## algorithms – The moving van problem: a mix of a knapsack and a bin-packing problem

This problem is a mix of the bin-packing and the knapsack problems. I call it “the moving van problem”: there is a moving van with a limit on the weight it can transport, and a set of boxes that you want to fill with valuable objects to be transported in the van. Both the boxes and the objects that are available can be choosen repeatedly and have weights. Also, the size of each box is bounded by above and below (their size limits), so you cannot put, for example, empty boxes in the van.

More:

• not all boxes accept all objects: the available objects and boxes are partitioned so that any object of a group can be saved in any box of the same group, but nowhere else;
• also, the value of each object doesn’t depend on itself but on the box you save it;
• lastly, the box size limit of all boxes of a same group are the same.

The problem is maximizing the transported value while respecting all of the above restrictions. Ah… almost forgot, the weights are multidimensional.

Formally:

1. I have one $$m$$-dimensional van whose capacity is $$overline{W}=(W_1, …, W_m)$$.
2. I have $$n$$ groups where each group $$G_i={B_i, O_i, V_i, l_i, u_i}$$:
1. Each $$overline{b_{ij}}in B_i$$ is the $$m$$-dimensional weight of the $$j$$-th box, and each $$overline{o_{ik}}in O_i$$ is the $$m$$-dimensional weight of the $$k$$-th object.
2. $$V_i$$ is a matrix of size $$|B_i|times|O_i|$$, where $$v_{ijk}$$ is the value given by the $$j$$-th box to the $$k$$-th object.
3. $$l_i$$ and $$u_i$$ are the box size limits that applies to every box of the group, and satisfy $$1leq l_ileq u_i$$.
4. Let’s call $$x_{ij}$$ the number of times the $$j$$-th box has been choosen.
5. Let’s call $$x_{ijk}$$ the total number of times the $$k$$-th object has been saved in $$j$$-th boxes.
6. Let’s call $$f_{ij} =sum_{k=1}^{|O_i|}x_{ijk}$$, the total number of items saved in $$j$$-th boxes, or its filled size.
3. The problem is then: begin{align} maxquad & sum_{i=1}^nsum_{j=1}^{|B_i|}sum_{k=1}^{|O_i|}v_{ijk}x_{ijk}\ text{s.t.}quad & sum_{i=1}^nsum_{j=1}^{|B_i|}left(overline{b_{ij}}x_{ij}+sum_{k=1}^{|O_i|}overline{o_{ik}}x_{ijk}right)leq overline{W}\ & forall i,~forall jleq|B_i|~:~l_ileq frac{f_{ij}}{x_{ij}}leq u_i\ end{align}

The question is, is there any formal problem with a similar formulation, or some way of transforming this problem to some other famous problem? Of course, my ultimate question is an efficient algorithm for this. I guess this problem is NP-hard because it’s almost an NP-hard problem inside another one.

Fortunately, my instances are not big so I’m even interested in exact algorithms to see if they behave well for me.

I have tried to vectorized a bit more the formulation of the problem and replace inner $$sum$$ by dot products, trying to see any pattern or resemblance with other knapsack problems, but those vectorizations attempts were confusing me even more and running me out of symbols to name things.

## Space-Complexity of Van Eck’s Sequence

What is the space-complexity of Van Eck’s sequence?

https://oeis.org/A181391

Input: $$x$$ (number in binary representation with $$n$$ bits)

Output: $$x$$th number in Van Eck’s sequence

Length of input: $$n$$

The naive algortithm would generate one entry of the sequence after the other and store all previous entries. Storing $$2^n$$ numbers of up to $$n$$ bits would need $$n2^n$$ bits. Maybe you are lucky and smaller numbers appear so often more than large numbers, that you only need constant space for each number. This would reduce space-complexity to $$O(2^n)$$. But can you do better than that?

## plaats van een gastenboek

Getrocknetes Schilf ist eine weitere coole Möglichkeit, Ihre rustikale Hochzeitsdekoration von der Natur inspiriert, stilvoll und unvergesslich zu machen. Das Schilf kann für verschiedene Projekte eingesetzt werden. Zunächst können Sie Trauben entweder einzeln oder in Kombination mit Blumen in Vasen legen. Solche Verzierungen sind erstaunliche diy Hochzeitstischdekorationen. Außerdem können sie sich in der Nähe des Ganges befinden.

