Solution Manual For Production And Operations Analysis, 6th Edition

Chapter 02 – Forecasting Forecasting Solutions To Problems From Chapter 2 2.1 Trend Seasonality Cycles Randomness 2.2 Cycles have repeating patterns that vary in length and magnitude. 2.3 a) b) c) 2.4 Marketing: New sales and existing sales forecasts. Causal models relating advertising to sales Accounting Interest rate forecasts; cost components, bad debts. Finance: Changes in stock market, forecast return on investment return from specific projects. Production: Forecast product demand (aggregate and individual), availability of resources, labor. 2.5 Time Series Regression or Causal Model Delphi Method a) Aggregate forecasts deal with item groups or families. b) Short term forecasts are generally next day or month; Long term forecasts may be for many months or years into the future. c) Causal models are based on relationship between predictor variables and other variables. Naive models are based on the past history of series only 2.6 The Delphi Method is a technique for achieving convergence of group opinion. The method has several potential advantages over the Jury of Executive Opinion method depending upon how that method is implemented. If the executives are allowed to reach a consensus as a group, strong personalities may dominate. If the executives are interviewed, the biases of the interviewer could affect the results. 2-1 Chapter 02 – Forecasting 2.7 Some of the issues that a graduating senior might want to consider when choosing a college to attend include: a) how well have graduates fared on the job market, b) what are the student attrition rates, c) what will the costs of the college education be and d) what part-time job opportunities might be available in the region. Sources of data might be college catalogues, surveys on salaries of graduating seniors, surveys on numbers of graduating seniors going on to graduate or professional schools, etc. 2.8 The manager should have been prepared for the consequences of forecast error. 2.9 It is unlikely that such long term forecasts are accurate. 2.10 This type of criteria would be closest to MAPE, since the errors measured are relative not absolute. It makes more sense to use a relative measure of error in golf. For example, an error of 10 yards for a 200 yard shot would be fine for most golfers, but a similar error for a 20 yard shot would not. 2.11 a) (26)(.1) + (21)(.1) + (38)(.2) + (32)(.2) + (41)(.4) = 35.1 b) (23)(.1) + (28)(.1) + (33)(.2) + (26)(.2) + (21)(.4) = 25.3 2.12 a) and b) Forecast Period Actual 80.5 73.5 77.5 107.5 98.5 87.5 100.0 78.5 79.5 95.0 3 4 5 6 7 8 9 10 11 12 72 83 132 65 110 90 67 92 98 73 (86 + 75)/2 (75 + 72)/2 etc = = c) (216)/10 (7175)/10 MAD MSE = = MAPE = = = 21.6 717.5 100  1  ei  n = 25.61 Di  2-2 et +8.5 -9.5 -54.5 42.5 -11.5 -2.5 +33.0 -13.5 -18.5 +22.0 Chapter 02 – Forecasting 2.13 Fcst 1 Fcst 2 Demand Err 1 Err 2 223 289 430 134 190 550 256 340 375 110 225 525 33 51 -55 -24 35 -25 46 20 -15 -2 75 35 e1/D*100 e2/D100 46 20 15 2 75 35 12.89062 15.0000 14.66667 21.81818 15.55556 4.761905 17.96875 5.88253 4.00000 1.81818 33.33333 6.66667 14.11549 (MAPE1) 11.61155 (MAPE2) It means that E(ei)  0. This will show up by considering n  ei i =1 A bias is indicated when this sum deviates too far from zero. 2.15 Using the MAD: 1.25 MAD = (1.25)(21.6) = 27.0 (Using s, the sample standard deviation, one obtains 28.23) 2.16 MA (3) forecast: MA (6) forecast: MA (12) forecast: 258.33 249.33 205.33 2-3 Er2^2 1089 2116 2601 400 3025 225 576 4 