Graphical Method for solving Mathematical Models

Introduction

Mathematical models are powerful tools used to represent real-world problems in a structured and analytical manner. By utilizing equations, inequalities, and variables, these models simplify complex issues, enabling effective problem-solving. Among various techniques for solving mathematical models, the graphical method is a visually intuitive approach, especially for optimization problems like linear programming (LP). Linear programming focuses on achieving the best outcome—such as maximizing profit or minimizing cost—within given constraints. The graphical method is particularly effective for problems involving two variables, as it provides a clear visualization of feasible solutions and optimal outcomes. This article explores the graphical method, detailing its application, advantages, and limitations while providing practical insights for solving optimization problems.

What is a Mathematical Model?

A mathematical model represents real-world problems through mathematical expressions such as equations, inequalities, or functions. These models simplify complex scenarios by identifying critical elements like variables and constraints, enabling structured analysis.

For example:

·        Resource Allocation: Determining the best way to distribute limited resources like time or money.

·        Production Optimization: Maximizing output while minimizing costs in manufacturing.

·        Transportation Problems: Optimizing logistics for cost-effective distribution of goods.

Overview of Linear Programming (LP)

Linear programming is a mathematical technique used to optimize an objective function, subject to a set of constraints. The objective function represents the goal, such as maximizing profit or minimizing cost, while constraints represent the limitations or conditions to be satisfied.

Key components of LP include:

·        Objective Function: The mathematical expression to be optimized.

E.g.- Z = 3x + 5y

·        Constraints: Inequalities that define the limits.

E.g.- x =2y <= 8

·        Decision Variables: Variables representing the choices.

E.g. x and y

When to Use the Graphical Method

The graphical method is suitable for solving LP problems with two decision variables. It provides a clear and intuitive way to visualize the problem, identify feasible solutions, and locate the optimal point. However, it is limited to two-variable problems because higher-dimensional spaces cannot be easily represented graphically.

Step-by-Step Procedure

Step 1: Define the Problem

·        Identify the objective function and constraints from the problem statement.

Step 2: Graph the Constraints

·        Treat each constraint inequality as an equation to plot lines on a 2D graph.

·        Identify the feasible region, where all constraints overlap, by shading appropriately.

Step 3: Plot the Objective Function

·        Represent the objective function as a line on the graph by assigning arbitrary values to test points.

·        Understand the direction of optimization (maximize or minimize) and adjust the line accordingly.

Step 4: Determine the Optimal Solution

·        Locate the vertices (corner points) of the feasible region.

·        Substitute these points into the objective function to find the maximum or minimum value.

Illustrative Example

Solution:

Convert each inequality into an equation for graphing,

            2x1 + 3x2 = 18

            X1 + 2x2 = 8

Plot these lines on a 2D plane, considering x1, x2 >=0.

Find the feasible region. Shade the area that satisfies all constraints. The feasible region will be the overlapping area that satisfies both 2x1 + 3x2 <= 18 and x1 + 2x2 <= 8, as well as x1, x2 >= 0.

Locate the corner points. Identify the vertices of the feasible region (intersection points of the constraint lines and axes). Here, the corner points are (0,0), (0,4), and (8,0).

Substitute the corner points into Z = 40x1 + 50 x2 and calculate the Objective Function at each corner point as follows.

Types of Solutions in Linear Programming

1.  Feasible Solution: A solution that satisfies all constraints, including non-negativity conditions. It represents any point within or on the boundary of the feasible region.

2.  Infeasible Solution: A solution that violates one or more constraints, often due to contradictory or overly restrictive conditions, resulting in no feasible region.

3.  Optimal Solution: A feasible solution that maximizes or minimizes the objective function. This typically occurs at a vertex of the feasible region.

4.  Multiple Solutions (Alternate Optimal Solutions): When the objective function is parallel to a constraint edge within the feasible region, multiple points on that edge yield the same optimal value..

5.  Unbounded Solution: When the feasible region is open and the objective function can increase or decrease indefinitely, leading to no finite optimal value.

6.  Degenerate Solution: A feasible solution where the number of binding constraints exceeds the number of variables, often resulting in a shared corner point.

Advantages of Graphical Method

·        Simple and easy to understand for beginners.

·        Visual representation aids comprehension of constraints and optimal solutions.

·        Ideal for two-variable problems.

Limitations

·        Restricted to LP problems with only two variables.

·        Inefficient for problems with more constraints or higher dimensions.

Applications of Graphical Methods

·   Resource Allocation: Efficient distribution of resources in manufacturing or services.

·        Profit Maximization: Determining optimal product mixes for higher profitability.

·        Cost Minimization: Reducing operational expenses in logistics and transportation.

Conclusion

The graphical method is a straightforward and effective tool for solving linear programming problems involving two variables. By visualizing the feasible region and identifying the optimal point, this method simplifies problem-solving and enhances understanding. While its applicability is limited to two-dimensional problems, it serves as an excellent foundation for exploring more advanced optimization techniques for complex models. Mathematical modeling remains a vital skill in decision-making, enabling informed choices in diverse fields such as business, engineering, and logistics.