Category Archives: Forecasting

The Application of Forecasting in Service Operations

Service managers are recognizing the important distribution that forecasting can make in improving both the efficiency and the level or service in a service operation. Point-of-sale (POS) equipment can now provide! the service manager with historical sales data in time increments that are as small as 15 minutes. The av availability of these data permits accurate forecasting of future sales in similar time increments. thereby permitting the service manager to schedule workers more efficiently. Davis and Berger” point out that. in addition to forecasting sales. such models also can forecast product usage which reduces the spoilage of perishable items that have a short shelf life. Forecasting is also an integral part of yield ensnarement (also known as revenue management). which is discussed in more detail in later hater In brief, yield management attempts to maximize the revenues of those service operations that have high fixed costs and small variable costs. Examples of such service business  include airlines. car rental agencies and hotels. The goal of yield management maximize capacity utilization en if it means offering large price discounts. when to fill available capacity. At the same
time the manager does not want III turn paying customer because the capacity had been previously sold to a discount accomplish the success fully the manage must be able! to forecast demand for different market segments.

neural Networks

neural networks represent a relatively new and growing area of forecasting. Unlike the more common statistical forecasting techniques such as time-series analysis and regression analysis neural networks simulate human learning. Thus. over time and with repeated use. neural networks can develop an understanding of the complex relationships that exist been input into a forecasting model and the outputs. For example. in a every ice operation input might include such factors as historic sales weather time of day day of week nth The output would be the number of customers that are expected to arrive on a e day and in a given time period. In addition neural networks perform computation

much faster than traditional forecasting technique-. For example. the Southern Company which is a utility company that provides electric to throughout the south currently uses neural networks to forecast short-term pow er requirements a week to 10 day s ahead. Previously only midterm forecasting. that is three months ahead was feasible with traditional forecasting techniques

Reliability of the Data

With causal relationship forecasting we are concerned with how much of the changes in the dependent variable are being “explained” by changes in the independent variable. This is measured by the variance. The greater the proportion of the variance that can be explained by the independent variable. the stronger the relationship. The coefficient of determination (2) measures the proportion of the variability in the dependent variable that can be explained by changes in the independent variable, and is calculated as follows:

where
v, = Actual value of Y that has been observed for a given value of X
Y = Arithmetic mean for all values of y
Si = Value of r corresponding to a given value of X that has been calculated from the
regression equation The relationship between these variables is shown in Exhibit 9.16. In the equation above. the first term in the numerator and the term in the denominator are the same

This term represents the total variation of the Y variable around the arithmetic mean Y. The second term in the numerator represents the error or that variation in the Y variable that cannot be explained by the regression equation. Thus the numerator represents that amount of variation that can be explained by the regression equation. and the denominator represents the total variation. As stated above. the coefficient of determination therefore measures the proportion of variation in Y that can be explained by changes in X. Another measure for evaluating the reliability of a regression forecast is the mean squared error (MSE). Using the same notation as above. the MSE is calculated as follows:

where n = the number of observations. The following example will demonstrate the use of both of these terms.

Causal Relationship Forecasting

Any independent variable, to be of value from a forecasting perspective, must be a leading indicator. For example, if the weather service or the Farmer's Almanac predicts that next winter is going to have an abnormally large number of snow terms, people would probably go out and buy snow shovels and snow blowers in the fall. Thus. the weather prediction or the Fanner's Almanac is said to be a leading indicator of the sale of snow shovels and snow blowers. These relationships between variables can be viewed as causal relationships-the occurrence of one event causes or influences the occurrence of the other. Running out of gas while driving down a highway. however, does not provide useful data to forecast that the car will stop. The car will stop. of course, but we would like to know enough in advance in order. to do something about it. A "low gas lever' warning light, for example. is a good leading indicator that forecasts that the car will stop shortly. The first step in causal relationship forecasting is to identify those occurrences that are really the causes of the change. Often 'leading indicators are not causal relationships but in some indirect way may suggest that some other things might happen. Other noncausal relationships just seem to exist a a coincidence. One tudy Some years ago showed that the amount of alcohol sold in Sweden was directly proportional to teacl crs' salaries. Presumably this was a spurious or non causal relationship. The following problem illustrates one example of how a forecast is developed using a causal relationship.

