iegr 506: industrial engineering principles ii iegr 361...
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IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7 Test
Name: --------------
Time: -- 30 Minutes--
For a problem whose feasible region and other information
is plotted below, set up the information for XB at C and
perform one iteration of revised simplex method.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
KEY:
First, let’s write the LP and its standard form. (10 pts)
Original LP Standard Form
Max 6X1+X2
ST:
-X1+X2<=4
2X1+X2<=6
5X1-2X2<=10
X1,X2>=0
Z-6X1- X2 =0
L1: - X1+ X2+S1 =4
L2: 2X1+ X2 +S2 =6
L3: 5X1-2X2 +S3=10
X1,X2,S1,S2,S3>=0
Extreme point C is the intersection of L1 and L2, so S1 and
S2 are our non-basic variables. Set up XB and B. (15 pts)
X1
B =
-1 1 0
XB = X2 2 1 0
S3 5 -2 1
Use Gauss-Jordan method to find B-inverse. (15 pts)
-1 1 0 1 0 0 -R1
2 1 0 0 1 0 2R1+R2
5 -2 1 0 0 1 5R1+R3
1 -1 0 -1 0 0 1/3 R2+R1
0 3 0 2 1 0 1/3 R2
0 3 1 5 0 1 - R2+R3
1 0 0 - 1/3 1/3 0
0 1 0 2/3 1/3 0
0 0 1 3 -1 1
B - 1 =
- 1/3 1/3 0
2/3 1/3 0
3 -1 1
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Perform matrix operations to check for optimality. X1 and
X2 are the only main variables and both are in XB so we do
not need to check cB*B-inv*A-c as it is zero.
Check cB*B-inv: (20 pts)
cB = 6 1 0
- 1/3 1/3 0
cB B - 1 = 6 1 0 * 2/3 1/3 0
3 -1 1
cB B - 1 = - 4/3 7/3 0
There is a negative value in cB*B-inv associated with S1.
So, S1 will enter the basis. To find the leaving variable
we need to find the minimum ratio between B-inv*b and the
first column of B-inv.
B - 1 b=
- 1/3 1/3 0 4 2/3
2/3 1/3 0 * 6 = 14/3
3 -1 1 10 16
Min {---, 7, 16/3} = 16/3 associated with S3. So, S3 enters
the basis replacing S3. Set up the new XB and B. (15 pts)
X1
B =
-1 1 1
XB = X2 2 1 0
S1 5 -2 0
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Use Gauss-Jordan method to find B-inverse. (15 pts)
-1 1 1 1 0 0 -R1
2 1 0 0 1 0 2R1+R2
5 -2 0 0 0 1 5R1+R3
1 -1 -1 -1 0 0 1/3 R2+R1
0 3 2 2 1 0 1/3 R2
0 3 5 5 0 1 - R2+R3
1 0 - 1/3 - 1/3 1/3 0 1/9 R3+R1
0 1 2/3 2/3 1/3 0 - 2/9 R3+R2
0 0 3 3 -1 1 1/3 R3
1 0 0 0 2/9 1/9
0 1 0 0 5/9 - 2/9
0 0 1 1 - 1/3 1/3
B - 1 =
0 2/9 1/9
0 5/9 - 2/9
1 - 1/3 1/3
Perform matrix operations to check for optimality. Again,
X1 and X2 are the only main variables and both are in XB so
we do not need to check cB*B-inv*A-c as it is zero.
