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I wonder if someone can help me on this.

In planning their production of two products, X and Y, a company has to take into account the demand for these products, as well as their internal production capacity. In addition they can (if necessary) buy in these products from a third party supplier.

For the forthcoming month demand is estimated to be 120 units for X and 150 units for Y. The company sells these products for $25 and $34 for X and Y respectively. The company can buy X from its third party supplier for $20 per unit, and Y for $24 per unit.

These products are produced on a single machine in the company. This machine costs $3 per hour to run when making X or Y and there are 175 working hours available in the forthcoming month on this machine for the production of X or Y. Producing one unit of X on the machine requires 4.5 hours, producing one unit of Y requires 6.5 hours. Technological constraints mean that the ratio of the number of units of Y produced on the machine to the number of units of X produced on the machine must be at least 1.3.

By formulating and solving an appropriate linear program determine (for the forthcoming month) how much of each product should be made and how much should be bought from the third party supplier.

I calculated it as follows:

profit for X produced is GBP 25 -($3 x 4.5)=11.5

Profit for Y produced is GBP 34-($3x6.5)=14.5

thus

maximise11.5x +14.5y

Constraints

1.3y-x=0

4.5x+6.5y=175

feasible region is at the vertex of the above curves

so

x=1.3y

4.5 (1.3y) + 6.5y =175

5.85y + 6.5y=175

12.35y=175

y=14.17 (to two decimal points)

x = 18.42 (to two decimal points)

so X bought = 120-14 = 106

Y bought = 150-18=132

but I am sure that I am wrong somewhere as I did not use buying cost of X and Y

I thought to calculate total profit of X as 11.5 (produced)+ 5 (bought)=16.5

Y 14.5 (produced) and 10 (bought)= 24.5

and rewrite

maximise

16.5x+24.5 y

but I am not sure...totally lost