1225 5625 625 1225 1523.5 1599.166 (MSE1 (MSE2) Err2 32.16666 (MAD2) 2.14 210 320 390 112 150 490 Er1^2 |Err1| 33 51 55 24 35 25 37.16666 (MAD1) Chapter 02 – Forecasting 2.17, 2.18, and 2.19. One-step-ahead Two-step-ahead Month Forecast Forecast Demand July August September October November December 205.50 225.25 241.50 250.25 249.00 240.25 149.75 205.50 225.25 241.50 250.25 249.00 223 286 212 275 188 312 e1 MAD = e2 -17.50 -60.75 29.50 -24.75 61.00 -71.75 -73.25 -80.50 13.25 -33.50 62.25 -63.00 44.2 54.3 The one step ahead forecasts gave better results (and should have according to the theory). 2.20 Month Demand MA(3) MA(6) July August September October November December 223 286 212 275 188 312 226.00 226.67 263.00 240.33 257.67 225.00 161.33 183.67 221.83 233.17 242.17 244.00 MA (6) Forecasts exhibit less variation from period to period. 2.21 An MA(1) forecast means that the forecast for next period is simply the current period’s demand. Month Demand MA(4) MA(1) Error Month Demand MA(4) MA(1) Error July August September October November December 223 286 212 275 188 312 205.50 225.25 241.50 250.25 249.00 240.25 280 223 286 212 275 188 57 -63 74 -63 87 -124 MAD = 78.0 (Much worse than MA(4)) 2-4 Chapter 02 – Forecasting Ft = Dt-1 + (1-)Ft-1 2.22 a) FFeb = (.15)(23.3) + (.85)(25) = 24.745 FMarch = (.15)(72.3) + (.85)(24.745) = 31.88 FApr = (.15)(30.3) + (.85)(31.88) = 31.64 FMay = (.15)(15.5) + (.85)(31.63) = 29.22 b) FFeb = (.40)(23.3) + (.60)(25) = 24.32 FMarch = 43.47 FApr = 38.20 FMay = 29.12 c) Compute MSE for February through April: Month Error (a) ( = .15) Error (b) ( = .40) Feb Mar Apr 47.45 1.56 16.13 47.88 13.17 22.70 838.04 993.74 MSE  2.23 = = .15 gave a better forecast Small  implies little weight is given to the current forecast and virtually all weight is given to past history of demand. This means that the forecast will be stable but not responsive. Large  implies that a great deal of weight is applied to current observation of demand. This means that the forecast will adjust quickly to changes in the demand pattern but will vary considerably from period to period. 2-5 Chapter 02 – Forecasting 2.24 a) Week MA(3) Forecast 4 5 6 7 8 17.67 20.33 28.67 22.67 21.67 b) and c Week ES(.15) Demand MA(3) |err| |err| 4 5 6 7 8 17.67 18.32 20.67 19.37 19.32 22 34 12 19 23 17.67 20.33 28.67 22.67 21.67 4.33 15.68 8.67 0.37 3.68 4.33 13.67 16.67 3.67 1.33 6.547540 MAD-ES 7.934 MAD-MA Based on these results, ES(.15) had a lower MAD over the five weeks d) It is the same as the exponential smoothing forecast made in week 6 for the demand in week 7, which is 19.37 from part c). 2.25 2 2   = = .286 N+1 7 a)  = b) N= c) 2 2 From Appendix 2-A  e = =1.12 2 − 2 −  2 −.05 = 39 .05 2 Hence 2.26 N= 2 = 1.1 Solving gives 2 −  = .1818 It is the same as the one step ahead forecast made at the end of March which is 31.64. 2-6 Chapter 02 – Forecasting 2.27 The average demand from Jan to June is 161.33. Assume this is the forecast for July. a) b) Month Forecast Aug Sept Oct Nov Dec 173.7 196.2 199.4 214.5 209.2 [.2(223) + (.8)(161.33)] etc. Month Demand ES(.2) (Error) Aug Sept Oct Nov Dec 286 212 275 188 312 173.7 196.2 199.4 214.5 209.2 112.3 15.8 75.6 26.5 102.8 MAD 66.6 MA(6) 183.7 221.8 233.2 242.2 244 (Error) 102.3 9.8 41.8 54.2 68.0 55.2 MA(6) gave more accurate forecasts. c) For  = .2 the consistent value of N is (2-)/ = 9. Hence MA(6) will be somewhat more responsive. Also the ES method may suffer from not being able to flush out “bad” data in the past. 3000 2000 1000 500 1 Jan 2 Feb 3 Mar 