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Moving Average, MAD, Tracking Signal Problems Problems (short) 1-2 1. Given the following data, compute 3-period moving average forecast for period 6? Period 1 2 3 4 5 Demand 73 68 65 72 67 (65+72+67)/3 = 68 2. Monthly sales for the past five months were as follows: April (15), May (20), June (18), July (22), August (20). Determine a September forecast, using a 4-period moving average. A) 16.5 B) 18.75 C) 19.1 D) 20.0 E) none of the above MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 2 Problems (short) 3-4 3. In order to increase the responsiveness of a forecast (i.e., respond quickly to the data changes) made using the moving average technique, the number of periods in the average should be: A) decreased B) increased C) multiplied by a larger alpha D) multiplied by a smaller alpha E)none of the above 4. Given the following data, Compute MAD and TS at the end of period 4. Sum |A-F| = 15 Period Actual Forecast A-F |A-F| MAD = Sum |A-F| /n 1 15 15 0 0 =15/4 = 3.75 2 10 6 4 4 Sum (A-F) = 9 3 16 8 8 8 TS = Sum(A-F)/MAD 4 9 12 -3 3 TS = 9/3.75 = 2.4 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 3 Problems (short) 5-6 5. Given forecast errors of 4, 8, and - 3, what is the MAD? A) 4 B) 3 C) 5 D) 6 E) 12 6. The sum of the forecast errors (SFE) and the mean absolute deviation (MAD) are calculated in each period. The values of SFE and MAD in the last period to be 46 and 21, respectively. Which of the following is the value of tracking signal in the last period? A) 0.6 B) 1.8 C) 2 D) 2.2 E) 2.5 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 4 Problems 7-8 7. You must choose between two alternative forecasting models. Both models have been used to prepare forecasts for a six-month period. Compute mean absolute deviation (MAD) for the forecasting model 2. Month Sales Forecasts A) 0.33 B) 2.0 C) 5.67 D) 9.51 E) none of the above 1 2 3 4 5 6 492 470 485 493 498 492 Model 1 488 484 480 490 497 493 Model 2 495 482 478 488 492 493 8. Which forecasting model would you recommend? What is the MAD for the recommended forecasting model? A) Model 1 with MAD of 4.67 B) Model 2 with MAD of 0.33 C) Model 1 with MAD of 3.39 D) Model 2 with MAD of 2.0 E) none of the above MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 5 Problem 9 9. I- to select the best forecasting technique II- to estimate the standard deviation of the forecast. III- to see if the forecast is within control limits IV- to see if the forecast does not show any specific pattern. A) the main two applications of MAD are I and II. The main two applications of Tracking Signal are III and IV. B) the main two applications of MAD are I and III. The main two applications of Tracking Signal are II and IV. C) the main two applications of MAD are I and IV. The main two applications of Tracking Signal are II and III. D) the main two applications of MAD are II and III. The main two applications of Tracking Signal are I and IV. E) none of the above MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 6 Problem 10 10. Given the following tracking signal graph A) the forecasting method overestimates the demand B) the forecasting method underestimates the demand C) the demand is very seasonal D) the forecasting method is moving average E) the forecasting method is exponential smoothing MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 7 Problem 11 11. The 5-period moving average in month 6 was 150 units. Actual demand in month 7 is 180 units. What is 6 period moving average in month 7? MA56 = (A6+A5+A4+A3+A2)/5 MA67 = (A7+A6+A5+A4+A3+A2)/6 MA56 = (A6+A5+A4+A3+A2)/5 = 150 A6+A5+A4+A3+A2 = 750 A7 = 180 MA67 = (A7+A6+A5+A4+A3+A2)/6 MA67 = (180+750)/6 = 155 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 8 Problem 12 12. Actual demand in month 8 is 160 units. The 4-period moving average in month 7 was 110 units. What is 5-period moving average in month 8? A) 100 MA47 = (A7+A6+A5+A4)/4 = 110 B) 110 C) 120 A7+A6+A5+A4 = 440 D) 140 A8 = 160 E) 150 MA58 = (A8+ A7+A6+A5+A4)/5 MA58 = (160+440)/5 = 120 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 9 Problem 13 13. Suppose the 5-period moving average in period 20 is equal to 800. Suppose period 16 demand is 850. Also suppose the demand for period 21 is 900. Compute 5-period moving average for period 21. MA520 = (A20+A19+A18+A17+A16)/5 =800 = ( =( +850)/5 =800 +850 =4000 MA521 = (A21+ A20+A19+A18+A17)/5 =??? = ( =( +A16)/5 =3150 +A21)/5 +900)/5 = (3150+900)/5 =810 MA521 = MA520 +(A21- A16)/5 MA521 = 800 +(900- 850) /5=810 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 10 Moving Average, from period t to t+1 Using the following data you can compute 4-period and 7-period moving averages in period 20. t 14 15 16 17 18 19 20 At 658 864 1110 634 855 738 910 (658+864+1110+634+855+738+910)/7 = 824.14 (634+855+738+910)/4 = 784.25 Now suppose you do not have the actual data. You only the demand for period 21 to be 800, 4-period moving average in period 20 to be 784.25, and 7-period moving average in period 20 to be 824.14. Can you compute 7-period moving average and 4 period moving average in period 21 without using the original data? MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 11 Problem 14: n-Period Moving Average from t to t+1 Let us first check how do we compute 7-period moving average using all the available data MA720 = (A14+A15+A16+A17+A18+A19+A20)/7 MA720 = (658+864+1110+634+855+738+910)/7 = 824.14 Demand for period 21 is 800 MA721 = (864+1110+634+855+738+910+800)/7 = 844.43 MA720 = (864+1110+634+855+738 +910)/7 + (658)/7 = 824.14 MA721 = (800 )/7 + ( 864+1110+634+855+738+900)/7=844.43 Therefore, we can compute MA721 , in the following simple way MA721 = MA720 +(A21- A14)/7 MA721 = 824. 14 +(800- 658) /7=844.43 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 12 Problem 15: n-Period Moving Average from t to t+1 Suppose 4-Period moving average in month 20 is 784.25, the actual demand for period 17 is 634, and the demand for period 21 is 800. Compute 4-period moving average for period 21 without using other data? MA4 20 = (634+855+738+910)/4 = 784.25 MA421 = (855+738+910+800)/4 = 825.75 Therefore, we can compute MA421 , in the following simple way MA421 = MA420 +(A21- A17)/4 MA421 = 784.25 +(800 - 634) /4=825.75 In general ( the data of period t minus the data of period t-n) / n In Moving Average forecasting always F(t+1) = MAt F22= MA21 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 13 Problem 16: MAD from t to t+1 Actual and forecast data are available from period 1 to period 10. In period 10: MAD = 110 and TS = 2.2, Forecast for period 11 is 1210 (F11=1210) , Actual demand in Period 11 is 1100 (A11=1100). Compute MAD in period 11. First Lets look at MAD in Period 10 MAD = Sum |A-F| /n 110 = Sum |A-F| /10 Sum |A-F| = 1100 Sum |A-F| from period 1 to 10 = 1100 In period 11, A-F = 1100 -1210 = -110 Sum |A-F| from period 1 to 11 = 1100 + 110 = 1210 MAD in period 11 = Sum |A-F| /11 MAD in period 11 = 1210/11 = 110 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 14 Problem 17: TS from t to t+1 Actual and forecast data are available from period 1 to period 10. In period 10: MAD = 110 and TS = 2.2, Forecast for period 11 is 1210 (F11=1210) , Actual demand in Period 11 is 1100 (A11=1100). MAD in Period 11 = 110. Compute TS in period 11. First Lets look at TS in Period 10 TS = Sum(A-F)/MAD 2.2 = Sum(A-F)/110 Sum (A-F) from period 1 to 10 = 242 (A-F) in period 11 = 1100-1210 = -110 Sum (A-F) from period 1 to 11 = 242 -110 = 132 TS = Sum(A-F)/MAD Sum(A-F) in Period 11 = 132 MAD in Period 11 = 110 TS in Period 11 = 132/110 = 1.2 MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 15 Problem 18: TS UCL and LCL What are the reasonable values for UCL and LCL in Tracking Signal? At is Actual and Ft is forecast of a random variable such as demand. Forecast error (A random Variable) Et =At-Ft has mean of 0. MAD provides an estimate for the standard deviation of Et. StdDev (Et) = 1.25 MAD. See for example http://www.estepsoftware.com/papers/madrsquare.pdf Et = Normal (0,1.25MAD) If x = Normal(µ,σ) Sum (x) = Normal(µ, √N σ) StdDev [Sum(Et)] = √N StdDev (Et) MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 16 Problem 18: TS UCL and LCL Et = Normal (0,1.25MAD) Sum (Et) = Normal (0, √N 1.25MAD) 3 ≥ (Et -0)/(N 1.25 MAD) ≥ -3. + 3n 1.25 ≥ (Et -0)/MAD ≥ - 3N 1.25. + 3.75N ≥ (Et -0)/MAD ≥ - 3.75N. Tracking Signal TS= Et/MAD with samples of size N is distributed normally around 0 with UCL = 3.75N and LCL =-3.75N MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 17 Problem 18: TS UCL and LCL Et = Normal (0,1.25MAD) Sum (Et) = Normal (0, √N 1.25MAD) 3 ≥ (Et -0)/N 1.25 MAD ≥ -3. + 3n 1.25 ≥ (Et -0)/MAD ≥ - 3N 1.25. + 3.75N ≥ (Et -0)/MAD ≥ - 3.75N. Tracking Signal TS= Et/MAD with samples of size N is distributed normally around 0 with UCL = 3.75N and LCL =-3.75N MA, MAD, TS Problems Ardavan Asef-Vaziri Jan-2015 18