Operations Management
FORECASTING FUNDAMENTALS Forecast: A prediction, projection, or estimate of some future activity, event, or occurrence. Types of Forecasts * Economic forecasts * Predict a variety of economic indicators, like money supply, inflation rates, interest rates, etc. * Technological forecasts * Predict rates of technological progress and innovation. * Demand forecasts * Predict the future demand for a company’s products or services.
Since virtually all the operations management decisions (in both the strategic category and the tactical category) require as input a good estimate of future demand, this is the type of forecasting that is emphasized in our textbook and in this course. TYPES OF FORECASTING METHODS
Qualitative methods: These types of forecasting methods are based on judgments, opinions, intuition, emotions, or personal experiences and are subjective in nature. They do not rely on any rigorous mathematical computations.
Quantitative methods: These types of forecasting methods are based on mathematical (quantitative) models, and are objective in nature. They rely heavily on mathematical computations.
QUALITATIVE FORECASTING METHODS
Qualitative Methods
Delphi
Method
Approach in which consensus agreement is reached among a group of experts
Sales Force Composite
Approach in which each salesperson estimates sales in his or her region
Executive
Opinion
Approach in which a group of managers meet and collectively develop a forecast
Market
Survey
Approach that uses interviews and surveys to judge preferences of customer and to assess demand
QUANTITATIVE FORECASTING METHODS
Quantitative Methods
Time-Series Models
Time series models look at past patterns of data and attempt to predict the future based upon the underlying patterns contained within those data.
Associative Models
Associative models (often called causal models) assume that the variable being forecasted is related to other variables in the environment. They try to project based upon those associations.
TIME SERIES MODELS
Model | Description | Naïve | Uses last period’s actual value as a forecast | Simple Mean (Average) | Uses an average of all past data as a forecast | Simple Moving Average | Uses an average of a specified number of the most recent observations, with each observation receiving the same emphasis (weight) | Weighted Moving Average | Uses an average of a specified number of the most recent observations, with each observation receiving a different emphasis (weight) | Exponential Smoothing | A weighted average procedure with weights declining exponentially as data become older | Trend Projection | Technique that uses the least squares method to fit a straight line to the data | Seasonal Indexes | A mechanism for adjusting the forecast to accommodate any seasonal patterns inherent in the data |
DECOMPOSITION OF A TIME SERIES
Patterns that may be present in a time series Trend: Data exhibit a steady growth or decline over time. Seasonality: Data exhibit upward and downward swings in a short to intermediate time frame (most notably during a year). Cycles: Data exhibit upward and downward swings in over a very long time frame. Random variations: Erratic and unpredictable variation in the data over time with no discernable pattern.
ILLUSTRATION OF TIME SERIES DECOMPOSITION
Hypothetical Pattern of Historical Demand
Demand
Time
TREND COMPONENT IN HISTORICAL DEMAND
Demand
Time
SEASONAL COMPONENT IN HISTORICAL DEMAND
Demand
Year 1 Year 2 Year 3 Time
CYCLE COMPONENT IN HISTORICAL DEMAND
Demand
Many years or decades Time
RANDOM COMPONENT IN HISTORICAL DEMAND
Demand
Time
DATA SET TO DEMONSTRATE FORECASTING METHODS
The following data set represents a set of hypothetical demands that have occurred over several consecutive years. The data have been collected on a quarterly basis, and these quarterly values have been amalgamated into yearly totals. For various illustrations that follow, we may make slightly different assumptions about starting points to get the process started for different models. In most cases we will assume that each year a forecast has been made for the subsequent year. Then, after a year has transpired we will have observed what the actual demand turned out to be (and we will surely see differences between what we had forecasted and what actually occurred, for, after all, the forecasts are merely educated guesses). Finally, to keep the numbers at a manageable size, several zeros have been dropped off the numbers (i.e., these numbers represent demands in thousands of units).
Year | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Total Annual Demand | 1 | 62 | 94 | 113 | 41 | 310 | 2 | 73 | 110 | 130 | 52 | 365 | 3 | 79 | 118 | 140 | 58 | 395 | 4 | 83 | 124 | 146 | 62 | 415 | 5 | 89 | 135 | 161 | 65 | 450 | 6 | 94 | 139 | 162 | 70 | 465 |
ILLUSTRATION OF THE NAÏVE METHOD
Naïve method: The forecast for next period (period t+1) will be equal to this period's actual demand (At).
In this illustration we assume that each year (beginning with year 2) we made a forecast, then waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | -- | There was no prior demand data on which to base a forecast for period 1 | 2 | 365 | 310 | From this point forward, these forecasts were made on a year-by-year basis. | 3 | 395 | 365 | | 4 | 415 | 395 | | 5 | 450 | 415 | | 6 | 465 | 450 | | 7 | | 465 | |
MEAN (SIMPLE AVERAGE) METHOD
Mean (simple average) method: The forecast for next period (period t+1) will be equal to the average of all past historical demands.
In this illustration we assume that a simple average method is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). At the end of year 1 we could start using this forecasting method. In this illustration we assume that each year (beginning with year 2) we made a forecast, then waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310.000 | From this point forward, these forecasts were made on a year-by-year basis using a simple average approach. | 3 | 395 | 337.500 | | 4 | 415 | 356.667 | | 5 | 450 | 371.250 | | 6 | 465 | 387.000 | | 7 | | 400.000 | |
SIMPLE MOVING AVERAGE METHOD
Simple moving average method: The forecast for next period (period t+1) will be equal to the average of a specified number of the most recent observations, with each observation receiving the same emphasis (weight).
In this illustration we assume that a 2-year simple moving average is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, after year 1 elapsed, we made a forecast for year 2 using a naïve method (310). Beyond that point we had sufficient data to let our 2-year simple moving average forecasts unfold throughout the years. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310 | This forecast was made using a naïve approach. | 3 | 395 | 337.500 | From this point forward, these forecasts were made on a year-by-year basis using a 2-yr moving average approach. | 4 | 415 | 380.000 | | 5 | 450 | 405.000 | | 6 | 465 | 432.500 | | 7 | | 457.500 | |
ANOTHER SIMPLE MOVING AVERAGE ILLUSTRATION
In this illustration we assume that a 3-year simple moving average is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, after year 1 elapsed, we used a naïve method to make a forecast for year 2 (310) and year 3 (365). Beyond that point we had sufficient data to let our 3-year simple moving average forecasts unfold throughout the years. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310 | This forecast was made using a naïve approach. | 3 | 395 | 365 | This forecast was made using a naïve approach. | 4 | 415 | 356.667 | From this point forward, these forecasts were made on a year-by-year basis using a 3-yr moving average approach. | 5 | 450 | 391.667 | | 6 | 465 | 420.000 | | 7 | | 433.333 | |
WEIGHTED MOVING AVERAGE METHOD
Weighted moving average method: The forecast for next period (period t+1) will be equal to a weighted average of a specified number of the most recent observations.
In this illustration we assume that a 3-year weighted moving average is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, after year 1 elapsed, we used a naïve method to make a forecast for year 2 (310) and year 3 (365). Beyond that point we had sufficient data to let our 3-year weighted moving average forecasts unfold throughout the years. The weights that were to be used are as follows: Most recent year, .5; year prior to that, .3; year prior to that, .2 Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310 | This forecast was made using a naïve approach. | 3 | 395 | 365 | This forecast was made using a naïve approach. | 4 | 415 | 369.000 | From this point forward, these forecasts were made on a year-by-year basis using a 3-yr wtd. moving avg. approach. | 5 | 450 | 399.000 | | 6 | 465 | 428.500 | | 7 | | 450.500 | |
EXPONENTIAL SMOOTHING METHOD
Exponential smoothing method: The new forecast for next period (period t) will be calculated as follows: New forecast = Last period’s forecast + (Last period’s actual demand – Last period’s forecast) (this box contains all you need to know to apply exponential smoothing) Ft = Ft-1 + (At-1 – Ft-1) (equation 1) Ft = At-1 + (1-)Ft-1 (alternate equation 1 – a bit more user friendly) Where is a smoothing coefficient whose value is between 0 and 1. | The exponential smoothing method only requires that you dig up two pieces of data to apply it (the most recent actual demand and the most recent forecast). An attractive feature of this method is that forecasts made with this model will include a portion of every piece of historical demand. Furthermore, there will be different weights placed on these historical demand values, with older data receiving lower weights. At first glance this may not be obvious, however, this property is illustrated on the following page. DEMONSTRATION: EXPONENTIAL SMOOTHING INCLUDES ALL PAST DATA
Note: the mathematical manipulations in this box are not something you would ever have to do when applying exponential smoothing. All you need to use is equation 1 on the previous page. This demonstration is to convince the skeptics that when using equation 1, all historical data will be included in the forecast, and the older the data, the lower the weight applied to that data. To make a forecast for next period, we would use the user friendly alternate equation 1: Ft = At-1 + (1-)Ft-1 (equation 1) When we made the forecast for the current period (Ft-1), it was made in the following fashion: Ft-1 = At-2 + (1-)Ft-2 (equation 2) If we substitute equation 2 into equation 1 we get the following: Ft = At-1 + (1-)[At-2 + (1-)Ft-2] Which can be cleaned up to the following: Ft = At-1 + (1-)At-2 + (1-)2Ft-2 (equation 3) We could continue to play that game by recognizing that Ft-2 = At-3 + (1-)Ft-3 (equation 4) If we substitute equation 4 into equation 3 we get the following: Ft = At-1 + (1-)At-2 + (1-)2[At-3 + (1-)Ft-3] Which can be cleaned up to the following: Ft = At-1 + (1-)At-2 + (1-)2At-3 + (1-)3Ft-3 If you keep playing that game, you should recognize that Ft = At-1 + (1-)At-2 + (1-)2At-3 + (1-)3At-4 + (1-)4At-5 + (1-)5At-6 ………. As you raise those decimal weights to higher and higher powers, the values get smaller and smaller. |
EXPONENTIAL SMOOTHING ILLUSTRATION
In this illustration we assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model. After the forecast was made, we waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. This set of forecasts was made using an value of .1 Year | Actual Demand (A) | Forecast (F) | Notes | 1 | 310 | 300 | This was a guess, since there was no prior demand data. | 2 | 365 | 301 | From this point forward, these forecasts were made on a year-by-year basis using exponential smoothing with =.1 | 3 | 395 | 307.4 | | 4 | 415 | 316.16 | | 5 | 450 | 326.044 | | 6 | 465 | 338.4396 | | 7 | | 351.09564 | |
A SECOND EXPONENTIAL SMOOTHING ILLUSTRATION
In this illustration we assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model. After the forecast was made, we waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. This set of forecasts was made using an value of .2 Year | Actual Demand (A) | Forecast (F) | Notes | 1 | 310 | 300 | This was a guess, since there was no prior demand data. | 2 | 365 | 302 | From this point forward, these forecasts were made on a year-by-year basis using exponential smoothing with =.2 | 3 | 395 | 314.6 | | 4 | 415 | 330.68 | | 5 | 450 | 347.544 | | 6 | 465 | 368.0352 | | 7 | | 387.42816 | |
A THIRD EXPONENTIAL SMOOTHING ILLUSTRATION
In this illustration we assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model. After the forecast was made, we waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. This set of forecasts was made using an value of .4 Year | Actual Demand (A) | Forecast (F) | Notes | 1 | 310 | 300 | This was a guess, since there was no prior demand data. | 2 | 365 | 304 | From this point forward, these forecasts were made on a year-by-year basis using exponential smoothing with =.4 | 3 | 395 | 328.4 | | 4 | 415 | 355.04 | | 5 | 450 | 379.024 | | 6 | 465 | 407.4144 | | 7 | | 430.44864 | |
TREND PROJECTION
Trend projection method: This method is a version of the linear regression technique. It attempts to draw a straight line through the historical data points in a fashion that comes as close to the points as possible. (Technically, the approach attempts to reduce the vertical deviations of the points from the trend line, and does this by minimizing the squared values of the deviations of the points from the line). Ultimately, the statistical formulas compute a slope for the trend line (b) and the point where the line crosses the y-axis (a). This results in the straight line equation Y = a + bX Where X represents the values on the horizontal axis (time), and Y represents the values on the vertical axis (demand).
