Demand Forecasting LEARNING OBJECTIVES IEEM Understand commonly used forecasting techniques. 2. Learn to evaluate forecasts

Demand Forecasting

IEEM 517

1

LEARNING OBJECTIVES

1. Understand commonly used forecasting techniques 2. Learn to evaluate forecasts

CONTENTS

• Motivation

• Forecast Evaluation

• Qualitative Methods

• Causal Models

• Time-Series Models

• Summary

3 long term Demand fulfillment Purchasing

Aggregate planning Demand

forecasting Inventory Distribution network design

PRODUCTION AND OPERATIONS MANAGEMENT

Product development medium term Distribution Facility location and layout Manufacturing Supply net-work design Partner selection Product portifolio Derivative product development Adaptions Supply contract design

Demand forecasting is

the starting point of all

planning and control!

WAL-MART EXPERIENCE – SITUATION 1996

Warner-Lambert Store 2500 . . . • Forecast errors of 60% • 25% savings possible if

forecast accuracy were improved (corresponding to $179 billion in US) • No less than 10,000 SKUs

per store

Store 1 Situation 1996

Wal-Mart

Other suppliers

5

WAL-MART EXPERIENCE – SITUATION NOW

• Collaborative forecasting and replenishment software installed • Initial forecasts generated by Wal-Mart

• Forecasts refined by Warner-Lambert • Inventory cost reduced by 70%

• Service levels improved from 96% to 99% • System adapted by others

SOME CHARACTERISTICS OF FORECASTS

• Forecasts are usually wrong; Knowledge of the forecast error makes forecasts more meaningful

• Aggregate forecasts are more accurate than individual forecasts • Short-term forecasts are more accurate than long-term forecasts • Choosing appropriate aggregation levels, time horizons, and forecasting

techniques is crucial 7

CONTENTS

• Motivation

• Forecast Evaluation

• Qualitative Methods

• Causal Models

• Time-Series Models

TWO FORECASTS

16 17 18 19 20

Aug-02 Sep-02 Oct-02 Nov-02 Dec-02

Actual Forecast 1 Forecast 2 Sales

• Which forecast is better? • How can we evaluate the forecasting performance? Forecast quality

9

FORECAST ERROR

Actual

Forecast 2 16 17 18 19 20 1 2 3 4 5 Sales ε1 ε2 ε3 ε4 ε5 Forecast error εt= ft- At A_t: Actual value in period t f_t: Forecast for period t A_t f_t

Mean absolute deviation

MAD =

MEASURES OF FORECAST ACCURACY

Mean squared error

MSE =

Mean absolute percentage error

MAPE =

∑

= T 1 t t

T

1

ε

∑

= T 1 t 2 t

T

1

ε

∑

= T 1 t t t

A

T

1

ε

11

MEASURING FORECAST ACCURACY – FORECAST 1

1 17.73 17.99 0.26 0.26 0.07 0.01 2 19.09 17.73 -1.36 1.36 1.84 0.07 3 19.64 19.09 -0.55 0.55 0.30 0.03 4 17.42 19.64 2.22 2.22 4.92 0.13 5 17.61 17.42 -0.19 0.19 0.03 0.01 t A_t f_t εt |εt| εt 2 εt A_t MAD =

MEASURING FORECAST ACCURACY – FORECAST 2

1 17.73 17.19 -0.54 0.54 0.29 0.03 2 19.09 17.48 -1.61 1.61 2.58 0.08 3 19.64 17.94 -1.69 1.69 2.86 0.09 4 17.42 18.11 0.69 0.69 0.47 0.04 5 17.61 18.16 0.56 0.56 0.31 0.03 MAD = MSE = MAPE = t A_t f_t εt |εt| εt 2 εt A_t 13

EVALUATIONS OF FORECASTS

MAD = 1.016 MSE = 1.302 MAPE = 5.4% MAD = 0.912 MSE = 1.430 MAPE = 5.0% Forecast 1 Forecast 2 Evaluation • Forecast 1 has smaller • Forecast 2 has smaller

