Five-Day IT Support Roster With Conflicting Constraints
An RRB's IT support desk must assign five officers-A, B, C, D and E-to five weekday shifts, Monday to Friday, with exactly one officer per day and each officer used once. The roster rules are: A must be on Wednesday; B must be on the day immediately after C; D cannot be on Monday or Friday; E must be before A; and C cannot be on Monday. The branch manager also wants to know whether the roster remains valid if D and E exchange days, and what happens if a new rule B cannot be on Friday is added. The desk handles ATM reconciliation, network tickets and CBS password resets, so an incorrect day assignment would leave a critical function uncovered. You are asked to prepare the roster and advise on the two changes before it is circulated.
1. Determine the only valid Monday-to-Friday roster.
Read the worked answer
A is fixed on Wednesday. The B-immediately-after-C block can be (C,B) on (Mon,Tue), (Tue,Wed), (Wed,Thu) or (Thu,Fri). C cannot be Monday, so (Mon,Tue) is out. B cannot be Wednesday because A is there, so (Tue,Wed) is out. C cannot be Wednesday because A is there, so (Wed,Thu) is out. The only remaining block is C on Thursday and B on Friday.
E must be before A, so E is Monday or Tuesday. D cannot be Monday or Friday. The two unfilled days are Monday and Tuesday, occupied by D and E. Since D cannot be Monday, D must be Tuesday and E must be Monday. The roster is therefore: Monday E, Tuesday D, Wednesday A, Thursday C, Friday B. Every rule is satisfied and each officer appears once.
2. If D and E exchange days, which rule is broken and is the roster otherwise valid?
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After the exchange, Monday is D, Tuesday is E, Wednesday is A, Thursday is C and Friday is B. Check each rule: A is on Wednesday, so that rule holds. B is immediately after C on Friday after Thursday, so that block holds. E is before A because Tuesday is before Wednesday, so that rule holds. C is not on Monday, so that rule holds. However, D is now on Monday, which directly violates the rule that D cannot be on Monday or Friday. The roster is therefore invalid, and the specific broken rule is the D placement rule.
3. What happens if the new rule B cannot be on Friday is added? Show the effect on feasibility.
Read the worked answer
The original constraints force B to Friday. Adding B cannot be on Friday leaves no possible placement for the C-B block. (Mon,Tue) is excluded by C cannot be Monday; (Tue,Wed) would put B on Wednesday, where A already sits; (Wed,Thu) would put C on Wednesday, where A already sits; and (Thu,Fri) would put B on Friday, which the new rule forbids. Therefore the constraint set becomes infeasible, not merely ambiguous.
To restore a roster, at least one existing rule must be relaxed-for example, allow C on Monday, allow A to move from Wednesday, or drop the immediate-after condition. The manager should be told that the new rule conflicts with the current rules rather than being given an invented roster.