For example, given the constraint from the linear program.In this way, all lower bound constraints may be changed to non-negativity restrictions.
George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator.
During 1946 his colleague challenged him to mechanize the planning process to distract him from taking another job.
It can be shown that for a linear program in standard form, if the objective function has a maximum value on the feasible region, then it has this value on (at least) one of the extreme points.
This in itself reduces the problem to a finite computation since there is a finite number of extreme points, but the number of extreme points is unmanageably large for all but the smallest linear programs.
First, for each variable with a lower bound other than 0, a new variable is introduced representing the difference between the variable and bound.
The original variable can then be eliminated by substitution.
Dantzig later published his "homework" as a thesis to earn his doctorate.
The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient.