Get the minimum value for multivariable

Question

I need a way to calculate the very minimum of each variable for a grade average.

Each grade variable have a weighing percentage (I don't know if it's the right term)

The weighing sum must be the total (100%)

The value it must be >=1 and <=7

For example

$$ 0.25x + 0.2y + 0.25z + 0.3u = 4 $$ or any number of variables $$ 0.15x + 0.2y + 0.25z + 0.15u + 0.25w = 4 $$

So I need the minimum value for each grade/variable to match 4

How can I do that?

Sorry if it's a dumb question

Thanks

Answer 1

$$\text{There are a lot of ways to define the minimum for the grades as a whole.}$$ $$\text{One way to define the magnitude for a set of values is distance from the origin.}$$ $$$$ $$\text{Use Lagrange Multipliers to compute the extreme values.}$$

$\text{Let } f(x, y, z, u) = x^2 + y^2 + z^2 + u^2$
$\text{Let } g(x, y, z, u) = 0.25x + 0.2y + 0.25z + 0.3u - 4 = 0$

$$\text{We wish to find the points of extreme values of }f \text{ about the constraint g.}$$

$$\vec{\bigtriangledown{f}} = \bigg(2x, 2y, 2z, 2u \bigg) $$ $$\vec{\bigtriangledown{g}} = \bigg(0.25, 0.2, 0.25, 0.3 \bigg)$$

$$\text{There exists }\lambda \text{ such that: } \vec{\bigtriangledown f} = \lambda \cdot \vec{\bigtriangledown g}$$

$$\begin{align} 2x &= 0.25\lambda\\ 2y &= 0.2\lambda\\ 2z &= 0.25\lambda\\ 2u &= 0.3\lambda \end{align}$$

$$\text{Thus, }\lambda = 8x = 10y = 8z = \frac{20}{3}u$$

$$\text{Now, we can solve for one of the variables, say } x \text{ about the constraint.}$$ $$0.25x + 0.2(0.8x) + 0.25x + 0.3(1.2x) - 4 = 0 \space \implies \space x = \frac{4}{1.02} \approx 3.92$$ $$\text{Using the equality with } \lambda \text{ to find } y \text{, } z \text{ and } u \text{ is left as an exercise for the reader.}$$

Answer 2

Suppose you want to compute the minimum for $x$. Just set all the other variables $y=z=u=w=7$ and then solve for $x$.

Answer 3

You need just solve the problem:

\begin{equation} \begin{array}{c} minimize \hspace{1cm} \sum \limits_{i=1}^{n} x_i \\ s.t. \hspace{2cm} \sum \limits_{i=1}^{n} \alpha_ix_i = \beta \\ \hspace{3cm} a_i \leq x_i \leq b_i, i=1,\ldots, n. \\ \end{array} \end{equation} Thus, you will find each minimal $x_i$ satisfying the constraints. This is an LP problem, which is easy to solve.

Remarks:

$(1) \hspace{3mm}\sum \limits_{i=1}^{n}\alpha_i = 1$, and $\alpha_i \geq 0$;

$(2) \hspace{3mm}$ Each $a_i > -\infty$, otherwise the problem has no solution, because some $x_j$ could diverge to $-\infty$.