Can someone explain to me why. You need a block chain for this? Why not just an evented database with a signature field and periodic hashes?
Imagine you want to track eggs through the supply chain from farm to table. Ultimately the trust of your system is rooted in the farmer and various other organizations that are updating the state of these eggs in the chain. If they make up fake data, the system isn't trustworthy.
Typically the solution for this is to only let trusted players into the system and have some sort of organization - perhaps the government - that vets users of the system and penalizes those that enter fake data. In this case, the trust of your blockchain-egg-supply-chain-tracking system is rooted in the organization accrediting and inspecting users of the system. If that's the case, you're better off trusting that organization to run a traditional database. You shouldn't use a blockchain.
We have been using analogues of the system described for literally centuries. A titles office is very like a block chain. Rather than currency being transferred between people, it's titles to land or buildings. Individuals do transactions by creating a transfer and giving it to the titles office to be blessed. The transfer is worth nothing until the titles office puts it's stamp on it. The titles office function is to create an immutable, consistent chain of ownership.
The formal name for the software equivalent is a "permissioned block chain". There is nothing particularly special about it - if you gave the problem to a bunch of 4th Uni Students, this is what you would expect to come back. The term "block chain" looks to be a marketing grab to me. I don't think it really applies - the only thing they have in common with crypto currencies is they record a chain of ownership. Crypto currencies have blocks so they can spread the high cost of proof or work over many transactions but there is no expensive proof of work in permissioned block chains, so they don't need blocks either.
All that aside, given we have been using these manual systems for centuries I don't think there is much doubt automating them with electronic equivalents would be result in something that is both faster and less costly. But whether it's a good idea for logistics chains is debatable. One thing the titles office and crypto currencies necessarily have in common is there there isn't a lot of "new stuff" entering the system - they are tracking existing stuff (the titles or currency), which they can easily verify by just checking the person authorising the transaction is the person recorded as the current owner by the chain. When new stuff does enter the system it is very tightly controlled. Crypto currency does it via mining. Titles are usually created by acts of parliament releasing land.
In these logistics systems new stuff is continually entering the system, and some separate external system has to vouch it is in fact a tray of A grade fruit or whatever. But I sure whoever is fogging their existing permissioned block chain implementation is telling every customer with a vaguely similar system it is a near perfect fit.