Ninety-Six Tonnes Before the Road Closes
A relief convoy must lift 96 tonnes of tarpaulins, food packets and medicines from a depot to a cut-off block headquarters 180 km away. The transport officer has 6 trucks of 8-tonne capacity and 6 trucks of 12-tonne capacity. Standing orders forbid loading any truck below 75 per cent of its rated capacity. Diesel costs ₹90 a litre; an 8-tonne truck runs 4 km per litre and a 12-tonne truck 3 km per litre. Drivers must take a 30-minute rest after every 3 hours of driving, and the convoy averages 45 km/h including traffic halts but excluding the mandated rest. Loading is completed by 05:00, and the convoy must clear the block headquarters gate before 13:00, because the road is closed to heavy vehicles afterwards. The officer wants the smallest fleet that carries the full 96 tonnes in one lift, the fuel bill for that fleet, and a fallback if three of the 12-tonne trucks are withdrawn for law-and-order duty.
1. What is the smallest number of trucks that can carry 96 tonnes in one lift, and how do you prove it is minimal?
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The 75 per cent floor means an 8-tonne truck must carry at least 6 t and a 12-tonne truck at least 9 t. Try eight trucks: the best mix is 6 twelves at 12 t (72 t) plus 2 eights at 8 t (16 t) = 88 t, and 5 twelves plus 3 eights gives 60 + 24 = 84 t. Both fall short of 96 t, so eight trucks cannot do it. Nine trucks work exactly: 6 twelves loaded at 12 t give 72 t, and 3 eights loaded at 8 t give 24 t, total 96 t, and every truck is at or above its floor. Since eight is impossible and nine is achievable, nine is minimal. Note that the binding constraint is tonnes, not the loading floor: full loading is what makes nine possible, and deliberately under-loading to 'balance' the fleet would push the requirement back to twelve trucks.
2. Compute the fuel bill for the nine-truck fleet and comment on which truck type is cheaper per tonne carried.
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Fuel per truck is distance divided by mileage. For a 12-tonne truck: 180 ÷ 3 = 60 litres, and six of them burn 360 litres. For an 8-tonne truck: 180 ÷ 4 = 45 litres, and three of them burn 135 litres. Total 495 litres at ₹90 a litre = ₹44,550. Per tonne lifted that is 44,550 ÷ 96 ≈ ₹464. The useful comparison is litres per tonne: a 12-tonne truck uses 60 litres for 12 t, about 5.0 litres per tonne, while an 8-tonne truck uses 45 litres for 8 t, about 5.6 litres per tonne. So the heavier truck is cheaper per tonne despite its poorer mileage, because it carries half again as much. Where both types are available and capacity permits, load the big trucks first; the fleet here is limited by availability, not by fuel economics.
3. Three 12-tonne trucks are withdrawn. Can the 96 tonnes still reach the block headquarters before 13:00 in one lift? What do you advise?
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Remaining capacity is 3 × 12 + 6 × 8 = 36 + 48 = 84 tonnes, a shortfall of 12 tonnes. A second shuttle by one 12-tonne truck adds 360 km, about 120 litres (₹10,800) and roughly eight hours of driving plus rest, returning around 19:00, well past the closure. The single lift itself is comfortable: departing 05:30 after loading, 180 km at 45 km/h is 4 hours, plus one 30-minute rest after the third hour, giving arrival about 10:00, three hours inside the deadline. My advice is to hire one civilian 12-tonne truck, load it to its full 12 tonnes, and keep the single-lift plan; if no civilian truck can be had by 05:00, drop 12 tonnes of the bulkiest non-priority item and send it by the next available road convoy rather than risk the whole load on a shuttle that cannot beat the closure.