## How Do You Find the Optimal Size Production Run?

Batch size problems are easily solved if the total cost function is given to you. Since this is a minimization problem at its heart, taking the derivative to find the critical point and then applying the first of second derivative test does the trick. However, if we need to find the total cost function the problem is more involved. In Section 12.4 I pointed out a strategy in which we can apply our pattern recognition skills and a table to come up with the total cost function. Let’s look at an example:

Problem Every year, Danielle Santos sells 59,520 cases of her Delicious Cookie Mix.  It costs her \$2 per year in electricity to store a case, plus she must pay annual warehouse fees of \$4 per case for the maximum number of cases she will store.  If it costs her \$747 to set up a production run, plus \$6 per case to manufacture a single case, how many production runs should she have each year to minimize her total costs?

Solution Start the table by labeling the columns: I have filled in the table with some examples of number of production runs and size of production run to establish the pattern in the last row. Now let’s fill in the first row. Examine each of the entries in the first row carefully. You can click on the table to view a larger image.

This is mostly the same as in the text. However, the warehouse fees are new. Since the fees are \$4 times the maximum stores in the warehouse at any time, we get 59520 ∙ 4.

Now let’s fill in the second row: I hope you see the pattern emerging. Let’s fill in the last row: The total cost function in ? is the sum of all of the individual costs or After simplifying this function, you can find its minimum by taking the derivative and setting equal it equal to zero. This will give you the critical number that you can then verify using the second derivative.

## How Do You Minimize The Surface Area Of A Container? As the maximization and minimization problems become more complicated, using a table to organize examples of what is being modeled becomes more and more useful. Let’s look at another example of how this might work.

Problem A rectangular container with a square base, an open top, and a volume of 8788 ft3 is to be constructed of sheet metal. Find the dimensions of the tank that has the minimum surface area.

## How Do You Maximize Profit?

As long as you have the profit function, you can find the maximum using the first derivative of the profit function.

Problem 1 The total profit P(x) (in thousands of dollars) from the sale of x units of a drug is

\$latex displaystyle P(x)=ln left( -{{x}^{3}}+3{{x}^{2}}+72x+1 right)\$

for x in [0, 10].

a. How many units should be sold to maximize profit?

b. What is the maximum profit? If you are given the revenue and cost functions, you’ll need to combine those to get the profit function first. Then you can use the first derivative to find the maximum.