CTL.SC1x -Supply Chain & Logistics Fundamentals

Exponential Smoothing: Level & Trend Data

MIT Center for Transportation & Logistics Treatment of History • Previous models differed in the amount of history considered, but were similar in their equal treatment of it.

n Cumulative & Moving Average – equal weighting to all observations

n Naïve – all weight to most recent observation

t t xi x ∑i=1 ∑i t 1 M i xˆ x xˆ = xˆ = = + − t,t+1 = t t,t+1 t t,t+1 M

• Is there something in between these extremes? • Should we treat historical data differently?

n The value of data degrades over time

n Weight the newer observations more than the older ones • This is what exponential smoothing does

n Each observation is weighted

n Weights decrease exponentially as they age xˆ = αx + 1−α xˆ 0 ≤ α ≤1 t,t+1 t ( ) t−1,t

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 2 Simple Exponential Smoothing xˆ = αx + 1−α xˆ 0 ≤ α ≤1 t,t+1 t ( ) t−1,t Obs. α=0.2 α=0.4 α=0.6 t 0.2 0.4 0.6 Recall that: t-1 0.16 0.24 0.24 t-2 0.128 0.144 0.096 xˆ = αx + 1−α xˆ t−1,t t−1 ( ) t−2,t−1 t-3 0.1024 0.0864 0.0384 t-4 0.08192 0.05184 0.01536 Expanding and collecting terms: t-5 0.065536 0.031104 0.006144 Weights attached to observations xˆ = αx + 1−α αx + 1−α xˆ for different alpha values t,t+1 t ( )( t−1 ( ) t−2,t−1) 2 xˆ = αx +α 1−α x + 1−α xˆ t,t+1 t ( ) t−1 ( ) t−2,t−1 Continuing to substitute: 2 3 xˆ = αx +α 1−α x +α 1−α x + 1−α xˆ t,t+1 t ( ) t−1 ( ) t−2 ( ) t−3,t−2

Which leads us to the general form: 0 1 2 3 xˆ = α 1−α x +α 1−α x +α 1−α x +α 1−α x . . . t,t+1 ( ) t ( ) t−1 ( ) t−2 ( ) t−3

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 3 Exponential Smoothing Weights

1 xˆ = αx + 1−α xˆ 0 ≤ α ≤1 t,t+1 t ( ) t−1,t 0.9

0.8

0.7 The alpha value indicates the value of “new” information versus “old” information. 0.6 As alpha approaches it limits: 0.5 • αè1 leads to fast smoothing (nervous, volatile, naïve) • αè0 leads to slow smoothing (calm, staid, cumulative) 0.4 Effective Weight Weight Effective

0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10 Age of Observation

Alpha=0.1 Alpha=0.3 Alpha=0.5 Alpha=0.7 Alpha=0.9

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 4 Simple Exponential Smoothing

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 5 Time Series Analysis • Simple Exponential Smoothing n Stationary demand – no trends or seasonality

n Value of observations degrade over time n Utilizes a smoothing constant (α) where 0 ≤ α ≤ 1 n In practice 0.1 ≤ α ≤ 0.3

Underlying Model: We can also think of this as error-correcting. xt = a + e t xˆ = αx + 1−α xˆ t,t+1 t ( ) t−1,t where: ˆ ˆ ˆ xt,t+1 = αxt + xt−1,t −αxt−1,t e ~ iid (μ=0 , σ2=V[e]) t xˆ = xˆ +α x − xˆ t,t+1 t−1,t ( t t−1,t ) Forecasng Model: ˆ ˆ xt,t+1 = xt−1,t +αet xˆ = αx + 1−α xˆ 0 ≤ α ≤1 New estimate is the old estimate t,t+1 t ( ) t−1,t plus some fraction of the most recent error.

