Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function. S=20L0.30 0.70 In this formula L represents the units f labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 60,000 tons of steel at minimum cost. min s.t. L, C ≥ 0 = 60,000 (b) Solve the optimization problem you formulated in part (a). What is the optimal solution value (in dollars)? Hint: Use the Multistart option as described in Appendix 8.1. Add lower and upper bound constraints of 0 and 5,000 for both L and C before solving. (Round your answers to three decimal places.) at (L, C) =
Let S represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (L) and the amount of capital used (C) by the following function. S=20L0.30 0.70 In this formula L represents the units f labor input and C the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100. (a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 60,000 tons of steel at minimum cost. min s.t. L, C ≥ 0 = 60,000 (b) Solve the optimization problem you formulated in part (a). What is the optimal solution value (in dollars)? Hint: Use the Multistart option as described in Appendix 8.1. Add lower and upper bound constraints of 0 and 5,000 for both L and C before solving. (Round your answers to three decimal places.) at (L, C) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let \( S \) represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (\( L \)) and the amount of capital used (\( C \)) by the following function:
\[
S = 20L^{0.30}C^{0.70}
\]
In this formula, \( L \) represents the units of labor input and \( C \) the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100.
(a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 60,000 tons of steel at minimum cost.
\[
\begin{align*}
\text{min} & \\
\text{s.t.} & \\
& = 60,000 \\
& L, C \geq 0
\end{align*}
\]
(b) Solve the optimization problem you formulated in part (a). What is the optimal solution value (in dollars)? *Hint: Use the Multistart option as described in Appendix 8.1. Add lower and upper bound constraints of 0 and 5,000 for both \( L \) and \( C \) before solving. (Round your answers to three decimal places.)*
\[
\$ \, \, \text{at} \, \, (L, C) = (\, \, \, \, \, \, )
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeaa8045-e0ae-4f8f-bfb9-81e4f69114f4%2F60c25514-0c57-4e50-b997-268ef6e6a31f%2Funz6bp6_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( S \) represent the amount of steel produced (in tons). Steel production is related to the amount of labor used (\( L \)) and the amount of capital used (\( C \)) by the following function:
\[
S = 20L^{0.30}C^{0.70}
\]
In this formula, \( L \) represents the units of labor input and \( C \) the units of capital input. Each unit of labor costs $50, and each unit of capital costs $100.
(a) Formulate an optimization problem that will determine how much labor and capital are needed in order to produce 60,000 tons of steel at minimum cost.
\[
\begin{align*}
\text{min} & \\
\text{s.t.} & \\
& = 60,000 \\
& L, C \geq 0
\end{align*}
\]
(b) Solve the optimization problem you formulated in part (a). What is the optimal solution value (in dollars)? *Hint: Use the Multistart option as described in Appendix 8.1. Add lower and upper bound constraints of 0 and 5,000 for both \( L \) and \( C \) before solving. (Round your answers to three decimal places.)*
\[
\$ \, \, \text{at} \, \, (L, C) = (\, \, \, \, \, \, )
\]
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