Check cB*B-inv: (20 pts)
cB = 6 1 0
0 2/9 1/9
cB B - 1 = 6 1 0 * 0 5/9 - 2/9
1 - 1/3 1/3
cB B - 1 = 0 17/9 4/9
Optimality conditions are satisfied. Optimal solution is
reached.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
B - 1 b=
0 2/9 1/9 4 22/9
0 5/9 - 2/9 * 6 = 10/9
1 - 1/3 1/3 10 16/3
22/9
cB B - 1 b= 6 1 0 * 10/9
= 142/9
16/3
X1=22/9, X2=10/9, Z=142/9
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 01
Name: --MOR--
Pickup Time and Date: -- R/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 3 X1 + 5 X2 + 3 X3 + 4 X4 + 6 X5 + 3 X6
SUBJECT TO
X1 + 3 X3 + X4 + 3X5 + 2 X6 <= 30
2 X1 + X2 + X3 + 2 X4 + X5 - X6 <= 40
X1 + 2 X2 + X3 + 3 X4 + 2 X5 + 2 X6 <= 30
3 X1 - X2 + 2 X3 + X4 + X5 <= 25
X1, X2, X3, X4, X5, X6 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 02
Name: --KYM --
Pickup Time and Date: -- R/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 6 X1 + 5 X2 + 2 X3 + 4 X4 + 6 X5 + 7 X6
SUBJECT TO
3 X1 + 2X2+ 4 X3 + X4 + 3 X5 + 2 X6 <= 60
2 X1 + 4 X2 + X3 + 2 X4 + X5 - X6 <= 48
-X1 + 2 X2 + 3 X3 + 3 X4 + 2 X5 + 2 X6 <= 45
X1 + X2 + X3 + 3X4 + X5 +2 X6<= 36
X1, X2, X3 X4, X5, X6 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 03
Name: --EDT --
Pickup Time and Date: -- R/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 4 X1 + 7 X2 + X3 + 7 X4 + 8 X5
SUBJECT TO
3 X1 + 2 X2 + 4 X3 + 3 X4 + 3 X5 <= 60
2 X1 + 4 X2 + X3 + 2 X4 + 5 X5 <= 48
X1 + 2 X2 + 3 X3 + 3 X4 + 2 X5 <= 45
X1 + X3 + 3 X4 + X5 <= 36
- X1 + 3 X2 + 2 X3 + 2 X4 + 2 X5 <= 25
X1, X2, X3 X4, X5, X6 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7 Problem 04
Name: --NGT --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 5X1 + 9 X2 + 3 X3 + 7 X4 + 3 X5 + 8 X6
ST
3 X1 + X2 + 3 X3 + X4 + 2 X6 <= 45
2 X1 + 3 X2 + X3 - 2 X4 + 2 X5 <= 40
- X1 + X2 + 4 X3 + X4 + X5 + 3 X6 <= 60
3 X2 + X3 + 2 X4 + 3 X5 + 2 X6 <= 36
2 X1 + 2 X3 + 3 X4 + 2 X5 + X6 <= 48
X1, X2, X3 X4, X5, X6 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 05
Name: --ARH --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 5X1 + 9 X2 + 8 X3 + 7 X4 + 9 X5 + 8 X6
ST
X1 + 2 X2 + 3 X3 + X4 + 3 X5 + 2 X6 <= 42
2 X1 + 3 X2 + X3 + 2 X4 + 2 X5 <= 54
X1 + X2 + 4 X3 - X4 + X5 + 3 X6 <= 50
3 X2 + X3 + 2 X4 + 3 X5 + 2 X6 <= 36
2 X1 + 2 X3 + 3 X4 + 2 X5 - X6 <= 38
X1, X2, X3 X4, X5, X6 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 06
Name: --CAF --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 10X1 + 9 X2 + 8 X3 + 7 X4 + 9 X5
ST
X1 + 2 X2 + 3 X3 + X4 + 3 X5 <= 42
2 X1 + 3 X2 + X3 + 2 X4 + 2 X5 <= 40
