1. Meaning of Constraints
Constraints are the restrictions or limits within which a decision must be made in a linear programming problem. They describe the boundaries created by limited resources such as materials, money, time, or labour.
In an LPP, constraints are written as linear equations or inequalities involving the decision variables.
Typical forms look like:
\( a_1x + a_2y \le b \)
\( a_1x + a_2y \ge b \)
\( a_1x + a_2y = b \)
2. Types of Constraints
Constraints usually fall into three common types depending on the meaning of the limitation.
2.1. Less Than or Equal To (\( \le \)) Constraints
These constraints represent limits that cannot be exceeded, such as maximum hours available or maximum material supply.
Example:
\( 3x + 2y \le 18 \)
This could mean that the total machine-time demand for making \( x \) and \( y \) units cannot exceed 18 hours.
2.2. Greater Than or Equal To (\( \ge \)) Constraints
These appear when something must meet a minimum requirement, like minimum production or minimum demand.
Example:
\( x + y \ge 20 \)
This could mean at least 20 units must be produced.
2.3. Equality Constraints (\( = \))
These occur when something must be exactly a certain amount. They are less common but appear in problems with fixed totals.
Example:
\( 2x + y = 10 \)
This might represent a fixed mixture requirement.
2.4. Non-Negativity Constraints
Since we cannot produce or use negative quantities, we always include:
\( x \ge 0, \quad y \ge 0 \)
3. Writing Constraints from Real Situations
To form constraints, the limitations in a real situation must be translated into inequalities involving decision variables.
Here is a simple process:
- Identify the resources that are limited (materials, time, labour).
- Write how each variable uses the resource.
- Set the total usage less than or equal to the available amount.
3.1. Example: Machine Time Limitation
Suppose product A requires 3 hours and product B requires 2 hours on a machine. The machine is available for 30 hours.
Let:
- \( x \) = units of A
- \( y \) = units of B
Constraint is:
\( 3x + 2y \le 30 \)
3.2. Example: Material Requirement
If each unit of A uses 1 kg of material and each unit of B uses 4 kg, and total material available is 40 kg:
\( x + 4y \le 40 \)
3.3. Example: Minimum Production Requirement
If at least 10 units must be produced in total:
\( x + y \ge 10 \)