Two machines a and b produce items at the rate of 50 per...

Two machines, A and B, produce items at the rate of 50 per hour and 40 per hour, respectively. Under a certain production plan, the total number of items needed is at least 1000 items, and the total number of man-hours available for running the machines is at most 24 hours. If the hourly cost is P100 for running machine A and P70 for machine B, determine the equation that will yield the most economical production program.

Transcript

00:01 So in this problem, we're working with two machines, a and b.

00:04 And they have different production rates and different costs.

00:07 And we need to make 1 ,000 items at least.

00:13 So the way this works is machine a has a production rate of 50 items per hour.

00:19 Machine b does 40 items per hour.

00:22 And their costs are that machine a costs $100 per item.

00:27 This is, i need to clarify this, right? this is per item and machine b is $70 an item and we only have 24 man hours available so this is we can set this up as a linear programming problem first of all we could do it on the production we can write well let's talk about what we're trying to do let's do that first okay so we're trying to minimize our z function which is our costs so that's a hundred times the items produced by a plus 70 times the items produced by b.

01:08 And this is going to be subject to these constraints.

01:20 First of all, the time on each of these machines, a and b are the number of items, has to be greater and equal to zero.

01:29 It can be zero, but it's going to be something positive or zero.

01:34 Then the production there gives us 50a plus 4 .4.

01:41 40 b has got to be at least a thousand items.

01:46 So we've got to be greater than equal to a thousand.

01:49 And then the 24 man hours available right here tells us that the time we spent on a plus the time we spend on b is at most 24.

02:01 It has to be less than or equal to 24.

02:04 So what we need to do is find the corner points here for this operating envelope.

02:11 So i'm going to call this equation 1 and this equation 2.

02:14 And try to solve this to figure out the corner points now here for a minute.

02:19 Okay, so let's take minus 50.