# solve the LLP. optimum value of the objective function, the basic feasible optimal vector

Fruity Ltd manufactures an orange flavoured soft drink called OrangeFiZZ by combining orange soda and orange juice. Each ounce of orange soda contains 0.5 oz of sugar and 1 mg of vitamin C. Each ounce of orange juice contains 0.25 oz of sugar and 3 mg of vitamin C. It costs Fruity 2 pence to produce an ounce of orange soda and 3 pence to produce an ounce of orange juice. Fruity’s marketing department has decided that each 10 oz bottle of OrangeFiZZ must contain at least 20 mg of vitamin C and at most 4 oz of sugar.

I have formulated this data. The variables are: $$x_1$$=number of ounces of orange soda in a bottle of OrangeFiZZ, $$x_2$$=The number of ounces of orange juice in a bottle of OrangeFiZZ

the objective function is $$2x_1+3x_2$$ as a minimum

the constraints are: $$0.5x_1+0.25x_2\leqslant 4$$ for the amount of sugar, $$x_1+3x_2\leqslant 20$$ for the Vitamin C, $$x_1+x_2\leqslant 10$$ for the 10oz in 1 bottle of OrangeFiZZ and $$x_1, x_2\geqslant 0$$.

I am stuck on the way I would go solve the LLP whilst also finding the optimum value of the objective function, the basic feasible optimal vector and the amounts of sugar and vitamin C required to achieve the optimum.

• First, rewrite $0.5x_1 + 0.25x_2\leqslant 4$ as $2x_1 + x_2\leqslant 16$ so you have all integer coefficients. This just makes the problem easier to deal with. – Math1000 Oct 28 '19 at 19:53
• okay I've done that but I'm still stuck on how to go around in finding the optimum value of the objective function, the basic feasible optimal vector and the amounts of sugar and vitamin C required to reach the optimum. – jon doe Oct 28 '19 at 19:57
• Are you sure you have formulated this correctly? All of your constraints are $\leqslant$ with positive coefficients, so the minimum would simply be $(0,0)$. I suspect at least one of your constraints must be a $\geqslant$ – Math1000 Oct 28 '19 at 19:58
• I think for the sugar and vitamin C constraint it might be the other way round because they said there should be at least 4 oz of sugar and 20 mg in each 10oz bottle of OrangeFiZZ. But I'm not too sure – jon doe Oct 28 '19 at 20:02
• It is at most 4oz of sugar, but at least 20mg of vitamin C. So the constraint with $20$ on the rhs should be a $\geqslant$. When I make that change, I get the optimal solution of $(0, 20/3)$ – Math1000 Oct 28 '19 at 20:04