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Optimization Problems

The primary idea in the business world is to maximize profit. However, it's not as simple as trying to sell as many products as possible. Other factors and costs go into a business, such as employee salaries, cost of production, cost of materials, and price of advertisement. Often, the answer to maximizing profit is not simply producing and selling as many products as possible.

Mathematical optimization can help find the answer that maximizes profit subject to the constraints of the real world. Optimization is one of the most interesting real-world applications of Calculus. This article will further define optimization, its other applications, and a method for solving simple optimization problems.

For a focus on business and economic-type optimization problems, see our article on Applications to Business and Economics

Optimization Problems Meaning

Mathematical optimization is the study of maximizing or minimizing a function subject to constraints, essentially finding the most effective and functional solution to a problem.

The constraints in optimization problems represent the limiting factors involved in the maximization/minimization problem. In our example of a business, the constraints would be the cost of labor, production, and advertisement. These constraints must be accounted for in our calculations as they can greatly influence the solution.

You've likely been learning and working through finding a function's extreme values (maximums and minimums). Optimization is a real-world application of finding and interpreting extreme values. Given an equation that models cost, we seek to find its minimum value, thus minimizing cost. Given an equation that models profits, we seek to find its maximum value, thus maximizing profit.

Applications and Types of Optimization Problems

In addition to the business application we've discussed, optimization is crucial in various other fields. Optimization can be as simple as a traveler seeking to minimize transportation time. We can also apply optimization in medicine, engineering, financial markets, rational decision-making and game theory, and packaging shipments.

Optimization is also heavily discussed in computer science. Program optimization, space and time optimization, and software optimization are crucial in writing and developing efficient code and software.

Optimization problems can be quite complex, considering all the constraints involved. Converting real-world problems into mathematical models is one of the greatest challenges. As you progress through higher-level math classes, you'll deal with more complex optimization problems with more constraints to consider. In Calculus, we'll start with smaller-scale problems with fewer constraints. However, the baseline procedure is similar for all optimization problems.

Method for Solving Optimization Problems in Calculus

Before we start working through optimization examples, we'll go through a general step-by-step method for working through these problems. Later on, we'll apply these steps as we work through real examples.

Step 1: Fully understand the problem

Optimization problems tend to pack loads of information into a short problem. The first step to working through an optimization problem is to read the problem carefully, gathering information on the known and unknown quantities and other conditions and constraints. It may be helpful to highlight certain values within the problem.

Step 2: Draw a diagram

To better visualize the problem, it might be helpful to draw a diagram, including labels of known values provided in the problem.

Step 3: Introduce necessary variables

Carefully declare variable names for values that are being maximized or minimized and other unknown quantities.

Step 4: Set up the problem by finding relationships within the problem

Use the known values and your declared variables to set up a function. You must set up your function in terms of these values and variables based on their relation to each other.

Step 5: Find the absolute extrema

There are a couple of methods for finding absolute extrema in optimization problems.

The Closed Interval Method

If the domain of your function is a closed interval, the Closed Interval Method may be a good way to compute absolute extrema.

This method involves finding all critical values within the interval by setting and solving for . Each critical point, as well as the endpoints of the interval, should be plugged in to . The absolute extrema are largest value and smallest value of at the critical points.

First Derivative Test

The First Derivative Test for Absolute Extrema Values states that for a critical point of a function on an interval:

• if for all and for all , then is the absolute maximum value of

• if for all and for all , then is the absolute minimum value of

In other words, if the function goes from increasing to decreasing, it is a maximum. If the function goes from decreasing to increasing, it is a minimum.

Constrained Optimization Problems Examples

Let's work through a common maximization problem.

Example 1

You are tasked with enclosing a rectangular field with a fence. You are given 400 ft of fencing materials. However, there is a barn on one side of the field (thus, fencing is not required on one side of the rectangular field). What dimensions of the field will produce the largest area subject to the 400 ft of fencing materials?

We will solve this problem using the method outlined in the article.

Step 1: Fully understand the problem

Let's draw the important information out from the problem.

We need to fence three sides of a rectangular field such that the area of the field is maximized. However, we only have 400 ft of fencing material to use. Thus, the perimeter of the rectangle must be less than or equal to 400 ft.

Step 2: Draw a diagram

Clearly, you don't have to be an artist to sketch a diagram of the problem!

The diagram of the fencing problem helps us to better visualize the problem - StudySmarter Original

Step 3: Introduce necessary variables

Looking at the diagram above, we've already introduced some variables. We'll let the height of the rectangle be represented by . We'll let the width of the rectangle be represented by .

So, we can calculate area and perimeter as

Step 4: Set up the problem by finding relationships within the problem

The fencing problem wants us to maximize area , subject to the constraint that the perimeter must be greater or less than 400 ft. Intuitively, we know that we should use all 400 ft of fencing to maximize the area.

