Graphical Solution of Linear Programming Problems

Linear programming is the simplest way of optimizing a problem. Through this method, we can formulate a real-world problem into a mathematical model. There are various methods for solving Linear Programming Problems and one of the easiest and most important methods for solving LPP is the graphical method. In Graphical Solution of Linear Programming, we use graphs to solve LPP.

We can solve a wide variety of problems using Linear programming in different sectors, but it is generally used for problems in which we have to maximize profit, minimize cost, or minimize the use of resources. In this article, we will learn about Solutions of Graphical solutions of linear programming problems, their types, examples, and others in detail.

Linear Programming

Linear programming is a mathematical technique employed to determine the most favorable solution for a problem characterized by linear relationships. It is a valuable tool in fields such as operations research, economics, and engineering, where efficient resource allocation and optimization are critical.

Now, let's learn about types of linear programming problems

Types of Linear Programming Problems

There are mainly three types of problems based on Linear programming:

• Manufacturing Problem: In this type of problem, some constraints like manpower, output units/hour, and machine hours are given in the form of a linear equation. And we have to find an optimal solution to make a maximum profit or minimum cost.

• Diet Problem: These problems are generally easy to understand and have fewer variables. Our main objective in this kind of problem is to minimize the cost of diet and to keep a minimum amount of every constituent in the diet. 

• Transportation Problem: In these problems, we have to find the cheapest way of transportation by choosing the shortest route/optimized path.

Some commonly used terms in linear programming problems are:

Objective function: The direct function of the form Z = ax + by, where a and b are constant, which is reduced or enlarged, is called the objective function. For example, if Z = 10x + 7y. The variables x and y are called the decision variables.

Constraints: The restrictions that are applied to a linear inequality are called constraints.

• Non-Negative Constraints: x > 0, y > 0 etc.
• General Constraints: x + y > 40, 2x + 9y ≥ 40 etc.

Optimization problem: A problem that seeks to maximization or minimization of variables of a linear inequality problem is called an optimization problem.

Feasible Region: A common region determined by all given issues, including the non-negative (x ≥ 0, y ≥ 0) constraint, is called the feasible region (or solution area) of the problem. The region other than the feasible region is known as the infeasible region.

Feasible Solutions: These points within or on the boundary region represent feasible solutions to the problem. Any point outside the scenario is called an infeasible solution.

Optimal (Most Feasible) Solution: Any point in the emerging region that provides the right amount (maximum or minimum) of the objective function is called the optimal solution.

➤NOTE

• If we have to find maximum output, we have to consider the innermost intersecting points of all equations.
• If we have to find the minimum output, we consider the outermost intersecting points of all equations.
• If there is no point in common in the linear inequality, then there is no feasible solution.

Graphical Solution of a Linear Programming Problem

We can solve linear programming problems using two different methods are,

1. Corner Point Methods
2. Iso-Cost Methods

Corner Point Methods

To solve the problem using the corner point method, you need to follow the following steps:

Step 1: Create a mathematical formulation from the given problem. If not given.

Step 2: Now plot the graph using the given constraints and find the feasible region.

Step 3: Find the coordinates of the feasible region(vertices) that we get from step 2.

Step 4: Evaluate the objective function at each corner point of the feasible region. Assume N and n denote the largest and smallest values of these points.

Step 5: If the feasible region is bounded, then N and n are the maximum and minimum values of the objective function. Or if the feasible region is unbounded, then:

• N is the maximum value of the objective function if the open half plane is obtained by the ax + by > N has no common point with the feasible region. Otherwise, the objective function has no solution.
• n is the minimum value of the objective function if the open half plane is obtained by the ax + by < n has no common point with the feasible region. Otherwise, the objective function has no solution.

Solved Examples of LPP using Corner Point Method

Example 1:Solve the given linear programming problems graphically: 

Maximize: Z = 8x + y

Constraints are,

• x + y ≤ 40
• 2x + y ≤ 60
• x ≥ 0, y ≥ 0

Solution:

Step 1: Constraints are,

• x + y ≤ 40
• 2x + y ≤ 60
• x ≥ 0, y ≥ 0

Step 2: Draw the graph using these constraints. 

