Let's say we have a kiosk inside gas station selling cigarettes and snacks outside a supermarket. There are limited space for each product. The kiosk only allows to get restocked from the supermarket, so there's no lead time to restock. The objective is to figure out what is the ideal BOH(balance on hand )value or (s,S), (r,Q) for each product to minimize the lost sales. There's no labor or any type of cost. We have the ability to monitor the BOH every day. Another question is that some people buy a carton of cigarette rather than an individual pack, and I think we need to treat them as different product. We can only restock once per day.

My thinking: since each product has fixed self space(could be changed based in the future), this space number becomes the S. The problems becomes to find the right safe stock s considering the demand(could be bootstrapped from history or forecast).

Is there any inventory model for this kind of problem? Is it too over-simplified?

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    $\begingroup$ If there's no lead time and no fixed cost for replenishment, then the optimal solution is to replenish the machine every time anyone buys any item from it. It's a continuous-review base-stock policy with base-stock level = shelf space for that product. So I think you are missing something in your model that prevents this. Typically that would be either a fixed cost or a reorder interval (e.g., we can only restock once per day or once per week). $\endgroup$ Commented Feb 8, 2021 at 2:09
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    $\begingroup$ If the shelf space for each item is fixed, then you just want to optimize each product individually. Carton and pack would be treated as separate items. There are models that optimize the shelf space, but those are more complicated than standard inventory models. $\endgroup$ Commented Feb 8, 2021 at 2:11
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    $\begingroup$ Also: What do you mean by BOH? Beginning on-hand inventory? $\endgroup$ Commented Feb 8, 2021 at 2:11
  • $\begingroup$ @LarrySnyder610 yes,at least according to fuseinventory.com/blog/inventory-planning-dictionary .@janicebaratheon if you really have no costs/restrictions at all and no lead time And want to minimize lost sales, it is intuitively and mathematically the best option to restock the moment your inventory drops below S. I also think you are missing something in your problem formulation. $\endgroup$ Commented Feb 8, 2021 at 9:40

1 Answer 1


I think you could use a simple continuous-review base-stock policy, as @LarrySnyder610 mentioned too. About the inventorying each product to minimize the lost sales, first of all, you will need to forecast demand for each product in a specific time bucket, (e.g. weekly sales). Based on, further, you would be able to calculate the inventory level and replenishment periods for each item. Also, are being able to capture customers favours based on their frequent orders to stock items.

As you said, you do not have any specific cost, such as labour or logistics costs, it would be easy to replenishment the kiosk storage area by your own at the end of the day. Indeed, you might be interesting to use shared storage policy in contrast fixed storage policy to increase sales flexibility.

Finally, to control the inventory level of each item, you would not need to use any specific software and all of the things might be applied in the separate sheet, like excel.

  • $\begingroup$ Thank you! Is there any example or model I can take a look at? How to track lost sales? We can actually just double stack the products as we want, and maybe leave the products on the floor if necessary. I think for the fast moving items, we can set the S based on the forecast, while follow (s,S) for slow moving items. $\endgroup$ Commented Feb 9, 2021 at 17:03
  • $\begingroup$ While ago, I did something like this but now, I really do not know can find that. I will try and if it was, I will update the answer. About the fast-moving items, what you mentioned would be useful. Would you say please, why you are trying to use some of the complicating models to calculate inventory for a small area which can be replenished ASAP? $\endgroup$
    – A.Omidi
    Commented Feb 9, 2021 at 18:29
  • $\begingroup$ I am in the discovering stage, and realized we have the capability of replenish say 2 times a day. Thank you! $\endgroup$ Commented Feb 9, 2021 at 18:54

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