Welcome! In this tutorial,
we calculate slack and surplus variables, identify binding constraints,
and write the linear programming model in standard form.
Let’s begin with this LP model. Here is a graph of the model showing the constraint
lines and the feasible region. The optimal solution occurs at the intersection
of the first two constraints, at the point X = 24, and Y = 12.
The constraints that touch or intersect at the optimal solution point are called “binding
constraints”. So the first 2 constraints are binding and
the third one is non-binding. Now let’s see what happens when we plug
the optimal solution point into the left hand side of each constraint.
For resource 1, plugging in X = 24 and Y = 12 gives 60. So the left side equals the right
side. This tells us that resource 1 was fully utilized
at optimal solution. That is, there is no slack.
The term Slack refers to the amount of a resource that is unused and it applies only to “less
than or equal to” constraints. It is basically the amount of “underutilization”
or amount “left over”. So for this constraint we have a slack of
zero. That is, all the available 60 units were utilized
at optimal solution. Plugging the optimal solution into resource
2, we have 120. That is, all available 120 units of 2 resource
were utilized so we have a slack of 0 again. Recall that the first 2 constraints are binding.
It follows that if a “less than or equal” constraint is binding, its slack value will
be 0. The third constraint has a “greater than
or equal sign”. On plugging in X, we 24 ≥ 10.
That is, we are required to produce at least 10 units of X and we produced 24 which is
14 more than 10.
Therefore, because this is a greater than constraint, we will say we have a surplus
of 14. “Surplus” simply refers to the amount
by which we exceed the minimum requirement and it applies only to “greater or equal
to” constraints.
Let‘s now write the LP model in standard form. That is, we convert the inequalities
into equalities by adding slack and surplus variables.
Since the first constraint is a “less than or equal” constraint, we introduce a slack
variable and add it to the left side of the constraint.
So for the first constraint we write X + 3Y + s1 = 60, where s1 is the slack variable.
To write the second constraint in standard form we add the slack variable s2 and change
the less than sign to equal sign. Since that the third constraint is a “greater
or equal” constraint, we introduce a surplus variable and subtract it from the left side
of the constraint. Now slack and surplus variables are always
non-negative. So the non-negativity constraints are written
to include the slack and surplus variables. The slack and surplus variables do not affect
the objective function in any way, so we usually just include them in the objective function
with 0 coefficients. The standard form of the LP is now complete.
Let’s see another example. Suppose the optimal solution to this LP model
occurs at X = 3, and Y = 4. We want to write it in standard form,
calculate slack and surplus values, and determine which constraints are binding.
Let’s first write it in standard form. The first constraint has a “greater or equal”
sign. So we introduce a surplus variable with a negative sign as we convert the inequality
to equality. The second constraint has a “less or equal”
sign. So we introduce a slack variable with a positive sign.
The 3rd constraint will also have a surplus variable with negative sign.
The 4th constraint is an equality constraint. Since the purpose of writing the model in
standard form is to convert inequalities to equalities, this constraint is already in
standard form because it contains equality already. So no slack or surplus variable is
needed for this fourth constraint. Here are the non-negativity constraints.
And here is the objective function. We now use this standard form of the constraints
to obtain their slack and surplus values. Recall that the optimal solution is X =3,
and Y = 4. So for the first constraint, we plug in the
optimal solution point to give 20 – s1 = 20. That is s1 = 0.
Plugging the optimal solution into the second constraint we have -2 + s2 = 6.
So s2 = 8. For the 3rd constraint, we have 3 + 4 – s3
= 4. So s3 =3. The fourth constraint has neither slack nor
surplus. So which of these four constrains are binding?
Constraint 1 is binding because it has a surplus of 0.
Constraint 2 is non-binding because it has a non-zero slack of 8.
Constraint 3 is non-binding because it has a slack of 3.
Constraint 4 is binding because it neither has a slack nor surplus. Or we just can say
that the slack or surplus is 0. In other words an equality constraint will
always be binding at optimal solution. Therefore, only constraints 1 and 4 are binding.
And there you have it. Thanks for watching.