For the demonstration data, computations for b and a reveal the following (NOTE: I will not require you to make the statistical calculations for b and a; these would be given to you. However, you do need to know what to do with these values when given to you.) b = 30 a = 295 Y = 295 + 30X This equation can be used to forecast for any year into the future. For example: Year 7: Forecast = 295 + 30(7) = 505 Year 8: Forecast = 295 + 30(8) = 535 Year 9: Forecast = 295 + 30(9) = 565 Year 10: Forecast = 295 + 30(10) = 595
STABILITY VS. RESPONSIVENESS IN FORECASTING
All demand forecasting methods vary in the degree to which they emphasize recent demand changes when making a forecast. Forecasting methods that react very strongly (or quickly) to demand changes are said to be responsive. Forecasting methods that do not react quickly to demand changes are said to be stable. One of the critical issues in selecting the appropriate forecasting method hinges on the question of stability versus responsiveness. How much stability or how much responsiveness one should employ is a function of how the historical demand has been fluctuating. If demand has been showing a steady pattern of increase (or decrease), then more responsiveness is desirable, for we would like to react quickly to those demand increases (or decreases) when we make our next forecast. On the other hand, if demand has been fluctuating upward and downward, then more stability is desirable, for we do not want to “over react” to those up and down fluctuations in demand.
For some of the simple forecasting methods we have examined, the following can be noted:
Moving Average Approach: Using more periods in your moving average forecasts will result in more stability in the forecasts. Using fewer periods in your moving average forecasts will result in more responsiveness in the forecasts.
Weighted Moving Average Approach: Using more periods in your weighted moving average forecasts will result in more stability in the forecasts. Using fewer periods in your weighted moving average forecasts will result in more responsiveness in the forecasts. Furthermore, placing lower weights on the more recent demand will result in more stability in the forecasts. Placing higher weights on the more recent demand will result in more responsiveness in the forecasts.
Simple Exponential Smoothing Approach: Using a lower alpha (α) value will result in more stability in the forecasts. Using a higher alpha (α) value will result in more responsiveness in the forecasts.
SEASONALITY ISSUES IN FORECASTING Up to this point we have seen several ways to make a forecast for an upcoming year. In many instances managers may want more detail that just a yearly forecast. They may like to have a projection for individual time periods within that year (e.g., weeks, months, or quarters). Let’s assume that our forecasted demand for an upcoming year is 480, but management would like a forecast for each of the quarters of the year. A simple approach might be to simply divide the total annual forecast of 480 by 4, yielding 120. We could then project that the demand for each quarter of the year will be 120. But of course, such forecasts could be expected to be quite inaccurate, for an examination of our original table of historical data reveals that demand is not uniform across each quarter of the year. There seem to be distinct peaks and valleys (i.e., quarters of higher demand and quarters of lower demand). The graph below of the historical quarterly demand clearly shows those peaks and valleys during the course of each year.
Mechanisms for dealing with seasonality are illustrated over the next several pages.
CALCULATING SEASONAL INDEX VALUES This is the way you will find seasonal index values calculated in the textbook. Begin by calculating the average demand in each of the four quarters of the year.
Col. 1 | Col. 2 | Col. 3 | Col. 4 | Col. 5 | Col. 6 | Year | Q1 | Q2 | Q3 | Q4 | Annual Demand | 1 | 62 | 94 | 113 | 41 | 310 | 2 | 73 | 110 | 130 | 52 | 365 | 3 | 79 | 118 | 140 | 58 | 395 | 4 | 83 | 124 | 146 | 62 | 415 | 5 | 89 | 135 | 161 | 65 | 450 | 6 | 94 | 139 | 162 | 70 | 465 | Avg. Demand Per Qtr. | (62+73+ 79+83+ 89+94) ÷ 6 = 80 | (94+110+ 118+124+ 135+139) ÷ 6 = 120 | (113+130+ 140+146+ 161+162) ÷ 6 = 142 | (41+52+ 58+62+ 65+70) ÷ 6 = 58 | |
Next, note that the total demand over these six years of history was 2400 (i.e., 310 + 365 + 395 + 415 + 450 + 465), and if this total demand of 2400 had been evenly spread over each of the 24 quarters in this six year period, the average quarterly demand would have been 100 units. Another way to look at this is the average of the quarterly averages is 100 units, i.e. (80 + 120 + 142 + 58)/4 = 100 units. But, the numbers above indicate that the demand wasn’t evenly distributed over each quarter. In Quarter 1 the average demand was considerably below 100 (it averaged 80 in Quarter 1). In Quarters 2 and 3 the average demand was considerably above 100 (with averages of 120 and 142, respectively). Finally, in Quarter 4 the average demand was below 100 (it averaged 58 in Quarter 4). We can calculate a seasonal index for each quarter by dividing the average quarterly demand by the 100 that would have occurred if all the demand had been evenly distributed across the quarters. This would result in the following alternate seasonal index values:
Year | Q1 | Q2 | Q3 | Q4 | Seasonal Index | 80/100 = .80 | 120/100 = 1.20 | 142/100 = 1.42 | 58/100 = .58 | A quick check of these alternate seasonal index values reveals that they average out to 1.0 (as they should). (.80 + 1.20 + 1.42 + .58)/4 = 1.000 USING SEASONAL INDEX VALUES
The following forecasts were made for the next 4 years using the trend projection line approach (the trend projection formula developed was Y = 295 + 30X, where Y is the forecast and X is the year number). Year | Forecast | 7 | 505 | 8 | 535 | 9 | 565 | 10 | 595 | If these annual forecasts were evenly distributed over each year, the quarterly forecasts would look like the following:
Year | Q1 | Q2 | Q3 | Q4 | Annual Forecast | Annual/4 | 7 | 126.25 | 126.25 | 126.25 | 126.25 | 505 | 126.25 | 8 | 133.75 | 133.75 | 133.75 | 133.75 | 535 | 133.75 | 9 | 141.25 | 141.25 | 141.25 | 141.25 | 565 | 141.25 | 10 | 148.75 | 148.75 | 148.75 | 148.75 | 595 | 148.75 | However, seasonality in the past demand suggests that these forecasts should not be evenly distributed over each quarter. We must take these even splits and multiply them by the seasonal index (S.I.) values to get a more reasonable set of quarterly forecasts. The results of these calculations are shown below. S.I. | .80 | 1.20 | 1.42 | .58 | | | | | | | | Year | Q1 | Q2 | Q3 | Q4 | Annual Forecast | 7 | 101.000 | 151.500 | 179.275 | 73.225 | 505 | 8 | 107.000 | 160.500 | 189.925 | 77.575 | 535 | 9 | 113.000 | 169.500 | 200.575 | 81.925 | 565 | 10 | 119.000 | 178.500 | 211.225 | 86.275 | 595 | If you check these final splits, you will see that the sum of the quarterly forecasts for a particular year will equal the total annual forecast for that year (sometimes there might be a slight rounding discrepancy). OTHER METHODS FOR MAKING SEASONAL FORECASTS
Let's go back and reexamine the historical data we have for this problem. I have put a little separation between the columns of each quarter to let you better visualize the fact that we could look at any one of those vertical strips of data and treat it as a time series. For example, the Q1 column displays the progression of quarter 1 demands over the past six years. One could simply peel off that strip of data and use it along with any of the forecasting methods we have examined to forecast the Q1 demand in year 7. We could do the same thing for each of the other three quarterly data strips.
