### Example 16.1

```Example 16.1
Forecasting Sales at Best Chips
Exponential Growth.xlsx
• The Best Chips Company produces and sells
potato chips throughout the country.
• Its sales have been growing steadily over the past
10 years, as shown on the next slide and in this
file.
• The company wants to predict its sales for the next
couple of years, assuming that the upward trend it
has observed in the past 10 years will continue in
the future.
• How should the company proceed?
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution
• We begin by using the Chart Wizard to create the
X-Y plot of Sales versus Year shown on the next
slide.
• Sales are clearly increasing through time, but it is
not absolutely clear whether they are increasing at
a constant rate, which would favor a linear
trendline, or at an increasing rate, which would
favor an exponential trendline.
• Therefore, we try fitting both of these.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
• To superimpose a linear trendline on any
scatterplot, select the chart and then More
Trendline Options from the
Trendline dropdown on the
Chart Tools Layout ribbon.
• This brings up the dialog box
shown here.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
• You can select any of six types of trendlines.
• For now, select the default Linear option. Also,
click on the Options tab and check the Display
equation box.
• The result is shown on the next slide.
• This figure shows the best-fitting straight line to the
points, and it indicates that the equation of this
straight line is y = 92,091x +1,168,200.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
• Here, y corresponds to sales and x corresponds to
year.
• The most important part of this equation is the
coefficient of x, 92,091. It implies that sales are
increasing by \$92,091 per year—if we believe that
the linear trendline provides a good fit.
• To obtain an exponential trendline, we go through
the same procedure except that we select the
Exponential option in the dialog box.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
• The resulting curve appears on the next slide.
• The equation for the curve is y 1.2278e0.0541x.
• The most important part of this equation is the
coefficient in the exponent, 0.0541.
• It implies that sales are increasing by
approximately 5.4% per year.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
• Which of these trendlines provides the better fit?
• We can proceed in two ways.
– First, we can “eyeball” it. Looking at the superimposed
trendlines, it appears that the exponential fit is slightly
better.
– The typical way to measure fits to a trendline through
time is to calculate the historical predictions from each
curve and the corresponding absolute percentage errors
(APEs).
• We find the predictions by plugging the year
indexes (1 to 10) into the trendline equations.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
• We then calculate the APE for each year from the
following equation.
• A measure of goodness-of-fit is then the average
of these APE values, denoted by MAPE (mean
absolute percentage error).
• This measure is quite intuitive.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
•
•
All of this is implemented and shown on the next
slide.
To create the predictions, APEs, and MAPEs,
proceed as follows:
1. Predictions. Calculate the predictions from the linear
trendline by entering the formula
=1168200+92091*A53 in cell D3 and copying it down
to cell D14. Similarly, calculate the predictions from the
exponential trendline by entering the formula
=1227762*EXP(0.0541*A3) in cell E3 and copying it
down to cell E14.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
2. APE values. Calculate all of the APE values at once
by entering the formula =ABS(\$B3-D53/\$B3in cell F3
and copying it to the range F3:G12.
3. MAPE values. Calculate the MAPE for each trendline
by entering the formula =AVERAGE(F3:F12) in cell
F16 and copying it to cell G16.
•
The MAPE values confirm that the exponential
trendline is slightly better than the linear trendline.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage
2009
Thomson/South-Western
Solution -- continued
• Exponential trendlines are often used in predicting
sales and other economic quantities.
• However, we urge caution with such predictions. It
is difficult for any company to sustain a given
percentage increase year after year.
Winston/Albright
Practical Management Science, Revised 3e
South-Western/Cengage