Nahmias Chapter 2

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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
2 e

c)

From Appendix 2-A σ Hence

2 = 1.1 2 −α

σ2 2 = =1.1σ2 2 −α
Solving gives α = .1818

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 α = .15 β = .10

S1 = αD1 + (1-α)(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.

2.34

a)
(1) Quarter 1 2 Demand 12 25 MA Centered MA (2) Centered MA on periods 42.440 42.440 Ratio (1)/(2) 0.2828 0.5891

41.25

16

Solutions For Chapter 2
3 4 5 6 7 8 9 10 11 12 76 52 16 32 71 62 14 45 84 47 41.25 42.25 44.00 42.75 45.25 44.75 48.00 51.25 47.50 42.25 44.00 42.75 45.25 44.75 48.00 51.25 47.50 41.750 43.125 43.375 44.000 45.000 46.375 49.625 49.375 49.500 49.500 1.8204 1.2058 0.3689 0.7272 1.5778 1.3369 0.2821 0.9114 1.6970 0.9494

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 we finally obtain the factors:
0.318 0.758 1.735 1.189

b)
Quarter 1 2 3 4 5 6 7 8 9 Demand 12 25 76 52 16 32 71 62 14 Factor 0.318 0.758 1.735 1.189 0.318 0.758 1.735 1.189 0.318 Deseasonalized Series 37.74 32.98 43.80 43.73 50.31 42.22 40.92 52.14 44.03

17

Production and Operations Analysis, Fourth Edition
10 11 12 45 84 47 0.758 1.735 1.189 59.37 48.41 39.53

c) d)

47.40

Must "re-seasonalize" the forecast from part (c) (47.40)(.318) = 15.07 V1 =...
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