Year | | Q1 | | Q2 | | Q3 | | Q4 | 1 | | 62 | | 94 | | 113 | | 41 | 2 | | 73 | | 110 | | 130 | | 52 | 3 | | 79 | | 118 | | 140 | | 58 | 4 | | 83 | | 124 | | 146 | | 62 | 5 | | 89 | | 135 | | 161 | | 65 | 6 | | 94 | | 139 | | 162 | | 70 | To illustrate, I have used the linear trend line method on the quarter 1 strip of data, which would result in the following trend line: Y = 58.8 + 6.0571X For year 7, X = 7, so the resulting Q1 forecast for year 7 would be 101.200 We could do the same thing with the Q2, Q3, and Q4 strips of data. For each strip we would compute the trend line equation and use it to project that quarter’s year 7 demand. Those results are summarized here: Q2 trend line: Y = 89.4 + 8.7429X; Year 7 Q2 forecast would be 150.600 Q3 trend line: Y = 107.6 + 9.8286X; Year 7 Q3 forecast would be 176.400 Q4 trend line: Y = 39.2 + 5.3714X; Year 7 Q4 forecast would be 76.800 Total forecast for year 7 = 101.200 + 150.600 + 176.400 + 76.800 = 505.000 These quarterly forecasts are in the same ballpark as those made with the seasonal index values earlier. They differ a bit, but we cannot say one is correct and one is incorrect. They are just slightly different predictions of what is going to happen in the future. They do provide a total annual forecast that is equal to the trend projection forecast made for year 7. (Don’t expect this to occur on every occasion, but since it corroborates results obtained with a different method, it does give us confidence in the forecasts we have made.) ASSOCIATIVE FORECASTING METHOD
Associative forecasting models (causal models) assume that the variable being forecasted (the dependent variable) is related to other variables (independent variables) in the environment. This approach tries to project demand based upon those associations. In its simplest form, linear regression is used to fit a line to the data. That line is then used to forecast the dependent variable for some selected value of the independent variable. In this illustration a distributor of drywall in a local community has historical demand data for the past eight years as well as data on the number of permits that have been issued for new home construction. These data are displayed in the following table: Year | # of new home construction permits | Demand for 4’x8’ sheets of drywall | 2004 | 400 | 60,000 | 2005 | 320 | 46,000 | 2006 | 290 | 45,000 | 2007 | 360 | 54,000 | 2008 | 380 | 60,000 | 2009 | 320 | 48,000 | 2010 | 430 | 65,000 | 2011 | 420 | 62,000 | If we attempted to perform a time series analysis on demand, the results would not make much sense, for a quick plot of demand vs. time suggests that there is no apparent pattern relationship here, as seen below.
ASSOCIATIVE FORECASTING METHOD (CONTINUED) If you plot the relationship between demand and the number of construction permits, a pattern emerges that makes more sense. It seems to indicate that demand for this product is lower when fewer construction permits are issued, and higher when more permits are issued. Therefore, regression will be used to establish a relationship between the dependent variable (demand) and the independent variable (construction permits).
The independent variable (X) is the number of construction permits. The dependent variable (Y) is the demand for drywall. Application of regression formulas yields the following forecasting model: Y = 250 + 150X If the company plans finds from public records that 350 construction permits have been issued for the year 2012, then a reasonable estimate of drywall demand for 2012 would be: Y = 250 + 150(350) = 250 + 52,500 = 52,750 (which means next year’s forecasted demand is 52,750 sheets of drywall)
MEASURING FORECAST ACCURACY
Mean Forecast Error (MFE): Forecast error is a measure of how accurate our forecast was in a given time period. It is calculated as the actual demand minus the forecast, or Et = At - Ft Forecast error in one time period does not convey much information, so we need to look at the accumulation of errors over time. We can calculate the average value of these forecast errors over time (i.e., a Mean Forecast Error, or MFE).Unfortunately, the accumulation of the Et values is not always very revealing, for some of them will be positive errors and some will be negative. These positive and negative errors cancel one another, and looking at them alone (or looking at the MFE over time) might give a false sense of security. To illustrate, consider our original data, and the accompanying pair of hypothetical forecasts made with two different forecasting methods.
Year | Actual Demand At | Hypothetical Forecasts Made With Method 1 Ft | Forecast Error With Method 1 At - Ft | Hypothetical Forecasts Made With Method 2 Ft | Forecast Error With Method 2 At - Ft | 1 | 310 | 315 | -5 | 370 | -60 | 2 | 365 | 375 | -10 | 455 | -90 | 3 | 395 | 390 | 5 | 305 | 90 | 4 | 415 | 405 | 10 | 535 | -120 | 5 | 450 | 435 | 15 | 390 | 60 | 6 | 465 | 480 | -15 | 345 | 120 | Accumulated Forecast Errors | 0 | | 0 | Mean Forecast Error, MFE | 0/6 = 0 | | 0/6 = 0 | | | | | | | | | | | | | Based on the accumulated forecast errors over time, the two methods look equally good. But, most observers would judge that Method 1 is generating better forecasts than Method 2 (i.e., smaller misses).
MEASURING FORECAST ACCURACY
Mean Absolute Deviation (MAD): To eliminate the problem of positive errors canceling negative errors, a simple measure is one that looks at the absolute value of the error (size of the deviation, regardless of sign). When we disregard the sign and only consider the size of the error, we refer to this deviation as the absolute deviation. If we accumulate these absolute deviations over time and find the average value of these absolute deviations, we refer to this measure as the mean absolute deviation (MAD). For our hypothetical two forecasting methods, the absolute deviations can be calculated for each year and an average can be obtained for these yearly absolute deviations, as follows:
Year | Actual Demand At | Hypothetical Forecasting Method 1 | Hypothetical Forecasting Method 2 | | | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | 1 | 310 | 315 | -5 | 5 | 370 | -60 | 60 | 2 | 365 | 375 | -10 | 10 | 455 | -90 | 90 | 3 | 395 | 390 | 5 | 5 | 305 | 90 | 90 | 4 | 415 | 405 | 10 | 10 | 535 | -120 | 120 | 5 | 450 | 435 | 15 | 15 | 390 | 60 | 60 | 6 | 465 | 480 | -15 | 15 | 345 | 120 | 120 | | Total Absolute Deviation | 60 | | | 540 | | Mean Absolute Deviation | 60/6=10 | | | 540/6=90 |
The smaller misses of Method 1 has been formalized with the calculation of the MAD. Method 1 seems to have provided more accurate forecasts over this six year horizon, as evidenced by its considerably smaller MAD. MEASURING FORECAST ACCURACY
Mean Squared Error (MSE): Another way to eliminate the problem of positive errors canceling negative errors is to square the forecast error. Regardless of whether the forecast error has a positive or negative sign, the squared error will always have a positive sign. If we accumulate these squared errors over time and find the average value of these squared errors, we refer to this measure as the mean squared error (MSE). For our hypothetical two forecasting methods, the squared errors can be calculated for each year and an average can be obtained for these yearly squared errors, as follows:
Year | Actual Demand At | Hypothetical Forecasting Method 1 | Hypothetical Forecasting Method 2 | | | Forecast Ft | Forecast Error At - Ft | Squared Error (At - Ft)2 | Forecast Ft | Forecast Error At - Ft | Squared Error (At - Ft)2 | 1 | 310 | 315 | -5 | 25 | 370 | -60 | 3600 | 2 | 365 | 375 | -10 | 100 | 455 | -90 | 8100 | 3 | 395 | 390 | 5 | 25 | 305 | 90 | 8100 | 4 | 415 | 405 | 10 | 100 | 535 | -120 | 14400 | 5 | 450 | 435 | 15 | 225 | 390 | 60 | 3600 | 6 | 465 | 480 | -15 | 225 | 345 | 120 | 14400 | | Total Squared Error | 700 | | | 52200 | | Mean Squared Error | 700/6 = 116.67 | | | 52200/6 = 8700 |
Method 1 seems to have provided more accurate forecasts over this six year horizon, as evidenced by its considerably smaller MSE. The Question often arises as to why one would use the more cumbersome MSE when the MAD calculations are a bit simpler (you don’t have to square the deviations). MAD does have the advantage of simpler calculations. However, there is a benefit to the MSE method. Since this method squares the error term, large errors tend to be magnified. Consequently, MSE places a higher penalty on large errors. This can be useful in situations where small forecast errors don’t cause much of a problem, but large errors can be devastating.
MEASURING FORECAST ACCURACY
Mean Absolute Percent Error (MAPE): A problem with both the MAD and MSE is that their values depend on the magnitude of the item being forecast. If the forecast item is measured in thousands or millions, the MAD and MSE values can be very large. To avoid this problem, we can use the MAPE. MAPE is computed as the average of the absolute difference between the forecasted and actual values, expressed as a percentage of the actual values. In essence, we look at how large the miss was relative to the size of the actual value. For our hypothetical two forecasting methods, the absolute percentage error can be calculated for each year and an average can be obtained for these yearly values, yielding the MAPE, as follows:
Year | Actual Demand At | Hypothetical Forecasting Method 1 | Hypothetical Forecasting Method 2 | | | Forecast Ft | Forecast Error At - Ft | Absolute % Error 100|At - Ft|/At | Forecast Ft | Forecast Error At - Ft | Absolute % Error 100|At - Ft|/At | 1 | 310 | 315 | -5 | 1.16% | 370 | -60 | 19.35% | 2 | 365 | 375 | -10 | 2.74% | 455 | -90 | 24.66% | 3 | 395 | 390 | 5 | 1.27% | 305 | 90 | 22.78% | 4 | 415 | 405 | 10 | 2.41% | 535 | -120 | 28.92% | 5 | 450 | 435 | 15 | 3.33% | 390 | 60 | 13.33% | 6 | 465 | 480 | -15 | 3.23% | 345 | 120 | 17.14% | | Total Absolute % Error | 14.59% | | | 134.85% | | Mean Absolute % Error | 14.59/6= 2.43% | | | 134.85/6= 22.48% |
Method 1seems to have provided more accurate forecasts over this six year horizon, as evidenced by the fact that the percentages by which the forecasts miss the actual demand are smaller with Method 1 (i.e., smaller MAPE).
