Muhammad Ade Endrawan

1.
max= 110*X11 + 110*X21 + 118*X12 +118*X22 - 100*A1 - 100*A2;
X11/5 + X12/5 <= 8 + L1;
X21/8 + X22/4 <= 8 + L2;
A1 <= 4;
A2 <= 4;

Global optimal solution found.

Objective value: 16840.00
Infeasibilities: 0.000000
Total solver iterations: 0
Elapsed runtime seconds: 0.03

Model Class: LP

Total variables: 6
Nonlinear variables: 0
Integer variables: 0

Total constraints: 5
Nonlinear constraints: 0

Total nonzeros: 14
Nonlinear nonzeros: 0

Variable Value Reduced Cost

X11 0.000000 8.000000
X21 96.00000 0.000000
X12 60.00000 0.000000
X22 0.000000 102.0000
A1 4.000000 0.000000
A2 4.000000 0.000000

Row Slack or Surplus Dual Price

1 16840.00 1.000000
2 0.000000 590.0000
3 0.000000 880.0000
4 0.000000 490.0000
5 0.000000 780.0000
2.
min = 4*X11+6*X12+8*X13+12*X14+50*(X11+X12+X13+X14)+
9*X21+6*X22+7*X23+11*X24+55*(X21+X22+X23+X24)+
8*X31+12*X32+3*X33+5*X34+62*(X31+X32+X33+X34);

X11 + X21 + X31 >= 80;

X12 + X22 + X32 >= 70;
X13 + X23 + X33 >= 60;
X14 + X24 + X34 >= 40;

X11 + X12 + X13 + X14 <= 120;

X21 + X22 + X23 + X24 <= 100;
X31 + X32 + X33 + X34 <= 140;

(0.08*(X11+X12+X13+X14)+0.06*(X21+X22+X23+X24)+ 0.04*(X31+X32+X33+X34)) /
(X11+X12+X13+X14+X21+X22+X23+X24+X31+X32+X33+X34)<= 0.05;
(0.05*(X11+X12+X13+X14)+0.04*(X21+X22+X23+X24)+0.03*(X31+X32+X33+X34)) /
(X11+X12+X13+X14+X21+X22+X23+X24+X31+X32+X33+X34)<= 0.04;

Local optimal solution found.

Objective value: 16060.00
Infeasibilities: 0.000000
Total solver iterations: 9
Elapsed runtime seconds: 0.03

Model Class: NLP

Total variables: 12
Nonlinear variables: 12
Integer variables: 0

Total constraints: 10
Nonlinear constraints: 2

Total nonzeros: 60
Nonlinear nonzeros: 24

Variable Value Reduced Cost

X11 15.00000 0.000000
X12 0.000000 5.000000
X13 0.000000 9.000000
X14 0.000000 11.00000
X21 25.00000 0.000000
X22 70.00000 0.000000
X23 0.000000 3.000000
X24 0.000000 5.000000
X31 40.00000 0.000000
X32 0.000000 7.000000
X33 60.00000 0.000000
X34 40.00000 0.000000

Row Slack or Surplus Dual Price

1 16060.00 -1.000000
 2 0.000000 -69.00000
3 0.000000 -66.00000
4 0.000000 -64.00000
5 0.000000 -66.00000
6 105.0000 0.000000
7 5.000000 0.000000
8 0.000000 4.000000
9 0.000000 125000.0
10 0.5000000E-02 0.000000

3.

max = 12*(ba+bc)+16*ca+26*dc-6*A-3*A C1;

ba + ca + C1 = A;
ba = 0.6*A;
ca + C1 = 0.4*a;
dc = 0.6*C1;
bc = 0.4*C1;

ba + bc <= 30;
ca <= 60;
dc <= 40;

3*A + C1 <= 200;

Global optimal solution found.

