beginalign textMaximize & fracP_totalA_total textsubject to & P_i leq S_i forall i & A_total leq A_max & P_total geq P_rated endalign

To maximize the objective function $\frac{P_{total}}{A_{total}}$, we can use linear programming techniques. Here's the setup for the linear programming problem:

Let $x_i$ represent the power output of device $i$, and let $P_i$ and $A_i$ be the power and area constraints for device $i$, respectively. Also, let $P_{total}$ and $A_{total}$ represent the total power output and total area used, respectively.

The linear programming problem can be formulated as follows:

Objective function: Maximize $\frac{P_{total}}{A_{total}}$

Subject to: $P_{total} = \sum_{i}^{} x_i$ $A_{total} = \sum_{i}^{} A_i$

Constraints: $x_i \leq S_i, \forall i$ $A_{total} \leq A_{max}$ $P_{total} \geq P_{rated}$

This formulation ensures that the power output of each device is within its specified limits, the total area used is within the maximum allowed area, and the total power output is at least the rated power.

By solving this linear programming problem, you can find the optimal values of $x_i$ that maximize the objective function while satisfying all the constraints.