MIE 477 - Session 8 Forecasting

Class Outline

◆ Methods for Seasonal Forecasts – Simple – Moving Average – Winter�s Method

◆ Tracking Signals ◆ Practical Considerations

Example 1

◆ A textbook publisher would like to estimate the sales of a popular textbook on PPC in the next quarter. He has compiled the history of sales over the last 4 years.

Quarter Year 1 Year 2 Year 3 Year 41 6 8 5 92 31 28 35 383 40 41 35 374 17 20 12 15

Sales (in thousands)

Plot Sales over Time

Seasonal Forecasts

◆ Seasonality corresponds to a pattern in the data that repeats at regular intervals, let�s say every N periods.

Forecasting For Seasonal Series

Σ ci = N

ci = 1.25 implies 25% higher than the baseline on avg. ci = 0.75 implies 25% lower than the baseline on avg.

Multiplicative seasonal factors: c1 , c2 , . . . , cN where i = 1 is first period of season, i = 2 is second period of the season, etc. N is length of a season. They represent the amount that the ith period in the season is above or below the overall average.

Simple Method for Estimating Seasonal Factors and Forecasting Stationary Series

1.  Compute the sample mean of the entire data set (should be at least several seasons of data).

2.  Divide each observation by the sample mean. (This gives a factor for each observation.)

3.  Average the factors for like periods in a season. (The resulting N numbers will exactly add to N and correspond to the N seasonal factors.)

4.  Forecast by multiplying overall sample mean with appropriate seasonal factor

Calculation of Seasonal Factors

Quarter Year 1 Year 2 Year 3 Year 41 6 8 5 92 31 28 35 383 40 41 35 374 17 20 12 15

Sales (in thousands)

Total 94 97 87 99 377Average 23.5 24.25 21.75 24.75 23.5625

Quarter Year 1 Year 2 Year 3 Year 41 0.25 0.34 0.21 0.382 1.32 1.19 1.49 1.613 1.70 1.74 1.49 1.574 0.72 0.85 0.51 0.64

RatioSeasonal Factor

0.301.401.620.68

4.00

Forecast7.0033.0038.2516.00

Moving Average Method

◆ The simple method does not account for changes in demand over time, or placing more importance in recent observations.

◆ Let�s now calculate seasonal factors accounting for the conditions that were present in the season where the observations occurred.

Moving Average Method

Quarter Q Sales1 62 313 404 175 86 287 418 209 510 3511 3712 1213 914 3815 3716 15

MA(4)

23.524

23.2523.524.2523.525.2524.2522.2523.252424

24.75

Centered MA23.6923.6923.7523.6323.3823.8823.8824.3824.7523.2522.7523.6324.0024.3824.1924.19

Ratio0.251.311.680.720.341.171.720.820.201.511.630.510.381.561.530.62

Est. Factor

0.291.391.640.67

Sum3.9860389

Factor0.291.391.650.67

4

Centered Q

2.53.54.55.56.57.58.59.510.511.512.513.514.5

Deseasonalizing a Series

◆  To remove seasonality from a series, simply divide each observation in the series by the appropriate seasonal factor.

◆  The resulting series will have no seasonality and may then be predicted using an appropriate method (Moving averages, Regression Analysis etc.).

◆  Once a forecast is made on the deseasonalized series, one then multiplies that forecast by the appropriate seasonal factor to obtain a forecast for the original series.

Calculate Forecast for Year 5 using MA(6)

Factor0.291.391.650.67

Deseasonalized Series20.4022.2824.3125.4027.2020.1224.9229.8817.0025.1622.4917.9330.5927.3122.4922.41

Year 5 Forecast

7.0233.2139.2715.98

MA(6) 23.87

Quarter Q Sales1 62 313 404 175 86 287 418 209 510 3511 3712 1213 914 3815 3716 15

Drawback

◆ As new data become available… – All factors have to be computed from scratch!

Example 2 ◆  The manager of the Stanley Steemer carpet cleaning company

needs a quarterly forecast of the number of customers expected next year. The carpet cleaning business is seasonal, with a peak in the third quarter and a trough in the first quarter. Following are the quarterly demand data from the past four years. The managers wants to forecast customer demand for each quarter of year 5, based on an estimate of total year 5 demand of 2,600 customers.

