Question

Name 1. Observations of the demand for a certain part stocked at a parts supply depot during the calendar year 1999 were Month Janua Demand 89 57 144 221 Februa March ril Ma June Jul August September October November December 280 223 286 212 275 188 312 Determine the forecasts for July though December of the year 1999 considering 2, 3, and 6- month moving averages Compute the MAD for the obtained forecasts. Which method gave you better results? Determine the forecast for January of the year 2000 using the method that resulted in higher accuracy a. b. c. 2. Using the data of the previous problem Determine the forecasts for January- December of the year 1999 using exponential smoothing with a smoothing constant equal to 0.15, 0.25, and 0.5 a. b Compute the MSE for the obtained forecasts. Which method gave you better results? c. Determine the forecast for January of the year 2000 using the method that gave you more accurate forecasts based on MSE

Answer 1

Answer 1)

Answer a)

2-month moving average

The moving averages (MA) forecast for the nth period is computed using the following formula Forecast period YI-1 The 2-month moving average forecasts are calculated in the table below: Period January February March April May Data 89 57 144 221 MA(2) Forecasts (89 + 57)12 73 (57144)/2 100.5 (144 221)/2 182.5 (221 + 177)/2 199 (177 280)/2 228.5 (280 + 223)/2 251.5 (223 + 286)/2 254.5 (286+ 212)/2 249 (212 + 275)/2 243.5 (275 + 188)/2 231.5 (188312)/2 250 280 223 286 212 275 188 312 ne July August September October November December January

Now, the following table shows the calculations of the corresponding error metrics MA(2) Forecasts Abs. % Error2 Period Data Abs. Error Error January 89 57 February 144-73 144 40,31% (144- 73)2 5041 144- 73= 71 March 144 73 221- 100.51 = 120.5 (221 100.5)2 14520.25 221-1005 221 100.5 April 54.52% 182.5 May 182.5)2 3.11% 182.5 5.5 30.25 (280 280-199 280 280一 1991 = 81 199)" = 6561 June 280 199 28.93% 223 223一 228.5| 5.5 July 228.5 223 228.5)2 2.47% 30.25 (286 286 251.51 286-251.5 August 251.5 251.5)2 = 1190.25 286 12.06% 34.5 212 212 254.51 212 254.5 2545)2 = 1806.25 September 20.05% 42.5 (275- 249) 676 275-249 275 249 26 249 October 275 9.45%

(188 243.5)- 188- 188-243.5 188 243.5 November 243.5 29.52% 3080.25 (312 6480.25 55.5 312 231.5 312 December 231.5 231.5) 25.8% 80.5 January 250 Therefore, we have that for the given time series data, the forecast for January using a 2-month moving average is 250). The mean average deviation is MAD = 52.25, the mean square error is MSE 394 1.58 and the mean average percentage error is MAPE 23.52%

3-month moving average

The moving averages (MA) forecast for the nth period is computed using the following formula: Forecast period n The 3-month moving average forecasts are calculated in the table below: Period January February March April May Data 89 57 144 221 MA(3) Forecasts (89 +57 1443 96.6667 (57 144 221)/3 140.6667 (144 + 221177/3- June 280 180.6667 (221177 280/3 226 (177 280 +223)y3 223 286 212 275 July August September October 226.6667 (280 223 286)/3 263 (223 286 212)3 240.3333 (286+212 275)13 188 November 257.6667 (212 275 188)/3 225 (275 + 188 +312y3 December 312 January 258.3333

Now, the following table shows the calculations of the corresponding error metrics MA(3) Forecasts Abs. % Error Period January February March Data Abs. Error Error2 89 57 144 221 (221 21-96.6667 April 221 96.6667 96.666796.6667) 56.26% 15458.7778 124.3333 _ -lem 20.53% 177-140.6667 May 140.6667 140.66671-140.6667)"- 36.3333 1320.1111 (280 280 2-180.6667 180.6667 180.6667)2- June 280 180.6667 35.48% 9867.1111 99.3333 (223 223 Jul 223 226 2267 = 9 (286- 3520.4444 226 3 1.35% 286- 286-226.6667 August 226.6667 286 226.6667 226.6667) 20.75% 59.3333 (212 263)2 = 212-263 212 263 51 212 September 263 24.06% 2601 (275 240.3333) 275 275-240.3333 275 240.3333 October 240.3333- 12.61% 34.6667 188- 69.6667 1201.7778 (188 183-257.6567 November 188 257.6667 257.6667| 257.6667 - 4853.4444 37.06%

