LINEAR PROGRAMMING –SIMPLEX METHOD


Objectives

Formulate the standardized LPP
b. Identify the use of slack, surplus and artificial variables
c. Identify the solving method
d. Formulate the initial simplex table
e. Calculate the tables for each iteration
f. Calculate the shadow costs/profits

Example-1
The LPP is maximize Z = 6x1 + 4x2
Subject to the constraints

Solution:

Introducing slack variables, the above problem becomes,
Maximize z = 4x1 + 3x2 + 0S1 + 0S2 + 0S3
Subject to



 




Example-2
                Maximize Z = 6x1 + 4x2
Subject to the constraints

Solution:

Convert inequality constraints into equality constraints by introducing slack
Variables.
 The LPP is
Max. Z = 6x1 + 4x2 + 0.S1 + 0.S2 + 0.S3
subject to
2x1 + x2 + S1 = 390
3x1 + 3x2 + S2 = 810
X2 + S3 = 200
x1, x2, S1, S2, S3 0


 





Example:3

Solve the following linear programming problem by simplex method.
Minimize Z = 16x1 + 16x2
Subject to



Solution:

Convert inequality constraints into equality constraints by introducing surplus and
artificial variables.
The LPP is minimize Z = 16x1 + 16x2 + 0.S1 + 0.S2 + MA1 + MA2
Subject to
2x1 + 4x2 – S1 + A1 = 3
3x1 + 2x2 – S2 + A2 = 4
x1, x2, S1, S2, A1, A2 0
from the following table we can conclude  that all the Zj – Cj 0.
With Min Z =22
 x1 =5/4,;
x2 =1/8

 
Previous University Examination Questions:


1)     Define a feasible region.
2)    Define a feasible solution
3)     What is a redundant constraint
4)    Define optimal solution
5)    What is the difference between feasible solution and basic feasible solution
6)    Define the following:
(a) Basic solution
(b) non-degenerate solution
(c) degenerate solution.
(a) Basic solution
7)    Define unbounded solution
8)    What are the two forms of a LPP?
        
    
9)      When does the simplex method indicate that the LPP has unbounded solution?
10)  What is meant by optimality?
11)  How will you find whether a LPP has got an alternative optimal solution or not, from the optimal simplex table?
12)  What are the methods used to solve an LPP involving artificial variables?
13)  Define artificial variable.
14)  When does an LPP possess a pseudo-optimal solution?
15)  What is degeneracy?                                       
16)  Solve Graphically Following LPP

Related Topics
Linear Programming Problem
Linear Programming: Graphical Method

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