Quantity-discount purchase decision when the cheap bracket carries a hidden defect rate
A pump assembler consumes 24 000 precision seal rings a year at a steady rate over 52 working weeks. Ordering cost is Rs 1 200 per purchase order and carrying cost is 18% of unit price per year. Supplier A quotes Rs 50 for 1–1 999 pieces, Rs 47.50 for 2 000–4 999 and Rs 46 above 5 000. Incoming inspection shows defect rates of 0.5% for the first two brackets but 5% for the top bracket, because that volume is run at A's second plant. Defectives are caught at goods-in, cannot be reworked, get no credit note and are scrapped; production genuinely needs 24 000 good rings a year. The warehouse holds at most 8 000 rings. Weekly demand has a standard deviation of 60 rings and A's lead time is 2 weeks with a standard deviation of 0.5 week. The buyer has already told the plant manager that the Rs 46 bracket will be taken because it is the cheapest line on the price list.
1. Compute the economic order quantity for each price bracket and the total annual cost, and identify the lowest-cost bracket on price data alone.
Read the worked answer
EOQ = √(2DS/iC) with D = 24 000, S = 1 200, i = 0.18. So 2DS = 57 600 000.
At Rs 50: EOQ = √(57 600 000/9.00) = √6 400 000 = 2 530 - outside the 1–1 999 bracket, so infeasible at that price. At the bracket edge Q = 1 999: purchase 12 00 000; ordering (24 000/1 999) × 1 200 = 14 407; holding 999.5 × 9.00 = 8 996 → Rs 12 23 403.
At Rs 47.50: EOQ = √(57 600 000/8.55) = √6 736 842 = 2 596, inside 2 000–4 999. purchase 11 40 000 + ordering (24 000/2 596) × 1 200 = 11 096 + holding 1 298 × 8.55 = 11 096 → Rs 11 62 192.
At Rs 46: EOQ = √(57 600 000/8.28) = 2 638, below the 5 000 break, so the bracket minimum Q = 5 000 must be used. purchase 11 04 000 + ordering 4.8 × 1 200 = 5 760 + holding 2 500 × 8.28 = 20 700 → Rs 11 30 460.
On price alone the Rs 46 bracket wins by Rs 31 732 a year over Rs 47.50, and the buyer's conclusion looks correct. Note the classic discount trade-off: at 5 000 the holding cost nearly doubles (Rs 20 700 vs Rs 11 096) but the purchase saving of Rs 36 000 more than pays for it.
2. Now bring the defect data in. Recompute the comparison on a good-unit basis and state the correct purchasing decision.
Read the worked answer
The purchase decision must be made on the cost of usable rings, because the plant needs 24 000 good pieces, not 24 000 pieces.
Effective unit cost: Rs 46/0.95 = Rs 48.42; Rs 47.50/0.995 = Rs 47.74. The cheap bracket is now the more expensive metal.
Total cost at the Rs 46 bracket: to receive 24 000 good rings you must order 24 000/0.95 = 25 263 pieces. Purchase = 25 263 × 46 = 11 62 105; orders = 25 263/5 000 = 5.05, use 6 orders to respect the bracket → 7 200, or 5.05 orders = 6 061 if the quantity is allowed to vary around the break; holding 2 500 × 8.28 = 20 700. Total ≈ Rs 11 88 900.
Total cost at the Rs 47.50 bracket: order 24 000/0.995 = 24 121 pieces; purchase = 11 45 729; ordering = (24 121/2 596) × 1 200 = 11 152; holding = 1 298 × 8.55 = 11 096. Total ≈ Rs 11 67 977.
Decision: stay in the 2 000–4 999 bracket at Rs 47.50; the apparent discount costs about Rs 21 000 a year more, before counting incoming-inspection effort, scrap handling and the risk of a defective ring reaching assembly. If the Rs 46 price is wanted, negotiate it with a 1% AQL clause on the first plant, or require the supplier to replace defectives free of charge so that the effective price falls back to Rs 46 and the discount becomes genuine.
3. Set the reorder point for a 95% service level (z = 1.645) and check whether the order quantity and safety stock you have chosen can live inside the 8 000-ring warehouse limit.
Read the worked answer
Demand during lead time: d = 24 000/52 = 461.5 rings/week, L = 2 weeks → mean LT demand = 923 rings.
Standard deviation of demand during lead time, with both demand and lead time variable: σ_DDLT = √(L·σ_D² + d²·σ_L²) = √(2 × 60² + 461.5² × 0.5²) = √(7 200 + 53 246) = √60 446 = 246 rings.
Safety stock = 1.645 × 246 = 405 rings. Reorder point = 923 + 405 = 1 328 rings.
Note how much of the uncertainty comes from lead-time variability (53 246) rather than demand variability (7 200): cutting the supplier's delivery spread is worth far more than forecasting ring demand better. Any attempt to trim stock should start with a lead-time agreement, not with the forecast.
Warehouse check: with Q = 2 596, maximum on hand = Q + SS = 2 596 + 405 = 3 001 rings, comfortably inside 8 000. Even the 5 000 bracket would give 5 405 rings, also inside the limit, so the storage constraint is not what decides this problem - the defect rate is. Ordering policy should be reviewed if the supplier's lead time is renegotiated, and the reorder point recalculated whenever σ_L changes.