312 225 87 312 December 225 225)2 27.88% 7569 January 258.3333 Therefore, we have that for the given time series data, the forecast for January using a 3-month moving average is 258.3333). The mean average deviation is MAD 62.74, the mean square error is MSE 5155.63 and the mean average percentage error is MAPE-26.22%.

6-month moving average

The moving averages (MA) forecast for the nth period is computed using the following formula: Forecast period 6 The 6-month moving average forecasts are calculated in the table below: Period January February March April May June Data 89 57 144 221 MA(6) Forecasts 280 (8957 144221+ 177 + 280)16 161.3333 Ju 223 (57 144 221177 280 +223V6 183.6667 August 286 (144221177 280 +223 + 286)/6 = 221.8333 September 212 (221 + 177 280+ 223+ 286 + 2126 233.1667 October 275 (177 280223+ 286212 +27516 242.1667 188 November (280 + 223 286 + 212+ 275 +188V6 244 312 December (223+286+ 212 275+188 312/6 249.3333 January

Now, the following table shows the calculations of the corresponding error metrics MA(6) Forecasts Abs. % Error Period Data Abs. Erroir rror2 89 57 144 221 January February March April May June 280 223 61.6667 286 102.3333 (223 223-1 223 July 161.3333 161.3333161.3333) 27.65% 3802.7778 286 286-183.6667 August 286 183.6667 183.6667 183.6667) 35.78% 10472.1111 212 221.8 (212 212-221.8333 8333221.8333) 96.6944 September 212 221.8333 4.64% 9.8333 275 41.8333 (275 233.1667 233166722461507 - 1750.0278 275-233.1667 233.1667 October 275 15.21% 188 (188 188-242.1667 188 242.1667 November 242.1667 242.1667)2 28.81% 54.1667 2934.0278 (312 244)2 312 24468 December 312 244 21.79% 4624 249.3333 January

Therefore, we have that for the given time series data, the forecast for January using a 6-month moving average is 249.3333). The mean average deviation is MAD 56.31, the mean square error is MSE-3946.61 and the mean average percentage error is MAPE-22.32%

Answer b)

MAD (2-month moving average) = Σ|Error|/N = (71+120.5+5.5+81+5.5+34.5+42.5+26+55.5+80.5)/10 = 52.25

MAD (3-month moving average) = Σ|Error|/N = (124.3333+36.3333+99.3333+3+59.3333+51+34.6667+69.6667+87)/9 = 62.74

MAD (6-month moving average) = Σ|Error|/N = (61.6667+102.3333+9.8333+41.8333+54.1667+68)/6 = 56.31

We can see that MAD value of 2-month moving average is lowest among the three forecasting methods. So, 2-month moving average will give the better result in comparison to other methods.

Answer c)

The forecast of January of the year 2000 using 2-month moving average method is 250 (See Part (a) for calculation)

Answer 2

Answer a)

Exponential smoothing (ES) forecast with smoothing constant α=0.15

The exponential smoothing (ES) forecast with smoothing constant α = 0.15 for the nth period is computed using the following formula The corresponding exponential smoothing forecasts with smoothing constant a calculated in the table below. 0.15 are ES(0.15) Forecasts 89 89 +0.15 (89 89) 89 890.15 (57 89)84.2 Period January February March Data 89 57 144 84.2 +0.15. (144 84.2) 93.17 0.15 (221 112.34450.15. (177 122.0428250.15 (280 April 221 93.17 May 9317) = 112.3445 June 280 112.3445) 122.0428 223 122042825) = 145.7364 145.73640125+0.15 157.3259 (223 145.73640125) August 286 157.32594106250.15 (286 1573259410625) = 176.627 September 212