ILLUSTRATION OF THE FOUR FORECAST ACCURACY MEASURES
Here is a further illustration of the four measures of forecast accuracy, this time using hypothetical forecasts that were generated using some different methods than the previous illustrations (called forecasting methods A and B; actually, these forecasts were made up for purposes of illustration). These calculations illustrate why we cannot rely on just one measure of forecast accuracy.
| | Hypothetical Forecasting Method A | Hypothetical Forecasting Method B | Year | Actual DemandAt | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | SquaredDeviation(At - Ft)2 | Abs. % Error |At-Ft|/At | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | SquaredDeviation(At - Ft)2 | Abs. % Error|At-Ft|/At | 1 | 310 | 330 | -20 | 20 | 400 | 6.45% | 310 | 0 | 0 | 0 | 0% | 2 | 365 | 345 | 20 | 20 | 400 | 5.48% | 365 | 0 | 0 | 0 | 0% | 3 | 395 | 415 | -20 | 20 | 400 | 5.06% | 395 | 0 | 0 | 0 | 0% | 4 | 415 | 395 | 20 | 20 | 400 | 4.82% | 415 | 0 | 0 | 0 | 0% | 5 | 450 | 430 | 20 | 20 | 400 | 4.44% | 390 | 60 | 60 | 3600 | 13.33% | 6 | 465 | 485 | -20 | 20 | 400 | 4.30% | 525 | -60 | 60 | 3600 | 12.90% | | | Totals | 0 | 120 | 2400 | 30.55% | Totals | 0 | 120 | 7200 | 26.23% | | | | MFE =0/6 =0 | MAD =120/6 =20 | MSE =2400/6 =400 | MAPE=30.55/65.09% | | MFE =0/6 =0 | MAD =120/6 =20 | MSE =7200/6 =1200 | MAPE=26.23/64.37% |
You can observe that for each of these forecasting methods, the same MFE resulted and the same MAD resulted. With these two measures, we would have no basis for claiming that one of these forecasting methods was more accurate than the other. With several measures of accuracy to consider, we can look at all the data in an attempt to determine the better forecasting method to use. Interpretation of these results will be impacted by the biases of the decision maker and the parameters of the decision situation. For example, one observer could look at the forecasts with method A and note that they were pretty consistent in that they were always missing by a modest amount (in this case, missing by 20 units each year). However, forecasting method B was very good in some years, and extremely bad in some years (missing by 60 units in years 5 and 6). That observation might cause this individual to prefer the accuracy and consistency of forecasting method A. This causal observation is formalized in the calculation of the MSE. Forecasting method A has a considerably lower MSE than forecasting method B. The squaring magnified those big misses that were observed with forecasting method B. However, another individual might view these results and have a preference for method B, for the sizes of the misses relative to the sizes of the actual demand are smaller than for method A, as indicated by the MAPE calculations. MONITORING FORECAST ACCURACY OVER TIME Tracking Signal: A tracking signal (T.S.) is a tool used to continually monitor the quality of our forecasting method as we progress through time. A tracking signal value is calculated each period and a determination is made as to whether it falls into an acceptable range. An upper limit and a lower limit will have been established for the tracking signal, and these values define the acceptable range. If the tracking signal drifts outside of the acceptable range, that is an indication that the forecasting method being used is no longer providing accurate forecasts. Tracking signals also help to indicate whether there is bias creeping into the forecasting process. Bias is a tendency for the forecast to be persistently under or persistently over the actual value of the data. Tracking signal is calculated as follows: Tracking signal = | Cumulative error | | MAD | Illustration of the computation of tracking signals to accompany a progression of hypothetical forecasts made over time some hypothetical forecasting method. (These forecasts were not made with any of the forecasting methods we illustrated – the forecasts were contrived to keep the numbers manageable.) Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10 | +1.00 | 2 | 365 | 371 | -6 | 4 | 6 | 16 | 8 | +.50 | 3 | 395 | 387 | 8 | 12 | 8 | 24 | 8 | +1.50 | 4 | 415 | 431 | -16 | -4 | 16 | 40 | 10 | –.40 | 5 | 450 | 460 | -10 | -14 | 10 | 50 | 10 | –1.40 | 6 | 465 | 461 | 4 | -10 | 4 | 54 | 9 | –1.11 | Keep in mind that each line in the above table would have been calculated in successive years. At the end of each year we can look back at the most recent year and compare the forecast we made with the actual demand that occurred. The next several pages show how these calculations would have unfolded through the years, and how they would have been plotted on a graph to determine whether our forecasting method still appeared to be working well. We now begin illustrating the computation and plotting of tracking signals to accompany the progression of forecasts made over time with hypothetical forecasting Method 1. In this illustration we will assume that the upper limit has been set at a value of 3, and the lower limit has been set at a value of -3. In practice these limits may be higher or lower than these values, and they do not necessarily need to have the same numerical value. The values for these limits are largely a function of how costly or disruptive inaccurate forecasts are. As we run through time, assume that the forecast made for year 1 was 300, and the subsequent demand that occurred in year 1 was 310. The tracking signal calculated and plotted after year 1 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 2 was 371, and the subsequent demand that occurred in year 2 was 365. The tracking signal calculated and plotted after year 2 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 3 was 387, and the subsequent demand that occurred in year 3 was 395. The tracking signal calculated and plotted after year 3 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 4 was 431, and the subsequent demand that occurred in year 4 was 415. The tracking signal calculated and plotted after year 4 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | 4 | 415 | 431 | -16 | (10)+(-6) +(8)+(-16) -4 | 16 | (10)+(6) +(8)+(16) 40 | 40/4 = 10 | -4/10 = –.40 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 5 was 460, and the subsequent demand that occurred in year 5 was 450. The tracking signal calculated and plotted after year 5 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | 4 | 415 | 431 | -16 | (10)+(-6) +(8)+(-16) -4 | 16 | (10)+(6) +(8)+(16) 40 | 40/4 = 10 | -4/10 = –.40 | 5 | 450 | 460 | -10 | (10)+(-6) +(8)+(-16) +(-10) -14 | 10 | (10)+(6) +(8)+(16) +(10) 50 | 50/5 = 10 | -14/10 = –1.40 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 6 was 461, and the subsequent demand that occurred in year 6 was 465. The tracking signal calculated and plotted after year 6 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | 4 | 415 | 431 | -16 | (10)+(-6) +(8)+(-16) -4 | 16 | (10)+(6) +(8)+(16) 40 | 40/4 = 10 | -4/10 = –.40 | 5 | 450 | 460 | -10 | (10)+(-6) +(8)+(-16) +(-10) -14 | 10 | (10)+(6) +(8)+(16) +(10) 50 | 50/5 = 10 | -14/10 = –1.40 | 6 | 465 | 461 | 4 | (10)+(-6) +(8)+(-16) +(-10)+(4) -10 | 4 | (10)+(6) +(8)+(16) +(10)+(4) 54 | 54/6 = 9 | -10/9 = –1.11 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
SOME NOTES ABOUT TRACKING SIGNALS
Tracking signal review Recall that the tracking signal is calculated as follows: Tracking Signal (TS) = | Cumulative Error | | MAD |
The cumulative error can be either positive or negative, since each forecast error (At - Ft) can be either positive or negative. If you forecast too low in any period (i.e., the forecast is below the demand), the forecast error will be positive. If you forecast too high in any period (i.e., the forecast is above the demand), the forecast error will be negative). The MAD will always be positive! Consequently, the tracking signal could end up being either a positive number or a negative number.
What to watch for If the tracking signal plots outside the acceptable range (i.e., above the upper limit or below the lower limit), that is an indication that things are no longer going well in our forecasting, and we should re-examine our method. However, even when the tracking signal plots between the upper limit and lower limit, this is not always an indication that things are going well in our forecasting. Consider the following: When we forecast, we expect to miss (i.e., make forecast errors). If we are unbiased in our forecasting approach, we should expect our forecast to be too high on some occasions, and too low on other occasions. Ultimately, if we are making reasonably accurate forecasts the cumulative error should fluctuate between positive and negative values, always hovering around zero. Suppose the TS is consistently plotting in the positive range. This would be an indication that we are consistently incurring a lot of positive forecast errors (i.e., forecasting too low). This would be an indication that bias has crept into our forecasting approach (i.e., a bias toward forecasting too low), and we should re-examine our forecasting approach. Suppose the TS is consistently plotting in the negative range. This would be an indication that we are consistently incurring a lot of negative forecast errors (i.e., forecasting too high). This would be an indication that bias has crept into our forecasting approach (i.e., a bias toward forecasting too high), and we should re-examine our forecasting approach. What are reasonable tracking signal limits How tight or how loose the tracking signal limits are set is a function of the consequences of forecast errors. The upper limit and the lower limit do not have to be the same distance from the zero mark. Suppose that it is not a big deal if we forecast too high (negative forecast error) and make too much of a product. We can hold the excess in inventory and sell it at a later date. However, if we forecast too low (positive forecast error) we will not have enough product to satisfy customer demand, and we are likely to lose customers to our competitors. In such a case, we would probably have a tight range on the positive side of our tracking signal graph, and a relatively loose range on the negative side of our tracking signal graph. Alternatively, suppose that if we forecast too high (negative forecast error) and make too much of a product, the cost consequences are severe, for this product has a short shelf life or becomes obsolete quickly, and excess inventory will quickly become worthless. However, if we forecast too low (positive forecast error) and do not have enough product to satisfy customer demand it is no big deal, for customers are willing to wait for later deliveries. In such a case, we would probably have a tight range on the negative side of our tracking signal graph, and a relatively loose range on the positive side of our tracking signal graph.
Since virtually all the operations management decisions (in both the strategic category and the tactical category) require as input a good estimate of future demand, this is the type of forecasting that is emphasized in our textbook and in this course. TYPES OF FORECASTING METHODS
Qualitative methods: These types of forecasting methods are based on judgments, opinions, intuition, emotions, or personal experiences and are subjective in nature. They do not rely on any rigorous mathematical computations.
Quantitative methods: These types of forecasting methods are based on mathematical (quantitative) models, and are objective in nature. They rely heavily on mathematical computations.
