Exercise 1.
(a) Find all optimal sequences for the scheduling problem 1 || wjCj
with the following jobs.
jobs
wj
pj
1
0
3
2
18
6
3
12
6
4
8
5
5
8
4
6
17
8
7
16
9
(b) Determine the effect of a change in p2 from 6 to 7 on the optimal
sequence.
1
Exercise 2.
Find the optimal sequence for the scheduling problem 1 | | hmax
with the following jobs.
jobs
pj
hj(Cj)
1
4
3Cj
2
8
77
3
12
C3 2
4
7
1.5C4
5
6
70+
C5
6
9
16C6
7
9
1.4C7
Calculate hmax for the obtained optimal schedule.
2
Exercise 3.
Solve by branch-and-bound the scheduling problem 1 | rj | Lmax
with the following jobs:
jobs
pj
rj
dj
1
6
0
8
2
18
0
42
3
12
0
44
4
10
14
24
5
10
25
90
6
17
25
85
7
16
50
68
3
Exercise 4.
Consider six jobs having processing times and due-dates given
in the table:
jobs
pj
dj
1
10
15
2
3
6
3
4
9
4
8
23
5
10
20
6
6
30
Find the optimal sequence for the scheduling problem 1 || Uj .
4
Exercise 5.
Find the optimal sequence for the scheduling problem 1 | Sjk | Cmax
where the sequence-dependent setup times have the form sjk = | ak - bj | .
The parameters are given in the table:
jobs
bj
aj
0
39
19
1
20
44
2
2
8
3
30
34
4
17
16
5
6
7
6
27
23
5
Exercise 1. Answer.
(a)
jobs 1
2
wj
0
18
pj
3
6
wj /pj 0
3
jobs 2
6
wj
18
17
Cj
6
14
wjCj = 1610
3
12
6
2
4
8
5
1.6
5
8
4
2
6
17
8
2.125
7
16
9
1.78
3
12
20
5
8
24
7
16
33
4
8
38
1
0
41
5
8
18
3
12
24
7
16
33
4
8
38
1
0
41
or
jobs 2
6
wj
18
17
Cj
6
14
wjCj = 1610
6
(b)
jobs
wj
pj
wj /pj
1
0
3
0
2
18
7
2.57
3
12
6
2
jobs
2
6
... optimal sequence is the same
...
...
...
...
7
Exercise 2. Answer.
J=
Cmax = 55
h1(55) = 165
h2(55) = 77
h3(55) = 3025
h4(55) = 82.5
h5(55) = 77.42
h6(55) = 88
h7(55) = 77
J = {2}
Cmax = 47
h1(47) = 141
h3(47) = 2209
h4(47) = 70.5
h5(47) = 76.86
h6(47) =75.2
h7(47) = 65.8
J = {2, 7}
Cmax = 38
h1(38) = 114
h3(38) = 1444
h4(38) = 57
h5(38) = 76.16
h6(38) = 60.8
J ={2, 7, 4, 6}
Cmax = 22
h1(22) = 66
h3(22) = 484
h5(22) = 74.69
J ={2, 7, 4, 6, 1}
Cmax = 18
h3(18) = 324
h5(18) = 74.24
J ={2, 7, 4, 6, 1, 5}
Cmax = 12
h3(12) = 144 = hmax
Optimal schedule: 3, 5, 1, 6, 4, 7, 2
J ={2, 7, 4}
Cmax = 31
h1(31) = 93
h3(31) = 3025
h5(31) = 75.57
h6(31) = 49.6
8
or
J=
Cmax = 55
h1(55) = 165
h2(55) = 77
h3(55) = 3025
h4(55) = 82.5
h5(55) = 77.42
h6(55) = 88
h7(55) = 77
J = {7}
Cmax = 46
h1(46) = 138
h2(46) = 77
h3(46) = 2116
h4(46) = 69
h5(46) =76.78
h6(46) = 73.6
J = {7, 4}
Cmax = 39
h1(39) = 117
h2(39) = 77
h3(39) = 1521
h5(39) = 76.24
h6(39) = 62.4
J ={7, 4, 6, 5}
Cmax = 24
h1(24) = 72
h2(24) = 77
h3(24) = 576
J ={7, 4, 6, 5, 1}
Cmax = 20
h2(20) = 77
h3(20) = 400
J ={7, 4, 6, 5, 1, 2}
Cmax = 12
h3(12) = 144 = hmax
J ={7, 4, 6}
Cmax = 30
h1(30) = 90
h2(30) = 77
h3(30) = 900
h5(30) = 75.48
9
Optimal schedule: 3, 2, 1, 5, 6, 4, 7
Exercise 3. Answer
jobs ordered by EDD: 1, 4, 2, 3, 7, 6, 5
*, *, *, *, *, *, *
1, *, *, *, *, *, *
1 [0, 6] L1 = -2
2 [6, 14] 10 left
4 [14, 24] L4 = 0
2 [24, 34] L2 = -8
3 [34, 46] L3 = 2
6 [46, 50] 13 left
7 [50, 66] L7 = -2
6 [66, 79] L6 = -6
5 [79, 89] L5 = -1
2, *, *, *, *, *, *
2 [0, 18] L2 = -24
1 [18, 24] L1 = 16 >2
3, *, *, *, *, *, *
2 [0, 12] L3 = -32
1 [12, 18] L1 = 10 >2
Lower bound = 2
10
1, *, *, *, *, *, *
1, 2, *, *, *, *, *
1 [0, 6] L1 = -2
2 [6, 24] L2 = -18
4 [24, 34] L4 = 10
3 [34, 46] L3 = 2
6 [46, 50] 13 left
7 [50, 66] L7 = -2
6 [66, 79] L6 = -6
5 [79, 89] L5 = -1
1, 3, *, *, *, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
4 [18, 28] L4 = 4
2 [28, 46] L2 = 4
6 [46, 50] 13 left
7 [50, 66] L7 = -2
6 [66, 79] L6 = -6
5 [79, 89] L5 = -1
Lower bound = 10
Lower bound = 4
1, 5, *, *, *, *, *
1 [0, 6] L1 = -2
5 [25, 35] L5 = -55
4 [35, 45] L4 = 21 > 4
...
