Solve each of the following linear programming problems by graphical method.
Maximize Z = 2x + 4y
Subject to :
x + y ≥ 8
x + 4y ≥ 12
x ≥ 3
y ≥ 2
Objective function is: Z = 2x + 4y
x + y ≥ 8
x + 4y ≥ 12
x ≥ 3
y ≥ 2
First convert the given inequations into corresponding equations and plot them:
x + y ≥ 8 → x + y = 8 (corresponding equation)
Two coordinates required to plot the equation are obtained as:
Put, x = 0 ⇒ y = 8 (0,8) - - - - first coordinate.
Put, y = 0 ⇒ x = 8 (8,0) - - - - second coordinate
Join them to get the line.
As we know, Linear inequation will be a region in the plane, and we observe that the equation divides the XY plane into 2 halves only, so we need to check which region represents the given inequation,
If the given line does not pass through origin then just put (0,0) to check whether inequation is satisfied or not. If it satisfies the inequation origin side is the required region else the other side is the solution.
Similarly, we repeat the steps for other inequation also and find the common region.
As, x + 4y ≥ 12 → x + 4y = 12
Put x = 0 ⇒ y = 3 coordinate - - - - - (0,3)
Put y = 0 ⇒ x = 12 coordinate - - - - - - (12,0)
y = 2 (line parallel to x - axis passing through (0,2))
x = 3 (line parallel to x – axis passing through (3,0))
Hence, we obtain a plot as shown in figure:
The shaded region in the above figure represents the region of feasible solution.
Now to maximize our objective function, we need to find the coordinates of the corner points of the shaded region.
We can determine the coordinates graphically our by solving equations. But choose only those equations to solve which gives one of the corner coordinates of the feasible region.
Solving x + y = 8 and x = 3 gives (3, 5)
Similarly, solve other combinations by observing graph to get other coordinates.
From the figure we have obtained coordinates of corners as:
(3,5) and (6,2)
Now we have coordinates of the corner points so we will put them one by one to our objective function and will find at which point it is maximum.
∵ Z = 2x + 4y
∴ Z at (3,5) = 2×(3) + 4×(5) = 26
Z at (6,2) = 2 × (6) + 4 × (8) = 20
Note: As the region is unbounded as we can’t say blindly that Z = 24 is maximum because there might be other points in the feasible region that may Make Z even greater.
So we need to check whether Z is maximum or not or Z greater than 24 or not.
For this we define inequation using the optimal function if the solution region of the inequation does not coincide with the feasible region, it means it has a maxima
Inequation : 2x + 4y > 24 or x + 2y > 12
Now we again plot the graph with the constraints and the above inequation
Clearly, x + 2y > 24 has solutions in feasible region.
This proves that the values of Z greater than 24 are possible.
∴ Optimal value of Z is not possible.
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