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I have a simple maximization problem. $\ 7x_1+9x_2+5x_3 => max$

With following constraints:

$\ 3x_1+ 4x_2+ 1.5x_3 <= 100 $

$\ 19x_1+ 12x_2+13x_3 <= 1000 $

$\ 8x_1+ 10x_2+9x_3 <= 500 $

$\ 2x_1+ 3x_2+1.9x_3 <= 75 $

After applying simplex method I receive following solution.

$\ x_1 = 9.45, x_3 = 47.14, d_2 =203.5, d_4 = -33.42 $

With some rounding involved, so it is a little rough. However when checking the solution it became apparent that condition number 4 has been broken.

I thought that I made some numerical mistake but online solver came up with the same solution.

It appears that for some reason the algorithm came up with optimal answer violating one of the conditions, which I though to be impossible.

So either the simplex does not work like I think it does or I made some mistake when defining the constraints.

Anyway I wonder what exactly happened and if there is a way how to prevent it from happening again.

Edit: $\ x_2$ is supposed to be multiplied by 9 in the problem.

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    Oh, I made a mistake writing the equation. Going to fix it. – 2017-01-17

1 Answers 1

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I have used this calculator and its calculation steps are the following:

Bas x1  x2  x3  x4  x5  x6  x7  b
x4  3   4   1.5 1               100
x5  19  12  13      1           1000
x6  8   10  9           1       500
x7  2   3   1.9             1   75
F   -7  -9  -5  0   0   0   0   0


Bas x1      x2  x3      x4     x5   x6  x7  b
x2  0.75    1   0.375   0.25    0   0   0   25
x5  10      0   8.5     -3      1   0   0   700
x6  0.5     0   5.25    -2.5    0   1   0   250
x7  -0.25   0   0.775   -0.75   0   0   1   0
F   -0.25   0   -1.625  2.25    0   0   0   225



Basis   x1         x2   x3      x4         x5   x6      x7          b
x2  0.870967741935  1   0   0.612903225806  0   0   -0.483870967742 25
x5  12.7419354839   0   0   5.22580645161   1   0   -10.9677419355  700
x6  2.1935483871    0   0   2.58064516129   0   1   -6.77419354839  250
x3  -0.322580645161 0   1   -0.967741935484 0   0   1.29032258065   0
F   -0.774193548387 0   0   0.677419354839  0   0   2.09677419355   225

Bas x1    x2    x3   x4    x5   x6    x7        b
x1  1   1.148   0   0.704   0   0   -0.556  28.703703
x5  0   -14.630 0   -3.740  1   0   -3.889  334.259
x6  0   -2.519  0   1.037   0   1   -5.556  187.037
x3  0   0.370   1   -0.7407 0   0   1.111   9.259259
F   0   0.888   0   1.2222  0   0   1.667   247.222

It satisfies the fourth condition:$2\cdot 28.703703+1.9\cdot 9.259259=74.9999981$

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    Ok so it appears that the process goes wrong in the second step. The calculator I used didn't consider $/ b_4 $ to be eligible for the check because it was not positive. The question is: is it possible to perform check on non positive RHS variable? – 2017-01-17
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    @user1561358 What do you mean ? Please be more specific. – 2017-01-17
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    I updated the comment. I hit the enter by accident. – 2017-01-17
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    @user1561358 The simplex rule refers to non-negative $b_i´s$. This is the rule. The rule cannot be changed by anybody. – 2017-01-17