Answers to Problems at the End of Chapter 10 10-97
10.120b (M|M|1):(GD|6|6) λ = 0.5/hr, μ = 2.5/hr, c = 1, K = 6. Lq = 1.1511.
10.121 (M|D|1):(GD|∞|∞)
10.122 (D|M|1):(GD|∞|∞)
10.123 (M|G|50):(GD|50|∞)
c
c
!
As shown on a following page, P50 = 0.00022 and L = 29.99.
10.124 Operator 1: (M|M|1):(GD|∞|∞)
nλnμnPnPnPn
# Waiting
(m)
mPn
0 3.0 0.0
P0P00.19184726 0 0.00000000
1 2.5 2.5
1.2P01.2P00.23021671 0 0.00000000
2 2.0 2.5
1.0P11.2P00.23021671 1 0.23021671
3 1.5 2.5
0.8P20.96P00.18417337 2 0.36834674
4 1.0 2.5
0.6P30.576P00.11050402 3 0.33151206
5 0.5 2.5
0.4P40.2304P00.04420161 4 0.17680643
6 0.0 2.5
0.2P50.04608P00.00884032 5 0.04420161
ΣPn = 5.21248P0Lq = 1.15108355
P0 = 0.19184726
Answers to Problems at the End of Chapter 10 10-98
Choose the more expensive, but faster operator.
10.125 Errors were introduced in this problem during the publication process. Service time is
supposed to be exponentially distributed with a mean of 3 minutes.
122727.0
)375.01(2
)454545.0()75.0(375.0
)1(!
)(
2
2
2
0
c
Pc
L
c
q
VC = $27(0.122727) + $18(2) = $39.31
c = 3
)470588.0()75.0(25.0
470588.0
2
)75.0(
75.01
)25.01(6
)75.0(
3
1
23
0
P
10.126 (M|M|1):(GD|K|K)
Since the number of machines assigned to an individual operator is to be determined, we
will seek to minimize the expected total cost per machine. The objective function to be
minimized can be written as
Answers to Problems at the End of Chapter 10 10-99
Next, let’s consider K = 4. As shown below,
Next, let’s consider K = 5. As shown below,
10.127 (M|M|c):(GD|30|30)
n
P0 Multipliers P nnP n(n- 1)Pn
0 4.0000 0 1.000000000 0.7063953 0.000000 —
1 2.6667 12 0.333333333 0.2354651 0.235465 0
2 1.3333 12 0.074074074 0.0523256 0.104651 0.052326
3 0 12 0.008230453 0.0058140 0.017442 0.01163
Sum = 1.41563786 L = 0.35756
P0 = 0.706395349 Lq = 0.06395
n
n
n
P0 Multipliers P nnP n(n- 1)Pn
0 5.3333 0 1.000000000 0.6138086 0.000000 —
1 4.0000 12 0.444444444 0.2728038 0.272804 0
2 2.6667 12 0.148148148 0.0909346 0.181869 0.090935
3 1.3333 12 0.032921811 0.0202077 0.060623 0.04042
4 0 12 0.003657979 0.0022453 0.008981 0.00674
Sum = 1.62917238 L = 0.52428
P0 = 0.613808588 Lq = 0.13809
n
n
n
P0 Multipliers P nnP n(n- 1)Pn
0 6.6667 0 1.000000000 0.5204664 0.000000 —
1 5.3333 12 0.555555556 0.2891480 0.289148 0
2 4.0000 12 0.246913580 0.1285102 0.257020 0.12851
3 2.6667 12 0.082304527 0.0428367 0.128510 0.08567
4 1.3333 12 0.018289895 0.0095193 0.038077 0.02856
5 0 12 0.018289895 0.0095193 0.047596 0.03808
Sum = 1.92135345 L = 0.76035
P0 = 0.520466445 Lq = 0.28082
n
n
Answers to Problems at the End of Chapter 10 10-100
Answers to Problems at the End of Chapter 10 10-101
10.122 (Continued)
θkθk+1 |θk – θk+1|
0.55000000 0.54881164 0.00118836
0.54881164 0.54794274 0.00086890
0.54794274 0.54730830 0.00063444
0.54730830 0.54684552 0.00046278
0.54684552 0.54650820 0.00033732
0.54650820 0.54626245 0.00024574
0.54626245 0.54608349 0.00017896
0.54608349 0.54595321 0.00013029
0.54595321 0.54585837 0.00009483
0.54585837 0.54578936 0.00006902
0.54578936 0.54573914 0.00005022
0.54573914 0.54570259 0.00003654
0.54570259 0.54567601 0.00002659
Answers to Problems at the End of Chapter 10 10-102
10.123 (Continued)
k Individual Terms in Sum
Pk
0 1.00000 9.36041E-14
1 30.00000 2.80812E-12
2 450.00000 4.21219E-11
3 4500.00000 4.21219E-10
4 33750.00000 3.15914E-09
5 202500.00000 1.89548E-08
6 1012500.00000 9.47742E-08
7 4339285.71429 4.06175E-07
8 16272321.42857 1.52316E-06
9 54241071.42857 5.07719E-06
10 162723214.28571 1.52316E-05
11 443790584.41558 4.15406E-05
12 1109476461.03896 0.000103852
