Fat Standardisation at a Milk Chilling Plant
A dairy plant at Kalyani sells one grade of pouch milk that must test 4.5% fat. On a given morning the chilling tank holds three raw streams: full-cream milk at 6.0% fat costing ₹58 per litre (up to 400 litres available), toned milk at 3.0% fat costing ₹48 per litre (up to 300 litres), and skim milk at 0.5% fat costing ₹30 per litre (only 120 litres left, and no more can be procured today). Pasteurisation and filling lose 2% of the input volume; only filled pouches are saleable. The plant intends to send 500 litres of raw blend through the line today. The trade price is ₹60 per litre of pouch milk and the day's fixed cost (power, labour, packing) is ₹3,000. The plant manager, who learnt the trade on the shop floor, wants to "just mix full-cream and toned in equal parts, because that is what we have always done". The owner wants the arithmetic before authorising the batch.
1. Work out the full-cream : toned blend ratio that gives exactly 4.5% fat, the raw-material cost of a 500-litre batch on that ratio, and the profit or loss at a trade price of ₹60 per saleable litre.
Read the worked answer
Alligation on fat. Let x litres of 6.0% milk and (500 − x) litres of 3.0% milk be used. Required fat = 4.5% of 500 = 22.5 L.
0.06x + 0.03(500 − x) = 22.5 → 0.03x = 7.5 → x = 250.
Ratio = 250 : 250 = 1 : 1. Alligation confirms it: (4.5 − 3.0) : (6.0 − 4.5) = 1.5 : 1.5.
Cost = 250 × 58 + 250 × 48 = ₹14,500 + ₹12,000 = ₹26,500.
Saleable output = 500 × 0.98 = 490 L, so raw-material cost per pouch litre = 26,500 ÷ 490 = ₹54.08.
Result at ₹60: revenue ₹29,400 − raw material ₹26,500 − fixed cost ₹3,000 = −₹100, a small loss. The "equal parts" habit does not even cover cost, so the manager's confidence is misplaced. Note also that the 2% line loss, not the fat ratio, is what pushes the unit cost above the naive ₹53 per litre.
2. The owner wants the cheapest 500-litre blend that still tests 4.5% fat, using no more of any stream than is available. Find it, and quantify the saving against the equal-parts blend.
Read the worked answer
Skim milk is the cheap source of fat-free solids, so push as much of it in as stock allows.
Ideal two-stream ratio (6.0% and 0.5%): (4.5 − 0.5) : (6.0 − 4.5) = 4 : 1.5 = 8 : 3, i.e. 500 × 3/11 = 136.4 L of skim. Stock is only 120 L, so the ideal is infeasible and we bind at 120 L of skim.
Fill the remaining 380 L with full-cream (x) and toned (380 − x): 0.06x + 0.03(380 − x) + 0.005(120) = 22.5 0.03x + 11.4 + 0.6 = 22.5 → 0.03x = 10.5 → x = 350, toned = 30.
Check fat: 21.0 + 0.9 + 0.6 = 22.5% L ✓. Check stock: 350 ≤ 400, 30 ≤ 300, 120 = 120 ✓.
Cost = 350 × 58 + 30 × 48 + 120 × 30 = ₹20,300 + ₹1,440 + ₹3,600 = ₹25,340.
Saving = 26,500 − 25,340 = ₹1,160 per batch (4.4% of raw-material cost), and cost per saleable litre falls to ₹51.71. A sub-optimal blend recipe is quietly destroying margin every single day.
3. Using the cheapest blend, find the break-even trade price per saleable litre and the price needed to earn ₹2,500 profit on the batch. Should the plant accept ₹60?
Read the worked answer
Total cost of the batch = raw material ₹25,340 + fixed ₹3,000 = ₹28,340 over 490 saleable litres.
Break-even price = 28,340 ÷ 490 = ₹57.84 per litre. Any trade price below this loses money on the day.
Profit at ₹60 = 490 × 60 − 28,340 = 29,400 − 28,340 = ₹1,060, a margin of 3.6% on cost. That is razor-thin: a 1% fat shortfall penalty, a rejected pouch lot or a ₹1,000 transport surprise wipes it out.
Price for ₹2,500 profit = (28,340 + 2,500) ÷ 490 = 30,840 ÷ 490 = ₹62.94 per litre.
Judgement: ₹60 is technically viable only because the blend recipe was optimised; it was not viable on the manager's equal-parts habit. Since the trade price is fixed, the real levers are the 2% line loss (every 1% recovered adds roughly ₹260 per batch), the skim-milk procurement limit, and packaging cost - not price. The owner should fix those before accepting the thin margin as normal.