Both solutions (optimal solution and optimal solution with crashing) is shown below
1. LINDO used to find optimal solution without crashing the project:
Code:
Minimize X11 - X1
st
X2 - X1 >= 0.5
X3 - X2 >= 1
X4 - X3 >= 0.8
X5 - X3 >= 1.4
x6 - X4 >= 1
X7 - X5 >= 1.2
X8 - X3 >= 1.5
X8 - X6 >= 0
X8 - X7 >= 0
X9 - X8 >= 0.4
X10 - X9 >= 1.4
X11 - X10 >= 0.5
END
Reports window (explanatory text indicated with *):
LP OPTIMUM FOUND AT STEP 10
OBJECTIVE FUNCTION VALUE
1) 6.400000
VARIABLE VALUE REDUCED COST
X11 6.400000 0.000000
X1 0.000000 0.000000
X2 0.500000 0.000000
X3 1.500000 0.000000
X4 3.100000 0.000000
X5 2.900000 0.000000
X6 4.100000 0.000000
X7 4.100000 0.000000
X8 4.100000 0.000000
X9 4.500000 0.000000
X10 5.900000 0.000000
*Dual price indicates a critical path when its value is -. Increasing the duration of a constraint with a dual price of - will increase the duration of the whole project by the same amount.
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 -1.000000
3) 0.000000 -1.000000
4) 0.800000 0.000000
5) 0.000000 -1.000000
6) 0.000000 0.000000
7) 0.000000 -1.000000
8) 1.100000 0.000000
9) 0.000000 0.000000
10) 0.000000 -1.000000
11) 0.000000 -1.000000
12) 0.000000 -1.000000
13) 0.000000 -1.000000
NO. ITERATIONS= 10
RANGES IN WHICH THE BASIS IS UNCHANGED:
*In the objective coefficient ranges block we see the amount by which each variable`s objective function coefficient may alter before becoming suboptimal. The current coefficient gives the variable`s objective function coefficient. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X11 1.000000 INFINITY 0.000000
X1 -1.000000 INFINITY 0.000000
X2 0.000000 INFINITY 0.000000
X3 0.000000 INFINITY 0.000000
X4 0.000000 0.000000 0.000000
X5 0.000000 INFINITY 0.000000
X6 0.000000 0.000000 0.000000
X7 0.000000 INFINITY 0.000000
X8 0.000000 INFINITY 0.000000
X9 0.000000 INFINITY 0.000000
X10 0.000000 INFINITY 0.000000
*In the righthand side ranges block we see the amount the amount by which each variable`s righthand side may alter before becoming suboptimal. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 0.500000 INFINITY 0.500000
3 1.000000 INFINITY 1.500000
4 0.800000 0.800000 INFINITY
5 1.400000 INFINITY 0.800000
6 1.000000 0.800000 INFINITY
7 1.200000 INFINITY 0.800000
8 1.500000 1.100000 INFINITY
9 0.000000 0.800000 INFINITY
10 0.000000 INFINITY 0.800000
11 0.400000 INFINITY 4.500000
12 1.400000 INFINITY 5.900000
13 0.500000 INFINITY 6.400000
--------------------------------------------------------------------------------------------------------------------------------------------------------
2. LINDO used to determine optimal solution when the project is crashed:
Code:
Minimize 20A + 200B + 0C + 0D + 35E + 80F + 100G + 20H + 20I + 0J
Subject to
A <= 0.1
B <= 0.4
C <= 0
D <= 0
E <= 0.3
F <= 0.1
G <= 0.3
H <= 0.1
I <= 0.5
J <= 0
A >= 0
B >= 0
C >= 0
D >= 0
E >= 0
F >= 0
G >= 0
H >= 0
I >= 0
J >= 0
X2 - X1 + A >= 0.5
X3 - X2 + B >= 1
X4 - X3 + F >= 0.8
X5 - X3 + D >= 1.4
X6 - X4 + G >= 1
X7 - X5 + E >= 1.2
X8 - X3 + C >= 1.5
X8 - X6 >= 0
X8 - X7 >= 0
X9 - X8 + H >= 0.4
X10 - X9 + I >= 1.4
X11 - X10 + J >= 0.5
X11 - X1 <= 5
Reports window (explanatory text added with *):
LP OPTIMUM FOUND AT STEP 24
OBJECTIVE FUNCTION VALUE
1) 104.5000
VARIABLE VALUE REDUCED COST
A 0.100000 0.000000
B 0.400000 0.000000
C 0.000000 0.000000
D 0.000000 0.000000
E 0.300000 0.000000
F 0.000000 80.000000
G 0.000000 100.000000
H 0.100000 0.000000
I 0.500000 0.000000
J 0.000000 0.000000
X2 0.400000 0.000000
X1 0.000000 0.000000
X3 1.000000 0.000000
X4 2.300000 0.000000
X5 2.400000 0.000000
X6 3.300000 0.000000
X7 3.300000 0.000000
X8 3.300000 0.000000
X9 3.600000 0.000000
X10 4.500000 0.000000
X11 5.000000 0.000000
*Dual price indicates a critical path when its value is -. Increasing the duration of a constraint with a dual price of - will increase the duration of the whole project by the same amount.