13 2560330294.70529 0.000239657
14 5486422060.08277 0.000513552
15 10972844120.16550 0.00102710
16 20574082725.31040 0.001925819
17 36307204809.37130 0.00339850
18 60512008015.61880 0.005664174
19 95545275814.13500 0.008943432
20 143317913721.20300 0.013415148
21 204739876744.57500 0.01916450
22 279190741015.33000 0.026133405
23 364161836106.95200 0.03408705
24 455202295133.69000 0.042608813
25 546242754160.42700 0.051130575
26 630280100954.33900 0.058996817
27 700311223282.59900 0.065552019
28 750333453517.07100 0.070234306
29 776207020879.72800 0.072656179
30 776207020879.72800 0.072656179
31 751168084722.31800 0.070312431
32 704220079427.17300 0.06591790
33 640200072206.52100 0.059925368
34 564882416652.81200 0.052875324
35 484184928559.55400 0.045321707
36 403487440466.29400 0.037768089
37 327151978756.45500 0.030622775
38 258277877965.62200 0.024175875
39 198675290742.78600 0.018596827
40 149006468057.09000 0.01394762
41 109029122968.60200 0.010205576
42 77877944977.57310 0.00728970
43 54333449984.35330 0.005085835
44 37045534080.24090 0.003467615
45 24697022720.16060 0.002311743
46 16106753947.93080 0.001507659
47 10280906775.27500 0.000962335
48 6425566734.54687 0.00060146
49 3934020449.72257 0.000368241
10.127 (Continued)
1.33333 12 c= 4
n
P0 Multipliers P nnP n(n-c )Pn
0 40.0000 0 1.000000000 0.0356268 0.000000 —
1 38.6667 12 3.333333333 0.1187561 0.118756 0
2 37.3333 24 5.370370370 0.1913293 0.382659 0
3 36.0000 36 5.569272977 0.1984156 0.595247 0
4 34.6667 48 4.176954733 0.1488117 0.595247 0
5 33.3333 48 3.016689529 0.1074751 0.537375 0.107475
6 32.0000 48 2.094923284 0.0746355 0.447813 0.149271
7 30.6667 48 1.396615523 0.0497570 0.348299 0.149271
8 29.3333 48 0.89228214 0.0317892 0.254313 0.12716
9 28.0000 48 0.54528353 0.0194267 0.174841 0.097134
10 26.6667 48 0.318082059 0.0113323 0.113323 0.067994
11 25.3333 48 0.176712255 0.0062957 0.069253 0.04407
12 24.0000 48 0.093264801 0.0033227 0.039873 0.026582
13 22.6667 48 0.046632401 0.0016614 0.021598 0.014952
14 21.3333 48 0.022020856 0.0007845 0.010983 0.007845
15 20.0000 48 0.009787047 0.0003487 0.005230 0.003835
16 18.6667 48 0.004077936 0.0001453 0.002325 0.001743
17 17.3333 48 0.001585864 0.0000565 0.000960 0.000734
18 16.0000 48 0.000572673 0.0000204 0.000367 0.000286
19 14.6667 48 0.000190891 0.0000068 0.000129 0.000102
20 13.3333 48 5.83278E-05 0.0000021 0.000042 3.32E-05
21 12.0000 48 1.62022E-05 0.0000006 0.000012 9.81E-06
22 10.6667 48 4.05054E-06 0.0000001 0.000003 2.6E-06
23 9.3333 48 9.00121E-07 0.0000000 0.000001 6.09E-07
24 8.0000 48 1.75023E-07 0.0000000 0.000000 1.25E-07
25 6.6667 48 2.91706E-08 0.0000000 0.000000 2.18E-08
26 5.3333 48 4.05147E-09 0.0000000 0.000000 3.18E-09
27 4.0000 48 4.50163E-10 0.0000000 0.000000 3.69E-10
28 2.6667 48 3.75136E-11 0.0000000 0.000000 3.21E-11
29 1.3333 48 2.08409E-12 0.0000000 0.000000 1.86E-12
30 0.0000 48 5.78914E-14 0.0000000 0.000000 5.36E-14
Sum = 28.06873189 L = 3.71865
P0 = 0.035626832 Lq = 0.79850
VC(4) = $538.09
VC(5) = $511.83
VC(6) = $517.55
n
n
Answers to Problems at the End of Chapter 10 10-104
10.128 (M|M|1):(GD|K|K)
We modify Problem 10.126 and seek to minimize the expected total cost per running
machine. The objective function to be minimized can be written as
VC(3) = [$90(0.06395) + $45(0.35756 – 0.06395) + $18]/(3 – 0.35756) = $13.99
10.129 (M|M|1):(GD|K|K)
We modify Problem 10.128 and let λ = 4/3/hr, μ = 12/hr, C1 =$75/hr, C2 = $75/hr, and C3
The optimum number of machines to assign an operator, 4, is the same as for Problem
10.126 and Problem 10.128.
