Yes, this problem can be formulated and solved using linear programming. The goal is to minimize the expression $h_1w_1 + h_1w_2 + h_2w_3$ given the constraints:

1. $h_1 \times w_3 = 2$
2. $h_2 \times w_1 = 4$
3. $h_2 \times w_2 = 4$
4. $h_1, h_2, h_3, w_1, w_2, w_3 \geq 0$

Steps to Solve the Problem:

1. Define the Objective Function: $\text{Minimize } Z = h_1w_1 + h_1w_2 + h_2w_3$

2. Constraints: $h_1 \times w_3 = 2 \quad \text{(Constraint 1)}$ $h_2 \times w_1 = 4 \quad \text{(Constraint 2)}$ $h_2 \times w_2 = 4 \quad \text{(Constraint 3)}$ $h_1, h_2, h_3, w_1, w_2, w_3 \geq 0$

3. Substitute the Constraints into the Objective Function: From constraint 1: $w_3 = \frac{2}{h_1}$
   From constraint 2: $w_1 = \frac{4}{h_2}$
   From constraint 3: $w_2 = \frac{4}{h_2}$

   The objective function becomes: $Z = h_1\left(\frac{4}{h_2}\right) + h_1\left(\frac{4}{h_2}\right) + h_2\left(\frac{2}{h_1}\right)$ Simplifying: $Z = \frac{8h_1}{h_2} + \frac{2h_2}{h_1}$

4. Optimization: To find the minimum value, we differentiate $Z$ with respect to $h_1$ and $h_2$, set the derivatives to zero, and solve for the optimal values of $h_1$ and $h_2$.

Solution via Substitution:

The expression $Z = \frac{8h_1}{h_2} + \frac{2h_2}{h_1}$ is not linear, but it can be minimized by recognizing it's a convex function in terms of $h_1$ and $h_2$. We use the symmetry or AM-GM inequality to find the minimum.

Using the AM-GM inequality: $\frac{8h_1}{h_2} + \frac{2h_2}{h_1} \geq 2\sqrt{ \left(\frac{8h_1}{h_2}\right)\left(\frac{2h_2}{h_1}\right)} = 2\sqrt{16} = 8$

The equality holds when: $\frac{8h_1}{h_2} = \frac{2h_2}{h_1}$

Solving this gives: $h_2^2 = 4h_1^2$

So $h_2 = 2h_1$.

Using the constraints, substitute $h_2 = 2h_1$ into $h_2 \times w_1 = 4$: $2h_1 \times w_1 = 4 \Rightarrow w_1 = \frac{2}{h_1}$ Similarly, for $w_2$: $w_2 = \frac{2}{h_1}$

Finally, $w_3 = \frac{2}{h_1}$, and the expression for $Z$ becomes: $Z = \frac{8h_1}{2h_1} + \frac{2(2h_1)}{h_1} = 4 + 4 = 8$

So, the minimum value of $Z$ is 8.

Conclusion:

The minimum value of $h_1w_1 + h_1w_2 + h_2w_3$ is $\boxed{8}$.

Would you like more details or have any questions?

---

Here are five related questions:

1. How does the AM-GM inequality help in optimizing non-linear functions?
2. What other methods can be used to minimize a non-linear function like $\frac{8h_1}{h_2} + \frac{2h_2}{h_1}$?
3. Can this problem be generalized for more complex systems?
4. How do we know the function $Z = \frac{8h_1}{h_2} + \frac{2h_2}{h_1}$ is convex?
5. What role does substitution play in solving optimization problems?

Tip: When dealing with constraints, always check if substituting them into the objective function simplifies the problem before proceeding to more complex methods.