Constraints
Identify and write constraints in linear programming: translate real-world limitations into inequalities that define the boundaries of the feasible region for optimization problems.
Key Concepts
The constraints are the set of linear equations or inequalities that limit the possible solutions. These are the rules of the game! You cannot spend more money than you have or use more materials than are available. They fence in your possible moves, defining the boundaries of what is achievable in the problem.
Example 1: A factory has at most 400 hours for labor and 500 pounds of material. The constraints are 2x + 5y ā¤ 400 and 4x + 3y ā¤ 500.
Example 2: A diet requires at least 50 grams of protein and no more than 20 grams of fat. The constraints are 10pā + 5pā ā„ 50 and 2fā + 4fā ā¤ 20.
Common Questions
What are constraints in linear programming?
Constraints are the inequalities that limit the values the variables can take in an optimization problem. Each constraint represents a real-world restriction such as a budget limit, available time, or resource capacity. Together the constraints define the feasible region.
How do you write a constraint inequality from a word problem?
Identify the limiting condition and the variables involved. Translate the verbal description into a mathematical inequality. For example, 'a factory can produce no more than 200 units per day' becomes x<=200 where x is the number of units produced daily.
What types of constraints always appear in standard linear programming problems?
Non-negativity constraints (x>=0, y>=0) are standard because most real-world quantities cannot be negative. Additional problem-specific constraints cover resource limits such as labor hours, material quantities, or budget caps. All constraints together bound the feasible region.