Metode grafik untuk menyelesaikan masalah pemrograman linier

Main Ideas / Concepts Covered

• Goal: Solve a linear programming (LP) problem using the graphical method.
• Reason the graphical method works: The LP has two decision variables: (x) and (y). With two variables, constraints can be graphed on a 2D Cartesian plane.

• Objective function:

  • Maximize [ z = 3x + 2y ]

• Constraints: Three inequality constraints are given:

  • Two are of the “(\le)” type.
  • The third is an axis/boundary-type line.
  • The solution region is found by determining which side of each line satisfies all inequalities simultaneously.

• Approach overview:

  1. Draw each constraint boundary line in the (x)-(y) plane.
  2. Determine the feasible (solution) region by finding the intersection that satisfies all inequalities.

  3. To maximize (z), draw isoprofit (isoline) lines: [ 3x + 2y = \text{constant} ]

  4. Move the isoprofit line outward in the direction of increasing (z) until it last touches the feasible region.

  5. The point of last contact is the optimal solution.

Method / Step-by-Step Instructions (Graphical Method)

Step 1: Set Up the Graphical Representation

• Choose axes: (x)-axis and (y)-axis on a Cartesian plane.
• For each constraint inequality, draw the corresponding boundary line:
  • Treat the inequality as equality first (use “=” to get the boundary).
• Example described in the video for drawing a constraint boundary:
  • Find intercepts with the axes ((x)-intercept and (y)-intercept).
  • Plot two points and draw the straight boundary line through them.

Step 2: Determine the Feasible Region

• For each constraint line, identify which side satisfies the inequality (the video uses shading).
• Feasible region = all points that satisfy every constraint at the same time.
• The feasible region appears as a shaded polygon/area.

Step 3: Construct Isoprofit Lines (Objective Contours)

• Start from the objective: [ z = 3x + 2y ]

• Form isoprofit lines by setting: [ 3x + 2y = k ] where (k) is a constant.

• Draw multiple lines for increasing values of (k) (e.g., examples like 60, 90, etc.).

• Each isoprofit line represents all ((x,y)) points that yield the same objective value (z).

Step 4: Move Toward the Maximum

• Since the problem is a maximization, move the isoprofit lines so that (z) increases.
• The optimal solution occurs where the isoprofit line last touches the feasible region (typically at a corner point).

Step 5: Identify the Optimal Corner Point and Compute Coordinates

• The optimal corner point (C) is found as the intersection of two active constraints (a corner of the feasible region).

• The video reports:

  • Optimal point: [ (x, y) = (20, 60) ]

  • Maximum objective value: [ z = 3(20) + 2(60) = 60 + 120 = 180 ]

  • Therefore:

    • Maximum (z = 180)

Key Lesson / Conclusion

• The graphical method is effective when there are two variables.
• The feasible region is obtained by graphing constraints and shading the overlapping area.
• The maximum objective value is found by sliding isoprofit lines until they last touch the feasible region.
• For 3 or more variables, graphical methods become impractical, and the simplex method is suggested next.

Speakers or Sources Featured

• Speaker: Video narrator/teacher (name not provided in subtitles).
• Sources: No explicit sources cited beyond the explanation of the mathematical method.