demand forecasting - note

7/27/2019 Demand Forecasting - Note

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National Institute of Technology Calicut Department of Mechanical Engineering

Demand Forecasting 1 March 2012

Forecasting

Forecasting is the projection or estimation of the occurrence of uncertain future events orlevel of activity.

Used for predicting Demand, Revenues, Costs, Profits, Prices, Technological changes, Environment

problems, Rainfall, etc.

Forecast is one input to many types of planning and control

Fig. 1 Master forecasting

Fig. 2 Functional forecasting

Forecasting usually involves the following considerations Item to be forecasted (products, product groups, assemblies, etc)

Top down or bottom up forecasting Forecasting techniques (quantitative or qualitative model)

Financial planning (financial aggregate, cash flow,

balance sheets, income statement)

Master scheduling (product output levels)

Production planning (aggregate output levels)

Market planning (product lines, pricing, and

promotionForecasting

Policy decisions (economic, social, political,

technological conditions)

Forecasting

Operations decisions (output scheduling and

control)

Plant decision (facility location and layout)

Process decision (process and methods)

Product design (product lines, services and market)


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Units of measure (Rs, units, weights, etc) Time interval (weeks, months, quarters, etc) Forecast horizons (how many time intervals to include) Forecasting components (levels, trends, seasonal, cycles and random variations) Forecast accuracy (error measurement) Exception reporting and special situations Revision of forecasting model parameters

Sales Forecasting

Sales forecasts are used to establish product levels, facilitate scheduling, set inventorylevels, determine manpower loading, make purchasing decisions, establish sales

conditions pricing and advertising, and financial planning cash budgeting and capital

budgeting Generally, sales forecast is used to estimate the demand of independent items Many environmental factors influence the demand for products and services of an

organisation.

Some major environmental factors are1. General business conditions and state of the economy.2. Competitor actions and reactions3. Governmental legislative actions4. Marketplace trend

a) Product life cycle

b) Style and fashion

c) Changing consumer demands

5. Technological innovations Presence of randomness preclude a perfect forecast Forecast for groups of items tend to be more accurate than forecast for individual items Error potential increases as time horizon of a forecast increases We are interested in estimating the level of future demand. Statistical techniques are used

to forecast.

Statistical methods use historical (past) data All statistical forecasting techniques assume to some extent that forces that have existed

in the past will persist in the future.

New product demand (with little or no history of past demand) rely more on subjectivephenomenon and solicitation of opinions

Direct survey approach asking prospective customers of their buying interest Indirect survey approach information from salesmen, wholesalers, area managers,

etc


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Comparison with substitute or comparable products Limited market test of the new product

Basic demand forecasting models

Time series analysis, soliciting opinions, economic indicators and econometric models These are short range forecasting models Generally, these forecasts give starting point for making the final forecast Final forecast usually requires an additional input in the form of judgment, intuition, and

experience and requires periodic review

Note on economic indicators and econometric models

Economic Indicators

Knowledge of one variable is used to predict the value of another (prediction byassociation)

Certain economic indicators are Gross domestic product (GDP), Personal income, Bank deposits, Freight car

loadings, etc

One or more of these indicators have relationship with the forecast variableEconometric Models

Involves a set of simultaneous equations that explains the interactions of variablesinvolved in a business situation

Attempt to show the relationships between relevant variables such as supply, demand,prices and purchasing power of the consumer

Time Series Analysis

Time series analysis predict future demand from past interval data. A time series is a set of time ordered observations on a variable during successive and

equal time periods

Period Jan Feb Mar Apr May Jun

Demand (in units) 75 70 82 76 87 90

The above table shows a time series. This table shows the past demand in successive and

equal interval of time

Period Jan Mar Apr May Jul Sep

Demand (in units) 75 70 82 76 87 90

This table is not representing a time series as it is not showing demand in equal interval of

time.

Interactive components: levels, trends, seasonal variations, cyclical variations, andrandom variations

Fig. 3 illustrates the various components of time series Levels indicates the scale or magnitude of a time series


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Random variations have no particular pattern and usually are without specific assignablecause

They represent all influences not included in trend, seasonal, and cyclical variations Erratic occurrence may be isolated and removed from the data, but there are no general

techniques for doing so

Averaging process will help to eliminate its influence Random variations are often referred to as noise, residuals, or irregular variationsVarious techniques in time series analysis

Last period demand Arithmetic average Simple moving average

Weighted moving average Exponentially weighted moving average (EWMA)

Simple exponentially weighted moving average Trend adjusted exponentially weighted moving average Seasonally adjusted exponentially weighted moving average Trend and Seasonally adjusted exponentially weighted moving average

