I ran my IPO whe I realized I needed help in quashing my competition in my main hub. I also didn't want to wait so late that it took a lot of subscribers to get to 50% of offered shares. So I went pretty early and ended up getting five planes which put me over the amount my competitor had, and after that, it's been cake squeezing him and others out.
There's not really a mathematical answer to this. You do it when it will benefit your company the most. If you go too early, you won't get much in your IPO. If you go too late, it might take several tries to get the IPO to take.
The only thing I don't like with the IPO is paying out $10+ million per week right after paying salaries. That's a huge chunk of money that takes me about a day and a half of saving up for, which means no spending money when I see $25 million in my bank. Hard to do that. The fake AS cash burns a hole in my pocket.
I think the disadvantages outweight the advantages by far, at least for your main airline...You are getting quiet an amount of money but you will have to pay it back every week...when there is growth in the game world people who didn't go for an IPO will make more money (as they dont have to pay a part of the money they make to other airlines) and expand faster than the people who decided in favor of an IPO. It may make sense for a subsidary though. I have for example a regional subsidary which wasn't founded with the goal to grow extremly fast, but rather to supplement the main company in smaller markets and to generate connecting passengers. So growth is limited at some point (as it will interfere with the main company if it grows too much). In this case it may make sense to go for an IPO to get some money in the holding which can be used to build terminals/buy and lease out aircrafts etc...
One of the main reasons I really did my IPO was the ease in shifting money around. It may seem counter-intuitive, but if you have a holding and a main airline subsidiary, you can run an IPO and 80% of 15% of profits would automatically go back to your holding.
Right now, my main airline is kicking out about $10-$12 million in dividends a week. My holding still has right about 80%, so that means $8-$10 million are going right back into my holding. No money lost. The other 20%, which comes out to be about $2-$3 million, is split up from other companies. However, I have another subsidiary that is going to be my main stock holder/subscriber in other companies. A lot of companies that participate in an IPO are wanting to sell their stock back as soon as it doubles in price. With the price still being low, I can use my other subsidiary to buy up the stock people want to sell (or use my holding company, too). That means more than 80% is coming back to my company. The net result is that I'm losing about $1-$2 million a week in profit, which amounts to about 2 to 4 hours of real time flights.
Considering that if I wanted to shift that money around in my subsidiaries by using the transfer-sell plane trick, and got one of the cheap planes to do it (so you aren't losing $20 million at the beginning), it would take at least that amount of time to transfer the money from my main airline back to my holding or another subsidiary.
Also, keep in mind you can transfer money with stocks between your holding and subsidiaries, so long as you use the current closing price. You can transfer more than the price of a cheap airplane at once.
I agree there are times you shouldn't run an IPO and keep your company private, but I happen to think that if you time the IPO right and infuse your company with needed capital at an important time, it can be well worth it.