The Vendor's Completion Certificate
You assist a district electoral office that must deliver 48,000 laminated identity cards within 12 working days. Three printing units are installed. Unit A prints 1,200 cards a day without interruption. Unit B prints 900 a day. Unit C prints 1,500 a day but is unavailable for its first two working days because of calibration, and loses every fourth working day after that to mandatory maintenance. Lamination is a separate stage: two operators together laminate 2,400 cards a day. A card may be laminated on the day after it is printed, so printing and lamination of different cards may overlap, but no card can be laminated before it is printed. The vendor has submitted a completion certificate claiming the full order can be finished inside 12 days with all three printers and both laminators running from day one. Last year's audit found that 6 per cent of printed cards were rejected at lamination and had to be reprinted.
1. Test the vendor's claim twice - once assuming zero wastage, once allowing the 6 per cent rejection rate. Show the arithmetic for both stages.
Read the worked answer
Printing, 12 days, zero wastage
- Unit A: 1,200 × 12 = 14,400
- Unit B: 900 × 12 = 10,800
- Unit C: available days 3–12 = 10 available days. Within those, the 4th and 8th working days are maintenance losses, so 8 productive days × 1,500 = 12,000
- Total printed = 14,400 + 10,800 + 12,000 = 37,200
Shortfall against 48,000 = 10,800 cards. The certificate already fails on printing alone, before any wastage argument.
Lamination, 12 days
- 2,400 × 12 = 28,800 against 48,000 → shortfall 19,200. This stage is the harder bottleneck.
Allowing 6 per cent rejection
Rejection happens at lamination, so the quantity that must enter the pipeline is the good output divided by the yield: 48,000 ÷ 0.94 = 51,063.8, i.e. 51,064 cards must be printed and laminated.
- Printing gap: 51,064 − 37,200 = 13,864
- Lamination gap: 51,064 − 28,800 = 22,264
The claim is not a near-miss that better scheduling could rescue; it is short by roughly a quarter of the order at the printing stage and nearly half at lamination.
2. The Director asks whether to accept the certificate or sanction additional capacity. Give your recommendation and state the minimum additional resource, assuming the 12-day deadline cannot move.
Read the worked answer
Recommendation: reject the certificate. The figures are not a rounding dispute, so accepting it would transfer a known default onto the office.
Minimum extra lamination capacity. Required rate = 51,064 ÷ 12 = 4,255.3 cards/day against an installed 2,400/day. One additional two-operator pair adds 2,400/day, taking capacity to 4,800/day, or 57,600 laminated in 12 days. Good output at 94 per cent = 54,144 ≥ 48,000. One pair is sufficient; a half-pair is not, because capacity is lumpy.
Minimum extra printing capacity. Required rate = 51,064 ÷ 12 = 4,255.3/day against 37,200 ÷ 12 = 3,100/day. Gap = 1,155.3 cards/day. A single uninterrupted 1,200/day unit closes it at 51,600 printed - a margin of only 536 cards, which is under one hour of one machine's output.
Judgement on that thin margin. Sanction a 1,500/day unit or authorise scheduled overtime rather than a 1,200/day unit, for three reasons: (i) the perfect 12-day run assumed for Units A and B is an assumption, not a guarantee; (ii) a single day's breakdown of the new laminator costs 2,400 cards, which the margin cannot absorb; (iii) rejected cards must be reprinted inside the same 12 days, so every rejection consumes printing capacity already fully committed. Attach a daily printed/laminated/rejected reconciliation to the sanction order, and inspect the certificate's arithmetic before it is filed - a vendor who miscalculated once will do so again.