Quarter Year 1 Year 2 Year 3 Year 41 45 70 100 1002 335 370 585 7253 520 590 830 11604 100 170 285 215

Total 1000 1200 1800 2200Average 250 300 450 550

Seasonal Series with a Trend

Triple Exponential Smoothing – Winters�s Method

◆  Predict seasonal series with a trend ✣ Easy to update as new data become available ◆  Assume model of the form

( )t t t tD G cµ ε= + ⋅ +

Base signal,

intercept at t=0

Trend component

Multiplicative seasonal

component

Winters�s Method at work

◆  A forecast made in period t for any future period t+τ is

( ),t t t t tF S G cτ ττ+ += + ⋅

–  Observe that the seasonal factor at the time forecasted needs to be applied, i.e.,

–  If iN<t+� ≤(i+1)N then tc τ+

t t iNc cτ τ+ + −=

Winters�s Method – 3 exponential smoothing equations

1.  The deseasonalized series.

2.  The trend

3.  The seasonal factors

( )( )1 11tt t t

t N

DS S Gc

α α − −−

⎛ ⎞= + − +⎜ ⎟

⎝ ⎠

( ) ( )1 11t t t tG S S Gβ β− −= − + −

( )1tt t N

t

Dc cS

γ γ −

⎛ ⎞= + −⎜ ⎟

⎝ ⎠Re-adjust the current seasonal factors to sum to N

Initialization of Winter�s Method

◆  At least two seasons of data (i.e. 2N periods) should be available

◆  First the initial slope is computed based on the difference of the average of the first vs. the last seasons.

◆  Then the initial intercept is computed using the average of the last season and the initial slope.

◆  Seasonal factors for each period for each of the seasons are computed and averaged over the seasons to obtain the initial values (after normalizing, i.e. sum = N).

◆  See Textbook pages 86-87 for the exact formulas.

Example 1 (Cont�d)

◆ Calculate the seasonal factors and the year 5 forecast for the Stanley Steemer carpet cleaning company using Winter�s Method.

Example – Winter�s Method

Year Quarter # Customers1 452 3353 5204 1001 702 3703 5904 1701 1002 5853 8304 2851 1002 7253 11604 215

1

2

3

4

Average250

300

450

550

G25

S587.5

TrendLine212.5237.5262.5287.5262.5287.5312.5337.5412.5437.5462.5487.5512.5537.5562.5587.5

Ratio 0.211.411.980.350.271.291.890.500.241.341.790.580.201.352.060.37

Est. Factor Factor0.23 0.231.35 1.361.93 1.950.45 0.46

Sum3.96 4.00

Then we are ready to calculate our forecast…

( ),t t t t tF S G cτ ττ+ += + ⋅

Year 5 Forecast141.79867.351293.54313.12

Measuring Bias - Tracking Signals ◆  Tracking signals may be useful for indicating forecast

bias. ◆  Forecast error at time t: et=Ft-Dt ◆  The smoothed values of the error and absolute error are

◆  The tracking signal then is

◆  If Tt is large then the bias is significant –  It depends on �"–  For � = 0.1 then Tt>0.51 ! significant bias

tt

t

ET

M=

1

1

(1 )(1 )

t t t

t t t

E e EM e M

β ββ β

−

−

= + −

= + −

Practical Considerations

◆  Graphing data helps identify existence of trend or seasonal fluctuations –  Statistical tests to assess significance of regression. –  Complex relationships, however, require more sophisticated methods,

such as in the Box-Jenkins Models. ◆  Box-Jenkins methods require substantial data history, use the

correlation structure of the data, and can provide significantly improved forecasts under some circumstances.

◆  Overly sophisticated forecasting methods can be problematic, especially for long term forecasting. –  Figure on the next slide.

◆  Average of forecasts from different methods tend to be more accurate than following a single method.

The Difficulty with Long-Term Forecasts

Sport Obermeyer (cont.)

Initial Forecast: Each data point represents the forecast and actual season sales for a particular parka.

Example. Parka A had an initial forecast of 2500 units and season sales of 510

Sport Obermeyer (cont.)

Updated forecast, incorporating first 20% of sales data

Final forecast, incorporating first 80% of sales data

Biggest Forecasting Error in History?

�Believing that the only reality lies in the quantifiable, and that decisions should be guided entirely by computer modeling and statistical analysis, leads inexorably to corporate stagnation and marketplace defeat by livelier risk-takers.�

Demand Forecast for the first PC

When would you use each type of forecasting method?

◆  Subjective

–  No much data/history available

◆  Causal methods (such as regression)

–  Historical data available

–  Relationship between internal/external factors and the factor to be forecasted can be identified

When would you use each type of forecasting method?

◆  Time series forecasting

–  Historical data available relative to one single factor - time

–  The dependent variable past pattern will replicate in the future

–  Addresses trends, seasonal patterns –  Appropriate for low level production planning decisions such

as inventory management when items need to be forecasted at the individual level. At the aggregate level, one may want to take into account causality (relationships to other factors).

Today�s Concepts

◆ Key Take-aways – The forecasting method needs to capture the patterns

in the data. » Trend » Seasonality » Causality

◆ Learning Objectives�–  Understand the available forecasting techniques and when to

use them –  Be able to account for seasonal factors when forecasting –  Be aware of the practical considerations in forecasting

Next Class

◆ Aggregate Planning ◆ Homework #3– Due ◆ Littlefield Report - Due