Five lots, five days: a dispatch sequence that stops working
The dispatch clerk at a depot must release five lots - paddy (P), rice (R), wheat (W), sugar (S) and chana (C) - one lot per working day from Monday to Friday, because the loading team and the single weighbridge can handle only one full truck a day. The manager's standing instructions are: (1) rice must move before paddy; (2) wheat must move on Thursday because its rake is scheduled that evening; (3) sugar cannot move on Monday or Friday, when the sugar godown is shut for stock-taking; (4) rice must move on the day immediately after sugar; (5) chana must move before wheat. On Tuesday morning the storekeeper reports a bag shortage, so sugar cannot be dispatched before Wednesday, though it may go later in the week. The clerk also hands you a schedule he prepared earlier: Monday paddy, Tuesday chana, Wednesday sugar, Thursday wheat, Friday rice.
1. Derive the one sequence that satisfies all five instructions, showing each deduction.
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Work from the tightest constraint, which is the linked pair in (3) and (4).
- Wheat occupies Thursday by (2).
- Sugar cannot be Monday or Friday by (3), and Thursday is gone, so sugar is Tuesday or Wednesday.
- If sugar were Wednesday, rice would be Thursday by (4), which contradicts wheat on Thursday. So sugar is Tuesday and rice is Wednesday.
- Rice must precede paddy by (1). Thursday is wheat, so paddy must be Friday.
- Chana must precede wheat by (5). Of the days still free, only Monday remains for chana.
The sequence is: Monday chana, Tuesday sugar, Wednesday rice, Thursday wheat, Friday paddy.
It is unique, not merely one of several. Test the alternatives: sugar on Wednesday breaks wheat-on-Thursday, and chana on any day after Monday would either collide with a fixed lot or break rule (5), because Tuesday, Wednesday and Thursday are already committed and chana before wheat cannot be satisfied on Friday. Uniqueness matters operationally, because it means any deviation the clerk proposes is a breach of an instruction and must be escalated, not adjusted on the floor.
2. With sugar now barred before Wednesday, does a valid sequence still exist? If not, identify what must be relaxed and give the resulting schedule.
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No valid sequence exists. Sugar cannot be Monday or Friday by (3) and cannot be before Wednesday by the new report, so sugar must be Wednesday. Rule (4) then puts rice on Thursday, and rule (2) puts wheat on Thursday. Two lots cannot share a day. The constraint set is infeasible, and no amount of reordering fixes it.
What to relax. The cheapest relaxation is rule (2), the wheat date, because a rake can be re-booked while a bag shortage cannot be conjured away and the sugar godown's closure is a physical fact.
- Sugar: Wednesday.
- Rice: Thursday.
- Paddy: Friday, satisfying rice before paddy.
- Chana and wheat fill Monday and Tuesday, with chana before wheat, so Monday chana and Tuesday wheat.
Result: Monday chana, Tuesday wheat, Wednesday sugar, Thursday rice, Friday paddy. This is again unique.
What to check before signing it. Moving wheat off Thursday has a cost outside the sequence: the rake scheduled that evening will be under-loaded and the depot may attract wagon detention or lose the allotment. So the relaxation should be confirmed with the traffic section in writing, and the alternative relaxation - dropping the immediate-after link and placing sugar on Wednesday with rice later in the week - should be priced against it. When a constraint set is infeasible, the correct answer is to say so and choose which constraint to buy out, not to produce a schedule that quietly breaks one.
3. Verify the clerk's earlier schedule - Monday paddy, Tuesday chana, Wednesday sugar, Thursday wheat, Friday rice - against each instruction, and say which check you would run first in future.
Read the worked answer
Check every rule, one at a time, rather than judging the schedule as a whole.
| Rule | Test | Result |
|---|---|---|
| (1) rice before paddy | rice Friday, paddy Monday | Violated |
| (2) wheat on Thursday | wheat is Thursday | Satisfied |
| (3) sugar not Monday or Friday | sugar is Wednesday | Satisfied |
| (4) rice immediately after sugar | sugar Wednesday, rice Friday | Violated |
| (5) chana before wheat | chana Tuesday, wheat Thursday | Satisfied |
The schedule fails two rules, so it is rejected outright. Note that a superficial check would have passed it: three of the five instructions are met, and the sequence looks orderly because each lot appears once and the week is fully occupied.
Run the linked pair first. Sugar-then-rice is a two-day block that consumes both the day of one lot and the immediately following day. Fixing it first either collapses the problem to three free days or, as in the second task, proves infeasibility at once. Checking single-lot rules first feels easier but hides the structural conflict, and the clerk's schedule is exactly what that checking order produces.