00:01
So in this problem, we're asked to solve this linear program by the simplex method, and we're given that the objective function p is 5x plus 4y, subject to these constraints, 3x plus 5y is less than equal to 78, 4x plus y is less than or equal to 36, and of course, x is greater than equal 0, and y is greater than equal to 0.
00:34
Okay, so we begin by setting up our initial table.
00:38
So for iteration one, we have our coefficients.
00:47
We have x, y, our first slack variable, a second slack variable, and our bound.
00:56
And so from the first constraint right here, i can load in the coefficients 3, 5, a slack variable, and that's going to be 78.
01:11
And i'll say that my first solution is zero.
01:14
Okay.
01:16
Now, from the second constraint here, i have coefficients for one.
01:23
I'll use the second slack variable, and that's got to be 36, and again, i'll say that i'm at zero.
01:31
So then my cj, my coefficients from my objective function, give me 5 -4 -0.
01:42
And so then the zj, calculating the zj here, where i take these times these, so i have three times zero plus four times zero is zero.
02:05
Same thing over here.
02:06
I have five times zero plus one times zero is zero.
02:14
Then the delta j's across here is 5 minus 0 which is 5 4 minus 0 which is 4 of course these are 0 0 right because 1 times 0 and 0 times 0 same thing over there okay so this is the greater delta and so i will go up on the x by that so then on iteration 2 again c x y first slack variable, second slack variable on a bound.
02:56
So i'll go 5 and 1 here.
03:03
And when i do, i'm going to divide everything else by 4.
03:06
So this is 1 4th, 0, 1 4th, 9.
03:13
And up here i'll have 0 and 0 by taking, if i call this equation 1, and this equation 2, i take equation 1, minus three times equation two, then what happens? well, i have five minus three -fourths gives me 17 -fourths because five times four is 20, so 20 minus three is 17.
03:42
And i end up with still a one on that slack variable because i'll have one minus zero.
03:57
Okay, then zero minus three over four.
04:02
Minus 3 over 4.
04:05
And then i have 78 minus 3 times 9, which is 27, so that's 51.
04:17
All right, and so then my cjs are still 5400.
04:26
Then using my c's and my values for x and y now, i get 5 here, and here i get 0 times 17 .040, 5.
04:42
Times 1 over 4, that's 5 .4th.
04:47
And over here, first slack variable, i have 0, because 1 times 0 and 5 times 0.
04:52
Then i have negative 3 4 times 0, and i have 5 over 4.
04:58
So then my delta j now, 5 minus 5 is 0.
05:04
4 minus 5 over 4.
05:10
Well, that's going to be 4 times 4 is 16.
05:13
So 16 times 5 is 11 over 4.
05:16
0 minus 0.
05:18
0...