Chapter 1, Problem 4PEA

Example 1.2 in the chapter described a scenario for new product sales that can be characterized by a formula called a Gompertz curve: ${\text{S = ae}}^{{\text{be}}^{\text{ct}}}$ Develop a spreadsheet for calculating sales using this formula for t = 0 to 160 in increments of 10 when a = 15,000, b = -8, and c = -0.05.

EXAMPLE 1.2 Three Forms of a Model

Models are usually developed from theory or observation and establish relationships between actions that decision makers might take and results that they might expect, thereby allowing the decision makers to evaluate scenarios or to predict what might happen. For example, the sales of a new product, such as a first-generation iPad, Android phone, or 3-D television, often follow a common pattern. We might represent this in one of the three following ways:

1. 1. A simple verbal description of sales might be: The rate of sales starts small as early adopters begin to evaluate a new product and then begins to grow at an increasing rate over time as positive customer feedback spreads. Eventually, the market begins to become saturated, and the rate of sales begins to decrease.
2. 2. A sketch of sales as an S-shaped curve over time, as shown in Figure 1.2, is a visual model that conveys this phenomenon.
3. 3. Finally, analysts might identify a mathematical model that characterizes this curve. Several different mathematical functions do this; one is called a Gompertz curve and has the formula: ${\text{S = ae}}^{{\text{be}}^{\text{ct}}}$, where S = sales, t = time, e is the base of natural logarithms, and a, b, and c are constants. Of course, you would not be expected to know this; that’s what analytics professionals do.