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Linear Programming

Sensitivity Analysis

or

Post Optimality Analysis

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Post Optimality Analysis Max Z = 25x1 + 40x2 + 50x3

S/t. 5x1 + 20x2 + 25x3 400

10x1 + 15x2 + 20x3 500

x1, x2, x3 0

25 40 50 0 0RHS

x1 x2 x3 w1 w2

40 x2 0 1 6/5 2/25 1/25 12

25 x1 1 0 1/5 3/25 4/25 32

0 0 3 1/5 12/5 1280

Final Table :

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[ 1 ] Change in Cjxj non-basic

xj basic

[ 2 ] Change in RHS

[ 3 ] Change in aijxj non-basicxj basic

[ 4 ] Addition of a new variable

[ 5 ] Addition of a new constraint

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Therefore in general :

(Ij)xj non-basic

25 40 50 0 0

RHSx1 x2 x3 w1 w2

40 x2 0 1 6/5 2/25 1/25 12

25 x1 1 0 1/5 3/25 4/25 32

0 0 3 1/5 12/5 1280

[ 1 ] ( a ) Cj w.r.t. xj non-basic

To retain optimality : 3 0

3

+

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25 40 50 0 0RHS

x1 x2 x3 w1 w2

40 x2 0 1 6/5 2/25 1/25 12

25 x1 1 0 1/5 3/25 4/25 32

0 0 3 1/5 12/5 1280

[ 1 ] ( b ) Cj w.r.t. xj basic

To retain optimality :

3+6 /5 0

1/5+2 /25 0

12/51 /25 0

5/2 60

+

+

+6 /5 +2 /25 1 /25

5/2

5/2

60

+12

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Therefore in general : for Cj w.r.t. xj basic

( ) ( )j j

j j

ij x ij x

I Ia a

Maxm

of

aij > 0

Minm

of

aij < 0

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25 40 50 0 0RHS

x1 x2 x3 w1 w2

40 x2 0 1 6/5 2/25 1/25 12

25 x1 1 0 1/5 3/25 4/25 32

0 0 3 1/5 12/5 1280To get for x2 :

3/(6/5), (1/5)/(2/25), (12/5)/(-1/25)

5/2,5/2 , 60

5/2 60

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25 40 50 0 0RHS

x1 x2 x3 w1 w2

40 x2 0 1 6/5 2/25 1/25 12

25 x1 1 0 1/5 3/25 4/25 32

0 0 3 1/5 12/5 1280To get for x1 :

3/(1/5), (1/5)/(-3/25), (12/5)/(4/25)

5/3,15 , -15

15 5/3

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Q.1 What will happen if C3

is changed to :

( i ) 45 ( ii ) 52 ( iii ) 55

Q.2 What will happen if C1 is changed to :

( i ) 15 ( ii ) 5 ( iii ) 30

Q.3 What will happen if C2 is changed to :

( i ) 35 ( ii ) 50 ( iii ) 110

Answer the following questions.

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To retain optimality :

32- 3 /25 0

150

800 / 3

150 800/3

12+ 2 /25 0

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Change in RHS.

Maxm

ofaij > 0

Minm

ofaij < 0

( ) ( )j

i i

ij ij x

R R

a a

Where aij Column corresponding to slack variable w.r.t. ith

constraint

For R1 12/(2/25), 32/( 3/25) = 150, 800/3

150, R1 800/3

For R2 12/( 1/25), 32/(4/25) = 300, 200

200, R1 300

To get range of R for which basic variables are retained :

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Change in aij w.r.t. non-basic variable

&

Addition of a new variable

2 1

2525 25

3 4 20

25 25

6

5

1

5

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What will happen if is changed to ?25

20

10

10

2 1

1025 25

3 4 10

25 25

0

1

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25 40 50 0 0RHS

x1 x2 x3 w1 w2

40 x2 0 1 0 2/25 1/25 12

25 x1 1 0 1 3/25 4/25 32

0 0 25 1/5 12/5 1280

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25 40 50 0 0RHS

x1 x2 x3 w1 w2

40 x2 0 1 0 2/25 1/25 12

50 x3 1 0 1 3/25 4/25 32

25 0 0 14/5 32/5 2080

x1 x2 x3 w1 w2 RHS

0 w1 150

50 x3 50

25 45 0 0 5 2500

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Thank you

For any Query or suggestion :Contact :Dr. D. B. Naik

Professor, Training & Placement SectionSardar Vallabhbhai National Institute ofTechnology, Surat

Ichchhanath, Surat395007, Gujarat.

Email: dbnaik_svr@yahoo.com

Phone no 0261-2255225 (Direct)