Quantitative Forecasting Models: From the Book Referenced Below:
The same is true relative to service settings where inventory is not used to buffer demand. Here capacity
availability relative to expected demand is the issue. If we can predict demand in a service setting very accurately,
then tactically all we need to do is ensure that we have the appropriate capacity in the short term. When demand is not
predictable, then excess capacity may be needed if servicing customers quickly is important.
Bear in mind that a perfect forecast is virtually impossible. Too many factors in the business environment cannot
be predicted with certainty. Therefore, rather than search for the perfect forecast, it is far more important to establish
the practice of continual review of forecasts and to learn to live with inaccurate forecasts. This is not to say that we
should not try to improve the forecasting model or methodology or even to try to influence demand in a way that
reduces demand uncertainty. When forecasting, a good strategy is to use two or three methods and look at them for
the commonsense view. Will expected changes in the general economy affect the forecast? Are there changes in our
customers’ behaviors that will impact demand that are not being captured by our current approaches? In this chapter,
we look at both qualitative techniques that use managerial judgment and also quantitative techniques that rely on
mathematical models. It is our view that combining these techniques is essential to a good forecasting process that is
appropriate to the decisions being made.
Forecasting can be classified into four basic types: qualitative, time series analysis, causal relationships,
and simulation. Qualitative techniques are covered later in the chapter. Time series analysis, the primary focus of this
chapter, is based on the idea that data relating to past demand can be used to predict future demand. Past data may
include several components, such as trend, seasonal, or cyclical influences, and are described in the following section.
Causal forecasting, which we discuss using the linear regression technique, assumes that demand is related to some
underlying factor or factors in the environment. Simulation models allow the forecaster to run through a range of
assumptions about the condition of the forecast. In this chapter we focus on qualitative and time series techniques
since these are most often used in supply chain planning and control.
Time series analysis
A forecast in which past demand data is used to predict future demand.
Components of Demand
In most cases, demand for products or services can be broken down into six components: average demand for the
period, a trend, seasonal element, cyclical elements, random variation, and autocorrelation. Exhibit 18.1 illustrates a
demand over a four-year period, showing the average, trend, and seasonal components and randomness around the
smoothed demand curve.
Cyclical factors are more difficult to determine because the time span may be unknown or the cause of the cycle
may not be considered. Cyclical influence on demand may come from such occurrences as political elections, war,
economic conditions, or sociological pressures.
Random variations are caused by chance events. Statistically, when all the known causes for demand (average,
trend, seasonal, cyclical, and autocorrelative) are subtracted from total demand, what remains is the unexplained
portion of demand. If we cannot identify the cause of this remainder, it is assumed to be purely random chance.
Autocorrelation denotes the persistence of occurrence. More specifically, the value expected at any point is highly
correlated with its own past values. In waiting line theory, the length of a waiting line is highly autocorrelated. That
is, if a line is relatively long at one time, then shortly after that time, we would expect the line still to be long.
When demand is random, it may vary widely from one week to another. Where high autocorrelation exists, the
rate of change in demand is not expected to change very much from one week to the next.
Trend lines are the usual starting point in developing a forecast. These trend lines are then adjusted for seasonal
effects, cyclical elements, and any other expected events that may influence the final forecast. Exhibit 18.2 shows four
of the most common types of trends. A linear trend is obviously a straight continuous relationship. An S-curve is
typical of a product growth and maturity cycle. The most important point in the S-curve is where the trend changes
from slow growth to fast growth or from fast to slow. An asymptotic trend starts with the highest demand growth at
the beginning but then tapers off. Such a curve could happen when a firm enters an existing market with the objective
of saturating and capturing a large share of the market. An exponential curve is common in products with explosive
growth. The exponential trend suggests that sales will grow at an ever-increasing rate—an assumption that may not
be safe to make.
ime series forecasting models try to predict the future based on past data. For example, sales figures collected for the
past six weeks can be used to forecast sales for the seventh week. Quarterly sales figures collected for the past several
years can be used to forecast future quarters. Even though both examples contain sales, different forecasting time
series models would likely be used.
