7.133 Bug in the help page of simplex,ratio, Maple V to Maple 8 (13.10.02)

7.133.1 PierLuigi Zezza

In the description of many commands of the simplex package one of the input is a set or list of equations (the constraints). The use of a list of equation can lead easily to wrong results as it is shown by the following trivial example

with(simplex): 
Warning, new definition for maximize 
Warning, new definition for minimize 
Constraints 
 
 > cnsts:={2*x[2]-  x[3]<=3,x[1]-3*x[2]-  x[3]<=2,x[1]+3*x[2]+x[3]<=5}; 
 
   cnsts := {2 x[2] - x[3] <= 3, x[1] - 3 x[2] - x[3] <= 2, 
 
         x[1] + 3 x[2] + x[3] <= 5} 
Objective function 
 
 > ob:=x[1]-3*x[2]+3*x[3]; 
 
                      ob := x[1] - 3 x[2] + 3 x[3] 
Slack variables 
 
 > eqs:=setup(cnsts); 
 
   eqs := {_SL1 = 3 - 2 x[2] + x[3], _SL2 = 2 - x[1] + 3 x[2] + x[3], 
 
          _SL3 = 5 - x[1] - 3 x[2] - x[3]} 
The pivot variable 
 
 > pivotvar(ob); 
 
                                  x[3] 
THE RATIO TEST 
 
 > ratio(eqs,%); 
 
                             {infinity, 5}
 

Since two of ratios are infinity we obtain a set of TWO elements (The pivot equation should be the third and it will never be the one !!!)

 > pivoteqn(eqs,%%); 
 
                   [_SL2 = 2 - x[1] + 3 x[2] + x[3]] 
 
 > eqs:=pivot(eqs,pivotvar(ob),pivoteqn(eqs,pivotvar(ob))); 
 
   eqs := {x[3] = -2 + x[1] - 3 x[2] + _SL2, 
 
          _SL1 = 1 - 5 x[2] + x[1] + _SL2, _SL3 = 7 - 2 x[1] - _SL2} 
 
 > basis(%)=cterm(%); 
 
                    {x[3], _SL1, _SL3} = {1, -2, 7}
 

and we end up with negative values !!!

7.133.2 Robert Israel (17.10.02)

On the contrary, the problem comes from the use of a _set_ of equations rather than a list. Since the ordering of elements of a set is arbitrary, and duplicate entries are removed, it does not really make much sense to use this on a set.

The description in the help page ?simplex,ratio is incorrect:

simplex[ratio] - return a list of ratios 
... 
    C - set of linear equations 
... 
The call ratio(C, x) returns a list of ratios 
...
 

It should say that C is a list or set of ratios, and ratio(C,x) returns a list or set (whichever type is C). And there should probably be a warning that if you use a set the results should not be expected to be in the same order as the constraints. Since there may be some rare cases where you would want to use this on a set (where you aren’t going to rely on the order of the results), I hesitate to say that the use of a set here should be an error. But probably it should at least result in a warning.