Thread: Help with Factory Planning Problem

Hi everyone!

I wonder if someone could help with this. I am new to this site, so I am not quite sure whether I am posting it in correct field.

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.

Calculate how much profit you make by producing and selling one unit of X, and calculate the same for Y. Also calculate how much profit you make by buying one unit of X from your third-party supplier and selling it, and calculate the same for Y. Since you are going to make a profit on any of these actions, you should try to meet the demand for both X and Y through some combination of these actions, but you have to prioritize based on what makes you the most money. I would suggest writing a function of total profit in terms of p and q, where p is the number of units of X produced, and q is the number of units of Y produced. Then 120 - p should be the number of units of X purchased from the third party supplier and 150 - q should be the number of units of Y purchased from the third party supplier. Then maximize the function.

Constraints are:

Thanks a lot for this.

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


iso-profit 11.5x +14.5y=180

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

Does anyone else have any suggestions/ideas on this matter, please?