Graphical Method of Solving Linear Programming Problems
It is not hidden that the simplex method is a well-studied and widely used method for solving Linear Programming problems. But as far as non-Linear Programming is concerned, such a universal method does not exist. With graphical methods, any optimization programming problems consisting of only two variables can easily be solved. These variables can be referred as x₁ and x₂ and with the help of these variables, most of the analysis can be done on a two-dimensional graph. Although we cannot generalize a large number of variables using a graphical approach, the basic concepts of Linear Programming in a two-variable context can be easily demonstrated. We can always turn to two-variable problems if the problems seem to be complicated and we find ourselves in a pool of questions. And we can always search for answers in a two-variable case using graphs, that is solving Linear Programming problems graphically.
The graphical approach wraps itself with another advantage and that is its visual nature. It provides us with a picture to get along with the algebra of Linear Programming. This picture can quench our thirst for understanding the basic definitions and possibilities. These reasons are proof that the graphical approach works smoothly with Linear Programming concepts.
Now, for solving Linear Programming problems graphically, we must two things:
Inequality constraints
And the objective function.
The Graphical Method of Solving Linear Programming problems is based on a well-defined set of logical steps. With the help of these steps, we can master the graphical solution of Linear Programming problems.
Methods used for Solving Linear Programming
Following two methods used for Graphical Method of Solving Linear Programming:
Extreme point solution method
Iso-profit (cost) performance line method to find the perfect solution to the LP problem.
Important Points
A set of variable values xj (j = 1, 2,3..., N) that satisfies the issues of the LP problem is said to form the solution to that LP problem.
Potential Solution Set of variable values xj (j = 1, 2,3.., N) that satisfies all the barriers and non-negative conditions of the LP problem at the same time is said to form a possible solution to that LP problem.
Unlikely solution A set of variable values xj (j = 1, 2,3..., N) that do not satisfy all the barriers and non-negative conditions of the LP problem at one time is said to form an impossible solution for that. LP problem.
With an m-group of simultaneous n (n> m) variables in the LP problem, the solution obtained by placing (n - m) zero-equal variables and solving the remaining m-variables of m is called the basic solution. of that LP problem. Variables (n - m) whose value did not appear in the basic solution are called non-core variables and the remaining variables of m are called the base variables.
A possible basic solution A possible solution to the LP problem and a basic solution is called a possible basic solution. That is, all basic changes take non-negative values.
The Basic Solution is two Types
Degenerate A basic solution that is possible is called degenerate if the value of at least one basic element is zero.
Non-degradable The basic practical solution is called non-degenerate if the value of all the basic variables is zero and positive.
Ways to Solve the LP Problem Law
Point Method
To resolve the issue using the existing point method you need to follow these steps:
Step 1: Create statistical design from a given problem. If not provided.
Step 2: Now plan the graph using the given obstacles and find a region that you can use.
Step 3: Find the possible location links (vertices) that we found in step 2.
Step 4: Now check the objective function in each of the possible location areas. Assume that N and n represent the largest and smallest values in these points.
Step 5: If the possible region is bound then N and n are the maximum and minimum values of the objective function. Or if the area is likely to have no boundaries then:
N is the highest value of the target function if the open half system is obtained with an ax + by> N without a common point in the possible area. If not, purpose work has no solution.
n a small amount of objective work if the open half system is found by the ax + by <n does not have a common point in the probable location. If not, purpose work has no solution.
Iso-Cost Method
The term iso-cost or iso-profit method provides a combination of points that produce the same cost/profit as any other combination in the same line. This is done by arranging the lines corresponding to the slope of the equation.
Follow the following steps for the Iso-cost method solution:
Step 1: Create a statistical design from a given problem. If not provided.
Step 2: Now plan the graph using the given obstacles and find a region that you can use.
Step 3: Now find the possible location links we found in step 2.
Step 4: Find the correct Z (objective function) and draw a line for this objective activity.
Step 5: If the objective function is high type, draw a line corresponding to the objective performance line and this line is farther from the root and has only one common point in the possible area. Or if the target function is a small type then draw a line that corresponds to the target performance line and this line is very close to the source and has one common point in the possible position.
Step 6: Now find the links to the common point we find in step 5. Now, this point is used to find the right solution and the amount of objective work.
Information for the Wooden Tables and Chairs Linear Programming Problem
Table 1 gives us the information for the Linear Programming problem. We can go step-by-step for solving the Linear Programming problems graphically.
Step 1) The aforementioned table can help us to formulate the problem. The bottom row will serve the objective function. The objective function of the company is to maximize unit profit. The woods and the laborers are the constraint set. The nonnegativity conditions are also stated.
Maximize Z = 6x2+8x2 (is the objective function)
Subject to: 30x1+20x2≤ 300 (300 bf available)
5x1+10x2 ≤ 110 (110 hours available)
x1+x2≥ 0 (non - negative conditions)
The two variables (wood and labor) in this problem, can be solved graphically.
Step 2) This is the graph plotting step. With the x-axis as the number of tables and y-axis as the number of chairs, we can find the two constraint lines. This can be found if we find the x and y-intercepts for the two constraint equations. But before that, we have to rewrite the constraint inequalities as equalities.
