How many kilograms of each type of fertilizer should the farmer use?

Question

A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 5% phosphoric acid, and type B which contains 6% nitrogen and 10% phosphoric acid. After testing the soil conditions of the field, it was found that atleast 14 kg of nitrogen and 14 kg of phosphoric acid is required for producing a good crop. The fertilizer of type A costs Rs.5 per kg and the type B costs Rs.3 per kg. How many kg of each type of the fertilizer should be used to meet the requirement at the minimum possible cost? Using L.P.P. solve the above problem graphically.

Answer 1

Your inequalities are right. Now see here how to solve the a linear program graphically. Below the picture shows the graph of your problem.

I got $Z_{min}$ min at two points (80,100) and (0, 700/3) the value of Zmin in both cases is 700 now how do we chose between them ?

As you see at the graph that after the shift the objective function lies direcly on the first constraint $y\geq \frac{700}3-\frac53\cdot x$. If we solve the objective function for y we get $y=\frac{z}3-\frac53\cdot x$

The reason why the objective function lies directly on the first constraint is that both have the same slope of $-\frac53$.

Thus the optimal solution is on every point on the constraint.

$(x^*,y^*)=\left(x, \frac{700}3-\frac53\cdot x\right)$, where $0\leq x\leq 80$

So both solutions you mentioned are valid. But all solutions in between them as well.