Rice-Blend Purchase Plan for a Kochi Hostel Mess
The Kochi hostel mess buys two rice blends. Each kilogram of blend X gives 3 units of protein and 1 unit of iron; each kilogram of blend Y gives 2 units of protein and 2 units of iron. The daily contract requires at least 60 units of protein and at least 40 units of iron across the blends purchased. Storage limits blend X to at most 20 kg per day. Blend X costs ₹30 per kg and blend Y ₹40 per kg.
The warden wants the cheapest daily purchase plan. A supplier offers blend X at ₹25 per kg on a monthly contract, and the accountant concludes that cheaper X will obviously drive the plan towards X alone. The warden is not convinced, and also wants to know which of the three limits actually bites at the optimum, since that decides how much storage she must rent.
1. Formulate the linear programming problem, find all corner points of the feasible region, and give the minimum-cost plan.
Read the worked answer
Let $x$ and $y$ be kilograms of X and Y per day. Minimise
$$Z=30x+40y$$
subject to $3x+2y\ge 60$ (protein), $x+2y\ge 40$ (iron), $x\le 20$, $x\ge 0$, $y\ge 0$.
Corner points:
- Intersection of the two nutrient lines: subtracting gives $2x=20$, so $x=10$ and $y=15$. Check: protein $30+30=60$, iron $10+30=40$. Feasible.
- $x=20$ with iron binding: $20+2y=40$, so $y=10$; protein $60+20=80\ge 60$. Feasible.
- $x=20$ with protein binding gives $y=0$, but iron $20<40$, so infeasible - discarded.
- $x=0$ with protein binding: $y=30$; iron $60\ge 40$. Feasible.
Evaluate: $Z(10,15)=300+600=900$; $Z(20,10)=600+400=1000$; $Z(0,30)=1200$.
Minimum cost ₹900 per day at 10 kg of X and 15 kg of Y. Both nutrient requirements are met exactly; no slack is wasted.
2. With blend X at ₹25 per kg, is the accountant right that the plan will shift towards X? Answer with numbers.
Read the worked answer
With X at ₹25, $Z=25x+40y$ and the feasible region is unchanged. Re-evaluate the same corners:
- $(10,15)$: $250+600=850$
- $(20,10)$: $500+400=900$
- $(0,30)$: $1200$
The optimum is still $(10,15)$, now at ₹850 per day. The accountant is wrong.
The reason is that the binding limits are nutritional, not budgetary. Blend Y is the cheaper source of iron (2 units per ₹40 against 1 unit per ₹30 for X), so iron forces Y into the mix regardless of X's price. Pushing towards X alone would need 20 kg of X, which supplies only 20 units of iron; adding the 10 kg of Y needed to reach 40 units costs $500+400=₹900$, worse than ₹850. A price cut lowers the bill but does not move the mix.
3. Which constraints are binding at the optimum, and what does that tell the warden about renting extra storage?
Read the worked answer
At $(10,15)$: protein $=3(10)+2(15)=60$ (binding), iron $=10+2(15)=40$ (binding), and $x=10<20$ (slack by 10 kg).
The storage cap is therefore non-binding, with 10 kg of unused allowance. Since the optimum lies strictly inside the $x\le 20$ limit, relaxing or tightening that limit within a reasonable range leaves both the optimal plan and the ₹850 cost unchanged - renting extra storage would buy nothing.
If the warden wants to cut costs further, she must attack a binding constraint: negotiate a cheaper source of iron, relax the iron or protein specification, or improve Y's price. Storage is the wrong lever.