Mathematical derivation of the Production Possibility Frontier What are the mathematical basics of production possibility frontier? How can I derivate it? Can I have an example for it?
The question is broad, but I believe there is plenty of literature that defines this concept in similarly broad terms.
The following is adapted from the Wikipedia on Pareto Efficiency,
which is the mathematical basis of the Production Possibilities Frontier.

There may be better definitions out there, but this one should probably work in a lot of cases:

The Production Possibilities Frontier, $P(Y)$, may be more formally described as follows. Consider a system with function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, where $X$ is a compact space of feasible decisions (including allocations of time and endowment goods) in the metric space $\mathbb{R}^n$, and $Y$ is the feasible set of criterion vectors (say, final goods and services) in $\mathbb{R}^m$, such that $Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}$.

We assume that the preferred directions of criteria values are known so that more of any good in $Y$ is better. A point $y^{\prime\prime} \in \mathbb{R}^m$ strictly dominates another point $y^{\prime} \in \mathbb{R}^m$, written as $y^{\prime\prime} &gt; y^{\prime}$, means that for each element index $i$, $y''_i \geq y'_i$ and there is at least one element $j$ such that $y_j'' &gt; y_j'$. The Pareto frontier is thus written as:

$P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} &gt; y^\prime, y^{\prime\prime} \neq y^\prime \; \} = \emptyset \}. $