Incorrect algorithm

[denisorehovsky]

I think this algorithm doesn't work properly. For example, for this data:

F = 300x1 + 250x2 + 450x3 --> max
15x1 + 20x2 + 25x3 <= 1200
35x1 + 60x2 + 60x3 <= 3000
20x1 + 30x2 + 25x3 <= 1500
250*x2 >= 500
x >= 0

Returns: [(0, 12.0), (2, 180.0), (4, 60.0)], 23400.0

But the answer should be:
F = 23060
x1 = 56, x2 = 2, x3 = 12.8

Also, I've checked some other data. And for this data, it doesn't work at all.

[j2kun]

The code indeed does not solve this correctly. I don't yet have a fix.

From what I can see, variable x2 is never pivoted. After inspecting the pivot choices (which are heuristics), I believe the pivot rules are to blame. I previously thought that improving "any" improvable column would work, but this is apparently incorrect. The solution that was reported (but had an indexing bug) was x1 = 60, x2=0, x3 = 12. This is obviously wrong because of the constraint that x2 >= 2.

If I force the code to first pivot on the tableau entry (1,1), then I let it solve to completion as using a better pivot choice heuristic (max of bottom row, min quotient), I get the right output:

[3.333333333333332, 0.0, 5.0, 1.0, -0.3333333333333333, 0.0, 0.0, 200.0]
[0.5833333333333334, 1.0, 1.0, 0.0, 0.016666666666666666, 0.0, 0.0, 50.0]
[2.5, 0.0, -5.0, 0.0, -0.5, 1.0, 0.0, 0.0]
[-145.83333333333334, 0.0, -250.0, 0.0, -4.166666666666667, 0.0, -1.0, -12000.0]
[154.16666666666666, 0.0, 200.0, 0.0, -4.166666666666667, 0.0, 0.0, -12500.0]
Next pivot index is=0,2 

Tableau after pivot:
[0.6666666666666664, 0.0, 1.0, 0.2, -0.06666666666666667, 0.0, 0.0, 40.0]
[-0.08333333333333304, 1.0, 0.0, -0.2, 0.08333333333333333, 0.0, 0.0, 10.0]
[5.833333333333332, 0.0, 0.0, 1.0, -0.8333333333333333, 1.0, 0.0, 200.0]
[20.833333333333258, 0.0, 0.0, 50.0, -20.833333333333336, 0.0, -1.0, -2000.0]
[20.83333333333337, 0.0, 0.0, -40.0, 9.166666666666668, 0.0, 0.0, -20500.0]

Next pivot index is=3,0 

Tableau after pivot:
[0.0, 0.0, 1.0, -1.4000000000000052, 0.6000000000000023, 0.0, 0.032000000000000105, 104.00000000000021]
[0.0, 1.0, 0.0, 0.0, -2.7755575615628914e-17, 0.0, -0.004, 1.9999999999999982]
[0.0, 0.0, 0.0, -13.000000000000048, 5.000000000000021, 1.0, 0.28000000000000097, 760.0000000000019]
[1.0, 0.0, 0.0, 2.400000000000009, -1.0000000000000038, 0.0, -0.048000000000000174, -96.00000000000036]
[0.0, 0.0, 0.0, -90.00000000000028, 30.000000000000117, 0.0, 1.0000000000000056, -18499.99999999999]

Next pivot index is=2,4 

Tableau after pivot:
[0.0, 0.0, 1.0, 0.1599999999999997, 0.0, -0.11999999999999995, -0.0015999999999999973, 12.800000000000011]
[0.0, 1.0, 0.0, -7.216449660063513e-17, 0.0, 5.551115123125759e-18, -0.003999999999999998, 2.000000000000002]
[0.0, 0.0, 0.0, -2.5999999999999983, 1.0, 0.19999999999999915, 0.05599999999999995, 151.99999999999974]
[1.0, 0.0, 0.0, -0.1999999999999993, 0.0, 0.1999999999999999, 0.007999999999999986, 55.99999999999996]
[0.0, 0.0, 0.0, -12.000000000000028, 0.0, -5.999999999999998, -0.6799999999999995, -23060.0]

Up to rounding error, you can see x3=12.8, x2=2, x1=56.

However, even with the updated pivot selection rule, the first chosen pivot is not (1,1). I will have to think more on this.

[j2kun]

Ah, I should be selecting the column according to the minimum positive entry in the last row. Fixing.