Exhibit 18.3 shows the time series models discussed in the chapter and some of their characteristics. Terms such
as short, medium, and long are relative to the context in which they are used. However, in business forecasting short
term usually refers to under three months; medium term, three months to two years; and long term, greater than two
years. We would generally use short-term forecasts for tactical decisions such as replenishing inventory or scheduling
employees in the near term, and medium-term forecasts for planning a strategy for meeting demand over the next six
months to a year and a half. In general, the short-term models compensate for random variation and adjust for shortterm changes (such as consumers’ responses to a new product). They are especially good for measuring the current
variability in demand, which is useful for setting safety stock levels or estimating peak loads in a service setting.
Medium-term forecasts are useful for capturing seasonal effects, and long-term models detect general trends and are
especially useful in identifying major turning points.
Which forecasting model a firm should choose depends on:
1.Time horizon to forecast
4. Size of forecasting budget
5.Availability of qualified personnel
Simple Moving Average When demand for a product is neither growing nor declining rapidly, and if it does not
have seasonal characteristics, a moving average can be useful in removing the random fluctuations for forecasting.
The idea here is to simply calculate the average demand over the most recent periods. Each time a new forecast is
made, the oldest period is discarded in the average and the newest period included. Thus, if we want to forecast June
with a five-month moving average, we can take the average of January, February, March, April, and May. When June
passes, the forecast for July would be the average of February, March, April, May, and June. An example using weekly
demand is shown in Exhibit 18.4. Here, 3-week and 9-week moving average forecasts are calculated. Notice how the
forecast is shown in the period following the data used. The 3-week moving average for week 4 uses actual demand
from weeks 1, 2, and 3.
A forecast based on average past demand.
Selecting the period length should be dependent on how the forecast is going to be used. For example, in the case
of a medium-term forecast of demand for planning a budget, monthly time periods might be more appropriate,
whereas, if the forecast were being used for a short-term decision related to replenishing inventory, a weekly forecast
might be more appropriate. Although it is important to select the best period for the moving average, the number of
periods to use in the forecast can also have a major impact on the accuracy of the forecast. As the moving average
period becomes shorter, and fewer periods are used, and there is more oscillation, there is a closer following of the
trend. Conversely, a longer time span gives a smoother response, but lags the trend.
The formula for a simple moving average is
Ft=At−1+At−2+At−3+⋯+At−nnFt = At−1 + At−2 + At−3 + ⋯ + At−nn
Ft = Forecast for the coming period
n = Number of periods to be averaged
At21 = Actual occurrence in the past period
At−2, At−3, and At−n = Actual occurrences two periods ago, three periods ago, and so on, up to n periods ago
A plot of the data in Exhibit 18.4 shows the effects of using different numbers of periods in the moving average.
We see that the growth trend levels off at about the 23rd week. The three-week moving average responds better in
following this change than the nine-week average, although overall, the nine-week average is smoother.
The main disadvantage in calculating a moving average is that all individual elements must be carried as data
because a new forecast period involves adding new data and dropping the earliest data. For a three- or six-period
moving average, this is not too severe. But plotting a 60-day moving average for the usage of each of 100,000 items
in inventory would involve a significant amount of data.
