Optimization in Several Variables

Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives. An optimization process that involves only a single variable is rather straightforward. After finding out the function $f(x)$ to be optimized, local maxima or minima at critical points can be easily found. (Of course, end points may have maximum/minimum values as well.) The same strategy applies for optimization with several variables. In this atom, we will solve a simple example to see how optimization involving several variables can be achieved.

Cardboard Box with a Fixed Volume

A packaging company needs cardboard boxes in rectangular cuboid shape with a given volume of 1000 cubic centimeters and would like to minimize the material cost for the boxes. What should be the dimensions $x$, $y$, $z$ of a box?

First of all, the material cost would be proportional to the surface area $S$ of the cuboid . Therefore, the goal of the optimization is to minimize a function $S(x,y,z) = 2(xy + yz+zx)$. The constraint in the case is that the volume is fixed: $V = xyz = 1000$.

Rectangular Cuboid

Mathematical optimization can be used to solve problems that involve finding the right size of a volume such as a cuboid.

We will first remove $z$ from $S(x,y,z)$. We can do this by using the constraint $z = \frac{1000}{xy}$. Inserting the expression for $z$ in $S(x,y,z)$ yields:

$\displaystyle{S(x,y,z) = 2\left(xy + \frac{1000}{x} + \frac{1000}{y}\right)}$ 

To find the critical points:

$\displaystyle{\frac{\partial S}{\partial x} = 2 \left(y - \frac{1000}{x^2} \right) = 0\\ \therefore y = \frac{1000}{x^2}}$

and

$\displaystyle{\frac{\partial S}{\partial y} = 2\left(x - \frac{1000}{y^2}\right) = 0\\ \therefore x = \frac{1000}{y^2}}$ 

Then, substituting in the expression found equal to $y$ above yields:

 $x^3 = 1000$

Therefore, we find that:

 $x=y=z=10$

That is to say, the box that minimizes the cost of materials while maintaining the desired volume should be a 10-by-10-by-10 cube.