# Stochastic Modeling for Inventory and Production Planning in the Paper Industry

Topics: Markov chain, Inventory, Markov decision process Pages: 71 (10858 words) Published: March 17, 2013
Stochastic Modeling for Inventory and Production
Planning in the Paper Industry
K. Karen Yin
Dept. of Bio-based Products, University of Minnesota, St. Paul, MN 55108

G. George Yin
Dept. of Mathematics, Wayne State University, Detroit, MI 48202

Hu Liu
3M Commerce Services, 3M Center, St. Paul, MN 55144
DOI 10.1002/aic.10251
Published online in Wiley InterScience (www.interscience.wiley.com).

Problem formulations and solution procedures of production planning and inventory management for manufacturing systems under uncertainties is discussed. Markov decision processes and controlled Markovian dynamic systems are used in the models. Considering an inventory problem in discrete time and formulating it by a ﬁnite-state Markov chain lead to a Markov decision process model. Using the policy-improvement algorithm yields the optimal inventory policy. In controlled dynamic system modeling, the random demand and capacity processes involved in planning are described by two ﬁnite-state continuoustime Markov chains. Such an approach enables us to embed the randomness in the differential equations of the system. The optimal production rates that minimize an expected cost are obtained by numerically solving the corresponding Hamilton–Jacobi– Bellman (HJB) equations. To overcome the so-called curse of dimensionality, frequently encountered in computation, we resort to a hierarchical approach. Illustrative examples using data collected from a large paper manufacturer are provided. © 2004 American Institute of Chemical Engineers AIChE J, 50: 2877–2890, 2004

Keywords: inventory management; production planning; Markov chain; optimal policy; hierarchical approach

Introduction
In the pulp and paper industry, the search for good inventory control policies and production plans started several decades ago. Many models have been developed and used (see
Leiviska, 1999 and references therein). Nevertheless, similar to ¨
the situation in many other industries, inventory management and production planning remain to be challenging problems to paper manufacturers.

© 2004 American Institute of Chemical Engineers

AIChE Journal

November 2004

Production planning and inventory management are crucial
components of any supply chain. Efﬁcient planning and inventory policies are necessary for the successful operation of any modern-day enterprise. It is well known that manufacturing
systems are subject to random events such as raw material
variation, demand ﬂuctuation, and equipment failures. The
dynamic and random nature of the demands makes their forecasting very difﬁcult or sometimes impossible. Despite the existence of the various models, very often managers cannot
ﬁnd a single one that is suitable to their needs. As a result, decisions on inventory have to be based on a combination of
experience, mathematical models, and even on the gut feeling of a few individuals, whereas production is planned following Vol. 50, No. 11

2877

an everyday practice without concerning the optimality. It is desirable to shift such experience-based decision making to an information-based decision making model. This will require a systematic use of historical data and a theoretically sound

mathematical model that is applicable to the real situation. This work is intended to contribute in this direction.
A large manufacturer/corporation usually has a certain number of plants/mills at different geographical locations. Having a collection of machines/equipment, each plant has its own inventory with a ﬁxed carrying cost. Characterized by its production rates for a given product type, each machine can produce multiple products. At any given time, the enterprise has a set of orders in hand. Each order speciﬁes a customer, a product, an order quantity, and a due date. Orders are fulﬁlled on one or more machines of speciﬁed plant(s). To minimize the...