Crawford Labs has increased its profitability dramatically by using models to allocate its sales force among the company's major drugs. To guide that effort, a nonlinear sales-response curve is estimated for each drug.
A sales-response curve relates the sale of a product to the effort expended to sell it, which in this case is measured by the number of calls made by the sales force. For low levels of effort, sales rise rapidly with increased effort, but eventually the sales response levels out. Two elements are needed to develop a sales-response curve for a particular product:
(1) Some data or managerial judgments on the sales at various levels of effort, and
(2) A family of S-shaped curves to fit the data. A suitable family of curves for Crawford is given by the function
Effort _______________ Estimated Sales
0% of current ............. 50% of current
25% ................................. 53%
50% ................................. 55%
75% ................................. 75%
100% .............................. 100%
125% ............................. 120%
150% ............................ 127%
175% ............................ 132%
200% ............................. 135%
a. Fit a sales-response curve to the Crawford data, using a least-squares criterion. (Use percentages relative to the base case for the Effort and Sales measures.) What are the best values of the parameters a, b, c, and d?
b. What sales does the model in (a) predict for an effort of 115 percent?