Forecast 1 Forecast 2

BIAS IN FORECASTS

-4 -2 0 2 4 0 5 10 15 Error εt t

• Forecast 1 seems biased • Forecast 2 seems

un-biased

Biased: Persistent

tendency for forecasts to be greater or smaller than the actual values (E(εt) >

0 or < 0)

Unbiased: E(εt) = 0

15

BIAS IN FORECASTS - CONTINUED

0 10 20 30 0 5 10 15 εi Forecast 1 Forecast 2 An alternative test: judge bias by plotting

εi for t = 1, 2, 3, … instead of plotting εt

Σ

_i=1t

•

Forecast 1 seems biased • Forecast 2 seems

un-Σ

i=1 t

REASON FOR BIAS IN FORECASTS

If relevant elements are not considered in the forecast, the forecast can become biased. These elements can include:

• Linear trend or non-linear trend • Seasonality

• External factors, such as promotion and advertisement

17

CONTENTS

• Motivation

• Forecast Evaluation

• Qualitative Methods

• Causal Models

• Time-Series Models

• Summary

QUALITATIVE METHODS

Qualitative Methods Sales Force Estimate Executive Opinion Market Research Delphi Method Application

• Used to generate forecasts if historical data are not available (e.g., introduction of new product)

• Used to modify forecasts generated by other approaches (e.g.,

considering information not included in quantitative methods)

19

SALES FORCE ESTIMATE

Rationale

Sales force is close to customer and has good information on future demands

Approach

Members of sales force periodically report their estimates. These estimates are then aggregated to generate the overall forecast

Main advantages

• Sales force knows customer well

• Sales territories are typically divided by district/region. Sales forecasts can be broken down correspondingly

SALES FORCE ESTIMATE – CONTINUED

• Bias of sales force

- Might have incentives to overestimate sales or underestimate sales - Might naturally be optimistic or pessimistic

• Sales force does not always have all information necessary to generate forecast - Features of products launched in future

- Preferences of customers in new market segments

Typical application Main drawbacks

Short-term and medium-term demand forecasting

21

EXECUTIVE OPINION

Rationale

Upper-level management has best information on latest product developments and future product launches

Approach

Small group of upper-level managers collectively develop forecasts

• Combine knowledge and expertise from various functional areas • People who have best information on future developments generate the

forecasts Main advantages

EXECUTIVE OPINION – CONTINUED

• Expensive

• No individual responsibility for forecast quality • Risk that few people dominate the group Typical applications

Main drawbacks

Short-term and medium-term demand forecasting

23

MARKET RESEARCH

Rationale

Ultimately, consumers drive demand Approach

Determine consumer interests by creating and testing hypotheses through data-gathering surveys:

1. Design questionnaire 2. Select customer sample

3. Conduct survey (e.g., telephone, mail, or interview) 4. Analyze information and generate forecast

MARKET RESEARCH – CONTINUED

• Expensive

• Require considerable knowledge and skills

• Sometimes validity not guaranteed due to low response rates: For mailed questionnaires response rate often < 30%

Typical application

• Systematic and fact-based approach • Excellent accuracy for short-term forecasts • Good accuracy for medium-term forecasts Main advantages

Main drawbacks

Short-term and medium-term demand forecasting

25

DELPHI METHOD

Rationale

Anonymous written responses encourage honesty and avoid that a group of experts are dominated by only a few members

Approach Coordinator sends initial questionnaire Each expert writes response (anonymous) Coordinator performs analysis Coordinator sends updated questionnaire Consensus reached? Coordinator summarizes forecast No Yes

DELPHI METHOD – CONTINUED

• Slow process

• Experts are not accountable for their responses

• Little evidence that reliable long-term forecasts can be generated with Delphi or other methods

• Long-term forecasting • Technology forecasting • Generate consensus

• Can forecast long-term trend without availability of historical data Main advantages

Main drawbacks

Typical application

27

LONG-TERM FORECASTS ARE OFTEN WRONG!

“The phonograph is not of any commercial value.”