Regression analysis (Linear forecasting technique) The time series contains interactive components. The models representing interactivecomponents of demand are classified as

o Multiplicative modelo Additive modelo Mixed model (partially additive, partially multiplicative)

The demand in period (t) for a multiplicative model is represented asDemand = (Trend) (seasonal) (cycle) (random)

Dt= b F c t

The demand in period (t) for a additive model is represented asDemand = level + trend + seasonal + cyclic + random

Dt = a + b t+ Ft+ Ct+ t

Generally, demand process can be modelled asDt= a + t (level model additive type)

Dt= a + b t+ t (trend model additive type)

Dt= (a + b t) Ft+ t

(mixed model type-

trend part is additive and seasonalpart is in multiplicative in form)


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Some Notations

Dt Actual demand for the period t

ft - forecast for the period t

Simple Moving Average This method is used to represent a demand process of type

Dt= a + t

That is, the demand is represented as a level with random noise.

Parameter a is not really known and is subjected to random changes from time to time. Using the simple moving average procedure we can get an estimate for a and it can be get

updated as time progresses.

Estimating procedure (updating procedure)

The procedure involves the determination of average of demand of lastNperiods. As new period demand observation is available, the old period demand data is removed

from average calculation.

Number of periods considered for average calculation is same but, demand dataconsidered for the calculation is different at different time periods.

This way of estimation is actually an updating procedure also.MAt,N = (Dt+Dt-1 +Dt-2 + .Dt-N+1)/N

Where,Nis the period of moving average,Dtis the actual demand at period tand

MAt,N is the moving average at period tbased on demand ofNperiods.

The estimate ofa, as of the end of the period tis represented asta =MAt,N

This estimate ofa results from minimizing the sum of squares of error over the precedingNperiod.

A slightly simple updating procedure for this method isMAt,N=MAt-1,N +

N

NtDtD

ta minimizes the standard error, ( )21

+=

=

t

Njtjajxs

Forecast equations areft+1 =MAt,N

ft+n =MAt,N (forecast for n period ahead)

Simple EWMA

Underlying demand model isDt= at+ t


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where, t is normally distributed with mean zero

A best estimates for atis the exponentially weighted smoothing average. The equation for simple exponential smoothing uses only two pieces of information: (1)

actual demand for the most recent period and (2) the most recent average

Let Xt= exponentially weighted moving average for the period t1)1( += ttt XDX

Forecast equation isft+1 =Xt

The above equation forXtcan be written as )( 11 += tttt XDXX That is, )( tttt fDfX +=

This equation indicates that using exponential average in one period as a forecast for thenext period; it is possible to revise the average upward or downward, depending on theforecast error.

Weights for the past data and for the initial average can be easily identified from theequation given below.

( ) 01

0

)1(1 XDXt

kt

kt

k

t +=

=

The weight for demand in a period kfrom now (t) is k)1( Expansion of exponentially weighted moving average equation

Xt= ])1()[1( 21 ++

ttt XDD = 2

2

1 )1()1( ++ ttt XDD

])1([)1()1( 322

1 +++= tttt XDDD

3

3

2

2

1 )1()1()1( +++= tttt XDDD

If exponential average is determined for third period, the weight for the demand ofvarious periods and the initial average can be clearly seen from the equation below

0

3

1

2

233 )1()1()1( XDDDX +++=

For the demand process, the best estimate ofat which minimize the following the sum ofdiscounted squares of residuals

S= ( )2

0

1

=

+

j

tjt

j aDd where, d= a distant factor (0 < d< 1)

The resulting estimate ofatsatisfies the following updating formation( ) 11 += ttt aDa

Average age of data in a simple EWMA is 1/period. In aNmonth moving average theaverage age of data is (N+1)/2

1/ = (N+1)/2

Relationship betweenN(period of moving average) and is1

2+

=N


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Initialization

When significant historical data exists, simply use the average demand in the first severalperiods as the initial estimate ofXt.

Forecast for future periodsft+1 =Xtft+n =Xt(forecast for n period ahead)

Trend Adjusted EWMA

Dt= at+ bt t+ t is the demand processEstimation procedure

Estimate for tb is Tt= (Xt-Xt-1) + (1- )Tt-1 Estimate for ta isXt= Dt+ (1-)ft The above equation is written in terms of forecast. A trend adjusted forecast is the sum of the best average and trend available at the current

time

Hence, the averageXtcan be written asXt= Dt+ (1-)(Xt-1+Tt-1)

Forecast equations areft+1 =Xt+ Tt

ft+n=Xt+ (n-1) Tt

Seasonally adjusted EWMA

Dt= at t+ t is a multiplicative demand processWhere,t= seasonal factor

Simple EWMA provides an estimate for at An estimate for tbe calculated by an index,It

t

t

t

X

DI =

Seasonal factors allow us to connect back and forth between periods of sales and theexponential average.

