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• Today– Chapter 7
• Homework 3– Online tomorrow– Due Friday February 26 before 5:00pm
• Next week Thursday– Finish Chapter 7 (forecast error measures)– Start with Chapter 8– Network design simulation assignment
Announcements
• What?– Tour the Staples Fulfillment Center in Brighton, CO– Informal Lunch-and-Learn– Up to 20 students with a Operations Management major
• When?– Weeks of March 15 or March 29– There is a fair amount of time involved in the activity
• Transit is close to an hour in each direction• Probably 2 hours onsite
• Interested?– Let me know (email) by the end of this week
Time Series Forecasting
Observed demand =
Systematic component + Random component
L Level (current deseasonalized demand)T Trend (growth or decline in demand)S Seasonality (predictable seasonal fluctuation)
The goal of any forecasting method is to predict the systematic component (Forecast) of demand and measure the size and
variability of the random component (Forecast error)
Summary: N-Period Moving Average Method
1. Estimate level• Take the average demand over the most recent N periods
• Lt = (Dt + Dt-1 + … + Dt-N+1) / N
2. Forecast• Forecast for all future periods is based on the current estimate of
level Lt
• Ft+n = Lt
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Forecast Ft+n = Lt
Summary: Simple Exponential Smoothing Method
1. Estimate level• The initial estimate of level L0 is the average of all historical
data• L0 = (∑i Di)/ n
• Revise the estimate of level for all periods using smoothing constant • Lt+1 = Dt+1 + (1 – )*Lt
2. Forecast• Forecast for future periods is
• Ft+n = Lt
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Forecast Ft+n = Lt
Summary: Holt’s Method (Trend Corrected Exponential Smoothing)
1. Estimate level and trend• The initial estimate of level L0 and trend T0 are obtained using
linear regression• =INTERCEPT(known_y’s, known_x’s)• =LINEST(known_y’s, known_x’s)
• Revise the estimates for all periods using smoothing constants and • Lt+1 = Dt+1 + (1 – )*(Lt + Tt)• Tt+1 = (Lt+1 – Lt) + (1 – )*Tt
2. Forecast• Forecast for future periods is
• Ft+n = Lt + nTt
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Forecast Ft+n = Lt + nTt
Summary: Winter’s Model (Trend and Seasonality Corrected Exp. Smoothing)1. Estimate level, trend, and seasonality
• The initial estimates of L0, T0, S1, S2, S3, and S4 are obtained from static forecasting procedure
• Revise the estimates for all periods using smoothing constants , and • Lt+1 = (Dt+1/St+1) + (1 – )*(Lt + Tt)
• Tt+1 = (Lt+1 – Lt) + (1 – )*Tt
• St+p+1 = (Dt+1/Lt+1) + (1 – )St+1
2. Forecast• Forecast for future periods is
• Ft+n = (Lt + nTt)*St+n
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Forecast Ft+n = (Lt + nTt)St+n
Components of an Observation
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Trend (T)
Forecast(F)
Ft+n = Lt + nTt
Holt’s method is appropriate when demand is assumed to have a level and a trend
Example: Holt’s Method
• An electronics manufacturer has seen demand for its latest MP3 player increase over the last six months– 8415, 8732, 9014, 9808, 10413, 11961
Demand Level Trend Forecast
t Dt Lt Tt Ft
1 8,4152 8,7323 9,0144 9,8085 10,4136 11,96178910
Determine initial levelL0 = INTERCEPT(y’s, x’s)
T0 = LINEST(y’s, x’s)
Example: Holt’s Method
• An electronics manufacturer has seen demand for its latest MP3 player increase over the last six months– 8415, 8732, 9014, 9808, 10413, 11961
Demand Level Trend Forecast
t Dt Lt Tt Ft7,367 673
1 8,4152 8,7323 9,0144 9,8085 10,4136 11,96178910
Demand Level Trend Forecast
t Dt Lt Tt Ft7,367 673
1 8,415 8,078 6812 8,732 8,756 6803 9,014 9,394 6724 9,808 10,040 6675 10,413 10,677 6616 11,961 11,401 67378910
