Corner of the Feasible Region Calculator
Feasible Region Calculator
Enter the coefficients and constants for your linear inequalities (assuming ‘≤’ and x, y ≥ 0). Leave fields blank for unused constraints.
x +
y ≤
x +
y ≤
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y ≤
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Corner Points (x, y):
| Corner No. | x-coordinate | y-coordinate |
|---|
Method: The calculator identifies the boundary lines from the given inequalities (including x ≥ 0 and y ≥ 0). It then finds the intersection points of every pair of these lines. Each intersection point is tested against ALL original inequalities. If a point satisfies all of them, it’s a corner of the feasible region.
What is a Corner of the Feasible Region?
In linear programming, a feasible region is the set of all possible points (combinations of variable values, e.g., x and y) that satisfy all the given constraints (linear inequalities). The corner of the feasible region, also known as a vertex or extreme point, is a point in the feasible region where two or more boundary lines of the constraints intersect. These corners are crucial because the optimal solution (maximum or minimum value of the objective function) for a linear programming problem always occurs at one or more of these corner points (if a solution exists and the region is bounded).
This corner of the feasible region calculator helps you identify these critical points for a system of two-variable linear inequalities. It is used by students, operations researchers, economists, and anyone dealing with optimization problems constrained by linear relationships. A common misconception is that every intersection of boundary lines is a corner, but only those intersections that also satisfy *all* other constraints are true corners of the feasible region.
Corner of the Feasible Region Formula and Mathematical Explanation
To find the corners of the feasible region defined by a set of linear inequalities in two variables (x and y), we follow these steps:
- Identify Boundary Lines: For each inequality (e.g., ax + by ≤ c), identify the corresponding boundary line (ax + by = c). Include the non-negativity constraints x ≥ 0 and y ≥ 0, which give boundary lines x = 0 and y = 0.
- Find Intersection Points: Take every possible pair of these boundary lines and solve them as a system of two linear equations to find their intersection point (x, y). For two lines a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the intersection is found by solving:
x = (c₁b₂ – c₂b₁) / (a₁b₂ – a₂b₁)
y = (a₁c₂ – a₂c₁) / (a₁b₂ – a₂b₁)
provided the determinant (a₁b₂ – a₂b₁) is not zero (lines are not parallel or coincident). - Check Feasibility: For each intersection point found, substitute its x and y values back into ALL the original inequalities (including x ≥ 0 and y ≥ 0). If the point satisfies every inequality, it is a corner point of the feasible region. If it violates even one inequality, it’s not a corner.
Our corner of the feasible region calculator automates this process for up to four user-defined inequalities plus x≥0, y≥0.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| aᵢ, bᵢ | Coefficients of variables x and y in the i-th constraint | Depends on context | Any real number |
| cᵢ | Constant term (right-hand side) of the i-th constraint | Depends on context | Any real number |
| x, y | Decision variables | Depends on context | Usually non-negative |
Practical Examples (Real-World Use Cases)
Example 1: Production Planning
A company produces two products, A and B. Product A requires 2 hours of machine time and 1 hour of labor. Product B requires 3 hours of machine time and 1 hour of labor. The company has 12 hours of machine time and 5 hours of labor available per day. We want to find the feasible production combinations.
Let x be the number of units of product A, and y be the number of units of product B. Constraints:
- Machine time: 2x + 3y ≤ 12
- Labor time: 1x + 1y ≤ 5
- Non-negativity: x ≥ 0, y ≥ 0
Using the corner of the feasible region calculator with a1=2, b1=3, c1=12, a2=1, b2=1, c2=5, we find the corners: (0, 0), (0, 4), (5, 0), and (3, 2). These represent the extreme feasible production plans.
Example 2: Diet Planning
A diet requires at least 4 units of vitamin C and 5 units of vitamin D. Food 1 provides 1 unit of C and 2 units of D per serving. Food 2 provides 1 unit of C and 1 unit of D per serving.
