2. Relaxed solution: x1 = 5.71, Z = 17.13
b. Relaxed solution: x1 = 10.5, x2 = 23.7, Z = 1,473
x1 = 5, x2 = 1.67, Z = 25 Relaxed solution: x1 = 10.8, x2 = 23, s1 = 2.6
x1 = 6, s1 = 2, Z = 18
Relaxed solution: x1 = 10, x2 = 24.5, s2 = 4 1 17.13
x1 ≤ 5
2 25
x1 ≥ 6
1 1,473
x 1 ≤ 23
3 18
2 1,460
x 1 ≥ 24
3 1,460
3. a.
The rounded down solution would not be optimal. 4. a.
b.
Relaxed solution: x1 = 6, x2 = 3.2, s2 = 4.8,
Z = 2,720
1 2,720
x1 ≤ 3
x1 ≥ 4
2 2,700
3 2,400
Relaxed solution: x1 = 6, x2 = 3, s1 = 2, s2 = 6,
Z = 2,700
Relaxed solution: x1 = 5, x2 = 4, s2 = 2, s3 = 1,
Z = 2,400 5. a.
b. Node 1: x2 25 20 15 10
x1 = 6.25 x2 = 0
5
Relaxed solution: x1 = 6, x2 = 1, s1 = 8, Z = 310
Infeasible
0
5
10 15 20
25
30
x1
UB = 312.5 (x1 = 6.25, x2 = 0) LB = 300 (x1 = 6, x2 = 0) 1 312.5
x1 ≤ 6
x1 ≥ 7
Node 2: x2 25
25
20
20
15
Optimal integer solution: x1 = 6 x2 = 1
10 2 310
UB = 310 (x1 = 6, x2 = 1) LB = 310 (x1 = 6, x2 = 1)
3 ⬁
5
Infeasible Optimal solution at node 2: x1 = 6 x2 = 1 Z = 310
6. a.
Relaxed solution: x2 = 10.71, s2 = 1.28, s2 = 5,
Z = 5,785.7
Relaxed solution: x1 = 0.63, x2 = 10, s1 = 1.39, s2 = 4.37,
Z = 5,778
Node 3: x2
0
5
10 15 20
25
x1
15 Infeasible 10 5 0
5
10 15 20
25
x1
1 5,785.7
x 2 ≤ 10
2 5,778
x 2 ≥ 11
3 ⬁
Relaxed solution: x1 = 0, x2 = 10, x3 = 1, s1 = 1,
s2 = 5, Z = 5,775
Relaxed solution: x1 = 1, x2 = 9.57, s1 = 1.43, s2 = 4,
s4 = .43, Z = 5,767.8
1 5,785.7
x 2 ≥ 11
x 2 ≤ 10
2 5,778
x1 = 0
4 5,775
7. a.
3 ⬁
x1 ≥ 1
15 5,767.8
Infeasible
Relaxed solution: x 1 = 3.33, x 2 = 5, s 1 = 3.33,
Z = 366.5
Relaxed solution: x 1 = 2.5, x 2 = 6, s 2 = 0.5,
Z = 365
1 372.9
x2 ≤ 5
2 366.5
x2 ≥ 6
3 365.0
Relaxed solution: x 1 = 3, x 2 = 5, s 1 = 4, s 2 = 1,
Z = 350
Relaxed solution: x 1 = 4, x 2 = 4, s 1 = 7, s 3 = 1,
Z = 360
1 372.9
x2 ≥ 6
x2 ≤ 5
2 366.5
x1 = 3
4 350.0
3 365.0
x1 ≥ 4
15 360.0
Relaxed solution: x1 = 2, x2 = 6.25, s2 = 0.5, s3 = 0.25,
Z = 350
1 372.9
x2 ≥ 6
x2 ≤ 5
2 366.5
x1 = 3
4 350.0
3 365.0
x1 ≥ 4
5 360.0
x1 = 2
6 350.0
The rounded down solution would not be optimal.
x1 ≥ 3
7 ⬁
Infeasible