Lesson 10
The formula for the moving average is:
Ft = w 1 A t -1 + w 2 A t - 2 + w 3 A t -3 + ... + w n A t - n n
wt = weight given to time period “t” occurrence (weights must add to one) ∑w
i=1
i =1
Weighted Moving Average Problem (1) Data
Question: Given the weekly demand and weights, what is the forecast for the 4th period
or Week 4?
Week Demand
Weights:
1 650
t-1 .5
2 678
3 720
t-2 .3
4 t-3 .2
Weighted Moving Average Problem (1) Solution
Week Demand Forecast
1 650
2 678
3 720
4 693.4
F4 = 0.5(720)+0.3(678)+0.2(650)=693.4
Note: More weight age would be given to recent most values.
Weighted Moving Average Problem (2) Data
Question: Given the weekly demand information and weights, what is the weighted
moving average forecast of the 5th period or week?
Week Demand
Weights:
1 820
t-1 0.7
2 775
3 680
t-2 0.2
4 655 t-3 0.1
Weighted Moving Average Problem (2) Solution
Week Demand Forecast
1 820
2 775
3 680
4 655
5 672
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F5 = (0.1)(755)+(0.2)(680)+(0.7)(655)= 672
Note: More weight age would be given to recent most values.
Exponential Smoothing Model
Ft = Ft-1 + a(At-1 - Ft-1)
Where:
Ft = Forcast value for thecomingt timeperiod
Ft - 1 = Forecast valuein 1 past timeperiod
At - 1 = Actualoccurancein thepast t time period
α = Alphasmoothingconstant
Exponential Smoothing Problem (1) Data
Question: Given the weekly demand data, what are the exponential smoothing
forecasts for periods 2-10 using a=0.10 and a=0.60?
Assume F1=D1
Week Demand
1 820
2 775
3 680
4 655
5 750
6 802
7 798
8 689
9 775
10
Exponential Smoothing Solution (1)
Week Demand 0.1 0.6
1 820 820.00 820.00
2 775 820.00 820.00
3 680 815.50 820.00
4 655 801.95 817.30
5 750 787.26 808.09
6 802 783.53 795.59
7 798 785.38 788.35
8 689 786.64 786.57
9 775 776.88 786.61
10 776.69 780.77
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Exponential Smoothing Problem (2) Data
Question: What are the exponential smoothing forecasts for periods 2-5 using Alpha
=0.5? Assume F1=D1
Week Demand
1 820
2 775
3 680
4 655
5
Exponential Smoothing Problem (2) Solution
Week Demand
1 820
2 775
3 680
4 655
5
F1=820+(0.5)(820-820)=820
F3=820+(0.5)(775-820)=797.75
Example 3 - Exponential Smoothing
Period Actual Alpha = 0.1 Error Alpha = 0.4 Error
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
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Common Nonlinear Trends
Parabolic
Exponential
Growth
• Parabolic Trends
• Concaved Upwards and Concaved Downwards
• The left and right arms are widening as the value increases or the parabola
is opening upwards.
• It represents the quadratic function
Linear Trend Equation
Ft = a + bt
Where:
• Ft = Forecast for period t
• t = Specified number of time periods
• a = Value of Ft at t = 0
• b = Slope of the line
n ∑ (ty) - ∑ t ∑ y
b =
n∑ t 2 - ∑ t 2
a = ∑
y - ∑t
n
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Linear Trend Equation Example
t y
Week t2 Sales ty
1 1 150 150
2 4 157 314
3 9 162 486
4 16 166 664
5 25 177 885
Σ t = 15 Σ t2 = 55 Σ y = 812 Σ ty =
2499
(Σ t)2 =
225
Linear Trend Calculation
5 (2499) - 15(812) 12495 -12180
b = = = 6.3
5(55) - 225 275 -225
812 - 6.3(15)
a = = 143.
5
y = 143.5 + 6.3t
Associative Forecasting
1. Predictor variables - used to predict values of variable interest
2. Regression - technique for fitting a line to a set of points
3. Least squares line - minimizes sum of squared deviations around the line
Forecast Accuracy
• Error - difference between actual value and predicted value
• Mean Absolute Deviation (MAD)
• Average absolute error
• Mean Squared Error (MSE)
• Average of squared error
• Mean Absolute Percent Error (MAPE)
• Average absolute percent error
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Simple Linear Regression Formulas for Calculating “a” and “b”
a = y - bx
b=
∑ xy - n( y)(x)
∑ x - n( x)
2 2
Simple Linear Regression Problem Data
Question: Given the data below, what is the simple linear regression model that can be
used to predict sales in future weeks?
Week Sales
1 150
2 157
3 162
4 166
5 177
Answer: First, using the linear regression formulas, we can compute “a” and “b”
Week Week*Week Sales Week*Sales
1 1 150 150
2 4 157 314
3 9 162 486
4 16 166 664
5 25 177 885
3 55 162.4 2499
Average Sum Average Sum
b=
∑ xy - n( y)(x ) = 2499 - 5(162.4)(3) = 63 = 6.3
∑ x - n( x )
2 2
55 − 5(9) 10
a = y - b x = 162.4 - (6.3)(3) = 143.5
The resulting regression model is:
Yt = 143.5 + 6.3x
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