Thomas A. Edison, 1880

“… photographic telegraphy permits transmission of facsimile of any form or writing or illustration…”

Jules Verne, 1863

“The telephone is a toy no one would want to use.”

CONTENTS

• Motivation

• Forecast Evaluation

• Qualitative Methods

• Causal Models

• Time-Series Models

• Summary

29

CAUSAL MODELS

Causal Models Linear Regression Non-linear Regression Application Used to forecast the

performance (demand, profit, etc.) of a business investment based on the observed data of existing and similar business activities

LINEAR REGRESSION: A SIMPLE EXAMPLE

0 250 500 0 10 20 Population Demand Population Demand 7 150 2 100 6 130 4 150 14 250 15 270 16 240 12 200 14 270 20 440 15 340 7 170 10 ? Population around the new store

A company is going to open a new store with nearby population of 10 thousands. The company would like to predict the daily demand. The company has collected data (i.e., nearby population and daily demand) of stores opened in other places.

Æ

31

LINEAR REGRESSION: GENERAL SETTING

• Have obtained data of existing and similar business activities

(y_k; x_1k, x_2k, …, x_mk) for k = 1, …, K, m: # predictive variables K: # existing activities demand population # competitors … store-size

Predicted Predictive

variable variables

LINEAR REGRESSION: OBJECTIVE

Objective

∑

K k=1

Find the coefficients b₀, b₁, …, b_mof the linear function y(x₁, x₂, …, x_m) = b₀+ b₁x₁+ b₂x₂+ … + b_mx_m such that the sum of squared errors

SSE = [y(x_1k, x_2k, …, x_mk) - y_k]2 is minimized

Idea

Find a linear function that represents the predicted variable y as a function of predictive variables x₁, x₂, …, x_mand best fits the observed data

33

LINEAR REGRESSION: ANALYSIS

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35

EXPRESSIONS FOR m = 1

When m = 1 (i.e., when there is only one predictive variable), the expression of b0and b1can be written as

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EXAMPLE: m = 1 (1)

b₀= 1,796 x 2,710 – 132 x 35,290 12 x 1,796 – 132 x 132 = 50.6 b₁= 12 x 35,290 – 132 x 2,710 12 x 1,796 – 132 x 132 = 15.9 Coefficients xk yk xk2 xkyk 7 150 49 1,050 2 100 4 200 6 130 36 780 4 150 16 600 14 250 196 3,500 15 270 225 4,050 16 240 256 3,840 12 200 144 2,400 14 270 196 3,780 20 440 400 8,800 15 340 225 5,100 7 170 49 1,190 132 2,710 1,796 35,290

Observed data and analysis

37

EXAMPLE: m = 1 (2)

250 500 Demand y(x) = b0+ b1x = 50.6 + 15.9x Question:

What demand would we expect from investing in a business with a nearby population 10 thousand ? Answer:

EXAMPLE: m = 2 (1)

x1k x2k yk x1k2 x2k2 x1kx2k x1kyk x2kyk 310 10,240 2.93 96,100 96,100 3,174,400 908 30,003 980 7,510 5.27 960,400 960,400 7,359,800 5,165 39,578 1,210 10,810 6.85 1,464,100 1,464,100 13,080,100 8,289 74,049 1,290 9,890 7.01 1,664,100 1,664,100 12,758,100 9,043 69,329 1,120 13,720 7.02 1,254,400 1,254,400 15,366,400 7,862 96,314 1,490 13,920 8.35 2,220,100 2,220,100 20,740,800 12,442 116,232 780 8,540 4.33 608,400 608,400 6,661,200 3,377 36,978 940 12,360 5.77 883,600 883,600 11,618,400 5,424 71,317 1,290 12,270 7.68 1,664,100 1,664,100 15,828,300 9,907 94,234 480 11,010 3.16 230,400 230,400 5,284,800 1,517 34,792 240 8,250 1.52 57,600 57,600 1,980,000 365 12,540 550 9,310 3.15 302,500 302,500 5,120,500 1,733 29,327 10,680 127,830 63.04 11,405,800 11,405,800 118,972,800 66,031 704,692

Observed data and analysis

39

EXAMPLE: m = 2 (2)

Question:

What sales volume (i.e., y) would we expect from a store with one thousand customers per day (i.e., x₁) and a size of one thousand square meters ((i.e., x₂)?

