Finally, the total demand for widgets is only 100,000 widgets a day.

So if you make any more than this,

they just won't sell, and that's no good for anybody.

So writing these constraints down in a reasonable way,

if we let W be the number of workers that we have.

And M the number of machines, we have a bunch of constraints.

The number of workers should be non-negative,

the number of machines should be between 0 and 100.

The number of workers needs to be at least twice the number of machines.

And then finally, 100,000 is at least 200 times the number of unoccupied workers.

That's W minus 2M, plus 600 times the number of the machines.

And so these constraints sort of constrain which allowable combinations we can have.

Now we can try and graph these constraints.

So here we've got a plane of possible values of M and

W that satisfy these constraints.

Now if we're just starting where M and W both need to be non negative,

we have this quadrant as the allowable values.

But when we require that M needs to be at least 100,

we're reduced to being in this strip here.

When we look at our constraint based on the total demand,

we find that M + W is at most 500.

And so we're now constrained to this region.

And we add the final constraint that the workers need to be at least

twice the number of machines.

We finally come to this diagram of possible configurations

of machines and workers that we can use.

What's next?

Profit, well suppose that profits are determined as follows.

Each widget that you make earns you a $1 but

each worker that you're hiring costs you $100 a day.

So the total profit that you get, then, in terms of dollars per day.

Well it's number of widgets, 200 workers minus twice machines,

plus 600 times number of machines, minus the total salaries you paid to workers,

100 times the number of workers.

So that's 100 times the number of workers plus 200 times the number of machines.

And if we want to plot that on our graph we can do it as follows.

So these lines that I've drawn are lines of equal profit.

There's a line with $30,000 a day, and then $40,000 a day, and

then $50,000 a day.

And sort of as you go from left to right, or

from bottom to top, you make more profit.