Zweitens kann Schilf für Ihren Hochzeitsbogen wie im Bild oben angewendet werden. Bei diesem Bild ist Schilf nur ein Akzent, aber Sie können es auch zum Hauptmaterial für Ihren Bogen machen. Drittens kann Schilf eine süße Ergänzung zu Ihrem Hochzeitsstrauß und dem Boutonniere eines Bräutigams sein. Darüber hinaus können Sie es für die Dekoration anderer Hochzeitsartikel wie ein Gästebuch, ein Hochzeitsschild oder einen Fotorahmen verwenden. Schalten Sie einfach Ihre Fantasie ein und verwenden Sie es, wo immer es schön aussieht.

Nach dem Hochzeitskleid sind die Brautjungfernkleider die nächstwichtigsten Outfits für eine Hochzeit. Während Smoking und Anzüge für Männer in Bezug auf Farboptionen und Stiloptionen ziemlich begrenzt sind, reichen Kleider für Brautjungfern von kurz und hell, zu lang und raffiniert, mit einer großen Auswahl dazwischen. Es ist wichtig, dass ein zusammenhängendes Thema ausgewählt wird, um dem Farbschema der Hochzeit zu folgen und die Damen in der Brautpartei als geplante Gruppe mit leicht identifizierbaren Mitgliedern anstelle einer zufälligen Auswahl von Frauen erscheinen zu lassen. Es ist auch eine Herausforderung, lila Brautjungfernkleider und -farben auszuwählen, die mehreren Frauen schmeicheln und bei Bedarf in einer Vielzahl von Größen erhältlich sind.

Farbe ist oft das verbindende Thema in einer Brautparty. Meistens tragen die Frauen, die mit der Braut aufstehen, identische Kleider in identischen Farben. Ein kreativerer Weg ist es, allen Frauen zu ermöglichen, genau die gleichen lila Brautjungfernkleider auszuwählen, jedoch in unterschiedlichem Stil. Lila Brautjungfernkleider in voller Länge mit einem Rock, der bis zum Boden reicht, sind eine häufige Option für Brautjungfern, insbesondere in formelleren Umgebungen, in denen das Brautkleid ebenfalls lang ist. Eine weitere Option sind die teelangen lila Brautjungfernkleider mit einem Rock, der von knapp über dem Knöchel bis knapp unter das Knie reicht. Cocktaillange lila Brautjungfernkleider sind eine praktikable dritte Option, deren Röcke über die Knie hinausragen. Im Allgemeinen schmeichelt ein Rock, der unter die Knie fällt, mehr Körpertypen als lila Brautjungfernkleider in Cocktaillänge. Berücksichtigen Sie außerdem den Ort für die Hochzeit. Ein kurzer Rock wäre in einer Kirche oder sogar in einer Kathedrale fehl am Platz, während für eine Hochzeit im Frühling oder Herbst im Freien die kurzen lila Brautjungfernkleider möglicherweise nicht warm genug sind. Die lila Brautjungfernkleider sollten jedoch für die gesamte Gruppe alle gleich lang sein.

Planen Sie, lila Brautjungfernkleider im Voraus zu kaufen und diese nach Möglichkeit direkt an die Brautjungfern zu versenden. Eine typische Faustregel ist, die lila Brautjungfernkleider etwa sechs Monate vor der Hochzeit zu bestellen, um eventuelle Sonderbestellungen und Änderungen zu berücksichtigen. Vor der Bestellung sollte jedoch jede Brautjungfer gemessen werden, um die richtige Kleidergröße sicherzustellen, da die Größentabellen je nach Designer variieren können.

.

## What’s the difference between VAN and FTP connections?

Not much to add, what’s the main difference between the two? I can’t seem to find easily accessible information anywhere.

## Which airlines accept Virtual Accounts (VAN) for payment?

Do you know which airlines accept Virtual Accounts (VAN) as form of payment, either by a corporate customer or a travel agent.

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