Mathematical Finance, Stochastic Analysis, and Machine Learning Seminar by Abdoulaye Thiam: On Inventory Control Problems with Uncertainty Quantification and a Case Study
Abdoulaye Thiam, Department of Applied Mathematics, Illinois Institute of Technology
On Inventory Control Problems with Uncertainty Quantification and a Case Study.
The inventory control problem is the optimal control problem for which a company must decide how much to order for its products to satisfy the demand in each time period. This problem can be modeled by stochastic dynamic programming introduced by Richard E. Bellman. More precisely, we study an optimal control problem in the form of a Bellman equation of decision making under uncertainty and find a policy that shows the optimal behavior. In practice, the assumption of known demand distribution conflicts with many real situations in which the inventory manager (IM) has some doubts about the distribution. In this presentation, we suppose the distribution to depend on an unknown parameter. However, the IM has an initial belief on the unknown parameter which we express by a probability density. This belief function will evolve through the Bayesian process in each time period. We introduce this learning method to study how this affects the inventory policy. First, we begin with a completely general belief function and analyze our problem through the functional Bellman equation resulting from the dynamic programming formulation of the infinite-dimensional inventory problem under consideration. We show, by a transformation, how our problem can be reduced to a linear one in the belief updating. By the monotonicity argument, we show that the value function is now a function that not only depends on the usual current inventory level but also on the latest belief function. We derive a functional equation for the derivative of the value function with respect to the inventory level, obtain it as the fixed point of the equation directly by an iterative approach, and recover the base stock. Our major contribution is to generalize the classical base stock policy which will now depend on the latest belief function updated from observing the demand realizations. We also validate the model considered by numerical simulation in the case where the demand comes from one of two possible distributions, but it’s not known which.
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Mathematical Finance, Stochastic Analysis, and Machine Learning Seminar