So, our problem becomes:

Since we seek to maximize the area, we must write the area in terms of the perimeter to achieve one single equation. In this example, we will write the area equation in terms of width, .

First, let's solve for the height, :

Now, plug into the area in terms of the width equation,

In this case, we solved for the variable h to write the area equation in terms of width. This is because solving for h does not yield a fractional answer, so it may be "easier" to work with for most students. It is entirely possible to solve for width and write the area equation in terms of height as well! Give it a try and see if you get the same answer!

Step 5: Find the absolute extrema

Now that we have a single equation containing all of the information from the problem, we want to find the absolute maximum of . We can define an interval for w so we can use the Closed Interval Method.

For starters, we know that w cannot be smaller than 0. If we let , according to our perimeter equation, we have

This tells us that if , the maximum width possible is 200. So our closed interval for is .

To apply the Closed Interval Method:

First, find the extrema of by taking the derivative and setting it equal to 0.

Second, plug in the critical values and identify the largest area.

So, the largest value of occurs at where .

We can confirm this using the First Derivative Test.

Graphing ...

We can apply the First Derivative Test to the graph of the derivative - StudySmarter Original

clearly only equals 0 at one point, . For all , is positive (above the x-axis). For all , is negative (below the x-axis). So, by the First Derivative Test, is the absolute maximum of .

Let's plug in to our perimeter equation to find out what h should be.

Therefore, to maximize the area enclosed by the fence subject to our material constraints, we should use a rectangle with a width of 100 ft and a height of 200 ft.

Example 2

Now, let's try a minimization problem.

You are tasked with building a can that holds 1 liter of liquid. To maximize profit, you must build the can such that the material used to build it is minimized. What is the minimum surface area of the can required?

Again, we will solve this problem using the method outlined in the article.

Step 1: Fully understand the problem

Let's draw the important information out from the problem.

We need to build a can that holds 1 liter of liquid while minimizing the material used to build it. Essentially, this means we need to minimize the can's surface area.

Step 2: Draw a diagram

With this diagram, we can better understand what the problem is asking us to do.

The diagram of the can problem helps us to better visualize the problem - StudySmarter

Step 3: Introduce necessary variables

Looking at the diagram above, we've already introduced some variables. We'll let the radius of the cylindrical can be represented by . We'll let the height of the cylinder be represented by . So, the volume of the cylinder is and the surface area of the cylinder is .

Step 4: Set up the problem by finding relationships within the problem

The can problem wants us to minimize the surface area subject to the constraint that the can must hold at least 1 liter. Intuitively, we know that to minimize surface area, we should build a can that holds 1 liter of liquid. However, since we are looking for a length measurement for and , we should convert liters into cubic centimeters. Thus, we should build a can that holds 1,000 cm3 of liquid.

So, our problem becomes:

Since we seek to minimize the surface area, we must write the area in terms of the volume to achieve one single equation.

First, let's solve for :

Now, plug into the area equation:

Step 5: Find the absolute extrema

Now that we have a single equation containing all the information from the problem, we want to find the absolute minimum of .

We know that . However, we do not have an upper bound for .

First, we'll find the extrema of by taking the derivative and setting it equal to 0.

Graphing the derivative:

We can apply the first derivative test to the graph of the derivative - StudySmarter Original

We can see at one point. We can confirm that the point is an absolute minimum for by applying the First Derivative Test. Looking at the graph, For all , is negative (below the x-axis). For all , is positive (above the x-axis). So, by the First Derivative Test, is the absolute maximum of .

Let's plug in to our volume equation to find out with should be.

So, to build a can that holds at least 1 liter, the minimum surface area required is

Optimization Problems - Key takeaways

• Mathematical optimization is the study of maximizing or minimizing a function subject to constraints, essentially finding the most effective and functional solution to a problem
• Optimization is a real-world application of finding and interpreting extreme values
• Solving optimization problems can seem daunting at first, but following a step-by-step procedure helps:
• Step 1: Fully understand the problem
• Step 2: Draw a diagram
• Step 3: Introduce necessary variables
• Step 4: Set up the problem by finding relationships within the problem
• Step 5: Find the absolute extrema
• To find the absolute extrema, use either the Closed Interval Method or the First Derivative Test

Optimization problems seek to maximize or minimize a function subject to constraints, essentially finding the most effective and functional solution to the problem.

A real-world example of an optimization problem is the idea of maximizing profits and minimizing cost within a business.

To solve an optimization problem, you must set up a function in terms of known values and variables. Then, find the extrema of the function by taking the derivative and evaluating.

Optimization problems can be seen in a variety of fields including business, medicine, engineering, financial markets, rational decision making and game theory, packaging shipments, and computer science.

Optimization problems involve maximizing or minimizing certain quantities. To determine if a problem is an optimization problem, carefully read the problem and look for language that suggests maximizing or minimizing.

