Scheduling the Audit Team
A compliance team of five - Priya, Quan, Rosa, Sam, and Tariq - must cover four audit sites (North, South, East, West) over Monday–Friday. Constraints:
- Each site needs exactly one auditor per day it is open; North and South are open all five days; East and West are open only Tuesday–Thursday.
- No auditor may cover more than one site per day.
- Priya is unavailable Monday and Friday.
- Quan refuses to work at North.
- Rosa must work at South on at least two days.
- Sam and Tariq cannot be at the same site on the same day.
- Tariq is available only Wednesday and Thursday.
1. Determine whether a valid schedule exists, and if so, construct one. If not, identify the binding constraint.
Read the worked answer
Count the demand. North (5 days) + South (5) + East (3) + West (3) = 16 auditor-days needed.
Count the supply. Priya 3 (Tue–Thu), Quan 5, Rosa 5, Sam 5, Tariq 2 (Wed–Thu) = 20 auditor-days available. Capacity is sufficient in aggregate.
Check the tight days. Monday: only Quan, Rosa, Sam available (Priya out, Tariq out) = 3 people, but North and South both need coverage = 2 needed. OK. Friday: same 3 available, 2 needed. OK. Tuesday: Priya, Quan, Rosa, Sam = 4 available; four sites open = 4 needed. Tuesday is exactly tight - all four must work, and Tariq cannot. Wednesday/Thursday: all five available, 4 needed. OK.
Apply exclusions. Quan cannot do North. On Tuesday, the four workers (Priya, Quan, Rosa, Sam) must fill North, South, East, West with Quan not at North. Feasible: Priya→North, Quan→East, Rosa→South, Sam→West. Sam and Tariq never co-located - Tariq isn't working Tuesday, fine.
Sample valid schedule (North/South/East/West):
- Mon: North=Sam, South=Rosa
- Tue: North=Priya, South=Rosa, East=Quan, West=Sam
- Wed: North=Priya, South=Rosa, East=Quan, West=Tariq
- Thu: North=Priya, South=Sam, East=Quan, West=Tariq
- Fri: North=Sam, South=Rosa
Verify: Rosa at South Mon, Tue, Wed, Fri = 4 days (≥2 ✓). Quan never at North ✓. Priya none Mon/Fri ✓. Tariq Wed/Thu only ✓. Sam and Tariq never same site (Wed: Sam not working; Thu: Sam at South, Tariq at West) ✓. A valid schedule exists.
2. Suppose Tariq's availability expands to include Tuesday. Does this change whether a schedule exists, and what does it reveal about the schedule's fragility?
Read the worked answer
Before: Tuesday was exactly tight - 4 available for 4 sites, so every Tuesday worker was forced. Any single Tuesday absence would have made the schedule impossible.
After: Tuesday supply becomes 5 (add Tariq) for 4 sites, creating one degree of slack. The schedule still exists (it already did), but now Tuesday has a buffer.
Reveals fragility: The original feasibility hinged on zero Tuesday absences. The schedule was feasible but not robust - a single sick day on Tuesday would break it. Expanding Tariq's availability converts a knife-edge constraint into a survivable one.
Model insight: A strong answer distinguishes feasibility (does a solution exist?) from resilience (how much disturbance can it absorb?). The exam rewards noticing that the Tuesday equality (supply = demand) is the real pressure point, not the aggregate totals, which looked comfortable at 20 vs 16. Aggregate slack hid a local bottleneck - a classic analytical trap.