QUALITATIVE FORECASTING METHODS
Qualitative Methods
Delphi
Method
Approach in which consensus agreement is reached among a group of experts
Sales Force Composite
Approach in which each salesperson estimates sales in his or her region
Executive
Opinion
Approach in which a group of managers meet and collectively develop a forecast
Market
Survey
Approach that uses interviews and surveys to judge preferences of customer and to assess demand
QUANTITATIVE FORECASTING METHODS
Quantitative Methods
Time-Series Models
Time series models look at past patterns of data and attempt to predict the future based upon the underlying patterns contained within those data.
Associative Models
Associative models (often called causal models) assume that the variable being forecasted is related to other variables in the environment. They try to project based upon those associations.
TIME SERIES MODELS
Model | Description | Naïve | Uses last period’s actual value as a forecast | Simple Mean (Average) | Uses an average of all past data as a forecast | Simple Moving Average | Uses an average of a specified number of the most recent observations, with each observation receiving the same emphasis (weight) | Weighted Moving Average | Uses an average of a specified number of the most recent observations, with each observation receiving a different emphasis (weight) | Exponential Smoothing | A weighted average procedure with weights declining exponentially as data become older | Trend Projection | Technique that uses the least squares method to fit a straight line to the data | Seasonal Indexes | A mechanism for adjusting the forecast to accommodate any seasonal patterns inherent in the data |
DECOMPOSITION OF A TIME SERIES
Patterns that may be present in a time series Trend: Data exhibit a steady growth or decline over time. Seasonality: Data exhibit upward and downward swings in a short to intermediate time frame (most notably during a year). Cycles: Data exhibit upward and downward swings in over a very long time frame. Random variations: Erratic and unpredictable variation in the data over time with no discernable pattern.
ILLUSTRATION OF TIME SERIES DECOMPOSITION
Hypothetical Pattern of Historical Demand
Demand
Time
TREND COMPONENT IN HISTORICAL DEMAND
Demand
Time
SEASONAL COMPONENT IN HISTORICAL DEMAND
Demand
Year 1 Year 2 Year 3 Time
CYCLE COMPONENT IN HISTORICAL DEMAND
Demand
Many years or decades Time
RANDOM COMPONENT IN HISTORICAL DEMAND
Demand
Time
DATA SET TO DEMONSTRATE FORECASTING METHODS
The following data set represents a set of hypothetical demands that have occurred over several consecutive years. The data have been collected on a quarterly basis, and these quarterly values have been amalgamated into yearly totals. For various illustrations that follow, we may make slightly different assumptions about starting points to get the process started for different models. In most cases we will assume that each year a forecast has been made for the subsequent year. Then, after a year has transpired we will have observed what the actual demand turned out to be (and we will surely see differences between what we had forecasted and what actually occurred, for, after all, the forecasts are merely educated guesses). Finally, to keep the numbers at a manageable size, several zeros have been dropped off the numbers (i.e., these numbers represent demands in thousands of units).
Year | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Total Annual Demand | 1 | 62 | 94 | 113 | 41 | 310 | 2 | 73 | 110 | 130 | 52 | 365 | 3 | 79 | 118 | 140 | 58 | 395 | 4 | 83 | 124 | 146 | 62 | 415 | 5 | 89 | 135 | 161 | 65 | 450 | 6 | 94 | 139 | 162 | 70 | 465 |
ILLUSTRATION OF THE NAÏVE METHOD
Naïve method: The forecast for next period (period t+1) will be equal to this period's actual demand (At).
In this illustration we assume that each year (beginning with year 2) we made a forecast, then waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | -- | There was no prior demand data on which to base a forecast for period 1 | 2 | 365 | 310 | From this point forward, these forecasts were made on a year-by-year basis. | 3 | 395 | 365 | | 4 | 415 | 395 | | 5 | 450 | 415 | | 6 | 465 | 450 | | 7 | | 465 | |
MEAN (SIMPLE AVERAGE) METHOD
Mean (simple average) method: The forecast for next period (period t+1) will be equal to the average of all past historical demands.
In this illustration we assume that a simple average method is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). At the end of year 1 we could start using this forecasting method. In this illustration we assume that each year (beginning with year 2) we made a forecast, then waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310.000 | From this point forward, these forecasts were made on a year-by-year basis using a simple average approach. | 3 | 395 | 337.500 | | 4 | 415 | 356.667 | | 5 | 450 | 371.250 | | 6 | 465 | 387.000 | | 7 | | 400.000 | |
SIMPLE MOVING AVERAGE METHOD
Simple moving average method: The forecast for next period (period t+1) will be equal to the average of a specified number of the most recent observations, with each observation receiving the same emphasis (weight).
In this illustration we assume that a 2-year simple moving average is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, after year 1 elapsed, we made a forecast for year 2 using a naïve method (310). Beyond that point we had sufficient data to let our 2-year simple moving average forecasts unfold throughout the years. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310 | This forecast was made using a naïve approach. | 3 | 395 | 337.500 | From this point forward, these forecasts were made on a year-by-year basis using a 2-yr moving average approach. | 4 | 415 | 380.000 | | 5 | 450 | 405.000 | | 6 | 465 | 432.500 | | 7 | | 457.500 | |
ANOTHER SIMPLE MOVING AVERAGE ILLUSTRATION
In this illustration we assume that a 3-year simple moving average is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, after year 1 elapsed, we used a naïve method to make a forecast for year 2 (310) and year 3 (365). Beyond that point we had sufficient data to let our 3-year simple moving average forecasts unfold throughout the years. Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310 | This forecast was made using a naïve approach. | 3 | 395 | 365 | This forecast was made using a naïve approach. | 4 | 415 | 356.667 | From this point forward, these forecasts were made on a year-by-year basis using a 3-yr moving average approach. | 5 | 450 | 391.667 | | 6 | 465 | 420.000 | | 7 | | 433.333 | |
WEIGHTED MOVING AVERAGE METHOD
Weighted moving average method: The forecast for next period (period t+1) will be equal to a weighted average of a specified number of the most recent observations.
In this illustration we assume that a 3-year weighted moving average is being used. We will also assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, after year 1 elapsed, we used a naïve method to make a forecast for year 2 (310) and year 3 (365). Beyond that point we had sufficient data to let our 3-year weighted moving average forecasts unfold throughout the years. The weights that were to be used are as follows: Most recent year, .5; year prior to that, .3; year prior to that, .2 Year | Actual Demand (At) | Forecast (Ft) | Notes | 1 | 310 | 300 | This forecast was a guess at the beginning. | 2 | 365 | 310 | This forecast was made using a naïve approach. | 3 | 395 | 365 | This forecast was made using a naïve approach. | 4 | 415 | 369.000 | From this point forward, these forecasts were made on a year-by-year basis using a 3-yr wtd. moving avg. approach. | 5 | 450 | 399.000 | | 6 | 465 | 428.500 | | 7 | | 450.500 | |
EXPONENTIAL SMOOTHING METHOD
Exponential smoothing method: The new forecast for next period (period t) will be calculated as follows: New forecast = Last period’s forecast + (Last period’s actual demand – Last period’s forecast) (this box contains all you need to know to apply exponential smoothing) Ft = Ft-1 + (At-1 – Ft-1) (equation 1) Ft = At-1 + (1-)Ft-1 (alternate equation 1 – a bit more user friendly) Where is a smoothing coefficient whose value is between 0 and 1. | The exponential smoothing method only requires that you dig up two pieces of data to apply it (the most recent actual demand and the most recent forecast). An attractive feature of this method is that forecasts made with this model will include a portion of every piece of historical demand. Furthermore, there will be different weights placed on these historical demand values, with older data receiving lower weights. At first glance this may not be obvious, however, this property is illustrated on the following page. DEMONSTRATION: EXPONENTIAL SMOOTHING INCLUDES ALL PAST DATA
Note: the mathematical manipulations in this box are not something you would ever have to do when applying exponential smoothing. All you need to use is equation 1 on the previous page. This demonstration is to convince the skeptics that when using equation 1, all historical data will be included in the forecast, and the older the data, the lower the weight applied to that data. To make a forecast for next period, we would use the user friendly alternate equation 1: Ft = At-1 + (1-)Ft-1 (equation 1) When we made the forecast for the current period (Ft-1), it was made in the following fashion: Ft-1 = At-2 + (1-)Ft-2 (equation 2) If we substitute equation 2 into equation 1 we get the following: Ft = At-1 + (1-)[At-2 + (1-)Ft-2] Which can be cleaned up to the following: Ft = At-1 + (1-)At-2 + (1-)2Ft-2 (equation 3) We could continue to play that game by recognizing that Ft-2 = At-3 + (1-)Ft-3 (equation 4) If we substitute equation 4 into equation 3 we get the following: Ft = At-1 + (1-)At-2 + (1-)2[At-3 + (1-)Ft-3] Which can be cleaned up to the following: Ft = At-1 + (1-)At-2 + (1-)2At-3 + (1-)3Ft-3 If you keep playing that game, you should recognize that Ft = At-1 + (1-)At-2 + (1-)2At-3 + (1-)3At-4 + (1-)4At-5 + (1-)5At-6 ………. As you raise those decimal weights to higher and higher powers, the values get smaller and smaller. |
EXPONENTIAL SMOOTHING ILLUSTRATION
In this illustration we assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model. After the forecast was made, we waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. This set of forecasts was made using an value of .1 Year | Actual Demand (A) | Forecast (F) | Notes | 1 | 310 | 300 | This was a guess, since there was no prior demand data. | 2 | 365 | 301 | From this point forward, these forecasts were made on a year-by-year basis using exponential smoothing with =.1 | 3 | 395 | 307.4 | | 4 | 415 | 316.16 | | 5 | 450 | 326.044 | | 6 | 465 | 338.4396 | | 7 | | 351.09564 | |
A SECOND EXPONENTIAL SMOOTHING ILLUSTRATION
In this illustration we assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model. After the forecast was made, we waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. This set of forecasts was made using an value of .2 Year | Actual Demand (A) | Forecast (F) | Notes | 1 | 310 | 300 | This was a guess, since there was no prior demand data. | 2 | 365 | 302 | From this point forward, these forecasts were made on a year-by-year basis using exponential smoothing with =.2 | 3 | 395 | 314.6 | | 4 | 415 | 330.68 | | 5 | 450 | 347.544 | | 6 | 465 | 368.0352 | | 7 | | 387.42816 | |
A THIRD EXPONENTIAL SMOOTHING ILLUSTRATION
In this illustration we assume that, in the absence of data at startup, we made a guess for the year 1 forecast (300). Then, for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model. After the forecast was made, we waited to see what demand unfolded during the year. We then made a forecast for the subsequent year, and so on right through to the forecast for year 7. This set of forecasts was made using an value of .4 Year | Actual Demand (A) | Forecast (F) | Notes | 1 | 310 | 300 | This was a guess, since there was no prior demand data. | 2 | 365 | 304 | From this point forward, these forecasts were made on a year-by-year basis using exponential smoothing with =.4 | 3 | 395 | 328.4 | | 4 | 415 | 355.04 | | 5 | 450 | 379.024 | | 6 | 465 | 407.4144 | | 7 | | 430.44864 | |
TREND PROJECTION
Trend projection method: This method is a version of the linear regression technique. It attempts to draw a straight line through the historical data points in a fashion that comes as close to the points as possible. (Technically, the approach attempts to reduce the vertical deviations of the points from the trend line, and does this by minimizing the squared values of the deviations of the points from the line). Ultimately, the statistical formulas compute a slope for the trend line (b) and the point where the line crosses the y-axis (a). This results in the straight line equation Y = a + bX Where X represents the values on the horizontal axis (time), and Y represents the values on the vertical axis (demand).