Weighted Moving Average Whereas the simple moving average assigns equal importance to each component of
the moving average database, a weighted moving average allows any weights to be placed on each element, provided,
of course, that the sum of all weights equals 1. For example, a department store may find that, in a four-month period,
the best forecast is derived by using 40 percent of the actual sales for the most recent month, 30 percent of two months
ago, 20 percent of three months ago, and 10 percent of four months ago. If actual sales experience was
Weighted moving average
A forecast made with past data where more recent data is given more significance than older data.
the forecast for month 5 would be
= 0.40(95) + 0.30(105) + 0.20(90) + 0.10(100)
= 38 + 31.5 + 18 + 10
The formula for a weighted moving average is
Ft = W1At−1 + W2At−2 + … + Wn At−n
W1 = Weight to be given to the actual occurrence for the period t − 1
W2 = Weight to be given to the actual occurrence for the period t − 2
Wn = Weight to be given to the actual occurrence for the period t − n
n = Total number of prior periods in the forecast
Although many periods may be ignored (that is, their weights are zero) and the weighting scheme may be in any order
(for example, more distant data may have greater weights than more recent data), the sum of all the weights must
n∑i=1Wi=1∑i = 1nWi = 1
Suppose sales for month 5 actually turned out to be 110. Then the forecast for month 6 would be
= 0.40(110) + 0.30(95) + 0.20(105) + 0.10(90)
= 44 + 28.5 + 21 + 9
Experience and trial and error are the simplest ways to choose weights. As a general rule, the most recent past is the
most important indicator of what to expect in the future, and, therefore, it should get higher weighting. The past
month’s revenue or plant capacity, for example, would be a better estimate for the coming month than the revenue or
plant capacity of several months ago.
However, if the data are seasonal, for example, weights should be established accordingly. Bathing suit sales in
July of last year should be weighted more heavily than bathing suit sales in December (in the Northern Hemisphere).
The weighted moving average has a definite advantage over the simple moving average in being able to vary the
effects of past data. However, it is more inconvenient and costly to use than the exponential smoothing method, which
we examine next.
Exponential Smoothing In the previous methods of forecasting (simple and weighted moving averages), the major
drawback is the need to continually carry a large amount of historical data. (This is also true for regression analysis
techniques, which we soon will cover.) As each new piece of data is added in these methods, the oldest observation is
dropped and the new forecast is calculated. In many applications (perhaps in most), the most recent occurrences are
more indicative of the future than those in the more distant past. If this premise is valid—that the importance of data
diminishes as the past becomes more distant—then exponential smoothing may be the most logical and easiest
method to use.
A time series forecasting technique using weights that decrease exponentially (1 – α) for each past period.
Exponential smoothing is the most used of all forecasting techniques. It is an integral part of virtually all
computerized forecasting programs, and it is widely used in ordering inventory in retail firms, wholesale companies,
and service agencies.
Exponential smoothing techniques have become well accepted for six major reasons:
1.Exponential models are surprisingly accurate.
2.Formulating an exponential model is relatively easy.
3.The user can understand how the model works.
4.Little computation is required to use the model.
5.Computer storage requirements are small because of the limited use of historical data.
6.Tests for accuracy as to how well the model is performing are easy to compute.
In the exponential smoothing method, only three pieces of data are needed to forecast the future: the most recent
forecast, the actual demand that occurred for that forecast period, and a smoothing constant alpha (β). This
smoothing constant determines the level of smoothing and the speed of reaction to differences between forecasts and
actual occurrences. The value for the constant is determined both by the nature of the product and by the manager’s
sense of what constitutes a good response rate. For example, if a firm produced a standard item with relatively stable
demand, the reaction rate to differences between actual and forecast demand would tend to be small, perhaps just 5 or
10 percentage points. However, if the firm were experiencing growth, it would be desirable to have a higher reaction
rate, perhaps 15 to 30 percentage points, to give greater importance to recent growth experience. The more rapid the
growth, the higher the reaction rate should be. Sometimes users of the simple moving average switch to exponential
smoothing but like to keep the forecasts about the same as the simple moving average. In this case, is approximated
by 2 ÷ (n + 1), where n is the number of time periods in the corresponding simple moving average.
Smoothing constant alpha (α)
The parameter in the exponential smoothing equation that controls the speed of reaction to differences between
forecasts and actual demand.