Estimate for ta isXt= ( ) 11

+ tmt

t XI

D

where, m is the number of periods in seasonal pattern (m = 12 for monthly data and m = 4

for quarterly data with an annual seasonal pattern)


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Estimate for tisIt= ( ) mtt

t IX

D+ 1

Forecast equation isft+1 =XtIt+1-mTrend & seasonally adjusted EWMA

Demand process is ttttt tbaD ++= )( Estimate for ta is

( ) )(1 11

++= ttmt

tt TX

I

DX

Estimate for tb is( ) ( )

111 +=

ttttTXXT

Estimate for tI is( ) mt

t

tt I

X

DI += 1

Forecast equation is( ) mtttt ITXf ++ += 11

Forecast Error Measurement

The forecast error measurement belongs to any one of the followingError estimate to know the magnitude of error

to get an idea on biasness of forecast

to get an idea on revision of parameters

Magnitude of Error (extent of error)

Mean Absolute Deviation (MAD)MAD =

n

tftDn

t

||

1

=

where, n is the number deviations available

Mean Square Error,MSE= 2)(1 n

tftDn

t

=

MSEpenalise deviations with large magnitude Standard deviation, ( )

=

=

n

t

ttr

n

fDS

1

22

2

The 2 in the denominator represents the number of degree of freedom. Sr=1.25MAD for error normally distributed.


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Bias measurement Direction

Running Average Forecast Error (RAFE)RAFE=

n

tftDn

t

)(

1

=

Revision

Tracking Signal (TS)MAD

RAFETS=

11 TS

The limiting conditions are achieved if all errors are positive or all errors are negative.Error updating procedure

= smoothing constant 1)1()( += tttt MADfDMAD 12 )1()( += tttt MSEfDMSE

Forecasting ProblemsMoving Average Methods

Twelve-month demand data of a product is given below. Use this data to develop forecasts

using three- and six-month moving averages, and three-month weighted moving average

(weights for data: 0.25, 0.25 and 0.5 for most recent) method.

TABLE 1 Demand dataMonth Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

Demand 450 440 460 510 520 495 475 560 510 520 540 550

TABLE 2 Three and Six-Month Moving Averages Used as Forecasts

Month

Demand

(Dt)

Three-Month

Moving

Average

(MA t)

Three-Month

Moving Average

Forecast*

(ft)

Six-Month

Moving

Average

(MA t)

Six-Month

Moving

Average

Forecast (ft)

January 450 - - - -

February 440 - - - -March 460 450 - - -

April 510 470 450 - -

May 520 497 470 - -

June 495 508 497 479 -

July 475 497 508 483 479

August 560 510 497 503 483

September 510 515 510 512 503

October 520 530 515 513 512

November 540 523 530 517 513

December 550 537 523 526 517

Note: The average at time tbecomes a forecast for time t+1*Usingftas the forecast for period t, ftis set equal to the most recently calculated moving

average,ft=MA 1t


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TABLE 3 Forecast Using Moving Average and Weighted Moving Average

Month

Demand

(Dt)

Three-Month

Moving

Average

(MA t)

Three-Month

Moving

Average

Forecast (ft)

Three -Month

Weighted Moving

Average

(0.25,0.25.0.50)

Most Recent (MA t)

Three-Month

Weighted

Moving

Average

Forecast (ft)

January 450 - - - -

February 440 - - - -

March 460 450 - 453 -

April 510 470 450 480 453

May 520 497 470 503 480

June 495 508 497 505 503

July 475 497 508 491 505

August 560 510 497 523 491

September 510 515 510 514 523

October 520 530 515 528 514

November 540 523 530 528 528December 550 537 523 540 528

Simple Exponential Smoothing Method

Determine the forecast from March to December for the demand data given in Table 1. Given

= 0.2 and initial average for March = 480. The last column of the Table 4 is the weights

given to various months when exponentially weighted average of December month is

calculated. The weight for a month can be calculated as ( )k 1 where, kvaries from 0 to(10-1) for December to March with December having a value of zero. That is, the value of k

is zero for the current month, 1 for just previous month and so on. Hence, the smoothing

expression can be written as

( ) ( )

=

+=

1

0

011t

k

tkt

kt XDX

where, tis the current month; here for December t= 10.