Here both the constraints are less than or equal to, so they satisfy the below region (towards origin). You can find the vertex of the feasible region by graph, or you can calculate using the given constraints:

•  x + y = 40 ...(i)
• 2x + y = 60 ...(ii)

Now multiply eq(i) by 2 and then subtract both eq(i) and (ii), we get
y = 20

Now, put the value of y in any of the equations, and we get
x = 20 

So the third point of the feasible region is (20, 20)

Step 3: To find the maximum value of Z = 8x + y. Compare each intersection point of the graph to find the maximum value

Points	Z = 8x + y
(0, 0)	0
(0, 40)	40
(20, 20)	180
(30, 0)	240

So the maximum value of Z = 240 at point x = 30, y = 0.

Example 2: One kind of cake requires 200 g of flour and 25g of fat, and another kind of cake requires 100 g of flour and 50 g of fat Find the maximum number of cakes that can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients, used in making the cakes.

Solution: 

Step 1: Create a table like this for easy understanding (not necessary).

	Flour(g)	Fat(g)
Cake of first kind (x)	200	25
Cake of second kind (y)	100	50
Availability	5000	1000

Step 2: Create a linear equation using inequality

• 200x + 100y ≤ 5000 or 2x + y ≤ 50
• 25x + 50y ≤ 1000 or x + 2y ≤ 40
• Also, x > 0 and y > 0

Step 3: Create a graph using the inequality (remember only to take positive x and y-axis)

Step 4: To find the maximum number of cakes (Z) = x + y. Compare each intersection point of the graph to find the maximum number of cakes that can be baked.

x	y	Z = (x+y)
0	20	20
20	10	30
25	0	25

Z is maximum at coordinate (20, 10). So the maximum number of cakes that can be baked is Z = 20 + 10 = 30.

Iso-Cost Method

The term iso-cost or iso-profit method provides the combination of points that produces the same cost/profit as any other combination on the same line. It is a line on the graph where every point gives the same value of the objective function Z = ax + by. This is done by plotting lines parallel to the slope of the equation.

To solve the problem using the Iso-cost method, you need to follow the following steps:

Step 1: Create a mathematical formulation from the given problem. If not given.

Step 2: Now, plot the graph using the given constraints and find the feasible region.

Step 3: Now, find the coordinates of the feasible region that we get from step 2.

Step 4: Find the convenient value of Z(objective function) and draw the line of this objective function.

Step 5: If the objective function is maximum type, then draw a line that is parallel to the objective function line, and this line is farthest from the origin and only has one common point with the feasible region. Or if the objective function is minimum type, then draw a line that is parallel to the objective function line, and this line is nearest to the origin and has at least one common point with the feasible region.

Step 6: Now get the coordinates of the common point that we find in step 5. Now, this point is used to find the optimal solution and the value of the objective function.

Solved Examples of Graphical Solution of LPP using Iso-Cost Method

Example 1: Solve the given linear programming problems graphically: 

Maximize: Z = 50x + 15y

Constraints are,

• 5x + y ≤ 100
• x + y ≤ 50
• x ≥ 0, y ≥ 0

Solution: 

Given,

• 5x + y ≤ 100
• x + y ≤ 50
• x ≥ 0, y ≥ 0

Step 1: Finding points 

We can also write as
5x + y = 100....(i)
x + y = 50....(ii)

Now we find the points 
so we take eq(i), now in this equation

When x = 0, y = 100
When y = 0, x = 20

So, the points are (0, 100) and (20, 0)

Similarly, in eq(ii)

When x = 0, y = 50
When y = 0, x = 50

So, the points are (0, 50) and (50, 0)

Step 2: Now, plot these points in the graph and find the feasible region.

Step 3: Now we find the convenient value of Z(objective function) 

So, to find the convenient value of Z, we have to take the lcm of the coefficient of 50x + 15y, i.e., 150. So, the value of Z is the multiple of 150, i.e., 300. Hence, 

50x + 15y = 300

Now we find the points 

Put x = 0, y = 20

Put y = 0, x = 6

draw the line of this objective function on the graph:

Step 4: As we know that the objective function is maximum type then we draw a line which is parallel to the objective function line and farthest from the origin and only has one common point to the feasible region.

