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Forecasting
- Forecasting is the projection or estimation of the occurrence of uncertain future events or level 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 Forecasting promotion
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)
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 inventory levels, 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 are
- General business conditions and state of the economy.
- Competitor actions and reactions
- Governmental legislative actions
- Marketplace trend a) Product life cycle b) Style and fashion c) Changing consumer demands
- 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 subjective phenomenon and solicitation of opinions Direct survey approach – asking prospective customers of their buying interest Indirect survey approach – information from salesmen, wholesalers, area managers, etc
- Trend identify the rate of growth or decline of a series over time (Long-term historical pattern of demand over time)
- Seasonal variations consists of annually recurring movements above and below the trend line o Demand fluctuates in a repetitive pattern from year to year o Seasonal periodic peaks and valleys should occur at the same time every year o Seasonal variations should be of larger magnitude than the random variations
- Cyclical variations are long term oscillations or swings about a trend line
- The cycles may or may not be periodic, but they often are the result of business cycles of expansion and contraction of economic activity over a number of years
- Business cycles may be due to one or more of the following: prosperity, recession, depression and recovery
- The cycles may vary with respect to the time of occurrence, the length of the phases, and the amplitude of the fluctuations
Fig. 3 Various components of a time series
Time (years)
Raw Data
Trend Component
Seasonal Component
Cyclic Component
Random Component
- Random variations have no particular pattern and usually are without specific assignable cause
- 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 variations
Various 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 interactive components of demand are classified as o Multiplicative model o Additive model o Mixed model (partially additive, partially multiplicative)
- The demand in period ( t ) for a multiplicative model is represented as
Demand = (Trend) (seasonal) (cycle) (random)
Dt = b × F × c × ε t
- The demand in period ( t ) for a additive model is represented as
Demand = level + trend + seasonal + cyclic + random
D t = a + b × t + Ft + Ct + ε t
- Generally, demand process can be modelled as
Dt = a + ε t – (level model – additive type)
Dt = a + b × t + ε t – (trend model – additive type)
Dt = ( a + b × t ) × Ft + ε t – (mixed model type- trend part is additive and seasonal
part is in multiplicative in form)
where, ε t is normally distributed with mean zero
- A best estimates for at is 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 t
Xt = α Dt +( 1 − α) Xt − 1
ft +1 = Xt
- The above equation for Xt can be written as Xt = Xt − 1 + α( Dt − Xt − 1 )
That is, X t = ft + α( Dt − ft )
- This equation indicates that using exponential average in one period as a forecast for the next period; it is possible to revise the average upward or downward, depending on the forecast error.
- Weights for the past data and for the initial average can be easily identified from the equation given below.
( ) (^0)
1
0
X 1 Dtk ( 1 ) tX
t k
k
t =^ α^ −α − + −^ α
−
=
∑
- The weight for demand in a period k from now ( t ) is α ( 1 − α ) k
- Expansion of exponentially weighted moving average equation
Xt = α Dt + ( 1 −α)[α Dt − 1 +( 1 − α) Xt − 2 ]
= α Dt + α( 1 −α) Dt − 1 +( 1 − α)^2 Xt − 2
= α Dt +α( 1 −α) Dt − 1 +( 1 −α)^2 [ Dt − 2 +( 1 − α) Xt − 3 ]
3
3 2
2
= α Dt +α( 1 −α) Dt − 1 +α( 1 −α) Dt − +( 1 − α) Xt −
- If exponential average is determined for third period, the weight for the demand of various periods and the initial average can be clearly seen from the equation below
0
3 1
2
X 3 =α D 3 +α( 1 −α) D 2 +α( 1 −α) D +( 1 − α) X
- For the demand process, the best estimate of at which minimize the following the sum of discounted squares of residuals
S= ( )
2
0
∑
∞
=
−
j
t j t d j^ D a where, d = a distant factor (0 < d < 1)
- The resulting estimate of at satisfies the following updating formation
a ˆ^ t = α Dt + ( 1 − α) a ˆ t − 1
- Average age of data in a simple EWMA is 1/ α period. In a N month moving average the average age of data is ( N +1)/ 1/ α = ( N +1)/
- Relationship between N (period of moving average) and α is 1
N
Initialization
- When significant historical data exists, simply use the average demand in the first several periods as the initial estimate of Xt.
