ISM 3530 – Spring 2005 Forecasting: Page 1 of 24TYPES OF FORECASTING METHODS Qualitative methods: These types of forecasting methods are based on judgments or opinions, and are subjective in nature. They do not rely on any mathematical computations. Quantitative methods: These types of forecasting methods are based on quantitative models, and are objective in nature. They rely heavily on mathematical computations. ISM 3530 – Spring 2005 Forecasting: Page 2 of 24 QUALITATIVE FORECASTING METHODS Qualitative Methods Executive Opinion Approach in which a group of managers meet and collectively develop a forecast. Market Research Approach that uses surveys and interviews to determine customer preferences and assess demand. Delphi Method Approach in which a forecast is the product of a consensus among a group of experts. 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. They try to project based upon those associations.ISM 3530 – Spring 2005 Forecasting: Page 3 of 24 QUANTITATIVE FORECASTING METHODS Quantitative forecasting methods can be divided into two categories: time series models and causal models. Causal Models Causal models assume that the variable being forecasted is related to other variables in the environment. . ISM 3530 – Spring 2005 Forecasting: Page 4 of 24 TIME SERIES MODELS Model Naïve Description Uses last period’s actual value as a forecast Simple Mean (Average) Uses an average of all past data as a forecast Uses an average of a specified number of the most recent observations. with each observation receiving the same emphasis (weight) Uses an average of a specified number of the most recent observations. with each observation receiving a different emphasis (weight) A weighted average procedure with weights declining exponentially as data become older An exponential smoothing model with a mechanism for making adjustments when strong trend patterns are inherent in the data A mechanism for adjusting the forecast to accommodate any seasonal patterns inherent in the data Technique that uses the least squares method to fit a straight line to the data Simple Moving Average Weighted Moving Average Exponential Smoothing Trend Adjusted Exponential Smoothing Seasonal Indexes Linear Trend Line . . Random: Erratic and unpredictable variation in the data over time. Seasonality: Data exhibit upward and downward swings in a short to intermediate time frame (most notably during a year). Trend: Data exhibit a steady growth or decline over time. with no growth or decline. Cycles: Data exhibit upward and downward swings in over a very long time frame.ISM 3530 – Spring 2005 Forecasting: Page 5 of 24 PATTERNS THAT MAY BE PRESENT IN A TIME SERIES Level or horizontal: Data are relatively constant over time. we may make slightly different assumptions about starting points to get the process started for different models. the forecasts are merely educated guesses).ISM 3530 – Spring 2005 Forecasting: Page 6 of 24 DATA SET TO DEMONSTRATE FORECASTING METHODS The following data set represents a set of hypothetical demands that have occurred over several consecutive years. these numbers represent demands in thousands of units).e. for. and these quarterly values have been amalgamated into yearly totals. Finally.. Then. to keep the numbers at a manageable size. For various illustrations that follow. Year 1 2 3 4 5 6 Quarter 1 20 58 40 104 116 136 Quarter 2 28 86 54 140 170 198 Quarter 3 34 104 72 174 210 246 Quarter 4 18 52 34 82 104 120 Total Annual Demand 100 300 200 500 600 700 . The data have been collected on a quarterly basis. several zeros have been dropped off the numbers (i. after all. In most cases we will assume that each year a forecast has been made for the subsequent year. 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. We then made a forecast for the subsequent year. . Actual Demand (At) 100 300 200 500 600 700 Year 1 2 3 4 5 6 7 Forecast (Ft) -100 300 200 500 600 700 Notes There was no prior demand data on which to base a forecast for period 1 From this point forward. these forecasts were made on a year-by-year basis. 