A factory makes tennis rackets and cricket bats.A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman's time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman's time.In a day,the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman,s time.
(ii)What number of rackets and bats must be made if the factory is to work at full capacity?
(iii)if the profit on a racket and on a bat is Rs20 and Rs10 respectively, find the maximum profit of the factory when it works at full capacity.
Solution and Explanation
(i)Let the number of rackets and the number of bats be made be x and y respectively.
The machine time is not available for more than 42 hours.
∴ 1.5x+3y≤42...(1) The craftsman's time is not available for more than 24 hours.
∴3x+y≤24...(2)
The factory is to work at full capacity.
Therefore, 1.5x+3y=42 3x+y=24
On solving these equations,we obtain x=4 and y=12
Thus,4 rackets and 12 bats must be made.
(i)The given information can be compilled in a table as follows.
| Tennis Racket | Cricket Bat | Availability | |
| Machine Time(h) | 1.5 | 3 | 42 |
| Craftsman's Time(h) | 3 | 1 | 24 |
∴1.5x+3y≤42 3x+y≤24 x,y≥0
The profit on a racket is Rs20 and on a bat is Rs10. ∴Z=20x+10y
The mathematical formulation of the given problem is
Maximize
Z=20x+10y...(1)
Subject to the constraints,
1.5x+3y≤42...(2)
3x+y≤24...(3)
x,y≥0...(4)
The feasible region determined by the system of constraints is as follows. The corner points are A(8,0),B(4,12),C(0,14),and O(0,0). The values of Z at these corner points are as follows.
Thus, the maximum profit of the factory when it works to it full capacity is Rs200.
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Concepts Used:
Linear Programming Problems
The Linear Programming Problems (LPP) is a problem that is concerned with finding the optimal value of the given linear function. The optimal value can be either maximum value or minimum value. Here, the given linear function is considered an objective function. The objective function can contain several variables, which are subjected to the conditions and it has to satisfy the set of linear inequalities called linear constraints.
Linear Programming Simplex Method
Step 1: Establish a given problem. (i.e.,) write the inequality constraints and objective function.
Step 2: Convert the given inequalities to equations by adding the slack variable to each inequality expression.
Step 3: Create the initial simplex tableau. Write the objective function at the bottom row. Here, each inequality constraint appears in its own row. Now, we can represent the problem in the form of an augmented matrix, which is called the initial simplex tableau.
Step 4: Identify the greatest negative entry in the bottom row, which helps to identify the pivot column. The greatest negative entry in the bottom row defines the largest coefficient in the objective function, which will help us to increase the value of the objective function as fastest as possible.
Step 5: Compute the quotients. To calculate the quotient, we need to divide the entries in the far right column by the entries in the first column, excluding the bottom row. The smallest quotient identifies the row. The row identified in this step and the element identified in the step will be taken as the pivot element.
Step 6: Carry out pivoting to make all other entries in column is zero.
Step 7: If there are no negative entries in the bottom row, end the process. Otherwise, start from step 4.
Step 8: Finally, determine the solution associated with the final simplex tableau.