules, there would have was consistent with their respective input-output character- been, in most cases, an astronomical number of feasible istics and with given resources. This would be no mean solutions to choose from. Incidentally, “Expert System” task since the military application had to be large scale, software which is very much in vogue today makes use of with thousands of items and activities. this ad hoc ground-rule approach. The activity analysis model I formulated would be All that I have related up to now about the early devel- described today as a time-staged, dynamic linear program opment took place before the advent of the computer, more with a staircase matrix structure. Initially there was no precisely, before in late 1946 we were aware that the com- objective function; broad goals were never stated explic- puter was going to exist. But once we were aware, the com- itly in those days because practical planners simply had no puter became vital to our mechanization of the planning waytoimplement such a concept. Noncomputability was process. So vital was the computer, that our group arranged the chief reason, I believe, for the total lack of interest in (in the late 1940’s) that the Pentagon fund the development optimization prior to 1947. of computers. To digress for a moment, I would like to say a few words A simple example may serve to illustrate the fundamen- about the electronic computer itself. To me, and I suppose tal difficulty of finding a solution to a planning problem to all of us, one of the most startling developments of all once it is formulated. Consider the problem of assigning time has been the penetration of the computer into almost 70 men to 70 jobs. Suppose a value or benefit v would ij every phase of human activity. Before a computer can be result if the ith man is assigned to the jth job. An activity intelligently used, a model must be formulated and good consists in assigning the ith man to the jth job. The restric- algorithms developed. To build a model, however, requires tion are: (i) each man must be assigned a job (there are 70 the axiomatization of a subject matter field. In time this such), and (ii) each job must be filled (also 70). The level axiomatization gives rise to a whole new mathematical dis- of an activity is either 1, meaning it will be used, or 0, cipline which is then studied for its own sake. Thus, with × meaning it will not. Thus there are 2 70 or 140 restric- each new penetration of the computer, a new science is × tions, 70 70 or 4900 activities with 4900 corresponding born. 0-1 decision variables xij . Unfortunately there are 70! dif- Von Neumann notes this tendency to axiomatize in his ferent possible solutions or ways to make the assignments paper on The General and Logical Theory of Automata.In xij . The problem is to compare the 70! solutions with one it he states that automata have been playing a continuously another and to select the one which results in the largest increasing role in science. He goes on to say: sum of benefits from the assignments. Now 70! is a big number, greater than 10100. Suppose we Automata have begun to invade certain parts of math- had a computer capable of doing a million calculations per ematics too, particularly but not exclusively mathematical physics or applied mathematics. The natural sys- second available at the time of the big bang fifteen billion tems (e.g., central nervous system) are of enormous years ago.

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