I would have replied, but I've not had enough experience to be able to give you any sound advice. Here goes anyway
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I tend to use all three settings depending on the situation. Minimise losses in attacks sometimes if I want to probe away at a position and not burn too much of the turn, ignore losses if I need to take a position or force a retreat , especially if I need to definitely force the retreat to cause evaporations (when enemy is surrounded or boxed in). Also use ignore losses on defenders quite often now, as they are less likely to retreat and as you say burn the opponent's turn.
In a recent game I had a severely weakened (in the red) infantry regiment entrenched in the mountains with an escarpment hexside, it was the last unit left in the line with the enemy on three hexsides. I left them on ignore losses to stall the enemy not expecting much. However, the enemy attacked in some force and my unit evaporated, but in the course of the attack it inflicted what I saw as hugely disproportionate losses. So terrain can allow you to use settings to your advantage.
Having said all that, I quite often when attacking go for the limit losses. If I don't want to force units forward into the hex I'm attacking I use limited attack. Just depends on the context, I don't know myself if limited attacks and full attacks make any difference in terms of effectiveness and supplies? I assume they do, I do have a hunch an attack pressed into a hex may give the defenders a harder time disengaging, especially with high recon in the attack, and so you may get more evaporations with weak retreating units, but this is only a hunch.
I think with a limited attack you lose less, and the defender loses less, and can escape more easily?
As for artillery in an indirect fire role, I was under the impression that it did not matter what setting in terms of burning time or causing casualties, that minimise, limit and ignore losses all gave you the same result. But maybe one of the more experienced players can clarify?
Anyway, that's about all I know, hope it helps
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