Automated Manufacturing Cost Surface Optimization (Systems & Industrial Engineering) Solution

Problem Overview

We evaluate the operational expenditure surface $C(x, y) = x^2 + 3xy + 2y^2$ at operating setpoint $P(4, 2)$ . We compute the gradient $\nabla C(4, 2)$ , determine the parameter vector of steepest descent $-\nabla C$ , find the optimal unit vector $\vec{u}_{\text{opt}}$ , calculate the sensitivity rate of cost reduction, and interpret the algorithm in terms of industrial gradient descent control.

Step-by-Step Mathematical Solution

1. Compute and Evaluate the Cost Gradient

State the Gradient Formula (defined in The Gradient Vector):

\nabla C(x, y) = \left\langle \frac{\partial C}{\partial x}, \frac{\partial C}{\partial y} \right\rangle

Compute the partial derivatives:

C_x = \frac{\partial}{\partial x}[x^2 + 3xy + 2y^2] = 2x + 3y

C_y = \frac{\partial}{\partial y}[x^2 + 3xy + 2y^2] = 3x + 4y

Evaluate at the current operational setpoint $P(4, 2)$ :

C_x(4, 2) = 2(4) + 3(2) = 8 + 6 = 14\text{ \$M/unit}

C_y(4, 2) = 3(4) + 4(2) = 12 + 8 = 20\text{ \$M/unit}

\nabla C(4, 2) = \langle 14, 20 \rangle\text{ \$M/unit}

2. Vector of Steepest Cost Reduction (Gradient Descent)

To reduce costs most rapidly, adjustments must move along the direction of steepest descent, given by the negative gradient:

\vec{v}_{\text{opt}} = -\nabla C(4, 2) = -\langle 14, 20 \rangle = \langle -14, -20 \rangle

Normalize to obtain the unit vector direction $\vec{u}_{\text{opt}}$ :

\|\vec{v}_{\text{opt}}\| = \sqrt{(-14)^2 + (-20)^2} = \sqrt{196 + 400} = \sqrt{596} = 2\sqrt{149} \approx 24.413

\vec{u}_{\text{opt}} = \frac{\langle -14, -20 \rangle}{2\sqrt{149}} = \left\langle -\frac{7}{\sqrt{149}}, -\frac{10}{\sqrt{149}} \right\rangle \approx \langle -0.573, -0.819 \rangle

3. Localized Sensitivity Rate

The rate of cost reduction achieved per unit step in parameter space along $\vec{u}_{\text{opt}}$ is:

D_{\vec{u}_{\text{opt}}} C(4, 2) = -\|\nabla C(4, 2)\| = -\sqrt{596} = -2\sqrt{149} \approx -24.41\text{ \$M per unit step}

4. Machine Learning & Gradient Descent in Automation

Mathematical Routine: Automated factory controllers update setpoints via the iterative gradient descent update: $P_{k+1} = P_k - \alpha \nabla C(P_k)$ where $\alpha > 0$ is the learning rate or actuator gain coefficient.
Physical Implementation: The vector $\langle -14, -20 \rangle$ informs the programmable logic controller (PLC) that raw material feed rate ( $x$ ) and conveyor velocity ( $y$ ) must both be decreased simultaneously, with conveyor speed trimmed slightly more aggressively (ratio of $20:14 = 10:7$ ). Taking steps along this negative gradient vector drives the manufacturing line down the steepest cost decline toward the global optimal operating minimum.