Abstract
This chapter originally appeared in Management Science, April–July 1955, Vol. 1, Nos. 3 and 4, pp. 197–206, published by The Institute of Management Sciences. This article was also reprinted in a special issue of {\em Management Science}, edited by Wallace Hopp, featuring the “Ten Most Influential Papers of Management Science´s First Fifty Years,” Vol. 50, No. 12, Dec., 2004, pp. 1764–1769. For this special issue George B. Dantzig provided the following commentary: “I am very pleased that my first paper on planning under uncertainty is being republished after all these years. It is a fundamental paper in a growing field.”
This chapter originally appeared in Management Science, April–July 1955, Vol. 1, Nos. 3 and 4, pp. 197–206, published by The Institute of Management Sciences. Copyright is held by the Institute for Operations Research and the Management Sciences (INFORMS), Linthicum, Maryland.
This chapter was also reprinted in a special issue of Management Science, edited by Wallace Hopp, featuring the “Ten Most Influential Papers of Management Science’s First Fifty Years,” Vol. 50, No. 12, Dec., 2004, pp. 1764–1769. For this special issue George B. Dantzig provided the following commentary:
“I am very pleased that my first paper on planning under uncertainty is being republished after all these years. It is a fundamental paper in a growing field.”
— George B. Dantzig
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Notes
- 1.
In some applications, however, it may not be desirable to minimize the expected value of the costs if the decision has too great a variation in the actual total costs. Markowitz (1953) in his analysis of investment portfolios develops a technique for computing for each possible expected value the minimum variance. This enables the investor to sacrifice some of his expectation to control his risks.
- 2.
No solution for this example will be given in this chapter. For this case perhaps the simplest approach is through the techniques of dynamic programming; see Bellman (1953).
- 3.
- 4.
Equation (1.8) should be viewed more generally than simply as a statement about the shortage and excess of supply. In fact, given any u j and d j, there is an infinite range of possible values of v j and s j satisfying (1.8). For example, v j might be interpreted as the amount obtained from some new source (perhaps at some premium price) and s j the amount not used. When the cost form is as in (1.9), it becomes clear that in order for C to be a minimum the values of v j and s j will have the more restrictive meaning above.
- 5.
Markowitz in his analysis of portfolios considers the interrelation of the variance with the expected value. See Markowitz (1953).
- 6.
A special case of the general model given in (1.20) is found in Example 4.
- 7.
The chance mechanism may be the “market,” the “weather.”
- 8.
The greatest lower bound instead of minimum is used to avoid the possibility that the minimum value is not attained for any admissible point \(X_2 \in \Omega_2\) or \(X_1 \in \Omega_1\). In case where the latter occurs, it should be understood that while there exists no X i where the minimum is attained, there exists X i for which values as close to minimum as desired are attained.
- 9.
This proof is along lines suggested by I. Glicksberg.
References
Bellman, R.: An introduction to the theory of dynamic programming. Report R-245, The RAND Corporation, June (1953)
Charnes, A., Lemke, C.E.: Minimization of non-linear separable functions, Graduate School of Industrial Administration, Carnegie Institute of Technology, May (1954)
Dantzig, G.B.: Notes on linear programming: Parts VIII, XVI,X—upper bounds, secondary constraints, and block triangularity in linear programming. Research Memorandum RM-1367, The RAND Corporation, October, 4 (1954)
Ferguson, A.R., Dantzig, G.B.: Notes on linear programming: Part XVI—the problem of routing aircraft—a mathematical solution. Research Memorandum RM-1369, The RAND Corporation, September, 1 (1954)
Markowitz, H.: Portfolio Selection. PhD thesis, The University of Chicago, Chicago, IL (1953)
Stigler, G.F.: The cost of subsistence. J. Farm Econ., 27, 303–314 May (1945)
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Dantzig, G.B. (2010). Linear Programming Under Uncertainty. In: Infanger, G. (eds) Stochastic Programming. International Series in Operations Research & Management Science, vol 150. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1642-6_1
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