Answers to Problems at the End of Chapter 10 10-105
10.130 (M|G|c):(GD|c|∞)
We modify Example 10.73 and let λ = 100/hr, μ = 4/hr, R = $6/hr, C5 =$1/hr, and
C6 = $20/hr. The optimum solution is 41 spaces.
10.131 (M|G|c):(GD|c|∞)
The total expected revenue is given by
c
PcTC(c )
35 0.011645823929 $60.04
36 0.008022496767 $53.25
37 0.005391381370 $48.59
38 0.003534424958 $45.60
39 0.002260535427 $43.86
40 0.001410841356 $43.03
41 0.000859529693 $42.85
42 0.000511363191 $43.10
43 0.000297215817 $43.64
44 0.000168844110 $44.36
45 0.000093793485 $45.20
Answers to Problems at the End of Chapter 10 10-106
10.132 (M|M|c):(GD|∞|∞)
λ = 4/hr, μ = 0.5/hr, C1 =$10,000/hr, and C3 = $2,500/hr. Based on Equation 10.187),
c
PcTC(c )
21 0.550400236816 –$0.36
22 0.529592991212 $13.37
30 0.367358992936 $119.38
40 0.184559445550 $233.90
50 0.054104472165 $301.31
51 0.045564047156 $303.99
52 0.037934646186 $305.86
53 0.031203634542 $306.92
54 0.025344007821 $307.19
55 0.020314759417 $306.72
56 0.016062155828 $305.54
57 0.012521863972 $303.73
58 0.009621761572 $301.34
59 0.007285168539 $298.44
60 0.005434184691 $295.11
70 0.000126855516 $249.89
80 0.000000722807 $200.00
90 0.000000001186 $150.00
100 0.000000000001 $100.00
110 0.000000000000 $50.00
120 0.000000000000 $0.00
Answers to Problems at the End of Chapter 10 10-107
10.133 (M|M|c):(GD|50|50) As shown, 7 unloading spaces are to be provided.
0.05 0.5 c= 6
n
P0 Multipliers P nnP n(n-c )Pn
0 2.50 0 1 0.0074129 0.000000 —
1 2.45 0.5 5 0.0370647 0.037065 0
2 2.40 1.0 12 0.0908085 0.181617 0
3 2.35 1.5 20 0.1452936 0.435881 0
4 2.30 2.0 23 0.1707199 0.682880 0
5 2.25 2.5 21 0.1570623 0.785312 0
6 2.20 3.0 16 0.1177968 0.706781 0
7 2.15 3.0 12 0.0863843 0.604690 0.086384
8 2.10 3.0 8 0.0619087 0.495270 0.12382
9 2.05 3.0 6 0.0433361 0.390025 0.130008
10 2.00 3.0 4 0.0296130 0.296130 0.118452
11 1.95 3.0 3 0.0197420 0.217162 0.09871
12 1.90 3.0 2 0.0128323 0.153988 0.076994
13 1.85 3.0 1 0.0081271 0.105653 0.05689
14 1.80 3.0 1 0.0050117 0.070164 0.040094
15 1.75 3.0 0 0.0030070 0.045106 0.027063
16 1.70 3.0 0 0.0017541 0.028066 0.017541
17 1.65 3.0 0 0.0009940 0.016898 0.010934
18 1.60 3.0 0 0.0005467 0.009841 0.00656
19 1.55 3.0 0 0.0002916 0.005540 0.00379
20 1.50 3.0 0 0.0001506 0.003013 0.002109
21 1.45 3.0 0 0.0000753 0.001582 0.00113
22 1.40 3.0 0 0.0000364 0.000801 0.000582
23 1.35 3.0 0 0.0000170 0.000391 0.000289
24 1.30 3.0 0 0.0000076 0.000183 0.000138
25 1.25 3.0 0 0.0000033 0.000083 6.29E-05
26 1.20 3.0 0 0.0000014 0.000036 2.76E-05
27 1.15 3.0 0 0.0000006 0.000015 1.16E-05
28 1.10 3.0 0 0.0000002 0.000006 4.66E-06
29 1.05 3.0 0 0.0000001 0.000002 1.78E-06
30 1.00 3.0 0 0.0000000 0.000001 6.52E-07
31 0.95 3.0 0 0.0000000 0.000000 2.26E-07
32 0.90 3.0 0 0.0000000 0.000000 7.45E-08
33 0.85 3.0 0 0.0000000 0.000000 2.32E-08
34 0.80 3.0 0 0.0000000 0.000000 6.82E-09
35 0.75 3.0 0 0.0000000 0.000000 1.88E-09
Sum = 134.89930182 L = 5.27418
P0 = 0.007412937 Lq = 0.80160
VC(6) = $23,015.96
VC(7) = $20,248.83
VC(8) = $20,929.21
n
n
Answers to Problems at the End of Chapter 10 10-108
10.134 (M(b)|M|1):(GD|∞|∞)
P = parts to be provided/hr = 125/hr
10.135 (M|M|c):(GD|N|∞)
λ = 15/mo, μ = 10/mo, C1 =$300, C3 = $15,000, C4 = $25,000, C5 = $5,000, and
C6 = $30,000. As shown below, c* = 2 and N* = 7.
