Nahmias Chapter 2
Production and Operations Analysis, Fourth Edition
Solutions To Problems From Chapter 2 2.12 a) and b)
Forecast (86 + 75)/2 (75 + 72)/2 etc = = 80.5 73.5 77.5 107.5 98.5 87.5 100.0 78.5 79.5 95.0 = =
Period 3 4 5 6 7 8 9 10 11 12 21.6 717.5
Actual 72 83 132 65 110 90 67 92 98 73
et +8.5 -9.5 -54.5 42.5 -11.5 -2.5 +33.0 -13.5 -18.5 +22.0
c)
MAD MSE
= =
(216)/10 (7175)/10
MAPE
=
100
1 n
∑D i ei
= 25.61
2.13
Fcst 1 Fcst 2 223 289 430 134 190 550 210 320 390 112 150 490
Demand 256 340 375 110 225 525
Err 1 33 51 -55 -24 35 -25
Err 2 46 20 -15 -2 75 35
Er1^2
Er2^2
|Err1| 33 51 55 24 35 25 37.16666 (MAD1)
1089 2116 2601 400 3025 225 576 4 1225 5625 625 1225 1523.5 1599.166 (MSE1 (MSE2)
Err2
46 20 15 2 75 35
e1/D*100
12.89062 19.92187 21.48437 9.375 13.67187 9.765625
e2/D∗100
17.96875 7.8125 5.859375 0.78125 29.29687 13.67187
14
Solutions For Chapter 2
32.16666 (MAD2)
14.51822 (MAPE1)
12.56510 (MAPE2)
2.15
Using the MAD: 1.25 MAD = (1.25)(21.6) = 27.0 (Using s, the sample standard deviation, one obtains 28.23)
2.17, 2.18, and 2.19.
One-step-ahead Month July August September October November December Forecast 205.50 225.25 241.50 250.25 249.00 240.25 Two-step-ahead Forecast 149.75 205.50 225.25 241.50 250.25 249.00 Demand 223 286 212 275 188 312 MAD = e1 -17.50 -60.75 29.50 -24.75 61.00 -71.75 44.2 e2 -73.25 -80.50 13.25 -33.50 62.25 -63.00 54.3
The one step ahead forecasts gave better results (and should have according to the theory).
2.21
An MA(1) forecast means that the forecast for next period is simply the current period's demand. Month Demand
Month July August September October November December
MA(4)
Demand 223 286 212 275 188 312
MA(1)
MA(4) 205.50 225.25 241.50 250.25 249.00 240.25 MAD =
Error
MA(1) 280 223 286 212 275 188 78.0 Error 57 -63 74 -63 87 -124
(Much worse than MA(4))
15
Production and Operations Analysis, Fourth Edition
2 2 ∝ α = = .286 N+1 7
2.25
a)
α = N=
b)
2 −α 2 −.05 = 39 ∝N= .05 α
Solutions To Problems From Chapter 2 2.12 a) and b)
Forecast (86 + 75)/2 (75 + 72)/2 etc = = 80.5 73.5 77.5 107.5 98.5 87.5 100.0 78.5 79.5 95.0 = =
Period 3 4 5 6 7 8 9 10 11 12 21.6 717.5
Actual 72 83 132 65 110 90 67 92 98 73
et +8.5 -9.5 -54.5 42.5 -11.5 -2.5 +33.0 -13.5 -18.5 +22.0
c)
MAD MSE
= =
(216)/10 (7175)/10
MAPE
=
100
1 n
∑D i ei
= 25.61
2.13
Fcst 1 Fcst 2 223 289 430 134 190 550 210 320 390 112 150 490
Demand 256 340 375 110 225 525
Err 1 33 51 -55 -24 35 -25
Err 2 46 20 -15 -2 75 35
Er1^2
Er2^2
|Err1| 33 51 55 24 35 25 37.16666 (MAD1)
1089 2116 2601 400 3025 225 576 4 1225 5625 625 1225 1523.5 1599.166 (MSE1 (MSE2)
Err2
46 20 15 2 75 35
e1/D*100
12.89062 19.92187 21.48437 9.375 13.67187 9.765625
e2/D∗100
17.96875 7.8125 5.859375 0.78125 29.29687 13.67187
14
Solutions For Chapter 2
32.16666 (MAD2)
14.51822 (MAPE1)
12.56510 (MAPE2)
2.15
Using the MAD: 1.25 MAD = (1.25)(21.6) = 27.0 (Using s, the sample standard deviation, one obtains 28.23)
2.17, 2.18, and 2.19.
One-step-ahead Month July August September October November December Forecast 205.50 225.25 241.50 250.25 249.00 240.25 Two-step-ahead Forecast 149.75 205.50 225.25 241.50 250.25 249.00 Demand 223 286 212 275 188 312 MAD = e1 -17.50 -60.75 29.50 -24.75 61.00 -71.75 44.2 e2 -73.25 -80.50 13.25 -33.50 62.25 -63.00 54.3
The one step ahead forecasts gave better results (and should have according to the theory).
2.21
An MA(1) forecast means that the forecast for next period is simply the current period's demand. Month Demand
Month July August September October November December
MA(4)
Demand 223 286 212 275 188 312
MA(1)
MA(4) 205.50 225.25 241.50 250.25 249.00 240.25 MAD =
Error
MA(1) 280 223 286 212 275 188 78.0 Error 57 -63 74 -63 87 -124
(Much worse than MA(4))
15
Production and Operations Analysis, Fourth Edition
2 2 ∝ α = = .286 N+1 7
2.25
a)
α = N=
b)
2 −α 2 −.05 = 39 ∝N= .05 α