Autoparts Inc. produces three types of parts' assemblies: (P,Q,R)and they each require (2,5,3) units of work on machine I and (2,2,4) units of work on machine II respectively. The two machines have respectively (160,190) units of work available per day. It is given that their profits are: (10,12,9) dollars respectively.
Assume that the production is given by x,y,z respectively and construct the following:
The profit function to be maximized is:?
The restriction imposed by machine I is:
? <= ?.
The restriction imposed by machine II is: ? <= ?.
Assume that the total production of the product P is required to be greater than or equal to the total production of the other two. Then we get the constraint: x >= ?.
To construct the profit function to be maximized, we need to multiply the production of each part assembly (x, y, z) with their respective profits (10, 12, 9), and then sum them up.
Profit function to be maximized = 10x + 12y + 9z
The restriction imposed by machine I is that the total work used on machine I should be less than or equal to the available work on machine I, which is 160 units. The work used on machine I for each part assembly is given as (2x, 5y, 3z).
Restriction imposed by machine I: 2x + 5y + 3z <= 160
Similarly, the restriction imposed by machine II is that the total work used on machine II should be less than or equal to the available work on machine II, which is 190 units. The work used on machine II for each part assembly is given as (2x, 2y, 4z).
Restriction imposed by machine II: 2x + 2y + 4z <= 190
Lastly, we are given that the total production of part assembly P should be greater than or equal to the total production of the other two part assemblies. Since P has production x, and total production of the other two is y + z, the constraint can be written as:
Constraint: x >= y + z