1. Introduction
In many linear programming problems, the conditions are not written as equalities but as inequalities. For example, a resource cannot be used more than a certain amount, or time available is at most a fixed number of hours. These conditions are called inequality constraints. They describe a whole region of possible values instead of a single line.
These notes collect all the basic ideas about inequality constraints: what they mean, how they are written in standard form, and how to draw them on a graph using straight lines and shaded regions.
2. What Is an Inequality Constraint?
An inequality constraint in linear programming is a condition on the decision variables written with symbols like \( \le, \ge, <, > \) instead of \( = \). It restricts the set of allowed solutions but still leaves many possible points.
General idea:
- Take variables like \( x \) and \( y \).
- Write a linear expression like \( ax + by \).
- Compare it with a constant \( c \) using \( \le, \ge, <, > \).
Any pair \( (x, y) \) that satisfies this comparison is said to satisfy the inequality constraint.
3. Types of Linear Inequalities
In linear programming, the constraints are usually linear inequalities in variables like \( x, y, z \). The most common types are:
- "Less than or equal to" type
- "Greater than or equal to" type
- Sometimes strict inequalities, though they are less common in practical models
3.1. Less Than or Equal To (\( \le \))
A constraint of the form
\( ax + by \le c \)
means that the left-hand side cannot be more than \( c \). In words, this is often read as "at most \( c \)".
Example:
\( 2x + 3y \le 12 \)
This can be interpreted as: the weighted combination \( 2x + 3y \) is allowed to go up to 12, but not beyond that. Any point with \( 2x + 3y = 12 \) or \( 2x + 3y < 12 \) satisfies the constraint.
3.2. Greater Than or Equal To (\( \ge \))
A constraint of the form
\( ax + by \ge c \)
means that the left-hand side must be at least \( c \). In words, this is often read as "not less than \( c \)".
Example:
\( x + 4y \ge 8 \)
This says that the combination \( x + 4y \) should be 8 or more. Any point with \( x + 4y = 8 \) or \( x + 4y > 8 \) satisfies the constraint.
3.3. Strict Inequalities (\( < \) and \( > \))
Strict inequalities are of the form:
- \( ax + by < c \)
- \( ax + by > c \)
They exclude the boundary line itself. In graphical form, the line is usually drawn as a dashed line to indicate that the points on the line are not included. In many linear programming problems, strict inequalities are replaced by non-strict forms (\( \le \) or \( \ge \)) because the optimal solution tends to lie on the boundary anyway.
4. Standard Form of an Inequality Constraint
When working with linear programming, it is convenient to write all inequalities in a clear and consistent way. This is called the standard form of an inequality constraint.
4.1. General Form \( ax + by \; (\le, \ge, <, >) \; c \)
A typical linear inequality in two variables can be written as:
\( ax + by \le c, \quad ax + by \ge c, \quad ax + by < c, \quad ax + by > c \)
where:
- \( a \) and \( b \) are real numbers (coefficients),
- \( x \) and \( y \) are variables,
- \( c \) is a real number (constant).
If needed, an inequality can be rearranged so that all variable terms are on one side and the constant is on the other side.
4.2. Meaning of Coefficients
In the inequality \( ax + by \le c \):
- \( a \) shows how strongly \( x \) contributes to the left-hand side.
- \( b \) shows how strongly \( y \) contributes.
- \( c \) is the upper or lower limit depending on the inequality sign.
Changing \( a \) or \( b \) changes the slope of the boundary line. Changing \( c \) shifts the line up, down, left, or right, which moves the boundary of the allowed region.
5. Graphical Representation of an Inequality
Each linear inequality in two variables represents a region on the plane. To understand the feasible region in a linear programming problem, it is important to know how to draw this region step by step.
5.1. Boundary Line
To draw the inequality, first convert it into an equation by replacing the inequality sign with an equal sign.
Example:
\( 2x + 3y \le 12 \Rightarrow 2x + 3y = 12 \)
This equation represents a straight line. One can find two easy points (for example, when \( x = 0 \) and when \( y = 0 \)) and join them to draw the boundary line.