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 6 ^ x t,t+1 Example Exp. Smoothing t xt Alpha Alpha Alpha Calculate the forecast for period 6 =0.7 =0.3 =0.1 from period 5 with alpha = 0.3: 1 109 109.0 109.0 109.0 2 92 97.1 103.9 107.3 3 98 97.7 102.1 106.4 xˆ5,6 = α x5 + 1−α xˆ4,5 ( ) ( ) 4 96 96.5 100.3 105.3 = .3 104 + 0.7 100.3 =101.4 5 104 101.8 101.4 105.2 ( )( ) ( )( ) 6 98 99.1 100.4 104.5 What is the forecast for period 12 7 109 106.0 103.0 104.9 from period 10 with alpha = 0.3? 8 99 101.1 101.8 104.3 9 94 96.1 99.4 103.3 (hint: it is the same as the forecast for period 13, 14, . . .) 10 96 96.0 98.4 102.6 110 108 106 104 102 100 98 96 94 92 90 1 2 3 4 5 6 7 8 9 10 11

Actual Alpha=0.7 Alpha=0.3 Alpha=0.1

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 7 Exponential Smoothing with Trend

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 8 Time Series: Non-Stationary Models

^ x t,t+1 Forecasts 170 Alpha = MA = 160

t xt 0.3 3 150 1 95 95.0 2 102 97.0 140 3 117 103.1 104.7 130 4 123 109.0 113.9 120 5 139 118.1 126.5 6 143 125.5 135.0 110

7 164 137.0 148.6 100 8 160 143.8 155.4 9 162 149.3 161.8 90 1 2 3 4 5 6 7 8 9 10 11 10 169 155.2 163.6 Actual Alpha=0.3 MA=3 Challenges: • Moving average & simple exponential smoothing models will always lag a trend • They only look at history to find the stationary level • Need to capture the ‘trend’ or ‘seasonality’ factors

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 9 Time Series Analysis a

• Exponential Smoothing for Level & Trend Demand rate time n Expand exponential smoothing to include trend

n Often referred to as Holt’s Method b n Uses smoothing constants ( ) where 0 ≤ ≤ 1 α,β α, β Demand rate Underlying Model: time

xt = a + bt + et

2 where: et ~ iid (µ=0 , σ =V[e]) ^ This is just x t-1,t Forecasting Model: xˆ = aˆ +τbˆ t,t+τ t t Updating Procedure:

ˆ ˆ ˆ The “old” trend - at = αxt + 1−α at−1 + bt−1 ( )( ) estimated trend These are estimates of the level and bˆ = β aˆ − aˆ + 1− β bˆ from last period trend components t ( t t−1) ( ) t−1 at end of time period t. The “new” trend - difference between this CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson:period Exponential and Smoothing last forperiod’s Level & Trend estimated level. 10 Forecasting Model Example ˆ ˆ ˆ xt,t+τ = at +τbt Suppose we are in time 101 Updating Procedure and we use alpha=0.3 and aˆ = αx + 1−α aˆ + bˆ t t ( )( t−1 t−1) beta=0.1. bˆ = β aˆ − aˆ + 1− β bˆ a) Forecast demand for t=102 t ( t t−1) ( ) t−1 b) Forecast demand for t=110 Part a) Estimating demand for t=102

1. Find a^ Data 101 aˆ 0.3 x 0.7 aˆ bˆ 101 = ( ) 101 +( ) 100 + 100 ^ ^ ^ ( ) t xt a t b t x t,t+1 = (0.3)(95.0) +(0.7)(90.0 +8.5) ≈ 97.5 100 92 90 8.5 98.5 ^ 2. Find b 101 101 95 97.5 8.4 105.9 bˆ 0.1 aˆ aˆ 0.9 bˆ 101 = ( )( 101 − 100 ) +( ) 100 Part b) Estimating demand for t=110 = 0.1 97.5− 90.0 + 0.9 8.5 ≈ 8.4 This means τ=9, so ( )( ) ( )( )

