When you are in charge of making the financial decisions for your company, it can be easy to get into a mode where you call the shots. The problem with this is how people may disagree with you and stray, bend, or push the limits which will ultimately make your job harder. Much like coaching a team you may be the one in charge, but the key is to get your players to do what you say, not just telling them to do it.
"Even if you don't take their advice they at least know you took the time to ask and consider their point of view."
Without learning to tactfully navigate personalities you might be in for more of a headache down the road unless if you make all the decisions. Asking for perspective by those effected by your decisions is a good start. Consider a budget cut made by the government where they don't consult teachers or listen to their needs or worries before setting a budget. There's likely to be protests and problems when a group feels under represented or neglected.
Even if you don't take their advice, they at least know you took the time to ask and consider their point of view. And if co-workers come asking why you made your decision the way you did you'll have a better answer to give them than if you didn't know their concerns in the first place.
Sometimes you will be expected to decide for everyone
The best thing to do here is to have met with people and departments in anticipation before hand. If you haven't, it may help to get a written communication or merely to meet with heads of the department. Call a quick meeting and ask concise questions. Let them know the timely nature and pressure you are under. If someone understands and can communicate the situation you've been put under it will help them cope with whatever circumstance your decisions put them in.
Should you meet with everyone at once or individually?
This can be a tricky situation. You only have a certain amount in a budget to parse out to varying departments. Getting them together can create a battle scenario where they are tempted to fight it out for every nickel and dime. The other (more desirable) outcome is that they see another team just like their own and understand sacrifices will need to be made on both sides. When done healthily, each side will understand the budgetary restraints they've been put under and will be less likely to place blame or be frustrated with you or someone else involved in the process. If it does turn competitive, remind them that you will have to decide and the goal is for it to be as favorable to everyone's long term success as possible. Stay open to suggestions and encourage people to propose alternative solutions to create room in the budget or save money in creative ways. You'd be surprised what the people who see what goes on in the day to day operations will come up with.
Do Not Make Promises - Especially Behind Closed Doors
Sometimes the pressure feels insurmountable. In these situations, it can be easy to want to please someone. Perhaps it's when dealing with a higher up and giving them news they don't want to hear. It could even be taking away something which has been granted for years but is no longer justifiable. You may be the bearer of bad news but avoiding this and promising anything you cannot deliver on doesn't look good on you in the long run. Even if you can come through for them, it isn't worth it if you don't do what's best for the company for the sake of avoiding confrontation.
No matter what you do during this process sit back and think about how you will look at operations in a day, week, or month. A tough decision now will become routine. People will adapt and develop a new sense of normalcy. The only way they are likely to continue pushing back is if they feel they had no say or consideration in the process.