IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Test Problem
Name: -------------Time: --30 Minutes -Consider the given LP below:
MIN -9X1+6X2+7X3
ST
-X1+2X2+4X3=0
2X1+X2+X3 <= 5
X1-5X2=7
X2+2X3<=1
3X1+X2+3X3=8
X1>=0, X2<=0
1. Set up the first legitimate tableau and identify the
entering and leaving variables. (40 pts)
2. Perform a partial iteration that has at least two rows
and the Z row of the new tableau. (30 pts).
3. Can the optimal solution be found in an easier fashion?
If so, show how and find the optimal solution. (30 pts)
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
KEY:
1. Set up the first legitimate tableau and identify the
entering and leaving variables. (40 pts)
Process the LP into standard form:
X2<=0, replace it with X2’ where X2’=-X2
X3 unrestricted, replace it with X3’ and X3” where
X3=X3’-X3”
Convert minimization problem into maximization by
multiplying Z by -1
–Z -9X1-6X2’+7X3’-7X3”
=0
- X1-2X2’+4X3’-4X3”
=0
2X1- X2’+ X3’- X3”+ S1
=5
X1+5X2’
=7
- X2’+2X3’-2X3”
+ S2 =1
3X1- X2’+3X3’-3X3”
=8
X1>=0
X2’>=0
X3’>=0
X3”>=0
S1>=0
S2>=0
Set up the Big-M by adding artificial variables to
constraints without slack variables, penalize the objective
function row:
–Z -9X1-6X2’+7X3’-7X3”
+MR1+MR2+MR3
- X1-2X2’+4X3’-4X3”
+ R1
2X1- X2’+ X3’- X3”+ S1
X1+5X2’
+ R2
- X2’+2X3’-2X3”
+ S2
3X1- X2’+3X3’-3X3”
+ R3
X1, X2’, X3’, X3”, S1, S2, R1, R2, R3
M>>0
=0
=0
=5
=7
=1
=8
>=0
Select R1, S1, R2, S2, and R3 as basic variables for the
first tableau but to legitimize the tableau, multiply rows
with artificial variables by -1 and add them to Z-row.
-Z+ (-3M-9) X1+ (-2M-6) X2’+ (-7M+7) X3’+ (7M-7) X3” = -15M
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
-Z
R1
S1
R2
S2
R3
-Z
1
0
0
0
0
0
X1
-3M-9
-1
2
1
0
3
X2’
-2M-6
-2
-1
5
-1
-1
X3’
-7M+7
4
1
0
2
3
X3”
7M-7
-4
-1
0
-2
-3
S1
0
0
1
0
0
0
S2
0
0
0
0
1
0
R1
0
1
0
0
0
0
R2
0
0
0
1
0
0
R3
0
0
0
0
0
1
RHS
-15M
0
5
7
1
8
Min
Ratio
0
5
--1/2
8/3
Solution:
X1=X2’=X3’=X3”=0
S1=5, S2=1, R1=0, R2=7, R3=8, Z=-15M
Not feasible; artificial variables in the basis and >0
Entering variable: X3’; most negative in the Z-row
Leaving variable: R1; min ratio
Pivot: 4
2. Perform a partial iteration that has at least two rows
and the Z row of the new tableau. (30 pts).
Pivot row (to be multiplied by reciprocal of pivot (1/4) is
selected and R2 row that does not need any additional
calculation (since its pivot column entry is zero) and the
Z-row (multiply X3’ row by 7M-7 and add to Z-row).
-Z
X3'
S1
R2
S2
R3
-Z
1
0
0
0
0
0
X1
(-19M-29)/4
-1/4
X2'
(-11M-5)/2
-1/2
X3'
0
1
X3"
0
-1
S1
0
0
S2
0
0
R1
(7M-7)/4
1/4
R2
0
0
R3
0
0
RHS
-15M
0
1
5
0
0
0
0
0
1
0
7
3. Can the optimal solution be found in an easier fashion?
If so, show how and find the optimal solution. (30 pts)
Both Big-M and Two-phase methods involve several steps to
reach a conclusion and although two-phase involves no M
calculations, still both methods are not that easy.
However, a close observation of the original problem
reveals an interesting fact. There are three variables in
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
this problem and constraint set also has three equations.
So, if this problem had any solution it would be the result
of solving those equations. However, after solving the
three equation-three unknown set of linear equations we
need to check for feasibility with respect to other
constraints.
-X1+2X2+4X3=0
X1-5X2=7
3X1+X2+3X3=8
Solve:
X1=2X2+4X3 (from first equation)
(2X2+4X3)-5X2=7  -3X2+4X3=7
3(2X2+4X3) +X2+3X3=8  7X2+15X3=8
7(-3X2+4X3=7)  -21X2+28X3=49
3(7X2+15X3=8)  21X2+45X3=24
--------------73X3=73  X3=1
-3X2+4(1)=7  -3X2=3  X2=-1
X1=2(-1)+4(1)=2
Check the solution against other constraints:
X1>0, X2<0 and X3 unrestricted (here positive) so special
constraints are OK.
2X1+X2+X3 <= 5  2(2)+(-1)+(1)=2<5
X2+2X3<=1  (-1)+2(1)=1=1
So this solution is feasible and optimal (feasible region
is just one point).
