Potato Supply Contract
A Galway food co-op needs 320 kg of usable potatoes each week for a school-lunch contract. Supplier A charges €1.85 per kg plus a fixed €45 per order, and its potatoes yield 4% waste when peeled; a written quotation confirms that a 350 kg order from A would cost €692.50. Supplier B charges €2.05 per kg, delivers free on orders of €600 or more and charges €35 otherwise, and its potatoes yield 1.5% waste. Both deliver in whole kilograms only, one order per week, and the co-op has no storage beyond the week's needs. The weekly potato budget is €700 including delivery. The buyer currently uses A because the headline unit price is lower.
1. Calculate the cheapest compliant weekly cost from each supplier for 320 kg usable, and state which supplier the co-op should use.
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Supplier A. Delivered weight must cover the waste: required order = 320 ÷ 0.96 = 333.33 kg, rounded up to whole kilograms → 334 kg. Cost = 334 × €1.85 = €617.90, plus €45 delivery = €662.90. (The quotation check confirms the pricing: 350 × €1.85 = €647.50, + €45 = €692.50.)
Supplier B. Required order = 320 ÷ 0.985 = 324.87 kg → 325 kg. Cost = 325 × €2.05 = €666.25. That exceeds €600, so delivery is free: €666.25.
Comparison. A is cheaper by €3.35 per week, and both sit inside the €700 budget. The buyer's instinct is right for this volume, but only just: the €45 fixed fee is the reason A wins, and it is a fixed cost that shrinks in relative terms as volume grows. Recommend A at current volume, with the position reviewed if the contract grows.
2. Find the smallest weekly usable requirement at which Supplier B becomes cheaper, showing working, and explain why the crossover is not a smooth one.
Read the worked answer
Compare on usable kilograms (call this N).
Ignoring rounding and the delivery threshold, A costs 1.85N/0.96 + 45 = 1.9271N + 45; B costs 2.05N/0.985 = 2.0812N. Setting them equal: 45 = 0.1541N, so N ≈ 292 kg. That is the first estimate.
But B's €35 delivery charge below €600 creates a step. B's order value reaches €600 at 600 ÷ 2.05 = 292.7 kg delivered → 293 kg, which yields 293 × 0.985 = 288.6 kg usable.
Test the boundary:
- N = 288: B needs 292.39 → 293 kg = €600.65, free delivery → €600.65. A needs 300 kg = €555.00 + €45 = €600.00. A cheaper by €0.65.
- N = 289: B needs 293.4 → 294 kg = €602.70, free delivery → €602.70. A needs 301.04 → 302 kg = €558.70 + €45 = €603.70. B cheaper by €1.00.
So B first wins at 289 kg usable per week. The crossover is not smooth because just below B's free-delivery threshold a whole €35 appears at once: at N = 287 B's order is 292 kg = €598.60, which takes the €35 charge and costs €633.60 against A's €598.15. That jump, not the unit price, is what sets the true crossover point.
3. The contract grows to 340 kg usable per week. Test both suppliers against the €700 budget and give two realistic changes the co-op could make.
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Supplier A. 340 ÷ 0.96 = 354.17 → 355 kg. 355 × €1.85 = €656.75, + €45 = €701.75 - €1.75 over budget.
Supplier B. 340 ÷ 0.985 = 345.18 → 346 kg. 346 × €2.05 = €709.30, free delivery (over €600) - €9.30 over budget.
Neither supplier fits the €700 envelope at 340 kg usable. Note also that with no storage, any surplus kilogram is itself waste, so the nominal waste rates understate the true loss and the real position is slightly worse than these figures.
Two changes that actually close the gap:
- Attack the waste rate. A is only €1.75 over. At 3% waste, A's order falls to 340 ÷ 0.97 = 350.52 → 351 kg, costing €649.35 + €45 = €694.35, inside budget. Simple changes - better blade maintenance, peeling to order, using peelings in stock - can plausibly buy 1 percentage point.
- Renegotiate the fixed charge, not the unit price. A 1-cent cut in A's unit price at 355 kg saves €3.55, enough to pass; so does cutting the €45 delivery fee to €43 or below, or consolidating to a fortnightly delivery if storage can be borrowed. Asking for a lower fixed fee is usually easier than a lower rate.
Splitting between suppliers does not help here: each would still need a near-full order, so the fixed costs and the €600 threshold would both bite.