Answer: x = 1,000 Coefficients Æ y(x₁, x₂) = -0.832 + 0.00473x₁+ 0.000175x₂           =                     704,692 66,031 63.04 b b b 500 1,410,143, 0 118,972,80 127,830 0 118,972,80 11,405,800 10,680 127,830 10,680 12 2 1 0

NON-LINEAR REGRESSION

Linear regression approaches can be often applied to non-linear regression with some modification. In this case, non-linear equations only need to be transformed to linear equations. Here we limit our consideration to some special forms of non-linear regression with one predictive variable. Non-linear equations Idea Exponential y = b₀exp(b₁x) Power y = b₀x^_b 1 Logarithmic y = b₀+ b₁ln(x) 41

EXAMPLE: y(x) = b

₀

exp(b

₁

x) (1)

EXAMPLE: y(x) = b

₀

exp(b

₁

x) (2)

x_k 0 2 4 0.00 0.50 1.00 xk yk yk = ln(yk) 0.50 3.0 1.099 0.70 6.0 1.792 0.80 8.3 2.116 0.90 12.2 2.501 0.92 14.1 2.646 0.95 16.3 2.791 0.96 17.8 2.879 0.97 19.0 2.944 0.98 21.2 3.054 0.99 24.6 3.203 ˜ _y k ˜ y = -1.04 + 4.09x˜ Observed data Figure in (x, y) space - linear˜

43

EXAMPLE: y(x) = b

₀

exp(b

₁

x) (3)

Coefficients in (x, y) space

0 10 20

0.00 0.25 0.50 0.75 1.00

Figure in (x, y) space – non-linear y_k

b0 =

b₁ =

Non-linear function Transformation Linear form

COMMONLY USED TRANSFORMATIONS

˜ Exponential y = ln(y) y = b0+ b1x y = b₀exp(b₁x) b₀= ln(b₀ ) Power y = b0x^b1 Logarithmic y = b₀+ b₁ln(x) ˜ ˜ ˜ 45

CONTENTS

• Motivation

• Forecast Evaluation

• Qualitative Methods

• Causal Models

• Time-Series Models

• Summary

TIME-SERIES MODELS

Time-series Models

Constant Level Models

Linear Trend Models

Seasonality Models Time Demand 47

TIME-SERIES MODELS

Naïve Forecast Moving Averages Exponential Smoothing Time-series Models Constant Level Models

Linear Trend Models

NAÏVE FORECAST

Idea

Main characteristics

• Easy to prepare and easy to understand

• Benchmark for more advanced forecasting approaches • Often low accuracy

The forecasts for future periods equal the actual value of the demand that is just observed in the current period

f_t+τ= A_t τ = 1, 2, … t: current period

τ: forecasting lag

A_t: demand observed in the current period t f_t+τ:forecast for a future period t + τ

49

MOVING AVERAGES (MA)

Main characteristics

• Puts equal weight on m most current observations • Lags behind trend, if any trend exists

• Requires expertise in choosing the value of m Idea

The forecasts for future periods equal the average value of demands of the previous m periods

m: number of periods to average demands

…

=

=

∑

_{=− +} +

_m

A

1,

2,

1

f

_{t τ} t_{i tm1 i}

τ

EXPONENTIAL SMOOTHING (ES)

Main characteristics

• More sensitive to recent observations if α is larger • Lags behind trend, if any trend exists

• Requires expertise in choosing the value of α Idea

The forecasts for future periods equal the exponentially weighted average of demands of previous periods

F_t = αAt+ (1 –α)Ft-1 f_t+τ= F_t τ = 1, 2, …

F_t : level estimate made at the end of period t α: smoothing parameter, 0 ≤ α ≤ 1

