In this section, we cover cases that might require a bit more attention when optimizing orders.
Special cases are often industry specific and we are happy to extend our solutions based on a need-be basis.
If you have any special requirements, feel free to let us know at . and we will be happy to open a discussion.
For instance, we have came across the need to support .

Special cases

While working with a company dealing with pahrmaceutical products, we have discovered the need to support plastic bags as packages.
What are prescription bags?
Skrym treats prescription bags as inelastic sleeves with opening on one side.
A bag has a base length and width with negligible height when not expanded.
However, as items are placed inside, the bag is extended in any-or-all directions causing it to also shorten in others (since it is inextensible).
Using it together with package optimization
Whether you need pure bag recommendation or a combined package and bag optimization, you should be able to use one of our .
If you need any helping setting up, shoot us an email on .

Prescription bags

Learn how Skrym can help with accurate order optimization

How Skrym Handles Bags
Skrym treats prescription bags as inelastic sleeves with opening on one side.
A bag has a base length and width with negligible height when not expanded, but this becomes more difficult when items are to be placed inside.
Ellipsoid Approximation
Just like in the case of rectangular boxes/packages, the items can be placed inside the bag in numerous possible configurations. Unlike in the case of rectangular boxes, the optimization space inside a bag can be freely deformed up to some limits.

For a single item order, this problem simplifies since one can check if the surface area of the item is less than that of the bag and if its dimensions can fit inside the bag. This can be formulated as a Lagrange multiplier problem that yields a system of equations.

As a simple workaround, Skrym approximates the fully expanded bag as an Ellipsoid and circumscribes (not entirely?) it with a cuboid which becomes the optimization space.
{/* NOTE: might want to check if the information in the collapsible is still up to date as we have made some small changes. Good to keep it internal for now. */}

A bag has a length  and width  where we set the length to the larger dimension.
When it is fully expanded without any elastic stretching, it should reach its maximum capacity in terms of volume, meaning that is the available volume for us to place items into.
Obtaining the exact volume of a fully expanded rectangular bag is, unfortunately, an unsolved problem (called the paper bag or teabag problem). Luckily, there is a decent approximation:

which we can use when approximating the inflated bag by an ellipsoid. Why can't we just find a cuboid with that volume? That's because we also need to make sure to respect the length and width of the bag.

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

An ellipsoid is a simple enough shape that seems to fill an inflated bag reasonable well. It can be easily characterized by its semi-major axes , which are halves of its maximum dimensions (diameters), with which we can also express its volume:

With this, we have enough information to create the ellipsoid, as long as we ensure that  and map , we can do:
func (pb Bag) CreateEllipsoid(heightCorrection float64, numberOfPoints int) Ellipsoid {
	pb.EnsureLengthIsGreater()

	a := (pb.Length / 2)
	b := (pb.Width / 2)
	c := (pb.Volume() - heightCorrection) / (4 / 3 * math.Pi * a * b)

	return Ellipsoid{SemiMajorAxisA: a, SemiMajorAxisB: b, SemiMajorAxisC: c, DeltaHeightRange: utl.LinSpace(0, math.Min(a, b)-c, numberOfPoints), Bag: pb}
}

The resulting ellipsoid tends to be smaller than the actual available space, as the  is a under-approximation itself. That is why the CreatineEllipsoid method supports a heightCorrection. Setting heightCorrection < 0 will cause the ellipsoid to increase its height.

This is a little more difficult because the surface area of an ellipsoid can also only be approximated, since it involves incomplete elliptic integrals.

In order to generate the cuboid for the optimization space, we would ideally want to find a maximal cuboid that preserves the surface area, volume and length, width and height constraints. As that is borderline impossible, we settle for circumscribing a cuboid around our ellipsoid.
However, the volume of a circumscribed cuboid,  is roughly . If we were to use an inscribed cuboid, the cuboid would however be too small, roughly . As a compromise, we settle for:
func (e Ellipsoid) CircumscribedBox() Container {
	return Container{
		BoundingBox: [3]float64{
			1.75 * e.SemiMajorAxisA,
			1.75 * e.SemiMajorAxisB,
			1.75 * e.SemiMajorAxisC,
		},
		Ellipsoid: e,
	}
}



We are always interested in exploring new problems that can be solved with the help
of optimization, and have experience from several different industry-specific challenges.
If you have a specific problem that you think could be solved with the help of optimization,
but that you do not see mentioned explicitly in this documentation, feel free to reach out to us at
.