2 production and operation problems. MSD 340
scotla
chapter_3.pptxCHAPTER 3 Forecasting
Outline
Definition
Forecast Accuracy
Types of Forecasts
Judgmental
Time series
Associative models
2
Forecast
A statement about the future value of a variable of
interest, such as demand. Equivalently, prediction
about the future.
- Not necessarily numerical, e.g. weather forecasts
3
Cautions
Assume a causal system
Future resembles the past
Forecasts rarely perfect because of randomness
Forecasts more accurate for groups vs. individuals.
Forecasting errors among items in a group usually have a canceling effect.
Extremes in a group cancel each other
Forecast accuracy decreases as time horizon for forecasts increases.
Ex. weather forecast
4
| Accounting | Cost/profit estimates |
| Finance | Cash flow and funding |
| Human Resources | Hiring/recruiting/training |
| Marketing | Market trend, pricing |
| MIS | IT/IS systems, computers |
| Operations | Schedules, workloads |
| Product/service design | New products or services |
Uses of Forecast
5
Steps in the Forecasting Process
Step 1. Determine purpose of forecast
Step 2. Establish a time horizon
Step 3. Select a forecasting technique
Step 4. Obtain, clean, analyze appropriate data
Step 5. Make the forecast
Step 6. Monitor the forecast
“The forecast”
6
Forecast Accuracy
7
Forecast Error
Error:
Difference between the actual value and the value that was predicted for a given period.
8
Errort = Actualt - Forecastt
Measures of Forecast Accuracy
Mean Absolute Deviation (MAD)
Mean Squared Error (MSE)
Mean Absolute Percent Error (MAPE)
MAD =
Actualt
Forecastt
n
2
MSE
=
Actualt
Forecastt)
-
1
n
(
MAPE =
Actualt
Forecastt
n
Actualt
× 100
9
Example 1
Period
1
2
3
4
5
6
7
8
Forecast
215
216
215
214
211
214
217
216
(A-F)
|A-F|
(A-F) 2
(|A-F|/Actual)×100
Actual
217
213
216
210
213
219
216
212
10
Example 1
Period
1
2
3
4
5
6
7
8
Forecast
215
216
215
214
211
214
217
216
(A-F)
2
-3
1
-4
2
5
-1
-4
-2
|A-F|
2
3
1
4
2
5
1
4
22
(A-F) 2
4
9
1
16
4
25
1
16
76
(|A-F|/Actual)×100
0.92
1.41
0.46
1.90
0.94
2.28
0.46
1.89
10.26
Actual
217
213
216
210
213
219
216
212
MAD=
׀A-F׀/ n = 22 / 8 = 2.75
MSE =
(A-F)2/ (n-1) = 76 / 7 = 10.86
MAPE =
10.26/8 = 1.28
11
Types of Forecasts
Judgmental - uses subjective inputs
Executive opinions
Sales force opinions
Consumer surveys
Time series - uses historical data assuming the future will be like the past
Associative models - uses explanatory variables to predict the future
12
Time Series Forecasting
13
Time series
Time-ordered sequence of observations taken at regular intervals
Data : measurements of demand (sales, earnings, profits, …)
Example:
Types of Variations in Time Series Data:
Trend - long-term movement in data
Seasonality - short-term regular variations in data
Cycles - wavelike variations of long-term
Irregular variations - caused by unusual circumstances
Random variations - caused by chance
| Year | 1998 | 1999 | 2000 | 2001 | 2002 |
| Sales (thousand units) | 78.7 | 63.5 | 89.7 | 93.2 | 92.1 |
14
Forecast Variations
Trend
Irregular variation
Cycles
Seasonal variations
Year 01
00
99
Cyclical
15
Time Series Methods
Naïve methods
Moving average
Weighted moving average
Exponential smoothing
Forecasting with a trend
Forecasting with seasonality
16
Time Series Forecasting
- Naïve Method
17
Naïve Forecast
The forecast for any period
=
The previous period’s actual value.
Uh, give me a minute....
We sold 250 wheels last
week.... Now, next week we should sell....
