Tuning a Delivery Van's Fuel Model
A courier firm models a van's fuel use as F(v) = 0.0004v^2 + 0.35v + 6 litres per 100 km, where v is the cruising speed in km/h, valid for 40 ≤ v ≤ 110. The firm pays 1.80 per litre. Drivers are paid 22 per hour, so labour cost per 100 km is 22·(100/v). The operations manager currently instructs drivers to cruise at 110 km/h because 'the trip finishes sooner'. A dispatcher claims slowing to 80 km/h always saves money on a 400 km route. The manager wants a defensible recommendation before the next contract renewal, and asks whether the dispatcher's claim is right for every route length.
1. Build the total cost per 100 km as a function of v and determine whether the dispatcher's 80 km/h advice is cost-optimal.
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Total cost per 100 km: C(v) = 1.80(0.0004v^2 + 0.35v + 6) + 2200/v. Simplify: C(v) = 0.00072v^2 + 0.63v + 10.8 + 2200/v. Differentiate: C'(v) = 0.00144v + 0.63 − 2200/v^2. Set C' = 0: 0.00144v^3 + 0.63v^2 − 2200 = 0. Test v = 80: 0.00144(512000) + 0.63(6400) − 2200 = 737.3 + 4032 − 2200 = 2569 > 0, so C' > 0 at 80; cost is still rising there, meaning the optimum is below 80. Test v = 70: 0.00144(343000) + 0.63(4900) − 2200 = 493.9 + 3087 − 2200 = 1381 > 0. Test v = 60: 0.00144(216000) + 0.63(3600) − 2200 = 311.0 + 2268 − 2200 = 379 > 0. Test v = 55: 0.00144(166375) + 0.63(3025) − 2200 = 239.6 + 1905.8 − 2200 = −54.6 < 0. So the root lies between 55 and 60, near 56 km/h. Since C'' = 0.00144 + 4400/v^3 > 0, this is a minimum. Conclusion: 80 km/h is not optimal; the cost-minimising speed is about 56 km/h, and the dispatcher's claim is directionally right (slower than 110 saves money) but numerically wrong.
2. The manager asks whether the optimal speed depends on route length. Explain how the model changes for a route of length L km, and state what this implies for the dispatcher's claim.
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For a route of L km, cost per 100 km is unchanged but the number of 100 km units is L/100. Total cost = (L/100)·[0.00072v^2 + 0.63v + 10.8 + 2200/v]. Because L/100 is a positive constant multiplier, it scales the total but does not move the minimising v. So the optimal cruising speed (≈56 km/h) is independent of route length in this model. The dispatcher's claim that 80 km/h 'always saves money' is therefore not supported: it saves relative to 110 km/h, but it is above the optimum, so a further reduction to about 56 km/h lowers cost on every route length. Caveat worth flagging: the model assumes a constant cruising speed and ignores fixed overheads, so the recommendation should be presented as a model output, not a guarantee.