51

EXPONENTIAL SMOOTHING: WHY EXPONENTIAL

The forecast equation can be re-written as: “The new level estimate is the weighted average of most recent observation and previous level estimate" F_t = αAt+ (1 –α)Ft-1

F_t-1 = αAt-1+ (1 –α)Ft-2

F_t-2 = αAt-2+ (1 –α)Ft-3

Repetitive substitution yields

F_t = αAt+ α(1 – α)At-1 + α2(1 –α)At-2+ ... _{"Declining set of weights} is put on all previous

WEIGHTS IN EXPONENTIAL SMOOTHING

0.00 0.05 0.10 0.15 0.20 t-8 t-7 t-6 t-5 t-4 t-3 t-2 t-1 t α = 0.20 Where the name

"Exponential Smoothing" Comes from current Weights 53

EXAMPLE

period Actual Naïve MA MA ES ES

m=2 m=3 α=0.2 α=0.3 1 11.00 2 10.00 11.00 11.00 11.00 3 8.00 10.00 10.50 10.80 10.70 4 12.00 8.00 9.00 9.67 10.24 9.89 5 7.50 12.00 10.00 10.00 10.59 10.52 6 11.50 7.50 9.75 9.17 9.97 9.62 7 12.50 11.50 9.50 10.33 10.28 10.18 8 12.00 12.50 12.00 10.50 10.72 10.88 9 10.00 12.00 12.25 12.00 10.98 11.21 10 8.50 10.00 11.00 11.50 10.78 10.85 11 10.00 8.50 9.25 10.17 10.33 10.14 One-step forecast τ = 1

COMPARING NAÏVE FORECAST, MA, AND ES

Assumptions

Forecast errors

• Demand in period t is generated by a level value µ plus a noise et • e_t is normally distributed with mean 0 and variance σ2

• e_t is independently distributed over periods • One-step forecast τ = 1 εt+1ES= Exponential smoothing εt+1MA= Moving averages εt+1NF= At– At+1 f_t+1= A_t Naïve Forecast Forecast errors Forecasts

∑

=− + + = t 1 m t i i t A m 1 f 1

A

)

-(1

f

t 1

∑

_i0 i t i ∞ = − +

=

α

α

1 t t 1 m t i Ai A m 1 + + − = −

∑

A

A

)

-(1

t 1 0 i t i i + ∞ = −

−

∑

α

α

55

FORECAST ERROR OF MA

[ ] [ ]

0 -m m 1 A E A E m 1 A A m 1 E ] E[ t1 t 1 m t i i 1 t t 1 m t i i MA 1 t = = − =     ₋ = _{=− +} + _{=− +} + +

∑

∑

µ µ ε

[ ]

[ ]

m 1 m m m 1 A Var A Var m 1 A A m 1 Var ] Var[ 2 2 2 2 1 t t 1 m t i i 2 1 t t 1 m t i i MA 1 t σ σ σ + = + = + =     ₋ = =− + + =− + + +

∑

∑

ε Mean Variance

PROPERTIES OF FORECAST ERROR OF MA

Distribution of forecast error

Property

Why would not we always choose a very large value for m?

Because we do not fully believe the assumption of stationary demands, i.e., that µ will not change over time

m 1 m 0, N ~ 2 MA 1 t     + + σ ε m 1 m+ _σ2

The variance of forecasting error gets smaller when m gets large

57

FORECAST ERROR OF ES

[

]

[ ] [ ]

0 -) -(1 A E A E ) -(1 A A ) -(1 E ] E[ i i 1 t i t i-i 1 t i t-i i ES 1 t = = − = − =

∑

∑

∑

∞ = + ∞ = + ∞ = + µ µ ε 0 0 0 α α α α α α Mean Variance

[

]

[ ]

[ ]

-2 2 ) -(1 A Var A Var ) -(1 A A ) -(1 Var ] Var[ 2 2 2 i 2i 1 t i t-i 2i 1 t i t-i i ES 1 t σ α σ σ α α α α α α = + = + = − =

∑

∑

∑

∞ = + ∞ = + ∞ = + 0 2 0 2 0 ε

PROPERTIES OF FORECAST ERROR OF ES

Distribution of forecast error

Property

Why would not we always choose a very small value for α?