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. and so on right through to the forecast for year 7.ISM 3530 – Spring 2005 Forecasting: Page 7 of 24 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). and so on right through to the forecast for year 7. then waited to see what demand unfolded during the year. Actual Demand (At) 100 300 200 500 600 700 Year 1 2 3 4 5 6 7 Forecast (Ft) -100 200 200 275 340 400 Notes There was no prior demand data on which to base a forecast for period 1 From this point forward. In this illustration we assume that each year (beginning with year 2) we made a forecast.ISM 3530 – Spring 2005 Forecasting: Page 8 of 24 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. . these forecasts were made on a year-by-year basis. We then made a forecast for the subsequent year. we made a forecast for year 2 using a naïve method (100). Actual Demand (At) 100 300 200 500 600 700 Year 1 2 3 4 5 6 7 Forecast (Ft) 200 100 200 250 350 550 650 Notes This forecast was a guess at the beginning. Then. these forecasts were made on a year-by-year basis. in the absence of data at startup. . This forecast was made using a naïve approach. after year 1 elapsed. In this illustration we assume that a 2-year simple moving average is being used. We will also assume that. with each observation receiving the same emphasis (weight). From this point forward.ISM 3530 – Spring 2005 Forecasting: Page 9 of 24 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. we made a guess for the year 1 forecast (200). Beyond that point we had sufficient data to let our 2-year simple moving average forecasts unfold throughout the years. we used a naïve method to make a forecast for year 2 (100) and year 3 (300). Then. We will also assume that. after year 1 elapsed. in the absence of data at startup. From this point forward. Beyond that point we had sufficient data to let our 3-year simple moving average forecasts unfold throughout the years. This forecast was made using a naïve approach. This forecast was made using a naïve approach.333 600 Notes This forecast was a guess at the beginning. Actual Demand (At) 100 300 200 500 600 700 Year 1 2 3 4 5 6 7 Forecast (Ft) 200 100 300 200 333. we made a guess for the year 1 forecast (200).ISM 3530 – Spring 2005 Forecasting: Page 10 of 24 ANOTHER SIMPLE MOVING AVERAGE ILLUSTRATION In this illustration we assume that a 3-year simple moving average is being used. . these forecasts were made on a year-by-year basis.333 433. . Beyond that point we had sufficient data to let our 3-year weighted moving average forecasts unfold throughout the years. these forecasts were made on a year-by-year basis. In this illustration we assume that a 3-year weighted moving average is being used. We will also assume that. we used a naïve method to make a forecast for year 2 (100) and year 3 (300). From this point forward. Then. year prior to that. year prior to that. The weights that were to be used are as follows: Most recent year. This forecast was made using a naïve approach. we made a guess for the year 1 forecast (200). . . in the absence of data at startup.3.ISM 3530 – Spring 2005 Forecasting: Page 11 of 24 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.5. after year 1 elapsed.2 Actual Demand (At) 100 300 200 500 600 700 Year 1 2 3 4 5 6 7 Forecast (Ft) 200 100 300 210 370 490 630 Notes This forecast was a guess at the beginning. This forecast was made using a naïve approach. . as follows: When we made the forecast for last period (Ft). you should recognize that Ft+1 = ∀ At + ∀ (1-∀ )At-1 + ∀ (1-∀ )2At-2 + ∀ (1-∀ )3At-3 + ∀ (1-∀ )4At-4 + ∀ (1-∀ )5At-5 ………. forecasts made with this model will include a portion of every piece of historical demand. This can be observed by expanding the above formula.ISM 3530 – Spring 2005 Forecasting: Page 12 of 24 EXPONENTIAL SMOOTHING METHOD Exponential smoothing method: The forecast for next period (period t+1) will be calculated as follows: Ft+1 = ∀ At + (1-∀ )Ft (equation 1) Where ∀ is a smoothing coefficient whose value is between 0 and 1. Furthermore. there will be different weights placed on these historical demand values. Although 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). with older data receiving lower weights. it was made in the