Your mention above "I would like to solve this equation for the lowest sum of a+b+c+d+e+f" pertained to your previous question at How to minimize the six z-scores in this linear equation and then calculate a probability? , but is not the correct objective function for the problem you are stating in this question.
Let $\Phi(x)$ denote the cumulative Normal distribution function.
Then your objective is to maximize $$(1-\Phi(a)) (1-\Phi(b)) (1-\Phi(c)) (1-\Phi(d)) (1-\Phi(e)) (1-\Phi(f))$$
Using the fact that $1-\Phi(x) = \Phi(-x)$ and taking the logarithm (which does not affect the optimal values of a,b,c,d,e,f), we arrive at the objective function $$log(\Phi(-a))+log(\Phi(-b))+log(\Phi(-c))+log(\Phi(-d))+log(\Phi(-e))+log(\Phi(-f))$$
which is to be maximized subject to the linear equality constraint on a,b,c,d,e,f and the lower and upper bound constraints on a,b,c,d,e,f.
For convenience, I chose to use CVX http://cvxr.com/cvx/, which uses an approximation log_normcdf(x) (so not exact) to $log(\Phi(x)$. Using other optimization software could eliminate this inaccuracy. CVX provides the globally optimal solution (subject to the use of approximation via log_normcdf(x)).
variables a b c d e f
% constraints follow
-1.42 <= a <= 2.69
-2.16 <= b <= 2.57
-2.18 <= c <= 2.67
-0.88 <= d <= 3.29
-1.03 <= e <= 0.87
-4.26 <= f <= 2.74
0 == 2.01106 - 0.00274*(34.647+24.24*a)-0.02059*(45.647+21.122*b)+1.37984*(2.05-0.206*c)-0.01176*(10.588+11.963*d)+0.00394*(118.29-21.097*e)-0.03552*(92.17+2.855*f)
The optimal [a,b,c,d,e,f] = [ -0.8615, 1.8084, 1.0194, -0.2159, -0.7163, -0.5573].
The approximation in log_normcdf(x) has manifested itself in a discrepancy in the CVX objective function of -6.5780 (actual value) vs. -6.4742 (as calculated using the approximation during the optimization).
Exponentiating back to the original objective function (i.e., before taking log), provides the objective value 1.3906e-03.
By contrast, if we use the values of a,b,c,d,e,f instead optimized to minimize a+b+c+d+e+f subject to the constraints, the objective value per the objective function which optimally produced 1.3906e-03, produces the value 1.3599e-04, which is of course a markedly inferior solution, as should not be a surprise given that it was based on optimizing a different objective.
I leave it as an exercise to the OP to redo the optimization using "exact" values of $\Phi(x)$ or $log(\Phi(x))$.
EDIT: I have now performed the optimization using the "exact" value of the Normal cdf (but could not use CVX to do this). I obtained a slight improvement in the optimal objective value from 1.3906e-03 using the log_normcdf approximation in CVX, to 1.4036e-03 based on using the exact value of the Normal cdf. Corresponding optimal [a,b,c,d,e,f] = [ -0.8676, 1.8278, 0.9082, -0.1279, -0.6724, -0.4829], which is essentially a refinement of the solution I obtained with CVX based on the log_normcdf approximation.