## Info

Here, V1 and V2 are the shadow prices for resources r1 and r2, respectively. Because r1 and r2 represent the quantities of the two resources available, the objective function measures the total implicit value of the resources available. Recalling the interpretation of a11 and a21 from the primal, it is obvious that a11V1 + a21V2 is the total value of inputs used to produce one unit of output Q1. Similarly, a12V1 + a22V2 is the total value of inputs used in production of a unit of output Q2, and a13 V1 + a23 V2 is the total value of inputs used in production of a unit of output Q3.

Finally, the primal and dual linear programming problems can be fully specified through the introduction of slack variables. Remember that with less-than-or-equal-to constraints, the left side of the constraint equation must be brought up to equal the right side. Thus, slack variables must be added to the left side of such constraint equations. With greater-than-or-equal-to constraints, the left side of the constraint equation must be brought down to equal the right side. Thus, slack variables must be subtracted from the left side of such constraint equations. With this, the full specification of the preceding primal and dual linear programs can be written as follows:

Primal Problem

Dual Problem

Minimize n*

subject to a11 Qj

+ a12Q2 + a13Q3

+

S1 =

r1

subject to a11 V1 + a21V2 -

L1

= n

a21Q1

+ a22Q2 + a23Q3

+

S2 =

r2

a12V1 + a22V2 -

L2

= n 