The dual is formulated entirely from the primal.

A primal maximization model with … constraints converts to a dual minimization model with Ú constraints, and vice versa.

The Dual

A-31

where

o f t a b l e s p r o d u c e d o f c h a i r s p r o d u c e d x 2 = n u m b e r x 1 = n u m b e r

This model represents the primal form. For a primal maximization model, the dual form is a minimization model. The dual form of this example model is

minimize Z_{d }= 40y_{1 }+ 216y_{2 }+ 240y_{3 }subject to

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

The specific relationships between the primal and the dual demonstrated in this exam- ple are as follows:

1.

The dual variables, y

_{1}, y_{2}, and y_{3}, correspond to the model constraints in the primal. For every constraint in the primal there will be a variable in the dual. For example, in this case the primal has three constraints; therefore, the dual has three decision variables.2.

The quantity values on the right-hand side of the primal inequality constraints are the objective function coefficients in the dual. The constraint quantity values in the primal, 40, 216, and 240, form the dual objective function: Z = 40y

_{1 }+ 216y_{2 }+ 240y_{3}.3.

The model constraint coefficients in the primal are the decision variable coefficients in the dual. For example, the labor constraint in the primal has the coefficients 2 and

4.

These values are the y

_{1 }variable coefficients in the model constraints of the dual: 2y_{1 }and 4y_{1}.4.

The objective function coefficients in the primal, 160 and 200, represent the model constraint requirements (quantity values on the right-hand side of the constraint) in the dual.

5.

Whereas the maximization primal model has … constraints, the minimization dual model has Ú constraints.

The primal–dual relationships can be observed by comparing the two model forms shown in Figure A-9.

Now that we have developed the dual form of the model, the next step is determining what the dual means. In other words, what do the decision variables y_{1}, y_{2}, and y_{3 }mean, what do the Ú model constraints mean, and what is being minimized in the dual objective function?

Interpreting the Dual Model The dual model can be interpreted by observing the simplex solution to the primal form of the model. The simplex solution to the primal model is shown in Table A-32.

# Interpreting this primal solution, we have

s 3 = 4 8 f t . 2 o f s t o r a g e s p a c e x 2 = 8 c h a i r s x 1 = 4 t a b Z = $2,240 profit l e s