1, 6, *, *, *, *, *
1 [0, 6] L1 = -2
6 [25, 42] L6 = -43
4 [42, 52] L4 = 28 > 4
...
1, 4, *, *, *, *, *
1 [0, 6] L1 = -2
4 [14, 24] L4 = 0
2 [24, 42] L2 = 0
3 [42, 54] L3 = 10>4
...
1, 7, *, *, *, *, *
1 [0, 6] L1 = -2
7 [50, 66] L7 = -2
4 [66, 76] L4 = 53 > 4
...
11
1, 3, *, *, *, *, *
1, 3, 2, *, *, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
2 [18, 36] L2 = -6
4 [36, 46] L4 = 22
6 [46, 50] 13 left
the same as before:
7 [50, 66] L7 = -2
6 [66, 79] L6 = -6
5 [79, 89] L5 = -1
1, 3, 6, *, *, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
6 [25, 42] L6 = -43
4 [42, 52] L4 = 28 >4
1, 3, 4, *, *, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
4 [18, 28] L4 = 4
2 [28, 46] L2 = 4
6 [46, 50] 13 left
the same as before:
7 [50, 66] L7 = -2
6 [66, 79] L6 = -6
5 [79, 89] L5 = -1
Lower bound = 4
1, 3, 5, *, *, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
5 [25, 35] L5 = -55
4 [35, 45] L4 = 21 >4
1, 3, 7, *, *, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
7 [50, 66] L5 = -2
12 >4
4 [66, 76] L4 = 53
1, 3, 4, *, *, *, *
1, 3, 4, 2, *, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
4 [18, 28] L4 = 4
2 [28, 46] L2 = 4
6 [46, 50] 13 left
the same as before:
7 [50, 66] L7 = -2
6 [66, 79] L6 = -6
5 [79, 89] L5 = -1
1, 3, 4, 5, *, *, *
1, 3, 4, 6, *, *, *
1, 3, 4, 7, *, *, *
Lower bound = 4
13
1, 3, 4, 2, *, *, *
1, 3, 4, 2, 5, *, *
1 [0, 6] L1 = -2
3 [6, 18] L3 = -26
4 [18, 28] L4 = 4
2 [28, 46] L2 = 4
5 [46, 56] L5 = -34
7 [56, 72] L7 = 4
6 [72, 89] L6 = 4
Lower bound = 4
1, 3, 4, 2, 6, *, *
1, 3, 4, 2, 7, *, *
non-preemptive schedule is obtained!
Schedule: 1, 3, 4, 2, 5, 7, 6
Lmax = 4
14
*, *, *, *, *, *, *
LB=2
1, 2
1
1, 3
2
1, 4
1, 5
3
1, 6
1, 7
LB=4
1, 3, 2
1, 3, 4
1, 3, 5
1, 3, 6
1, 3, 7
LB=4
1, 3, 4, 2
LB=4
1, 3, 4, 2, 5
LB=4
1, 3, 4, 2, 5, 7
LB=4
1, 3, 4, 2, 5, 7, 6
15
Exercise 4. Answer
J =  , Jd =  , JC = {1,...,6}
j*=2
J = {2} , Jd =  , JC = {1, 3, 4, 5, 6}, t=3 < 6 = d2
j*=3
J = {2, 3} , Jd =  , JC = {1, 4, 5, 6}, t=7 < 9 = d3
j*=1 J = {2, 3, 1} , Jd =  , JC = {4, 5, 6}, t=17 > 15 = d1
k*=1 J = {2, 3} , Jd = {1}, t=7
j*=5
J = {2, 3, 5} , Jd = {1}, JC = {4, 6}, t=17 < 20 = d5
j*=4 J = {2, 3, 5, 4} , Jd = {1}, JC = {6}, t=25 > 23 = d4
k*=5 J = {2, 3, 4} , Jd = {1, 5}, t=15
j*=6
J = {2, 3, 4, 6} , Jd = {1, 5}, JC = , t=25 < 30 = d6
optimal schedule 2, 3, 4, 6, 1, 5
Uj = 2
16
Exercise 5. Answer
a3=44
b6=39
a5=34
6
b4=27
a4=23
a6=19
a2=16
a0=8
b1=6
b0=2
0
2
b5=30
b3=20
b2=17
1
3
5
4
a1=7
17
a3=44
b6=39
a5=34
6
b4=27
a4=23
a6=19
a2=16
a0=8
b1=6
b0=2
0
2
b5=30
b3=20
b2=17
1
3
5
4
a1=7
18
C* I(0, 1) = 2 max ( (6-7), 0 ) = 0
C* I(1, 2) = 2 max ( (16-8), 0 ) = 16
C* I(2, 3) = 2 max ( (19-17), 0 ) = 4
C* I(3, 4) = 2 max ( (23-20), 0 ) = 6
C* I(4, 5) = 2 max ( (30-27), 0 ) = 6
C* I(5, 6) = 2 max ( (39-34), 0 ) = 10
arcs
R1, 2
R2, 3
R3, 4
R4, 5
bj
b1=6
b2=17
b3=20
b4=27
a*(j)
a0=8
a1=16
a3=19
a4=23
group
1
2
2
2
** = * I(1,2) I(2,3) I(3,4) I(5,6)
cost
0126345
5 + 10 + 2 + 5 + 3 + 7 + 22 = 54
19