How can you estimate the value of a component part where you only know the value of a whole? I'm not sure how to frame this question, or where to start. I'm new to data analytics, but looking to develop skills and knowledge.

An example of what I'm asking is if you have a retailers sales data where you have the total value of a given transaction, and details of each item related to a given transaction but don't have price data for each individual item, is it possible to estimate the value of each individual item given a large enough transaction data set?

The analogy breaks down a little for the use case I'm actually considering, as under this retailer example an item would likely have a fixed price. However, in my use case each item has a known arbitrary tariff value but an unknown actual value. We would only know an aggregated actual value in conjunction with other items which may be grouped under the same "transaction". Tariff and actual values will likely have a strong linear correlation with some variability, though this isn't known as actual values for individual items are not captured.

Hope this makes sense! Was wondering how define this problem, what approach would you take to a problem like this? And any links to related reading materials would be much appreciated.
Clearly this problem doesn't always have a unique solution, but if you are interested in finding one possible solution you could try a simple genetic algorithm simulation:


Each individual gene represents an item from the list of all possible items. 
Each gene/item is assigned a price randomly at first (gene expression)
When a mutation is applied to a gene/item, its price is slightly modified randomly.
A crossover causes a "child gene" to take as value the mean of its two "parents genes".


This setting means that every individual in a population consists of all the items being assigned a particular price. At each generation each individual/assignment is evaluated by applying the prices assignment to the actual data and then measuring the error compared to the actual prices. Finally the top N individuals/assignments which perform the best are selected as parents for the next generation. Eventually the population should converge to realistic prices assignments.

I think this is a perfect case for a genetic algorithm because the evaluation of a potential price assignment is a very simple calculation, so there is no major efficiency issue repeating the process over many generations (as opposed to many problems where evaluation is prohibitively expensive).