176.62704990312+0.15 275 (212 176.62704990312) October 181.933 181.93299241766+0.15 (275 181.93299241766) - 195.893 188 November 195.89304355501 0.15 (188 195.89304355501) 194.7091 312 December 194.70908702176 +0.15 (312 194.70908702176)- 212.3027 January Now, the following table shows the calculations of the corresponding error metrics ES(0.15) Forecasts Abs. % Error Error2 Period Data Abs. Error 89-89 (89 89)0 89-891-0% 89 89 January 89 0 57-89 57-89 32 February 89 57 57 50.14% 89) 1024 (144- 84.2) 3576.04 144-84.2 144- 84.2 59.8 84.2 144 41.53% March 144 (221 221 93.17 221-93.17 April 221 93.17 93.17) 163405089 57.84%

177-112.3445- 36.53% 112.3445 112.344512.3445) May 64.6555 4180.3337 (280- 122.0428122.0428) 280 290-122.042 280 June 280 122.0428 56.41% 157.9572 223 77.2636 286 24950.4691 (223 145.7364) 223-145.7364 Jul 223 145.7364 145.7364- 34.65% 5969.6637 (286 288-157.3259 44.99% 286 157.3259 157.3259157.3259) August 16557.0134 128.6741 212 35.373 275 93.067 188 7.893 312 (212 212-176.627 212 176.627 176.627) September 212 176.627 16.69% 1251.2456 (275- 275-181 933 275 October 181.933 181.933 181.933)2 33.84% 8661.4679 (188- 62.3001 (312 188-195 893 195.893195.893)2 195.893 188 November 188 4.2% 312-194.7091 312 194.7091 194.7 December 194.7091 37,59% 117.2909 13757.1583 January 212.3027 Therefore, we have that for the given time series data, the forecast for January using an exponential smoothing (ES) forecasting method with smoothing constant a0.15 is 212.3027). The mean average deviation is MAD 75.15, the mean square error is MSE 8027.52 and the mean average percentage error is MAPE-35.03%

Exponential smoothing (ES) forecast with smoothing constant α=0.25

0.25 for the nth period is The exponential smoothing (ES) forecast with smoothing constant a computed using the following formula The corresponding exponential smoothing forecasts with smoothing constant α = 0.25 are calculated in the table below Period January February March Data 89 57 144 221 ES(0.25) Forecasts 89 89 0.25 (89 89) 89 89 0.25 (57 -89) 81 81 0.25 (144 81) - April 96.75 96.75+0.25. (221 127.81250.25 (177 140.1093750.25 (280 May 96.75) 127.8125 June 280 127.8125) 140.1094 July 223 140.109375) = 175.082 175.082031250.25 (223 175.08203125) August 286 187.0615 187.0615234375-0.25 211.7961 (286 187.0615234375) September 212

211.79614257812+0.25 (212 211.79614257812)- 211.8471 275 October 211.84710693359 +0.25 (275 211.84710693359) - 227.6353 November 188 227.63533020020.25 312 (188 227.6353302002)- 217.7265 December 217.726497650150.25 (312 217.72649765015)- 241.2949 January Now, the following table shows the calculations of the corresponding error metrics ES(0.25) Forecasts Abs. % Error Period Data Abs. Error Error2 89-89- (89 89)0 89-89 89 : 0% 89 89 January 57-89 50.14% 57-89 (57 89) 1024 February 89 57 32 144-81 144 81 63 221 (144- 81) 3969 March 81 144 43.75% 221-96.75 96.75 221 56.22% April 221 96.75) 15438.0625 96.75 124.25 177-127.8125- 27.79% 127.8125) May 127.8125 127.8125 2419.4102 49.1875

