Chapter 11
Solved Problems
1
Exhibit 11.2 Example Linear and Nonlinear Trend Patterns
2
Exhibit 11.3 Seasonal Pattern of Home Natural Gas Usage
3
Exhibit Extra Trend and Business Cycle Characteristics
(each data point is 1 year apart)
4
Exhibit 11.4
Call Center Volume
Example of a time
series with trend
and seasonal
components:
5
Exhibit 11.5 Chart of Call Volume
6
Basic Concepts in Forecasting
•
Forecast error is the difference between the observed value of the time
series and the forecast, or At – Ft .
Mean Square Error (MSE)
MSE =
Σ(At – Ft )2
T
[11.1]
Mean Absolute Deviation Error (MAD)
MAD =
‫׀‬At – Ft ‫׀‬
[11.2]
T
Mean Absolute Percentage Error (MAPE)
MAPE =
Σ‫(׀‬At – Ft )/At ‫ ׀‬X 100
[11.3]
T
7
Exhibit 11.6 Forecast Error of Example Time Series Data
8
Solved Problem
Develop three-period and four-period moving-average forecasts
and single exponential smoothing forecasts with a = 0.5. Compute
the MAD, MAPE, and MSE for each. Which method provides a
better forecast?
Period
Demand
Period
Demand
1
86
7
91
2
93
8
93
3
88
9
96
4
89
10
97
5
92
11
93
6
94
12
95
9
Solved Problem
98
96
94
92
90
Moving
Average
Forecasts
88
86
84
82
80
1
2
3
4
5
6
7
8
9
10
11
Period
Based on these error metrics (MAD, MSE, MAPE), the 3-month moving average is
the best method among the three.
10
12
Exhibit 11.7 Summary of 3-Month Moving-Average Forecasts
11
Exhibit 11.8 Milk-Sales Forecast Error Analysis
12
Single Exponential Smoothing
• Single Exponential Smoothing (SES) is a
forecasting technique that uses a weighted
average of past time-series values to forecast the
value of the time series in the next period.
Ft+1 = At + (1 – )Ft
= Ft +  (At – Ft)
[11.5]
13
Exhibit 11.9 Summary of Single Exponential Smoothing Milk-Sales
Forecasts with α = 0.2
14
Exhibit 11.10 Graph of Single Exponential Smoothing Milk-Sales Forecasts
with α = 0.2
15
Regression as a Forecasting Approach
• Regression analysis is a method for building a
statistical model that defines a relationship between
a single dependent variable and one or more
independent variables, all of which are numerical.
Yt = a + bt


(11.7)
Simple linear regression finds the best values of a and b using the
method of least squares.
Excel provides a very simple tool to find the best-fitting regression model
for a time series by selecting the Add Trendline option from the Chart
menu.
16
Exhibit 11.11 Factory Energy Costs
17
Exhibit 11.12
Format Trendline
Dialog Box
18
Exhibit 11.13 Least-Squares Regression Model for Energy Cost Forecasting
19
Exhibit 11.14 Gasoline Sales Data
20
Exhibit 11.15 Chart of Sales versus Time
21
Exhibit 11.16 Multiple Regression Results
22