following fashion: Ft = ∀ At-1 + (1-∀ )Ft-1 (equation 2) If we substitute equation 2 into equation 1 we get the following: Ft+1 = ∀ At + (1-∀ )[∀ At-1 + (1-∀ )Ft-1] Which can be cleaned up to the following: Ft+1 = ∀ At + ∀ (1-∀ )At-1 + (1-∀ )2Ft-1 (equation 3) We could continue to play that game by recognizing that Ft-1 = ∀ At-2 + (1-∀ )Ft-2 (equation 4) If we substitute equation 4 into equation 3 we get the following: Ft+1 = ∀ At + ∀ (1-∀ )At-1 + (1-∀ )2[∀ At-2 + (1-∀ )Ft-2] Which can be cleaned up to the following: Ft+1 = ∀ At + ∀ (1-∀ )At-1 + ∀ (1-∀ )2At-2 + (1-∀ )3Ft-2 If you keep playing that game. . the values get smaller and smaller. .ISM 3530 – Spring 2005 Forecasting: Page 13 of 24 As you raise those decimal weights to higher and higher powers. Then. for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model.1 Actual Demand Forecast (At) (Ft) Notes 100 300 200 500 600 700 200 190 201 200.ISM 3530 – Spring 2005 Forecasting: Page 14 of 24 EXPONENTIAL SMOOTHING ILLUSTRATION In this illustration we assume that. Year 1 2 3 4 5 6 7 . in the absence of data at startup. and so on right through to the forecast for year 7. since there was no prior demand data. This set of forecasts was made using an ∀ value of . From this point forward.81 267.729 310. After the forecast was made.9561 This was a guess. we waited to see what demand unfolded during the year. these forecasts were made on a year-by-year basis.9 230. we made a guess for the year 1 forecast (200). We then made a forecast for the subsequent year. Then. we made a guess for the year 1 forecast (200). we waited to see what demand unfolded during the year. We then made a forecast for the subsequent year. From this point forward. This set of forecasts was made using an ∀ value of .ISM 3530 – Spring 2005 Forecasting: Page 15 of 24 A SECOND EXPONENTIAL SMOOTHING ILLUSTRATION In this illustration we assume that.56 330.0384 This was a guess. After the forecast was made. for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model.048 404. these forecasts were made on a year-by-year basis. Year 1 2 3 4 5 6 7 .2 262.2 Actual Demand Forecast (At) (Ft) Notes 100 300 200 500 600 700 200 180 204 203. in the absence of data at startup. and so on right through to the forecast for year 7. since there was no prior demand data. Year 1 2 3 4 5 6 7 .4 Actual Demand Forecast (At) (Ft) Notes 100 300 200 500 600 700 200 160 216 209. This set of forecasts was made using an ∀ value of . for each subsequent year (beginning with year 2) we made a forecast using the exponential smoothing model. and so on right through to the forecast for year 7. We then made a forecast for the subsequent year.456 541. these forecasts were made on a year-by-year basis. From this point forward. in the absence of data at startup.76 435. After the forecast was made.6 325.2736 This was a guess. Then. we made a guess for the year 1 forecast (200).ISM 3530 – Spring 2005 Forecasting: Page 16 of 24 A THIRD EXPONENTIAL SMOOTHING ILLUSTRATION In this illustration we assume that. we waited to see what demand unfolded during the year. since there was no prior demand data. Ultimately. computations for b and a reveal the following b = 120 a = -20 Y = -20 + 120X This equation can be used to forecast for any year into the future. and does this by minimizing the squared values of the deviations of the points from the line). the statistical formulas compute a slope for the trend line (b) and the point where the line crosses the y-axis (a). 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. This results in the straight line equation Y = a + bX Where X represents the values on the horizontal axis (time). For example: Year 7: Forecast = -20 + 120(7) = 820 Year 8: Forecast = -20 + 120(8) = 940 Year 10: Forecast = -20 + 120(10) = 1180 .ISM 3530 – Spring 2005 Forecasting: Page 17 of 24 LINEAR TREND LINE Linear trend line method: This method is a version of the linear regression technique. For the demonstration data. the approach attempts to reduce the vertical deviations of the points from the trend line. and Y represents the values on the vertical axis (demand). 120 1. Then compute an average value for the seasonal indexes for each quarter (average for the numbers in columns 2. 