Scheduling a document job to a deadline under a cost cap
A document-production unit has won a contract that requires 72 equal-sized chapters to be formatted before a fixed filing date. Three typists are available. Typist A alone would need 12 full working days, typist B alone 18 days and typist C alone 24 days. Their charge-out rates are Rs. 3,000, Rs. 2,200 and Rs. 1,800 per day respectively, billed pro rata for part-days worked. Chapters can be split between typists at any point, so several typists can work on the job simultaneously and work is never lost in a handover. The client's deadline is 8 working days from today and the engagement letter caps total typing cost at Rs. 39,000. All three typists are currently free of other commitments, and the unit's manager must decide who works on what before the job starts.
1. With all three typists available for the full period, find the cheapest way to finish within the 8-day deadline, and state the cost and the headroom against the cap.
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Express daily output in chapters using the LCM of 12, 18 and 24, which is 72:
- A: 6 chapters/day, cost Rs. 3,000 → Rs. 500 per chapter
- B: 4 chapters/day, cost Rs. 2,200 → Rs. 550 per chapter
- C: 3 chapters/day, cost Rs. 1,800 → Rs. 600 per chapter
The cheapest plan loads the cheapest-per-chapter resource first, subject to the 8-day cap on each person.
Give A all 8 days: 48 chapters, Rs. 24,000. Remaining 24 chapters go to B (next cheapest) at 4/day = 6 days, Rs. 13,200.
Total: 48 + 24 = 72 chapters, elapsed 8 days, cost Rs. 37,200 - Rs. 1,800 under the cap.
The trap is to put everyone on the job at once: all three together do 13 chapters/day and finish in 5.54 days, but that costs 7,000 × 5.54 ≈ Rs. 38,800, and a mixed A+C plan takes exactly 8 days at Rs. 38,400. Running the cheapest resource to its full limit beats finishing early, because speed is not rewarded here - only the deadline is binding.
2. A is now confirmed available for only the first 5 days. Find the cheapest feasible plan and state whether it stays within the Rs. 39,000 cap.
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A can now contribute at most 5 × 6 = 30 chapters, cost Rs. 15,000.
After filling A, 42 chapters remain. B, the next cheapest, works all 8 days: 8 × 4 = 32 chapters, Rs. 17,600. Running total 62 chapters.
The last 10 chapters must go to C at 3/day = 3.33 days, Rs. 1,800 × 3.33 = Rs. 6,000.
Total: 30 + 32 + 10 = 72 chapters, cost 15,000 + 17,600 + 6,000 = Rs. 38,600 - Rs. 400 inside the cap.
Check the alternative of shading B down to 7 days (28 chapters) so C takes 14: C would need 4.67 days at Rs. 8,400, total Rs. 38,800, which is worse, confirming that B must be used to its 8-day limit before C is touched. The plan is feasible but carries only Rs. 400 of contingency, so any slippage on A's five days is unfunded.
3. The client moves the deadline to 6 working days while A remains limited to 5 days. Is the job deliverable within the Rs. 39,000 cap? Show the arithmetic.
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Maximum output in 6 days with A capped at 5: A 30 + B (6 × 4 = 24) + C (6 × 3 = 18) = 72 chapters - exactly the requirement, with nothing to spare.
Feasibility therefore forces A to 5 days, B to 6 days and C to 6 days simultaneously. Cost: 15,000 + 13,200 + 10,800 = Rs. 39,000.
The job is deliverable, but only exactly at the cap, with zero cost headroom and zero schedule float. Any of the following breaks it: A losing half a day, a chapter needing rework, or C being unavailable on any of the six days (the shortfall cannot be made up, because B and A are both already at their limits).
A defensible answer states the plan, confirms the Rs. 39,000 exactly, and flags that the correct commercial response is to seek either one extra day or a cost variation before accepting the instruction - not to accept it and hope. An answer that only says 'yes, it fits' misses the judgement the question is testing.