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Sensitivity Analysis

A-35

T h e c o e f f i c i e n t s i n t h e o b j e c t i v e f u n c t i o n w i l l b e r e p r e s e n t e d s y m b o l i c a l l y a s ( t h e s a m e c notation used in the simplex tableau). Thus, c1 = 160 and c2 = 200. Now, let us consider a h a n g e i n o n e o f t h e v a l u e s b y a n a m o u n t . F o r e x a m p l e , l e t u s c h a n g e b y c 1 = 1 6 0 ¢ c j c j

¢ = 90. In other words, we are changing c1 from \$160 to \$250. The effect of this change on the solution of this model is shown graphically in Figure A-10.

Figure A-10 A change in c1

x2

20

15

10

c 1 = 2 5 0

A

B

5

c1 = 160

C

0

D

5

10

15

20

x1

The sensitivity range for a cj value is the range of values over which the current optimal solution will remain optimal.

Originally, the solution to this problem was located at point B in Figure A-10, where x1 = 4 and x2 = 8. However, increasing c1 from \$160 to \$250 shifts the slope of the objec- tive function so that point C (x1 = 8, x2 = 4) becomes the optimal solution. This demon- strates that a change in one of the coefficients of the objective function can change the optimal solution. Therefore, sensitivity analysis is performed to determine the range over w h i c h c a n b e c h a n g e d w i t h o u t a l t e r i n g t h e o p t i m a l s o l u t i o n . c j

T h e r a n g e o f t h a t w i l l m a i n t a i n t h e o p t i m a l s o l u t i o n 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 c the optimal simplex tableau. The optimal simplex tableau for our furniture company example is shown in Table A-33. j

200 160 0

x x s

2 1 3

8 4 48

0 1 0

1 0 0

1/2 -1>2 6

-1>18 1/9 -2

0 0 1

z c -z

j

j

j

2,240

160 0

200 0

20 - 20

20/3 -20>3

0 0

0

0

s2

s3

Basic

160

200

0

Variables

Quantity

x1

x2

s1

Table A-33 The Optimal Simplex Tableau

c j

¢ is added to cj in the optimal simplex tableau.

n First, consider a ¢ change for c1. This will change the c1 value from c1 = 160 to c1 = 160 + ¢, as shown in Table A-34. Notice that when c1 is changed to 160 + ¢, the e w v a l u e i s i n c l u d e d n o t o n l y i n t h e t o p r o w b u t a l s o i n t h e l e f t - h a n d c o l u m n . T h i s i s c j c because x1 is a basic solution variable. Since 160 + ¢ is in the left-hand column, it j

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