Operation Research Questions and Solutions

c Operation Research

Assignment c Question one Maximize Z = 16X1 + 15X2 Subject to:

40X1 + 31X2 ч 124 -X1 + X2 X1

ч1 ч3

X1,X2 ш 0 (a)c Solve the problem using graphical method: c
c 
cc
c c
c 
c c
c c cc c ccc c
cc c c c 
ccccccccccc c
c 
c c
c c c
c c ccc c
c c c c 
cccccccccccccc cc 
c cc c cc c cc cc c c c c cc c

(b)c Confirm the number of corner points feasible solution using the following formula: è



  




Where (m) and (n) are the number of equations and number of variables respectively c 
cc
cc
c c
c c

cccc


ccccccccc c
c

ccccccccccccccccccc

c
ccccccccccc cc

c c

c ›c c cc ›c Yc cc ›c Yc cc cc ›c Y c c
c c
c ›c  c c
c cc ›c

Y c c
c cc

c 

 ccX
    c c c c c c c c

Question Two: Solve the above linear programming problem using the simplex Method: Îc Îc Îc Îc Îc

Maximize Z = 16X1 + 15X2 40X1 + 31X2 ч 124 -X1 + X2 ч 1 X1 ч 3 X1,X2 ш 0

subject to:

c (1)c Start the simplex tableau from the following data: Ôc c

cc
c cc c

!" #c$ !Yc


cc
cc
c c
c 
ccccccccc c
c 
ccccccccccc cc c lasic

Z

X1

X2

S1

S2

S3

Soln.

Z


c


c


c

c

c

c

c

z-row

S1

c

c


c


c

c

c


c

S1-row

S2

c


c


c

c


c

c


c

S2-row

S3

c


c

c

c

c


c

c

S3-row

›c ƒ!Ycc# %c X1,X2cc& !c# %c ›c lc# %'c S1, S2, S3)c c c ntering value (X1)'c(!%cc (c# %c) (c (c! c *c!++Y cYc (cÔc !)cc 
c

lasic

ntering (X1)

Solution

S1

c


c

S2


c


c


c ,Y! cY, #c

S3


c

c

c Yc

Conclusion'cX1cY  cYcS3c%#c c

Ratio = (soln/entering)  

cc

c *%c!+c
c ccY c !cc c! Y c lc At point (l):c (c

Y cc# %cc c $ !Yc cc !cc& !c#%cYc !c (c%#Y,c# %c c c c c c -Y  c .#! c !%Y lasic

Z

X1

X2

S1

S2

S3

Soln.

Z


c


c


c

c

c

c

c

S1

c

c


c


c

c

c


c

S2

c


c


c

c


c

c


c

S3

c

c Îc .#! c%Y c c
c c c


c

c

c

c


c

c

/#c .#! c

!)c

6ivot row:
c 
cY  Y,c# %c)%%c %cc%#Y,c# %c c c ƒ)c#! c !)c Y)cc
c c0

Y c#! c !)c 

Y ccc1c#! c%Y c c
cc Other rows including (Z) : ƒ)c !)c c0

Y c !)c c  c#! c!%Yc!++Y ccY)c#! c !)c

Z

X1

X2

S1

S2

S3

Soln.


 
c c


 

c


 
c

 
c

 
c

 

c

 
c

lasic

Z S1 S2 X1


c

c


c

c

c


c

2c

 c

 
c


 c


 c

 c

 
c


 c

c

c


c


c

c

c

c

 
c


 

c


 
c

 
c


 
c

 

c


 
c

c

c


c

c


c


c

c

3
c


3
c

3
c

3
c

3
c


3
c

3
c

c


c

c

c

c


c

c

c ›c ƒ)clc# %'c X1, S2, S3)c ›c ƒ)cY!Yc# %' (X2,S1)c c

lasic

ntering (X2)

Solution

S1


c

c

S2


c

c

c ,Y! c

X1

c

c

4c ,Y! c

Conclusion'cX2cY  cYcS1c%#c c Îc ƒ)c#! c%Y c c
c c c

Ratio = (soln/entering) § 

c Yc

lasic

Z X2 S2 X1

Z

X1

X2

S1

S2

S3

Soln.


 
c c

 
c


 

c

 

3
c

 
c


 
3
c

2 
3
c


c

c

c


3
c

c


3
c


23
c

3
c

3
c


3
c


3
c

3
c

3
c

3
c

c

c


c


3
c

c

3
c

3
c


c


c




c



3
c



c



3
c


3
c

c

c

c


3
c


c

5
3
c


3
c

 c


 c

 c

 c

 c


 c

 c

c


c

c

c

c


c

c

c ÎcÔ !)c!++Y c%Y
c) (c (cY!Ycc# %c
cc c%%c! #c (Ycthis tableau is optimal  X1 = 3 X2 = 4/31 Z = 1548/31 6c!%cc (c! c!Y c