Final Optimization Problems Quiz

Question

What is mathematical optimization?

Mathematical optimization is the study of maximizing or minimizing a function subject to constraints, essentially finding the most effective and functional solution to a problem

Show question

Question

Think of an example of an optimization problem. What quantities are being maximized or minimized? What constraints may apply?

One example of an optimization problem is the desire to maximize profits in the business world. However, some constraints may apply such as the cost of labor, materials to build a product, the cost of advertisements...

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Question

What mathematical concept in Calculus does optimization rely on?

Finding absolute extrema

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Question

Absolute extrema can be found using...

The Closed Interval Method or the First Derivative Test

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Question

State the result of the First Derivative Test.

The First Derivative Test states that for a critical point c of a function on an interval:

• if f'(x) > 0 for all x < c and f'(x) < 0 for all x > c, then f(c) is the absolute maximum value of f(x)
• if f'(x) < 0 for all x < c and f'(x) > 0 for all x < c, then f(c) is the absolute minimum value of f(x)

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Question

How is the Closed Interval Method applied in optimization problems?

• Define a domain
• Take the derivative of the function and set it equal to 0 to find local extrema
• Plug extrema found by setting the derivative equal to 0 and the end points in to the function
• The point that produces the largest value is the absolute maximum on the interval while the point that produces the smallest value is the absolute minimum on the interval

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Question

Let's say we're maximizing the area of a garden. Which equations might be important for this problem?

Area equation for the shape of the garden

Perimeter equation for the shape of the garden

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Question

Optimization problems typically ask you to ___________ or __________ some quantity.

Minimize or maximize

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Question

Think of some phrases that might signify the problem is asking you to find a maximum.

• "Find the largest..."
• "What dimensions will give the greatest..."

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Question

Think of some phrases that might signify the problem is asking you to find a minimum.

• "Find the minimum..."
• "What is the smallest..."

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Question

Just as you would do when solving for an extreme value, to solve an optimization problem, set the _____ derivative of your equation equal to _________.

first, 0

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Question

It turns out, there are some business and economic problems that you can model and solve as optimization problems in calculus. These types of problems typically involve either:

Maximizing revenue.

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Question

Solving business/economic optimization problems almost __ requires you to find the marginal cost or marginal revenue, and occasionally both.

Always.

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Question

What is the definition of optimization?

Optimization is finding the maximum or minimum values of a given quantity, or finding when the maximums or minimums occur.

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Question

So, what quantities do you optimize in a business or economic problem?

In these cases, you are usually tasked with either:

• maximizing revenue,

• maximizing profits, or

• minimizing costs.

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Question

True or False? The first couple of steps in any optimization problem (business, economic, or otherwise) are always the same.

True.

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Question

What are the basic steps to solve optimization problems?

1. Identify the quantity you need to optimize.

2. Identify the feasible domain.

3. Find critical points.

4. Choose from $$3$$ possible optimization methods:

• The Closed Interval Method

• The First Derivative Test

• The Second Derivative Test

5. Once you have completed your optimization, there are $$2$$ questions you must ask yourself:

1. Does my answer make sense?

2. Did I answer the correct question?

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Question

The cost function is defined as:

The cost function, $$C(x)$$, is the cost of producing $$x$$ units of a product.

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Question

The marginal cost function is defined as:

The marginal cost function, $$C'(x)$$, is the rate of change of the cost with respect to the units of a product.

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Question

The demand function (also called the price function) is defined as:

The demand function, $$p(x)$$, is the price per unit that a company can charge if it sells $$x$$ units of a product.

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Question

The revenue function is defined as:

The revenue function, $$R(x)$$, is calculated by multiplying the number of units sold by the price at which they were sold,

$R(x) = x \cdot p(x).$

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Question

The marginal revenue function is defined as:

The marginal revenue function, $$R'(x)$$, is the rate of change of revenue with respect to the number of units sold,

$R'(x) = p(x) + x \cdot p'(x).$

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Question

The profit function is defined as:

The profit function, $$P(x)$$, is calculated by subtracting the cost from the revenue,

$P(x) = R(x) - C(x).$

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Question

The marginal profit function is defined as:

The marginal profit function, $$P'(x)$$, is the rate of change of profit with respect to number of units sold,

$P'(x) = R'(x) - C'(x).$

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Question

What quantity do you need to optimize in the following example?

Given the revenue function (in dollars):

$R(x) = 900x^{\frac{2}{3}} - 60x$

Find the number of units, $$x$$, that maximizes the revenue. What is the maximum revenue?

Revenue.

• Optimize by maximizing it.

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Question

What quantity do you need to optimize in the following example?

Given the cost function (in dollars):

$C(x) = 0.004x^{3} +20x + 1000$

What is the number of units, $$x$$, that minimizes the average cost per unit, $$\bar{c}(x)$$?

The average cost per unit.

• Optimize by minimizing it.

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