3. linear programming- graphical solution
DESCRIPTION
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3
Linear Programming- Graphical Solution
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LtdGraphical Solutions to A LPP
Working procedure for graphical method
Step 1: Formulate the appropriate LPP.
Step 2: Draw the graph of the LPP.
Step 3: Obtain a feasible region (a region which is common to all the constraints of the LPP which is a convex region).
Step 4: Obtain the solution points (the corner points of the feasible region)
Step 5: Calculate the values of objective function at the solution points.
Step 6: For maximisation problem, the optimum solution is the solution point which gives the maximum value of the objective function and for minimisation problems the optimum solution is the solution point that gives the minimum value of the objective function.
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LtdGraphical Solutions to A LPP
Maximize Z = 40x1 + 35x2
Subject to 2x1 + 3x2 ≤ 60 4x1 + 3x2 ≤ 96
x1, x2 ≥ 0
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10 20 30
10
20
30
x1
x2
P
FR
Q
R
Iso-profit Lines
Profit = Rs 1120
Profit = Rs 280
Profit = Rs 1000
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LtdGraphical Solutions to A LPP
Point x1 x2 Z
O 0 0 0
A 0 20 700
B 18 8 1000
C 24 0 960
Optimal Solution (unique)
Optimal Solution (unique)
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LtdGraphical Solutions to A LPP
Working procedure for graphical method
Step 1: Formulate the appropriate LPP.
Step 2: Draw the graph of the LPP.
Step 3: Obtain a feasible region (a region which is common to all the constraints of the LPP which is a convex region).
Step 4: Obtain the solution points (the corner points of the feasible region)
Step 5: Calculate the values of objective function at the solution points.
Step 6: For maximisation problem, the optimum solution is the solution point which gives the maximum value of the objective function and for minimisation problems the optimum solution is the solution point that gives the minimum value of the objective function.
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LtdGraphic Solution:
Max ProblemMaximize Z = 40x1 + 35x2
Subject to 2x1 + 3x2 ≤ 60 4x1 + 3x2 ≤ 96
x1, x2 ≥ 0
A
B
CO 10 20 30
10
20
30
x1
x2
Point x1 x2 Z
O 0 0 0
A 0 20 700
B 18 8 1000
C 24 0 960
Optimal Solution (unique)
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LtdGraphic Solution: Max Problem
(using Iso-profit Lines)
10 20 30
10
20
30
x1
x2
P
FR
Q
R
Iso-profit Lines
Profit = Rs 1120
Profit = Rs 280
Profit = Rs 1000
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LtdGraphic Solution:
Min ProblemMinimize Z = 40x1 + 24x2
Subject to 20x1 + 50x2 ≥ 4800 80x1 + 50x2 ≥ 7200 and x1 x2 ≥ 0
Point x1 x2 Z
P 0 144 3456
Q 40 20 3520
R 240 8 9600
Optimal Solution
36
72
108
144
Feasible Region
P
Q
R
60 120 180 240
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Bounded and Unbounded Feasible Regions
FR
FR
Bounded Feasible Region
Unbounded Feasible Region
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Redundant ConstraintsMinimize Z = 40x1 + 35x2
Subject to x1 + x2 ≤ 40 4x1 + 3x2 ≤ 96 2x1 + 3x2 ≤ 60 x1, x2 ≥ 0
x1 + x2 = 40
4x1 + 3x2 = 96
2x1 + 3x2 = 60
x1
x2 Redundant Constraint
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LtdBinding and Non-binding
Constraints Binding Constraint: If the LHS is equal to RHS
when optimal values of the decision variables are substituted in to the constraint
Non-binding Constraint: If LHS ≠ RHS on such substitution of optimal values
For min problem solution,20×0 + 50×144 = 7200 ≠ 4800 (RHS)80×0 + 50×144 = 7200 (=RHS)
Binding constraintNon-Binding constraint
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Solutions to LPPs
Unique Optimal Solution
Multiple Optimal Solutions
Infeasibility: No feasible solution
Unbounded Solution
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LtdGraphical Solutions to A LPP
Maximize Z = 40x1 + 35x2
Subject to 2x1 + 3x2 ≤ 60 4x1 + 3x2 ≤ 96
x1, x2 ≥ 0
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The solution space (feasible region) is ABCDE. The co-ordinates of the corner points of the feasible region are obtained and is given in the above figure. The values of the objective function at the corner points are z = x1 – 2x2
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Since the problem is of maximisation type the optimal solution is x1 = 5, x2 = 2 with maximum of z = 1
Redundant constraint: In a given LPP if any constraint does not affect the feasible region, then the constraint is said to be a redundant constraint.
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Infinite number of solutions: In some cases the maximum or minimum value of z occurs at more than one corner point of a feasible region. A point on the line joining that points will also give the same maximum/minimum value of z. Thus, there are infinite number of optimal solutions for this LPP.
An LPP having more than one optimal solution is said to have alternate or multiple optimal solutions. That is the resources can be combined in more than one way to maximise the profit.
Unbounded solution: When the value of decision variables in linear programming is permitted to increase infinitely without violating the feasibility condition, then the solution is said to be unbounded. Here the objective function value can also be increased indefinitely.
In graphical method if the feasible region is unbounded then we have to find the value of the objective function at the known corner points. If there are some points in the feasible region which give greater / lesser value of the objective function then we conclude that LPP has an unbounded solution.
No feasible solution: If there is no feasible region, we conclude that there exists no feasible solution to the given LPP.