Electric Delivery Van Highway Cruising Optimization (Mechanical Engineering)

In automotive and mechanical systems engineering, operating fleet vehicles requires balancing vehicle operational physics against commercial labor logistics. In commercial electric vehicle (EV) fleets, aerodynamic drag forces scale quadratically with speed ($F_D = \frac{1}{2}\rho C_D A v^2$), causing battery electrical power demand to scale cubically ($P \propto v^3$). Consequently, high cruising speeds rapidly deplete battery energy and inflate charging costs. Conversely, driving too slowly increases total trip duration, escalating hourly driver wages and vehicle leasing overhead.

A mechanical fleet engineer for a logistics distribution company in Southern Nevada is optimizing the highway cruise speed for a fleet of class-4 electric cargo vans.

• The delivery vans transport palletized freight between a fulfillment center in North Las Vegas and a distribution terminal near Primm over a round-trip highway transit distance of $D = 240\text{ miles}$.
• Fleet telemetry, commercial driver wages, and vehicle lease amortization cost a combined fixed rate of $w = \$45$ per operating hour.
• At a constant highway cruising velocity $v$ (in miles per hour, $\text{mph}$), the battery electrical energy consumption rate costs:
  $$R_{\text{energy}}(v) = \frac{v^2}{80}\text{ dollars per hour}$$
• Highway safety regulations and commercial fleet speed governors restrict the allowable cruising velocity to the closed interval $v \in [45, 75]\text{ mph}$.

Cost ($)
  ^
  |        / Total Operating Cost C(v)
  |       /
  |  \   /     * Optimal Cruising Speed (v = 60 mph)
  |   \ /
  |    *-----------
  |   / \          \ Hourly Labor Overhead = $10,800 / v
  |  /   \          \
  | /     \          \ Battery Charging Cost = 3v
  +----------------------------------------------------> Speed v (mph)
     45              60                     75

1. Formulate the total trip transit time $t(v)$ and construct the single-variable total trip cost objective function $C(v)$.
2. Differentiate $C(v)$ and locate the stationary cruising speed $v^*$.
3. Apply the Closed Interval Method across the legal domain $[45, 75]\text{ mph}$ to confirm the absolute minimum operating cost.
4. Calculate the separate labor cost and battery charging cost at the optimal speed, verifying the Cost Equipartition Theorem.
5. Explain how the fleet dispatch manager applies this model if driver wages increase due to overtime premiums.