Five Vendor Audits in Five Days
The retail chain's audit team has one slot per day from Monday to Friday and must place five external vendor audits-F, G, H, J and K-without overlap. The rules agreed with vendors are: H must be before J; K must be immediately after G; F cannot be on Monday or Friday; J cannot be on Wednesday; and there must be exactly one audit between F and H. The compliance officer proposes the order H, J, F, G, K. The operations manager objects that because the rules do not fix every position directly, at least one other valid order must exist, and asks for a check before the calendar is sent to vendors. Each vendor has confirmed only the five weekdays, so no audit can be moved to Saturday.
1. Test the compliance officer's proposed order H, J, F, G, K against every rule. Is it valid?
Read the worked answer
Check each rule:
- H before J: H is Monday, J is Tuesday. Valid.
- K immediately after G: G is Thursday, K is Friday. Valid.
- F not on Monday or Friday: F is Wednesday. Valid.
- J not on Wednesday: J is Tuesday. Valid.
- Exactly one audit between F and H: H is Monday and F is Wednesday, with J on Tuesday between them. Valid.
The proposed order satisfies all five rules.
2. The operations manager claims at least one other valid order must exist. Determine whether the order is unique. Show your case analysis.
Read the worked answer
The order is unique. Treat G and K as a two-day block GK.
GK can occupy (1,2), (2,3), (3,4) or (4,5). F and H must be two positions apart, because exactly one audit lies between them. F cannot be position 1 or 5.
- GK at (1,2): remaining slots 3,4,5. The only F/H pair two apart is (3,5), leaving J at 4. But then H would be 3 and F 5, and F cannot be Friday; the reverse puts H at 5, leaving no J after H. Impossible.
- GK at (2,3): remaining slots 1,4,5. No two slots are two apart without overlapping GK. Impossible.
- GK at (3,4): remaining slots 1,2,5. No F/H pair two apart. Impossible.
- GK at (4,5): remaining slots 1,2,3. The F/H pair must be (1,3). H must be before J, so H = 1, F = 3 and J = 2. This gives H, J, F, G, K.
So the compliance officer's order is the only valid one.
3. If the rule 'exactly one audit between F and H' is relaxed to 'at least one audit between F and H', give one valid order different from H, J, F, G, K and verify it.
Read the worked answer
One valid order is H, G, K, F, J.
Verification:
- H before J: Monday before Friday. Valid.
- K immediately after G: G is Tuesday, K is Wednesday. Valid.
- F not on Monday or Friday: F is Thursday. Valid.
- J not on Wednesday: J is Friday. Valid.
- At least one audit between F and H: H is Monday and F is Thursday, with G and K between them. Valid.
This order would not satisfy the original 'exactly one' rule, because there are two audits between F and H, but it satisfies the relaxed rule.