I am a master of time saving and efficiency.
This is the most inefficient system I have ever heard of. Here is how I would solve this.
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Get a big load of hay. Everyone buys in so it’s not your cash up front. Believe me, your boarders will like this much better than having to go find handfuls of bales every couple weeks. Plus it will be way cheaper for you all too. Get hay guy on a schedule, have boarders leave checks or pay in advance. If boarders want extra hay or a different type, then they have to arrange for it themselves.
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Hay guys will deliver. And stack. You do not need to lift a finger. Agree to either pay more for stacking, or have an unloading party with your boarders.
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No more baggies. What a pain for everyone! Use scoops with the quart markings. Write the feed amount on the scoop or bin lid. You are fighting a losing battle with this one and there’s no point.
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If someone runs out of feed, make it a big fee. Big enough so they never, ever want to run out. Big enough so you don’t begrudge them.
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Your barn, your rules. It’s up to you to define them and enforce them.
Good luck, it sounds like you have an awesome place!!!