Math Problem Statement
i have a problem Give h1, h2, w1, w2, w3 >= 0 h1w2 = 2 h2w1 = 4 h2w3 = 4 find minimum of (h1+h2)(w1+w2+w3) Can you solve with linear programming ?
Solution
Yes, we can solve this problem using linear programming. The goal is to minimize the expression , given the constraints:
and
Reformulating the Problem
To solve this problem with linear programming, we'll introduce some new variables and constraints.
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Objective function: We want to minimize . Let's denote:
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Constraints:
These are not linear constraints, but we can transform this problem using the fact that we can express either in terms of , or vice versa. Let's express each in terms of using the given equations:
From the two expressions for :
Thus, , and we can rewrite as:
Now, substitute these into the objective function:
Simplify to:
Converting to a Linear Program
Since the objective function is not linear, we can't directly apply linear programming. Instead, let's find the critical points by minimizing this function using calculus methods.
Calculating the Derivatives
We will find the partial derivatives with respect to and and set them to zero to find critical points.
Let me calculate the partial derivatives and find the minimum.The solution to the partial derivatives yields . This suggests that for the minimum value, the optimal condition occurs when .
Substituting into the Objective Function
Let's substitute into the expressions for , , and the objective function:
Thus,
Conclusion
The minimum value of is 18, which occurs when:
Would you like more details on the steps, or do you have any further questions?
Related Questions:
- What is linear programming and when is it used?
- How do you handle non-linear constraints in optimization problems?
- Can this problem be solved with different methods, like using Lagrange multipliers?
- What are the properties of critical points in multivariable functions?
- How does the choice of variables affect the solution in optimization problems?
Tip: Understanding how to convert non-linear constraints into linear ones can help simplify complex problems.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Calculus
Formulas
Objective function minimization
Theorems
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Suitable Grade Level
Advanced High School / College