For the demonstration data, computations for b and a reveal the following (NOTE: I will not require you to make the statistical calculations for b and a; these would be given to you. However, you do need to know what to do with these values when given to you.) b = 30 a = 295 Y = 295 + 30X This equation can be used to forecast for any year into the future. For example: Year 7: Forecast = 295 + 30(7) = 505 Year 8: Forecast = 295 + 30(8) = 535 Year 9: Forecast = 295 + 30(9) = 565 Year 10: Forecast = 295 + 30(10) = 595
STABILITY VS. RESPONSIVENESS IN FORECASTING
All demand forecasting methods vary in the degree to which they emphasize recent demand changes when making a forecast. Forecasting methods that react very strongly (or quickly) to demand changes are said to be responsive. Forecasting methods that do not react quickly to demand changes are said to be stable. One of the critical issues in selecting the appropriate forecasting method hinges on the question of stability versus responsiveness. How much stability or how much responsiveness one should employ is a function of how the historical demand has been fluctuating. If demand has been showing a steady pattern of increase (or decrease), then more responsiveness is desirable, for we would like to react quickly to those demand increases (or decreases) when we make our next forecast. On the other hand, if demand has been fluctuating upward and downward, then more stability is desirable, for we do not want to “over react” to those up and down fluctuations in demand.
For some of the simple forecasting methods we have examined, the following can be noted:
Moving Average Approach: Using more periods in your moving average forecasts will result in more stability in the forecasts. Using fewer periods in your moving average forecasts will result in more responsiveness in the forecasts.
Weighted Moving Average Approach: Using more periods in your weighted moving average forecasts will result in more stability in the forecasts. Using fewer periods in your weighted moving average forecasts will result in more responsiveness in the forecasts. Furthermore, placing lower weights on the more recent demand will result in more stability in the forecasts. Placing higher weights on the more recent demand will result in more responsiveness in the forecasts.
Simple Exponential Smoothing Approach: Using a lower alpha (α) value will result in more stability in the forecasts. Using a higher alpha (α) value will result in more responsiveness in the forecasts.
SEASONALITY ISSUES IN FORECASTING Up to this point we have seen several ways to make a forecast for an upcoming year. In many instances managers may want more detail that just a yearly forecast. They may like to have a projection for individual time periods within that year (e.g., weeks, months, or quarters). Let’s assume that our forecasted demand for an upcoming year is 480, but management would like a forecast for each of the quarters of the year. A simple approach might be to simply divide the total annual forecast of 480 by 4, yielding 120. We could then project that the demand for each quarter of the year will be 120. But of course, such forecasts could be expected to be quite inaccurate, for an examination of our original table of historical data reveals that demand is not uniform across each quarter of the year. There seem to be distinct peaks and valleys (i.e., quarters of higher demand and quarters of lower demand). The graph below of the historical quarterly demand clearly shows those peaks and valleys during the course of each year.
Mechanisms for dealing with seasonality are illustrated over the next several pages.
CALCULATING SEASONAL INDEX VALUES This is the way you will find seasonal index values calculated in the textbook. Begin by calculating the average demand in each of the four quarters of the year.
Col. 1 | Col. 2 | Col. 3 | Col. 4 | Col. 5 | Col. 6 | Year | Q1 | Q2 | Q3 | Q4 | Annual Demand | 1 | 62 | 94 | 113 | 41 | 310 | 2 | 73 | 110 | 130 | 52 | 365 | 3 | 79 | 118 | 140 | 58 | 395 | 4 | 83 | 124 | 146 | 62 | 415 | 5 | 89 | 135 | 161 | 65 | 450 | 6 | 94 | 139 | 162 | 70 | 465 | Avg. Demand Per Qtr. | (62+73+ 79+83+ 89+94) ÷ 6 = 80 | (94+110+ 118+124+ 135+139) ÷ 6 = 120 | (113+130+ 140+146+ 161+162) ÷ 6 = 142 | (41+52+ 58+62+ 65+70) ÷ 6 = 58 | |
Next, note that the total demand over these six years of history was 2400 (i.e., 310 + 365 + 395 + 415 + 450 + 465), and if this total demand of 2400 had been evenly spread over each of the 24 quarters in this six year period, the average quarterly demand would have been 100 units. Another way to look at this is the average of the quarterly averages is 100 units, i.e. (80 + 120 + 142 + 58)/4 = 100 units. But, the numbers above indicate that the demand wasn’t evenly distributed over each quarter. In Quarter 1 the average demand was considerably below 100 (it averaged 80 in Quarter 1). In Quarters 2 and 3 the average demand was considerably above 100 (with averages of 120 and 142, respectively). Finally, in Quarter 4 the average demand was below 100 (it averaged 58 in Quarter 4). We can calculate a seasonal index for each quarter by dividing the average quarterly demand by the 100 that would have occurred if all the demand had been evenly distributed across the quarters. This would result in the following alternate seasonal index values:
Year | Q1 | Q2 | Q3 | Q4 | Seasonal Index | 80/100 = .80 | 120/100 = 1.20 | 142/100 = 1.42 | 58/100 = .58 | A quick check of these alternate seasonal index values reveals that they average out to 1.0 (as they should). (.80 + 1.20 + 1.42 + .58)/4 = 1.000 USING SEASONAL INDEX VALUES
The following forecasts were made for the next 4 years using the trend projection line approach (the trend projection formula developed was Y = 295 + 30X, where Y is the forecast and X is the year number). Year | Forecast | 7 | 505 | 8 | 535 | 9 | 565 | 10 | 595 | If these annual forecasts were evenly distributed over each year, the quarterly forecasts would look like the following:
Year | Q1 | Q2 | Q3 | Q4 | Annual Forecast | Annual/4 | 7 | 126.25 | 126.25 | 126.25 | 126.25 | 505 | 126.25 | 8 | 133.75 | 133.75 | 133.75 | 133.75 | 535 | 133.75 | 9 | 141.25 | 141.25 | 141.25 | 141.25 | 565 | 141.25 | 10 | 148.75 | 148.75 | 148.75 | 148.75 | 595 | 148.75 | However, seasonality in the past demand suggests that these forecasts should not be evenly distributed over each quarter. We must take these even splits and multiply them by the seasonal index (S.I.) values to get a more reasonable set of quarterly forecasts. The results of these calculations are shown below. S.I. | .80 | 1.20 | 1.42 | .58 | | | | | | | | Year | Q1 | Q2 | Q3 | Q4 | Annual Forecast | 7 | 101.000 | 151.500 | 179.275 | 73.225 | 505 | 8 | 107.000 | 160.500 | 189.925 | 77.575 | 535 | 9 | 113.000 | 169.500 | 200.575 | 81.925 | 565 | 10 | 119.000 | 178.500 | 211.225 | 86.275 | 595 | If you check these final splits, you will see that the sum of the quarterly forecasts for a particular year will equal the total annual forecast for that year (sometimes there might be a slight rounding discrepancy). OTHER METHODS FOR MAKING SEASONAL FORECASTS
Let's go back and reexamine the historical data we have for this problem. I have put a little separation between the columns of each quarter to let you better visualize the fact that we could look at any one of those vertical strips of data and treat it as a time series. For example, the Q1 column displays the progression of quarter 1 demands over the past six years. One could simply peel off that strip of data and use it along with any of the forecasting methods we have examined to forecast the Q1 demand in year 7. We could do the same thing for each of the other three quarterly data strips.
