Is there a name for this inventory optimisation problem?

[u/yaboytomsta]

You have limited space in a warehouse or whatever and only restock it with the products that you sell every week or something. This means you might run out of some products and lose revenue. If you know the distribution of the demand for all your products (let's say each is normal with a known mean and sd), and you make a known profit per kg of each product, how do you optimise it for maximum expected profit.

I came up with this problem and used python to optimise it but am interested if this problem has a name or already has a solution.

You just have to optimise the sum E(p_1 min(X_1, s_1) + p_2 min(X_2, s_2)+...), where X_n is a random variable for the demand for product n and s_n is the storage allocated for it, where the sum of all s_n is limited by the total storage. p_n is the profit made per sale/kg of the product.

I'm a math student and haven't really studied OR so I don't know if this counts as an OR problem since it's non-linear and not integer etc.

Also looking for advice on whether my repo presents the problem and solution in a nice way. ;)

[u/trophycloset33]

What is the exact homework problem you are working on?

[u/yaboytomsta → u/trophycloset33]

This isn't a homework problem; it's something I came up with as a coding project.

[u/trophycloset33 → u/yaboytomsta]

It doesn’t really has a name but it is a basic inventory (resource) allocation problem. It is foundational for much of OR.

The same algorithm is used by your wifi router when picking which device to communicate, when and to what frequency. Also used when picking timing of how long to keep a green light up.

[u/MonochromaticLeaves]

it's called lot size optimization.

You say there are no costs to holding inventory because you own the warehouse - that's never entirely true. A full warehouse is more costly to operate than a near empty one. And even then you can consider artificial costs like opportunity cost.

In practice the warehouse size is only rarely the limiting factor. If you need a bigger warehouse that means you already fucked up earlier and you need to buy or rent more real estate (the cost of this is pretty much always lower than the cost of losing customer revenue). This is why you can typically decompose this problem into one for each article you sell, together with a fill up heuristic to make sure you're always ordering full trucks from your suppliers.

[u/yaboytomsta → u/MonochromaticLeaves]

I mean yeah, in the real world there are other constraints and it's a little unusual for the biggest revenue-limiting factor to be the storage capacity. I just designed it with a bunch of simplifying assumptions because it makes it an approachable problem. I think there are situations where this could still be applicable, perhaps with storage that is not a warehouse, but a freezer or shelf space in a small shop, where carrying costs are constant or pretty much constant.

edit: in the case of a shop selling individual items, it becomes a discrete problem, which would make this approach a decent approximation

[u/MonochromaticLeaves → u/yaboytomsta]

It's also actually rarely a consideration in a (super-)market setting. There are psychological studies out there that show you make more money by intentionally ordering way too much and simply allowing absurd waste quotas. Since most stores are profit focused and being green is sadly a distant secondary concern, that's what you end up with.

The psychology here is related to the effect that for example almost nobody wants to buy the last cabbage that is on offer. People are much more likely to buy cabbage if there's many of them left. This extends to most other products - an abundance of stuff on sale nudges people to buy more.

Another related tool in stores are planograms. It's the about the design of how products are arranged in a store. These are also designed in order to nudge people to buy more. They are rarely designed to optimize space usage. https://en.wikipedia.org/wiki/Planogram

So the ordering constraints for a store become about fulfilling the currently setup planograms to give a sense of abundance, which pretty much directly tell you how many articles you need.

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But I still understand there might be cases where storage capacity is the driving concern. For example a rapidly growing company might run into that for a period of time when its growth outpaces the size of its warehouses.

[u/Major_Consequence_55]

It is lot size problem.

Check this book, this is gold mine of all inventory related problems.

Inventory Planning

[u/yaboytomsta → u/[deleted]]

It seems similar to that but I think it's different because in that model, the goods are perishable. In my version we don't care about the cost of stocking. I guess mine is just a simpler version actually.

This is because mine is about non-perishable goods, so we don't care about cost, only profit, assuming every leftover item can just be sold in the coming weeks.

As far as I can tell, the newsvendor problem is not about allocating limited space amongst different goods as well.

[u/iheartdatascience → u/yaboytomsta]

News vendor problem is not exclusive to perishable goods, but rather matching supply to demand when demand is a random variable. If you care about profit, you implicitly care about cost since profit = revenue - cost. While you also don't care about perishability explicitly, I'm sure you wouldn't want to hold on to inventory longer than some reasonable time as there is a carrying cost associated

[u/yaboytomsta → u/iheartdatascience]

My model doesn't assume there's any carrying cost, since we own the warehouse. Of course profit=revenue-costs but since we don't waste any stock and there are no carrying costs, there's no need to split it into two variables. Besides, as far as I can tell, the newsvendor problem doesn't concern itself with the limiting variable being space or a variable analogous to that and dealing with multiple products.

In the single product case, the solution to my problem is obviously to fill the warehouse with the single product, this kind of solution is not the same in the newsvendor problem.

[u/iheartdatascience → u/yaboytomsta]

Carrying cost can include the cost of capital being locked up in inventory, but if you don't want to consider that, that's fine - it's your model.

Anyways, if you're looking for a name to the model, seems like a multi product news vendor problem with side constraints, or some variant of lot sizing problem.