A variable is ineligible to be substituted out of a problem if (a) it is subject to any bounds or integrality conditions, or (b) it has already appeared on the right-hand side of a constraint that was used to make a substitution.

Case (a) includes bounds or integrality conditions specified in the variable’s declaration, and also bounds that are added by AMPL’s presolve phase. Thus turning presolve off (by setting `option presolve 0`

) may permit a greater number of substitutions.

For constraints indexed over a set, the incidence of case (b) may depend on the ordering of the set. Consider for example the constraints

```
var x {1..10};
subj to step {j in 1..9}: x[j] = x[j+1] + 1;
```

The first several constraints generated are

```
step[1]: x[1] = x[2] + 1
step[2]: x[2] = x[3] + 1
step[3]: x[3] = x[4] + 1
step[4]: x[4] = x[5] + 1
```

Constraint `step[1]`

may be used to substitute for `x[1]`

. Constraint `step[2]`

may not be used to substitute for `x[2]`

, however, because `x[2]`

has already appeared on the right-hand side of a constraint, namely `step[1]`

, that was used to make a substitution. Similarly, `step[3]`

may be used to substitute for `x[3]`

, but `step[4]`

may not be used to substitute for `x[4]`

, and so forth. Only the odd-numbered constraints `step[j]`

are eliminated by substitution in this case.

If instead you write the declaration with the ordered index set reversed,

```
subj to step {j in 9..1 by -1} x[j] = x[j+1] + 1;
```

then the constraints are generated as

```
step[9]: x[9] = x[10] + 1
step[8]: x[8] = x[9] + 1
step[7]: x[7] = x[8] + 1
step[6]: x[6] = x[7] + 1
```

and case (b) does not occur. All of the constraints `step[j]`

can be eliminated, and every variable except `x[10]`

is substituted out as a result.

Close attention to formulation may thus be necessary to get the substitutions that you want. Set option `show_stats 1`

to see how many substitutions are being made, and use AMPL’s constraint expansion commands to see the constraints (in order) before and after substitution.