I'm bringing a word problem/conceptual problem to the math stack exchange. Feel free to edit the title of this question if it does not reflect the following word problem/conceptual problem.

What I want to do is maximize the value called "health" subject to the constraint of budget. "Health" is determined by a number of "micronutrients". (Say twenty different kinds of micronutrients.) For each micronutrient, you can have a certain percentage of your daily value up to a 100%, and there is no penalty for exceeding the daily value but there is no "health" advantage for exceeding this value either.

In the grocery store there are "products" with attributes, aka "micronutrients". One product may have (for daily values) (90% vitamin A, 10% vitamin C, 0% calcium... and so on) another product may have different values. Each product has a price, and we can simplify by assuming this price is "per serving".

1) What I am interested in is finding how to describe the "healthiest" possible solution for a given average daily budget, algorithmically and as a general set of equations.

2) I am certain that there must be a description missing in my account for how "health" relates to the (twenty) micronutrients. I am interested in some general formulas that have been used elsewhere in mathematics that might be applicable to this situation too.


"Health is determined by" is the key phrase in your question. If we can say that health is a SUM, like

$$ H = w_1 + w_2 + \ldots + w_{20} $$ where $w_i$ is the fraction of the RDA, but limited to a maximum of $1$, then this is a constrained optimization problem, and pretty well adapted to standard techniques like the simplex method.

If "Health" is some more complex function of the nutrients, then you need much more sophisticated methods in general.

But the question of "what kind of function of the micronutrient components is the health function?" is not a mathematics question, but a biology question. The kinds of combinations that have been useful in other areas of mathematics are not relevant: the actual biology of the system you're studying determines the equations you need to work with. Once you've got those, you can decide whether to approximate them with some other sort of function to make finding an approximate solution easier or not. But I strongly suggest that you don't start out trying to approximate the underlying model. Experience has shown that this is generally not a good idea.

There's a general principle here: "Approximate the solution, not the problem."


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