x^ =97.5+(9)8.4 = 173.1 101,110 ^ 3. Find x 101,102 =97.5+8.4 = 105.9

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 11 Example

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 12 Spreadsheets are in resources Example: #VMI1984 link for this video You need to develop monthly forecasts (in pallets) for item #VMI1984 that seems to have an upward trend. Looking at past year’s data, you ^ have determined that α=0.25 and β= 0.10. Your estimated level (a 0) in ^ January (t=0) is 28 pallets/month and the estimate of trend (b 0) is 1.35. a) Using exponential smoothing, estimate a forecast for February This is easy – just plug in the numbers. ^ ^ ^ x J,F=a J + (1)(b J) = 28 + (1)(1.35) = 29.35 pallets b) It is now the end of February and demand was 27 pallets. What is your forecast for March?

=D5+E5 =C6-F5

=$B$1*C6+(1-$B$1)*F5 =$B$2*(D6-D5)+(1-$B$2)*E5

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 13 Spreadsheets are in resources Example: #VMI1984 link for this video c) Build a spreadsheet for “next month” estimates for the next 8 months.

Actual Demand “Next Month” Forecasts

How good are the forecasts? Look at MSE=(1/n)Σe2, but which one? We will need an estimate of the the forecast error for finding safety stock. Keep a current running update of the MSE – using exponential smoothing. Select an omega where 0.01≤ ω ≤0.1

2 MSE = ω x − xˆ + 1−ω MSE =$H$1*(C14-F13)^2+(1-$H$1)*I13 t ( t t−1,t ) ( ) t−1 CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 14 Damped Trends

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 15 Damped Trends • Problems with trend terms

n Trends do not continue unchanging indefinitely

n Constant linear trends can lead to over-forecasting

n This is especially true for longer forecast horizons • Damped trend model

n Slight modification to exponential smoothing model

n Select dampening parameter phi, where 0≤ ϕ ≤ 1

n If ϕ = 1, this is just a linear trend

τ Forecasting Model xˆ = aˆ + φ ibˆ t,t+τ t ∑i=1 t

aˆ = αx + 1−α aˆ +φbˆ Updating t t ( )( t−1 t−1) Procedure bˆ = β aˆ − aˆ + 1− β φbˆ t ( t t−1) ( ) t−1

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 16 Spreadsheets are in resources Same Example link for this video Build a spreadsheet for “next month” estimates for the next 8 months using a damped trend.

=D6+$B$3*E6

=$B$1*C15+(1-$B$1)*(D14+$B$3*E14) =$B$2*(D15-D14)+(1-$B$2)*$B$3*E14

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 17 • ff Comparing Linear versus Damped

38 Next Month Forecasts Comparing MSE: 36 Linear: 4.52

34 Damped: 3.80

32 Results are data 30 specific, obviously Monthly Demand Demand Monthly

28 Actual Linear Damped

26 Jan Feb Mar Apr May Jun Jul Aug Sep Oct

42 Forecast @ EOM Jan 40

Nine month forecast 38

made at EOM Jan. 36

34

Note the tapering effect 32

of the damped model. Demand Monthly 30

28

26 Feb Mar Apr May Jun Jul Aug Sep Oct

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 18 Key Points from Lesson

CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 19 Key Points xˆ = αx + 1−α xˆ t,t+1 t ( ) t−1,t • Exponential Smoothing Models

n Simple Model (level) ˆ ˆ ˆ xt,t+τ = at +τbt

n Holt Model (level & trend) aˆ = αx + 1−α aˆ + bˆ t t ( )( t−1 t−1) bˆ = β aˆ − aˆ + 1− β bˆ t ( t t−1) ( ) t−1

• Other Smoothing Models

n MSE Trending – for use in inventory models (ω)

n Damped Trends – tapering effect (ϕ) • Core Concepts:

n Value of information degrades over time

n Balance of using both old & new information CTL.SC1x - Supply Chain and Logistics Fundamentals Lesson: Exponential Smoothing for Level & Trend 20 CTL.SC1x -Supply Chain & Logistics Fundamentals Questions, Comments, Suggestions? Use the Discussion!

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