Optimisation with lifespan of products after moving

Our team is trying to solve an optimization problem related to resource management with 7 item types, 4 departments, and 168 time steps. In this problem, we must decide on actions such as moving and buying to meet market demand while optimizing profit. However, every time we buy a number of new products, these products will have a life expectancy, e.g., 50. This means they will be dismissed after their life expectancy is reached.

A significant problem arises when we move products from one department to another (since each pair of item and department has different selling prices, and the demand depends on these pairs). The issue lies in the life expectancy. When we decide to move a certain number of items from, for example, Department 1 (D1) to Department 2 (D2), we don't know how many items with what remaining lifespan we should move. However, if we add lifespan as a variable, it will increase the number of variables and constraints so much that we cannot solve the problem in a reasonable time.

We would like to use column generation to solve this problem, but we don't know how to formulate the optimization problem. Can anyone help with ideas or provide relevant keywords to approach this issue? We would really appreciate it.