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

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

Graph is:

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.


1 Answer 1


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.

  • 1
    $\begingroup$ 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 $W=20000/11=1818.2$) $\endgroup$
    – EhsanK
    Commented Oct 18, 2019 at 4:26
  • $\begingroup$ thank you for the perfect answer and explanation! $\endgroup$
    – Mark K
    Commented Oct 18, 2019 at 4:27
  • 1
    $\begingroup$ You're very welcome. Glad it helped. $\endgroup$
    – EhsanK
    Commented Oct 18, 2019 at 4:29

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