how frequently should shipments of the deodorant be ordered to minimize the total cost?

Question

Please show equation used and steps.
The manager of an Eckerd drug store expects to sell 600 bottles of a certain deodorant each year. Each bottle costs the manager $4.00 and the ordering fee is $30 per shipment. In addition, it costs .90 cents per year to store each bottle. Assuming that the deodorant sells at a uniform rate throughout the year and that each shipment arrives just as the last bottle from the previous shipment ids sold, how frequently should shipments of the deodorant be ordered to minimize the total cost?
If a graph was made with cost function as a function of x (where x represents the frequency of shipments). How would this be done?
Please show equation used and steps.

Answer 1

Let the number of shipments per year be N so the time between shipments is T = 1/N.

Let Cs be the cost of one shipment, so the total cost per year is N x Cs

Next let's compute Cs:

The cost of the shipment is the purchase cost Cp plust the storage (or inventory) cost of Ci.

The purchase cost is the ordering fee plus the price per bottle P times the number of bottles Bs.

The number of bottles Bs should be the total number of bottles needed per year divided by the number of shipments.

We know the total number needed per year so we can compute Bs in terms of N. We also know the ordering fee and so can get an equation for Cp in terms of N.

Each bottle sits around for an average of T/2. We know the storage cost per bottle per unit time, the number of bottles as a function of N, and T as a function of N so we can compute Ci as a function of N.

With Cp and Ci as functions of N, we can get Cs = Cp + Ci as a function of N, and then total cost = N x Cs as a function of N.

Then it is just a matter of plugging in the numbers for various values of N.

Answer 2

The manager will have to pay 600*4 = $2400 per year for the deodorant no matter how many shipments there are. This is a fixed cost.

Now, suppose he gets only one shipment per year. That shipment will be for 600 bottles. He only has to pay one ordering fee, but he has to pay for storage for 600 bottles, so his variable costs for this case are

1×30 + 600×0.9 = $570

Suppose instead he gets two shipments per year. Then there will be two ordering fees, but he only needs to pay for storage for 300 bottles. For this case, his variable costs are

2×30 + (600/2)×0.9 = $330

For 3 shipments a year, the variable costs are

3×30 + (600/3)×0.9 = $270

Do you see the pattern?

Forgetting for the moment that the number of shipments must be (I'm assuming) a positive integer, write down, based on the above, what the variable costs for x shipments per year would be. Then add the fixed yearly cost to that, and you have the total cost as a function of the number of shipments.

If you differentiate the cost function with respect to x, you will get that the total cost is minimized when x = 3√2. Because this is not an integer, you will need to look at the total cost for the integer just above and just below 3√2; whichever of these results in the lower cost is the choice that minimizes total cost, subject to the condition that an integer number of shipments must be in a year. Your graph will corroborate that looking on either side of the "true" minimum is the correct strategy. (And the cost difference between the "true" minimum and the integer minimum is about 56 cents.)