# Graphical method in linear programming

This page describes the graphical method to solve a linear program. The formulation is as follows.

\begin{alignat}{2} \max &\quad Z = 200W + 100B\\ \text{s.t.} &\quad 1W + 0.8B &&\leq 4000\\ &\quad 0.004W + 0.001B &&\leq 10\\ &\quad W, B &&\geq 0\end{alignat}

The solution given is:

Co-ordinates of the optimum point are approximately 1850 W and 2750 B (1850, 2750).

What would be an easy way to calculate the optimal solution in addition to an estimate seen from graph (rather than the simplex method)? Thank you.

In the example you shared with two variables and two constraints, as you're seeing from the graph, your solution lies at the intersection of the two constraints (not considering the non-negativity constraints). So, just solve that system of equations and you get the values for $$B$$ and $$W$$ (which should give you $$B = \frac{30000}{11}$$ and $$W = \frac{20000}{11}$$.)
If you want to go further and try the optimum values in your objective function, there are four corner points: $$(0,0), (3000, 0), (0, 5000), (\frac{20000}{11}, \frac{30000}{11})$$. Place these values in your objective function and you get the objective value of each point as well.
• If it makes it easier, multiply the first equation by 10 and the second one by 1000 to get rid of the decimal points in the coefficients of $B$ and $W$. Then solve that system of linear equations. i.e. $10W+8B=40,000; 4W+B=10,000$. You'll get the above results (or if you prefer the decimal values, $B=30000/11=2727.3$ and $W=20000/11=1818.2$)