Scheduling Five Site Inspections Under Conflicting Constraints
Five site inspections-Foundation (F), Brickwork (B), Column (C), Slab (S) and Plumbing (P)-must be completed on five consecutive working days, Monday to Friday, one per day. The following restrictions apply: (i) Foundation must be completed before Column; (ii) Slab must be on the day immediately after Column; (iii) Brickwork cannot be on Monday or Friday; (iv) Plumbing must be after Slab; and (v) Column cannot be on Tuesday. The assistant prepares a schedule but it is not shown to you. You are asked to verify whether the restrictions produce a single valid sequence, and to advise what happens if the Slab inspection is postponed to Friday because the concrete is not ready. The executive engineer wants the inspection calendar fixed today and warns that a wrong sequence will leave the slab unapproved at the end of the month.
1. Determine the unique valid sequence with reasoning, or show why none exists.
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Column and Slab must be consecutive, with Slab immediately after Column. Test the possible pairs. (C,S) = (1,2): Foundation cannot be before Column, so impossible. (2,3): Column would be Tuesday, forbidden by (v). (4,5): Plumbing must be after Slab, but there is no day after Friday, so impossible. That leaves (3,4): Column on Wednesday and Slab on Thursday. Foundation must be before Column, so F is Monday or Tuesday. Plumbing must be after Slab, so P is Friday. Brickwork must occupy the remaining day among Monday and Tuesday and cannot be Monday, so B is Tuesday and F is Monday. The unique sequence is Monday F, Tuesday B, Wednesday C, Thursday S, Friday P.
2. Identify which one of the five restrictions is redundant, and prove it.
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The Friday part of restriction (iii) is redundant: Brickwork could not have been placed on Friday even without that prohibition. Suppose B were on Friday. Then C and S must be consecutive among the remaining days. (C,S) = (4,5) is impossible because Friday is occupied by B. (C,S) = (2,3) is forbidden because C would be Tuesday. (C,S) = (1,2) is impossible because F must be before C. (C,S) = (3,4) leaves no day after S for P, because Friday is taken by B. Every possibility fails, so B cannot be Friday from the other restrictions alone. By contrast, the Monday part is not redundant: the sequence B, F, C, S, P satisfies all rules except 'B cannot be Monday', so that part does real work.
3. If Slab must move to Friday, what is the consequence and what is the least change needed to keep a valid schedule?
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If S is Friday, then C must be Thursday. Rule (iv) requires P after S, but there is no working day after Friday. Therefore the five restrictions become inconsistent and no Monday-to-Friday schedule exists. The least disruptive change is either to allow P on Saturday after the Friday slab inspection, or to relax rule (iv) and schedule P earlier. If a Saturday inspection is not acceptable, the revised rule must explicitly permit Plumbing before Slab; even then, several sequences become possible, so a further priority rule is needed. Simply moving S to Friday while keeping 'P after S' is not a valid calendar.