In this proposal we plan to study solution methodologies for general multi-echelon systems with possible stochastic lead-times, economies of scale, production and transportation capacities, and demand occurring at each stage or node of the system. The main objective is in developing novel state of the art solution methodologies for these complex systems. General multi-echelon systems are modeled as stochastic dynamic programs, which are solved by approximate dynamic programming techniques.
One of the main contributions of this proposal is the notion of multivariate piecewise linear functions, which are called the ridge functions. These functions are easy to encode and several properties of these functions are easily stated. The key component of the dynamic programming algorithm is a suitable value function approximation. We first embed ridge functions in the approximate dynamic programming algorithm and then we plan to study properties typical for inventory systems. An important feature of the algorithm is its robustness and generality. For example, the algorithm can easily handle stochastic lead-times, capacity constraints, economies of scale, and correlated demand, in addition to general materials flows. We also modify the algorithm for a different type of problems like the vendor managed inventory problem, where the time between two consecutive decisions is non stationary.
The main impact is in showing the applicability of ridge functions in operations research and management. They are particularly appealing in approximating complex functions in an underlying mathematical programming model. Per se, they do not introduce nonlinearities. Perhaps even a bigger impact is in solving complex multi-echelon inventory and production systems, where among other features the materials flows are arbitrarily. We firmly believe that such a global view of the proposed algorithm can have a significant impact on the current practice.
