Schedule a Pick Up — Ext. In most cases, this comes down to the way you might ship, store or produce the products you sell. But what about the way you package those products?

Package optimization of your products for supply chain management can actually save your companies both time and money. Most companies leave packaging decisions to designers and marketing experts who understand how to make the product fly off the shelves. Unfortunately, this can add time and costs to the supply chain process. By making small tweaks to the way a product is packaged, they hope to save time and money in producing and transporting products.

Package optimization has many different benefits, from not wasting money on unnecessary materials to creating a more sustainable package.

The materials you use for packaging could be holding your supply chain down.

**Optimization Problem #5 - Max Volume of a Box Made From Square of Material**

Optimizing your packaging means making smarter decisions about the packaging of the product itself and the shipping materials you use to get the product from Point A to Point B. See if you can find alternatives, such as using hot-melt instead of tapethat can save you both time and money. Through package optimization, you should look at each and every material you use in the packaging process. Creating plastic containers or cardboard boxes can have serious consequences on our environment.

With so many companies looking for ways they can go green, package optimization can be one of the best ways to promote sustainable business practices. Companies can look for ways to use less materials and reduce their carbon footprint.

Whether you choose to use a thinner plastic or recycled cardboard, there are many different ways to make your package more sustainable. Not only does this include the size and weight of the product itself, but also the packaging it is in. Optimizing your product package allows you to eliminate the unnecessary excess you could be wasting your money on. When looking for the appropriate way to package your items, you need to consider both complexity and efficiency.

To reduce complexity, you want to have as many similar packages as possible, but to improve efficiency, you want each item to have its best package. For companies with a wide variety of products, this can be difficult to balance.

Finding the optimal level of complexity and efficiency is crucial. When you find the right balance, you can lower your shipping costs, save space and weight with each shipment.

Unnecessary costs hurt everyone. From your company to your customer, spending money on wasteful practices means everyone is getting a bad deal. Get the latest industry advice on logistics, transportation management, best practices, trends, tips and more! All rights reserved. To subscribe to our blog, enter your email address below and stay on top of things. We'll email you with a confirmation of your subscription.

Learn More: Contact Cerasis. Share on facebook. Share on twitter. Share on linkedin. Share on pinterest. Share on reddit. Share on email. Share on print. Megan Ray Nichols. Megan Ray Nichols is a freelance science writer interested in engineering, technology, and other science disciplines.One of the key applications of finding global extrema is in optimizing some quantity, either minimizing or maximizing it.

For example, suppose you wanted to make an open-topped box out of a flat piece of cardboard that is 25" long by 20" wide. You cut a square out of each corner, all the same size, then fold up the flaps to form the box, as illustrated below. Suppose you want to find out how big to make the cut-out squares in order to maximize the volume of the box.

This applet will illustrate the box and how to think about this problem using calculus. The applet shows the flat piece of cardboard in the upper left, and a 3D perspective view of the folded box on the lower left. Move the x slider to adjust the size of the corner cutouts and notice what happens to the box. When x is small, the box is flat and shallow and has little volume.

When x is large, the box it tall and skinny, and also has little volume. Somewhere in between is a box with the maximum amount of volume.

Obviously, the smallest x can be is zero, which corresponds to not cutting out anything at all.

### 4.5: Optimization Problems

What is the largest possible value for xand why? The volume of the box, since it is just a rectangular prism, is length times width times height. The height is just the size of the corner cut out x in this problem. The length and width of the bottom of the box are both smaller than the cardboard because of the cut out corners. The graph of this function is shown in the upper right corner. As you move the x slider, the corresponding point moves along the graph, and the volume for that particular x value is also shown in the upper corner of the graph.

Prior to calculus, you might have solved this problem by graphing it on a calculator and finding the highest point on the graph. But, you can do better by finding the derivative of the volume function, setting this equal to zero and solving to find the critical points, determining which is a local maximum, and lastly comparing the volume at this point with the volume at the endpoints which we don't really need to do in this problem, since the volume is zero at the two ends of the relevant domain for x.

It is easier for most people to find the derivative by first exanding the volume formula into and then finding the derivative, which is Setting this equal to zero and solving e. The first of these is outside the allowable values for xso the solution is the second. In the applet, the derivative is graphed in the lower right graph. Note that the derivative crosses the x axis at this value, and goes from positive to negative, indicating that this critical point is a local maximum. At the bottom of the applet are input fields for the length and width of the cardboard.

Play around with different values to see how it affects the solution and the shape of the volume function. Note that this applet automatically computes the limits for the graphs i. The applet also displays the formula for the volume in terms of xLand W as well as the formula for the derivative, but it computes the derivative without expanding i. Home Contact About Subject Index. This device cannot display Java animations. The above is a substitute static image See About the calculus applets for operating instructions.

