Question:

A cottage industry manufactures pedestal lamps and wooden shades,each requiring the use of a grinding/cutting machine and a sprayer.It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp.It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade.On any day the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours.The profit from the sale of a lamp is Rs5 and that from a shade is Rs3.Assuming that the manufacturer can sell all the lamps and shades that he produces,how should he schedule his daily production in order to maximize his profit?

Updated On: Sep 20, 2023
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Solution and Explanation

Let the cottage manufacture industry x pedestal lamps and y wooden shades.

Therefore, x≥0 and y≥0 The given information can be compiled in a table as follows.

 LampsShadesAvailability
Grinding/Cutting Machine(h) 2112
Sprayer(h)3220

Therefore, the constraints are 2x+y≤12 3x+2y≤20 Total profit,Z=5x+3y

The mathematical formulation of the given problem is
Maximize Z=5x+3y...(1)

Subject to the constraints,
2x+y≤12...(2)
3x+2y≤20....(3)
x,y≥0....(4)

The feasible region determined by the system of constraints is as follows.

feasible region determined by the system of constraints

The corner points are A(6,0),B(4,4)and C(0,10).
The value of Z at these corner points are as follows.

Corner pointsZ=5x+3y 
A(6,0)30 
B(4,4)32→Maximum
C(0,10)30 

Thus, the manufacturer should produce 4 pedestal lamps and 4 wooden shades to maximize his profits.

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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.