4 Apr Month 2-7 5 May 6 Jun Chapter 02 – Forecasting a) “Eyeball” estimates: slope = 2750/6 = 458.33, intercept = -500. b) Regression solution obtained is Sxy = (6)(28,594) – (21)(5667) = 52,557 Sxx = (6)(91) – (21)2 = 105 b = Sxy S xx = 52, 577 = 500.54 105 a = D − b (n + 1) / 2 = -.807.4 c) Regression equation  = -807.4 + (500.54)t D t Month Forecasted Usage July (t = 7) Aug (t = 8) Sept (t = 9) Oct (t = 10) Nov (t = 11) Dec (t = 12) 2696 3197 3698 4198 4699 5199 d) One would think that peak usage would be in the summer months and the increasing trend would not continue indefinitely. 2.29 a) Month Forecast Month Forecast Jan Feb Mar Apr May June 5700 6200 6700 7201 7702 8202 July Aug Sept Oct Nov Dec 8703 9203 9704 10,204 10,705 11,206 (note that these are obtained from the regression equation  t = 807.4 + 500.54 t with t = 13, 14,. . . .) D The total usage is obtained by summing forecasted monthly usage. Total forecasted usage for 1994 = 101,431 2-8 Chapter 02 – Forecasting b) Moving average forecast made in June = 944.5/mo. Since this moving average is used for both one-step-ahead and multiple-step-ahead forecasts, the total forecast for 1994 is (944.5)(12) = 11,334.) c ) 1200 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec The monthly average is about 1200 based on a usage graph of this shape. This graph assumes peak usage in summer months. The yearly usage is (1200)(12) = 14,400 which is much closer to (b), since the moving average method does not project trend indefinitely. 2-9 Chapter 02 – Forecasting 2.30 From the solution of problem 24, a) slope = 500.54 value of regression in June = -807.4 + (500.54)(6) = 2196 S0 = 2196 G0 = 500.54  = .15  = .10 S1 = D1 + (1-)(S0 + G0) = (.15)(2150) + (.85)(2196 + 500.54) = 2615 G1 = (.1)[2615 – 2196] + (.9)(500.54) = 492.4 S2 = (.15)(2660) + (.85)(2615 + 492.4) = 3040 G2 = .1 [3040 – 2615] + (.9) (492.4) = 485.7 b) One-step-ahead forecast made in Aug. for Sept. is S2 + G2 = 3525.7 Two-step-ahead forecast made in Aug for Oct is S2 + G2 = 3040 + 2(485.7) = 4011.4 c) S1 + 5(G1) = 2615 + 5(492.4) = 5077. 2.31 This observation would lower future forecasts. Since it is probably an “outlier” (nonrepresentative observation) one should not include it in forecast calculations. 2.32 Both regression and Holt’s method are based on the assumption of constant linear trend. It is likely in many cases that the trend will not continue indefinitely or that the observed trend is just part of a cycle. If that were the case, significant forecast errors could result. 2.33 Month Yr 1 2 3 4 5 6 7 8 9 10 12 18 36 53 79 134 112 90 66 45 1 Yr 16 14 46 48 88 160 130 83 52 49 2 Dem1/Mean Dem2/Mean 0.20 0.31 0.61 0.90 1.34 2.27 1.90 1.53 1.12 0.76 0.27 0.24 0.78 0.81 1.49 2.71 2.20 1.41 0.88 0.83 2-10 Avg (factor)” 0.24 0.27 0.70 0.86 1.42 2.49 2.05 1.47 1.00 0.80 Chapter 02 – Forecasting 11 12 23 21 14 26 Totals 689 726 0.39 0.36 0.24 0.44 0.31 0.40 12 We used the Quick and Dirty Method here. The average demand for the two years was (689 + 726)/2 = 707.5. 