The equation for a single exponential smoothing forecast is simply
Ft = Ft−1 + α(At−1 − Ft−1)
= The exponentially smoothed forecast for period t
= The exponentially smoothed forecast made for the prior period
= The actual demand in the prior period
= The desired response rate, or smoothing constant
This equation states that the new forecast is equal to the old forecast plus a portion of the error (the difference between
the previous forecast and what actually occurred).
To demonstrate the method, assume that the long-run demand for the product under study is relatively stable and
a smoothing constant (α) of 0.05 is considered appropriate. If the exponential smoothing method were used as a
continuing policy, a forecast would have been made for last month. Assume that last month’s forecast (Ft21) was 1,050
units. If 1,000 actually were demanded, rather than 1,050, the forecast for this month would be
= Ft−1 + (At−1 − Ft−1)
= 1,050 + 0.05(1,000 − 1,050)
= 1,050 + 0.05(−50)
= 1,047.5 units
Because the smoothing coefficient is small, the reaction of the new forecast to an error of 50 units is to decrease the
next month’s forecast by only 21/2 units.
When exponential smoothing is first used for an item, an initial forecast may be obtained by using a simple
estimate, like the first period’s demand, or by using an average of preceding periods, such as the average of the first
two or three periods.
Single exponential smoothing has the shortcoming of lagging changes in demand. Exhibit 18.5 presents actual
data plotted as a smooth curve to show the lagging effects of the exponential forecasts. The forecast lags during an
increase or decrease, but overshoots when
a change in direction occurs. Note that the higher the value of alpha, the more closely the forecast follows the actual.
To more closely track actual demand, a trend factor may be added. Adjusting the value of alpha also helps. This is
termed adaptive forecasting. Both trend effects and adaptive forecasting are briefly explained in following sections.
Exponential Smoothing with Trend Remember that an upward or downward trend in data collected over a
sequence of time periods causes the exponential forecast to always lag behind (be above or below) the actual
occurrence. Exponentially smoothed forecasts can be corrected somewhat by adding in a trend adjustment. To correct
the trend, we need two smoothing constants. Besides the smoothing constant α, the trend equation also uses
a smoothing constant delta (δ). Both alpha and delta reduce the impact of the error that occurs between the actual
and the forecast. If both alpha and delta are not included, the trend overreacts to errors.
Smoothing constant delta (δ)
An additional parameter used in an exponential smoothing equation that includes an adjustment for trend.
To get the trend equation going, the first time it is used the trend value must be entered manually. This initial
trend value can be an educated guess or a computation based on observed past data.
The equations to compute the forecast including trend (FIT) are
Ft = FITt−1 + α(At−1 − FITt−1)
Tt = Tt−1 + δ(Ft + FITt−1)
FITt = Ft + Tt
= The exponentially smoothed forecast that does not include trend for period t
= The exponentially smoothed trend for period t
= The forecast including trend for period t
= The forecast including trend made for the prior period
= The actual demand for the prior period
= Smoothing constant (alpha)
= Smoothing constant (delta)
To make an exponential forecast that includes trend, step through the equations one at a time.
Using equation 18.4 make a forecast that is not adjusted for trend. This uses the previous forecast and previous actual d
Using equation 18.5 update the estimate of trend using the previous trend estimate, the unadjusted forecast just made, a
Make a new forecast that includes trend by using the results from steps 1 and 2.
Exponential smoothing requires that the smoothing constants be given a value between 0 and 1. Typically fairly small
values are used for alpha and delta in the range of .1 to .3. The values depend on how much random variation there is
in demand and how steady the trend factor is. Later in the chapter, error measures are discussed that can be helpful in
picking appropriate values for these parameters.
Linear Regression Analysis Regression can be defined as a functional relationship between two or more
correlated variables. It is used to predict one variable given the other. The relationship is usually developed from
observed data. The data should be plotted first to see if they appear linear or if at least parts of the data are linear. Linear
regression refers to the special class of regression where the relationship between variables forms a strai ...
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