18
Some Notation:
Today’s temperature is 63 → AToday = 63
Forecast for tomorrow → FTomorrow = 63
| Ft | Forecast at time t |
| At | Actual observation at time t |
19
Naïve Method
Example 2: Naïve Forecasts
Forecast for period t is the actual value for period t-1: Ft = At-1.
| Period t | Actual Demand | Forecast |
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
20
Solution to Example 2
| Period t | Actual Demand | Forecast |
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
-
42
40
43
40
41
39
46
44
45
38
40
21
Time Series Forecasting
- Averaging
22
Moving Average
Moving average – A technique that averages a number of recent actual values, updated as new values become available.
Ft = MAn =
n
At-i
i = 1
n
t = an index that corresponds to periods.
n = Number of periods (data points) in the moving
average period.
At = Actual value in period t.
MAn = Forecast based on most-recent n periods.
Ft = Forecast for time period t.
Where
23
Example 3: Moving Average
Find moving average with n = 5.
| Period t | Actual Demand | Forecast |
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
24
Solution to Example 3
Start from F6 (forecast for period 6).
| Period t | Actual Demand | Forecast |
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
-
-
-
-
-
41.2
40.6
41.8
42.0
43.0
42.4
42.6
42+40+43+40+41
5
=
46+44+45+38+40
5
=
25
Moving Averages
New value becomes available?
Drop oldest value from total
Add newest value to total
Recalculate average (divide by n)
26
Lag Increases with Periods
27
MA3
MA5
Average the last 3 actual values
Average the last 5 actual values
The less numbers used to get average the more likely line will be straighter.
Moving Averages
Fewer data points (n )
More responsive to real changes
More responsive to random variations
28
Weighted moving average – more recent values in a series are given more weight in computing the forecast
Reminder that Simple Moving Average:
Weighted Moving Averages
Ft = WMAn
=
wiAt-i
i = 1
n
w1 + w2 + + wn = 1
wi =
n
1
n
MAn
=
n
At-i
i = 1
=
At-i
n
1
i = 1
n
29
w1 ≥ w2 ≥ ≥ wn
Weighted Moving Average
Moving Average
Pros: Easy to compute and easy to understand
Cons: All values in the average are weighted equally
Weighted Moving Average
Similar to moving average
Assigns more weight to recent observed values
Idea: most recent observations are better indicators of future
More responsive to changes
Selection of weights is arbitrary, but weights must add to one. The values for the weights are always given.
30
Example 4: Weighted Moving Average
Find weighted moving average using
Ft =0.4At-1 + 0.3At-2 + 0.2At-3 + 0.1At-4.
| Period t | Actual Demand | Forecast |
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
31
Solution to Example 4
Start from F5 (forecast for period 5).
| Period i | Actual Demand | Forecast |
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
-
-
-
-
-
41.1
41.0
40.2
42.3
43.3
44.3
42.1
40.8
0.1(42)+.2(40)+.3(43)+.4(40)
=
=
0.1(39)+.2(46)+.3(44)+.4(45)
32
.4 goes to the most recent weight and .1 the furthest. When taking the previous four numbers the 4th one is multiplied by .4 and the first gets .1
Shown solutions of Example 3 and 4
30
32
34
36
38
40
42
44
46
48
1
2
3
4
5
6
7
8
9
10
11
12
Observed
MA
WMA
33
Exponential Smoothing
Current forecast = Previous forecast + (Actual - Previous forecast)
Ft = Ft-1 + (At-1 - Ft-1)
Ft = Forecast for period t
Ft-1 = Forecast for period t-1
= Smoothing constant
At-1 =Actual demand or sales for period t-1
where
34
Example 5: Exponential Smoothing
Period (t)
Actual (At)
Ft (α = 0.1)
Error (A-F)
Ft ( α = 0.4)
1
42
2
40
3
43
4
40
5
41
6
39
7
46
8
44
9
45
10
38
11
40
12
Error (A-F)
Hint: To calculate Ft, you need Ft-1 and At-1
For initial forecast, you can use naïve approach
35
Solution to Example 5
For example: α = 0.1
A1 = 42 → F2 = 42 (Naïve)
A2 = 40 → F3 = F2 + α (A2 - F2)
= 42 + 0 .1 × (40 - 42) = 41.8
A3 = 43 → F4 = F3 + α (A3 - F3)
= 41.8 + 0 .1 × (43 - 41.8) = 41.92
Ft = Ft-1 + (At-1 - Ft-1) = (1 – ) Ft-1 + At-1
36
Solution to Example 5 (Cont.)