Wood
30x1+20x2= 300
Setting x₂ = 0 to solve x₁
30x₁ = 300
x₁ =300/30 = 10 tables (Wood used to make tables)
Next:
Setting x₁ = 0 to solve x₂
20x₂ = 300
x₂ =300/20 = 15 chairs (Wood used to make chairs)
Labor
50x1+10x2= 110
Setting x₂ = 0 to solve x₁
5x₁ = 110
x₁ = 110/5= 22 tables(labors used to make tables)
Setting x₁ = 0 to solve x₂
10x₂ = 110
x₂ = 110/10= 11 chairs (labors used to make chairs)
Now, plot the wood constraint line (x1= 10 and x2=15) and labor constraint line (x1=22
and x2=11)
(Image will be uploaded soon)
Step 3) To check the valid side for both constraint lines use the origin (0,0).
30(0) + 20(0) < 300 is the valid side of the wood constraint line. In the same way 5(0) + 10(0) < 110 also is a valid side of the labor constraint line. Now, draw the arrows indicating the valid side of each constraint line.
(Image will be uploaded soon)
Step 4) Identify the feasible region which is the area on the valid side of both constraint lines.
Step 5) Find x1 and x2 using Z = 48 and 72.
In the first case, the values will be x1 = 8 and x2= 12
In the second case, the values will be x1= 6 and x2= 9
Plot the objective function lines when Z = 48 and Z = 72.
The two objective function lines move away from the origin (0,0), Z increases.
(Image will be uploaded soon)
Step 6) Find the most attractive corner.
(Image will be uploaded soon)
Step 7) Calculate the coordinates and find the values of x and y.
Therefore, according to the company’s optimal solution four tables and nine chairs can be manufactured.
Step 8) finally, determine the value of the objective function for the optimal solution by plugging in the number of tables and chairs and solve for Z:
Z =$6(4) +$8(9) =$96
Thus, by producing four tables and nine chairs we can achieve the maximum profit of $96.
FAQs on Graphical Method Linear Programming
1. What is the graphical method in Linear Programming (LPP)?
The graphical method is a visual technique used to solve Linear Programming Problems that involve two decision variables. It works by plotting the linear inequality constraints on a graph to identify a feasible region. The optimal solution (either maximum or minimum) is then found by locating the corner point of this region that best satisfies the objective function.
2. What are the basic steps to solve a Linear Programming Problem using the graphical method?
To solve an LPP graphically, you should follow these key steps as per the CBSE/NCERT syllabus for the 2025-26 session:
- Formulate the LPP: Define the objective function (e.g., Maximise Profit Z = ax + by) and the inequality constraints from the given problem.
- Plot the Constraints: Treat each inequality as an equation (e.g., x + y ≤ 10 becomes x + y = 10) and draw these lines on a graph.
- Identify the Feasible Region: Determine the area on the graph that satisfies all constraints simultaneously. This common area is the feasible region.
- Find the Corner Points: Identify the coordinates of the vertices (corner points) of the feasible region.
- Evaluate the Objective Function: Substitute the coordinates of each corner point into the objective function to find its value at each point.
- Determine the Optimal Solution: The point that gives the maximum or minimum value, as required by the problem, is the optimal solution.
3. What is a 'feasible region' in the context of the graphical method?
A feasible region is the solution space on the graph that contains all the possible points that satisfy every single constraint of the Linear Programming Problem, including the non-negativity constraints (x ≥ 0, y ≥ 0). Any point inside or on the boundary of the feasible region is a feasible solution, while the optimal solution is always found at one of its corner points.
4. Why is the optimal solution to an LPP always found at a corner point of the feasible region?
The optimal solution is found at a corner point (or vertex) because these points represent the extreme values of the feasible region. The objective function is a straight line, and as you move this line parallel to itself across the feasible region, it will touch the region's boundary. The last point it touches before leaving the region will be one of the corners, and this is where the function's value will be at its maximum or minimum for that region.
5. What is the difference between a bounded and an unbounded feasible region?
The key difference lies in whether the region is enclosed or extends infinitely:
- A bounded feasible region is one that is completely enclosed by the constraint lines. In this case, both a maximum and a minimum value for the objective function will exist at its corner points.
- An unbounded feasible region extends indefinitely in at least one direction. For an unbounded region, an optimal solution may or may not exist. A maximum or minimum value exists only if the moving objective function line is eventually constrained by a corner point in the direction of optimisation.
6. Why can't the graphical method be used for problems with more than two variables?
The primary limitation of the graphical method is its dimensionality. It relies on plotting constraints on a two-dimensional (x-y) plane, which can only represent two decision variables. For a problem with three variables (e.g., x, y, and z), you would need a three-dimensional graph, which is very difficult to draw and interpret. For problems with more than three variables, a graphical representation is impossible, and more advanced techniques like the Simplex method are required.
7. What does it mean if an LPP has no feasible solution?
An LPP has no feasible solution when the constraints are contradictory and there is no single point or region on the graph that satisfies all of them simultaneously. Graphically, this means there is no overlapping common area for the constraints. For example, one constraint might require x > 5 while another requires x < 3, which is impossible to satisfy at the same time.
8. How do you handle a problem with multiple optimal solutions using the graphical method?
A Linear Programming Problem has multiple optimal solutions if the objective function line is parallel to one of the boundary lines of the feasible region. When this occurs, the objective function will have the same optimal value at two adjacent corner points. In such a case, every point on the line segment connecting these two corner points is also an optimal solution, leading to an infinite number of solutions.
9. What are some real-world applications of Linear Programming that can be solved graphically?
Linear Programming is used to find the best outcome in a mathematical model. For simple two-variable scenarios, its applications include:
- Production Planning: Determining the number of two different products to manufacture to maximize profit, given constraints like labour hours and raw materials.
- Diet Problems: Creating a diet plan from two different food sources to meet nutritional requirements at a minimum cost.
- Resource Allocation: Allocating two types of resources (e.g., advertising budget between two media channels) to achieve maximum reach.
