Scheduling Six Village Inspections with Travel and Time Constraints
Six villages - A, B, C, D, E and F - must be inspected from Monday to Saturday, one village per day. Distances from block headquarters are A 12 km, B 18 km, C 25 km, D 9 km, E 30 km and F 15 km. Constraints: C cannot be on Monday; D must be before B; E must be exactly two days after A; F cannot be on Saturday. The VDO also wants the village with the longest travel, E, not to fall on Thursday, the day of the weekly video conference. A complaint from village C is marked urgent, so C should be as early as possible after Monday. The office has one motorcycle that gives 35 km per litre, and petrol costs ₹96 per litre. The VDO returns to headquarters after each day's inspection.
1. Determine one valid inspection order and explain how it satisfies each constraint.
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One valid order is: Monday A, Tuesday C, Wednesday E, Thursday D, Friday F, Saturday B. Checks:
- C is not Monday; it is Tuesday, the earliest possible day after Monday, so the urgency condition is met.
- D (Thursday) is before B (Saturday).
- E is exactly two days after A: Monday → Wednesday.
- F is Friday, not Saturday.
- E is Wednesday, not Thursday, so the video conference day is free of the longest journey. The order also places the urgent complaint early in the week without breaking the E-after-A rule.
2. Using your order, calculate the total kilometres ridden during the week and the fuel cost at 35 km per litre and ₹96 per litre.
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Daily round trip = 2 × one-way distance. A: 2 × 12 = 24 km C: 2 × 25 = 50 km E: 2 × 30 = 60 km D: 2 × 9 = 18 km F: 2 × 15 = 30 km B: 2 × 18 = 36 km Total distance = 24 + 50 + 60 + 18 + 30 + 36 = 218 km. Fuel used = 218 ÷ 35 = 6.2286 litres approximately. Fuel cost = 6.2286 × ₹96 = ₹597.94, i.e. about ₹598. The answer assumes no extra diversion and that the VDO returns to headquarters every day as stated.
3. If the video conference is shifted to Wednesday, does your original order still work? If not, give a revised valid order or explain why none exists.
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No. The original order puts E on Wednesday, which would clash with the shifted video conference. A revised valid order is: Monday F, Tuesday C, Wednesday A, Thursday D, Friday E, Saturday B. Checks:
- C is Tuesday, not Monday, and is as early as possible after Monday.
- A is Wednesday, so E is Friday, exactly two days later.
- E is Friday, not Wednesday, so the conference day is clear.
- D (Thursday) is before B (Saturday).
- F is Monday, not Saturday. The order is valid. Total travel remains 218 km because the same six villages are visited with the same round trips, so fuel cost remains about ₹598.