Rostering Five Mandi Inspections Across Five Days Under Six Fixed Instructions
The District Mandi Board must inspect five mandis - P (12,000 quintal storage), Q (18,000), R (25,000), S (30,000) and T (40,000) - on five working days, Monday to Friday, one mandi per day. Five officers A, B, C, D and E each inspect exactly one mandi, one on each day. The Board's written instructions are:
- A inspects P, and P is inspected on Monday.
- C inspects on Wednesday.
- The officer who inspects T is scheduled on the day immediately after C.
- B inspects the mandi with 18,000 quintal capacity, and not on Friday.
- D inspects on Thursday.
- E does not inspect S.
A trader's association claims that the 30,000-quintal mandi was inspected before the 18,000-quintal mandi, and that the officer inspecting R did so on Friday. The Board wants the roster settled before the inspection week begins.
1. Derive the complete roster and test both claims made by the traders' association.
Read the worked answer
Derivation
- A inspects P on Monday (instruction 1).
- T is inspected the day immediately after C, and C is Wednesday (2, 3), so T falls on Thursday.
- D inspects on Thursday (5), so D inspects T.
- Left over: officers B and E, days Tuesday and Friday, mandis Q (18,000), R (25,000), S (30,000).
- B inspects the 18,000 mandi and not on Friday (4), so B inspects Q on Tuesday.
- E is left with Friday and either R or S; E does not inspect S (6), so E inspects R on Friday, and C inspects S on Wednesday.
Roster: Monday A-P (12,000); Tuesday B-Q (18,000); Wednesday C-S (30,000); Thursday D-T (40,000); Friday E-R (25,000).
Claims
- 'The 30,000 mandi was inspected before the 18,000 one' - false: S is Wednesday and Q is Tuesday, so the 18,000 mandi came first.
- 'R was inspected on Friday' - true: E inspects R on Friday.
2. A sixth mandi U (22,000 quintals) is added and must be inspected on Saturday by one of the five officers, applying the rule that no officer inspects on two consecutive days. Who cannot be assigned, and who can?
Read the worked answer
Excluded: E. E inspects on Friday, and Saturday is the next consecutive day, so E is barred by the stated rule.
Available: A, B, C and D. None of their days (Monday, Tuesday, Wednesday, Thursday) is consecutive with Saturday.
But the arithmetic does not close. The original design gives each officer exactly one inspection on one day. Adding a Saturday mandi means somebody must take two mandis. The cleanest reading is that A, B, C or D may take Saturday as a second assignment, which the consecutive-day rule allows, because the rule bars only back-to-back days and not a second inspection as such.
If the Board insists on the senior-most officer A for U, A is free on paper (Monday to Saturday is not consecutive), but Monday's P then has no inspector, because all five officers are already committed. The Board must either accept a double assignment or post a sixth officer; it cannot have both 'exactly one inspection each' and a Saturday inspection.
3. On Tuesday morning B reports sick. A clerk proposes moving B's Q inspection to Friday and E's R inspection to Tuesday. Is that permissible under the six instructions? What should the supervisor do instead?
Read the worked answer
Not permissible. Instruction 4 says B does not inspect on Friday. Moving B to Friday breaks that instruction directly. Swapping only the mandis and not the days changes nothing, because the conflict is in the officer-day pair, not in which mandi is inspected.
What the supervisor should do
- Record the absence and the infeasibility in writing rather than quietly amending the roster; a schedule that breaks a written instruction is a finding waiting to happen in an audit.
- Assign a substitute officer F to Q on Tuesday, keeping B's own inspection to be made good later, and issue an amended roster stating exactly which instruction has changed.
- If no substitute is available, hold Q over to Saturday alongside U, with the same officer taking both, and record that as a variation approved by the Board - not a swap invented at the counter.
The examinable point is that a roster becomes infeasible, not merely inconvenient, when a hard constraint collides with a substitution; the fix is to name the relaxed rule on the record.