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Home  >>  CBSE XII  >>  Math  >>  Linear Programming
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One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

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  • To solve a Linear Programming problem graphically, first plot the constraints for the problem. This is done by plotting the boundary lines of the constraints and identifying the points that will satisfy all the constraints. One we graphically plot the area bounded by the constraints, it’s easy to see which points satisfy all constraints. This common region determined by all the constraints including non-negative constraints of a linear programming problem is called the $\textbf{Feasible Region (or solution region).}$
  • Now, any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an $\textbf{Optimal Solution}$. We see that every point in the feasible region satisfies all the constraints, and there are infinitely many points.
  • Since we know from theory that the optimal value must occur at a corner point (vertex) of the feasible region, calculate the objective function values associated with the coordinates of all the extreme points. This method is called the $\textbf{Corner Point Method}$.
  • If the feasible region is bounded (if it can be enclosed), the point with the best objective function value is the best optimal solution. If the feasible region is unbounded (means that the feasible region does extend indefinitely in any direction), the then a maximum or a minimum value of the objective function may not exist. However, if it exists, it must occur at a corner point of the feasible region, which can be calculated.
We are given 5kg of flour and 1 kg of fat. Let x be the number of Cakes of Type-1 and y be the number of cakes of Type-2 that we can make. Our problem is to maximize x + y.
Clearly, x, y ≥ 0. Let us construct the following table from the given data (all numbers converted from kg to g)
  Cake1 Cake2 Requirements
Flour A 200 100 5000
Flour B 25 50 1000
Since we have 5000g of Flour and 1000g of fat, we have to maximize x + y, given the following constraints:
$(1): 200x + 100y \leq5000 \to 2x+y \leq 50$
$(2): 25x + 50y \leq 1000 \to x+2y \leq 40$
$(3): x \leq 0, \; (4): y \leq 0$
$\textbf{Plotting the constraints}:$
Plot the straight lines 2x+y = 50 and x+2y = 40.
First draw the graph of the line 2x+y = 50. If x = 0, y = 50 and if y = 0, x = 25. So, this is a straight line between (0,50) and (25,0).
At (0,0), in the inequality, we have 0 + 0 = 0 which is≤ 0. So the area associated with this inequality is bounded towards the origin.
Similarly, draw the graph of the line x + 2y = 40. If x = 0, y = 20 and if y =0, x = 40. So, , this is a straight line between (0,20) and (40,0).
At (0,0), in the inequality, we have 0 + 0 = 0 which is ≤ 0. So the area associated with this inequality is bounded towards the origin.
$\textbf{Finding the feasible region}$:
We can see that the feasible region is bounded and in the first quadrant.
On solving the equations 2x + y = 50 and x + 2y = 40, we get,
2 (40 – 2y) + y = 50 $\to$ 80 – 4y + y = 50 $\to$ -2y = -30 $\to$ y = 10.
If y = 10, then x = (50 – 10)/2 = 20.
$\Rightarrow x = 20, y = 10$
Therefore the feasible region has the corner points (0,0), (0,20), (20, 10), (25,0) as shown in the figure.
$\textbf{Solving the objective function using the corner point method}$:
The values of Z at the corner points are calculated as follows:
Corner Point x+y
O (0,0) 0
C(0,2) 20
E(20,10) 30 (Max Value)
B(25,0) 25
$\textbf{The maximum number of cakes we can make is 30, 20 of one kind and 10 of another. }$

 

answered Apr 17, 2013 by balaji.thirumalai
 

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