simplex method for solving lp problems with two variables
TRANSCRIPT
Simplex MethodSimplex Method for solving LP problems for solving LP problems with two variableswith two variables
1- Introduction Graphical method presented in last chapter is fine for 2
variables. But most LP problems are too complex for simple graphical procedures.
The Simplex Method:o is appropriate for problems with more than 2 variables;o uses algebra rules, to find optimal solutions; o is an algorithm and is a series of steps that will
accomplish a certain task.
In this chapter will introduce the simplex method for problems with 2 variables.
2- Simplex algorithm through an example:
Assume one company producing flair furniture: Tables (T) and Chairs (C). The following table provides the information available:
Department
Production Available
Working
Hours
Tables Chairs
Carpentry 4 3 240
Painting & varnishing
2 1 100
Profit 7SR 5SR
Formulation:
• Decision variables:T = Number of tablesC = Number of chairs
• Objective function:Maximize Z = 7T + 5C
• Constraints:2T + 1C 100 (Painting & varnishing) 4T + 3C 240 (Carpentry)T, C 0 (non-negativity constraints)
Graphical method solutionN
um
ber
of
Ch
airs 100
80
60
40
20
0 20 40 60 80 100 T
C
Number of Tables
FeasibleRegion
4T + 3C
2T + 1C
Optimal Solution:
T=30, C=40 Z= 410 SR
11stst Step: Step: Built initial Simplex tableau Less-than-or-equal-to constraints (≤) are converted
to equations by adding a variable called “slack variable”.
o Slack variables represent unused resources.
For the flair furniture problem, define the slacks as:o S1 = unused hours in the painting department
o S2 = unused hours in the carpentry department
The constraints are now written as:o 2T + 1C + S1 = 100
o 4T + 3C + S2 = 240
Slack variables not appearing in an equation are added with a coefficient of 0.This allows all the variables to be monitored at all times.
The final Simplex equations appear as:o 2T + 1C + 1S1 + 0S2 = 100
o 4T + 3C + 0S1 + 1S2 = 240
o T, C, S1, S2 0
The slacks are added to the objective coefficient with 0 profit coefficients. The objective function, then, is: Max. Z= 7T + 5C + 0S1 + 0S2
11stst Step, continued Step, continued
Initial Simplex tableau
T C S1 S2 Quantity (Q)
S1 2 1 1 0 100
S2 4 3 0 1 240
Z 7 5 0 0 0
We start by a basic solution Z=0.
Non-Basic variables
Bas
ic
vari
able
s
22ndnd Step: Step: Entering variable
Choose one entering variable from non-basic variables (T or C) for which we have the largest positive coefficient in the objective function. Here the entering variable will be T and the corresponding column is called pivot column.
T C S1 S2 Quantity (Q)
S1 2 1 1 0 100S2 4 3 0 1 240Z 7 5 0 0 0
Pivot column
33rd rd Step: Step: Leaving variable
Choose one leaving variable from the basic variables (S1 or S2) for which we have the smallest value of quantities (Q) divided by items of pivot column:
for S1 we have 100/2=50for S2 we have 240/4=60
Then the leaving variable will be S1 and the corresponding row is called pivot row.
33rd rd Step: Step: Leaving variable
T C S1 S2 (Q)
S1 2 1 1 0 100S2 4 3 0 1 240Z 7 5 0 0 0
Pivot column
Pivot row
Pivot item
44th th Step: Step: Pivoting
The pivoting is the changing of simplex tableau values as follow:
The entering variable (T) takes the place of leaving variable (S1).
All items of pivot column are =0 except pivot item =1.All items of pivot row are divided by pivot item.Other items of the tableau are calculated as follow:
A B
C D
Pivot column
Pivot rowNew A = A – B*C/D
44th th Step: Step: Pivoting
The New value of the objective function is calculated as follow:
Capacity of pivot rowZ’ = Z + Largest coefficient * Pivot item
If all coefficients of objective function are negatives or equal to zero the optimal solution is found. Otherwise go to step2.
44th th Step: Step: Pivoting
T C S1 S2 Q
S1 2 1 1 0 100S2 4 3 0 1 240
Z 7 5 0 0 0
Pivot column
Pivot row
Pivot item
Pivoting
44th th Step: Step: Pivoting
All coefficients of objective function aren’t negatives or equal to zero, the optimal solution is not found. Then go to step2.
T C S1 S2 Q
T 1 1/2 1/2 0 50S2 0 1 -2 1 40Z 0 3/2 -7/2 0 350
22ndnd Step: Step: Entering variable
The new largest positive coefficient in the objective function is 3/2 then the entering variable will be C.
33rd rd Step: Step: Leaving variable
The new smallest value of quantities (Q) divided by items of pivot column is 40/1=40 then the leaving variable will be S2 .
