Q.A dietician wants to prepare a diet that provides at least 8 units of vitamin A and at least 10 units of vitamin B daily. Food $F_1$ costs Rs 6 per unit and gives 2 units of vitamin A and 1 unit of vitamin B; food $F_2$ costs Rs 4 per unit and gives 1 unit of vitamin A and 2 units of vitamin B. Formulate this as a linear programming problem and solve it graphically to find the minimum cost.

Formulate the LPP with $x,y$ = units of $F_1,F_2$, minimise cost $Z=6x+4y$ subject to the two vitamin requirements, then find and evaluate the corner points of the (unbounded) feasible region.

Formulation. Let $x$ = units of $F_1$, $y$ = units of $F_2$. Cost to minimise:

Vitamin A requirement: $2x+y\ge8$. Vitamin B requirement: $x+2y\ge10$. Non-negativity: $x\ge0,\ y\ge0$.

Corner points. On the x-axis ($y=0$): $2x\ge8\Rightarrow x\ge4$; $x\ge10$ (from $x+0\ge10$). Both must hold, so the larger value governs: $x=10$, giving corner $(10,0)$ (checking: $2(10)+0=20\ge8$, true).

On the y-axis ($x=0$): $y\ge8$ (from $0+y\ge8$); $2y\ge10\Rightarrow y\ge5$. The larger governs: $y=8$, giving corner $(0,8)$ (checking: $0+2(8)=16\ge10$, true).

Intersection of the two lines. Solving $2x+y=8$ and $x+2y=10$: from the first, $y=8-2x$; substituting,

so the lines meet at $(2,4)$, the third corner point.

Evaluating $Z=6x+4y$ at each corner:

For a minimisation problem with $\ge$ constraints, students often stop as soon as they find the smallest corner value without performing the open-half-plane check needed because the region is unbounded — for a MAXIMISATION problem on an unbound …