Lagrange multiplier example, part 1

Khan Academy

15 Nov 201607:50

EducationalLearning

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TLDRThis instructional video explores the application of the Lagrange multiplier technique to maximize revenue in a widget production company. It discusses labor and steel costs, introduces a revenue model dependent on labor hours and steel tons, and sets a budget constraint. The video guides through visualizing the problem in an 'hs' plane, explaining how to find the revenue contour tangent to the constraint curve. It delves into calculating gradients of the revenue and constraint functions, setting them proportional to each other with a common multiplier, and promises to solve the equations in a follow-up video.

Takeaways

🏭 The company in question produces widgets and has main costs in labor and steel.
💰 Labor costs are $20 per hour and steel costs are $2,000 per ton.
📈 The revenue model is a function of labor hours and steel tons, with the formula being 100 * (hours of labor)^(2/3) * (tons of steel)^(1/3).
💼 The company has a budget constraint of $20,000 for labor and steel expenses.
🔍 The goal is to maximize revenue within the given budget using the revenue model.
📚 The Lagrange multiplier technique is introduced as the method to solve this optimization problem with a constraint.
📉 The constraint is formulated as 20 * hours of labor + 2000 * tons of steel = $20,000.
📝 The function g(h, s) represents the budget constraint, where h is hours of labor and s is tons of steel.
📊 The problem is visualized in the hs-plane, with the constraint forming a line and revenue contours representing different revenue levels.
🔍 The solution involves finding the point where the revenue contour is tangent to the constraint line, indicating maximum revenue under the budget.
✂️ The gradient of the revenue function r is calculated, involving partial derivatives with respect to hours and tons of steel.
🔄 The gradient of the constraint function g is also calculated, which is simpler due to its linear nature.
🔗 The gradients of r and g are set proportional to each other, introducing the Lagrange multiplier λ to represent this relationship.
🔑 Two equations are derived from setting the gradients proportional, one for each component of the gradients, involving λ.

Q & A

What is the main subject of the video script?
-The main subject of the video script is the application of the Lagrange multiplier technique to maximize revenue in a hypothetical widget production company, given labor and steel costs and a budget constraint.
What are the two main costs associated with the production of widgets in the script?
-The two main costs associated with the production of widgets are labor costs at $20 per hour and steel costs at $2,000 per ton.
How is the revenue function defined in the script?
-The revenue function is defined as 100 times the hours of labor to the power of 2/3 multiplied by the tons of steel to the power of 1/3.
What is the budget constraint for the company in the script?
-The budget constraint for the company is $20,000, which is the maximum amount they are willing to spend on labor and steel.
What technique is suggested for solving the problem of maximizing revenue with a given constraint?
-The Lagrange multiplier technique is suggested for solving the problem of maximizing revenue with a given constraint.
How is the constraint function represented in the script?
-The constraint function is represented as the sum of labor costs (20 times the number of hours) and steel costs (2,000 times the tons of steel), which must equal the budget of $20,000.
What is the role of the Lagrange multiplier in this context?
-The Lagrange multiplier is a proportionality constant that ensures the gradient of the revenue function is proportional to the gradient of the constraint function at the point of tangency.
What does the gradient represent in the context of this script?
-In the context of this script, the gradient represents the direction of the steepest ascent of the revenue function and is used to find the point of tangency with the constraint function.
What is the first step in applying the Lagrange multiplier technique as described in the script?
-The first step is to compute the gradient of the revenue function with respect to both labor hours and tons of steel.
How are the gradients of the revenue and constraint functions related in the Lagrange multiplier technique?
-The gradients of the revenue and constraint functions are related by being proportional to each other, with the proportionality constant being the Lagrange multiplier.
What is the significance of finding the point of tangency between the revenue contour and the constraint line?
-The point of tangency is significant because it represents the optimal allocation of resources (labor and steel) that maximizes revenue while adhering to the budget constraint.

Outlines

00:00

📊 Introduction to the Revenue Maximization Problem

The video script introduces a business scenario where a company produces widgets with labor and steel as the main costs. The labor costs $20 per hour, and steel costs $2,000 per ton. The revenue model is given as a function of labor hours and steel tons, aiming to maximize revenue within a $20,000 budget. The script sets up the problem for solving using the Lagrange multiplier technique, which is suitable for optimization problems with constraints. The constraint is formulated as the sum of labor and steel costs not exceeding the budget, and the function g(h, s) is introduced to represent this constraint. The script also discusses the visualization of the problem in the hs-plane, where the constraint forms a line and the revenue function forms contours, with the goal of finding the point of tangency between the highest revenue contour and the constraint line.

05:00

🔍 Calculating Gradients and Setting Up the Lagrange Multiplier

The script continues by explaining the process of finding the optimal solution using the Lagrange multiplier method. It involves calculating the gradient of the revenue function R with respect to labor hours (h) and steel tons (s), resulting in two partial derivatives that represent the rate of change of revenue for each input. The gradient of the constraint function g is also calculated, which is simpler due to the linear nature of g. The gradients of R and g are then set proportional to each other, introducing the Lagrange multiplier (λ) as the constant of proportionality. Two equations are derived from setting the gradients equal, which will be used to solve for the optimal values of h, s, and λ in the next part of the explanation. The script concludes with the setup of these equations, leaving the detailed solution for the subsequent video.

