subject: Linear Programming Graphical Solution [print this page] Linear programming graphical is represented by Z which denotes for certain conditions for the variable, the importance task for the LPP is to optimize the value whether it is maximize or minimize.
Linear programming structure:
The standard form of the linear programming problem is used to develop the procedure for solving a general programming problem.
A general LPP is of the form
Max (or min) Z = c1x1 + c2x2 + +cnxn
x1, x2, ....xn are called decision variable.
Examples for Linear Programming Graphical Solution:
Examples:
1. To determine the feasible solution of an LPP, we have the following steps.
Let us consider to find the feasible solution for the decorative item
Z = 50x + 18y ... (1)
Subject to the constraints
2x + y 'less than or equal to' 100
x + y 'less than or equal to' 80
x greater than or equal to0, y 'greater than or equal to' 0
Step 1:
Since x0, y0, we consider only the first quadrant of the xy - plane
Step 2:
We draw straight lines for the equation
2x+ y = 100 ... (2)
x + y = 80
To determine two points on the straight line 2x + y = 100
Put y = 0, 2x = 100
x = 50
(50, 0) is a point on the line (2)
Put x = 0 in (2), y =100
(0, 100) is the other point on the line (2)
Plotting these two points on the graph paper draw the line which represent the line
2x + y =100.
This line divides the 1st quadrant into two regions, say R1 and R2. Choose a point say (1, 0) in R1. (1, 0) satisfy the in equation 2x + y 100. Therefore R1 is the required region for the constraint 2x + y 100.
Similarly draw the straight line x + y = 80 by joining the point (0, 80) and (80, 0). Find the required region say R1', for the constraint x + y 80.
The intersection of both the region R1 and R1' is the feasible solution of the LPP. Therefore every point in the shaded region OABC is a feasible solution of the LPP, since this point satisfies all the constraints including the non-negative constraints of graphical .
2. To find the optimal solution of an LPP:
The optimal solution to a LPP is determined by Corner Point Method.
The method includes the following steps
Step 1:
Find the feasible region of the LLP.
Step 2:
Find the co-ordinates of each vertex of the feasible region.
The above co-ordinates can be obtained by solving the equation of the lines.
Step 3:
At both corner points find the value of the objective function.
Step 4:
Identify the corner point at which the value of the objective function is maximum (or minimum depending on the LPP)
The co-ordinates of above vertex is the optimal solution and the value of Z is the optimal value.
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Example:
Determine the optimal solution for the above problem of decorative item dealer whose objective function is Z = 50x + 18y.
In the graph, the corners of the feasible region are
O (0, 0), A (0, 80), B(20, 60), C(50, 0)
At (0, 0) Z = 0
At (0, 80) Z = 50 (0) + 18(80)
= Rs. 1440
At (20, 60), Z = 50 (20) +18 (60)
= 1000 + 1080 = Rs.2080
At (50, 0) Z = 50 (50) + 18 (0)
= Rs. 2500.
Since our object is to maximize Z and Z has maximum at (50, 0) the optimal solution is
x = 50 and y = 0.
The optimal value is Rs. 2500.
Linear Programming Graphical Solution Applications:
Application Areas of Linear Programming graphical
Linear Programming has much application area. They are,
1. Transportation Problem
2. Military Applications
3. Operation of System of Dams
4. Personnel Assignment Problem
5. Manufacturing plants
Limitations of Linear Programming:
(a). Linear programming is applicable only to problems which can be expressed as equations of straight lines. This technique cannot be used when constraints or objective functions are not linear in real life situations.
(b). The term uncertainty, weather conditions are not taken into consideration.
(c). It can not accept an integer as the solution.
(d). These technique may give practical valued answer which is not desirable.
(e). It deals only one objective problems in real life.