1
42
2
40
42
-2.00
42
-2
3
43
41.8
1.20
41.2
1.8
4
40
41.92
-1.92
41.92
-1.92
5
41
41.73
-0.73
41.15
-0.15
6
39
41.66
-2.66
41.09
-2.09
7
46
41.39
4.61
40.25
5.75
8
44
41.85
2.15
42.55
1.45
9
45
42.07
2.93
43.13
1.87
10
38
42.36
-4.36
43.88
-5.88
11
40
41.92
-1.92
41.53
-1.53
12
41.73
40.92
Period (t)
Actual (At)
Ft (α = 0.1)
Error (A-F)
Ft ( α = 0.4)
Error (A-F)
Ft = Ft-1 + (At-1 - Ft-1)
37
Picking a Smoothing Constant α
Actual
.1
.4
35
40
45
50
1
2
3
4
5
6
7
8
9
10
11
12
Period
Demand
38
Time Series Forecasting
- Trend
39
Linear Trend Equation
Ft = Forecast for period t
t = Specified number of time periods from t = 0
a = Value of Ft at t = 0
b = Slope of the line
Ft = a + b t
0 1 2 3 4 5 t
Ft
40
Calculating a and b
n = Number of periods
y = Value of the time series
t = Specified number of time periods from t = 0
t
n
-
b
=
n
(ty)
-
y
t
2
(
t)
2
a
=
y
-
b
t
n
41
Example 6:
Calculator sales for a California-based firm over the last 10 weeks are shown in the following table.
| Week (t) | y | yt | t2 |
1
2
3
4
5
6
7
8
9
10
700
724
720
728
740
742
758
750
770
775
700
1448
2160
2912
3700
4452
5306
6000
6930
7750
1
4
9
16
25
36
49
64
81
100
55
7407
41358
385
42
Solution to Example 6
Plot the data, and visually check to see if a linear trend line would be appropriate.
n = 10, t = 55, y =7407, yt = 41358, t2 = 385
b
=
10(41358)
-
55(7407)
10(385)
-
55(55)
=
413580
-
407385
3850
-
3025
≈
7.51
y = 699.40 + 7.51t
a
=
7407 -
7.51(55)
10
≈
699.40
43
Solution to Example 6 (Cont.)
660
680
700
720
740
760
780
800
1
2
3
4
5
6
7
8
9
10
Observed
Trend line
44
Solution to Example 6 (Cont.)
Then determine the equation of the trend line, and predict sales for weeks 11 and 12.
y11 =699.40 + 7.51(11) = 782.01
y12 =699.40 + 7.51(12) = 789.51
45
Time Series Forecasting
- Seasonality
46
Techniques for Seasonality
Example :
Winter and summer sports equipment
Rush hour traffic occurs twice a day
Theaters and Restaurants often experience weekly demand pattern
Banks may experience daily and monthly seasonal variation.
Regularly repeating upward or downward movements in time series values
47
Techniques for seasonality
Seasonality: Expressed in terms of the amount that actual values deviate from the average (or trend) value of the series
Additive: seasonality is expressed as a quantity, which is added or subtracted from the average to incorporate seasonality.
Multiplicative: seasonality is expressed as a percentage of the average amount, which is used to multiply the value of a series to incorporate seasonality.
seasonal percentages = seasonal relatives = seasonal indexes
1.20 sales 20% above average 0.90 sales 10% below average
48
Additive Model and Multiplicative Model
Seasonal Relative
49
Example 7: Seasonality
A manager wants to predict the quarterly demand for period 15 and 16, which are the 2nd and 3rd quarters of a particular year. Demand series consists of both trend and seasonality. The trend portion is Ft = 124 + 7.5t. Quarter relatives are Q1 = 1.20, Q2 = 1.10, Q3 = 0.75, and Q4 = 0.95.
Trend equation: Ft = 124 + 7.5t
t
y
Q1
Q2
Q3
Q4
50
Example 7 (P 92)
The trend values at t = 15 and t = 16:
F15 = 124 + 7.5(15) = 236.5
F16 = 124 + 7.5(16) = 244.0
Incorporating seasonality
Period 15: 236.5(1.10) = 260.15
Period 16: 244.0(0.75) = 183.00
51
Example : Compute seasonality relatives (one period)
Day
Tues
Wed
Thur
Fri
Sat
Sun
Mon
Demand
67
75
82
98
90
36
55
Average
71.86
Seasonal Index
67/71.86
75/71.86
82/71.86
98/71.86
90/71.86
36/71.86
55/71.86
52
Associative Forecasting
53
Simple Linear Regression (SLR)
Find linear relationship between the predictor and the predicted.