2 3 4 5 6 7 8 9 10 11 12
2 0.188235294 0.162436548 0.147295742 0.137671417 0.131239987 0.126797406 0.123657963 0.121403543 0.119765945 0.118566448 0.117682476
3 0.210526316 0.198757764 0.193353474 0.190760060 0.189489267 0.188860199 0.188547229 0.188391132 0.188313181 0.188274230
4 0.227353464 0.223336968 0.221867129 0.221320915 0.221116777 0.221040322 0.221011666 0.221000921 0.220996892
5 0.224128874 0.223179258 0.222895941 0.222811086 0.222785642 0.222778010 0.222775720 0.222775033
6 0.223336968 0.223139866 0.223090644 0.223078342 0.223075267 0.223074498 0.223074306
7 0.223168002 0.223131828 0.223124078 0.223122417 0.223122061 0.223121985
8 0.223136349 0.223130415 0.223129303 0.223129094 0.223129055
9 0.223131074 0.223130195 0.223130049 0.223130024
10 0.223130283 0.223130165 0.223130147
11 0.223130967 0.223130952
Calculations for P0
c
Answers to Problems at the End of Chapter 10 10-109
10.136a (M|D|1):(GD|∞|∞). In the production process, the infinity symbols were changed to 4.
2
2 3 4 5 6 7 8 9 10 11 12
2 $145,529.41 $118,249.22 $104,304.91 $97,282.21 $94,241.92 $93,689.45 $94,766.74 $96,950.64 $99,905.62 $103,408.13 $107,304.87
3 $123,552.63 $101,523.14 $94,119.64 $93,166.77 $95,250.48 $98,807.19 $103,089.30 $107,731.33 $112,552.61 $117,463.32
4$110,746.00 $102,662.17 $102,875.03 $106,096.39 $110,431.78 $115,182.92 $120,089.66 $125,054.70 $130,041.59
5$114,708.46 $115,409.90 $119,127.64 $123,743.65 $128,628.53 $133,594.01 $138,583.65 $143,580.55
6$129,533.26 $133,386.54 $138,100.21 $143,028.66 $148,010.77 $153,006.30 $158,005.19
7$148,154.16 $152,896.77 $157,841.63 $162,829.81 $167,827.28 $172,826.74
8$167,842.39 $172,792.46 $177,783.10 $182,781.34 $187,781.01
9$187,781.22 $192,772.68 $197,771.25 $202,771.02
10 $207,770.61 $212,769.30 $217,769.11
11 $227,768.99 $232,768.81
12 $247,768.76
c
N
Answers to Problems at the End of Chapter 10 10-110
below.
10.136c (M|D|1):(GD|∞|∞). In the production process, the infinity symbols were changed to 4.
This is supposed to be an (M|D|1):(GD|∞|∞) queue.
Answers to Problems at the End of Chapter 10 10-111
SECTION 10.10
10.137 The primary challenge in developing such a simulation model is to keep track of the
position of each crane so that “blocking” (i.e., one crane cannot move because it is
10.138 Simulation has extensive applications in facility location and layout. A full treatment of
parts flow, and resource availability.
10.140 Simulation may play an important role in the design of AS/R systems and order picking
10.141 Simulation may be used to model cases where the load attempts (or the number of parts
10.142 Left as an exercise for the reader, who may wish to try the following reference as a
10.143 There are many papers on this topic. The following represents only a sample list
1994 Winter Simulation Conference, Buena Vista, FL, USA.
10.144 There are many papers on this topic. The following represents only a sample list
identified through a computer-based keyword search:
1995.
Chan, F. T.S., Jayaprakash, B., and Tang, N. K.H., “Design of automated cellular
10.145 Solution depends on example choice.
10.146 Solution depends on problem choice.