Let x be servings of Food 1, y be servings of Food 2. Constraints:
- Vitamin C: x + y ≥ 4 (or -x – y ≤ -4)
- Vitamin D: 2x + y ≥ 5 (or -2x – y ≤ -5)
- Non-negativity: x ≥ 0, y ≥ 0
If our calculator was set for ‘≥’, we would input these. For our current ‘≤’ calculator, we’d rewrite as -x – y ≤ -4 and -2x – y ≤ -5. The corners define the boundary of minimum food servings.
How to Use This Corner of the Feasible Region Calculator
- Enter Constraints: For each linear inequality of the form ax + by ≤ c, enter the values of ‘a’, ‘b’, and ‘c’ into the corresponding input fields (a₁, b₁, c₁ for the first, etc.). The calculator assumes the ‘≤’ relationship and x ≥ 0, y ≥ 0. If you have fewer than 4 constraints, leave the fields for the extra ones blank.
- Calculate: Click the “Calculate Corners” button.
- View Results: The calculator will display:
- The number of corner points found.
- A list of the (x, y) coordinates of each corner point.
- A table summarizing the corners.
- A graph showing the boundary lines and the corner points. The feasible region (for ‘≤’ and x,y≥0) is generally the area bounded by the lines and the axes near the origin.
- Interpret: The corner points are the vertices of the feasible region. If you have an objective function to optimize (maximize or minimize), you would evaluate it at these corner points to find the optimal solution.
Key Factors That Affect Corner of the Feasible Region Results
- Coefficients (aᵢ, bᵢ): These determine the slopes of the boundary lines. Changing them rotates the lines, altering the shape and corners of the feasible region.
- Constants (cᵢ): These determine the intercepts of the boundary lines, shifting them parallel to themselves. Changes can expand or shrink the feasible region.
- Inequality Signs (≤, ≥, =): Our calculator assumes ‘≤’ for user inputs and ‘≥’ for non-negativity. Different signs define different half-planes, drastically changing the feasible region. For ‘≥’ constraints, the feasible region is often unbounded away from the origin.
- Number of Constraints: More constraints generally lead to a smaller or more complex feasible region, potentially with more corners, or even an empty region.
- Redundant Constraints: If one constraint is implied by others, it doesn’t affect the feasible region or its corners.
- Parallel Constraints: If boundary lines are parallel, they might not intersect to form a corner within the region defined by other constraints, or one might make the other redundant.
Understanding how these factors influence the output of the corner of the feasible region calculator is key to correctly modeling and solving linear programming problems.
Frequently Asked Questions (FAQ)
A: This calculator is designed for ‘≤’ inequalities and x≥0, y≥0. To handle ‘≥’, you can multiply the inequality by -1 (e.g., x + y ≥ 5 becomes -x – y ≤ -5). For ‘=’, you essentially have two constraints (e.g., x+y=5 is x+y≤5 and x+y≥5 or -x-y≤-5).
A: It could mean the feasible region is empty (no points satisfy all constraints simultaneously) or unbounded in a way that the standard corner-finding method with the given constraints didn’t yield intersections satisfying all conditions. Check your constraints for contradictions (e.g., x ≤ 1 and x ≥ 2).
A: This corner of the feasible region calculator uses a graphical approach and finds intersections in a 2D plane. Problems with more variables require more complex methods like the Simplex algorithm, which operate algebraically and are harder to visualize directly with a simple 2D graph.
A: Yes. If the constraints don’t fully enclose an area (e.g., only x ≥ 0, y ≥ 0, x + y ≥ 5), the feasible region can extend infinitely. The calculator will still find corners formed by the intersecting boundary lines that satisfy all conditions.
A: The calculator detects near-parallel lines (determinant close to zero) and won’t find a unique intersection point between them. They won’t form a corner together unless they are coincident and intersect other lines.
A: The calculations are done using standard floating-point arithmetic. For well-conditioned problems, the results are quite accurate. Small rounding errors can occur.
A: The Fundamental Theorem of Linear Programming states that if an optimal solution exists for a linear programming problem, it will occur at one or more of the corner points of the feasible region (or along a line segment connecting two corners if multiple optimal solutions exist).
A: No, this calculator finds corners in the continuous feasible region. Integer programming requires that the variables take integer values, which involves more complex algorithms.