Predictor variable - used to predict values of variable interest, sometimes called independent variable
Predicted variable - Dependent variable
Regression - technique for fitting a straight line to a set of points
Objective - obtain an equation of a straight line (least square line) that minimizes the sum of squared vertical deviations of data points from the line.
54
Some Examples of SLR
Based on identification of related variables that can be used to predict values of the variable of interest.
Sales of mountain bikes in an area may be related to the percentage of the young population living in that area.
Sales of Harley-Davidson motorcycles is related to mid-aged men population. Average age of H-D owners is 46.
Ice cream sales can be related to temperature.
Home depot bases sales forecasts on mortgage refinancing rates, and smaller rates imply higher sales.
Increase in energy cost leads to price increases in products and services
55
y
What is a straight line?
y = predicted (dependent) variable
x = predictor (independent) variable
b = slope of the line
a = value of y when x = 0
(the height of line at the y intercept)
y = a + bx
y
x
x
0
a
b>0
b<0
56
Graphical Interpretation of SLR
y = a + bx
57
Which straight line is a better fit?
L2: y = a2 + b2x
L1: y = a1 + b1x
L3: y = a3 + b3x
58
Graphical Interpretation of SLR
Dependent variable
Independent variable
x
y
Estimate of
y from
regression
equation
Actual
value
of y
Value of x used
to estimate y
Deviation,
or error
Regression
equation:
y = a + bx
y = dependent variable
x = independent variable
a = y-intercept of the line
b = slope of the line
59
Computing a and b
Given n data points, find the intercept a and slope b to
60
One Example: SLR Model Seems Reasonable
A straight line is fitted to a set of sample points.
Computed relationship
y = 5.06 + 1.593 x
61
Correlation
Correlation (r) between variables: The strength and direction of relationships between two variables
1.00 means changes in one variable are always matched by changes in the other.
-1.00 means increases in one variable are always matched by decreases in the other.
A correlation close to zero (0) means little linear relationship.
The square of the correlation coefficient, i.e., r2, provides a measure of the percentage of variability in the values of y that is explained by the independent variable.(80% or more: the independent variable is a good predictor of the values of dependent variable)
62
Recap
Forecasting
Forecast Error
Mean Absolute Deviation
Mean Squared Error
Mean Absolute Percent Error
Judgmental Forecast
Time Series Data
63
Trend
Seasonality
Associative Forecast
Predictor Variables
Regression
Correlation
2
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Sheet1
| Period | Actual | MA3 | MA5 | ||
| 1 | 700 | ||||
| 2 | 675 | ||||
| 3 | 690 | ||||
| 4 | 730 | 688 | |||
| 5 | 740 | 698 | |||
| 6 | 650 | 720 | 707 | ||
| 7 | 800 | 707 | 697 | ||
| 8 | 790 | 730 | 722 | ||
| 9 | 775 | 747 | 742 | ||
| 10 | 700 | 788 | 751 | ||
| 11 | 780 | 755 | 743 | ||
| 12 | 800 | 752 | 769 | ||
| 13 | 760 | 769 | |||
| x | y | xy | x2 | y2 | |
| 7 | 15 | 105 | 49 | 225 | |
| 2 | 10 | 20 | 4 | 100 | |
| 6 | 13 | 78 | 36 | 169 | |
| 4 | 15 | 60 | 16 | 225 | |
| 14 | 25 | 350 | 196 | 625 | |
| 15 | 27 | 405 | 225 | 729 | |
| 16 | 24 | 384 | 256 | 576 | |
| 12 | 20 | 240 | 144 | 400 | |
| 14 | 27 | 378 | 196 | 729 | |
| 20 | 44 | 880 | 400 | 1936 | |
| 15 | 34 | 510 | 225 | 1156 | |
| 7 | 17 | 119 | 49 | 289 | |
| 132 | 271 | 3529 | 1796 | 7159 |
Sheet1
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