Mindmap

Keywords

💡Lagrange Multiplier

The Lagrange Multiplier is a mathematical method used for finding the local maxima and minima of a function subject to equality constraints. In the context of the video, it is used to maximize the revenue function within the given budget constraint. The script describes how the gradient of the revenue function becomes proportional to the gradient of the constraint function, with the proportionality constant being the Lagrange Multiplier.

💡Widgets

In the script, 'widgets' are used as a placeholder for a product that the company produces. They represent any small trinket or item that consumers might enjoy buying. The concept is used to illustrate the economic principles being discussed without focusing on a specific product, allowing the audience to understand the broader application of the revenue model.

💡Labor Costs

Labor costs refer to the expenses incurred by a company for the workforce involved in the production process. In the video script, labor costs are specified as $20 per hour, which is a direct expense for the company making widgets. These costs are a critical part of the budget constraint in the optimization problem presented.

💡Steel Costs

Steel costs are the expenses associated with the raw material used in production, which in this case is steel. The script mentions a cost of $2,000 for every ton of steel used in the production of widgets. These costs, along with labor costs, contribute to the total budget that the company has to maximize its revenue.

💡Revenue Model

A revenue model is a representation of how a company expects to generate income from its business activities. The script describes a specific revenue model as a function of labor hours and steel tons, which is used to predict the earnings from producing widgets. The model is essential for understanding how to maximize revenue within the given constraints.

💡Budget Constraint

The budget constraint is a limit on the amount of money that can be spent on certain activities, in this case, production costs. The script outlines a budget of $20,000 that the company must adhere to while trying to maximize revenue. This constraint is central to the optimization problem being discussed.

💡Gradient

In the context of the video, the gradient refers to the vector of partial derivatives of a function, which points in the direction of the greatest rate of increase of the function. The script explains that the gradient of the revenue function must be proportional to the gradient of the constraint function at the point of tangency to find the maximum revenue.

💡Partial Derivative

A partial derivative is the derivative of a function with respect to one of its variables, while the other variables are held constant. The script calculates the partial derivatives of the revenue function with respect to labor hours and steel tons, which are necessary to find the gradient of the function.

💡Contour

In the script, a contour represents a level set of the revenue function, indicating all points where the revenue is the same. The contours are used to visualize the revenue levels and to find the point where the revenue contour is tangent to the budget constraint, which corresponds to the maximum revenue within the budget.

💡Tangency

Tangency in the script refers to the point where a contour of the revenue function just touches the budget constraint line without crossing it. This point is significant because it represents the maximum revenue that can be achieved within the budget, as explained using the Lagrange multiplier technique.

💡Optimization Problem

An optimization problem seeks to find the best solution within a set of possible solutions, often subject to constraints. In the video, the optimization problem is to maximize the revenue from producing widgets given the labor and steel costs and a fixed budget. The script uses the Lagrange multiplier technique to solve this problem.

Highlights

Introduction of a company producing widgets with labor and steel as main costs.

Labor costs are $20 per hour and steel costs are $2,000 per ton.

Revenue model is presented as a function of labor hours and tons of steel.

Revenue function is defined as 100 times labor hours to the power of 2/3 multiplied by steel tons to the power of 1/3.

The goal is to maximize revenue with a budget constraint of $20,000.

Introduction of the Lagrange multiplier technique for optimization problems with constraints.

Explanation of the budget constraint as a linear function of labor hours and steel tons.

The concept of maximizing revenue by hitting the budget constraint exactly.

Naming the budget constraint function as 'g of h, s'.

Visualizing the problem in the 'h s' plane with constraints and revenue contours.

Desire to find the revenue contour tangent to the constraint curve.

Explanation of the gradient of the revenue function 'r' and its components.

Calculation of the gradient of the constraint function 'g'.

Setting the gradients of 'r' and 'g' proportional to each other with the Lagrange multiplier.

Derivation of the equations using the gradients and the Lagrange multiplier.

Simplification of the equations to solve for the optimal labor hours and steel tons.

Teaser for the next video to work through the solution details.

Transcripts

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Takeaways

Q & A

What is the main subject of the video script?

What are the two main costs associated with the production of widgets in the script?

How is the revenue function defined in the script?

What is the budget constraint for the company in the script?

What technique is suggested for solving the problem of maximizing revenue with a given constraint?

How is the constraint function represented in the script?

What is the role of the Lagrange multiplier in this context?

What does the gradient represent in the context of this script?

What is the first step in applying the Lagrange multiplier technique as described in the script?

How are the gradients of the revenue and constraint functions related in the Lagrange multiplier technique?

What is the significance of finding the point of tangency between the revenue contour and the constraint line?