Objective value: 1318.889
Infeasibilities: 0.000000
Total solver iterations: 1
Elapsed runtime seconds: 0.05

Model Class: LP

Total variables: 7
Nonlinear variables: 0
Integer variables: 0

Total constraints: 10
Nonlinear constraints: 0

Total nonzeros: 24
Nonlinear nonzeros: 0

Variable Value Reduced Cost

BA 3.333333 0.000000
BC 26.66667 0.000000
CA 2.222222 0.000000
DC 40.00000 0.000000
A 5.555556 0.000000
C1 66.66667 0.000000
A 172.2222 0.000000
 Row Slack or Surplus Dual Price
1 1318.889 1.000000
2 0.000000 16.00000
3 0.000000 -11.66667
4 0.000000 0.000000
5 0.000000 -1.222222
6 0.000000 4.333333
7 0.000000 7.666667
8 57.77778 0.000000
9 0.000000 27.22222
10 116.6667 0.000000

4.
min = 30000*(A+B+C+D)+30*(E+F+G);

A + B + D >= 4000/500;
A + B + C >= 2000/500;
A + C + D >= 3000/500;
B + C + D >= 10000/500;

500*A + 500*B + 500*C + 500*D + E >=3400;

500*A + 500*B + 500*C + 500*D + E + F >= 2000;
500*A + 500*B + 500*C + 500*D + F + G >= 3000;
500*A + 500*B + 500*C + 500*D + G >= 10000;

A >= 0;
B >= 0;
C >= 0;
D >= 0;

Global optimal solution found.

Objective value: 600000.0
Infeasibilities: 0.000000
Total solver iterations: 3
Elapsed runtime seconds: 0.05

Model Class: LP

Total variables: 7
Nonlinear variables: 0
Integer variables: 0

Total constraints: 13
Nonlinear constraints: 0

Total nonzeros: 45
Nonlinear nonzeros: 0

Variable Value Reduced Cost

A 0.000000 15000.00
B 14.00000 0.000000
C 6.000000 0.000000
 D 0.000000 0.000000
E 0.000000 30.00000
F 0.000000 30.00000
G 0.000000 0.000000

Row Slack or Surplus Dual Price

1 600000.0 -1.000000
2 6.000000 0.000000
3 16.00000 0.000000
4 0.000000 0.000000
5 0.000000 -15000.00
6 6600.000 0.000000
7 8000.000 0.000000
8 7000.000 0.000000
9 0.000000 -30.00000
10 0.000000 0.000000
11 14.00000 0.000000
12 6.000000 0.000000
13 0.000000 0.000000

5.
min = 3*1000*X11 + 2*1100*X12 + 2*1200*X13 + 1500*X14+
4*800*X21 + 3*900*X22 + 3*1000*X23 + 2*1000*X24+
5*600*X31 + 5*800*X32 + 4*800*X33 + 2*900*X34 +
40*S1 + 50*S2 + 4*S3 + 70*S4;

X11+X12+X13+X14 <= 5;
X21+X22+X23+X24 <= 8;
X31+X32+X33+X34 <= 10;

3*50*X11 + 4*30*X21 + 5*20*X31 + S1 <= 1000;

2*50*X12 + 3*30*X22 + 5*20*X32 + S2 <= 2000;
2*50*X13 + 3*30*X23 + 4*20*X33 + S3 <= 900;
50*X14 + 2*30*X24 + 2*20*X34 + S4 <= 1200;

Global optimal solution found.

Objective value: 0.000000
Infeasibilities: 0.000000
Total solver iterations: 0
Elapsed runtime seconds: 0.16

Model Class: LP

Total variables: 16
Nonlinear variables: 0
Integer variables: 0

Total constraints: 8
Nonlinear constraints: 0

Total nonzeros: 44
Nonlinear nonzeros: 0
 Variable Value Reduced Cost
X11 0.000000 3000.000
X12 0.000000 2200.000
X13 0.000000 2400.000
X14 0.000000 1500.000
X21 0.000000 3200.000
X22 0.000000 2700.000
X23 0.000000 3000.000
X24 0.000000 2000.000
X31 0.000000 3000.000
X32 0.000000 4000.000
X33 0.000000 3200.000
X34 0.000000 1800.000
S1 0.000000 40.00000
S2 0.000000 50.00000
S3 0.000000 4.000000
S4 0.000000 70.00000

Row Slack or Surplus Dual Price

1 0.000000 -1.000000
2 5.000000 0.000000
3 8.000000 0.000000
4 10.00000 0.000000
5 1000.000 0.000000
6 2000.000 0.000000
7 900.0000 0.000000
8 1200.000 0.000000

6.
min = 30*C1 + 35*C2 + 33*C3;

C1 <= 30;
C2 <= 30;
C3 <= 30;

C1 + C2 + C3 >= 50;

(2500*C1+1500*C2+1600*C3)<= 2000*(C1+C2+C3);

Local optimal solution found.

