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The sensitivity range for a qi value is the range of values over which the right-hand-side values can vary without changing the solution variable mix, including slack variables and the shadow prices.

Table A-36 The Initial Simplex Tableau

A ¢ in a qi value is duplicated in the si column in the final tableau.

Sensitivity Analysis

A-39

s 1 a n d s 2 = 0 s 3 = 1 2 Z = $2,360

Thus, a change in a qi value can change the values of the optimal solution. At some point an increase or decrease in qi will change the variables in the optimal solution mix, including the slack variables. For example, if q2 increases to 240 board feet, then the optimal solution point will be at

s 1 a n d s 2 = 0 s 3 = 0 x 2 = 6 . 6 7 x 1 = Z = $2,400 6 . 6 7

Notice that s3 has left the solution; thus, the optimal solution mix has changed. At this point, where q2 = 240, which is the upper limit of the sensitivity range for q2, the shadow price will also change. Therefore, the purpose of sensitivity analysis is to determine the range for qi over which the optimal variable mix will remain the same and the shadow price will remain the same.

A s i n t h e c a s e o f t h e v a l u e s , t h e r a n g e f o r c a n b e d e t e r m i n e d d i r e c t l y f r o m t h e o p t i - q i c mal simplex tableau. As an example, consider a ¢ increase in the number of labor hours. The model constraints become j

2 4 x 1 + 1 2 x 2 2 4 0 + 0 ¢ 1 8 x 1 + 1 8 x 2 2 1 6 + 0 ¢ 2 x 1 + 4 x 2 4 0 + 1 ¢

Notice in the initial simplex tableau for our example (Table A-36) that the changes in the quantity column are the same as the coefficients in the s1 column.

s

1

s

2

s

3

c j

0 0 0

Basic Variables

c j - z j z j

160

Quantity

x1

40 + 1¢

2

216 + 0¢

18

240 + 0¢

24

0

0

160

200

0

x2

s1

4

1

18

0

12

0

0

0

200

0

0

0

s2

s3

0

0

1

0

0

1

0

0

0

0

This duplication will carry through each subsequent tableau, so the s1 column values will duplicate the ¢ changes in the quantity column in the final tableau (Table A-37).

In effect, the ¢ changes form a separate column identical to the s1 column. Therefore, to determine the ¢ change, we need only observe the slack (si) column corresponding to the model constraint quantity (qi) being changed.

Recall that a requirement of the simplex method is that the quantity values not be nega- tive. If any qi value becomes negative, the current solution will no longer be feasible and a new variable will enter the solution. Thus, the inequalities

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