Slack variables are an essential component of linear programming and acquiring proficiency in their utilization can significantly improve your ability to solve problems. These variables are inserted into inequalities to convert them into equalities and are employed in linear programming designs to accommodate excess resources or slack.
What are Slack Variables?
Imagine you’re working with a system of inequalities and you want to find the optimal solution. You can convert these inequalities into equalities by adding what is known as “slack” variables. These variables measure the difference between the resources used and the total resources available, highlighting the “slack” or “surplus” resource.
How to Add Slack Variables
The process of introducing slack variables is fairly straightforward. For each inequality, add a slack variable to the left-hand side to make it equal to the right-hand side. For example, if you have the inequality:
x1 + 2×2 <= 10
You would introduce a slack variable (s1) like so:
x1 + 2×2 + s1 = 10
It’s important to note that the value of a slack variable is always non-negative (s1 >= 0), as it represents the unused portion of a resource.
Using Slack Variables in Linear Programming
Now, let’s see how we can apply slack variables in a simple linear programming problem. Suppose we have the following problem:
Maximize Z = 3×1 + 2×2
Subject to:
x1 + 2×2 <= 10
3×1 + 2×2 <= 18
x1, x2 >= 0
We can transform the constraints into equalities by adding slack variables:
x1 + 2×2 + s1 = 10
3×1 + 2×2 + s2 = 18
Now, we can solve this system of equalities using your preferred method, such as the Simplex method, to find the optimal values of x1, x2, s1, and s2 that will maximize Z.
Conclusion
Slack variables are powerful tools in linear programming. They allow us to convert inequalities into equalities, making them easier to work with. By understanding how to use slack variables effectively, you can solve complex optimization problems with ease.