2.34 a) (1) MA Centered MA (2) Centered MA on periods Ratio (1)/(2) 42.440 42.440 0.2828 0.5891 41.750 1.8204 43.125 1.2058 43.375 0.3689 44.000 0.7272 45.000 1.5778 46.375 1.3369 49.625 0.2821 49.375 0.9114 Quarter Demand 1 2 12 25 3 76 4 52 41.25 5 16 42.25 6 32 44.00 7 71 42.75 8 62 45.25 9 14 44.75 10 45 48.00 11 84 51.25 49.500 1.6970 12 47 47.50 49.500 0.9494 41.25 42.25 44.00 42.75 45.25 44.75 48.00 51.25 47.50 The four seasonal factors are obtained by averaging the appropriate quarters (1, 5, 9 for first; 2, 6, 10 for the second, etc.) One obtains the following seasonal factors 0.3112 0.7458 1.6984 1.1641 The sum is 3.9163. Norming the factors by multiplying each by 4 = 1.0214 3, 9163 2-11 Chapter 02 – Forecasting we finally obtain the factors: 0.318 0.758 1.735 1.189 b) 2.35 Quarter Demand Factor Deseasonalized Series 1 2 3 4 5 6 7 8 9 10 11 12 12 25 76 52 16 32 71 62 14 45 84 47 0.318 0.758 1.735 1.189 0.318 0.758 1.735 1.189 0.318 0.758 1.735 1.189 37.74 32.98 43.80 43.73 50.31 42.22 40.92 52.14 44.03 59.37 48.41 39.53 c) 47.40 d) Must “re-seasonalize” the forecast from part (c) (47.40)(.318) = 15.07 a) V1 = (16 + 32 + 71 + 62)/4 = 45.25 V2 = (14 + 45 + 84 + 47)/4 = 47.5 1. G0 = (V2 – V1)/N = 0.5625 2. S0 = V2 + G0 (N-1/2) = 47.5 + (0.5625)(3/2) = 48.34 3. ct = Dt Vi N + 1/ 2 − j G0 -2N+1 =  t  0 c-7 = 16 = 0.36 45.25 − (5/ 2 − 1)(..56) c-6 = 32 = 0.71 45.25 − (5/ 2 − 2)(.56) 2-12 Chapter 02 – Forecasting c-5 = 71 = 1.56 43.25 − (5/ 2 − 3)(.56) c-4 = 62 = 1.35 45.25 − (5/ 2 − 4)(.56) c-3 = 14 = 0.30 47.5 − (5/ 2 − 1)(.56) c-2 = 45 = 0.95 47.5 − (5/ 2 − 2)(.56) c-1 = 84 = 1.76 47.5 − (5/ 2 − 3)(.56) c0 = 47 = 0.97 47.5 − (5/ 2 − 4)(.56) (c7 + c3)/2 = .33 (c6 + c2)/2 = .83 (c5 + c1)/2 = 1.66 (c4 + c0)/2 = 1.16 Sum = 3.98 Norming factor = 4/3.9 = 1.01 Hence the initial seasonal factors are: b) c-3 = .33 c-1 = 1.67 c-2 = .83 c-0 = 1.17  = 0.2,  = 0.15,  = 0.1, D1 = 18 S1 = (D1/c-3) + (1-)(S0 + G0) = 0.2(18/0.33) + 0.8(48.34 + 0.56) = 50.03 G1 = (S1 – S0) + (1 – ) = G0 = 0.1(50.03 – 48.34) + 0.9(0.56) = 0.70 c1 = (D1/S1) + (1-)c3 = 0.15(18/50.03) + 0.85(0.33) 2-13 Chapter 02 – Forecasting = .3345 c) Forecasts for 2nd, 3rd and 4th quarters of 1993 F1,2 = [S1 + G1]c2 = (50 + .70)0.83 = 42.08 F1,3 = [S1 + 2G1]c3 = (50 + 2(.70))1.67 = 85.84 F1,4 = [S1 + 3G1]c4 = (50 + 3(.70))1.17 = 60.96 2.36 Period Dt Forecast Forecast from from 30(d) et  31(c)  et  1 2 3 4 51 86 66 35.8 82.4 56.5 15.2 3.6 9.5 42.08 85.84 60.96 8.92 0.16 5.04 MAD = 9.43 MAD = 4.71 MSE = 111.42 MSE = 35.00 Hence we conclude that Winter’s method is more accurate. 2.37 S1 = 50.03 G1 = 0.67  = 0.2  = 0.15  = 0.1 D1 = 18 D2 = 51 D3 = 85 D4 = 66 S2 = 0.2(51/0.83) + 0.8(50.03 + 0.70) = 52.87 G2 = 0.1(52.87 – 50.03) + 0.9(0.70) = 0.914 S3 = 0.2(86/1.67) + 0.8(52.87 + 0.914) = 53.33 G3 = 0.1(53.33 – 52.85) + 0.9(0.885) = 0.8445 S4 = 0.2(66/1.17) + 0.8(53.33 + 0.8445) = 54.62 G4 = 0.1(54.62 – 53.33) + 0.9(0.8445) = 0.8891 c1 = (.15)[18/50] + (0.85)(.33) = .3345  .34 c2 = (.15)[51/52.85] + 0.85(0.83) = .8502  .85 c3 = (.15)(86/53.29) + 0.85(1.67) = 1.6616  1.66 c4 = (.15)(66/54.59) + 0.85(1.17) = 1.1758  1.18 The sum of the factors is 4.02. Norming each of the factors by multiplying by 4/4.02 = .995 gives the final factors as: c1 = .34 2-14 Chapter 02 – Forecasting c2 = .84 c3 = 1.65 c4 = 1.17 The forecasts for all of 1995 made at the end of 1993 are: F4,9 = [S4 + 5G4]c1 = [54.62 + 5(0.89)]0.34 = 20.08 F4,10 = [S4 + 6G4]c2 = [54.62 + 6(0.89)]0.84 = 50.37 F4,11 = [S4 + 7G4]c3 = [54.62 + 7(0.89)]1.65 = 100.40 F4,12 = [S4 + 8G4]c4 = [54.62 + 8(0.89)]1.17 = 72.24 2.42. ARIMA(2,1,1) means 2 autoregressive terms, one level of differencing, and 1 moving average term. The model may be written ut = a0 + a1ut −1 + a2ut −2 +  t − b1 t −1 where ut = Dt − Dt −1 . Since ut = (1 − B) Dt , we have a) (1 − B) Dt = a0 + (a1 B + a2 B 2 )(1 − B) Dt + (1 − b1B) t b) Dt = a0 + (a1B + a2 B 2 )Dt + (1 − b1B) t c) Dt − Dt −1 = a0 + a1 ( Dt −1 − Dt −2 ) + a2 ( Dt −2 − Dt −3 ) +  t − b1 t −1 or Dt = a0 + (1 + a1 ) Dt −1 − a1Dt −2 + a2 ( Dt −2 − Dt −3 ) +  t − b1 t −1 2.43. ARIMA(0,2,2) means no autoregressive terms, 2 levels of differencing, and 2 moving average terms. The model may be written as wt = b0 +  t − b1 t −1 − b2 t −2 Where wt = ut − ut −1 and ut = Dt − Dt −1 . Using backshift notation, we may also write wt = (1 − B)2 Dt , so that we have for part a) a) (1 − B) 2 Dt = b0 + (1 − b1B − b2 B 2 ) t b)  2 Dt = b0 + (1 − b1 B − b2 B 2 ) t c) Dt − 2Dt −1 + Dt −2 = b0 +  t − b1 t −1 − b2 t −2 or Dt = 2Dt −1 − Dt −2 + b0 +  t − b1 t −1 − b2 t −2 2.44. The ARMA(1,1) model may be written Dt = a0 + a1Dt −1 − b1 t −1 +  t . If we substitute for Dt −1 , Dt −2 ,… one can easily see this reduces to a polynomial in ( t ,  t −1 ,…) and if we substitute for  t ,  t −1 ,… we see that this reduces to a polynomial in Dt −1 , Dt −2 ,… . . 2-15 Chapter 02 – Forecasting 2.45 a) 1400 – 1200 = 200 200/5 = 40 Change = -40 (He should decrease the forecast by 40.) b) (0.2)(0.8)4 = 0.08192 200(0.08192) = 16.384 16.384) 2.46 Change = -16.384 (He should decrease the forecast by From Example 2.2 we have the following: Forecast (ES(.1)) Observed Error (et) Quarter Failures 2 3 4 5 6 7 250 175 186 225 285 305 200 205 202 201 203 211 -50 +30 +16 -24 -82 -94 8 190 220 +30 Using MADt =  |et| + (1 -)MADt-1, we would obtain the following values: MAD1 = 50 (given) MAD2 = (.1)(50) + (.9)(50) = 50.0 MAD3 = (.1)(30) + (.9)(50) = 48.0 MAD4 = (.1)(16) + (.9)(48) = 44.8 MAD5 = (.1)(24) + (.9)(44.8) = 42.7 MAD6 = (.1)(82) + (.9)(42.7) = 46.6 MAD7 = (.1)(94) + (.9)(46.6) = 51.3 MAD8 = (.1)(30) + (.9)(51.3) = 49.2 The MAD obtained from direct computation is 46.6, so this method gives a pretty good approximation after eight periods. It has the important advantage of not requiring the user to save past error values in computing the MAD. 2.47 c1 = 0.7 c2 = 0.8 c3 = 1.0 c4 = 1.5 2-16 Chapter 02 – Forecasting 2.48 Dept yr 1 yr 2 yr 3 ratio 1 ratio 2 ratio 3 Management Marketing Accounting Production Finance Economics 835 620 440 695 380 1220 956 540 490 680 425 1040 774 575 525 624 410 1312 1.20 0.89 0.63 1.00 0.55 1.75 1.37 0.78 0.70 0.98 0.61 1.49 1.11 0.83 0.75 0.90 0.59 1.88 average 1.23 0.83 0.70 0.96 0.58 1.71 6 Mean pages over all fields and years = 696.72. The multiplicative factors in the final column give the percentages above or below the grand mean when multiplied by 100. 2.49 a) and b) Month Sales MA(3 Error Abs Err Sq Err Per Err 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 238 220 195 245 345 380 270 220 280 120 110 85 135 145 185 219 240 420 520 410 380 320 290 240 217.67 220.00 261.67 323.33 331.67 290.00 256.67 206.67 170.00 105.00 110.00 121.67 155.00 183.00 214.67 293.00 393.33 450.00 436.67 370.00 330.00 -27.33 -125.00 -118.33 53.33 111.67 10.00 136.67 96.67 85.00 -30.00 -35.00 -63.33 -64.00 -57.00 -205.33 -227.00 -16.67 70.00 116.67 80.00 90.00 27.33 125.00 118.33 53.33 111.67 10.00 136.67 96.67 85.00 30.00 35.00 63.33 64.00 57.00 205.33 227.00 16.67 70.00 116.67 80.00 90.00 747.11 15625.00 14002.78 2844.44 12469.44 100.00 18677.78 9344.44 7225.00 900.00 1225.00 4011.11 4096.00 3249.00 42161.78 51529.00 277.78 4900.00 13611.11 6400.00 8100.00 11.16 36.23 31.14 19.75 50.76 3.57 113.89 87.88 100.00 22.22 24.14 34.23 29.22 23.75 48.89 43.65 4.07 18.42 36.46 27.59 37.50 86.62 MAD 10547.47 MSE 38.31 MAPE 2-17 Chapter 02 – Forecasting 2.49 c) Month 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Sales 238 220 195 245 345 380 270 220 280 120 110 85 135 145 185 219 240 420 520 410 380 320 290 240 MA(6 Error 270.50 0.50 275.83 55.83 275.83 -4.17 