(280 280 20-140.1004 280 June 280 140.1094 140.1094 140.1094) 49.96% 19569.387 139.8906 223 47.918 286 (223 223-175.082 175.082) 175.082 Jul 223 175.082 21.49% 2296.1317 (286 1286-187.0615 - 615187.0615)- August 286 187.0615 187.0615- 98.9385 34.59% 9788.8221 212 0.2039 275 (212 0.0416 (275 212-211.7961 212 0.1% 211.7961 211.7961) September 212 211.7961 273-211.8471 275 211.8471)"- October 211.8471 211.8471 22.96% 3988.2879 63.1529 188 39.6353 312 94.2735 (188 188-227.6353 188 188 227.6353 227.6353227.6353) November 21.08% 1570.9594 (312- 8887.4932 312-217.7265 312 217.7265 217.7265217.7265)- December 312 30.22% 241.2949 January Therefore, we have that for the given time series data, the forecast for January using an exponential smoothing (ES) forecasting method with smoothing constant α = 0.25 is 241.2949). The mean average deviation is MAD 62.7, the mean square error is MSE 5745.97 and the mean average percentage error is MAPE 30.36%

Exponential smoothing (ES) forecast with smoothing constant α=0.5

0.5 for the nth period is The exponential smoothing (ES) forecast with smoothing constant a computed using the following formula The corresponding exponential smoothing forecasts with smoothing constant a in the table below: 0.5 are calculated Period January February March Data 89 57 144 221 ES(0.5) Forecasts 89 89+0.5. (89 89)=89 89 0.5.(57 89) 73 73+0.5. (144 73)- April 108.5 108.5+0.5 (221 164.75 +0.5 (177 170.875+0.5 (280 225.4375 0.5 (223 224.218750.5 (286 May 108.5) = 164.75 June 280 164.75) = 170.875 July 223 170.875) 225.4375 225.4375) 224.2188 224.21875) 255.1094 255.109375 +0.5. (212 August 286 September 212 275 October 255.109375) 233.5547

233.55468750.5 (275 233.5546875) = 254.2773 188 November 254.277343750.5. (188 254.27734375) 221.1387 December 312 221.1386718750.5 (312 221.138671875)- 266.5693 January Now, the following table shows the calculations of the corresponding error metrics ES(0.5) Forecasts Abs. % Error Error2 Period Data Abs. Error (89一 89)20 89 89 189-891 = 0% 89 89 January 89 0 57-8 57 89 57 89 February 89) 1024 50.14% 32 144-T3 144 49.31% (144 144 March 144 73 73) 5041 73 71 221 112.5 221-10.- 221 108.5 108.5)- 12656.25 April 108.51 = 50.9% 177-164.75 - 164.75 May 164.75 164.75) 6.92% 12.25 280 109.125 150.0625 (280 280-170.875- 38.97% June 280 170.875 170.875)"- 11908.2656 280 170.875|

223 2.4375 286 (223 5.9414 (286 223-225.4375 1.09% 225.4375 225.43751-225.4375)"- Jul 223 1286-22421881- 2188224.2188)- 286 August 224.2188 21.6% 61.7813 212 43.1094 3816.9229 (212 212-255.1094 212 255.1094 255.1094 September 212 255.1094 20.33% 1858.4182 275 (275- 275-233.5347 15.07% 275 233.5547 October 2335547 233.5547) 41.4453 188 66.2773 312 1717.7139 (188 188-254.2773 254.2773 254.2773)- 4392.6863 188 188 35.25% November 254.2773 (312 312-221.1387| 312 29.12% 312 221.1387 221.1387 221.1387)- 8255.7809 December 90.8613 January 266.5693 Therefore, we have that for the given time series data, the forecast for January using an exponential smoothing (ES) forecasting method with smoothing constant a0.5 is 266.5693). The mean average deviation is MAD 53.57, the mean square error is MSE 4235.59 and the mean average percentage error is MAPE 27.06%

Answer b)

MSE (α=0.15) = Σ|Error|²/N = 8027.52 (See Answer 2 Part (a) for calculation)

MSE (α=0.25) = Σ|Error|²/N = 5745.97 (See Answer 2 Part (a) for calculation)

MSE (α=0.5) = Σ|Error|²/N = 4235.59 (See Answer 2 Part (a) for calculation)

We can see that MSE value of Exponential smoothing (ES) forecast with smoothing constant α=0.5 is lowest among the three forecasting methods. So, ES forecast with smoothing constant α=0.5 will give the better result in comparison to other methods.

Answer c)

The forecast of January of the year 2000 using ES method with smoothing constant α=0.5 is 266.5693 (See Part (a) for calculation)