Surplus Variables

The optimal solution to the M&D Chemicals problem shows that the desired total production of $A+B=350$ gallons has been achieved by using all available processing time of $2A+1B=2(250)+1(100)=600$ hours. In addition, note that the constraint requiring that product A demand be met has been satisfied with $A=250$ gallons. In fact, the production of product A exceeds its minimum level by $250-125=125$ gallons. This excess production for product A is referred to as surplus. In linear programming terminology, any excess quantity corresponding to $a$ constraint is referred to as surplus.

Recall that with $a$ constraint, a slack variable can be added to the left$-$hand side of the inequality to convert the constraint to equality form. With $a$ constraint, a surplus variable can be subtracted from the left$-$hand side of the inequality to convert the constraint to equality form. Just as with slack variables, surplus variables are given a coefficient of zero in the objective function because they have no effect on its value. After including two surplus variables, $S_1$ and $S_2,$ for the $$ constraints and one slack variable, $S_3,$ for the $$ constraint, the linear programming model of the M&D Chemicals problem becomes

Min $2A+3B+0S_1+0S_2+0S_3,$

$s.t.$

$,1A-1S_1,125$

$,1A+1B-1S_2,=350$

$,2A+1B,+1S_3,=600$

$,A,B,S_1,S_2,S_{3}0$

All the constraints are now equalities. Hence, the preceding formulation is the standardform representation of the M&D Chemicals problem. At the optimal solution of $A=250$ and $B=100,$ the values of the surplus and slack variables are as follows:

Constraint

Demand for product A

Total production

Processing time

Value of Surplus or Slack Variables

$S_1=125$

$S_2=0$

$S_3=0$

Refer to Figures $2\mathrm{.}15$ and $2\mathrm{.}16.$ Note that the zero surplus and slack variables are associated with the constraints that are binding at the optimal solution $-$ that is$,$ the total production and processing time constraints. The surplus of $125$ units is associated with the nonbinding constraint on the demand for product $A.$

In the Par, Inc., problem all the constraints were of the $$ type, and in the M&D Chemicals problem the constraints were a mixture of $$ and $$ types. The number and types of constraints encountered in a particular linear programming problem depend on