TABLE 4 Simple Exponential Smoothing Forecast

Month

Demand

(D t)

Smoothed Average

( tX )Forecast

(ft) Weightsa

March 460 476.00 480 0.027

April 510 482.80 476 0.034

May 520 490.24 483 0.042June 495 491.19 490 0.052

July 475 487.95 491 0.066

August 560 502.36 488 0.082

September 510 503.89 502 0.102

October 520 507.11 504 0.128

November 540 513.69 507 0.160

December 550 520.95 514 0.200aAt the end of December,XDEC implicitly applies these weights to the sales from March through

December. To see this, calculateXDEC= 0.2(550)+0.16(540)+0.128(520)++0.027(460) = 520.95


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Trend adjusted exponential smoothing method

Twelve-month demand from March to February is given in Table 5. Determine the forecast

for these months using trend adjusted exponential smoothing method.

Given =0.2; =0.2; T0=9;X0= 480

TABLE 5 Trend adjusted exponential smoothed forecastMonth Demand

(Dt)

Simple

exponential

averagea

(Xt)

Trend adjusted

exponential

average (Xt)

Trend

(Tt)

Forecast

(ft)

480.00 9.00

March 460 476.00 483.20 7.84 489.00

April 510 482.80 494.83 8.60 491.04

May 520 490.24 506.74 9.26 503.43

June 495 491.19 511.80 8.42 516.00

July 475 487.95 511.18 6.61 520.22

August 560 502.36 526.23 8.30 517.79

September 510 503.89 529.62 7.32 534.53October 520 507.11 533.55 6.64 536.94

November 540 513.69 540.16 6.63 540.20

December 550 520.95 547.43 6.76 546.79

January 555 527.76 554.35 6.79 554.19

February 569 536.01 562.72 7.11 561.15aGiven for comparison purposes. Note how the simple exponential average lags the upward

trend.

Seasonally Adjusted Exponential Smoothing Method

Monthly demand data for three years is given in Table 6. Use the first two years data to

determine monthly seasonal index and determine monthly forecast of third year.TABLE 6 Demand data for a seasonal product

Month

Demand

2006 2007 2008

January 80 100 95

February 75 85 75

March 80 90 90

April 90 110 105

May 115 131 120

June 110 120 117

July 100 110 102

August 90 110 98

September 85 95 95

October 75 85 75

November 75 85 85

December 80 80 75

Consider the average demand of years 2006 and 2007 as initial average (X0) to start

exponential smoothing forecast. Given =0.2, = 0.05

Seasonal index calculation

Calculate the average demand for each month (eg:- Average demand of January = January

month demands in 2006 + 2007)

Average these to get average monthly demand


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Seasonal index of January = Average demand of January divided by average monthly

demand

Similarly calculate seasonal index of all other month.

TABLE 7 Sample Seasonal Index Computation

Month

DemandAverage

Demanda

SeasonalIndex

(It)2006 2007

January 80 100 90 0.957

February 75 85 80 0.851

March 80 90 85 0.904

April 90 110 100 1.064

May 115 131 123 1.309

June 110 120 115 1.223

July 100 110 105 1.117

August 90 110 100 1.064

September 85 95 90 0.957

October 75 85 80 0.851

November 75 85 80 0.851

December 80 80 80 0.851aAverage monthly demand: 1128/12=94

TABLE 8 Computation of Seasonalized Forecast

MonthDemand(Dt) 2008

Deseasonalized

Demand(Dt/(It-12))

Average

(Xt)X 0 =94Forecast

(ft)

Old

Seasonal

Factor(It-12)

New

Seasonal

Factor(It)

January 95 99.27 95.05 89.96 0.957 0.959

February 75 88.13 93.67 80.88 0.851 0.848

March 90 99.56 94.85 84.68 0.904 0.906

April 105 98.68 95.62 100.92 1.064 1.066

May 120 91.67 94.83 125.17 1.309 1.307

June 117 95.67 95.00 115.98 1.223 1.223

July 102 91.32 94.26 106.11 1.117 1.115

August 98 92.11 93.83 100.29 1.064 1.063

September 95 99.27 94.92 89.80 0.957 0.959

October 75 88.13 93.56 80.78 0.851 0.849

November 85 99.88 94.82 79.62 0.851 0.853

December 75 88.13 93.48 80.69 0.851 0.849

Problem from Economic Indicators

The General Manager of a building materials production plant feels the demand for plaster

board shipments may be related to the number of construction permits issued in the district

during the previous quarter. The manager has collected the data shown in the accompanying

table. Derive a regression forecasting equation and determine a point estimate for plaster

board shipments when the number of construction permits is 30


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Construction permits 15 9 40 20 25 25 15 35

Plaster board shipments 6 4 16 6 13 9 10 16

Solution

Consider construction permit as independent variable (X) and plaster board shipments asdependent variable (Y) and establish a linear relationship.

Let the linear relationship be Y= aX+ b

Normal equations to find a and b are

nbXaY +=

+= xbXaYX2

Forecasting equation is 915.0395.0 == XY

The point estimate for the plaster board shipments is 12.765 13

demand forecasting - note

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