4.2 What defines an optima?

In ``nice problems'' ...

Go back to the retail display example:

Consider the question: maximize sales of this new product.

Let's assume this time the quadratic relationship between display feet and sales:

$$\fbox{$S = -12.40 + 108.77\,L - 8.49\,L^2$ }$$

 

Table 1: Relationship between sales of the new product and the number of feet dedicated to its display

Display Feet	Sales ($)
1	87.88
2	171.18
3	237.50
4	286.84
5	319.20
6	334.58
7	332.98
8	314.40
9	278.84
10	226.30

Optima (in nice problems) are characterized by the derivatives of the function (objective function) being equal to zero.

Approach

Find the function you want to optimize.

Differentiate it.

Set the derivative equal to zero.

Solve for the quantity of interest.

Use these rules to find the number of shelf feet needed to maximize sales of the new product.

Find the function you want to optimize.

$$\fbox{$S = -12.40 + 108.77\,L - 8.49\,L^2$ }$$

Differentiate it.

$$\fbox{$\frac{dS}{dL} = 108.77 - 2 \times 8.49\,L$ }$$

Set the derivative equal to zero.

$$\fbox{$0 = 108.77 - 2 \times 8.49\,L$ }$$

Solve for the quantity of interest.

$$\fbox{$L = 6.405$ }$$

So about six and a half feet maximizes the sales of the new product.