290.00 170.00 269.17 159.17 230.00 145.00 180.83 45.83 158.33 13.33 145.83 -39.17 130.00 -89.00 146.50 -93.50 168.17 -251.83 224.00 -296.00 288.17 -121.83 332.33 -47.67 364.83 44.83 381.67 91.67 390.00 150.00 Abs Err Sq Err Per Err 0.50 55.83 4.17 170.00 159.17 145.00 45.83 13.33 39.17 89.00 93.50 251.83 296.00 121.83 47.67 44.83 91.67 150.00 0.25 3117.36 17.36 28900.00 25334.03 21025.00 2100.69 177.78 1534.03 7921.00 8742.25 63420.03 87616.00 14843.36 2272.11 2010.03 8402.78 22500.00 0.19 25.38 1.49 141.67 144.70 170.59 33.95 9.20 21.17 40.64 38.96 59.96 56.92 29.72 12.54 14.01 31.61 62.50 86.63 MAD 14282.57 MSE 42.63 MAPE MA(6) has about the same MAD and higher MSE and MAPE. 2.50 Month 1 2 3 4 5 6 7 8 9 10 11 12 13 Sales 238 220 195 245 345 380 270 220 280 120 110 85 135 ES(.1) 225 226.30 225.67 222.60 224.84 236.86 251.17 253.06 249.75 252.77 239.50 226.55 212.39 Error -13.00 6.30 30.67 -22.40 -120.16 -143.14 -18.83 33.06 -30.25 132.77 129.50 141.55 77.39 Abs Err 13.00 6.30 30.67 22.40 120.16 143.14 18.83 33.06 30.25 132.77 129.50 141.55 77.39 2-18 Sq Err 169.00 39.69 940.65 501.63 14437.78 20489.51 354.47 1092.65 915.07 17629.15 16769.56 20035.72 5989.65 Per Err 5.46 2.86 15.73 9.14 34.83 37.67 6.97 15.03 10.80 110.65 117.72 166.53 57.33 Alpha 0.1 Chapter 02 – Forecasting 14 15 16 17 18 19 20 21 22 23 24 145 185 219 240 420 520 410 380 320 290 240 204.65 198.69 197.32 199.49 203.54 225.18 254.67 270.20 281.18 285.06 285.56 59.65 13.69 -21.68 -40.51 -216.46 -294.82 -155.33 -109.80 -38.82 -4.94 45.56 59.65 13.69 21.68 40.51 216.46 294.82 155.33 109.80 38.82 4.94 45.56 3558.55 187.37 470.05 1641.27 46855.50 86915.99 24128.54 12056.10 1507.01 24.39 2075.31 41.14 7.40 9.90 16.88 51.54 56.70 37.89 28.89 12.13 1.70 18.98 79.18 MAD 11616.03 MSE 36.41 MAPE The error turns out to be a declining function of  for this data. Hence,  = 1 gives the lowest error. 2.51 a) Year 1 2 3 4 5 6 7 8 (Yi) Sales ($100,000) (Xi) Births Preceding Year 6.4 8.3 8.8 5.1 9.2 7.3 12.5 2.9 3.4 3.5 3.1 3.8 2.8 4.2 Obtain  Xi – 23.7,  Yi = 57.6,  XiYi = 201.29  Xi = 81.75,  Yi = 507.48 2 2 Sxx = 10.56 Sxy = 43.91 b = SXY = 4.158 SXX a = y – bx = -5.8 Hence Yt = – 5.8 + 4.158Xt-1 is the resulting regression equation. b) Y10 = -5.8 + (4.158)(3.3) = 7.9214 (that is, $792,140) 2-19 Chapter 02 – Forecasting c) Year 1 2 3 4 5 6 7 8 9 10 US Births (in 1,000,000) (Xi) Forecasted Births Using ES(.15) 2.9 3.4 3.5 3.1 3.8 2.8 4.2 3.7 3.2 3.3 3.2 3.4 3.4 3.4 Hence, forecasted births for years 9 and 10 is 3.4 million. d) Yt = -5.8 + 4.158 Xt-1 Xt-1 = 3.4 million in years 8 and 9. Substituting gives Yt = -5.8 + (4.158)(3.4) = 8.3372 for sales in each of years 9 and 10. Hence the forecast of total aggregate sales in these years is (8.3372)(2) = 16.6744 or $1,667,440. 2.52 Month 1 2 3 4 5 6 Ice cream Sales 325 335 172 645 770 950 Xi Yi Month Ice Cream Sales a) Sum Avg = Park Attendees 880 976 440 1823 1885 2436 XiYi 1 2 3 4 5 6 325 335 172 645 770 950 325 670 516 2580 3850 5700 = 21 3.5 3197. 532.8 13641 Sxx = 105 Sxy = 14709 2-20 Chapter 02 – Forecasting b = Sxy/Sxx = 140.1 a = Y – bX = 42.5 Y30 = 42.5 + (30)(140) = $4245.1 We would not be very confident about this answer since it assumes the trend observed over the first six months continues into month 30 which is very unlikely. Xi Park attendees b) Sum Avg = = Yi Ice Cream Sales XiYi 880 976 440 1823 1885 2436 325 335 172 645 770 950 286000 326960 75680 1175835 1451450 2314200 8440 1406.666 3197 532.8333 5630125 Sxx = 17,153,756 Sxy = 6,798,070 b = Sxy/Sxx = 0.396302 a = Y -bX = 24.6316 Hence the resulting regression equation is: Yi = -24.63 + 0.4Xi 2-21 Chapter 02 – Forecasting c) 6000 5000 Attendees 4000 3000 2000 1000 2 4 6 8 10 12 14 16 18 20 Months Readng the values from the curve: X12  5100 X13  5350 X14  5600 X15  5800 X16  5900 X17  5950 X18  5980 Using the regression equation Yi = -24.63 + 0.4Xi derived in part (b) we obtain the ice cream sales predictions below. Month Attendees Predicted Ice Cream Sales 12 13 14 15 16 17 18 5100 5350 5600 5800 5900 5950 5980 2015.37 2115.37 2215.37 2295.37 2335.37 2355.37 2367.37 2-22 Chapter 02 – Forecasting 2.53 The method assumes that the “best”  based on a past sequence of specific demands will be the “best”  for future demands, which may not be true. Furthermore, the best value of the smoothing constant based on a retrospective fit of the data may be either larger or smaller than is desirable on the basis of stability and responsiveness of forecasts. 2.54 Year Demand S sub t 0 1981 0.2 6.44 1982 4.3 12.16 1983 8.8 17.33 1984 18.6 23.08 1985 34.5 30.68 1986 68.2 43.65 1987 85.0 58.37 1988 58.0 65.81 G sub t 8 7.69 7.29 6.87 6.64 6.84 8.06 9.39 9.00 Forecast alpha 0.2 8.00 14.13 19.46 24.19 29.72 37.51 51.71 67.77 beta 0.2 |error| error^2 7.80 9.83 10.66 5.59 4.78 30.69 33.29 9.77 60.84 96.59 113.58 31.30 22.85 941.74 1108.00 95.37 14.05 MAD 308.78 MSE The forecast error appears to decrease with decreasing values of  and . That is,  =  = 0 appears to give the lowest value of the forecast error. 2.55 a) We are given in problem 22 that the forecast for January was 25. Hence e1 = 25-23.3 = 1.7 = E1 and M1 = |e1 | = 1.7 as well. Hence 1 = 1. FFeb = (1)(23.3) + (0)(25) = 23.3 e2 = 23.3 – 72.2 = -48.9 E2 = (.1)(-48.9)(.9)(1.7) = -3.36 M2 = (.1)(48.9) + (.9)(1.7) = 6.42 2 = 3.36/6.42 = .5234 FMarch = (.5234)(72.2) + (.4766)(23.3) = 48.73 e3 = 48.73 – 30.3 = 18.43 E3 = (.1)(18.43) + (.9)(-3.36) = -3.024 M3 = (.1)(18.43) + (.9)(6.42) = 7.621 3 = 3.024/7.621 = .396 ~ .40 FApr = (.40)(30.3) + (.60)(48.73) = 41.358 2-23 Chapter 02 – Forecasting Comparison of Methods Month Demand ES(.15) |Error| Trigg-Leach |Error| Feb March April 72.2 30.3 15.5 24.745 31.87 31.63 47.5 1.6 16.1 23.3 48.7 41.4 48.9 18.4 25.9 Obviously Trigg-Leach performed much worse for this 3-month period than did ES(.12). (The respective MAD’s are 21.7 for ES and 31.1 for Trigg-Leach.) b) Consider only the period July to December as in problem 36. As in part (a) 7 = 1. Use E6 = 567.1 – 480 = 87. F7 = 480 e7 = 480 – 500 = -20 E7 = (.2)(-20) + (.8)(87) = 65.6 M7 = (.2)(20) + (.8)(87) = 73.6 7 = 65.6/73.6 = .89 F8 = (.89)(500) + (.11)(480) = 498 e8 = 498 – 950 = -452 E8 = (.2)(-452) + (.8)(65.6) = -37.9 M8 = (.2)(452) + (.8)(73.6) = 149.3 8 = 37.9/149.3 = .25 F9 = (.25)(950) + (.75)(498) = 611 e9 = 611 – 350 = 261 E9 = (.2)(261) + (.8)(-37.9) = 21.9 M9 = (.2)(261) + (.8)(149.3) = 171.6 9 = 21.9/171.6 = .13 F10 = (.13)(350) + (.87)(620) = 584.9 e10 = 584.9 – 600 = -15.1 E10 = (.2)(-15.1) + (.8)(21.9) = 14.5 M10 = (.2)(17.8) + (.8)(171.6) = 140.8 10 = 14.5/140.8 = .10 F11= (.10)(600) + (.90)(584.9) = 586.4 e11 = 586.4 – 870 = -283.6 E11 = (.2)(-283.6) + (.8)(14.5) = -45.1 M11 =(.2)(283.6) + (.8)(140.8) = 169.4 2-24 Chapter 02 – Forecasting 11 = 45.1/169.4 = .27 F12 = (.27)(870) + (.73)(586.4) = 663.0 Performance Comparison Month Demand 7 8 9 10 11 12 500 950 350 600 870 740 MAD = Trigg-Leach Forecast 480 498 611 585 586 663 |Error| 20 452 261 15 284 77 185 The MAD for ES(.2) from problem 36 was 194.5. Hence Trigg-Leach was slightly better for this problem. c) Trigg-Leach will out-perform simple exponential smoothing when there is a trend in the data or a sudden shift in the series to a new level, since  will be adjusted upward in these cases and the forecast will be more responsive. However, if the changes are due to random fluctuations, as in part (a), Trigg-Leach will give poor performance as the forecast tries to “chase” the series. 