A LinkedHashMaP is ordered based on the insertion order. If you want that the natural order of a LinkedHashMap reflects a sort criteria of some objects, then you first have to establish that order on these objects, before then adding the objects in that order to a fresh LinkedHashMap instance.
In other words: you can't change the order on an existing LinkedHashMap - because the order was fixed when you added the elements to that map initially! Or to be precise: you could only do that by extending the LinkedHashMap class and changing its behavior to break its contract. But that doesn't make sense. You are simply using the wrong data structure to implement your requirement.
If you want a Map that re-orders based on a sorting criteria, you should be looking into a TreeMap for example!
Long story short: you have to differentiate between "order based on insertion order" and "order as result of sorting criteria". You want the later, and LinkedHashMap is the wrong kind of map for that.