In Case $4-4,$ Julie Murphy developed a naive model that combined seasonal and trend estimates $($similar to Equation $4\mathrm{.}5).$ One of the major reasons why she chose this naive model was its simplicity. Julie knew that her father, Glen, would need to understand the forecasting model used by the company.

It is now October of $2002,$ and a lot has changed. Glen Murphy has retired. Julie has completed sev$-$ eral business courses, including business forecasting, at the local university. Murphy Brothers Furniture built a factory in Dallas and began to manufacture its own line of furniture in October $1995.$

Monthly sales data for Murphy Brothers Furni$-$ ture from $1996$ to October $2002$ are shown in Table $5-10.$ As indicated by the pattern of these data demonstrated in Figure $5-13,$ sales have grown dra$-$ matically since $1996.$ Unfortunately, Figure $5-13$ also demonstrates that one of the problems with demand is that it is somewhat seasonal. The company$$s gen$-$ eral policy is to employ two shifts during the sum$-$ mer and early fall months and then work a single shift through the remainder of the year. Thus, sub$-$ stantial inventories are developed in the late sum$-$ mer and fall months until demand begins to pick up in November and December. Because of these

production requirements, Julie is very anxious to prepare short$-$term forecasts for the company that are based on the best available information concern$-$ ing demand.

For forecasting purposes, Julie has decided to use only the data gathered since $1996,$ the first full year Murphy Brothers manufactured its own line of furniture $($Table $5-10).$ Julie can see $($Figure $5-13)$ that her data have both trend and seasonality. For this reason, she decides to use a time series decom$-$ position approach to analyze her sales variable.

Since Figure $5-13$ shows that the time series she is analyzing has roughly the same variability throughout the length of the series, Julie decides to use an additive components model to forecast. She runs the model Yt $=$ Tt $+$ St $+$ It$.$A summary of the results is shown in Table $5-11.$ Julie checks the auto$-$ correlation pattern of the residuals $($see Figure $5-14)$ for randomness. The residuals are not random, and the model does not appear to be adequate.

Julie is stuck. She has tried a naive model that combined seasonal and trend estimates, Winters$$ exponential smoothing, and classical decomposition. Julie finally decides to adjust seasonality out of the data so that forecasting techniques that canno