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 180.000000
3) 0.000000 0.000000
4) 0.000000 0.000000
5) 0.000000 200.000000
6) 0.000000 165.000000
7) 0.100000 0.000000
8) 0.300000 0.000000
9) 0.000000 180.000000
10) 0.000000 180.000000
11) 0.000000 200.000000
12) 0.100000 0.000000
13) 0.400000 0.000000
14) 0.000000 0.000000
15) 0.000000 0.000000
16) 0.300000 0.000000
17) 0.000000 0.000000
18) 0.000000 0.000000
19) 0.100000 0.000000
20) 0.500000 0.000000
21) 0.000000 0.000000
22) 0.000000 -200.000000
23) 0.000000 -200.000000
24) 0.500000 0.000000
25) 0.000000 -200.000000
26) 0.000000 0.000000
27) 0.000000 -200.000000
28) 0.800000 0.000000
29) 0.000000 0.000000
30) 0.000000 -200.000000
31) 0.000000 -200.000000
32) 0.000000 -200.000000
33) 0.000000 -200.000000
34) 0.000000 200.000000
NO. ITERATIONS= 24
RANGES IN WHICH THE BASIS IS UNCHANGED:
*In the objective coefficient ranges block we see the amount by which each variable`s objective function coefficient may alter before becoming suboptimal. The current coefficient gives the variable`s objective function coefficient. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
A 20.000000 180.000000 INFINITY
B 200.000000 INFINITY 165.000000
C 0.000000 INFINITY 0.000000
D 0.000000 200.000000 INFINITY
E 35.000000 165.000000 INFINITY
F 80.000000 INFINITY 80.000000
G 100.000000 INFINITY 100.000000
H 20.000000 180.000000 INFINITY
I 20.000000 180.000000 INFINITY
J 0.000000 200.000000 INFINITY
X2 0.000000 INFINITY 0.000000
X1 0.000000 INFINITY 0.000000
X3 0.000000 165.000000 0.000000
X4 0.000000 0.000000 0.000000
X5 0.000000 165.000000 0.000000
X6 0.000000 0.000000 0.000000
X7 0.000000 180.000000 0.000000
X8 0.000000 180.000000 0.000000
X9 0.000000 180.000000 0.000000
X10 0.000000 200.000000 0.000000
X11 0.000000 200.000000 0.000000
*In the righthand side ranges block we see the amount the amount by which each variable`s righthand side may alter before becoming suboptimal. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 0.100000 0.400000 0.000000
3 0.400000 INFINITY 0.000000
4 0.000000 INFINITY 0.000000
5 0.000000 0.400000 0.000000
6 0.300000 0.400000 0.000000
7 0.100000 INFINITY 0.100000
8 0.300000 INFINITY 0.300000
9 0.100000 0.400000 0.000000
10 0.500000 0.400000 0.000000
11 0.000000 0.400000 0.000000
12 0.000000 0.100000 INFINITY
13 0.000000 0.400000 INFINITY
14 0.000000 0.000000 INFINITY
15 0.000000 0.000000 INFINITY
16 0.000000 0.300000 INFINITY
17 0.000000 0.000000 INFINITY
18 0.000000 0.000000 INFINITY
19 0.000000 0.100000 INFINITY
20 0.000000 0.500000 INFINITY
21 0.000000 0.000000 INFINITY
22 0.500000 0.000000 0.400000
23 1.000000 0.000000 0.400000
24 0.800000 0.500000 INFINITY
25 1.400000 0.000000 0.400000
26 1.000000 0.500000 INFINITY
27 1.200000 0.000000 0.400000
28 1.500000 0.800000 INFINITY
29 0.000000 0.500000 INFINITY
30 0.000000 0.000000 0.400000
31 0.400000 0.000000 0.400000
32 1.400000 0.000000 0.400000
33 0.500000 0.000000 0.400000
34 5.000000 0.400000 0.000000
1. LINDO used to find optimal solution without crashing the project:
Code:
Minimize X11 - X1
st
X2 - X1 >= 0.5
X3 - X2 >= 1
X4 - X3 >= 0.8
X5 - X3 >= 1.4
x6 - X4 >= 1
X7 - X5 >= 1.2
X8 - X3 >= 1.5
X8 - X6 >= 0
X8 - X7 >= 0
X9 - X8 >= 0.4
X10 - X9 >= 1.4
X11 - X10 >= 0.5
END
Reports window (explanatory text indicated with *):
LP OPTIMUM FOUND AT STEP 10
OBJECTIVE FUNCTION VALUE
1) 6.400000
VARIABLE VALUE REDUCED COST
X11 6.400000 0.000000
X1 0.000000 0.000000
X2 0.500000 0.000000
X3 1.500000 0.000000
X4 3.100000 0.000000
X5 2.900000 0.000000
X6 4.100000 0.000000
X7 4.100000 0.000000
X8 4.100000 0.000000
X9 4.500000 0.000000
X10 5.900000 0.000000
*Dual price indicates a critical path when its value is -. Increasing the duration of a constraint with a dual price of - will increase the duration of the whole project by the same amount.
