@id = sort { 
    ( my $a_ = $a ) =~ s/=.*//s;
    ( my $b_ = $b ) =~ s/=.*//s;
    $lid->{$a_} cmp $lid->{$b_}
} @id;
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Update: Oops, added missing underscores as per Re^5: How do I sort an array by hash value?

Thanks. I really appreciate your help.
Here is how I failed:

@id = sort { $lid->{split(/=/, $a)} cmp $lid->{split(/=/, $b)} } @id;
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Your approach doesn't seem to sort the array unless I change it from

my $a_ = $a
my $b_ = $b
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to

$a = $a
$b = $b
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But if I do that then the values of the array get cut off at the '=' sign (I understand why).
What does the 'my $a_ = $a' do?

It should be as ikegami had it, except the
    $lid->{$a} cmp $lid->{$b}
statement should be
    $lid->{$a_} cmp $lid->{$b_}
(note addition of the trailing underscores).

The  my $a_ = $a; step is so the values in the array aliased by the 'magical' variables  $a and  $b are not changed by the substitutions (i.e., the change you noted).

@id = sort { $lid->{split(/=/, $a)} cmp $lid->{split(/=/, $b)} } @id;

What does split return in scalar context...

Here is how I failed:

@id = sort { $lid->{split(/=/, $a)} cmp $lid->{split(/=/, $b)} } @id;
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Near to success:

@id = sort { $lid->{(split(/=/, $a))[0]} cmp $lid->{(split(/=/, $b))[0
+]} } @id;
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This line has to evaluate $a and $b each time sort does a comparison, which isn't efficient. I guess that ikegami's line neither is.

Should be used a temporary translation hash? Something like:

my %temp = map { $_ => (split(/=/, $_, 2))[0] } @id;
@id = sort { $lid->{$temp{$a}} cmp $lid->{$temp{$b}} } @id;
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