3 X1 + X2 + 2 X3 + 4 X4 + X5 <= 44
X1 + X2 + 4 X3 - X4 + X5 <= 50
3 X2 + X3 + 2 X4 + 3 X5 <= 36
2 X1 + X2 + 2 X3 + 3 X4 + 2 X5 <= 38
X1, X2, X3 X4, X5 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 07
Name: --KEM --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 3X1 + 9 X2 + 7 X3 + 7 X4 + 2 X5
ST
X1 + 2 X2 + X3 + 4 X4 + 3 X5 <= 42
2 X1 + 3 X2 + X3 + 2 X4 + 2 X5 <= 50
2 X1 + X2 + 2 X3 + 4 X5 <= 34
5 X1 - X2 + 4 X3 - X4 + X5 <= 30
3 X2 + X3 + 2 X4 + 3 X5 <= 46
2 X1 + X2 + 2 X3 + 3 X4 + 2 X5 <= 48
X1, X2, X3, X4, X5 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 08
Name: --CHV --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 3X1 + 4 X2 + 5 X3 + 5 X4 + 6 X5
ST
X1 + 2 X2 + X3 + 4 X4 + 3 X5 <= 22
2 X1 + 3 X2 + 3 X3 + 2 X4 + 2 X5 <= 25
2 X1 + X2 + 2 X3 + 4 X5 <= 24
5 X1 - X2 + 4 X3 - X4 + 2 X5 <= 30
3 X2 + X3 + 2 X4 + 3 X5 <= 26
2 X1 + X2 + 2 X3 + 3 X4 + 2 X5 <= 28
X1, X2, X3, X4, X5 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 09
Name: --ELT --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 9X1 + 8 X2 - 2 X3 + 7 X4 + 6 X5
ST
X1 + 2 X2 + X3 + 4 X4 + 3 X5 <= 28
2 X1 + 3 X2 + 3 X3 + 2 X4 + 2 X5 <= 35
2 X1 + X2 + 2 X3 + X5 <= 26
5 X1 + X2 + 4 X3 + X4 + 2 X5 <= 34
3 X1 + 3 X2 - X3 + 2 X4 + 3 X5 <= 36
2 X1 + X2 + 2 X3 + 3 X4 + X5 <= 34
X1, X2, X3 X4, X5 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 10
Name: --MOS --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 9X1 + 11 X2 + 12 X3 + 7 X4 + 8 X5
ST
4 X1 + 2 X2 + X3 + 4 X4 + 3 X5 <= 58
5 X1 + 3 X2 + 3 X3 + 2 X4 + 2 X5 <= 64
2 X1 + 2 X2 + 2 X3 + X4 - X5 <= 26
X1 + X2 + 4 X3 + X4 + 2 X5 <= 54
3 X1 + 3 X2 + X3 + 2 X4 + X5 <= 42
2 X1 + 4 X2 + 2 X3 + 3 X4 + X5 <= 30
X1, X2, X3, X4, X5 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 11
Name: --JUO --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 9X1 + 11 X2 + 10 X3 + 13 X4 + 8 X5
ST
4 X1 + 2 X2 + X3 + X4 + 3 X5 <= 54
X1 + 3 X2 + 3 X3 + 2 X4 + 2 X5 <= 54
2 X1 + 2 X2 + 2 X3 + X4 - X5 <= 46
X1 + X2 + 4 X3 + X4 + 2 X5 <= 54
3 X1 + 3 X2 + X3 + 2 X4 + X5 <= 48
2 X1 + 4 X2 + 2 X3 + 3 X4 + 3 X5 <= 50
X1, X2, X3, X4, X5 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 12
Name: --ABS --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 9X1 + 11 X2 + 10 X3 + 13 X4 + 8 X5 + 6 X6 + X7
ST
4 X1 + 2 X2 + X3 + 3 X4 + 3 X5 + 4 X6 + 2 X7 <= 64
X1 + 3 X2 + 3 X3 + 2 X5 + X6 <= 16
2 X1 + 2 X2 + 2 X3 + X4 - X5 + 3 X7 <= 27
3 X2 + 4 X3 + 2 X4 + X5 + 2 X6 - X7 <= 48
2 X1 + 4 X2 + 2 X3 + 3 X4 + 3 X5 - X6 + 2 X7 <= 50
