orecastA method often used in practice is the na$$ve forecast, whereby the forecast for the next period $1$Ft $+12$equals the demand for the current period $1$Dt$2.$ So if the actual demand for Wednesday is $35$ customers, the forecasted demand for Thursday is $35$ customers. Despite its name, the na$$ve forecast can perform well.The na$$ve forecast method may be adapted to take into account a demand trend. The increase $($or decrease$)$ in demand observed between the last two periods is used to adjust the current demand to arrive at a forecast. Suppose that last week the demand was $120$ units and the week before it was $108$ units. Demand increased $12$ units in $1$ week, so the forecast for next week would be $120+12=132$ units.The na$$ve forecast method also may be used to account for seasonal patterns. If the demand last July was $50,000$ units, and assuming no underlying trend from one year to the next, the forecast for this July would be $50,000$ units. The method works best when the horizontal, trend, or seasonal patterns are sta$-$ble and random variation is small.Horizontal Patterns: Estimating the AverageWe begin our discussion of statistical methods of time$-$series forecasting with demand that has no apparent trend, seasonal, or cyclical patterns. The horizon$-$tal pattern in a time series is based on the mean of the demands, so we focus on forecasting methods that estimate the average of a time series of data. The forecast of demand for any period in the future is the average of the time se$-$ries computed in the current period. For example, if the average of past demand calculated on Tuesday is $65$ customers, the forecasts for Wednesday, Thursday, and Friday are $65$ customers each day.Consider Figure $8\mathrm{.}5,$ which shows patient arrivals at a medical clinic over the past $28$ weeks. Assuming that the time series has only a horizontal and random pattern, one approach is simply to calculate the average of the data. However, this approach has no adaptive quality if there is a trend, seasonal, or cyclical pattern. The statistical techniques that do have an adaptive quality in estimating the average in a time series are $(1)$ simple moving averages, $(2)$ weighted moving averages, and $(3)$ exponential smoothing. Another option is the simple average, but it has no adaptive capability.Simple Moving Averages The simple moving average method simply involves calculating the average demand for the n most recent time periods and using it as the forecast for future time periods. For the next period, after the demand is known, the oldest demand from the previous average is replaced with the most recent demand and the average is recalculated. In this way, the n most recent demands are used, and the average moves from period to period.Specifically, the forecast for period t $+1$ can be calculated at the end of period t $($after the actual demand for period t is known$)$ asFt $+1=$ Sum of last n demandsn $=$ Dt $+$ Dt $-1+$ Dt $-2+$ g $+$ Dt $-$ n $+1$nwhereDt $=$ actual demand in period tn $=$ total number of periods in the averageFt $+1=$ forecast for period t $+1051015202530$Patient arrivalsWeek$350370390410430450$FIGURE $8\mathrm{.}5$Weekly Patient Arrivals at a Medical ClinicEXAMPLE $8\mathrm{.}3$ Using the Moving Average Method to Estimate Average Demanda. Compute a three$-$week moving average forecast for the arrival of medical clinic patients in week $4.$ The numbers of arrivals for the past $3$ weeks were as follows:Week Patient Arrivals$140023803411$b$.$ If the actual number of patient arrivals in week $4$ is $415,$ what is the forecast error for week $4?$c$.$ What is the forecast for week $5?$