# A-42

# Module A

# The Simplex Solution Method

The value 360 can be eliminated, because q_{2 }cannot exceed 240. Thus, the range over which the basic solution variables will remain the same is

180 … q_{2 }… 240

# The range for q_{3 }is

192 … q_{3 }6 q

The upper limit of q means that q_{3 }can increase indefinitely (without limit) without changing the optimal variable solution mix in the shadow price.

Sensitivity analysis of constraint quantity values can be used in conjunction with the dual solution to make decisions regarding model resources. Recall from our analysis of the dual solution of the Hickory Furniture Company example that

y 3 = $ 0 , m a r g i n a l v a l u e o f s t o r a g e s p a c e y 2 = $ 6 . 6 7 , m a r g i n a l v a l u e o f w o o d y 1 = $ 2 0 , m a r g i n a l v a l u e o f l a b o r

The shadow prices are valid only within the sensitivity range for the right-hand-side values.

Because the resource with the greatest marginal value is labor, the manager might desire to secure some additional hours of labor. How many hours should the manager get? Given that the range for q_{1 }is 32 … q_{1 }… 48, the manager could secure up to an additional 8 hours of labor (i.e., 48 total hours) before the solution basis changes and the shadow price also changes. If the manager did purchase 8 more hours, the solution values could be found by observing the quantity values in Table A-37:

s 3 = 4 8 + 6 ¢ x 1 = 4 - ¢ > 2 x 2 = 8 + ¢ > 2

# Since ¢ = 8,

x_{2 }= 8 + (8)>2

= 12 x_{1 }= 4 - (8)>2

=0 s_{3 }= 48 + 6(8)

= 96

Total profit would be increased by $20 for each extra hour of labor:

Z = $2,240 + 20¢ = 2,240 + 20(8) = 2,240 + 160 = $2,400

In this example for the Hickory Furniture Company, we considered only … constraints in determining the sensitivity ranges for q_{i }values. To compute the q_{i }sensitivity range, we observed the slack column, s_{i}, since a ¢ change in q_{i }was reflected in the s_{i }column. However, recall that with a Ú constraint we subtract a surplus variable rather than adding a slack variable to form an equality (in addition to adding an artificial variable). Thus, for a Ú constraint we must consider a - ¢ change in q_{i }in order to use the s_{i }(surplus) column to perform sensitivity analysis. In that case sensitivity analysis would be performed exactly