Demand Level Trend Forecast
t Dt Lt Tt Ft7,367 673
1 8,415 8,078 681 80402 8,732 8,756 680 87593 9,014 9,394 672 94364 9,808 10,040 667 100665 10,413 10,677 661 107076 11,961 11,401 673 113387 120748910
Demand Level Trend Forecast
t Dt Lt Tt Ft7,367 673
1 8,415 8,078 681 80402 8,732 8,756 680 87593 9,014 9,394 672 94364 9,808 10,040 667 100665 10,413 10,677 661 107076 11,961 11,401 673 113387 120748 127479 1342010 14094
Determine initial levelL0 = INTERCEPT(y’s, x’s)
T0 = LINEST(y’s, x’s)
Determine levelsLt+1 = Dt+1 + (1 – )*(Lt + Tt)
Tt+1 = (Lt+1 – Lt) + (1 – )*Tt
ForecastFt+n = Lt + nTt = 0.1, = 0.2
Example: Tahoe Salt
• Demand forecasting using Holt’s method
0
10,000
20,000
30,000
40,000
50,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Actual
Forecast (Holt)
Components of an Observation
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Seasonality (S)
Forecast(F)
Ft+n = (Lt + Tt)St+n
Example: Winter’s Model
• A theme park has seen the following attendance over the last eight quarters (in thousands)– 54, 87, 192, 130, 80, 124, 265, 171 Determine initial levels
L0 = From static forecast
T0 = From static forecast
Si,0 = From static forecast
ForecastFt+1 = (Lt + Tt)St+1
Determine levelsLt+1 = (Dt+1/St+1)+ (1 – )*(Lt + Tt)
Tt+1 = (Lt+1 – Lt) + (1 – )*Tt
St+p+1 = (Dt+1/Lt+1) + (1 – )*St+1
Demand Level Trend Seasonal ForecastFactor
t Dt L T Si Ft
1 542 873 1924 1305 806 1247 2658 171
Example: Tahoe Salt
• Demand forecast using Winter’s method
0
10,000
20,000
30,000
40,000
50,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Actual
Forecast (Winter)
Static Versus Adaptive Forecasting Methods
• Static– Dt: Actual demand
– L: Level– T: Trend– S: Seasonal factor
– Ft: Forecast
• Adaptive– Dt: Actual demand
– Lt: Level
– Tt: Trend
– St: Seasonal factor
– Ft: Forecast
Components of an Observation
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Seasonality (S)
Forecast(F)
Ft+n = (Lt + Tt)St+n
Example: Static Method
• A theme park has seen the following attendance over the last eight quarters (in thousands)– 54, 87, 192, 130, 80, 124, 265, 171
Determine initial levelL = INTERCEPT(y’s, x’s)
T = LINEST(y’s, x’s)
Determine deason. demandDt = L + Tt
Determine seasonal factorsSt = Dt / Dt
Determine seasonal factorsSi =AVG(St)Forecast
Ft = (L + T)Si
Demand Level Trend Deseason. Seasonal Seasonal ForecastDemand Factor Factor
t Dt L T Dt_bar Si_bar Si Ft
1 542 873 1924 1305 806 1247 2658 171
Demand Level Trend Deseason. Seasonal Seasonal ForecastDemand Factor Factor
t Dt L T Dt_bar Si_bar Si Ft59.3 17.3
1 542 873 1924 1305 806 1247 2658 171
Demand Level Trend Deseason. Seasonal Seasonal ForecastDemand Factor Factor
t Dt L T Dt_bar Si_bar Si Ft59.3 17.3
1 54 76.62 87 93.93 192 111.24 130 128.55 80 145.86 124 163.17 265 180.48 171 197.7
Demand Level Trend Deseason. Seasonal Seasonal ForecastDemand Factor Factor
t Dt L T Dt_bar Si_bar Si Ft59.3 17.3
1 54 76.6 0.702 87 93.9 0.933 192 111.2 1.734 130 128.5 1.015 80 145.8 0.556 124 163.1 0.767 265 180.4 1.478 171 197.7 0.86
Demand Level Trend Deseason. Seasonal Seasonal ForecastDemand Factor Factor
t Dt L T Dt_bar Si_bar Si Ft59.3 17.3
1 54 76.6 0.70 0.632 87 93.9 0.93 0.843 192 111.2 1.73 1.604 130 128.5 1.01 0.945 80 145.8 0.556 124 163.1 0.767 265 180.4 1.478 171 197.7 0.86
Demand Level Trend Deseason. Seasonal Seasonal ForecastDemand Factor Factor
t Dt L T Dt_bar Si_bar Si Ft59.3 17.3
1 54 76.6 0.70 0.63 48.02 87 93.9 0.93 0.84 79.23 192 111.2 1.73 1.60 177.74 130 128.5 1.01 0.94 120.65 80 145.8 0.55 91.46 124 163.1 0.76 137.67 265 180.4 1.47 288.28 171 197.7 0.86 185.5
Static Forecasting Method
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Demand
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Demand
Lin. Reg.
