Scheduling the Five-Day Pre-Placement Workshop
The coordinator of a five-day pre-placement workshop at a degree college must fix the order of five sessions - Quantitative (Q), Reasoning (R), Verbal (V), Computer Awareness (C) and Current Affairs (U) - one per day, Monday to Friday (slots 1 to 5). Four constraints were printed in the student handbook: (i) Q must be on the day immediately before R; (ii) V cannot be on day 1 or day 2, because both mornings are reserved for the principal's address; (iii) C must come after R; (iv) U must be on day 1. The visiting faculty member for V is now free only on Tuesday (slot 2) and cannot come on any other day. The coordinator will not move U, will not add a sixth day, and will relax exactly one constraint. A student representative has meanwhile proposed the order R, Q, C, U, V and claims it meets all four constraints.
1. Does the student representative's proposed order R, Q, C, U, V satisfy all four constraints? Identify every violation.
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Test each constraint against the order R(1), Q(2), C(3), U(4), V(5).
- (i) Q immediately before R: here R is on day 1 and Q on day 2, so Q comes after R, not before it. Violated.
- (ii) V not on day 1 or day 2: V is on day 5. Satisfied.
- (iii) C after R: C is on day 3, R on day 1. Satisfied.
- (iv) U on day 1: U is on day 4. Violated.
So the proposal breaks two constraints, not zero: the Q–R adjacency and the fixed position of U. The representative has probably checked only that the five session names all appear once, which every ordering does. Two violations also mean the proposal is not a near-miss that one edit can repair - moving U to day 1 pushes everything else and would still leave Q after R. The coordinator should reject it outright rather than try to patch it.
2. The visiting faculty for V can come only on Tuesday (slot 2). Which single constraint should the coordinator relax, and what schedule results? Compare the alternatives.
Read the worked answer
Relax constraint (ii), the ban on V in slots 1 and 2. Keeping U on day 1, slots 3–5 must hold Q, R and C with Q immediately before R and C after R. The only fit is Q = 3, R = 4, C = 5, giving the order U, V, Q, R, C.
Compare the other relaxations:
- Relax (iii), C after R: with V = 2 and U = 1, both Q3-R4-C5 and C3-Q4-R5 become legal. Two valid schedules mean a second circular and confusion, so this is a worse fix.
- Relax (i), the adjacency: Q before R without adjacency again forces Q3-R4-C5, so the same timetable is bought at the cost of the intended pedagogy (reasoning built directly on quantitative basics).
- Relax (iv), U on day 1: nothing is gained. With V = 2 and Q, R consecutive, C must still take slot 5 and U still falls in slot 1.
Relaxing (ii) is therefore the minimum change: it costs only the principal's morning slot, which can shift to an afternoon or to day 3.
3. If no constraint is relaxed at all, list every valid order and state the earliest slot V can occupy.
Read the worked answer
Fix U at slot 1. Slots 2–5 are filled by Q, R, C, V with Q immediately before R. The consecutive pair can sit at (2,3), (3,4) or (4,5).
- Pair at (2,3): slots 4 and 5 take C and V. C after R holds in both arrangements, and V is legal at 4 or at 5. Two orders: U, Q, R, V, C and U, Q, R, C, V.
- Pair at (3,4): slots 2 and 5 take C and V. V cannot be at 2, so V = 5 and C = 2 - but then C comes before R, breaking (iii). No order.
- Pair at (4,5): slots 2 and 3 take C and V. V cannot be at 2, so V = 3 and C = 2, again putting C before R. No order.
So exactly two orders survive: U, Q, R, V, C and U, Q, R, C, V. In both, Q, R and U occupy the same days, and the earliest day V can fall on is day 4 - which is precisely why the Tuesday-only visiting faculty cannot be accommodated without a relaxation.