A Note on the Use of Nonparametric Statistics in Stochastic Linear Programming

An ordinary linear programming problem is formulated as <disp-formula><tex-math><![CDATA[$$\mbox{Maximize} z = cz$$]]></tex-math></disp-formula> under the constraints <disp-formula><tex-math><![CDATA[\begin{eqnarray*} Ax \leqq b,\\ x \geqq 0, \end{eqnarray*}]]></tex-math></disp-formula> where A is a matrix with m rows and n columns, x and c are column vectors with n elements, and b is a column vector with n elements. The theory of stochastic linear programming first suggested by Tintner [Tintner, G. 1955. Stochastic linear programming with applications to agricultural economics. Sympos. Linear Programming, Vol. 1. National Bureau of Standards, Washington, D. C., 197 ff.] uses the following approach. The elements of b, c and the matrix A are assumed to be random variables with a known probability distribution. Two possible ways of deriving the distribution of z are known as direct and indirect methods. In this article, some nonparametric statistics were applied to test the difference between the distributions derived by the direct and indirect methods. The non-parametric tests include the Kolmogorov-Smirnov statistic and Alfred Rényi's statistics.