Introductory parts of Simplex Method, Write LPP in canonical form, Flow chart of simplex method

1. Linear Programming Methods
or
Linear Programming Problems Using Simplex
Method
Presented by
Dr. Devyanee K. Nemade
Assistant Professor
Department of Agricultural Economics and
Statistics, Dr. PDKV, Akola

2. Simplex Method

3. Introduction:
In previous chapter we have seen that, if a LPP
involves only two variables then it can be solved by using
graphical method. But if a given LPP involves more than
two variables then graphical method fails to solve it.
In this chapter we will learn a method known as
simplex method by which we can solve LPP with two or
more than two variables.
Simplex method is also called simplex technique or
simplex algorithm was develop by G.B. Dantzig, an
American Mathematician. Thus method is applicable
when the LPP is expressed in standard form.

4. Continue……………..
The Canonical Form:
The LPP is said to be in canonical form if,
a. The objective function is of the maximization type.
b. All constraints are ≤ type &
c. All the decision variables are non negative

5. Flow Charts of Simplex Method
Set up initial Simplex table
Compute Zj and Cj – Zj values.
Minimize Maximum
Cj – Zj
This Sol. is Cj – Zj
–ve value exist Optimal +ve value exist
Select key column with Select key column with largest
largest –ve value +ve value
Select key Row with Min Xb/aij
Identify key element at the intersection of key row and key column
or
The key the starting equation in balance and provide mathematical trick
for getting starting solution.
Is it LP problem Max Or Min
type?

6. Steps to Convert LPP in Standard Form:
A. Covert the given general LPP into standard LPP.
• Objective function : LPP must be maximized. If it is to be
minimized then we convert it into a problem of maximization
by Max Z = - Min Z.
• Check all the decision variable are greater than zero.
• Express the problem in standard form by introducing slack
or surplus variable to convert the inequality constraints into
equation.
• The slack & surplus variables denotes unused amount.
Therefore we put coefficient of such variable as zero in the
objective function.
• All the values of right hand side must be positive.

7. Continue……………
B. Write the values of initial basic feasible solution.
C. Write the standard form LPP into matrix form.
D. Construct the initial simplex table.
E. Calculate the values of Zj - Cj and check the basic feasible
solution of optimality.
i. Zj - Cj
ii. If all (Zj - Cj )is the optimal solution will obtained.
iii. If at least one (Zj - Cj ) ≥ 0, is –ve then indicate by an arrow and
their column is called key column.

8. 1. Convert the following LPP to canonical form,
Maximize Z = 5X1 + 3X2
Subject to
X1 - 3X2 = 2
- X1 + X2 ≥ 1
Non – negative constraint
X1 & X2 ≥ 0
Solution:
Canonical form of given LPP is as follows:
Maximize Z = 5X1 + 3X2
Subject to
- X1 + 3X2 ≤ - 2
X1 - X2 ≤ −1
X1 & X2 ≥ 0

9. 2. Convert the following LPP to canonical form,
Maximize Z = 3X + 5Y
Subject to
X - 3Y = 4
- X + Y ≥ 1
Non – negative constraint
X & Y ≥ 0
Solution:
Canonical form of given LPP is as follows:
Maximize Z = 3X + 5Y
Subject to
- X1 + 3Y ≤ - 4
X - Y ≤ −1
X & Y ≥ 0

10. 3. Convert the following LPP to canonical form,
• Maximize Z = 4X1 + 2X2+ 3X3
• Subject to
2X1 + 4X2 ≤ 5
2X1 + 2X2+ X3 ≥ - 4
Non – negative constraint
X1, X2 & X3 ≥ 0
Solution:
Maximize Z = 4X1 + 2X2+ 3X3
2X1 + 4X2 ≤ 5
- 2X1 - 2X2 - X3 ≤ 4
Non – negative constraint
X1, X2 & X3 ≥ 0