minimize/maximize

Question

Use linear programming to find the ordered pairs (x,Y) that maximize and minmize the objective equation P=-2x+3y subject to the foloowing constaints: x ≥l to 0,, y ≥ to 0, x+2y≤4, and x-y≤1. List the manixmum and minimum value, respectively

Answer 1

The strategy in linear programming is to find the corner points of the polygon of constraint.  In this problem the constraining polygon is a quadrilateral with corner points:

  (0,0)  ;  (1,0) ; (2,1)  ; (0,2)

 

   The P values at these points are:

      0  ;      2    ;   7     ;   6

 

  Therefore the maximum is P =7  occurring at (2,1)   and

   the minimum   is              P = 0  occurring at (0,0)

 

  The basic idea is that the max (or min) of a linear objective function (such as P =  2x + 3y)

  will be on the boundary of the constraining region.   In this problem the constraint region is a polygon and the objective function has a different value at each corner.  This implies that the maximum ( or minimum) must be

at a corner point.