i have a problem
Give h1, h2, h3, w1, w2, w3 >= 0
h1*w3 = 2
h2*w1 = 4
h2*w2 = 4
find minimum of h1*w1 + h1*w2 + h2*w3
Can you solve with linear programming ?

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?

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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.

Learn how to solve an optimization problem using linear programming techniques. This problem involves minimizing a linear expression subject to constraints using methods such as substitution and the AM-GM inequality. Suitable for advanced mathematics students.

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