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My question is about using "at the same time" in a sentence. Should we include the subject after "at the same time"? For example:

The element is chosen in order to improve the method and at the same time, (it should) reduce the delay and prevent collision.

Should I include "it should" in the first sentence?

Is it also possible to write the sentence in this way:

The element is chosen in order to improve the method while reducing the delay and preventing collision.

If so, which one is better?

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When you ask what's better, you are asking a question of style, in other words how to convey your message in the clearest possible way. The style you choose depends on what information you have to impart and the audience for that information. Only those familiar with both will be able to offer the most useful advice. Others, like me, can offer alternatives.

The topic at hand is the introduction of a new element into some system. I don't know whether it's a new engine part or a new software module, but that doesn't matter. What does is why the element was chosen and what its effect is likely to be.

If the element was chosen primarily to improve the method, and reducing delay and preventing collision will be a welcome but secondary consideration, then it makes sense for the latter considerations to appear in a subordinate clause:

[1a] The element is chosen in order to improve the method, which should reduce the delay and prevent collision.

If the kind of delay is known to your audience, then the definite article the is appropriate. If not, drop it: "reduce delay and prevent collision."

On the other hand, if the element was chosen with all three considerations in mind, then it makes sense that all three should appear as equals in a parallel construction:

[1b] The element is chosen in order to improve the method, reduce the delay, and prevent collision.

Before I deal with the question of where to place the phrase "at the same time," let me ask whether it's necessary to place it in the sentence at all. Timing might be an issue -- only you can tell me. It's possible that the introduction of some elements could prevent delay and collision, but only after a while. If that's the case, but your element prevents these conditions immediately, then "at the same time" in 2a makes sense. Otherwise, why not simply say also, as in 2b?

[2a] The element is chosen in order to improve the method, which will at the same time reduce the delay and prevent collision.
[2b] The element is chosen in order to improve the method, which will also reduce the delay and prevent collision.

If you need the construction, you have two choices for placement, before the auxiliary word will (3a) and after (3b):

[3a] which at the same time will reduce the delay and prevent collision
[3b] which will at the same time reduce the delay and prevent collision

I can see only the subtlest of differences. Notice that a post-position is not available:

[3c] * which will reduce the delay and prevent collision at the same time

because this means preventing delay and collision at the same time, and you mean preventing both delay and collision at the same time as improving the method.

Now to the word should, which in this case has the modal meaning (i.e., one not associated with tense and time) of expectation. This stands in contrast to will, which carries certainty. If you think there's some possibility that delay and collision might increase with your new element, then use should. If you think that it's inevitable that delay and collision will decrease, use will.

The selection of voice is important. You have chosen the passive voice in 4a, but you may transpose to active as in 4b.

[4a] The element is chosen ....
[4b] The team chose the element ....

Version 4a has the advantage of making the element the subject, which keeps the focus on the element. It also has the advantage (from the chooser's point of view) of keeping the responsible party or parties anonymous. If the element fails to bring the anticipated benefits, one can envision a second report that says, "Mistakes were made."

Finally, I have never seen a case where is was advisable to keep the words "in order" introductory to an infinitive of purpose (here to improve). In short (and a spirit of jest), in order to improve one's writing, just say "to improve" and leave the "in order" out.

Needless to say, judgment on matters of style are subjective, and you will easily find editors who disagree.

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