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 01
Name: --MOR -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -9X1+6X2
ST
-5X1+9X2>=7
-2X1+7X2 <= 11
X1+5X2>=2
-10X1+5X2=24
7X1+8X2<=5
X1<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 02
Name: --ARH -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 9X1-6X2
ST
5X1-9X2>=8
2X1-7X2 <= 11
10X1-3X2=14
-7X1-8X2<=5
-X1-2X2>=1
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 03
Name: --NGT -Pickup Time and Date: --W/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 14X1-15X2
ST
-3X2>=4
7X1-5X2 <= 34
-3X1-5X2<=6
3X1-5X2>=13
4X1+X2>=5
5X1-6X2<=30
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 04
Name: --MIF -Pickup Time and Date: --W/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 7X1-10X2
ST
-2X1-7X2 <= 9
X1-3X2>=1
-10X1-3X2=9
7X1-8X2<=10
-5X1-9X2>=8
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 05
Name: --NGT -Pickup Time and Date: --W/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -14X1+15X2
ST
-7X1-5X2 <= 34
-3X1-5X2>=13
-4X1+X2>=5
3X1-5X2<=6
-5X1-6X2=30
X1<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 06
Name: --ARH2 -Pickup Time and Date: --R/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -7X1+18X2
ST
-7X1+3X2 <= 24
-5X1+9X2=30
3X1+5X2<=3
-4X1+5X2>=11
-4X1-X2>=5
X1<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 07
Name: --MOR2 -Pickup Time and Date: --R/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -7X1-27X2
ST
X1-10X2>=5
7X1-3X2 <= 24
-3X1-6X2=5
4X1+X2>=7
5X1-9X2<=30
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 08
Name: --BBA -Pickup Time and Date: --R/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 15X1-4X2
ST
5X1-9X2>=8
3X1-7X2 <= 13
-5X1-8X2>=3
10X1-3X2=14
-7X1-9X2<=12
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 09
Name: --EDT -Pickup Time and Date: --R/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -15X1+7X2
ST
-5X1-9X2>=8
-3X1-7X2 <= 13
5X1-8X2>=3
7X1-9X2<=12
-10X1-3X2=14
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 10
Name: --CHV -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -13X1+9X2
ST
-5X1-9X2>=14
-9X1+7X2 <= 10
X1-5X2>=2
-11X1-3X2=27
5X1-9X2<=5
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 11
Name: --ELT -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -13X1+9X2
ST
-4X1+7X2=8
-8X1-5X2 <= 7
2X1+7X2>=2
-9X1+4X2>=10
7X1+8X2<=4
X1<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 12
Name: --OBM -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 11X1+17X2
ST
-4X1+7X2=26
-8X1-5X2 <= 7
2X1+7X2>=5
-9X1+4X2>=10
7X1+8X2<=4
X1<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 13
Name: --NGT2 -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -12X1+17X2
ST
-5X1-9X2>=11
-2X1-7X2 <= 11
2X1-7X2>=2
-10X1-3X2=22
7X1-9X2<=12
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 14
Name: --CHS -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 12X1-9X2
ST
-5X1-9X2>=11
-2X1-7X2 <= 11
2X1-7X2>=2
-10X1-3X2=22
7X1-9X2<=4
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 15
Name: --MOI -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -11X1+10X2
ST
-5X1+9X2>=11
-3X1+7X2 <= 11
2X1+7X2>=2
7X1+9X2<=3
-10X1+X2=11
X1<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 16
Name: --KYM -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -10X1+17X2
ST
-5X1+9X2>=4
-3X1+7X2 <= 9
2X1+9X2>=10
7X1+9X2<=16
10X1-X2=7
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 17
Name: --ADC -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 11X1-13X2
ST
3X1-7X2 <= 3
-5X1-8X2>=2
-9X1+4X2=3
-7X1-9X2<=7
5X1-9X2>=0
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 18
Name: --JUO -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -12X1+9X2
ST
X1-9X2=8
-5X1-9X2>=1
-3X1-7X2 <= 8
5X1-8X2>=2
7X1-9X2<=7
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 19
Name: --ABS -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 15X1-14X2
ST
-5X1+9X2=14
-7X1+3X2 <= 8
3X1+7X2<=8
-X1+10X2>=11
-8X1-X2>=1
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 20
Name: --KEM -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -15X1+17X2
ST
7X1-3X2 <= 17
5X1-9X2=18
-3X1-6X2<=0
X1-10X2>=8
4X1+X2>=6
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 21
Name: --WER -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -19X1+17X2
ST
-5X1+9X2>=15
-2X1+7X2 <= 11
X1+5X2>=3
-10X1+5X2=22
7X1+8X2<=0
X1<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 22
Name: --JOD-Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 15X1-11X2
ST
-5X1-9X2>=15
-2X1-7X2 <= 11
X1-5X2>=3
-12X1-7X2=27
7X1-8X2<=1
X2<=0
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 23
Name: --MIF2 -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN -15X1-11X2
ST
-6X1-7X2 <= 27
3X1-5X2=4
-3X1-5X2>=15
-5X1+6X2<=1
-4X1+X2>=6
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.
IEGR 516: Industrial Engineering Principles II
IEGR 361: Introduction to Linear Programming
Fall 2014
M. Salimian
Topic 5
Problem 24
Name: --EDT2 -Pickup Time and Date: --M/6:00pm-Due: in 24 hours
1. Solve the following LP using simplex method.
MIN 15X1-11X2
ST
-5X1+9X2>=8
-3X1+7X2 <= 8
5X1+8X2>=7
-10X1+3X2=5
7X1+9X2<=11
2. Plot the feasible region (using MAPLE)
3. Identify the extreme points associated with each tableau
on the plot and show the progress path from starting point
to the optimal solution.
4. Use LINDO to solve the problem and verify your final
tableau solution.