You can compute an appropriately sorted list of indices, and then reorder both arrays using those indices:
#!/usr/bin/perl -l
my @array_items = ("dog", "desk" ,"cow");
my @array_items_weight = ("20" , "10", "150");
my @i = sort { $array_items[$a] cmp $array_items[$b] } 0..$#array_item
+s;
@array_items = @array_items[@i];
@array_items_weight = @array_items_weight[@i];
print "@array_items"; # cow desk dog
print "@array_items_weight"; # 150 10 20
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