X1, X2, X3, X4, X5, X6, X7 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 13
Name: --OBM --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 8X1 + 13 X2 - X3 + 9 X4 + X5 + 11 X6 + 6 X7
ST
3 X1 + X2 + X3 + 4 X4 + 3 X5 + 4 X6 + 2 X7 <= 64
X1 + 3 X2 + 3 X3 + 5 X5 + X6 <= 46
2 X1 + X2 + X3 + X4 - X5 + 3 X7 <= 27
3 X2 + X3 + 2 X4 + X5 + 2 X6 - X7 <= 48
2 X1 + X2 + 4 X3 + X4 + 2 X5 - X6 + 2 X7 <= 50
X1, X2, X3, X4, X5, X6, X7 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 14
Name: --JOD --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 5X1 + 9 X2 + 3 X3 + 7 X4 + 4 X5 + 6 X6
ST
3 X1 + X2 + 3 X3 + X4 + 4 X5 + 2 X6 <= 45
2 X1 + 3 X2 + 4 X3 - 2 X4 + 2 X5 <= 20
- X1 + X2 + 4 X3 + X4 + X5 + 3 X6 <= 10
3 X2 + X3 + 2 X4 + 3 X5 + 2 X6 <= 36
2 X1 + 2 X3 + 3 X4 + 2 X5 + X6 <= 48
X1, X2, X3, X4, X5, X6 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 15
Name: --WER --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 9X1 + 11 X2 + 10 X3 + 10 X4 + 8 X5 + 6 X6 + 14 X7
ST
4 X1 + 2 X2 + X3 + 3 X4 + 3 X5 + 4 X6 + 2 X7 <= 64
5 X1 + 3 X2 + 3 X3 + 2 X5 + X6 <= 16
2 X2 + 2 X3 + X4 - X5 + 3 X7 <= 27
3 X2 + 4 X3 + 2 X4 + X5 + 2 X6 - X7 <= 48
2 X1 + 4 X2 + 2 X3 + 3 X4 + 3 X5 - X6 + 2 X7 <= 50
X1, X2, X3, X4, X5, X6, X7 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 16
Name: --BBA --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 11 X1 + 14 X2 + 15 X3 + 13 X4 + 12 X5
ST
4 X1 + 2 X2 + X3 + X4 + 3 X5 <= 50
X1 + 3 X2 + 3 X3 + 2 X4 + 2 X5 <= 54
2 X1 + 2 X2 + 2 X3 + X4 - X5 <= 26
X1 + X2 + 4 X3 + X4 + 2 X5 <= 54
3 X1 + 3 X2 + X3 + 2 X4 + X5 <= 48
2 X1 + 4 X2 + 2 X3 + 3 X4 - X5 <= 50
X1, X2, X3, X4, X5 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 17
Name: --ADC --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 7 X1 + 4 X2 + 6 X3
ST
4 X1 + 2 X2 + 4 X3 <= 23
X1 + 3 X2 + X3 <= 14
2 X1 + X2 + 2 X3 <= 6
X1 + 3 X2 + X3 <= 14
3 X1 + 3 X2 + X3 <= 18
2 X1 + 4 X2 + 2 X3 <= 20
3 X1 + 5 X2 <= 5
X1, X2, X3 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.
IEGR 506: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 7
Problem 18
Name: --MIF --
Pickup Time and Date: -- M/6:00pm--
Due: in 24 hours
1. Solve the following LP using revised simplex method.
MAX 7 X1 + 14 X2 + 11 X3
ST
- X1 + 2 X2 + 4 X3 <= 23
X1 - 3 X2 + X3 <= 14
2 X1 + X2 + 2 X3 <= 16
X1 + 3 X2 + 3 X3 <= 14
3 X1 + X3 <= 8
2 X1 + 4 X2 + 2 X3 <= 20
3 X1 + 5 X2 <= 5
X1, X2, X3 >= 0
2. Use EXCEL SOLVER to solve the problem and verify your
solution.
3. Use Gauss-Jordan method to calculate the B inverse of
the final tableau.