Hi.
Using Ivan’s example of yogurt I am thinking about small local businesses. A small farmer produces milk, sells it to an small milk product producer who then sells in the local farmers market (just an example). Or even add a few more intermediaries to this chain.
First, does everyone have to buy into the blockchain system?
Second, if I am the farmer what would prevent me saying my milk is organic and inputting that on the blockchain, even if it is not.
Third, what would be the incentive for truth is this example?
Bitcoin is primarily used for transfering value and the supply chain was just an example what it can do. There are other projects that tackle issues of supply chain, one of them being Origin Trail. I sugest you contact them and research by yourself.
I will try to answer your questions as much as I know.
In case you want to put a supply chain on the blockchain, all parties would have to be involved.
I’m not sure how projects that tackle supply chain problems deal with inaccurate data being inserted on the blockchain in the first place.
I guess reputation. 
Haha! Oh yeah, tells you about my morals, forgot about reputation! However, if reputation was the incentive than we wouldn’t need a trustless system.
Thanks for the answers and info Alko89