Every Number I Guess Means Arithmetic (E.N.I.G.M.A.)

We have the following addition:

$\begin{array}{rrrrr} & a & b & c & d \\ + & & e & f & g \\ + & & & h & i \end{array}$

Before we worry about maximising $\overline{abcd}$ let us minimise the result of the addition. As there are $9$ distinct non-zero digits we know that they are the numbers $1$ through $9$ . To minimise the result of this addition, we need to put the smallest digits in the places that are most significant (i.e. the places to the left that represent the greatest powers of $10$ ). Therefore, we first place $a = 1$ , followed by $b,e = 2,3$ and $c,f,h = 4,5,6$ and $d,g,i = 7,8,9$ . These numbers can be interchanged within these tuples and it won't make any change to the resulting minimum sum.

Now to maximise $\overline{abcd}$ we need to select the greatest possible digit available for each of $a,b,c,d$ . Doing so, we get that $\overline{abcd} = \boxed{1369}$ . (Also, the minimum possible value of the sum is $1674$ )