Because we do not fully believe the assumption of stationary demands, i.e., that µ will not change over time and we would sometimes prefer to give recent observations more weight than older observations

-2 0, N ~ 2 ES 1 t     + _ασ ε 2 -2 2 σ α 2

The variance of forecasting error gets smaller when α gets small

59

TIME-SERIES MODELS

Double Exponential Smoothing (Holt) Regression Analysis Time-series Models Constant Level Models

Linear Trend Models

DOUBLE EXPONENTIAL SMOOTHING (DES)

Main characteristics

• More sensitive to recent observations if α and β are larger • Captures trend

• Requires expertise in choosing the values of α and β Idea

The forecasts for future periods contain both the level estimate and the trend estimate based on the demands of previous periods

F_t = αAt+ (1 –α)(Ft-1+ Tt-1) T_t = β(Ft- Ft-1) + (1 –β)Tt-1 f_t+τ= F_t + τTt τ = 1, 2, … F_t : level estimate made at the end of period t T_t : trend estimate made at the end of period t

α: smoothing parameter for the level estimate, 0 ≤ α ≤ 1 β: smoothing parameter for the trend estimate, 0 ≤ β ≤ 1

61

INTERPRETATION OF PARAMETERS

⇒ Weighted average of just observed demand and one-step forecast made in the previous period

⇒ Weighted average of current estimate of slope and pervious estimate of slope F_t = αAt+ (1 –α)(Ft-1+ Tt-1) T_t = β(Ft- Ft-1) + (1 –β)Tt-1 f_t+τ= F_t + τTt A_t F_t F_t-1+T_t-1 F_t-1 T_t-1 T_t

EXAMPLE

period Actual Ft Tt One-step Two-step Three-step

τ= 1 τ= 2 τ= 3 1 11.0 11.00 0.00 2 12.0 11.20 0.04 11.00 3 13.5 11.69 0.13 11.24 11.00 4 15.0 12.46 0.26 11.82 11.28 11.00 5 17.0 13.57 0.43 12.72 11.95 11.32 6 18.0 14.80 0.59 14.00 12.97 12.08 7 17.5 15.81 0.67 15.39 14.43 13.23 8 19.0 16.49 15.98 14.86 9 20.0 18.21 0.86 17.16 16.57 10 22.0 19.66 0.98 19.07 17.83 11 25.0 21.51 1.15 20.64 19.94 12 25.0 23.13 1.25 22.67 21.62 20.80 α = 0.2, β = 0.2 1,2,3-step forecasts made at the end of period 8 Level and trend estimates

made at the end of period 8

63

REGRESSION ANALYSIS

Causal approach Time-series approach

LINEAR TIME-SERIES REGRESSION

Main characteristics

• Requires expertise in choosing the value of K Idea

The optimal values of b₀and b₁that best fits the demands of the previous K periods are

K: number of periods to fit demands Thus the forecasts are

f_t+τ= b0 + b1(K + τ) τ = 1, 2, … 1) K(K A 2 1 K kA 12 b 1)b (K 2 1 -A K 1 b 2 K 1 k K k t K 1 k K k t 1 1 K 1 k K k t 0 − + − = + =

∑

∑

∑

= +− = +− = +− 65

EXAMPLE

period Actual b0 b1 One-step Two-step Three-step

τ= 1 τ= 2 τ= 3 1 11.0 2 12.0 3 13.5 4 15.0 7.25 2.25 5 17.0 7.50 2.75 18.50 6 18.0 9.42 2.58 21.25 20.75 7 17.5 13.33 1.42 22.33 24.00 23.00 8 19.0 20.42 24.92 26.75 9 20.0 15.50 1.25 21.83 27.50 10 22.0 13.58 2.42 21.75 23.25 11 25.0 13.17 3.33 25.67 23.00 12 25.0 15.50 3.00 29.83 28.08 24.25 K = 4

