1. 10CS661 OPERATION RESEARCH
Engineered for Tomorrow

2. UNIT 3 SIMPLEX METHOD 2
Engineered for Tomorrow

3. Simplex Method
Simplex: a linear-programming algorithm that can solve problems having more than two decision variables.
The simplex technique involves generating a series of solutions in tabular form, called tableaus. By inspecting the bottom row of each tableau, one can immediately tell if it represents the optimal solution. Each tableau corresponds to a corner point of the feasible solution space. The first tableau corresponds to the origin. Subsequent tableaus are developed by shifting to an adjacent corner point in the direction that yields the highest (smallest) rate of profit (cost). This process continues as long as a positive (negative) rate of profit (cost) exists.
Subject Name: OPERATION RESEARCH
Subject Code:10CS661
Prepared By: Mrs.G.Annapoorani,Mrs.Prameladevi,Mrs.Sindhuja
Department: CSE
Date:

4. Simplex Method
Simplex: a linear-programming algorithm that can solve problems having more than two decision variables.
The simplex technique involves generating a series of solutions in tabular form, called tableaus. By inspecting the bottom row of each tableau, one can immediately tell if it represents the optimal solution. Each tableau corresponds to a corner point of the feasible solution space. The first tableau corresponds to the origin. Subsequent tableaus are developed by shifting to an adjacent corner point in the direction that yields the highest (smallest) rate of profit (cost). This process continues as long as a positive (negative) rate of profit (cost) exists.
TOPICS COVERED
Adapting to other model forms
Post optimality Analysis
Computer Implementation
Foundation of the Simplex Method

5. 1.ADAPTING TO OTHER MODEL FORMS
Equality Constraints
Negative Right Hand Sides
Functional Constraints in ≥ form
Minimization problems
There is another way to handle these constraints, with an artificial variable.

6. 1.1 “BIG M METHOD”
Solve the following linear programming problem by using the simplex method:
Min Z =2 X1 + 3 X2
S.t.
½ X1 + ¼ X2 ≤ 4
X1 + 3X2  20
X1 + X2 = 10
X1, X2  0

7. Contd… Solution Step 1: standard form Min Z, s.t.
Z – 2 X1 – 3 X2 - M A1 -M A = 0
½ X1 + ¼ X2 + S = 4
X1 + 3X S2 + A = 20
X1 + X A2 = 10
X1, X2 ,S1, S2, A1, A2  0
Where: M is a very large number

8. Contd…
Notes
M, a very large number, is used to ensure that the values of A1 and A2, …, and An will be zero in the final (optimal) tableau as follows:
1. If the objective function is Minimization, then A1, A2, …, and An must be added to the RHS of the objective function multiplied by a very large number (M).
Example: if the objective function is Min Z = X1+2X2, then the obj. function should be Min Z = X1 + X2+ MA1 + MA2+ …+ MAn OR
Z – X1 - X2- MA1 - MA2- …- MAn = 0
2. If the objective function is Maximization, then A1, A2, …, and An must be subtracted from the RHS of the objective function multiplied by a very large number (M).
Example: if the objective function is Max Z = X1+2X2, then the obj. function should be Max Z = X1 + X2- MA1 - MA2- …- MAn
Z - X1 - X2+ MA1 + MA2+ …+ MAn = 0
N.B.: When the Z is transformed to a zero equation, the signs are changed

9. Contd… Step 2: Initial tableau Basic variables X1 2 X2 3 S1 S2 A1 M A2S2 A1 M A2
RHS 1 4 -1 20 10 Z -2 -3 -M
Note that one of the simplex rules is violated, which is the basic variables A1, and A2 have a non zero value in the z row; therefore, this violation must be corrected before proceeding in the simplex algorithm as follows.