Year | | Q1 | | Q2 | | Q3 | | Q4 | 1 | | 62 | | 94 | | 113 | | 41 | 2 | | 73 | | 110 | | 130 | | 52 | 3 | | 79 | | 118 | | 140 | | 58 | 4 | | 83 | | 124 | | 146 | | 62 | 5 | | 89 | | 135 | | 161 | | 65 | 6 | | 94 | | 139 | | 162 | | 70 | To illustrate, I have used the linear trend line method on the quarter 1 strip of data, which would result in the following trend line: Y = 58.8 + 6.0571X For year 7, X = 7, so the resulting Q1 forecast for year 7 would be 101.200 We could do the same thing with the Q2, Q3, and Q4 strips of data. For each strip we would compute the trend line equation and use it to project that quarter’s year 7 demand. Those results are summarized here: Q2 trend line: Y = 89.4 + 8.7429X; Year 7 Q2 forecast would be 150.600 Q3 trend line: Y = 107.6 + 9.8286X; Year 7 Q3 forecast would be 176.400 Q4 trend line: Y = 39.2 + 5.3714X; Year 7 Q4 forecast would be 76.800 Total forecast for year 7 = 101.200 + 150.600 + 176.400 + 76.800 = 505.000 These quarterly forecasts are in the same ballpark as those made with the seasonal index values earlier. They differ a bit, but we cannot say one is correct and one is incorrect. They are just slightly different predictions of what is going to happen in the future. They do provide a total annual forecast that is equal to the trend projection forecast made for year 7. (Don’t expect this to occur on every occasion, but since it corroborates results obtained with a different method, it does give us confidence in the forecasts we have made.) ASSOCIATIVE FORECASTING METHOD
Associative forecasting models (causal models) assume that the variable being forecasted (the dependent variable) is related to other variables (independent variables) in the environment. This approach tries to project demand based upon those associations. In its simplest form, linear regression is used to fit a line to the data. That line is then used to forecast the dependent variable for some selected value of the independent variable. In this illustration a distributor of drywall in a local community has historical demand data for the past eight years as well as data on the number of permits that have been issued for new home construction. These data are displayed in the following table: Year | # of new home construction permits | Demand for 4’x8’ sheets of drywall | 2004 | 400 | 60,000 | 2005 | 320 | 46,000 | 2006 | 290 | 45,000 | 2007 | 360 | 54,000 | 2008 | 380 | 60,000 | 2009 | 320 | 48,000 | 2010 | 430 | 65,000 | 2011 | 420 | 62,000 | If we attempted to perform a time series analysis on demand, the results would not make much sense, for a quick plot of demand vs. time suggests that there is no apparent pattern relationship here, as seen below.
ASSOCIATIVE FORECASTING METHOD (CONTINUED) If you plot the relationship between demand and the number of construction permits, a pattern emerges that makes more sense. It seems to indicate that demand for this product is lower when fewer construction permits are issued, and higher when more permits are issued. Therefore, regression will be used to establish a relationship between the dependent variable (demand) and the independent variable (construction permits).
The independent variable (X) is the number of construction permits. The dependent variable (Y) is the demand for drywall. Application of regression formulas yields the following forecasting model: Y = 250 + 150X If the company plans finds from public records that 350 construction permits have been issued for the year 2012, then a reasonable estimate of drywall demand for 2012 would be: Y = 250 + 150(350) = 250 + 52,500 = 52,750 (which means next year’s forecasted demand is 52,750 sheets of drywall)
MEASURING FORECAST ACCURACY
Mean Forecast Error (MFE): Forecast error is a measure of how accurate our forecast was in a given time period. It is calculated as the actual demand minus the forecast, or Et = At - Ft Forecast error in one time period does not convey much information, so we need to look at the accumulation of errors over time. We can calculate the average value of these forecast errors over time (i.e., a Mean Forecast Error, or MFE).Unfortunately, the accumulation of the Et values is not always very revealing, for some of them will be positive errors and some will be negative. These positive and negative errors cancel one another, and looking at them alone (or looking at the MFE over time) might give a false sense of security. To illustrate, consider our original data, and the accompanying pair of hypothetical forecasts made with two different forecasting methods.
Year | Actual Demand At | Hypothetical Forecasts Made With Method 1 Ft | Forecast Error With Method 1 At - Ft | Hypothetical Forecasts Made With Method 2 Ft | Forecast Error With Method 2 At - Ft | 1 | 310 | 315 | -5 | 370 | -60 | 2 | 365 | 375 | -10 | 455 | -90 | 3 | 395 | 390 | 5 | 305 | 90 | 4 | 415 | 405 | 10 | 535 | -120 | 5 | 450 | 435 | 15 | 390 | 60 | 6 | 465 | 480 | -15 | 345 | 120 | Accumulated Forecast Errors | 0 | | 0 | Mean Forecast Error, MFE | 0/6 = 0 | | 0/6 = 0 | | | | | | | | | | | | | Based on the accumulated forecast errors over time, the two methods look equally good. But, most observers would judge that Method 1 is generating better forecasts than Method 2 (i.e., smaller misses).
MEASURING FORECAST ACCURACY
Mean Absolute Deviation (MAD): To eliminate the problem of positive errors canceling negative errors, a simple measure is one that looks at the absolute value of the error (size of the deviation, regardless of sign). When we disregard the sign and only consider the size of the error, we refer to this deviation as the absolute deviation. If we accumulate these absolute deviations over time and find the average value of these absolute deviations, we refer to this measure as the mean absolute deviation (MAD). For our hypothetical two forecasting methods, the absolute deviations can be calculated for each year and an average can be obtained for these yearly absolute deviations, as follows:
Year | Actual Demand At | Hypothetical Forecasting Method 1 | Hypothetical Forecasting Method 2 | | | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | 1 | 310 | 315 | -5 | 5 | 370 | -60 | 60 | 2 | 365 | 375 | -10 | 10 | 455 | -90 | 90 | 3 | 395 | 390 | 5 | 5 | 305 | 90 | 90 | 4 | 415 | 405 | 10 | 10 | 535 | -120 | 120 | 5 | 450 | 435 | 15 | 15 | 390 | 60 | 60 | 6 | 465 | 480 | -15 | 15 | 345 | 120 | 120 | | Total Absolute Deviation | 60 | | | 540 | | Mean Absolute Deviation | 60/6=10 | | | 540/6=90 |
The smaller misses of Method 1 has been formalized with the calculation of the MAD. Method 1 seems to have provided more accurate forecasts over this six year horizon, as evidenced by its considerably smaller MAD. MEASURING FORECAST ACCURACY
Mean Squared Error (MSE): Another way to eliminate the problem of positive errors canceling negative errors is to square the forecast error. Regardless of whether the forecast error has a positive or negative sign, the squared error will always have a positive sign. If we accumulate these squared errors over time and find the average value of these squared errors, we refer to this measure as the mean squared error (MSE). For our hypothetical two forecasting methods, the squared errors can be calculated for each year and an average can be obtained for these yearly squared errors, as follows:
Year | Actual Demand At | Hypothetical Forecasting Method 1 | Hypothetical Forecasting Method 2 | | | Forecast Ft | Forecast Error At - Ft | Squared Error (At - Ft)2 | Forecast Ft | Forecast Error At - Ft | Squared Error (At - Ft)2 | 1 | 310 | 315 | -5 | 25 | 370 | -60 | 3600 | 2 | 365 | 375 | -10 | 100 | 455 | -90 | 8100 | 3 | 395 | 390 | 5 | 25 | 305 | 90 | 8100 | 4 | 415 | 405 | 10 | 100 | 535 | -120 | 14400 | 5 | 450 | 435 | 15 | 225 | 390 | 60 | 3600 | 6 | 465 | 480 | -15 | 225 | 345 | 120 | 14400 | | Total Squared Error | 700 | | | 52200 | | Mean Squared Error | 700/6 = 116.67 | | | 52200/6 = 8700 |
Method 1 seems to have provided more accurate forecasts over this six year horizon, as evidenced by its considerably smaller MSE. The Question often arises as to why one would use the more cumbersome MSE when the MAD calculations are a bit simpler (you don’t have to square the deviations). MAD does have the advantage of simpler calculations. However, there is a benefit to the MSE method. Since this method squares the error term, large errors tend to be magnified. Consequently, MSE places a higher penalty on large errors. This can be useful in situations where small forecast errors don’t cause much of a problem, but large errors can be devastating.
MEASURING FORECAST ACCURACY
Mean Absolute Percent Error (MAPE): A problem with both the MAD and MSE is that their values depend on the magnitude of the item being forecast. If the forecast item is measured in thousands or millions, the MAD and MSE values can be very large. To avoid this problem, we can use the MAPE. MAPE is computed as the average of the absolute difference between the forecasted and actual values, expressed as a percentage of the actual values. In essence, we look at how large the miss was relative to the size of the actual value. For our hypothetical two forecasting methods, the absolute percentage error can be calculated for each year and an average can be obtained for these yearly values, yielding the MAPE, as follows:
Year | Actual Demand At | Hypothetical Forecasting Method 1 | Hypothetical Forecasting Method 2 | | | Forecast Ft | Forecast Error At - Ft | Absolute % Error 100|At - Ft|/At | Forecast Ft | Forecast Error At - Ft | Absolute % Error 100|At - Ft|/At | 1 | 310 | 315 | -5 | 1.16% | 370 | -60 | 19.35% | 2 | 365 | 375 | -10 | 2.74% | 455 | -90 | 24.66% | 3 | 395 | 390 | 5 | 1.27% | 305 | 90 | 22.78% | 4 | 415 | 405 | 10 | 2.41% | 535 | -120 | 28.92% | 5 | 450 | 435 | 15 | 3.33% | 390 | 60 | 13.33% | 6 | 465 | 480 | -15 | 3.23% | 345 | 120 | 17.14% | | Total Absolute % Error | 14.59% | | | 134.85% | | Mean Absolute % Error | 14.59/6= 2.43% | | | 134.85/6= 22.48% |
Method 1seems to have provided more accurate forecasts over this six year horizon, as evidenced by the fact that the percentages by which the forecasts miss the actual demand are smaller with Method 1 (i.e., smaller MAPE).
