Inventory control problem

The inventory control problem is a type of problem encountered within the field of optimal control.

Concepts

One issue is infrequent large orders vs. frequent small orders. Calculating shipping costs, volume discounts, storage costs, and capital costs, this can be figured with mathematical precision. Basically, how much money do you wish to have tied up in inventory?

A second issue is having the needed merchandise on hand in order to make sales during the appropriate buying season(s). A classic example is a toy store pre-Christmas. If one does not have the items on the shelves, one will not make the sales. And the wholesale market is not perfect. There can considerable delays, particularly with the most popular toys. So, the entrepreneur or business manager will buy on spec. Another example is a furniture store. If there is a six week, or more, delay for customers to get merchandise, some sales will be lost. And yet another example is a restaurant, where a considerable percentage of the sales are the value-added aspects of food preparation and presentation, and so it is rational to buy and store somewhat more to reduce the chances of running out of key ingredients. With all these examples, the situation often comes down to these two key questions: How confident are you that the merchandise will sell, and how much upside is there if it does?

And a third issue comes from the whole philosophy of Just In Time, which argues that the costs of carrying inventory have typically been under-estimated, both the direct, obvious costs of storage space and insurance, but also the harder-to-measure costs of increased variables and complexity, and thus decreased flexibility, for the business enterprise.

Equations

The mathematical approach exists in two variations, and is typically formulated as follows:A store has, at time $k$, $x\_k$ items in stock. It then orders (and receives) $u\_k$ items, and sells $w\_k$ items, where $w$ follows a given probability distribution. Thus : $x\_\{k+1\}\; =\; x\_k\; +\; u\_k\; -\; w\_k$. The store has costs that are related to the number of items in store and the number of items ordered: :$c\_k\; =\; c(x\_k,\; u\_k)$.The store wants to select $u\_k$ in an optimal way, i.e. to minimize:$sum\_\{k=0\}^\{infty\}\; c\_k$.xxxxxxx

Best of Both Worlds

And this is of course what is recommended, that an enterprise use both the judgment and experience of the business person and the most relevant equations and data.