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 -1.000000
3) 0.000000 -1.000000
4) 0.800000 0.000000
5) 0.000000 -1.000000
6) 0.000000 0.000000
7) 0.000000 -1.000000
8) 1.100000 0.000000
9) 0.000000 0.000000
10) 0.000000 -1.000000
11) 0.000000 -1.000000
12) 0.000000 -1.000000
13) 0.000000 -1.000000
NO. ITERATIONS= 10
RANGES IN WHICH THE BASIS IS UNCHANGED:
*In the objective coefficient ranges block we see the amount by which each variable`s objective function coefficient may alter before becoming suboptimal. The current coefficient gives the variable`s objective function coefficient. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X11 1.000000 INFINITY 0.000000
X1 -1.000000 INFINITY 0.000000
X2 0.000000 INFINITY 0.000000
X3 0.000000 INFINITY 0.000000
X4 0.000000 0.000000 0.000000
X5 0.000000 INFINITY 0.000000
X6 0.000000 0.000000 0.000000
X7 0.000000 INFINITY 0.000000
X8 0.000000 INFINITY 0.000000
X9 0.000000 INFINITY 0.000000
X10 0.000000 INFINITY 0.000000
*In the righthand side ranges block we see the amount the amount by which each variable`s righthand side may alter before becoming suboptimal. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 0.500000 INFINITY 0.500000
3 1.000000 INFINITY 1.500000
4 0.800000 0.800000 INFINITY
5 1.400000 INFINITY 0.800000
6 1.000000 0.800000 INFINITY
7 1.200000 INFINITY 0.800000
8 1.500000 1.100000 INFINITY
9 0.000000 0.800000 INFINITY
10 0.000000 INFINITY 0.800000
11 0.400000 INFINITY 4.500000
12 1.400000 INFINITY 5.900000
13 0.500000 INFINITY 6.400000
--------------------------------------------------------------------------------------------------------------------------------------------------------
2. LINDO used to determine optimal solution when the project is crashed:
Code:
Minimize 20A + 200B + 0C + 0D + 35E + 80F + 100G + 20H + 20I + 0J
Subject to
A <= 0.1
B <= 0.4
C <= 0
D <= 0
E <= 0.3
F <= 0.1
G <= 0.3
H <= 0.1
I <= 0.5
J <= 0
A >= 0
B >= 0
C >= 0
D >= 0
E >= 0
F >= 0
G >= 0
H >= 0
I >= 0
J >= 0
X2 - X1 + A >= 0.5
X3 - X2 + B >= 1
X4 - X3 + F >= 0.8
X5 - X3 + D >= 1.4
X6 - X4 + G >= 1
X7 - X5 + E >= 1.2
X8 - X3 + C >= 1.5
X8 - X6 >= 0
X8 - X7 >= 0
X9 - X8 + H >= 0.4
X10 - X9 + I >= 1.4
X11 - X10 + J >= 0.5
X11 - X1 <= 5
Reports window (explanatory text added with *):
LP OPTIMUM FOUND AT STEP 24
OBJECTIVE FUNCTION VALUE
1) 104.5000
VARIABLE VALUE REDUCED COST
A 0.100000 0.000000
B 0.400000 0.000000
C 0.000000 0.000000
D 0.000000 0.000000
E 0.300000 0.000000
F 0.000000 80.000000
G 0.000000 100.000000
H 0.100000 0.000000
I 0.500000 0.000000
J 0.000000 0.000000
X2 0.400000 0.000000
X1 0.000000 0.000000
X3 1.000000 0.000000
X4 2.300000 0.000000
X5 2.400000 0.000000
X6 3.300000 0.000000
X7 3.300000 0.000000
X8 3.300000 0.000000
X9 3.600000 0.000000
X10 4.500000 0.000000
X11 5.000000 0.000000
*Dual price indicates a critical path when its value is -. Increasing the duration of a constraint with a dual price of - will increase the duration of the whole project by the same amount.
ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 180.000000
3) 0.000000 0.000000
4) 0.000000 0.000000
5) 0.000000 200.000000
6) 0.000000 165.000000
7) 0.100000 0.000000
8) 0.300000 0.000000
9) 0.000000 180.000000
10) 0.000000 180.000000
11) 0.000000 200.000000
12) 0.100000 0.000000
13) 0.400000 0.000000
14) 0.000000 0.000000
15) 0.000000 0.000000
16) 0.300000 0.000000
17) 0.000000 0.000000
18) 0.000000 0.000000
19) 0.100000 0.000000
20) 0.500000 0.000000
21) 0.000000 0.000000
22) 0.000000 -200.000000
23) 0.000000 -200.000000
24) 0.500000 0.000000
25) 0.000000 -200.000000
26) 0.000000 0.000000
27) 0.000000 -200.000000
28) 0.800000 0.000000
29) 0.000000 0.000000
30) 0.000000 -200.000000
31) 0.000000 -200.000000
32) 0.000000 -200.000000
33) 0.000000 -200.000000
34) 0.000000 200.000000
NO. ITERATIONS= 24
RANGES IN WHICH THE BASIS IS UNCHANGED:
*In the objective coefficient ranges block we see the amount by which each variable`s objective function coefficient may alter before becoming suboptimal. The current coefficient gives the variable`s objective function coefficient. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
A 20.000000 180.000000 INFINITY
B 200.000000 INFINITY 165.000000
C 0.000000 INFINITY 0.000000
D 0.000000 200.000000 INFINITY
E 35.000000 165.000000 INFINITY
F 80.000000 INFINITY 80.000000
G 100.000000 INFINITY 100.000000
H 20.000000 180.000000 INFINITY
I 20.000000 180.000000 INFINITY
J 0.000000 200.000000 INFINITY
X2 0.000000 INFINITY 0.000000
X1 0.000000 INFINITY 0.000000
X3 0.000000 165.000000 0.000000
X4 0.000000 0.000000 0.000000
X5 0.000000 165.000000 0.000000
X6 0.000000 0.000000 0.000000
X7 0.000000 180.000000 0.000000
X8 0.000000 180.000000 0.000000
X9 0.000000 180.000000 0.000000
X10 0.000000 200.000000 0.000000
X11 0.000000 200.000000 0.000000
*In the righthand side ranges block we see the amount the amount by which each variable`s righthand side may alter before becoming suboptimal. The allowable increase shows the increase possible for the linear programme to remain optimal. The allowable decrease shows the decrease possible for the linear programme to remain optimal.
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 0.100000 0.400000 0.000000
3 0.400000 INFINITY 0.000000
4 0.000000 INFINITY 0.000000
5 0.000000 0.400000 0.000000
6 0.300000 0.400000 0.000000
7 0.100000 INFINITY 0.100000
8 0.300000 INFINITY 0.300000
9 0.100000 0.400000 0.000000
10 0.500000 0.400000 0.000000
11 0.000000 0.400000 0.000000
12 0.000000 0.100000 INFINITY
13 0.000000 0.400000 INFINITY
14 0.000000 0.000000 INFINITY
15 0.000000 0.000000 INFINITY
16 0.000000 0.300000 INFINITY
17 0.000000 0.000000 INFINITY
18 0.000000 0.000000 INFINITY
19 0.000000 0.100000 INFINITY
20 0.000000 0.500000 INFINITY
21 0.000000 0.000000 INFINITY
22 0.500000 0.000000 0.400000
23 1.000000 0.000000 0.400000
24 0.800000 0.500000 INFINITY
25 1.400000 0.000000 0.400000
26 1.000000 0.500000 INFINITY
27 1.200000 0.000000 0.400000
28 1.500000 0.800000 INFINITY
29 0.000000 0.500000 INFINITY
30 0.000000 0.000000 0.400000
31 0.400000 0.000000 0.400000
32 1.400000 0.000000 0.400000
33 0.500000 0.000000 0.400000
34 5.000000 0.400000 0.000000