Max z = $2x_{1} + 3x_{2} + 4x_{3} + x_{4} + 7x_{5} + 5x_{6} $ Subject to
x$_{1} + x_{2} ≤ 1 $
x$_{3} + x_{4} ≤ 1 $
x$_{5} + x_{6} ≤ 1 $
x$_{1} + x_{3} + x_{5} = 1 $
x$_{2} + x_{4} + x_{6} = 1$
As we can notice, the cost coefficients will be the part of the new transportation cost matrix. We will add slack variables say $x_{7}$ , $x_{8}$ and $x_{9}$ to the first three equations, now we have our constraints.
x$_{1} + x_{2} + x_{7} = 1 $
x$_{3} + x_{4} + x_{8} = 1 $
x$_{5} + x_{6} + x_{9} = 1 $
x$_{1} + x_{3} + x_{5} = 1 $
x$_{2} + x_{4} + x_{6} = 1$
According to them, the value of supply and demand for each row and column should be 1. If $x_{ij}$ denotes the entry at i-th row and j-th column, then our entries should be something like - $D_1 = 1, D_2 = 1, D_3 = 1$
$S_1=1$ $x_{11} = 2, x_{12} = 3, x_{13} = 4$
$S_2=1$ $x_{21} = 1, x_{22} = 7, x_{23} = 5$
$S_3=1$ $x_{31} = 0, x_{32} = 0, x_{33} = 0$
The third row is of course a dummy row. Now, we can use Least Cost Method(LCM) or Vogel's Approximation Method(VAM) to find the answer. However, the problem is of maximization type, so first we will convert it to minimization type, by subtracting the greatest element from each (except the dummy row). The new problem should be -
$D_1 = 1, D_2 = 1, D_3 = 1$
$S_1=1$ $x_{11} = 5, x_{12} = 4, x_{13} = 3$
$S_2=1$ $x_{21} = 6, x_{22} = 0, x_{23} = 2$
$S_3=1$ $x_{31} = 0, x_{32} = 0, x_{33} = 0$
I got til here. Now I solved this via both VAM and LCM. But, I got the answer as 11, however, when I solved the same LPP via the Big-M simplex method, I got it as 10. I cross checked my calculations and methods, but there was nothing wrong in them. My doubt is - Why aren't the answers matching?
