Program Linier Metode Grafik Part 1

Introduction

In this tutorial, we will explore the graphical method of solving linear programming problems, as demonstrated in the video "Program Linier Metode Grafik Part 1" by Dewinta Ayuni. This method is valuable for optimizing resources in various fields such as economics, engineering, and operations research. By the end of this guide, you will understand how to set up and graphically solve linear programming problems.

Step 1: Understanding Linear Programming

• Definition: Linear programming is a mathematical technique for maximizing or minimizing a linear function, subject to certain constraints.
• Components:
  • Objective Function: The function to be maximized or minimized.
  • Constraints: The inequalities that restrict the values of the variables.

Step 2: Setting Up the Problem

• Identify Variables: Determine the decision variables relevant to your problem.
• Formulate the Objective Function: Write the function you want to maximize or minimize. For example:
  • Maximize Z = 3x + 4y
• Identify Constraints: List the restrictions in the form of inequalities. For example:
  • x + 2y ≤ 8
  • 3x + y ≤ 9
  • x ≥ 0
  • y ≥ 0

Step 3: Graphing the Constraints

• Draw the Axes: Set up a coordinate system with x and y axes.
• Plot Each Constraint:
  • Convert each inequality to an equation (e.g., x + 2y = 8).
  • Find the intercepts and plot them on the graph.
  • Shade the feasible region that satisfies the inequality.

Practical Tips:

• Use a ruler for straight lines to ensure accuracy.
• Check where each line intersects to aid in plotting.

Step 4: Identifying the Feasible Region

• Intersection Points: Determine where the constraint lines intersect. These points are potential solutions.
• Feasible Region: The area where all constraints overlap is the feasible region. This area represents all possible solutions that satisfy the constraints.

Step 5: Evaluating the Objective Function

• Corner Points: Identify the corner points (vertices) of the feasible region.
• Calculate the Objective Function:
  • Evaluate the objective function at each corner point.
  • For example, if the corner points are (0,4), (2,3), and (3,0):
    • Z(0,4) = 3(0) + 4(4) = 16
    • Z(2,3) = 3(2) + 4(3) = 18
    • Z(3,0) = 3(3) + 4(0) = 9

Step 6: Determining the Optimal Solution

• Select the Maximum or Minimum Value: Based on your objective (maximize or minimize), choose the corner point that gives the best value.
• Solution Interpretation: The coordinates of this point provide the optimal values of the decision variables.

Conclusion

In this tutorial, we have covered the fundamental steps of solving a linear programming problem using the graphical method. You learned how to set up the problem, graph the constraints, identify the feasible region, and evaluate the objective function. As a next step, practice with different linear programming problems to strengthen your understanding and application of this method.