44th th Step: Step: Pivoting
T C S1 S2 QT 1 1/2 1/2 0 50S2 0 1 -2 1 40Z 0 3/2 -7/2 0 350
Pivot column
Pivot row
Pivot item
44th th Step: Step: Pivoting
T C S1 S2 QT 1 0 3/2 -1/2 30C 0 1 -2 1 40Z 0 0 -1/2 -3/2 410
All coefficients of objective function are now equal to zero. then the optimal solution is found: T=30, C=40; Z=410
Using Simplex method, solve the LP problems modeled in chapter1.
3- Applications
The mathematical model of problem is:
Max Z= 3x1 + 5x2
S.t. x1 4 (Plant 1 constraint)
2x2 12 (Plant 2 constraint)3x1+ 2x2 18 (Plant 3 constraint)x1, x2 0
3.1- Solving of problem1
11stst Step: Step: Initial Simplex tableau
We start by a basic solution Z=0.
Non-Basic variables
Bas
ic
vari
able
s
X1 X2 S1 S2 S3 Quantity (Q)
S1 1 0 1 0 0 4
S2 0 2 0 1 0 12
S3 3 2 0 0 1 18
Z 3 5 0 0 0 0
22ndnd Step: Step: Entering variable
The largest positive coefficient in the objective function is 5 then the entering variable will be X2.
33rd rd Step: Step: Leaving variable
The smallest value of quantities (Q) divided by items of pivot column is 12/2=6 then the leaving variable will be S2 .
44th th Step: Step: Pivoting
Pivot column
Pivot row
Pivot item X1 X2 S1 S2 S3 (Q)
S1 1 0 1 0 0 4
S2 0 2 0 1 0 12
S3 3 2 0 0 1 18
Z 3 5 0 0 0 0
44th th Step: Step: Pivoting
X1 X2 S1 S2 S3 (Q)
S1 1 0 1 0 0 4
X2 0 1 0 1/2 0 6
S3 3 0 0 -1 1 6
Z 3 0 0 -5/2 0 30
All coefficients of objective function aren’t negatives or equal to zero, the optimal solution is not found. Then go to step2.
22ndnd Step: Step: Entering variable
The new largest positive coefficient in the objective function is 3 then the entering variable will be X1.
33rd rd Step: Step: Leaving variable
The smallest value of quantities (Q) divided by items of pivot column is 6/3=2 then the leaving variable will be S3 .
44th th Step: Step: Pivoting
Pivot column
Pivot row
Pivot item X1 X2 S1 S2 S3 (Q)
S1 1 0 1 0 0 4
X2 0 1 0 1/2 0 6
S3 3 0 0 -1 1 6
Z 3 0 0 -5/2 0 30
44th th Step: Step: Pivoting
All coefficients of objective function are now equal to zero. then the optimal solution is found. X1=2, X2=6, Z=36.
X1 X2 S1 S2 S3 (Q)
S1 0 0 1 1/3 -1/3 2
X2 0 1 0 1/2 0 6
X1 1 0 0 -1/3 1/3 2
Z 0 0 0 7/2 -1 36
The mathematical model of problem is:
Max Z= 10x1 + 15x2
S.t. 2x1 + 4x2 100 ( aluminum constraint) 3x1 + 2x2 80 ( steel constraint)
x1 , x2 0
3.2- Solving of problem2
11stst Step: Step: Initial Simplex tableau
X1 X2 S1 S2 Quantity (Q)
S1 2 4 1 0 100
S2 3 2 0 1 80
Z 10 15 0 0 0
We start by a basic solution Z=0.
Non-Basic variables
Bas
ic
vari
able
s
22ndnd Step: Step: Entering variable
The largest positive coefficient in the objective function is 15 then the entering variable will be X2.
33rd rd Step: Step: Leaving variable
The smallest value of quantities (Q) divided by items of pivot column is 100/4=25 then the leaving variable will be S1 .
44th th Step: Step: Pivoting
Pivot column
Pivot row
Pivot item X1 X2 S1 S2 Q
S1 2 4 1 0 100
S2 3 2 0 1 80
Z 10 15 0 0 0
44th th Step: Step: Pivoting
X1 X2 S1 S2 Q
X2 1/2 1 1/4 0 25
S2 2 0 -1/2 1 30
Z 5/2 0 15/4 0 375
All coefficients of objective function aren’t negatives or equal to zero, the optimal solution is not found. Then go to step2.
22ndnd Step: Step: Entering variable
The new largest positive coefficient in the objective function is 5/2 then the entering variable will be X1.
33rd rd Step: Step: Leaving variable
The smallest value of quantities (Q) divided by items of pivot column is 30/2=15 then the leaving variable will be S2.
44th th Step: Step: Pivoting
Pivot column
Pivot row
Pivot item
X1 X2 S1 S2 Q
X2 1/2 1 1/4 0 25
S2 2 0 -1/2 1 30
Z 5/2 0 15/4 0 375
44th th Step: Step: Pivoting
All coefficients of objective function are now equal to zero, the optimal solution is found: X1=15, X2=17.5, Z= 412.5.
X1 X2 S1 S2 Q
X2 0 1 3/8 -1/4 17.5
X1 1 0 -1/4 1/2 15
Z 0 0 30/8 -5/4 412.5