10. Contd…
To correct this violation before starting the simplex algorithm, the elementary row operations are used as follows:
New (Z row) = old (z row) ± M (A1 row) ± M (A2 row)
In our case, it will be positive since M is negative in the Z row, as following:
Old (Z row): M M
M (A1 row): M M M M o 20M
M (A2 row): M M M 10M
New (Z row):2M-2 4M M M
It becomes zero

11. Contd… The initial tableau will be: Basic variables X1 2 X2 3 S1 S2 A1S2 A1 M A2
RHS
1/2
1/4 1 4 -1 20 10 Z
2M-2
4M-3 -M
30M
Since there is a positive value in the last row, this solution is not optimal
The entering variable is X2 (it has the most positive value in the last row)
The leaving variable is A1 (it has the smallest ratio)

12. Contd… First iteration Basic variables X1 2 X2 3 S1 S2 A1 M A2 RHSS2 A1 M A2
RHS
5/12 1
1/12
-1/12
7/3
1/3
-1/3
20/3
2/3
10/3 Z
2/3M-1
1/3M-1
1-4/3M
20+10/3M
Since there is a positive value in the last row, this solution is not optimal
The entering variable is X1 (it has the most positive value in the last row)
The leaving variable is A2 (it has the smallest ratio)

13. Contd… Second iteration Basic variables X1 X2 S1 S2 A1 A2 RHS 1 -1/81
-1/8
1/8
-5/8
1/4
-1/2
1/2 5
3/2 Z
½-M
3/2-M 25
This solution is optimal, since there is no positive value in the last row. The optimal solution is:
X1 = 5, X2 = 5, S1 = ¼
A1 = A2 = 0 and Z = 25

14. Contd…
In the final tableau, if one or more artificial variables (A1, A2, …) still basic and has a nonzero value, then the problem has an infeasible solution.
All other notes are still valid in the Big M method.
In the final tableau, if one or more artificial variables (A1, A2, …) still basic and has a nonzero value, then the problem has an infeasible solution
If there is a zero under one or more nonbasic variables in the last tableau (optimal solution tableau), then there is a multiple optimal solution.
When determining the leaving variable of any tableau, if there is no positive ratio (all the entries in the pivot column are negative and zeroes), then the solution is unbounded.

15. 1.2. Two-Phase Method
Initialization: Add artificial variables to get an obvious initial solution for the artificial problem.
The objective of this phase is to find a BFS to the real problem.
We minimize the artificial problem.
Example:Artificial Problem
Minimize Z = X X6
0.3X X2 +X = 27
0.5X X X = 6
0.6X X X5 + X6 = 6
The Real problem:Minimize Z = 0.4X X2
0.5X X = 6
0.6X X X = 6

16. Contd… Min Z = X4 + X6 becomes Max -Z = – X4 – X6
Use elementary row operations to zero these out.(E.g. add or subtract multiples of the other rows until the coefficients on X4 and X6 are zero.
That gives us
–Z -1.1X X2 +X5 = -12 We perform the usual simplex maximization and
find a BFS (6,6,0,3,0,0,0)
Min Z = X X6 becomes Max -Z = – X4 – X6 Use elementary row operations to zero these out.(E.g. add or subtract multiples of the other rows until the coefficients on X4 and X6 are zero.

17. Contd…

18. Start from the final tableau of phase 1
Contd…
Start from the final tableau of phase 1
Drop the artificial variables
Substitute the phase 2 objective function
Restore proper form by eliminating the basic variables X1 and X2 from the objective row (use row operations)
Solve the phase 2 problem

19. Phase 2 Optimal Solution: X1 = 7.5, X2 = 4.5, X3 (slack)=0,
X5 (surplus) = 0.3

20. Contd…
Artificial variable method can construct a BFS when an obvious one is not available.
Using either the big M method or the two phase method allows simplex to work.
Possible pitfall: There may be no feasible solutions to the real problem, but by constructing artificial variables we may allow the simplex to proceed and report an “optimal solution”
When this happens, one of the artificial variables will have a positive value in the final solution.

