How to formulate a storage component?

Let’s say I have a drink for several customers, and I know their demand. Besides, I also have a storage tank for the drink during times when demand exceeds supply. The storage tank's size is not yet known, but we need to determine what the minimum size of the storage tank needs to be. Unfortunately, the storage tank also comes with a cost per time unit. We want to maximize the profits. I am currently a bit stuck at the storage component without the model blowing up. We define $$c_{i,t}$$ as a decision variable, whether we are going to serve customer $$i$$ at time $$t$$. Also, we define $$D_{i,t}$$ as a parameter, which are the demands for customer $$i$$ at time $$t$$. Also, the storage cost of the tank is $$S_{t}$$. Basically, our objective function would be

$$\max \sum_{t} c_{i,t} \cdot D_{i,t} - q_{t} \cdot S_{t}$$

where $$q_{t}$$ is the current content of the storage tank. However, if we would like to define $$q_{t}$$, we need to sum up all the previous inflows and outflows to the storage tank, which is cumulative over the previous periods. This blows up the model, I think. What would be a better way to model this?

A big assumption in this is that the more liquid you store, the higher the cost. However, for simplicity's sake, let's assume $$S_{t}$$ is the price per unit storage.

PS: Other additions to speed up to solve this model or anything else would be highly appreciated. Suggestions and improvements are highly appreciated.

• Dear @Snowflake, I suggest to consider the Wagner-Whitin model. By means of some arrangements, this model fits your problem. By the way, I have two questions: first q. can the demand $D_{it}$ be dispatched partially? Second q,i s the price of seller the same for every customer? en.wikipedia.org/wiki/Dynamic_lot-size_model – marco tognoli Mar 19 at 19:45
• You have not indicated how supply is handled. Are you resupplied every period, or just when an order is placed? Is there a lead time to get replenishment orders? Is the cost of the drink constant regardless of reorder quantity? Is there a cost to place orders (separate from the cost of the drink itself)? – prubin Mar 19 at 21:15