Freezing a five-branch visit itinerary when the brief is under-specified
The district marketing officer of a regional rural bank must visit five branches - A, B, C, D and E - exactly once each, Monday to Friday. The regional office has fixed three conditions: (i) branch C can be visited neither on Monday nor on Friday; (ii) branch D must be visited on the day immediately after branch B; (iii) branch A must be visited before branch E, but not on the day immediately before it. Travel between any two branches consumes a full working day, so no two branches can be clubbed on the same day. Branch C's agricultural credit camp is fixed for Thursday, and branch E's staff review meeting is held only on Fridays. The cluster head wants a single frozen itinerary by this evening and has offered to add one extra condition of the officer's choice, provided it is operationally meaningful and not merely a restatement of the three fixed rules.
1. List every itinerary that satisfies the three fixed conditions, showing briefly how the impossible placements are eliminated, and state which branch falls on the same day in all valid itineraries.
Read the worked answer
Treat B and D as one block, BD, because D must follow B immediately. The block can occupy days (1,2), (2,3), (3,4) or (4,5).
- Block on (1,2): remaining days 3,4,5 for A, C, E. C cannot be day 5, and A must precede E without being adjacent. The only surviving order is B, D, A, C, E.
- Block on (2,3): remaining days 1,4,5. C cannot be day 1 or 5, so C = 4. Then A = 1 and E = 5 (not adjacent). Surviving order: A, B, D, C, E.
- Block on (3,4): remaining days 1,2,5. C must be day 2 (not 1 or 5). A = 1, E = 5. Surviving order: A, C, B, D, E.
- Block on (4,5): remaining days 1,2,3. Either C = 2 (forcing A = 1, E = 3, which are adjacent) or C = 3 (forcing A = 1, E = 2, also adjacent). Nothing survives.
Three itineraries remain, and branch C is on Thursday in all three. That is the only day that can be told to the branch with certainty today.
2. The cluster head suggests adding one of these conditions: (a) A is visited on Wednesday; (b) E is visited on Friday; (c) C is visited on Tuesday; (d) B is visited on Monday. Which one would NOT produce a unique itinerary, and what does that tell you about how to phrase the extra condition?
Read the worked answer
Test each against the three survivors: B-D-A-C-E, A-B-D-C-E and A-C-B-D-E.
- (a) A on Wednesday: only B-D-A-C-E qualifies (in the other two A is Monday). Unique.
- (c) C on Tuesday: only A-C-B-D-E qualifies. Unique.
- (d) B on Monday: only B-D-A-C-E qualifies (B is Tuesday and Wednesday in the others). Unique.
- (b) E on Friday: all three itineraries already end with E on Friday, because E could never sit on day 1 (A must precede it) and day 5 is the only remaining slot once C is barred from Monday and Friday.
So (b) is a redundant clause: it is already implied by the fixed rules, adds no information, and would leave the officer with three equally valid plans. The lesson is that an extra condition must discriminate between the surviving candidates, so it should be phrased around a day that the three plans assign differently - A, B or C - not around E.
3. The cluster head then insists that the branch with the weakest deposit mobilisation, which is C, must be covered on Monday. Is the itinerary feasible? What should the officer advise?
Read the worked answer
No. Branch C cannot be placed on Monday under any combination of the three fixed rules: it is forced to Thursday in every valid plan. The officer should say this plainly rather than produce a plan that quietly breaks a rule, because the itinerary is also the record the regional office audits against.
The officer should then offer the cheapest relaxation and show its cost. Dropping the 'not on the day immediately before' clause on A and E is the least damaging, since it removes a spacing preference rather than a hard sequencing or precedence rule. With that clause relaxed, C on Monday becomes possible and A and E can sit together on two consecutive days, which is operationally acceptable if the officer accepts back-to-back travel. If the cluster head refuses to relax anything, the alternative is to move the credit camp, not the visit plan, and the officer should ask him to confirm that in writing.