Lumber Cut Plan With Defects
A carpenter is building shelving frames and needs 14 pieces of 2x4 cut to 22 1/2 in. Each saw cut removes 1/8 in. of kerf, and both ends of every board from the yard are already square, so only the cuts between pieces count against the board length. The yard sells 8 ft 2x4s at $3.85 and 10 ft 2x4s at $4.70. Anything shorter than 12 in. left over is scrap, not stock. The rack is picked over: roughly half the 8 ft boards have knots that force him to lose about 10 in. of usable length, and he cannot tell which boards those are until he handles them. He wants the cheapest plan that still finishes the job.
1. For each stock length, determine how many 22 1/2 in. pieces fit on one board, then the minimum number of boards and the cost to get all 14 pieces.
Read the worked answer
Four pieces need 4 × 22.5 = 90 in. plus 3 kerfs (0.375) = 90.375 in. Five pieces need 112.5 + 4 kerfs (0.5) = 113.0 in.
8 ft (96 in.): 90.375 ≤ 96, but 113.0 > 96 → 4 pieces per board. 14 ÷ 4 = 3.5 → 4 boards, capacity 16. Cost: 4 × $3.85 = $15.40
10 ft (120 in.): 113.0 ≤ 120 → 5 pieces per board. 14 ÷ 5 = 2.8 → 3 boards, capacity 15. Cost: 3 × $4.70 = $14.10
The longer stock is both cheaper and yields more pieces per board, because the fixed 22.5 in. remainder at the end of a short board is too short to use.
2. Now apply the knot problem to the 8 ft plan. Does it still work, and what does that tell you?
Read the worked answer
A knotted 8 ft board yields 96 − 10 = 86 usable inches.
- 3 pieces: 67.5 + 0.25 = 67.75 ≤ 86 ✓
- 4 pieces: 90.375 > 86 ✗
So a defective 8 ft board gives only 3 pieces. With 4 boards and half of them bad (2 boards): 3 + 3 + 4 + 4 = 14 pieces exactly - zero margin, and that assumes he can identify the defectives before cutting. Because the 8 ft plan has zero slack even in the best case (16 pieces against 14 needed), a single misjudged knot means a second trip to the yard.
The 10 ft plan carries 15 pieces of capacity against 14 needed, and one bad board at 120 − 10 = 110 in. still yields 4 pieces (90.375 ≤ 110), leaving 4 + 5 + 5 = 14. It absorbs one defect without a trip.
3. Make the buy recommendation and justify it on more than just price.
Read the worked answer
Buy three 10 ft boards for $14.10.
Reasons, in order of weight:
- Lower cost - $1.30 less than the 8 ft plan, despite the higher unit price.
- Defect tolerance - one knotted 10 ft board still yields 4 pieces (110 in. usable), so the job completes; the 8 ft plan has zero margin and fails on the first bad board.
- Trip risk - a second yard run costs fuel and time that dwarfs $1.30, so the plan with slack is worth more than its sticker price.
- Offcut value - the 8 ft plan leaves 5 5/8 in. scraps (pure waste), while the 10 ft plan leaves about 7 in. per board; neither is reusable, but the 10 ft plan makes more of the material it buys.
The general rule: when a plan's yield exactly equals its demand, price it against the failure case, not the ideal case.