I'm working on a problem I think I have figured out, but would like to see what you think.
Toys for Kids produces two types of toys (for boys and girls) by their 40 kilos of plastic material available. A boy's toy requires twice as much plastic as a girl's toy. Each kilogram of plastic yields 4 girls' toys. Each boys' toy sells for $33 whereas each girls' toy sells for $32. One kilogram of plastic needs five hours to produce a boys' toy, and ten hours of labor to produce a girls' toy. Up to 300 hours of labor is available at the cost of $12 an hour.
Formulate as an optimisation problem. State the decision variables, objective function and constraints
I've set it up like this:
Decision variables: x1= number of boys toys produced
x2 = number of girls toys produced
Objective function: Each kilo of plastic produces 4 girls toys' which means they will make (4*32=$128). It will cost them $120 (10 hours*$12/hr) in labour. Thus $8 profit (x2)
Each kilo of plastic produces 2 boys toys (requires twice as much, can only produce half as much per kilo of plastic) They will make (2*33=$66). It will cost them $60 in labour (5*$12). Thus $6 profit. (x1)
Thus objective function is to max profit; max z = 6x1+8x2
And constraints are: 5x1+10x2 <= 300
& 2X1+4X2 <= 40 (This one I'm not 100% sure about. My reasoning is that each kilo produces 4 girls toys, the boys toys need twice as much plastic so for each unit can only contain half.. Should the x1 coefficient be different?)
x1,x2 >= 0
My solution (I won't bother putting it here) told me to make 20 boys toys and 0 girls toys for a profit of $120.00. Not sure if this makes sense, to make no girls toys.
Anyways, if anybody could help me out would be most appreciative.