The graphical method for two-variable linear programming problems is a visual approach to solving optimization challenges. It involves plotting constraints, identifying feasible regions, and evaluating corner points to find the optimal solution for maximizing profit or minimizing cost.

This method provides a clear way to understand the relationship between decision variables and constraints. By plotting the feasible region and evaluating the objective function at key points, we can visually determine the best solution for our optimization problem.

Graphical Method for Two-Variable Linear Programming Problems

Plotting feasible regions

• Decision variables identified x and y represent quantities to optimize (production units)
• Constraints plotted on coordinate plane form boundaries of feasible region
  • Lines drawn for each constraint equation $ax + by = c$
  • Shading applied to areas satisfying inequalities (≤, ≥)
• Intersection of all constraint regions creates feasible solution space
• Feasible region labeled clearly shows all possible solutions

Corner points of feasible regions

• Intersection points of constraint lines found using simultaneous equations
• Constraint line intersections with axes determined setting x or y to zero
• Corner points (vertices) listed as coordinate pairs (x, y)
• Algebraic methods calculate exact coordinates
  1. Solve systems of equations for intersecting lines
  2. Find x-intercepts and y-intercepts of constraint lines
  3. Check all points for feasibility within constraints

Objective function evaluation

• Objective function expressed $Z = ax + by$ for profit or cost
• Corner point coordinates substituted into objective function
• Objective value calculated for each feasible corner point
• Table created organizing corner points and corresponding objective values
  • Columns: Corner Point, x-coordinate, y-coordinate, Objective Value
  • Rows: Each feasible corner point and its calculated data

Optimal solution identification

• Problem type determined maximization (profit) or minimization (cost)
• Objective function values compared at all corner points
• Highest value selected for maximization problems (max profit)
• Lowest value chosen for minimization problems (min cost)
• Optimal solution verified examining nearby points in feasible region
• Special cases considered:
  • Multiple optimal solutions occur when objective function parallel to constraint
  • Unbounded solutions arise when feasible region extends infinitely in improvement direction