The box volume problem The applet shows the flat piece of cardboard in the upper left, and a 3D perspective view of the folded box on the lower left.

Explore At the bottom of the applet are input fields for the length and width of the cardboard.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. The minimum exists, at a place in "the middle.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 6 years, 8 months ago. Active 6 years, 2 months ago. Viewed 7k times. What am I doing wrong? Gamma Function 4, 8 8 gold badges 24 24 silver badges 47 47 bronze badges. Brooklynn Larson Brooklynn Larson 11 1 1 silver badge 5 5 bronze badges. To allow us to provide the most useful answers, please show us what you've tried already.

Also, if a diagram is needed to understand the problem, you'd best include it. As it is, we have no way to even guess. I'm wondering if I just set it up wrong in the first place. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook.

Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. The Overflow Blog. The Overflow How many jobs can be done at home?If the rectangular field has notional sides x and ythen it has area:. The length of fencing required, if x is the letter that was arbitrarily assigned to the side to which the dividing fence runs parallel, is:.

It matters not that the farmer wishes to divide the area into 2 exact smaller areas. Assuming the cost of the fencing is proportional to the length of fencing required, then:. To optimise cost, using the Lagrange Multiplier lambdawith the area constraint :. A farmer wants to fence an area of 6 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle.

How can he do this so as to minimize the cost of the fence? Jun 17, Related questions How do you find two numbers whose difference is and whose product is a maximum? How do you find the dimensions of a rectangle whose area is square meters and whose How do you find the dimensions of the rectangle with largest area that can be inscribed in a Question b1.

The fencing for the north and south How do you find the volume of the largest right circular cone that can be inscribed in a sphere How do you find the dimensions of a rectangular box that has the largest volume and surface area What are the dimensions of a box that will use the minimum amount of materials, if the firm How do you find the dimensions that minimize the amount of cardboard used if a cardboard box See all questions in Solving Optimization Problems.

Impact of this question views around the world. You can reuse this answer Creative Commons License.One common application of calculus is calculating the minimum or maximum value of a function.

For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume. In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter.

The basic idea of the optimization problems that follow is the same. We have a particular quantity that we are interested in maximizing or minimizing.

However, we also have some auxiliary condition that needs to be satisfied. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. However, what if we have some restriction on how much fencing we can use for the perimeter? In this case, we cannot make the garden as large as we like. A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides Figure. What is the maximum area?

Then the area of the garden is. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. We use the same process to look for a maximum or a minimum.

## Optimization Open Box Problem?

At this point, we use the Second Derivative Test SDT determine if this critical point corresponds to a minimum or a maximum. What size square should be cut out of each corner to get a box with the maximum volume? Solution: Step 0: Let x be the side length of the square to be removed from each corner Figure.

Then, the remaining four flaps can be folded up to form an open-top box. The remaining flaps are folded to form an open-top box.

Step 1: We are trying to maximize the volume of a box. Therefore, the volume of the box is. Watch a video about optimizing the volume of a box.One common application of calculus is calculating the minimum or maximum value of a function.

For example, companies often want to minimize production costs or maximize revenue. In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume.

In this section, we show how to set up these types of minimization and maximization problems and solve them by using the tools developed in this chapter. The basic idea of the optimization problems that follow is the same.

We have a particular quantity that we are interested in maximizing or minimizing. However, we also have some auxiliary condition that needs to be satisfied. For example, in Figurewe are interested in maximizing the area of a rectangular garden. Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger. However, what if we have some restriction on how much fencing we can use for the perimeter? In this case, we cannot make the garden as large as we like.

A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides Figure.

Given ft of wire fencing, determine the dimensions that would create a garden of maximum area.

What is the maximum area? Figure 1. Then the area of the garden is. Therefore, the constraint equation is. Thus, we can write the area as. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. We do not know that a function necessarily has a maximum value over an open interval. However, we do know that a continuous function has an absolute maximum and absolute minimum over a closed interval.

Therefore, we consider the following problem:. These extreme values occur either at endpoints or critical points. Figure 2. Determine the maximum area if we want to make the same rectangular garden as in Figurebut we have ft of fencing.One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function.

In the following example, you calculate the maximum volume of a box that has no top and that is to be manufactured from a inch-byinch piece of cardboard by cutting and folding it as shown in the figure. What dimensions produce a box that has the maximum volume? If a manufacturer can sell bigger boxes for more money, and he or she is making a million boxes, you better believe he or she will want the exact answer to this question:.

Express the thing you want maximized, the volume, as a function of the unknown, the height of the box which is the same as the length of the cut. Find the critical numbers of V h in the open interval 0, 15 by setting its derivative equal to zero and solving.

And because this derivative is defined for all input values, there are no additional critical numbers. So, 5 is the only critical number. Thus, the dimensions of the desired box are 5 inches by 20 inches by 20 inches.

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