5 Q4 .777 Q1 . 3 Q2 28 86 54 140 170 198 Col. 4 Q3 1.688 Col.360 1.406 Q3 1. 3 Q2 1.832 . Col.800 .773 . and 5) Col. 3. Col. 6 Annual Demand 100 300 200 500 600 700 Col. 4 Q3 34 104 72 174 210 246 Col.122 Col.080 1.720 .ISM 3530 – Spring 2005 Forecasting: Page 18 of 24 CALCULATING SEASONAL INDEX VALUES Begin by dividing the total annual demand by 4 (which is the number of periods in the year) to see what the quarterly demand would have been if the annual demand had been distributed evenly throughout the year (this is column 7 below). 1 Year 1 2 3 4 5 6 Col.387 1.397 Col. divide the actual quarterly demand by the column 7 average. 7 Annual/4 25 75 50 125 150 175 For each quarter in each year.693 .793 Col.680 .440 1.133 1.773 . . This is demonstrated on the next page. 2 Q1 .686 Q4 . This gives a measure of how each quarter's demand compared to a uniform split (each computation is a seasonal index for the quarter and year in question).131 Q2 1.120 1.392 1. 7 Annual/4 25 75 50 125 150 175 Any subsequent yearly forecast can be broken down into quarterly forecasts using these seasonal index values. 5 Q4 18 52 34 82 104 120 Col.147 1. 4. 2 Q1 20 58 40 104 116 136 Col.400 1.656 . 1 Year 1 2 3 4 5 6 Avg.800 .693 . 6 Annual Demand 100 300 200 500 600 700 Col. Year 7 8 9 10 Forecast 820 940 1060 1180 If these annual forecasts were evenly distributed over each year.I.122 Q3 205 235 265 295 1.695 182. .793 Q2 205 235 265 295 1. We must take these even splits and multiply them by the seasonal index (S.337 202.I.466 328.991 263.688 Annual/4 205 235 265 295 However.232 Q4 141.388 370.490 186. the quarterly forecasts would look like the following: Annual Forecast 820 940 1060 1180 Year 7 8 9 10 S.962 Q3 286.ISM 3530 – Spring 2005 Forecasting: Page 19 of 24 USING SEASONAL INDEX VALUES The following forecasts were made for the next 4 years using the linear trend line approach (the trend line formula developed was Y = -20 + 120X. Q1 205 235 265 295 .827 Q2 229.979 If you check these final splits. The results of these calculations are shown below. Annual Forecast 820 940 1060 1180 Year 7 8 9 10 Q1 162. 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).397 Q4 205 235 265 295 .648 297.269 210. where Y is the forecast and X is the year number). seasonality in the past demand suggests that these forecasts should not be evenly distributed over each quarter.053 161.305 330.310 412.) values to get a more reasonable set of quarterly forecasts.048 233. 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.8 + 23. For example.ISM 3530 – Spring 2005 Forecasting: Page 20 of 24 OTHER METHODS FOR MAKING SEASONAL FORECASTS Let's go back and reexamine the historical data we have for this problem.3714X For year 7. 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. I have used the linear trend line method on the quarter 1 strip of data. the Q1 column displays the progression of quarter 1 demands over the past six years. which would result in the following trend line: Y = -2.800 . X = 7. Year 1 2 3 4 5 6 Q1 20 58 40 104 116 136 Q2 28 86 54 140 170 198 Q3 34 104 72 174 210 246 Q4 18 52 34 82 104 120 To illustrate. so the resulting Q1 forecast for year 7 would be 160. Application of regression formulas yields the following forecasting model: Y = 92.e.. The textbook illustrates an example in which the sales for a product seem to be related to the amount of money spent on advertising. and the sales that corresponded to these expenditures.870) .15X If the company plans to spend $53. The following table shows varying amounts that have been spent on advertising on past occasions. In its simplest form. Advertising Expenditure (thousands of $) 32 52 50 55 Sales Dollars (thousands of $) 130 151 150 158 The independent variable (X) is the advertising expenditure.15(53) = 153. The dependent variable (Y) is the sales dollars.83 + 1. This approach tries to project demand based upon those associations.000 on advertising in a particular year (i. That line is then used to forecast the dependent variable for some selected value of the independent variable. X = 53). linear regression is used to fit a line to the