TIME-SERIES MODELS

Triple Exponential Smoothing (Winters) Regression Analysis (not covered in class) Time-series

Models

Constant Level Models

Linear Trend Models

Seasonality Models

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SEASONALITY IN TOY INDUSTRY

20% 30% 40% Quarter one Quarter two Quarter three Quarter four Percentage of annual demand

TRIPLE EXPONENTIAL SMOOTHING (TES)

• Each season (e.g., a year) contains N periods (e.g., 4 quarters or 12

months)

• A seasonal factor c_trepresents how much demand in period t is above/below overall average

• The underlying model is

A_t= (b₀+ b₁t)c_t, c_t= c_t+N, c₁+ c₂+ … + c_N= N

• Example: N = 4, c₁= 0.65, c₂= 0.75, c₃= 1.3, and c₄= 1.3 Assumptions

seasonal factor demand de-seasonalized_{part (linear trend)}

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TRIPLE EXPONENTIAL SMOOTHING (TES)

Main characteristics Idea

The forecasts for future periods contain the de-seasonalized level estimate, the de-seasonalized trend estimate, and seasonal-factor estimate based on the demands of previous periods

F_t = αAt/ct-N+ (1 –α)(Ft-1+ Tt-1) T_t = β(Ft- Ft-1) + (1 –β)Tt-1 c_t = γAt/Ft+ (1 –γ)ct-N

f_t+τ= (Ft + τTt ) ct+τ-N τ = 1, 2, …, N

F_t : de-seasonalized level estimate made at the end of period t T_t : de-seasonalized trend estimate made at the end of period t c_t : seasonal-factor estimate made for period t

α: smoothing parameter for the level estimate, 0 ≤ α ≤ 1 β: smoothing parameter for the trend estimate, 0 ≤ β ≤ 1

INITIALIZATION PROCEDURE

1. Select 1 season (i.e., N periods) of data for initialization 2. For period N (the last period of the first season), set

F_N= (A₁+ A₂+ … + A_N)/N TN= 0

c_i= A_i/F_N

3. Start forecasting for period N+1

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EXAMPLE: INITIALIZATION

Sales Level Trend Season-fac One-step estimate estimate estimate forecast

t At Ft Tt ct Q1-95 1 44.6 0.925 Q2-95 2 46.7 0.968 Q3-95 3 50.5 1.047 Q4-95 4 51.1 48.23 0.00 1.060 Q1-96 5 44.59 Quarter

EXAMPLE: RESULT

Sales Level Trend Season-fac One-step estimate estimate estimate forecast

t At Ft Tt ct Q1-95 1 44.6 0.925 Q2-95 2 46.7 0.968 Q3-95 3 50.5 1.047 Q4-95 4 51.1 48.23 0.00 1.060 Q1-96 5 48.2 49.00 0.15 0.936 44.59 Q2-96 6 47.4 49.12 0.15 0.967 47.57 Q3-96 7 50.2 49.01 0.10 1.043 51.58 Q4-96 8 49.2 48.57 -0.01 1.051 52.07 Q1-97 9 47.6 49.02 0.08 0.943 45.46 Q2-97 10 50.2 49.67 0.19 0.976 47.50 Q3-97 11 54.7 50.39 0.30 1.051 51.99 Q4-97 12 52.7 50.59 0.28 1.049 53.28 Q1-98 13 50.1 51.30 0.37 0.950 47.99 Q2-98 14 52.3 52.06 0.44 0.982 50.44 Q3-98 15 54.8 52.43 0.43 1.050 55.20 Q4-98 16 54.7 55.47 Quarter 73

CONTENTS

• Motivation

• Forecast Evaluation

• Qualitative Methods

• Causal Models

• Time-Series Models

• Summary

SUMMARY

• Demand planning/forecasting is the starting point of all planning

• The performance of forecasting approach can be evaluated based on various metrics

- MAD - MSE - MAPE

• Various forecasting approaches exist. Which one is appropriate depends on the situation. The approaches covered in class can be classified as

- Qualitative methods, - Causal models, or - Time-series models

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ANNOUNCEMENTS