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A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs. 5760.00 to invest and has space for at most 20 items. An electronic sewing machine costs him Rs. 360.00 and a manually operated sewing machine Rs. 240.00. He can sell an Electronic Sewing Machine at a profit of Rs. 22.00 and a manually operated sewing machine at a profit of Rs. 18.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a linear programming problem and solve it graphically. Keeping the rural background in mind, justify the 'values' to be promoted for the selection of the manually operated machine.

Asked by Topperlearning User 31st July 2016, 7:53 PM
Answered by Expert
Answer:

Suppose the dealer buys x fans and y sewing machines. Since the dealer has space for at most 20 items.  Therefore,

                   

          A fan costs Rs. 360 and a sewing machine costs Rs. 240.  Therefore, total cost of x fans and y sewing machine is Rs. (360 x  + 240y).  But the dealer has only Rs.5760 to invest.

          Therefore,

                   

          Since the dealer can sell all the items that he can buy and the profit on a fan is Rs.22 and on a sewing machine the profit is of Rs. 18.  Therefore, total profit on selling x fans and y sewing is of Rs. (22 x + 18 y).

          Let Z denote the total profit.  Then, Z = 22x + 18y.

          Clearly, x comma y space greater or equal than space 0

          Thus, the mathematical formulation of the given problem is

          Maximize 

          Subject to

                   

                        

                   

          and 

          To solve this LPP graphically, we first convert the inequations into equations and draw the corresponding lines. The feasible region of the LPP is shaded in Figure. The corner points of the feasible region OA2 PB1 are (0, 0), A2 (16, 0), P(8, 12) and B1 (0, 20).

 

 

 

          These points have been obtained by solving the corresponding intersecting lines, simultaneously.

          The values of the objective function Z at corner – points of the feasible region are given in the following table.

         

Point (x, y)

Value of the objective function

Z = 22 x+ 18y

O(0 0)

 

A2 (16, 0)

 

P(8, 12)

 

B1 (0, 20)

 

 

          Clearly, Z is maximum at x = 8 and y = 12.  The maximum value of Z is 392.

          Hence, the dealer should purchase 8 fans and 12 sewing machines to obtain the maximum  profit under given conditions.

Answered by Expert 31st July 2016, 9:53 PM
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