Is there any reason why all Cola cans have very similar dimensions (height and radius)?

We believe so and in this post we will provide you an explanation based on results from a simplified multi-objective optimization performed using Nexus.

### Contents:

- Formulating the problem
- Formulating the problem as a Multi-Objective Optimization
- Solving the problem within Nexus
- Assessing Results

#### Formulating the problem

What would you do if you would be in charge of the production of the beverage-can for a multi-billion company such as Coca-Cola or Pepsi? Well, very likely you will be asked to minimize the unit-cost of each produced can.

Clearly the real-world problem is much more complex, but to make it the simplest we can, the easiest way to achieve this is to use as less aluminum-alloy as possible. To make ever simpler, just suppose that you cannot reduce the thickness of the aluminum-alloy laminate further.

The best this you can do is thus to reduce the overall frontal area of the can, by also assuring the required volume (let’s say 33cl).

This translate in an optimization problem where the can height and radius are your variables, the objective is to minimize the frontal area of the can (i.e. to minimize the amount of aluminum-alloy and hence to reduce production costs), simultaneously satisfying a constraint on the minimum volume.

Let’s put this in formulas:

- 50mm < height < 150mm
- 10mm < radius < 80mm
- min( Area )

being: Area = 2*pi*radius*height + 2*pi*radius^2 - Volume > 33cc

being: Volume = pi*radius^2 * height

Solving the optimization problem (with Nexus) and rounding the final solution to a variable precision of 1e-3, we obtain the following result:

- height = 75.00mm
- radius = 37.42mm
- Area = 26435mm^2
- Volume = 33cc

Note that the volume is exactly the expected one, therefore not additional reduction on the overall area are expected. Obtained results are not so different from the real-world shape. Please note that the real shape has smoothed corners and accounts for many other factors that we completely neglected in this post. >> Back To Top

#### Formulating the problem as a Multi-Objective Optimization

Alternatively, we can search for a trade-off (if exists) between the minimum area and the maximum volume that can be contained with it. Note that the solution identified above should be already part of this larger set of solutions and more precisely it should be the solution capable to contain 33cc of fluid.

This is a typical example of a multi-objective optimization, where two conflicting objectives are optimized simultaneously. The idea here is not to find a single optimal solution, but a set of feasible solutions that represent the optimal compromise between the objectives. We call this set of solution the Pareto Set. A more formal (and rigorous) definition of the Pareto set can be found here.

Our multi-objective optimization problem can thus be formulated as it follows:

- 50mm < height < 150mm
- 10mm < radius < 80mm
- min( Area )

being: Area = 2*pi*radius*height + 2*pi*radius^2 - max( Volume )

being: Volume = pi*radius^2 * height

Note that this first variant of the problem does not actually constrain the minimum volume of the can to be higher than 33cc.

However, we can always add a constraint to request a minimum volume. Note this will actually limit the feasible solutions reducing the identified

- Volume > 33c

being: Volume = pi*radius^2 * height

>> Back To Top

#### Solving the problem within Nexus

In order to solve the problem above within Nexus, we start creating a new project and we move into the Flowchart module.

From there we define the two design variables (

**radius**and

**height**) with their relevant domain. Then we define two expressions, one to compute the

**Area**and the other one to compute the

**Volume**. Finally we define the two objective functions and a constraint, i.e.

**MaxVol**to maximize the**Volume****MinArea**to minimize the**Area****CnsVol**to constrain the**Volume**to be larger than 33cc

The resulting flowchart is reported on the right. We will use these two latter to define a MOGA (Multi-Objective Genetic Algorithm) procedure within the taskflow.

The optimization results are reported on the left. The chart shows all the analyzed solutions (gray color) against the Pareto ones (rainbow color). Note that no Pareto solutions have been identified for volume below 33cc. This is because of the constraint imposed on the volume.

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#### Assessing Results

We have used Optimization Theory to provide a rationale top-level explanation of the shape (height and radius) of beverage cans of 33cc. This has been done within Nexus using a MOGA (multi-objective Genetic Algorithm) capable to identify the Pareto Set of the optimal can-shapes varying the minimum value.

>> Back To Top