Objective value: 1583.333
Infeasibilities: 0.000000
Extended solver steps: 5
Best multistart solution found at step: 1
Total solver iterations: 25
Elapsed runtime seconds: 0.28

Model Class: NLP

 Total variables: 3
Nonlinear variables: 3
Integer variables: 0

Total constraints: 6
Nonlinear constraints: 1

Total nonzeros: 12
Nonlinear nonzeros: 3

Variable Value Reduced Cost

C1 22.22222 0.000000
C2 0.000000 1.666667
C3 27.77778 0.000000

Row Slack or Surplus Dual Price

1 1583.333 -1.000000
2 7.777778 0.000000
3 30.00000 0.000000
4 2.222222 0.000000
5 0.000000 -31.66667
6 0.000000 0.1666667

7.
min = 140000*tb + 80000*ts1 + 80000*ts2;

140000*tb + 80000*ts1 <= 28000000;

120000*tb + 120000*ts1 >= 20000000 + (0.16*20000000);

80000*ts2 <= 20000000;

1200000*tb + 120000*ts2 = 20000000 + (0.16*20000000);

tb + ts1 + ts2 >= 200;

Global optimal solution found.

Objective value: 0.1712000E+08
Infeasibilities: 0.000000
Total solver iterations: 2
Elapsed runtime seconds: 0.03

Model Class: LP

Total variables: 3
Nonlinear variables: 0
Integer variables: 0

Total constraints: 6
Nonlinear constraints: 0

Total nonzeros: 13
 Nonlinear nonzeros: 0

Variable Value Reduced Cost

TB 18.66667 0.000000
TS1 174.6667 0.000000
TS2 6.666667 0.000000

Row Slack or Surplus Dual Price

1 0.1712000E+08 -1.000000
2 0.1141333E+08 0.000000
3 0.000000 -0.5000000E-01
4 0.1946667E+08 0.000000
5 0.000000 -0.5000000E-01
6 0.000000 -74000.00

9.
max = 150*b + 200*w + 230*p + 35*m;

0.3*b + 0.1*m = 4000;

b = 0.8*w;
w = 0.95*p;
m >= 25;
p >= 25;
b >= 25;

w >= 25;

Global optimal solution found.

Objective value: 9368421.
Infeasibilities: 0.000000
Total solver iterations: 0
Elapsed runtime seconds: 0.03

Model Class: LP

Total variables: 4
Nonlinear variables: 0
Integer variables: 0

Total constraints: 7
Nonlinear constraints: 0

Total nonzeros: 13
Nonlinear nonzeros: 0

Variable Value Reduced Cost

S1 13333.33 0.000000
S2 16666.67 0.000000
 S3 17543.86 0.000000
S4 0.000000 199.2105

Row Slack or Surplus Dual Price

1 9368421. 1.000000
2 0.000000 2342.105
3 0.000000 -552.6316
4 0.000000 -242.1053
5 13308.33 0.000000
6 16641.67 0.000000
7 17518.86 0.000000
Global optimal solution found.
Objective value: 9363441.
Infeasibilities: 0.000000
Total solver iterations: 0
Elapsed runtime seconds: 0.04