Example The Carpet City Store has kept records of its sales (in square yards) each year, along with the number of permits that were issued for new houses in its area. Carpet City's operations manager believes that forecasting carpet sales is possible if the number of new housing permits is known for that year.

First, the data are plotted on Exhibit 9.15, with
X =Number of housing permits
Y =Sales of carpeting in square yards
Solution Since the points appear to be in a straight line, the manager decides to use the linear relation hip Y = a + bX. We solve this problem by using the least quarts method of solving for a and b using the equations presented earlier in this chapter, we obtain the following forecasting equation:

.

Now, suppose that there are 25 new housing permits granted in 2002. The 2002 sales  forecast would therefore be:
Y = 5,576.0 +387.2(25) = 15,256.0 square yards
In this problem, the lag between filing the permit with the appropriate agency and the new homeowner coming to Carpet City to buy carpet makes a causal relationship feasible for forecasting.

Measurement of Error

Several of the common terms used to describe the degree of error associated with forecasting are standard error, mean squared error (or variance), and mean absolute deviation. In addition. tracking signals may be used to indicate the existence of any positive or negative bias in the forecast. Standard error is discussed in the section on linear regression later in the chapter. Since the standard error is the square root of a function, it is often more convenient to use the function itself. This is called the mean square error, or variance.

The mean absolute deviation (MAD) was at one time very popular but subsequently was ignored in favor of the standard deviation and standard error measures. In recent years however MAD has made a comeback because of its simplicity and usefulness in obtaining tracking signals. MAD is the average error in the forecasts. u ing absolute values. It is valuable because MAD, like the tan dard deviation, measures the dispersion (or variation) of observed values around some expected value. MAD is computed using the differences between the actual demand and the forecast demand without regard to whether it is negative or positive. It therefore is equal to the sum of the absolute deviations divided by the number of data points or stated in equation form:

When the errors that occur in the forecast are normally distributed (which is usually assumed to be the case), the mean absolute deviation relate to standard division as

Conversely, I MAD 0.8 standard deviation The standard deviation is the larger measure. If the MAD for a set of points was found to be 60 units. then the standard deviation would be 75 units. And. in the usual statistical manner, if control limits were set at ±3 standard deviations (or ±3.75 MADs). then 99.7 percent of the points would fall within these limits. (See Exhibit 9.9.)

A tracking signal is a measurement that indicates whether the forecast average is keeping pace with any genuine upward or downward changes in demand. As used in forecasting the tracking signal is the number of mean absolute deviations that the forecast value is above or below the actual occurrence. Exhibit 9.9 shows a normal distribution with a mean of zero and a MAD equal to one. Thus if we compute a tracking signal and find it equal to 2 we can conclude that the forecast model is providing forecasts that are quite a bit above the mean of the actual occurrences. A tracking signal can be calculated using the arithmetic sum of forecast deviations divided by the mean absolute deviation, or

It is important to note that while the MAD, being an absolute value, is always positive, the tracking’signal can take on positive and negative values. Exhibit 9.10 illustrates the procedure for computing MAD and the tracking signal for a six-month period where the forecast had been set at a constant 1.000 and the actual demands that occurred are as shown. In this example, the forecast, Rutherford average, was off by 66.7 units and the tracking signal was equal to 3.3 mean absolute deviations. . We can obtain a better interpretation of the MAD and tracking signal by plotting the points on a graph. While not completely legitimate from a sample size standpoint. we plotted each month in Exhibit 9.11 to show the drifting of the tracking signal. Note that it drifted from -1MAD to +3.3 MADs. This occurred because the actual demand was greater than the forecast in four of the six periods. If the actual demand doesn’t fall below the forecast to offset the continual positive RSFE, the tracking signal would continue to rise and we ‘ould conclude that the assumption that demand ~ 1,000 is a bad forecast. When the tracking signal exceeds a pre established limit (for example, ±2.0 or ±3.0), the manager should n eider changing the forecast model or the value of ex. Acceptable limits for the tracking signal depend on the size of the demand being forecast (high-volume or high-revenue items should be monitored frequently) and the amount