Linear Programming
Set 1
a) Solve the following linear programming problem using the graphical method.
Maximize Z=9x+10y
Subject to:
2x–4y≥16
6x+y≤24
x+9y≤12
y≤4
x,y≥0
b) The following is the final simplex for a linear programming problem:
A,B and C represent the number of litres of grey,black and red paint to be mixed
together while S1,S2 and S3 are the slack variables for resource 1,resource 2 and
resource 3 rspectively.
a) Complete the above tableu.
Cj
Solution mix
10
15
12
0
0
0
Quantity
A
B
C
S1
S2
S3
B
2
8
0
4
0
7
600
S2
4
0
0
2
1
2
850
C
6
0
1
9
0
5
450
Zj
Cj–Zj
b) Determine the optimal solution and maximum profit.
c) Formulate the dual model for a given linear programming model using variable Y.
Maximize Z=15x+10y+8z
Subject to:
4x+8y+5z≤20
3x+2y+z≤18
X+y+7z≤26
X,y,z≥0
d) Interpret shadow price for each resource.
Set 2
a) Solve the following linear programming problem using the corner point method.
Minimize cost=5x+8y
Subject to:
x≤10
y≤20
5x+10y≥40
X+y≤35
X,y≥0
b) The optimal simplex tableau of a linear programming problem for the production
of X1,X2 and X3 is given below,.The objective of the problem is to maximize profit
(RM) based on three constraints relating to 3 resources.S1,S2 and S3 are the slack
associated with resources 1,resources 2 and resources 3 respectively.
a) Complete the above tableau.
b) If there is any other alternative?
c) State the optimal solution including the total profit
d) Maximize Z=60x+45y+80Z
Subject to:
X+145y+290z≤1800
65x+45y+89z≤1600
X+y+z≤42
X1,X2,X3≥0
Write the dualof the given primal problem.
Cj
Solution Mix
50
64
78
0
0
0
Quantity
X1
X2
X3
S1
S2
S3
S1
80
75
0
65
1
0
300
S2
50
0
20
0
58
1
500
X3
1
0
0
0
0
35
8
Zj
Cj–Zj
NETWORK
Kental Teguh Sdn Bhd introducing a new products as shown in the following table:
Activity
Immediate
Predecessor
Time (weeks)
Optimistic
Most likely
Pessimistic
A
–
5
1
3
B
–
8
4
6
C
B
5
3
4
D
C
6
7
E
2
3
E
1
3
E
1
2
9
6
7
I
1
2
3
J
2
5
8