For non-strict inequalities (\( \le \) or \( \ge \)), the boundary line is usually drawn as a solid line because points on the line are included in the solution set.
5.2. Shaded Half-Plane
After drawing the boundary line, the inequality tells which side of the line to shade. The shaded region is called a half-plane.
Example again with \( 2x + 3y \le 12 \):
All points that satisfy the inequality lie on one side of the line \( 2x + 3y = 12 \). The collection of these points forms a half-plane that includes the boundary line.
For \( \ge \) type inequalities, the shading lies on the opposite side compared to \( \le \), but the process is the same.
5.3. Test Point Method
A simple way to decide which side of the boundary to shade is the test point method.
- Choose a point that is not on the line, often \( (0, 0) \) if it is not on the boundary.
- Substitute this point into the inequality.
- If the inequality is true, shade the region containing that point. If it is false, shade the other side.
Example: For \( 2x + 3y \le 12 \), test the point \( (0, 0) \).
\( 2(0) + 3(0) = 0 \le 12 \)
The statement is true, so the side containing \( (0, 0) \) is the solution region.
6. Intersection of Multiple Inequalities
In a full linear programming problem, there is usually more than one inequality constraint. The final feasible region is the set of points that satisfy all the inequalities at the same time. Geometrically, this is the intersection of several half-planes.
6.1. Overlapping Regions
Plot each inequality one by one, shading the appropriate half-plane each time. The feasible region is where all the shaded regions overlap. Points in this overlapping region satisfy every constraint.
If after plotting all constraints there is no overlapping region, then there is no common solution and the set of inequalities is infeasible.
6.2. Example Illustration
Consider the system:
- \( x + y \le 6 \)
- \( x \ge 0 \)
- \( y \ge 0 \)
Steps:
- Draw the line \( x + y = 6 \) and shade the region below or on it (because of \( \le \)).
- Draw the vertical line \( x = 0 \) and keep only the region to the right (because of \( x \ge 0 \)).
- Draw the horizontal line \( y = 0 \) and keep only the region above (because of \( y \ge 0 \)).
The overlapping region is a triangular area bounded by the axes and the line \( x + y = 6 \). Every point in this triangle satisfies all three inequality constraints.
7. Real-Life Meaning of Common Inequalities
Inequality constraints arise naturally in everyday situations. They express ideas like "not more than", "at least", "within the limit", etc. A few typical interpretations are listed here for quick understanding.
7.1. Resource Limits
Suppose a certain activity uses \( 2 \) units of one resource per unit of \( x \) and \( 3 \) units per unit of \( y \). If at most 100 units of the resource are available, the constraint is:
\( 2x + 3y \le 100 \)
This means the total resource used by both activities together must not cross 100 units.
7.2. Budget or Cost Restrictions
If one unit of an item costs \( 50 \) and another costs \( 30 \), and the total money available is at most \( 500 \), then the cost constraint becomes:
\( 50x + 30y \le 500 \)
This inequality ensures that the total spending remains within the available budget.
7.3. Time or Capacity Limits
If each unit of \( x \) takes \( 2 \) hours and each unit of \( y \) takes \( 1 \) hour, and there are at most \( 40 \) hours available, the time constraint is:
\( 2x + y \le 40 \)
This inequality tells that the total time required for all activities must not exceed the available hours.
8. Quick Reference Table
| Inequality Form | Verbal Meaning | Graphical Idea |
|---|---|---|
| \( ax + by \le c \) | At most \( c \) | Half-plane including the line, shaded on the side where values are smaller |
| \( ax + by \ge c \) | At least \( c \) | Half-plane including the line, shaded on the side where values are larger |
| \( ax + by < c \) | Strictly less than \( c \) | Half-plane without the boundary line (dashed line) |
| \( ax + by > c \) | Strictly greater than \( c \) | Half-plane without the boundary line (dashed line) |
These patterns repeat in almost every linear programming problem, so keeping this small table in mind makes it easier to read and interpret inequality constraints.