Step 5: We have a common point that is (12.5, 37.5) with the feasible region. So, now we find the optimal solution of the objective function:

Z = 50x + 15y

Z = 50(12.5) + 15(37.5)

Z = 625 + 562.5

Z = 1187.5

Thus, the maximum value of Z with the given constraint is 1187.

Example 2: Solve the given linear programming problems graphically: 

Minimize: Z = 20x + 10y

Constraints are,

• x + 2y ≤ 40
• 3x + y ≥ 30
• 4x + 3y ≥ 60
• x ≥ 0, y ≥ 0

Solution: 

Given,

• x + 2y ≤ 40
• 3x + y ≥ 30
• 4x + 3y ≥ 60
• x ≥ 0, y ≥ 0

Step 1: Finding points 

We can also write as 

l₁ = x + 2y = 40 ....(I)
l₂ = 3x + y = 30 ....(ii)
l₃ = 4x + 3y = 60 ....(iii)

Now we find the points 

So we take eq(i), now in this equation

When x = 0, y = 20
When y = 0, x = 40

So, the points are (0, 20) and (40, 0)

Similarly, in eq(ii)

When x = 0, y = 30
When y = 0, x = 10

So, the points are (0, 30) and (10, 0)

Similarly, in eq(iii)

When x = 0, y = 20
When y = 0, x = 15

So, the points are (0, 20) and (15, 0)

Step 2: Now, plot these points in the graph and find the feasible region.

Step 3: Now we find the convenient value of Z(objective function) 

So let us assume z = 0

20x + 10y = 0

x = -1/2y

draw the line of this objective function on the graph:

Step 4: As we know that the objective function is minimum type then we draw a line which is parallel to the objective function line and nearest from the origin and has at least one common point to the feasible region.

This parallel line touches the feasible region at point A. So now we find the coordinates of point A:

As you can see from the graph, at point A, l2 and l3 lines intersect, so we find the coordinate of point A by solving these equations:

l₂ = 3x + y = 30 ....(iv)

l₃ = 4x + 3y = 60 ....(v)

Now multiply eq(iv) with 4 and eq(v) with 3, we get

12x + 4y = 120 

12x + 9y = 180 

Now, subtracting both equations, we get coordinates (6, 12)

Step 5: We have a common point that is (6, 12) with the feasible region. So, now we find the optimal solution of the objective function:

Z = 20x + 10y

Z = 20(6) + 10(12)

Z = 120 + 120

Z = 240

Thus, the minimum value of Z with the given constraint is 240.

Practice Questions on Graphical Solution of LPP

Question 1. Maximize Z=4x+3y subject to:

• 2x + y ≤ 8
• x + 2y ≤ 8
• x, y ≥ 0

Question 2. Minimize Z=5x + 7y subject to:

• x + y ≥ 6
• 2x + 3y ≤ 12
• x, y ≥ 0

Question 3. Maximize Z = 2x + 5y subject to:

• x + 4y ≤ 2
• 3x + y ≤ 9
• x, y ≥ 0

Question 4. Minimize Z=3x + 4y subject to:

• x + y ≤ 5
• 2x + 3y ≥ 12
• x, y ≥ 0

Question 5. Maximize Z= 6x + 4y subject to:

• x + 2y ≤ 10
• 2x + y ≥ 8
• x, y ≥ 0

Question 6. Minimize Z= 2x + 3y subject to:

• x + y ≥ 4
• x − y ≤ 2
• x, y ≥ 0

Question 7. Maximize Z = 7x + 5y subject to:

• x + 3y ≤ 15
• 2x + y ≥ 6
• x, y ≥ 0

Question 8. Minimize Z = x + 4y subject to:

• x + y ≤ 7
• x − 2y ≥ 3
• x, y ≥ 0

Question 9. Maximize Z = 8x + 2y subject to:

• 3x + y ≤ 18
• x + 2y ≥ 10
• x, y ≥ 0

Question 10. Minimize Z= 3x + 5y subject to:

• 2x + y ≥ 7
• x + 3y ≤ 12
• x, y ≥ 0

Conclusion

The graphical method for solving linear programming problems is a powerful visualization tool for problems with two variables. By plotting constraints and identifying the feasible region, one can find the optimal solution by evaluating the objective function at the corner points. This method not only provides insights into the problem but also helps in understanding the impact of each constraint on the solution. However, for problems with more than two variables, other techniques such as the Simplex method are required.