0

In my thesis, I'm dealing with SLAP (storage location assignment problem) -- which is finding optimal distribution of products to location slots in a generic warehouse. My aim was to implement multiple evolution algorithms (GA, DE, ABC & PSO) to solve this problematic and do a comparison. Since the mentioned problematic (SLAP) is also combinatoric problem with similar constraints and characteristics as TSP, I did research on TSP problematic solved using these algorithms, and applied it on my problem, e.g. based on those papers: GA/PSO, DE, ABC. These approaches were mapped on my problem and works really well. Now, my supervisor told me to create a mathematical proof of TSP - SLAP problematics equivalence or at least some mathematical mapping, he is not doing research in this field so he cannot help me.

To summarize my problem and why I think it is similar to (nearly same as) TSP:

  • In TSP, we need to use all cities in the chromosome (solution in general). In my problem, we need to use all articles in the solution.
  • In TSP, we cannot have duplicates (we cannot go through one city more then once). In my problem, we also cannot have duplicates, as it would imply that more products are stored within one location slot.

I did some research and found out both SLAP and TSP are NP-Complete problems. It would be possible to create mathematical proof that my problem is NP-Complete. It is both calculated in polynomical time and some well known NP-Complete problem, such as halting problem can be reduced to this problem. Although I don't think this would help me much.

(Pseudo-)formal description: Let's suppose we have $N$ products, and $M$ location slots, where $M$ >= $N$ (as a input). We need to distribute all products to the locations slots, in a way which decreases customer order processing time the most (i.e. all orders has to be processed ASAP). The objective function is duration of order processing and is evaluated as running a simulation of a warehouse (with given distribution of products to slots). So the output is the best possible allocation of the products to slots, which can be expresses simply as a vector of integers, where position in vector specifies article, and the number on the position specifies location slot (into which the product is placed).

Is there any way I can create a proof that these two problems are equivalent or atleast some mathematical mapping from my problem to TSP? Thank you.

If required, I can provide further data, such as chromosomes encoding in GA, ... (but it didn't seem relevant to the proof from my point of view).

kocica
  • 101
  • 2
  • 1
    You need to do it the other way around: reduce a hard problem (like TSP) to SLAP. It would help to have a formal description of the SLAP decision problem, i.e., what is the input and the output? – Juho Dec 03 '20 at 19:50
  • @Juho I added some description but its not really formal, but I hope you understand it. Also thank you for your input, I fixed it in the question :) – kocica Dec 03 '20 at 20:05
  • https://cs.stackexchange.com/q/1240/755, https://cs.stackexchange.com/q/11209/755 – D.W. Dec 03 '20 at 21:00
  • 1
    We still need a precise formalization of the problem statement. Until then it is not meaningful to discuss whether it is NP-complete or not. – D.W. Dec 03 '20 at 21:01
  • You say that SLAP is already known to be NP-complete. Why is that reference insufficient to show the equivalence of TSP and SLAP? You mention that your supervisor is "not doing research in this field". Does he mean something different from the usual polynomial-time reduction framework when speaks of "mathematical mapping"? Or, is he looking specifically for a reduction from TSP to SLAP? In which case, I would wonder what additional info is being sought, if SLAP is already known to be NP-complete. – mhum Dec 04 '20 at 04:25
  • I'm sorry if I wasn't clear. I'm not trying to do reduction, I just mentioned it would not probably help me, but it was the only thing I was able to find. I really do not know how to provide formal description of SLAP. Could you, please, provide me some simple example of mathematical proof of some other problem (even made out) and TSP equivalence or mapping? – kocica Dec 04 '20 at 08:07
  • Hm, this raises more questions than it answers. I don't know what kind of TSP equivalence or mapping you might be looking for if not the usual kind of reductions (e.g.: for NP-completeness proofs). I can't help but think there may be some kind of conceptual or terminology mismatch here, especially if your supervisor does not work in this field. As for formal descriptions, you can find a list of such descriptions here, drawn mainly from Garey & Johnson. – mhum Dec 04 '20 at 16:43
  • Thank you a lot for the provided resources, I'll go through them. As you said, there is probably some kind of terminology mismatch. In human words, he just want to create mathematical proof or at least formally written text, that my problem (SLAP) is essentially same or at least similar to the TSP, and solution for TSP can therefore be applied on SLAP. If only reduction can provide such verification, I'll do it :) Thanks again. – kocica Dec 04 '20 at 18:56

0 Answers0