Scheduling Five Branch Audits Under Sequencing Rules
A branch's internal audit cell must complete five branch audits-P, Q, R, S, and T-on five consecutive working days, Monday to Friday, one per day. The scheduling rules are: (i) R must be audited immediately after Q; (ii) S must be audited after R but before T; (iii) P cannot be scheduled on Monday or Friday; (iv) T cannot be scheduled on Wednesday; and (v) P must be audited after S. The audit team has five auditors, each assigned to one branch, so the sequence is the only variable. The branch heads have confirmed availability only on their scheduled days. A last-minute request asks whether the Wednesday audit can be swapped with the Thursday audit without violating any rule.
1. Determine the only possible Monday-to-Friday schedule and show how each rule eliminates alternatives.
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Because R comes immediately after Q, the possible (Q, R) pairs are (Mon, Tue), (Tue, Wed), (Wed, Thu), and (Thu, Fri). Rule (ii) requires S after R and T after S, so at least two days must remain after R.
- (Thu, Fri): no day remains after R for S and T → impossible.
- (Wed, Thu): only Friday remains, but both S and T must follow R → impossible.
- (Tue, Wed): S and T must occupy Thursday and Friday, leaving P on Monday. Rule (iii) forbids P on Monday → impossible.
- (Mon, Tue): S and T must be chosen from Wednesday, Thursday, and Friday, with S before T; P takes the remaining day. If S = Wednesday and T = Thursday, P = Friday, which rule (iii) forbids. If S = Wednesday and T = Friday, P = Thursday, but rule (v) requires P after S - Thursday is after Wednesday, so this works. If S = Thursday and T = Friday, P = Wednesday, but then P is before S, violating rule (v).
Thus the only valid order is: Monday Q, Tuesday R, Wednesday S, Thursday P, Friday T. Check: R is immediately after Q; S is after R and before T; P is neither Monday nor Friday; T is not Wednesday; and P is after S.
2. If the Wednesday and Thursday audits are swapped, which rule is violated? Is there any alternative swap that would work?
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Original schedule: Mon Q, Tue R, Wed S, Thu P, Fri T.
Swapping Wednesday and Thursday gives: Mon Q, Tue R, Wed P, Thu S, Fri T.
Check each rule:
- R immediately after Q: still true.
- S after R and before T: Thursday is after Tuesday and before Friday - true.
- P not on Monday or Friday: P is on Wednesday - true.
- T not on Wednesday: T is on Friday - true.
- P after S: P is on Wednesday, S is on Thursday, so P is before S - violated.
So the swap fails rule (v). No other single swap of adjacent days works either: swapping Monday/Tuesday breaks R-immediately-after-Q; swapping Tuesday/Wednesday moves R away from Q; swapping Thursday/Friday puts T on Thursday but P on Friday, violating the Friday restriction on P; and swapping Wednesday with any non-adjacent day either breaks the Q-R adjacency or the S-before-T ordering. The only compliant sequence remains Q, R, S, P, T.
3. If T's branch head becomes unavailable on Friday and T must be audited on Thursday instead, can the schedule be adjusted without breaking the Q-R adjacency? Explain.
Read the worked answer
If T must move to Thursday, the original Thursday slot (P) and Friday slot (T) must be rearranged. One candidate is Mon Q, Tue R, Wed S, Thu T, Fri P. But rule (iii) forbids P on Friday, so this fails. Another candidate is Mon Q, Tue R, Wed P, Thu T, Fri S - but S must be before T, and here S is after T, so it fails.
A more systematic check: Q-R must remain adjacent. If Q-R stay on Mon-Tue, the remaining days are Wed, Thu, Fri for S, P, T with S before T and P after S. With T fixed on Thursday, S must be Wednesday (the only day before Thursday among the remaining slots), leaving P on Friday - forbidden. If Q-R move to Tue-Wed, then Monday must be S or P; T on Thursday leaves Friday for the other. But S must be before T, so S can be Monday or Friday? Friday is after T, so S must be Monday. Then P is Friday, forbidden. If Q-R move to Wed-Thu, T cannot be Thursday because R occupies it. If Q-R move to Thu-Fri, T cannot be Thursday because Q occupies it. Therefore no adjustment satisfies all rules with T on Thursday. The branch must either relax the Friday restriction on P or move T back to Friday; the Q-R adjacency itself cannot be preserved alongside the other constraints in this case.