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

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

Formulation. Let $x$ = units of $F_1$, $y$ = units of $F_2$. Minimise $Z=6x+4y$. Vitamin A requirement: $2x+y\ge8$. Vitamin B requirement: $x+2y\ge10$. Non-negativity: $x\ge0,y\ge0$.

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

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

Intersection of the two lines. Solving $2x+y=8$ and $x+2y=10$: from the first, $y=8-2x$; substituting, $x+2(8-2x)=10 \Rightarrow x+16-4x=10 \Rightarrow -3x=-6 \Rightarrow x=2,\ y=8-4=4$ — the third corner point, $(2,4)$.

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

Stopping as soon as the smallest corner value is found, without performing the open-half-plane check needed because the region is unbounded, is a common gap — for a maximisation problem on an unbounded region, by …