ILLUSTRATION OF THE FOUR FORECAST ACCURACY MEASURES
Here is a further illustration of the four measures of forecast accuracy, this time using hypothetical forecasts that were generated using some different methods than the previous illustrations (called forecasting methods A and B; actually, these forecasts were made up for purposes of illustration). These calculations illustrate why we cannot rely on just one measure of forecast accuracy.
| | Hypothetical Forecasting Method A | Hypothetical Forecasting Method B | Year | Actual DemandAt | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | SquaredDeviation(At - Ft)2 | Abs. % Error |At-Ft|/At | Forecast Ft | Forecast Error At - Ft | Absolute Deviation |At - Ft| | SquaredDeviation(At - Ft)2 | Abs. % Error|At-Ft|/At | 1 | 310 | 330 | -20 | 20 | 400 | 6.45% | 310 | 0 | 0 | 0 | 0% | 2 | 365 | 345 | 20 | 20 | 400 | 5.48% | 365 | 0 | 0 | 0 | 0% | 3 | 395 | 415 | -20 | 20 | 400 | 5.06% | 395 | 0 | 0 | 0 | 0% | 4 | 415 | 395 | 20 | 20 | 400 | 4.82% | 415 | 0 | 0 | 0 | 0% | 5 | 450 | 430 | 20 | 20 | 400 | 4.44% | 390 | 60 | 60 | 3600 | 13.33% | 6 | 465 | 485 | -20 | 20 | 400 | 4.30% | 525 | -60 | 60 | 3600 | 12.90% | | | Totals | 0 | 120 | 2400 | 30.55% | Totals | 0 | 120 | 7200 | 26.23% | | | | MFE =0/6 =0 | MAD =120/6 =20 | MSE =2400/6 =400 | MAPE=30.55/65.09% | | MFE =0/6 =0 | MAD =120/6 =20 | MSE =7200/6 =1200 | MAPE=26.23/64.37% |
You can observe that for each of these forecasting methods, the same MFE resulted and the same MAD resulted. With these two measures, we would have no basis for claiming that one of these forecasting methods was more accurate than the other. With several measures of accuracy to consider, we can look at all the data in an attempt to determine the better forecasting method to use. Interpretation of these results will be impacted by the biases of the decision maker and the parameters of the decision situation. For example, one observer could look at the forecasts with method A and note that they were pretty consistent in that they were always missing by a modest amount (in this case, missing by 20 units each year). However, forecasting method B was very good in some years, and extremely bad in some years (missing by 60 units in years 5 and 6). That observation might cause this individual to prefer the accuracy and consistency of forecasting method A. This causal observation is formalized in the calculation of the MSE. Forecasting method A has a considerably lower MSE than forecasting method B. The squaring magnified those big misses that were observed with forecasting method B. However, another individual might view these results and have a preference for method B, for the sizes of the misses relative to the sizes of the actual demand are smaller than for method A, as indicated by the MAPE calculations. MONITORING FORECAST ACCURACY OVER TIME Tracking Signal: A tracking signal (T.S.) is a tool used to continually monitor the quality of our forecasting method as we progress through time. A tracking signal value is calculated each period and a determination is made as to whether it falls into an acceptable range. An upper limit and a lower limit will have been established for the tracking signal, and these values define the acceptable range. If the tracking signal drifts outside of the acceptable range, that is an indication that the forecasting method being used is no longer providing accurate forecasts. Tracking signals also help to indicate whether there is bias creeping into the forecasting process. Bias is a tendency for the forecast to be persistently under or persistently over the actual value of the data. Tracking signal is calculated as follows: Tracking signal = | Cumulative error | | MAD | Illustration of the computation of tracking signals to accompany a progression of hypothetical forecasts made over time some hypothetical forecasting method. (These forecasts were not made with any of the forecasting methods we illustrated – the forecasts were contrived to keep the numbers manageable.) Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10 | +1.00 | 2 | 365 | 371 | -6 | 4 | 6 | 16 | 8 | +.50 | 3 | 395 | 387 | 8 | 12 | 8 | 24 | 8 | +1.50 | 4 | 415 | 431 | -16 | -4 | 16 | 40 | 10 | –.40 | 5 | 450 | 460 | -10 | -14 | 10 | 50 | 10 | –1.40 | 6 | 465 | 461 | 4 | -10 | 4 | 54 | 9 | –1.11 | Keep in mind that each line in the above table would have been calculated in successive years. At the end of each year we can look back at the most recent year and compare the forecast we made with the actual demand that occurred. The next several pages show how these calculations would have unfolded through the years, and how they would have been plotted on a graph to determine whether our forecasting method still appeared to be working well. We now begin illustrating the computation and plotting of tracking signals to accompany the progression of forecasts made over time with hypothetical forecasting Method 1. In this illustration we will assume that the upper limit has been set at a value of 3, and the lower limit has been set at a value of -3. In practice these limits may be higher or lower than these values, and they do not necessarily need to have the same numerical value. The values for these limits are largely a function of how costly or disruptive inaccurate forecasts are. As we run through time, assume that the forecast made for year 1 was 300, and the subsequent demand that occurred in year 1 was 310. The tracking signal calculated and plotted after year 1 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 2 was 371, and the subsequent demand that occurred in year 2 was 365. The tracking signal calculated and plotted after year 2 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 3 was 387, and the subsequent demand that occurred in year 3 was 395. The tracking signal calculated and plotted after year 3 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 4 was 431, and the subsequent demand that occurred in year 4 was 415. The tracking signal calculated and plotted after year 4 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | 4 | 415 | 431 | -16 | (10)+(-6) +(8)+(-16) -4 | 16 | (10)+(6) +(8)+(16) 40 | 40/4 = 10 | -4/10 = –.40 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 5 was 460, and the subsequent demand that occurred in year 5 was 450. The tracking signal calculated and plotted after year 5 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | 4 | 415 | 431 | -16 | (10)+(-6) +(8)+(-16) -4 | 16 | (10)+(6) +(8)+(16) 40 | 40/4 = 10 | -4/10 = –.40 | 5 | 450 | 460 | -10 | (10)+(-6) +(8)+(-16) +(-10) -14 | 10 | (10)+(6) +(8)+(16) +(10) 50 | 50/5 = 10 | -14/10 = –1.40 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Continuing with our movement through time, assume that the forecast made for year 6 was 461, and the subsequent demand that occurred in year 6 was 465. The tracking signal calculated and plotted after year 6 would be as follows:
Year | At | Ft | At - Ft | Cum. Error | |At - Ft| | Total |At - Ft| | MAD | T.S. | 1 | 310 | 300 | 10 | 10 | 10 | 10 | 10/1 = 10 | 10/10 = +1.00 | 2 | 365 | 371 | -6 | (10)+(-6) 4 | 6 | (10)+(6) 16 | 16/2 = 8 | 4/8 = +.50 | 3 | 395 | 387 | 8 | (10)+(-6) +(8) 12 | 8 | (10)+(6) +(8) 24 | 24/3 = 8 | 12/8 = +1.50 | 4 | 415 | 431 | -16 | (10)+(-6) +(8)+(-16) -4 | 16 | (10)+(6) +(8)+(16) 40 | 40/4 = 10 | -4/10 = –.40 | 5 | 450 | 460 | -10 | (10)+(-6) +(8)+(-16) +(-10) -14 | 10 | (10)+(6) +(8)+(16) +(10) 50 | 50/5 = 10 | -14/10 = –1.40 | 6 | 465 | 461 | 4 | (10)+(-6) +(8)+(-16) +(-10)+(4) -10 | 4 | (10)+(6) +(8)+(16) +(10)+(4) 54 | 54/6 = 9 | -10/9 = –1.11 | Tracking Signal | | | | | | | | | | | | | | | | | | | 4 | | | | | | | | | | | | | | | | | | | | 3 | | | | | | | | Upper | Limit | | | | | | | | | | | 2 | | | | | | | | | | | | | | | | | | | | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Year | -1 | | | | | | | | | | | | | | | | | | | | -2 | | | | | | | | | | | | | | | | | | | | -3 | | | | | | | | | | | | | | | | | | Lower | Limit | -4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
SOME NOTES ABOUT TRACKING SIGNALS
Tracking signal review Recall that the tracking signal is calculated as follows: Tracking Signal (TS) = | Cumulative Error | | MAD |
The cumulative error can be either positive or negative, since each forecast error (At - Ft) can be either positive or negative. If you forecast too low in any period (i.e., the forecast is below the demand), the forecast error will be positive. If you forecast too high in any period (i.e., the forecast is above the demand), the forecast error will be negative). The MAD will always be positive! Consequently, the tracking signal could end up being either a positive number or a negative number.
What to watch for If the tracking signal plots outside the acceptable range (i.e., above the upper limit or below the lower limit), that is an indication that things are no longer going well in our forecasting, and we should re-examine our method. However, even when the tracking signal plots between the upper limit and lower limit, this is not always an indication that things are going well in our forecasting. Consider the following: When we forecast, we expect to miss (i.e., make forecast errors). If we are unbiased in our forecasting approach, we should expect our forecast to be too high on some occasions, and too low on other occasions. Ultimately, if we are making reasonably accurate forecasts the cumulative error should fluctuate between positive and negative values, always hovering around zero. Suppose the TS is consistently plotting in the positive range. This would be an indication that we are consistently incurring a lot of positive forecast errors (i.e., forecasting too low). This would be an indication that bias has crept into our forecasting approach (i.e., a bias toward forecasting too low), and we should re-examine our forecasting approach. Suppose the TS is consistently plotting in the negative range. This would be an indication that we are consistently incurring a lot of negative forecast errors (i.e., forecasting too high). This would be an indication that bias has crept into our forecasting approach (i.e., a bias toward forecasting too high), and we should re-examine our forecasting approach. What are reasonable tracking signal limits How tight or how loose the tracking signal limits are set is a function of the consequences of forecast errors. The upper limit and the lower limit do not have to be the same distance from the zero mark. Suppose that it is not a big deal if we forecast too high (negative forecast error) and make too much of a product. We can hold the excess in inventory and sell it at a later date. However, if we forecast too low (positive forecast error) we will not have enough product to satisfy customer demand, and we are likely to lose customers to our competitors. In such a case, we would probably have a tight range on the positive side of our tracking signal graph, and a relatively loose range on the negative side of our tracking signal graph. Alternatively, suppose that if we forecast too high (negative forecast error) and make too much of a product, the cost consequences are severe, for this product has a short shelf life or becomes obsolete quickly, and excess inventory will quickly become worthless. However, if we forecast too low (positive forecast error) and do not have enough product to satisfy customer demand it is no big deal, for customers are willing to wait for later deliveries. In such a case, we would probably have a tight range on the negative side of our tracking signal graph, and a relatively loose range on the positive side of our tracking signal graph.
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