21. 2.Post-Optimality Analysis
Re-optimization
Shadow prices
Sensitivity analysis
Parametric programming

22. Contd… Reoptimization:
If we have found an optimal solution for one version of an LP and then want to
Run A slightly different version, it is inefficient to start from scratch. Re-optimization
Involves deducing how changes in the original model get carried into the final
solution (our original optimal solution). The final tableau is then revised in
light of the new information and the iterations start from there. If the tableau that
emerges is not feasible, a dual method can be applied. )
Sensitivity analysis
Sensitivity parameters are those that can’t be changed without changing
the optimal solution.If a shadow price on a constraint is positive, then an increase
in the RHS of that constraint will change the optimal solution. If it is zero, there
is a range over which it can be changed with no change in the optimal solution.
Also, if a reduced cost is non-zero, there will be a range over which the related
Cj can be changed without affecting the solution

23. Contd… Parametric programming
Related to sensitivity analysis Involves the systematic study of how the optimal solution changes as many parameters change simultaneously over some range.
Trade-offs in parameter values can be studied, e.g. some resources (bi) can be increased if others are reduced.
Shadow Prices:
It is the change in the optimum value of the objective function per unit increase of the resources or the shadow prices of a resource is the unit price that is equal to increase in profit to be realized by one additional unit of the resource.

24. 3.Computer Implementation
Hand calculations of simplex method are difficult.The problem can be solved on desktop computer,personnel computer as well as work stations. The matrix form is the most suitable form to use in computers.
Software/Packages used are:
1. CPLEX
2. LINDO
It is the computer evolution that has made possible the wide spread application of linear programming in recent years.

25. 4.Foundations of the Simplex Method
Simplex method is an algebraic procedure
However, its underlying concepts are geometric
Understanding these geometric concepts helps before going into their algebraic equivalents
With two decision variables, the geometric concepts are easy to visualize
Should understand the generalization of these concepts to higher dimensions
Maximize Z = c1x1 + c2x2 + … + cnxn
subject to a11x1 + a12x2 + … + a1nxn ≤ b1
a21x1 + a22x2 + … + a2nxn ≤ b2 …
am1x1 + am2x2 + … + amnxn ≤ bm
x1 ≥ 0, x2 ≥ 0, …, xn ≥ 0

26. Contd… Constraint boundary equation
For any constraint, obtain by replacing its , =,  by =
It defines a “flat” geometric shape: hyperplane
Corner-point solution
Simultaneous solution of n constraint boundary equations
Property 1:
If there is exactly one optimal solution, then it must be a corner-point feasible solution
If there are multiple optimal solutions (and a bounded feasible region), then at least two must be adjacent corner-point feasible solutions
Property 2:
There are only a finite number of corner-point feasible solutions

27. Contd… Constraint boundary equation
For any constraint, obtain by replacing its , =,  by =
It defines a “flat” geometric shape: hyperplane
Corner-point solution
Simultaneous solution of n constraint boundary equations
Property 1:
If there is exactly one optimal solution, then it must be a corner-point feasible solution
If there are multiple optimal solutions (and a bounded feasible region), then at least two must be adjacent corner-point feasible solutions
Property 2:
There are only a finite number of corner-point feasible solutions

28. Assignment-3
Explain the typical steps in post optimality analysis in linear programming.
Solve the following LPP using Big M method.
Maximize Z= 3x1+5x2
STC x1<=4
2x2<=12
3x1+2x2 =18
x1,x2>=0
Minimize Z= 0.4x1+0.5x2
STC 0.3x x2 <=2.7
0.5x1+0.5x2 =6
0.6x x2 >=6

29. Solve the following LPP using Big M method.
Minimize Z= 2x1+x2
STC x1 + 2x2 <=4
3x1+x2 =3
4x1 + 3x2 >=6
x1,x2>=0
Solve the following LPP using Two phase method
Maximize Z= 4x1+x2
STC 3x1 + x2 =3
x1+2x2 <=4

30. Solve the following LPP using Two phase method
Minimize Z= 7.5x1-3x2
STC 3x1 - x2- x3 > =3
x1 - x2 + x3 >=2
x1,x2, x3>=0
Maximize Z= 3x1-x2
STC 2x1 + x2 > =2
x1 +3x2 <=2
x2 <=4
x1,x2>=0