FORECASTING DEMAND
5.4 FORECASTING METHODS
5.4.2 Time Series Methods
A time series is a set of observations of a variable at regular intervals over time. In decomposition analysis, the components of a time series are generally classified as trend T, cyclical C, seasonal S, and random or irregular R. (Note: Autocorrelation effects are sometimes included as an additional factor.)
Time series are tabulated or graphed to show the nature of the time dependence. The forecast value (Ye) is commonly expressed as a multiplicative or additive function of its components; examples here will be based upon the commonly used multiplicative model.
Yc = T. S. C. R multiplicative model (5.1)
Yc = T + S + C + R additive model (5.2)
where T is Trend, S is Seasonal, C is Cyclical, and R is Random components of a series.
Trend is a gradual long-term directional movement in the data (growth or decline).
Seasonal effects are similar variations occurring during corresponding periods, e.g., December retail sales. Seasonal can be quarterly, monthly, weekly, daily, or even hourly indexes.
Cyclical factors are the long-term swings about the trend line. They are often associated with business cycles and may extend out to several years in length.
Random component are sporadic (unpredictable) effects due to chance and unusual occurrences. They are the residual after the trend, cyclical, and seasonal variations are removed.
SUMMARYOF FORECASTING METHODS
M e t h o d Description Time Relative
horizon cost Opinion and judgment (qualitative)
Sales force composites Estimates from field sales people are aggregated SR-MR L-M Executive opinion (and/or panels) Marketing, finance, and production managers jointly SR-LR L-M
prepare forecast
contd.
Field sales and product-line management Estimates from regional sales people are reconciled MR M with national projections from product-line managers
Historical analogy Forecast from comparison with similar product SR-LR L-M previously introduced
Delphi Experts answer a series of questions (anonymously), LR M-H
receive feedback, and revise estimates
Market surveys Questionnaires/interviews for data to learn about MR-LR H consumer behaviour
Time series (quantitative)
Naive Forecast equals latest value or latest plus or minus SR L
some percentage
Moving average Forecast is average of n most recent periods SR L
(can also be weighted)
Trend projection Forecast is linear, exponential, or other projection MR-LR L of past trend
Decomposition Time series is divided into trend, seasonal, cyclical, SR-LR L and random components
Exponential smoothing Forecast is an exponentially weighted moving average, SR L where latest values carry most weight
Box-Jenkins A time-series-regression model is proposed, statistically MR-LR M-H tested, modified, and retested until satisfactory
Associative (quantitative)
Regression and correlation Use one or more associate variables to forecast via SR-MR M-H (and leading indicators) a least-squares equation (regression) or via a close
association (correlation) with an explanatory variable
Econometric Use simultaneous solution of multiple regression SR-LR H
equations that relate to broad range of economic activity Time series (quantitative)
Naive Forecast equals latest value or latest plus or SR L
minus some percentage
Moving average Forecast is average of n most recent periods SR L
(can also be weighted)
Trend projection Forecast is linear, exponential, or other projection of MR-LR L past trend
Decomposition Time series is divided into trend, seasonal, cyclical, SR-LR L and random components
Exponential smoothing Forecast is an exponentially weighted moving SR L
average, where latest values carry most weight
Box-Jenkins A time-series-regression model is proposed, MR-LR M-H
statistically tested, modified, and retested until satisfactory
Associative (quantitative)
Regression and correlation Use one or more associate variables to forecast via SR-MR M-H (and leading indicators) a least-squares equation (regression) or via a close
association (correlation) with an explanatory variable
Econometric Use simultaneous solution of multiple regression SR-LR H
equations that relate to broad range of economic activity
Key: L = low, M = medium, H = high, SR = short range, MR = medium range, LR = long range.
FORECASTING PROCEDUREFOR USING TIME SERIES
Following are the steps in time series forecasting:
1. Plot historical data to confirm relationship (e.g., linear, exponential).
2. Develop a trend equation (T) to describe the data.
3. Develop a seasonal index (SI, e.g., monthly index values).
4. Project trend into the future (e.g., monthly trend values).
5. Multiply trend values by corresponding seasonal index values.
6. Modify projected values by any knowledge of:
(C) Cyclical business conditions, (R) Anticipated irregular effects.
Trend: Three methods for describing trend are: (1) Moving average, (2) Hand fitting, and (3) Least squares.
1. MOVINGAVERAGE
A centered moving average (MA) is obtained by summing and averaging the values from a given number of periods repetitively, each time deleting the oldest value and adding a new value. Moving averages can smooth out fluctuations in any data, while preserving the general pattern of the data (longer averages result in more smoothing). However, they do not yield a forecasting equation, nor do they generate values for the ends of the data series.
MA = Number of Period
∑
xA weighted moving average (MAw) allows some values to be emphasized by varying the weights assigned to each component of the average. Weights can be either percentages or a real number.