Model Class: LP

Total variables: 4
Nonlinear variables: 0
Integer variables: 0

Total constraints: 8
Nonlinear constraints: 0

Total nonzeros: 14
Nonlinear nonzeros: 0

Variable Value Reduced Cost

B 13325.00 0.000000
W 16656.25 0.000000
P 17532.89 0.000000
M 25.00000 0.000000

Row Slack or Surplus Dual Price

1 9363441. 1.000000
2 0.000000 2342.105
3 0.000000 -552.6316
4 0.000000 -242.1053
5 0.000000 -199.2105
6 17507.89 0.000000
7 13300.00 0.000000
8 16631.25 0.000000
10. (a)
max = 130*X1l + 150*X1s + 90*X1n +
275*X2l + 300*X2s + 100*X2n +
180*X3l + 225*X3s + 140*X3n +
200*X4l + 120*X4s + 160*X4n;

X1l + X2l + X3l + X4l <= 2;

X1s + X2s + X3s + X4s <= 2;
X1n + X2n + X3n + X4n <= 2;

Global optimal solution found.

Objective value: 1470.000
Infeasibilities: 0.000000
Total solver iterations: 0
Elapsed runtime seconds: 0.05

Model Class: LP

Total variables: 12
Nonlinear variables: 0
Integer variables: 0

Total constraints: 4
Nonlinear constraints: 0

Total nonzeros: 24
Nonlinear nonzeros: 0

Variable Value Reduced Cost

X1l 0.000000 145.0000
X1s 0.000000 150.0000
X1n 0.000000 70.00000
X2l 2.000000 0.000000
X2s 2.000000 0.000000
X2n 0.000000 60.00000
X3l 0.000000 95.00000
X3s 0.000000 75.00000
X3n 0.000000 20.00000
X4l 0.000000 75.00000
X4s 0.000000 180.0000
X4n 2.000000 0.000000

Row Slack or Surplus Dual Price

1 1470.000 1.000000
2 0.000000 275.0000
3 0.000000 300.0000
4 0.000000 160.0000
(b)

min = 130*X1l + 150*X1s + 90*X1n +

275*X2l + 300*X2s + 100*X2n +
180*X3l + 225*X3s + 140*X3n +
200*X4l + 120*X4s + 160*X4n;

X1l + X2l + X3l + X4l >= 1;

X1s + X2s + X3s + X4s >= 1;
X1n + X2n + X3n + X4n >= 1;

Global optimal solution found.

Objective value: 340.0000
Infeasibilities: 0.000000
Total solver iterations: 0
Elapsed runtime seconds: 0.03

Model Class: LP

Total variables: 12
Nonlinear variables: 0
Integer variables: 0

Total constraints: 4
Nonlinear constraints: 0

Total nonzeros: 24
Nonlinear nonzeros: 0

Variable Value Reduced Cost

X1l 1.000000 0.000000
X1s 0.000000 30.00000
X1n 1.000000 0.000000
X2l 0.000000 145.0000
X2s 0.000000 180.0000
X2n 0.000000 10.00000
X3l 0.000000 50.00000
X3s 0.000000 105.0000
X3n 0.000000 50.00000
X4l 0.000000 70.00000
X4s 1.000000 0.000000
X4n 0.000000 70.00000

Row Slack or Surplus Dual Price

1 340.0000 -1.000000
2 0.000000 -130.0000
3 0.000000 -120.0000
4 0.000000 -90.00000
11.

max = W1 + W2 ;

A1 + A2 = W1 + W2;
2*A1 + 8*A1 + 16*A2 + 28*A2 - 30*B2 = 0;
2*A1 = W1;
2*A2 = W2;
B1 <= 25;
B2 <= 25;
A1 <= 20;
A2 <= 20;

Model Class: LP

Total variables: 6
Nonlinear variables: 0
Integer variables: 0

Total constraints: 9
Nonlinear constraints: 0

Total nonzeros: 17
Nonlinear nonzeros: 0

Variable Value Reduced Cost

W1 20.58824 0.000000
W2 29.41176 0.000000
B1 25.00000 0.000000
B2 25.00000 0.000000
A1 10.29412 0.000000
A2 14.70588 0.000000

Row Slack or Surplus Dual Price

1 50.00000 1.000000
2 0.000000 -1.000000
3 0.000000 0.000000
4 0.000000 0.000000
5 0.000000 0.000000
6 0.000000 1.000000
7 0.000000 1.000000
8 9.705882 0.000000
9 5.294118 0.000000

12.