Clinic Night-Shift Roster Under Seniority and Rest Rules
A 30-bed private clinic in Rohtak runs three shifts a day: Morning, Evening and Night. Five doctors-Dr Rao (R), Dr Sen (S), Dr Iyer (I), Dr Bhatt (B) and Dr Khan (K)-cover one shift each per day, and no doctor may take two shifts on the same day. The roster must satisfy these rules:
- Every shift must have exactly one doctor.
- At least one of Dr Rao or Dr Sen, the two senior consultants, must be on Night duty every day.
- Dr Iyer cannot work the Night shift.
- No doctor may work Night duty on two consecutive days.
- If Dr Bhatt works Night, Dr Bhatt must be completely off duty the following day.
- Dr Khan works Morning shifts only.
The clinic has drafted this three-day roster:
- Monday: Morning K, Evening B, Night R
- Tuesday: Morning K, Evening R, Night S
- Wednesday: Morning K, Evening S, Night I
The administrator suspects the roster is invalid. She also needs a corrected three-day roster that changes as few shift assignments as possible.
1. Identify every rule violated by the draft roster. Explain each violation with reference to the specific day and doctor.
Read the worked answer
The draft violates two rules, both on Wednesday:
- Rule 3 is violated: Dr Iyer (I) is assigned Wednesday Night. Rule 3 says Dr Iyer cannot work Night.
- Rule 2 is violated: Wednesday Night is covered by Dr Iyer, who is not one of the two seniors. Neither Dr Rao nor Dr Sen is on Wednesday Night, so there is no senior consultant on Night duty that day.
The other rules are satisfied:
- Rule 1: each shift has exactly one doctor.
- Rule 4: Night duties are R (Mon), S (Tue), I (Wed); no doctor has consecutive nights.
- Rule 5: Dr Bhatt never works Night, so the following-day rest rule is not triggered.
- Rule 6: Dr Khan is on Morning duty only.
So the administrator is right that the roster is invalid, but the problem is concentrated in one assignment.
2. Propose a corrected roster that satisfies all rules and changes the fewest possible assignments from the draft. State exactly which assignments you change and why no fewer changes are possible.
Read the worked answer
Change one assignment: replace Dr Iyer with Dr Rao on Wednesday Night.
Corrected roster:
- Monday: Morning K, Evening B, Night R
- Tuesday: Morning K, Evening R, Night S
- Wednesday: Morning K, Evening S, Night R
Check:
- Wednesday Night now has a senior (R), satisfying Rule 2.
- Dr Iyer is no longer on Night, satisfying Rule 3.
- Dr Rao's Night duties are Monday and Wednesday, not consecutive, so Rule 4 is satisfied.
- Dr Bhatt still never works Night, and Dr Khan remains on mornings only.
Why fewer changes are impossible: at least one change is required because the draft violates the rules. This single change fixes both Wednesday violations at once. Changing Wednesday Night to Dr Sen would fail Rule 4 because Dr Sen worked Tuesday Night. Changing it to Dr Bhatt would fail Rule 2 because Dr Bhatt is not senior. Changing it to Dr Khan would fail both Rule 2 and Rule 6. Therefore Dr Rao is the only valid one-change fix.
3. Suppose Dr Iyer is unavailable on Wednesday and Dr Khan can work only mornings. Can the clinic keep the same Monday and Tuesday assignments and still satisfy all rules on Wednesday? If yes, give the Wednesday roster; if no, explain which rule must be relaxed.
Read the worked answer
Yes, the clinic can keep Monday and Tuesday unchanged.
On Wednesday, Morning must be Dr Khan (Rule 6 and availability). The Night shift must contain a senior: either Dr Rao or Dr Sen. Dr Sen worked Tuesday Night, so assigning Dr Sen to Wednesday Night would violate Rule 4. Therefore Wednesday Night must be Dr Rao.
Dr Rao's Night duties would then be Monday and Wednesday, which are not consecutive. The Evening shift can be Dr Sen or Dr Bhatt:
- Option 1: Morning K, Evening S, Night R
- Option 2: Morning K, Evening B, Night R
Both satisfy all rules. Dr Bhatt is not on Night, so Rule 5 is not triggered. Dr Iyer is not used, so Rule 3 is irrelevant. Therefore no rule needs to be relaxed; the clinic only needs to avoid putting Dr Sen on Wednesday Night.