Five-Day Cooperative Audit Schedule Under Constraints
The district cooperative bank must schedule five branch audit visits from Monday to Friday, one visit per day, using four officers-A, B, C and D. A cannot audit on Monday or Friday. B must audit on the day immediately after C. D must audit at least twice. C cannot audit on Wednesday. Each visit covers a different branch-P, Q, R, S and T. Branch P has the largest cash balance and must be audited before branch T, which has the smallest. B cannot audit P. The current proposal is: Monday-C at Q; Tuesday-B at P; Wednesday-A at R; Thursday-D at S; Friday-D at T. The chief auditor wants either a reason to reject the plan or the smallest possible branch reassignment that preserves all officer-day assignments.
1. Test the proposal against every constraint. Which rule is violated, and does any other rule also fail? Show the check.
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Check each rule:
- A on Wednesday: allowed; A is barred only on Monday and Friday.
- C on Monday and B on Tuesday: B is immediately after C, so this rule passes.
- D on Thursday and Friday: D audits at least twice, so this passes.
- C on Monday: C is not on Wednesday, so this passes.
- Branches P and T: P is Tuesday and T is Friday, so P is before T; this passes.
- B cannot audit P: B is assigned to P on Tuesday. This fails. Branch uniqueness also passes: Q, P, R, S and T are all used once. Therefore the only violation is B-at-P. The plan must be rejected unless branch assignments are changed; the officer-day pattern itself can be retained.
2. Keeping the same officers on the same days, give one minimal branch reassignment that fixes the proposal. Verify the corrected schedule.
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One minimal fix is to swap branch P and branch S between B on Tuesday and D on Thursday. Corrected schedule: Monday C-Q; Tuesday B-S; Wednesday A-R; Thursday D-P; Friday D-T. Verification: A remains Wednesday; B immediately follows C; D still audits twice; C is not Wednesday; P is Thursday and T is Friday, so P is before T; B is not P. Only two branch entries change, which is the minimum because the original violation is the B-P pairing and B must be moved to another branch. A swap preserves the officer-day pattern and avoids changing D's two-audit requirement.
3. If B must audit S and C must audit Q, is a valid schedule possible? Give one or prove not.
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Yes. Start with C=Q. C cannot be Wednesday, and B must be the next day at S. Choose C on Monday and B on Tuesday. A cannot be Monday or Friday, so place A on Wednesday at R. D must audit at least twice; the remaining days are Thursday and Friday, so D takes both. Assign D to P on Thursday and T on Friday, which satisfies P before T and B not P. Schedule: Monday C-Q; Tuesday B-S; Wednesday A-R; Thursday D-P; Friday D-T. All constraints hold, so a valid schedule exists. This also shows the earlier corrected schedule can be adapted if the bank fixes B and C to those branches.