Predicting which cars and systems were fixed from the parts that were ordered Interesting problem from the auto industry. Wondering if anyone has suggestions on a good approach. Auto manufacturer sells cars to independent dealers who sells to end-customers. When cars break down, the dealer fixes them and orders parts from the manufacturer but seldom gives them info on which car was fixed or even what system is being fixed. Some parts for example are used in multiple systems and multiple cars. The manufacturer would like to know about the durability of each part in each system of each car among other things. In general, they would like to assign a car and system to each of these servicings. What's the best way to do that?
I figure this is a probabilistic problem that might benefit from some kind of graphical model. I also suspect the problem arises in other areas and might have some good known solutions. 
The do know which parts are used in each system of each car. Sometimes they are given complete information.
 A: I like the answers given here already (AdamO, Daniel, Dave31415), and I'll try to synthesize them and add a bit from my experience:
You mention: 

The manufacturer would like to know about the durability of each part in each system of each car among other things.

This leads me to think of a two part solution:


*

*First, map all possible parts in a one-to-many relationship with which car model and problem is solved in the repair. You'll need both experts and actual repair documentation. This can be done with a complete survey by amending the required documentation. Alternatively, you can randomly sample part orders and manually follow up on the individual repair (ie: call the repair shop) - eventually, you'll have a representative sample. Both are very time-intensive.

*Next, you build a dataset that takes actual repairs and builds lifecycle data (average, median, plot the distribution, etc) on a per-model-problem basis. This will likely demand at least 30 data points per part per problem per model. That's a lot of data to gather, suggesting that going the "required documentation" route would be better than hiring a fleet of interns. The added benefit is that this analysis can be automated and can update over time as new problems/parts/models are added.


I'm not sure you need to actually "predict" anything - gathering the dataset above will be sufficient. You can simply look at actual lifespans of parts in the models. You could then extrapolate to the lifespan of those parts in current service. Maybe "predict" can simply mean "use the median lifespan".
Overall, this might be too low-profile of a project to actually justify the expense of gathering these data. What if, instead, you brought in experienced mechanics from the community and from dealerships and simply asked them which parts fail and when? They might even be able to give you a great hypothesis on "why" they fail, not just which and how often. They are your real experts.
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A: For the problem you describe, you could use the Proportional Hazards survival model as developed by Cox. Consider each car an object, which may obtain a major failure (the event, as sampled in this application of the Proportional Hazards model).  
You can define the model by
\begin{equation}
\lambda(t \mid {\it {\bf X}}_i) = \lambda_0(t)\,exp(\beta_1\,X_{i,1},\beta_2\,X_{i,2},\ldots\,\beta_{p}\,X_{i,p})
\end{equation}
where $\lambda_0$ is the average survival time and each parameter $\beta_j$ an explanatory variable which influences how long each car remains on the road (the different spare parts in your prediction problem). Each parameter $\beta_j$ indicates the weight of each indicator variable $X_{i,\,j}$, actually a boolean being $1$ when part $j$ was replaced in the car, and $0$ when not.
A: The correct design is to collect better data. The question does not align with the current resources, so build them. Set up data sharing agreements with the shops who contract work with the dealer, use the VIN to part failures from purchase based on miles and age of the model. 
At best, you can perform an ecological study. Use parts' purchase volumes to predict future purchase volumes as a function of number of units sold in a region. So for instance, if Model A is a 2015 model, you can see how many 2014, 2015, 2016, ... sales were made to the dealer to try to predict the fleet volumes and their ages, then contrast that with volumes sold in other regions.
