Workload Assessment (Forecasting)
3.2 Time Series and Extrapolation
A time series is a stream of data that represents the past measurements. Each event (observation of demand) is time-tagged so that it is known where it is located in the series of data. The time series consists of data recorded at different time periods such as weekly or daily for the variable, which could be units produced or demands received. Forecasters attempt to predict the next value or set of values that will occur at a future time. In time-series analysis, external causes are not brought into the pic- ture. The pattern of the series is considered to be time-dependent. That is why the definition of American Production and Inventory Control Society (APICS) states
“the values of the variables are functions of the time periods.” APICS now lists itself as The Association for Operations Management. Extrapolation is the process of moving from observed data (past and present) to the unknown values of future points. The extrapolation of time series is one of the main functions of forecasting.
The data in time series may consist of several different kinds of variations.
Important among them are random variations, an increasing or decreasing trend, and seasonal variations. Random variations (see Figure 3.1) occur because the demand is seldom constant at a given level. Minor variations do occur from one
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Figure 3.1 Time series with random variations.
period to another. In many instances, there are no specific assignable causes for these random variations. The random variations are a result of the economic envi- ronment and the marketplace within which an organization is operating.
In addition to random variations, the time-series data may exhibit an increas- ing or decreasing trend (see Figure 3.2 for an increasing trend and Figure 3.3 for a decreasing trend). If the time series shows an increasing linear trend, as in Figure 3.3, there is a constant rate of change (increasing) with the increase in time. Linear decreases can also occur. Nonlinear trend lines occur where the rate of change is
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Figure 3.2 Time series with random variations and increasing trend.
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Figure 3.3 Time series with random variations and decreasing trend.
geometric. In such cases, it is possible to extrapolate the curve by eye, but the rate of change can also be calculated, and projections can be made mathematically.
The time series may also exhibit seasonal (or cyclical) variations. Figure 3.4 shows seasonal variations coupled with random variations, whereas Figure 3.5 com- bines all three components—random variations, an increasing trend, and seasonal variation. For example, one may observe seasonal variations in demand for resort hotels and home heating oil.
There are other categories of time series that are useful to know about. For example, step functions that look like stairs—going up or going down might also be present in a time series. It is hoped that the forecast can predict when the next
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Figure 3.4 Time series with random variations and seasonal variations.
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Figure 3.5 Time series with random variations, seasonal variations, and increas- ing trend.
step will occur and how far up or down it goes. If they occur regularly, a good estimate can be made of the period between steps and the timing of future steps.
Step functions are characteristic of systems that can only change in given quanti- ties. Thus, if sales must be made in lots of 100 units, then each change in demand will occur in hundred-unit steps. If December sales were 500, the sales for January could be 400, 500, or 600, that is, it could step up or down, or stay the same as the December sales of 500.
Time-series data can also reflect erratic bursts called impulses. If these spikes appear from time to time and some pattern can be associated with them, they could be added to the list of what can be extrapolated or projected into the future.
Short-term cycles can be piggybacked on long-term cycles. All kinds of com- binations are possible. The key point is that cycles, trends, and steps are the basic pallet for the development of forecasting models.
The term “time-series analysis” deserves explanation. Time-series analysis is
“analysis of any variable classified by time, in which the values of the variable are functions of the time periods.” Time-series analysis deals with using knowledge about cycles, trends, and averages to forecast future events. It is always useful to try to find causal links between well-known cycles, such as the seasons, and demand patterns that are being tracked. Using symbols, a forecast is to be made from some series of numbers x1, x2, …, xn. The numbers might be monthly sales figures for the past year.
The question is: What use can be made of these numbers to indicate the monthly sales figures for next year? How will the forecasted time series be used? Production schedules will be drawn up to satisfy demand. Orders will be placed for inventory to be purchased. Workers will be hired and trained, or let go as the production crew is downsized. Rented space will be increased or decreased, or it will stay the same. Many decisions that are interrelated require believable forecasts before they can be made.
Time-series analysis uses statistical methods, including trend and cycle analy- ses, to predict future values based on the history of sales or whatever the time series describes. In the following sections, we will discuss moving average (MA), weighted moving average (WMA), exponential smoothing (ES), and trend analysis methods for forecasting and extrapolation. How to make forecasts when seasonal variations are present is also discussed.