MAwt =
(Wt X) Wt
∑ ∑
I LLUSTRATION 1: Shipments (in tons) of welded tube by an aluminum producer are shown below:
Year 1 2 3 4 5 6 7 8 9 10 11
Tons 2 3 6 10 8 7 12 14 14 18 19
(a) Graph the data, and comment on the relationship. (b) Compute a 3-year moving average, plot it as a dotted line, and use it to forecast shipments in year 12. (c) Using a weight of 3 for the most recent data, 2 for the next, and 1 for the oldest, forecast shipments in year 12.
18
Year Shipments (tonns) 3-year moving total 3-year moving average
1 2 -
-(a) The data points appear relatively linear. (b) See Table 5.1 for computations and Fig. 5.2 for plot of the MA. The MA forecast for year 12 would be that of the latest average, 17.0 tons.
(c) MAwt = ( ) (1) (14) (2) (18) (3) (19)
A hand fit or freehand curve is simply a plot of a representative line that (subjectively) seems to best fit the data points. For linear data, the forecasting equation will be of the form:
Yc = a + b (X) (signature)
where Yc is the trend value, a is the intercept (where line crosses the vertical axis), b is the slope (the rise, ∆y, divided by the run, ∆x), and X is the time value (years, quarters, etc.). The “signature”
identifies the point in time when X = 0, as well as the X and Y units.
I LLUSTRATION 2: (a) Use a hand fit line to “develop a forecasting equation for the data in Fig. 5.2. State the equation, complete with signature. (b) Use your equation to forecast tube shipments for year 12.
(a) Select points some distance apart. A straight line connecting the values for years 3 and 8 might be a good
Free hand representation of the data. From this we can determine the slope and intercept:
:
Intercept: a = 0.5 tons (Note: This is the estimated Y value at X = 0 from graph.) Equation: YC = 0.5 + 1.6X (Yr 0 = 0, X = yrs, Y = tons) (b) For year 12: Yc =0.5 + 1.6(12) = 19.7 tons.
3. LEASTSQUARES
Least squares are a mathematical technique of fitting a trend to data points. The resulting line of best fit has the following properties: (1) the summation of all vertical deviations about it is zero, (2) the summation of all vertical deviations squared is a minimum, and (3) the line goes through the means X and Y. For linear equations, the line of best fit is found by the simultaneous solution for a and b of the following two normal equations:
∑
Y = na+b∑
X∑
XY = a∑
X +b∑
X2The above equations can be used in the form shown above and are used in that form for regression. However, with time series, the data can also be coded so that
∑
X = 0. Two terms then dropout, and the equations are simplified to:∑
Y = na a=∑
nY∑
XY = b∑
X2 =∑ ∑
X2
b XY .
To code the time series data, designate the center of the time span as X = 0 and let each successive period be ±1 more unit away. (For an even number of periods, use values of ±0.5, 1.5, 2.5, etc.).
I LLUSTRATION 3: Use the least square method to develop a linear trend equation for the data from illustration 1. State the equation and forecast a trend value for year 16.
Table 5.2
Year X year coded Y shipments (tons) XY X2
1 – S 2 – 10 25
Seasonal indexes: A seasonal index (SI) is a ratio that relates a recurring seasonal variation to the corresponding trend value at the given time. In the ratio-to-moving average method of calculation monthly (or quarterly) data are typically used to compute a 12-month (or 4-quarter) moving average.
(This dampens out all seasonal fluctuations.) Actual monthly (or quarterly) values are then divided by the moving average value centered on the actual month. In the ratio-to-trend method, the actual values are divided by the trend value centered on the actual month. The ratios obtained for several of the same months (or quarters) are then averaged to obtain the seasonal index values. The indexes can be used to obtain seasonalized forecast values, Ysz (or to deseasonalize actual data). Ysz = (SI) Yc.
I LLUSTRATION 4: Snowsport International has experienced low snowboard sales in July, as shown in Table 5.3. Using the ratio-to-trend values, calculate a seasonal index value for July and explain its meaning.
Table 5.3
Yr 5 Yr 6 Yr 7 Yr 8 Yr 9 Yr 10 Yr 11 Yr l2
July actual sales 22 30 18 26 45 36 40
July trend value, Yc 170 190 210 230 250 270 290
Ratio (actual ÷ trend) 0.13 0.16 0.09 0.11 0.18 0.13 0.14 Total = 0.94.
(a) A third row has been added to Table 5.3 to show the ratio of actual to trend values for July. Using a simple average, the July index is SIJuly = 0.94 ÷7 = 0.13. This means that July is typically only 13 per cent of the trend value for July in any given year. Winter months are likely quite high.
I LLUSTRATION 5: The forecasting equation for the previous example, centered in July of year 4 with X units in months, was Yc = 1800 + 20X (July 15, Yr 4 = 0, X = mo, Y = units/yr).
* Use this equation and the July seasonal index of 0.13 to compute (a) the trend (deseasonalized) value for July of year 12 and (b) the forecast of actual (seasonalized) snowboard sales in July of year 12.
(a) July of year 12 is (8)(12) = 96 months away from July of year 4, so the/trend value is:
Yc = 1800 + 20(96) = 3,240 units/yr or 3,240 units/yr 12 mo/yr = 310 units/mo (b) The actual (seasonalized) forecast is Ysz = (SI) Yc = (0.13)(310) = 40 units.