data.83 + 1. the forecast for sales will be: Y = 92.ISM 3530 – Spring 2005 Forecasting: Page 21 of 24 CAUSAL MODELS METHOD Causal models assume that the variable being forecasted (the dependent variable) is related to other variables (independent variables) in the environment.87 (which means sales projections are $153. . and looking at them alone might give a false sense of security. consider our original data. or E t = At .Ft -5 -10 5 10 15 -15 0 Hypothetical Forecasts Made With Method 2 Ft 160 390 110 620 540 580 Forecast Error With Method 2 At . But. To illustrate. clearly Method 1 is generating better forecasts than Method 2.Ft -60 -90 90 -120 60 120 0 Based on the accumulated forecast errors over time. the accumulation of the Et values is not very revealing. and the accompanying pair of hypothetical forecasts made with two different forecasting methods. These positive and negative errors cancel one another. Unfortunately. Hypothetical Forecasts Actual Made With Demand Method 1 Year At Ft 1 100 105 2 300 310 3 200 195 4 500 490 5 600 585 6 700 715 Accumulated Forecast Errors Forecast Error With Method 1 At .ISM 3530 – Spring 2005 Forecasting: Page 22 of 24 MEASURING FORECAST ACCURACY Forecast error: Forecast error is a measure of how accurate our forecast was in a given time period. so we need to look at the accumulation of errors over time. It is calculated as the actual demand minus the forecast. for some of them will be positive errors and some will be negative.Ft Forecast error in one time period does not convey much information. the two methods look equally good. For our hypothetical two forecasting methods.Ft| 5 10 5 10 15 15 60 60/6=10 Hypothetical Forecasting Method 2 Forecast Ft 160 390 110 620 540 580 Forecast Error At .Ft| 60 90 90 120 60 120 540 540/6=90 Total Absolute Deviation Mean Absolute Deviation Clearly Method 1 has provided more accurate forecasts over this six year horizon. as follows: Hypothetical Forecasting Method 1 Actual Demand Year At 1 100 2 300 3 200 4 500 5 600 6 700 Forecast Ft 105 310 195 490 585 715 Forecast Error At . we refer to this deviation as the absolute deviation.Ft -60 -90 90 -120 60 120 Absolute Deviation |At . When we disregard the sign and only consider the size of the error. .Ft -5 -10 5 10 15 -15 Absolute Deviation |At . we refer to this measure as the mean absolute deviation (MAD). regardless of sign).ISM 3530 – Spring 2005 Forecasting: Page 23 of 24 MEASURING FORECAST ACCURACY Mean Absolute Deviation (MAD): To eliminate the problem of positive errors canceling negative errors. If we accumulate these absolute deviations over time and find the average value of these absolute deviations. the absolute deviations can be calculated for each year and an average can be obtained for these yearly absolute deviations. as evidenced by its considerably smaller MAD. a simple measure is one that looks at the absolute value of the error (size of the deviation. If it drifts outside of the acceptable range.00 -1.ISM 3530 – Spring 2005 Forecasting: Page 24 of 24 MONITORING FORECAST ACCURACY OVER TIME Tracking Signal: A tracking signal (T. Bias is a tendency for the forecast to be persistently under or persistently over the actual value of the data. -1.5 6. and a determination is made as to whether it falls into an acceptable range (much like we saw with control charts).5 9 10 T. Each period a tracking signal value is calculated.) is a tool used to continually monitor the quality of our forecasting method as we progress through time. Tracking signals help to indicate whether there is bias creeping into the forecasting process.00 -2. that is an indication that the forecasting method being used is no longer providing accurate forecasts.Ft| 5 15 20 30 45 60 Year 1 2 3 4 5 6 At 100 300 200 500 600 700 Ft 105 310 195 490 585 715 At .Ft| 5 10 5 10 15 15 MAD 5 7.67 7.S. Tracking signal is calculated as follows: Tracking signal = algebraic sum of forecast errors (ASFE) MAD Illustration of the computation of tracking signals to accompany the progression of forecasts made over time with hypothetical forecasting Method 1. Total |At .Ft -5 -10 5 10 15 15 ASFE -5 -15 -10 0 15 0 |At .S.67 0 .50 0 +1.