- Forecast for future periods
ft +1 = Xt ft + n = Xt (forecast for n period ahead)
Trend Adjusted EWMA
- Dt = at + bt × t + ε (^) t is the demand process
Estimation procedure
- Estimate for bt is Tt = β ( Xt - Xt -1) + (1- β ) Tt -
- Estimate for at is Xt = αDt + (1- α ) 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 average Xt can be written as
Xt = αDt + (1- α )( Xt -1+ Tt -1)
ft +1 = Xt + Tt ft + n = Xt + ( n -1) Tt
Seasonally adjusted EWMA
- Dt = at × δt + εt is a multiplicative demand process
Where, δt = seasonal factor
- Simple EWMA provides an estimate for at
- An estimate for^ δt^ be calculated by an index,^ It
t
t t (^) X
D
I =
- Seasonal factors allow us to connect back and forth between periods of sales and the exponential average.
- Estimate for at is
Xt = α ( 1 )^ − 1
−
t (^) X I
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)
Bias measurement – Direction
- Running Average Forecast Error ( RAFE )
RAFE =
n
n Dt ft t
( ) 1
− ∑ =
Revision
MAD
RAFE
TS =
− 1 ≤ TS ≤ 1
- The limiting conditions are achieved if all errors are positive or all errors are negative.
Error updating procedure
- ω = smoothing constant
- MADt = ω( Dt − ft )+( 1 − ω) MADt − 1
- MSEt = ω( Dt − ft )^2 +( 1 − ω) MSEt − 1
Forecasting Problems
Moving 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 data Month 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 t becomes a forecast for time t+ 1 *Using ft as the forecast for period t , ft is set equal to the most recently calculated moving average, ft = MA (^) t − 1
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 540 523 530 528 528 December 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 α = 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 α ( 1 − α) k where, k varies 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
t
k
t t k
k
Xt α α D α X
where, t is the current month; here for December t = 10.
TABLE 4 Simple Exponential Smoothing Forecast
Month
Demand ( D (^) t )
Smoothed Average ( X (^) t )
Forecast ( ft ) Weightsa March 460 476.00 480 0. April 510 482.80 476 0. May 520 490.24 483 0. June 495 491.19 490 0. July 475 487.95 491 0. August 560 502.36 488 0. September 510 503.89 502 0. October 520 507.11 504 0. November 540 513.69 507 0. December 550 520.95 514 0. aAt the end of December, X DEC implicitly applies these weights to the sales from March through December. To see this, calculate X (^) DEC = 0.2(550)+0.16(540)+0.128(520)+…+0.027(460) = 520.
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
Demand Average Demanda
Seasonal Index ( It )
January 80 100 90 0. February 75 85 80 0. March 80 90 85 0. April 90 110 100 1. May 115 131 123 1. June 110 120 115 1. July 100 110 105 1. August 90 110 100 1. September 85 95 90 0. October 75 85 80 0. November 75 85 80 0. December 80 80 80 0. aAverage monthly demand: 1128/12=
TABLE 8 Computation of Seasonalized Forecast
Month
Demand ( Dt ) 2008
Deseasonalized Demand ( Dt /( It- 12 ))
Average ( Xt ) X 0 =
Forecast ( ft )
Old Seasonal Factor ( I t- 12 )
New Seasonal Factor ( It ) January 95 99.27 95.05 89.96 0.957 0. February 75 88.13 93.67 80.88 0.851 0. March 90 99.56 94.85 84.68 0.904 0. April 105 98.68 95.62 100.92 1.064 1. May 120 91.67 94.83 125.17 1.309 1. June 117 95.67 95.00 115.98 1.223 1. July 102 91.32 94.26 106.11 1.117 1. August 98 92.11 93.83 100.29 1.064 1. September 95 99.27 94.92 89.80 0.957 0. October 75 88.13 93.56 80.78 0.851 0. November 85 99.88 94.82 79.62 0.851 0. December 75 88.13 93.48 80.69 0.851 0.
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
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 as dependent variable ( Y ) and establish a linear relationship.
Let the linear relationship be Y = aX + b
Normal equations to find ‘ a ’ and ‘ b ’ are
Y = a ∑ X + nb
∑ X^ Y =^ a ∑ X + b ∑ x
2
Forecasting equation is Y = 0. 395 X = 0. 915
The point estimate for the plaster board shipments is 12.765 ≈ 13
