The sensitivity range for a q_{i }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 q_{i }value is duplicated in the s_{i }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 q_{i }value can change the values of the optimal solution. At some point an increase or decrease in q_{i }will change the variables in the optimal solution mix, including the slack variables. For example, if q_{2 }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 s_{3 }has left the solution; thus, the optimal solution mix has changed. At this point, where q_{2 }= 240, which is the upper limit of the sensitivity range for q_{2}, the shadow price will also change. Therefore, the purpose of sensitivity analysis is to determine the range for q_{i }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 s_{1 }column.

s

1

s

2

s

3

c j

0 0 0

Basic Variables

c j - z j z j

160

Quantity

x_{1 }

40 + 1¢

2

216 + 0¢

18

240 + 0¢

24

0

0

160

200

0

x_{2 }

s_{1 }

4

1

18

0

12

0

0

0

200

0

0

0

s_{2 }

s_{3 }

0

0

1

0

0

1

0

0

0

0

This duplication will carry through each subsequent tableau, so the s_{1 }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 s_{1 }column. Therefore, to determine the ¢ change, we need only observe the slack (s_{i}) column corresponding to the model constraint quantity (q_{i}) being changed.

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