By Hans-Berndt Brinkmann, Dieter Puppe
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Extra info for Abelsche und exakte Kategorien Korrespondenzen
A0 then the elements of the elementary pivot matrix are ( Pi( i( ' %1/ai0 Pki( ' ak0/ai0 , kÖ i ( ( ( Suppose we update the inverse in the Joe's van example problem using product form of the inverse. In the first pivot, after Xfancy has been identified to enter the problem in row 2, then we replace the second column in an identity matrix with a column with one over the pivot element (the element in the second row of B-1a0 divided by the pivot element elsewhere. Since B-1a0 equals P1 ' copyright Bruce A.
This maximization is subject to inequality constraints involving M resources so that A is an MxN matrix giving resource use coefficients by the X's, and b is an Mx1 vector of right hand side or resource endowments. We further constrain X to be non-negative in all elements. It is common to convert the LP inequality system to equalities by adding slack variables. These variables account for the difference between the resource endowment (b) and the use of resources by the variables (AX) at no cost to the objective function.
This shows the value of x0 which causes the I*th basic variable to become zero. Now since x0 must be nonnegative then we need only consider cases in which a basic variable is decreased by increasing the nonbasic variable. This restricts attention to cases where (B-1 a0 )I is positive. Thus, to preserve nonnegativity of all variables, the maximum value of x0 is x0 ' 6 (B & 1b)i / (B & 1a0)i > for all i where (B & 1a0)i > 0 The procedure is called the minimum ratio rule of linear programming. Given the identification of a nonbasic variable, this rule gives the maximum value the entering variable can take on.