The Simplex Solution Method
Now let us look again at the dual form of the model: minimize Zd = 40y1 + 16y2 + 240y3 subject to
The dual variables equal the marginal value of the resources, the shadow prices.
y 1 , y 2 , y 3 , Ú 0 4 y 1 + 1 8 y 2 + 1 2 y 3 Ú 2 0 0 2 y 1 + 1 8 y 2 + 2 4 y 3 Ú 1 6 0
Given the previous discussion on the value of the model resources, we can now define the decision variables of the dual, y1, y2, and y3, to represent the marginal values of the resources:
y 3 = m a r g i n a l v a l u e o f 1 f t . 2 o f s t o r a g e s p a c e = $ 0 y 2 = m a r g i n a l v a l u e o f 1 b d . f t . o f w o o d = $ 6 . 6 7 y 1 = m a r g i n a l v a l u e o f 1 h r . o f l a b o r = $ 2 0
The dual provides the decision maker with a basis for deciding
how much to pay for more resources.
Use of the Dual The importance of the dual to the decision maker lies in the information it provides about the model resources. Often the manager is less concerned about profit than about the use of resources because the manager often has more control over the use of resources than over the accumulation of profits. The dual solution informs the manager of the value of the resources, which is important in deciding whether to secure more resources and how much to pay for these additional resources.
If the manager secures more resources, the next question is, How does this affect the original solution? The feasible solution area is determined by the values forming the model constraints, and if those values are changed, it is possible for the feasible solution area to change. The effect on the solution of changes to the model is the subject of sensitivity analysis, the next topic to be presented here.
In this section we will show how sensitivity ranges are mathematically determined using the simplex method. While this is not as efficient or quick as using the computer, close examination of the simplex method for performing sensitivity analysis can provide a more thorough understanding of the topic.
Changes in Objective Function Coefficients To demonstrate sensitivity analysis for the coefficients in the objective function, we will use the Hickory Furniture Company example developed in the previous section. The model for this example was formulated as
maximize Z = $160x1 + 200x2 subject to
o f l a b o r o f w o o d o f s t o r a g e s p a c e x 1 , x 2 Ú 0 2 4 x 1 + 1 2 x 2 … 2 4 0 f t . 2 1 8 x 1 + 1 8 x 2 … 2 1 6 b d . f t . 2 x 1 + 4 x 2 … 4 0 h r .
n u m b e r o f t a b l e s p r o d u c e d n u m b e r o f c h a i r s p r o d u c e d x 2 = x 1 =