Free Tutorials, Linux Command, Source Code Architecture,  Software Engineering, Intelligent Systems, RDBMS, Computer Accounting,  Operations Research, Discrete Mathematics, Network, SAD Lay Networks Lay Networks
Computer Science Networking Operating Systems Linux and Unix Source Code Script & Languages Protocols Glossary
Web laynetworks.com
Google
 


Operations Research

Que. 3        Determine the optimum solutions for the following L.P.s by enumerating all the basic solutions : -

Maximize  Z = 2x1 - 4x2 + 5x3 - 6x4 
          s.t.     x1 + 4x2 - 2x3 + 8x4 £
                   -x1 + 2x2 + 3x3 + 4x4 £ 1
                   x1, x2, x3, x4 ³ 0

Ans.

Now by using slack variables s1 and s2 the above L.P model can be expressed as :- 

Maximize Z = 2x1 - 4x2 + 5x3 - 6x
          s.t.     x1 + 4x2 - 2x3 + 8x4 + s1 = 2
                   - x1 + 2x2 + 3x3 + 4x4 + s2 = 1
                   x1, x2, x3, x4, s1, s2 ³ 0

now here the number of basic solution = nCm

where n = number of variables = 6
         m = number of constraints = 2
\
number of basic solutions = nC
                                      = 6C2
                                                6!

                                                                = (6-2)! * 2!

                                                6*5*4*3*2*1

                                                                = (4*3*2*1) * (2*1)

                                      = 15

Now, two variables may be selected as a basic variable, thus the set of basic variables can be expressed as :- 

Number of basic variables = {(x1,x2), (x1,x3), (x1,x4), (x1, s1) (x1, s2) (x2, x3) (x2, x4) (x2,s1) (x2,s2) (x3, x4) (x3,s1) (x3,s2) (x4, s1) (x4,s2) (s1, s2)}
Now if we take x1 and x2 as basic variables :

We get the basic solutions as : 

Basic variables are x1 and x2.

Non basic variables are x3 = 0, x4 = 0, s1 = 0, and s2=0.

 By placing these all values in our L.Ps model we get

   x1 + 4x2 - 2x3 + 8x4 + s1 = 2
\ x1 + 4x2 - 2(0) + 8(0) + 0 = 2
= x1 + 4x2 = 2 ------------------------------------------------------(I)

and -          x1 + 2x2 + 3x3 + 4x4 + s2 = 1
\
- x1 + 2x2 + 3(0) + 4(0) + 0 =1

\
= -x1 + 2x2 = 1 ----------------------------------------------------(II)

Now by adding equation no (I) in equation (II) we get

X2 = ½ ---------------------------------------------------------(1)

By placing the value of x2 in equation(I) we get

x1 + 4x2 = 2
x1 + 4 (1/2) = 2
\
x1 = 0.------------------------------------------------------(2)

Therefore basic solutions:
X1=0, x2 = ½ and Z = -2 

In this way we can obtain other remaining basic solutions which are mentioned in the table bellow.

SR.

No.

Basic Variables

 

1st Variable      2nd Variable

Basic Solutions

 

1st solution       2nd solutions

Value of object function(Z)

1.  

x1

x2

0

½

-2

2.  

x1

x3

8

3

31

3.  

x1

x4

0

1/4

-(3/2)

4.  

x1

s1

-1

3

-2

5.  

x1

s2

2

3

4

6.  

x2

x3

½

0

-2

7.  

x2

x4

0

1/4

-(3/2)

8.  

x2

s1

½

0

-2

9.  

x2

s2

½

0

-2

10.   

x3

x4

0

1/4

-(3/2)

11.   

x3

s1

1/3

8/3

5/3

12.   

x3

s2

-1

4

-5

13.   

x4

s1

1/4

0

-(3/2)

14.   

x4

s2

1/4

0

-(3/2)

15.   

s1

s2

2

1

0

From the above table we can see that the basic solutions and the basic variables in which we get from the serial no 4 and 12 are negative because the value of the variables are negative. 

We can also see from serial no 2 that when the value of x1 = 8, and the value of x3 = 3 we get the maximum value of the objective function (i.e. Z = 31). 
Hence the solution for the above mentioned L.P models is as follows :

x1 = 8, 
x3 = 3 and 
Z = 31 (Maximum value)



Top

Back
Next
FDDI Frequently Asked Questions (FAQ), The function and frame format of FDDI,Aloha,Comparative analysis between two types of ATM Switches,Knockout Switch,Barcher-Banyan Switch,Various popular standards for compressing multimedia data,Distributed Multimedia Survey: Standards, ASCII to hex value chart,Comparative analysis - TCP - UDP, Addressing Formats and QoS parameters, Bellman Ford's Algorithm Lay networks, free, java, java script, asp, vb, linux, ignou, tutorial, Unix commands, System Analysis, System Design, Ipv6, quiz, download, free, Computer Architecture, Object Oriented System, Relational Database Management Systems, Object Oriented System, Operating Systems, Software Engineering, Communications and Networks, Discrete Mathematics, Intelligent Systems, Operations Research, Accounting and Finance on Computersmca, networking, protocols, glossary, assignment, project, tma, programming source code, programming, source code, unix, free
 
Book Mark/Share this site at BlinkBits BlinkList Blogmarks co.mments Delicious Digg Fark Furl it! Google Ma.gnolia Netvouz NewsVine RawSugar Reddit Shadows Simpy Stumble Technorati YahooMyWeb

Copyright © 2000- 2007 Lay Networks All rights reserved. 
This website is best viewed in Firefox 1.0.1 above.

Web Hosting sponsored by Customized Software Company India
Web Site Designed by Web Designing, Flash Animation, Multimedia Presentations, Broacher/catalogue designing, Web Promotion 
Refer to your freind About Us Legal IGNOU Contact Us Feedback Donate to laynetworks.com Download Management Tutorials Tutorials History Search here