There are several sites with simple graphics to illustrate the optimal solution to the EOQ model. Here is one:
https://en.wikipedia.org/wiki/Economic_order_quantity
It discusses the balance between the cost of purchasing and the cost of storing items, and tries to balance these costs optimally. It is a rather "academic" model because of the strong assumptions made, but it is utilized very often in order to illustrate the importance of incorporating all costs involved, and how they are balanced.
The convexity/concavity bit comes into play because an optimal solution to the EOQ model can easily be derived under the assumption that the total cost is convex, according to the simple analysis made on the web-page above.
It should be said that in reality inventory models are more complex than this - for example, we have most often more than one item, we have several types of capacity limitations, and so on, but it is a rather instrumental illustration nevertheless.
The role of concavity and convexity is to find the optimum function.
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