Supply chain does seem as one of the best fits for enterprise private ethereum blockchain. As far as I understand blockchain technology, its mandatory that no party gain more than 50% computation power, otherwise you loose all the trust giving features of blockchain (one party could alter the block chain or create wrong transaction). Hence, if only two companies or one company run a blockchain, you have to trust that they don't fire up some mining nodes and mine blocks which contain false information. Please correct me if I'm wrong.

Don't buy into the useless hype; there's no good use of blockchain inside a company. All that blockchain does is to timestamp documents (tell which of the documents was issued earlier). There are much more efficient cryptographic protocols for that, the difference is that they need a trusted service to work.