Nahmias Chapter Selected Question Solutions

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SOLUTIONS TO SELECTED PROBLEMS FROM NAHMIAS’ BOOK
CHAPTER 2
FORECASTING

2.13 Fcst 1 Fcst 2 Demand Err 1 Err 2 Er1^2 Er2^2 |Err1|

223 210 256 33 46 1089 2116 33
289 320 340 51 20 2601 400 51
430 390 375 -55 -15 3025 225 55
134 112 110 -24 -2 576 4 24
190 150 225 35 75 1225 5625 35
550 490 525 -25 35 625 1225 25
1523.5 1599.166 37.16666
(MSE1 (MSE2) (MAD1)

(Err2( (e1/D(*100 (e2/D((100

46 12.89062 17.96875
20 15.00005.88253
15 14.66667 4.00000
2 21.81818 1.81818
75 15.55556 33.33333
35 4.761905 6.66667

32.16666 14.11549 11.61155
(MAD2) (MAPE1) (MAPE2)

2.14It means that E(ei)  (  0. This will show up by considering

[pic]

A bias is indicated when this sum deviates too far from zero.

2.16 MA (3) forecast:    258.33
MA (6) forecast:    249.33
MA (12) forecast:    205.33

2.17, 2.18, and 2.19.

One-step-ahead Two-step-ahead
Month Forecast Forecast Demand    e1   e2  

July 205.50 149.75 223 -17.50 -73.25
August 225.25 205.50 286 -60.75 -80.50
September 241.50 225.25 212 29.50 13.25
October 250.25 241.50 275 -24.75 -33.50
November 249.00 250.25 188 61.00 62.25
December 240.25 249.00 312 -71.75 -63.00

MAD =  44.2 54.3

The one step ahead forecasts gave better results (and should have according to the theory).

2.20Month Demand MA(3) MA(6)

July 223 226.00 161.33
August 286 226.67 183.67
September 212 263.00 221.83
October 275 240.33 233.17
November 188 257.67 242.17
December 312 225.00 244.00

MA (6) Forecasts exhibit less variation from period to period.

2.21An 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 223 205.50 280 57
August 286 225.25 223 -63
September 212 241.50 286 74
October 275 250.25 212 -63
November 188 249.00 275 87
December 312 240.25 188 -124

MAD = 78.0
(Much worse than MA(4))

2.35a)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 = [pic]-2N+1 = ( t ( 0

c-7 = [pic] = 0.36

c-6 = [pic] = 0.71

c-5 = [pic] = 1.56

c-4 = [pic] = 1.35

c-3 = [pic] = 0.30

c-2 = [pic] = 0.95

c-1 = [pic] = 1.76

c0 = [pic] = 0.97

(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:

c-3 = .33 c-1 = 1.67

c-2 = .83 c-0 = 1.17

b)( = 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 =...
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