Graphical solutions

Understanding Graphical Solutions

• Graphical solutions in linear programming provide a visual representation of all feasible solutions to a problem.
• Graphs are typically constructed in a two-dimensional space when dealing with two decision variables.
• The axes of the graph represent the decision variables, whereas the space within the graph represents the feasible solutions according to constraints.
• The optimal solution can often be determined graphically by examining the intersection points, or corners, of the feasible region.

Constructing Graphs

• Begin by plotting the equations or inequalities corresponding to the constraints on a two-dimensional plane.
• Remember that each linear inequality or equation will form a line on the graph.
• Use a test point often the origin (0,0), to determine the direction of the inequality i.e., which side of the line constitutes solutions that satisfy the inequality.
• The feasible region is the area on the graph where all constraints are simultaneously satisfied, typically represented by a shaded or hatched region.

Identifying Optimal Solutions

• Optimal solutions are often found at the vertices or corners of the feasible region because they represent a set of conditions that lead to the maximum or minimum value of the objective function.
• For maximisation problems, the optimal solution is at a corner of the feasible region where the objective function has the highest possible value.
• For minimisation problems, the optimal solution is the corner where the function has the lowest possible value.
• Iso-profit (for maximisation problems) or iso-cost (for minimisation problems) lines also help in identifying the optimum point. These are lines parallel to the objective function that move away from the origin for maximisation problems and towards the origin for minimisation problems.
• If the problem has multiple optimal solutions, the segment connecting the corresponding points will lie wholly within the feasible region as well.

Concept of Redundant Constraints

• In some cases, constraints may not affect the feasible region or the optimal solution. Such constraints are called redundant constraints.
• Graphically, a redundant constraint may be represented by a line that does not intersect the feasible region or alters its shape or size.

Remember that graphical solutions provide a simple and intuitive way to understand and solve linear programming problems. They are particularly helpful when dealing with two decision variables, but for more complex cases involving more variables, alternative methods like the simplex method may be required.