How to determine what components make up a complex mixture based on a summary of that mixture? Any wine can be made up of many components. Each component is basically  a bunch of grapes from a specific place and time, so each component has three attributes: vintage, varietal, and appellation. 
While grapes come in to a winery as single components (eg. 2016 Russian River Valley Cabernet Sauvignon), a wine may end up composed of many components (after blending).
Given a summary of a finished wine's properties, how can you find a set of components — and their percentage contribution to the finished wine — that add up to that wine?
Here's what we have:
VINTAGE       %           VARIETY           %         APPELLATION               %
2014        98.00         Cab Sauv.       78.95       Russian River Valley    37.02
2015         1.50         Petit Sirah      9.12       Lake County             35.81
2016         0.50         Syrah            7.34       Sonoma Coast            22.79
                          Petit Verdot     4.42       Sonoma                   4.07
                          Sauv. Blanc      0.14       Napa                     0.31
                          Concentrate      0.03

Ideally we are able to find the smallest set of components, but any set of components that could solve this would be acceptable.
What is the answer(s) for this specific data set, and what is the right approach for this problem in general?

Your problem is essentially to solve $15$ linear equations and $180$ linear inequalities in $90$ unknowns. $181$ of these are easy to understand: 


*

*The sum of all the proportions adds to $1$ (or $100$, if you prefer percentages, but I will stick with $1$). 

*Each proportion is nonnegative. 

*Each proportion is less than or equal to $1$.


The remaining $14$ equations, say we index them by $i=1,\dots,14$, look like $\sum_{j \in S_i} p_j=b_i$, where $S_i$ is the set of wine combinations with the attribute $i$ (e.g. being 2014), $p_j$ is the proportion of wine $j$, and $b_i$ is the known fraction of the final wine which has attribute $i$. 
So for example, taking the lexicographic ordering on the set of wine combinations, the first equation reads $\sum_{i=1}^{30} p_i=0.98$, and represents the fact that $98\%$ of your final wine is from 2014.
The solution to this system is highly non-unique; it has $78$ free parameters. (Three of the equations are redundant, one for each wine property.) You may seek out the most sparse possible solution (i.e. the one with the largest number of zeros), though I don't know how to easily do this.
