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The Optimal Simplex Tableau with c2 = 200 + ¢

c

j

Basic Variables

200 + ¢ 160 0

x x s

2 1 3

Table A-35

c j - z j z j

8

0

1

1/2

-1>18

0

4

1

0

-1>2

1/9

0

48

0

0

6

-2

1

2,240 + 8¢

160

200 + ¢

20 + ¢/2

20/3 - ¢/18

0

0

0

-20 - ¢/2

-20>3 + ¢/18

0

160

200 + ¢

0

0

0

Quantity

x1

x2

s1

s2

s3

Sensitivity Analysis

A-37

A s b e f o r e , t h e s o l u t i o n s h o w n i n T a b l e A - 3 5 w i l l r e m a i n o p t i m a l a s l o n g a s t h e c j - row values remain negative or zero. Thus, for the solution to remain optimal, we must have z j

  • -

    20 - ¢>2 0

and

  • -

    20>3 + ¢>18 0

Solving these inequalities for ¢ gives

  • -

    20 - ¢>2 0

    • -

      ¢>2 20 ¢ Ú -40

and

  • -

    20>3 + ¢>18 0 ¢>18 20>3 ¢ … 120

Thus, ¢ Ú -40 and ¢ … 120. Since c2 = 200 + ¢, we have ¢ = c2 - 200. Substi- tuting this value for ¢ in the inequalities yields

¢ Ú -40 c2 - 200 Ú -40 c2 Ú 160

and

¢ … 120 c2 - 200 120 c2 320

Therefore, the range of values for c2 over which the solution will remain optimal is 160 c2 320 The ranges for both objective function coefficients are as follows:

1 6 0 c 2 3 2 0 1 0 0 c 1 2 0 0

However, these ranges reflect a possible change in either c1 or c2, not simultaneous changes in both c1 and c2. Both of the objective function coefficients in this example were f o r b a s i c s o l u t i o n v a r i a b l e s . D e t e r m i n i n g t h e s e n s i t i v i t y r a n g e f o r a d e c i s i o n v a r i a b l e t h a t c j

is not basic is much simpler. Because it is not in the basic variable column, the ¢ change

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