OPTIMIZATION: Minimizing Packaging Costs

In summary, the box can be constructed at minimum cost if the material costs $0.30/square foot, $0.10/square foot, and $0.20/square foot.
  • #1
EmilyGia47
2
0

Homework Statement


A rectangular box is to have a square base and a volume of 20 ft3. If the material for the base costs $0.30/square foot, the material for the sides costs $0.10/square foot, and the material for the top costs $0.20/square foot, determine the dimensions of the box that can be constructed at minimum cost.


Homework Equations


Optimization.


The Attempt at a Solution


I know basic optimization... I will take the derivative to find a candidate for a relative minimum, and then verify it... But I REALLY need help trying to find the formula that I will use. I cannot seem to derive it from the information given. Any help is appreciated.
 
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  • #2
Start out by labeling the variables that you need to determine the shape of the box. Then figure out formulas for the areas of the top and sides in terms of those variables. Then use that to write down a cost function to minimize. If you still have questions after you've shown us that much, I'm sure someone will help. You've got to at least START the problem.
 
  • #3
My attempt...

The volume of the box would be found by x2y.

So the cost function should be...

C(x) = .30x * .10y * .20z

or

C(x) = .30x2 * .10y * .20z

Then I would optimize... But I am having trouble determining if I have derived the correct equation...

Can anyone help?
 
  • #4
You don't have a correct cost function. Ok, so x is the length of the base and y is the height. I'm just guessing about that since you didn't say so. I have no idea what z is supposed to be. What are i) the area of the base and top and ii) what is the area of the sides? Multiply them by cost per unit area and add them up. Then you will have a correct cost function.
 

FAQ: OPTIMIZATION: Minimizing Packaging Costs

What is optimization and why is it important in minimizing packaging costs?

Optimization is the process of finding the best possible solution to a problem. In the context of minimizing packaging costs, it involves finding the most efficient and cost-effective way to package products. This is important because packaging costs can significantly impact a company's overall expenses and profitability.

How can optimization help reduce packaging costs?

Optimization can help reduce packaging costs by identifying areas where packaging materials or processes can be improved. This can include using more cost-effective materials, reducing excess packaging, or streamlining the packaging process to reduce labor costs. By finding the most efficient packaging solution, optimization can help minimize overall packaging costs.

What factors should be considered in optimizing packaging costs?

When optimizing packaging costs, several factors should be considered, including the type of product being packaged, the desired level of protection for the product, the size and weight of the product, and the target market or customer preferences. Other factors may include transportation requirements, regulatory compliance, and sustainability goals.

What techniques or tools can be used for optimization in packaging?

There are several techniques and tools that can be used for optimization in packaging. These include software programs that use mathematical algorithms to analyze packaging options and determine the most cost-effective solution. Other techniques may involve conducting packaging audits to identify areas for improvement, using lean manufacturing principles to streamline the packaging process, and exploring alternative packaging materials or designs.

How can optimization in packaging benefit both the company and the environment?

Optimization in packaging can benefit both the company and the environment by reducing costs and minimizing waste. By finding the most efficient packaging solutions, companies can save money on materials, labor, and transportation costs. This can also lead to a reduction in the amount of packaging waste generated. Additionally, optimizing packaging can help companies meet sustainability goals and reduce their environmental impact.

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