Static Forecasting Method
• Deseasonalize demand– Demand that would have been observed in the
absence of seasonal fluctuations
• Periodicity p– The number of periods after which the seasonal cycle
repeats itself• 12 months in a year• 7 days in a week• 4 quarters in a year• 3 months in a quarter
Deseasonalize demand
• Periodicity p is odd • Periodicity p is even
Demand Deseason.Demand
t Dt
1 8,0002 13,0003 23,000 19,7504 34,000 20,6255 10,000 21,2506 18,000 21,7507 23,000 22,5008 38,000 22,1259 12,000 22,62510 13,000 24,12511 32,00012 41,000
Demand Deseason.Demand
t Dt
1 8,0002 13,000 14,6673 23,000 15,3334 10,000 17,0005 18,000 17,0006 23,000 17,6677 12,000 16,0008 13,000 19,0009 32,000101112
Static Forecasting Method
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Demand
Deseason.
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Demand
Deseason.
Deseason. Lin. Reg.
Example: Tahoe Salt
• Demand forecast using Static forecasting method
0
10,000
20,000
30,000
40,000
50,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Actual
Forecast (Static)
Summary: Static Forecasting Method
1. Estimate level and trend• Deseasonalize the demand data• Estimate level L and trend T using linear regression
• Obtain deasonalized demand Dt
2. Estimate seasonal factors• Estimate seasonal factors for each period St = Dt /Dt
• Obtain seasonal factors Si = AVG(St) such that t is the same season as i
3. Forecast• Forecast for future periods is
• Ft+n = (L + nT)*St+n
0
500
1000
1500
2000
2500
3000
3500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Quarter
Dem
and
Forecast Ft+n = (L + nT)St+n
Forecast Forecast error
Time Series Forecasting
Observed demand =
Systematic component + Random component
L Level (current deseasonalized demand)T Trend (growth or decline in demand)S Seasonality (predictable seasonal fluctuation)
The goal of any forecasting method is to predict the systematic component (Forecast) of demand and measure the size and
variability of the random component (Forecast error)
1) Characteristics of Forecasts
• Forecasts are always wrong!– Forecasts should include an expected value and a
measure of error (or demand uncertainty)• Forecast 1: sales are expected to range between 100
and 1,900 units• Forecast 2: sales are expected to range between 900
and 1,100 units
Examples
8000
9000
10000
1 2 3 4 5 6 7 8 9 10 11 12
Demand
Forecast
0
10000
20000
30000
40000
1 2 3 4 5 6 7 8 9 10 11 12
Demand
Forecast
0
10000
20000
30000
40000
50000
1 2 3 4 5 6 7 8 9 10 11 12
Demand
Forecast
800000
900000
1000000
1 2 3 4 5 6 7 8 9 10 11 12
Demand
Forecast
Measures of Forecast Error
Measure Description
Error Forecast – Actual Demand
Mean Square Error (MSE) Estimate of the variance
Mean Absolute Deviation (MAD)
Estimate of the standard deviation of the random component
Mean Absolute Percentage Error (MAPE)
Absolute error as a percentage of the demand
Tracking signal Ratio of bias and MAD
Recommended