A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs17.50 per package on nuts and Rs7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs17.50 per package on nuts and Rs7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?
Solution and Explanation
Let the manufacturer produce x packages of nuts and y packages of bolts.
Therefore, x≥0 and y≥0 The given information can be compiled in a table as follows.
| Nuts | Bolts | Availability | |
| Machine(A)h | 1 | 3 | 12 |
| Machine(B)h | 3 | 1 | 12 |
The profit of packaging of nuts is Rs17.50 and on the packaging of bolts is Rs7.
Therefore, the constraints are x+3y≤12 3x+y≤12 Total profit,Z=17.5x+7y
The mathematical formulation of the given problem is
Maximise Z=17.5x+7y.....(1)
Subject to the constraints,
x+3y≤12...(2)
x,y≥0...(4)
The feasible region determined by the system of constraints is as follows.
The corner points are A(4,0),B(3,3),C(0,4).
The values of Z at these corner points are as follows.
The maximum value of Z is Rs.73.50 at(3,3).
Thus, 3 packages of nuts and 3 packages of bolts should be produced each day to get the maximum profit of Rs.73.50.
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Notes on Linear Programming Problem
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.