2.56 Given information:  = .2,  = 0.2, and  = 0.2 S10 = 120, G10 = 14 c10 = 1.2 c9 = 1.1 c8 = 0.8 c7 = 0.9 a) F11 = (S10 + G10)c7 = (120 + 14)(0.9) = 120.6 b) D11 = 128 S11 = (D11/c7) + (1 – )(S10 + G10) = 135.6 G11 = (S11 – S10) + (1 – )G10 = 14.3 c11 = (D11/S11) + (1-)c7 = .909 2-25 Chapter 02 – Forecasting 11  C = 4.009 t t=8 The factors are normed by multiplying each by 1/4.009 = .9978 They will not change appreciably. F11,13 = (S11 + 2G11)C9 = (135.6 + (2)(14.3))1.1 = 180.6 Xi Yi XiYi 1 2 3 4 5 6 7 8 9 10 11 649.8 705.1 772.0 816.4 892.7 963.9 1015.5 1102.7 1212.8 1359.3 1472.8 649.8 1410.2 2316.0 3265.6 4463.5 5783.4 7108.5 8821.6 10915.2 13593.0 16200.8 66 6 10,963.0 996.64 74,527.6 2.57 a) Sum = Avg = Sxy = n  i Di − (n)(n + 1) i=1 = (11)(74,527.6) = D i 2 (11)(12 ) 2 (10,963.0) = 96,245.6 SXX = n (n +1)(2n + 1) n (n +1) (11 )(12 )(23) ((11) (12) ) = 1210 − = − 6 4 6 4 2 b = 2 Sxy S xx = 2 2 96, 245.6 = 79.54 1210 − − 10,963.0 66 a = Y− b X = = 519.4 − (79.54) 11 11 Initialization for Holt’s Method S0 = regression line in year 11 (1974) = 519.4 + (11)(79.54) = 1394.34 2-26 2 2 Chapter 02 – Forecasting Updating Equations G0 = slope of regression line = 79.54 Si = Di + (1 -)(Si-1 + Gi+1) Gi =  (Si – Si-1) + (1 -)Gi-1 GI Si |Error| |Error|2 Obs Yr Di 1 1975 1598.4 1498.78 82.03 F0,1= S0+G0= 1473.88 2 1976 1782.8 1621.21 86.07 F1,2= S1+G1= 1580.81 201.99 40798.18 3 1977 1990.9 1764.01 91.74 F2,3= S2+G2= 1707.28 283.62 80439.38 4 1978 2249.7 1934.54 99.62 F3,4= S3+G3= 1855.75 393.95 155198.35 5 1979 2508.2 2128.97 109.10 F4,5= S4+G4= 2034.16 474.04 224714.16 6 1980 2732.0 2336.86 118.98 F5,6= S5+G5= 2238.07 493.93 243966.72 7 1981 3052.6 2575.20 130.92 F6,7= S6+G6= 2455.84 596.76 356126.04 8 1982 3166.0 2798.08 140.11 F7,8= S7 +G7= 2706.11 459.89 211502.67 9 1983 3401.6 3030.88 149.38 F8,9= S8 +G8= 2938.20 463.40 214740.75 10 1984 3774.7 3299.15 161.27 F9,10 =S9+G9= 3180.26 594.44 353357.64 124.52 Totals MAD = 408.6, 15505.23 4086.54 1896349.11 MSE = 189,634.9 b) Year GNP % MA(6) Forecast GNP |Error| ES(.2) 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 0.2 1976 47.9 1977 45.2 649.8 705.1 772.0 816.4 892.7 963.9 1015.5 1102.7 1212.8 1359.3 1472.8 1598.4 8.51% 9.49% 5.75% 9.35% 7.98% 5.35% 8.59% 9.98% 12.08% 8.35% 8.53% 8.72% 1601.3 2.9 8.54% 1598.6 1782.8 11.54% 8.81% 1739.3 43.5 8.54% 1734.9 1990.9 11.67% 9.84% 1958.3 32.6 9.14% 1945.7 2-27 Forecast GNP |Error| Chapter 02 – Forecasting 1978 66.8 1979 26.4 1980 40.8 1981 41.2 1982 208.0 1983 54.6 1984 73.1 2249.7 13.00% 10.36% 2197.1 52.6 9.65% 2182.9 2508.2 11.49% 10.86% 2494.0 14.2 10.32% 2481.8 2732.0 8.92% 10.76% 2778.2 46.2 10.55% 2772.8 3052.6 11.73% 10.86% 3028.6 24.0 10.23% 3011.4 3166.0 3.71% 10.98% 3387.9 221.9 10.53% 3374.0 3401.6 7.44% 10.30% 3492.0 90.4 9.16% 3456.2 3774.7 10.97% 9.71% 3731.9 42.8 8.82% 3701.6 = 57.1 *MAD *MAD = 60.4 The moving average and exponential smoothing forecasts based on percentage increases are more accurate than Holt’s method. c) One would expect that a causal model might be more accurate. Large-scale econometric models for predicting GNP and other fundamental economic time series are common. 2-28