Suppose I wish alter the units of a variable like X1 (e.g. changing it from kilograms to metric tons, requires multiplying all coefficients for X1 by 1000). Thus, I substitute X1' which is X1 divided by the scaling factor (X1/SC1). I also adjust the ai1's and c1 such that they are multiplied by the scaling factor
ai1' = SC1 * ai1 c1' = SC1 * c1
and the problem becomes
Notes:
| • | The solutions to the problems before and after scaling are mathematically related. http://agecon2.tamu.edu/people/faculty/mccarl-bruce/mccspr/new17.pdf 1 and Spreen show that such scaling divides the optimal variable value by the scaling factor and multiplies the reduced cost or variable marginal by that factor. |
| • | Note in doing such scaling that all coefficients for a variable have a common denominator in terms of units (resource use per unit of X1) both before and after scaling, so every coefficient associated with X1 needs to be multiplied by the same scaling factor to preserve denominator homogeneity of units. |