379819 Inventory Optimization for Production Network Flexibility
The industrial production of chemicals is characterized by complex networks of unit operations that transform raw materials into products. In these networks, material flows are constrained by the production capacity of the processing units, and the performance of the whole system often depends on the capacity of some limiting processes. The rigorous design of production networks is intended to coordinate the production capacity across all units under normal operation, but the variability in processing conditions often creates temporal bottlenecks that affect network performance.
It is a common practice in industry to allocate storage units at different stages of the production network in order to decouple the production of successive sections. The role of inventory is to buffer temporal mismatches among supply availability, processing rates, and demand. Different types of inventories are used for each purpose: raw material inventories hedge against variability in supply, intermediate inventories protect against variability in processing capacity, and final product inventories hedge against variability in demand. The importance of intermediate inventories resides in their ability to reduce processing units’ interdependence by delaying the formation of bottlenecks and increasing capacity utilization. In this framework, inventory management strategies can have a significant impact in the performance of the production network.
Inventory management includes all decisions related to the replenishment and depletion of inventories. In the context of production networks, inventory decisions are closely related to production decisions because most processing units are simultaneously internal suppliers and consumers. Furthermore, inventory replenishment requires using available capacity in the upstream processing units, which reinforce the role of capacity constraints in production networks.
A realistic representation of variability in production networks implies adopting a probabilistic description of the corresponding process parameters. The resulting mathematical model integrates the stochastic processes describing variability, the production decisions limited by capacity constraints, and the history dependent levels of inventory. The characterization of these stochastic inventory levels in capacitated production networks is a difficult task. They depend on the other random processes involved in the system and on the inventory management decisions.
The optimal inventory management strategy for a single storage unit subject to capacitated replenishment and uncertain demand has been proved to follow a modified base-stock policy . The policy consists of a set of rules intended to maintain the inventory at the base-stock level; however, this is not always possible because inventory replenishments are constantly limited by upstream production capacity. Using a base-stock policy in integrated production networks not only determines the inventory management, but also the production strategy that maximizes utilization when the inventory is below the base-stock level.
Adopting base-stock policies reduces the inventory management problem to finding the optimal base-stock level for each inventory location and to determine replenishment priorities among them. However, the resulting formulation is very challenging even for the simplest probabilistic descriptions of the stochastic parameters. In essence, the formulation includes stochastic differential equations that describe the rate of change of the inventory levels as functions of stochastic parameters and the base-stock policies.
In order to overcome the difficulties associated with such a complex formulation, a discrete-time version of the model is used. The model is based on the principles of discrete-event simulation, which is a widely used method to evaluate inventory management strategies. The idea is to include the discrete time model together with base-stock policies in an optimization problem that determines base-stock levels and replenishment priorities. In this framework, the variability in supply, processing capacity, and demand can be represented with sample-paths of the corresponding stochastic processes during long time horizons.
The sample-path optimization approach has the advantage that it provides an approximation of the optimal solution of the stochastic model using as a basis the deterministic problem given by random samples. Additionally, it has the flexibility to analyze production networks of complex topology and arbitrary characterizations of the stochastic processes.
The proposed methodology is implemented in two examples with variability in supply, production capacity, and demand. The first example is a single production-inventory system; it is used to illustrate the implementation of the base-stock policy and the solution method. The second example is a production network with processing units in series and parallel, and multiple inventory locations. The results show the importance of inventory management in the performance of production networks and the effectiveness of the proposed approach.
 Federgruen, A., and P. Zipkin. 1986. An Inventory Model with Limited Production Capacity and Uncertain Demands I. The Average-Cost Criterion. Mathematics of Operations Research, 11, 193.