# Allocating Items to a Limited Number of Slots

#### sumantra

##### New Member
Hi,

Let's say that I have a small retail shop that has slots available for only 5 types of products.

However, I also have a warehouse located behind my store in which I can store my inventory. Let's assume that I always keep an inventory of 10 different types of products.

I now want to figure out which 5 of those 10 types of products should be shown in my store and in what order (i.e. which product should be present in the first slot, which in the second slot etc.) so that my total sales are maximized. One of the constraints is that a particular product cannot be present in more than 1 slot. Therefore, what this means is that at any time, I will be showing exactly 5 products to my customers and I would not be able to show the remaining 5 products to my customers.

Let's assume that I currently have no idea which products are more popular among customers etc. We should also assume that the ordering of the 5 products that are actually shown to customers (i.e. which product is present in which slot) is important in terms of customers deciding whether or not they want to buy anything from me.

Obviously, the total number of ways 10 products can be allocated in 5 slots is "10 P 5" (i.e. permutation of 10 over 5) which is more than 25,000. And, one way to find out which is the best permutation would be to conduct an experiment in which every time a customer comes to my store, I present him with a different permutation. Once the first 25,000 customers have visited my store so that each permutation has been seen once, I can then repeat the same experiment with the next group of 25,000 customers and continue doing this for a while. Once I have enough data, I can then conclude which permutation would be maximizing my sales.

However, for obvious practical reasons, running such a large experiment is infeasible because it would take me years before I have enough data.

Is there any technique in statistics that will allow me to maybe test a small sub-set of all these permutations and then mathemtically predict which is the best one (without the best one necessarily having been present in the actual test)?

I am not sure whether Design of Experiments is going to be helpful here - but if you think it will, can you point me to the specific technique in DOE which would apply to my situation?

Thanks, and if you need any clarifications before you can answer my question, please let me know.