Automated Manufacturing Cost Surface Optimization (Systems & Industrial Engineering)

In automated manufacturing, robotics, and industrial systems engineering, supervisory controllers run continuous optimization routines to balance operating expenses. At an automated packaging facility in North Las Vegas, operations engineers monitor total operational expenditures $C(x, y)$ (measured in millions of dollars per year, $\$M/\text{yr}$) as a function of two coupled parameters: raw material feed rate $x$ (normalized index) and automated conveyor line velocity $y$ (normalized index).

The empirical cost surface is modeled by the coupled quadratic equation:

The production facility is currently operating at configuration setpoint $P(4, 2)$.

1. Compute the cost gradient vector $\nabla C(x, y)$ and evaluate it at the current setpoint $P(4, 2)$.
2. To minimize operational expenses as rapidly as possible, in what parameter direction should the control system adjust $(x, y)$? State both the vector of steepest descent and its corresponding unit vector $\vec{u}_{\text{opt}}$.
3. Determine the localized sensitivity rate (rate of cost reduction per unit step in parameter space) achieved along that optimal path.
4. Machine Learning & Gradient Descent in Automation: Explain how gradient descent optimization algorithms ($P_{k+1} = P_k - \alpha \nabla C(P_k)$) utilize this calculation in real-time PLC automation loops to guide factory operations away from high-cost bottlenecks.