BA982: Inventory Theory¶

Instructor: Kevin Shang
Class Time: Friday, 1-4 PM
Location: McKinley Seminar Room

Github Page      Fuqua Page

Textbooks¶

  • [Zipkin] Zipkin, P. Foundations of Inventory Management
  • [Porteus] Porteus, E. Foundations of Stochastic Inventory Theory
  • Referenced papers below

Grading Policy¶

  • Individual Assignments [50%]
  • Attendance and Class Participation [10%]
  • Paper Presentation [10%]
  • Final Project and Presentation [40%]

Individual Assignments¶

  • From class 2 to class 7, I will assign problems as your homework for you to complete in a week. These problems are mainly extensions of the class lecture.

Paper Presentation¶

  • In each class from 8 to 11, one student will present a paper selected by me. The presentation should cover the paper in detail, including the main ideas, theorems, and the key steps in the proofs.

Final Project and Presentation¶

  • Each student is required to complete a research paper as the final project. The paper must include a clear motivation, a comprehensive literature review, and some preliminary analysis [30%]. Students are also required to present their work during the final class session [10%]. A list of potential open questions is provided at the end of the page.

AI Policy¶

  • Students are permitted to use AI to aid their understanding of the course content. They may also use AI tools to generate ideas and assist in analyzing homework assignments. However, students are expected to critically evaluate AI-generated responses and verify their accuracy. Additionally, students must disclose how they have used AI in their assignments and final projects.

Class Schedule¶

  • Class 1 (8/29): Introduction
  • Class 2 (9/5): Determinisitc Demand Systems
  • Class 3 (9/12): Single-Stage Systems: Finite-Horizon Models
  • Class 4 (9/19): Single-Stage Systems: Infinite-Horizon Models
  • Class 5 (9/26): Serial Systems: Finite-Horizon Models
  • Class 6 (10/3): Serial Systems: Infinite-Horizon Models
  • Class 7 (10/17): Distribution Systems
  • Class 8 (10/24): Inventory Systems with Financial Flows
  • Class 9 (10/31): Data-Driven Inventory Models: Offline (Supervised) Learning
  • Class 10 (11/7): Data-Driven Inventory Models: Online Learning
  • Class 11 (11/14): Reinforcement Learning on Inventory Applications
  • Class 12 (11/21): Project Presentations

Class 1: Introduction¶

  • A motivation example
  • EOQ Model
    • With/w.o. backlogging
    • Stationary demand rate
    • Nonstationary demand rate
  • Newsvendor model

Readings:

  • [Zipkin] 3.1-3.6, 6.1-6.2.2
  • Shang, K. 2011, Introduction, Tutorial in Operations Research

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Class 2: Deterministic Demand Systems¶

  • Power-of-two policies

Readings:

  • [Zipkin] 5.3.
  • Roundy, R. 1985. 98%-effective integer-ratio lot sizing for one-warehouse multi-retailer systems. Management Science 31 1416–1430.
  • Chen, F. 1998. Stationary policies in multiechelon inventory systems with deterministic demand and backlogging. Operations Research 46 S26-S34.

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Class 3: Single-Stage Systems: Finite-Horizon Models¶

  • Single-stage system without fixed order costs
  • Single-stage system with fixed order costs
  • Myopic policies
  • Markov-modulated demand model
  • Lost sales system

Readings:

  • Section 2 of Clark, A. and H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Management Science 6 475-490.
  • [Zipkin] 9.1-9.6
  • [Porteus] 4.2, 7.1
  • Iglehart, D., S. Karlin. 1962. Optimal policy for dynamic inventory process with nonstationary stochastic demands. Chapter 8 in K. Arrow, S. Karlin, H. Scarf (eds.), Studies in Applied Probability and Management Science, Stanford, CA.
  • Zipkin, P. 2008. On the structure of lost-sales inventory models. Operations Research 56 937-944.

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Class 4: Single-Stage Systems: Infinite-Horizon Models¶

  • Base-stock policies
  • (r,Q) policies
  • (s,S) policies

Readings:

  • [Zipkin] 6.2.3-6.2.5; 6.4-6.6
  • Zheng, Y., A. Federgruen. 1991. Finding optimal (s,S) policies is about as simple as evaluating a single policy. Operations Research 39 654-665.
  • Zheng, Y. S. 1992. On properties of stochastic inventory systems. Management Science 38 87–103.

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Class 5: Serial Systems: Finite-Horizon Models¶

  • Finite-horizon model without fixed ordering costs
  • Finite-horizon model with fixed ordering costs
  • Lost-sales system

Readings:

  • Clark, A. and H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Management Science 6 475-490.
  • Shang, K. 2012. Single-stage approximations for optimal policies in serial inventory systems with non-stationary demand. Manufacturing & Service Operations Management 14 414-422.
  • Huh, W.T., G. Janakiraman. 2010. On the structure of serial inventory systems with lost sales. Operations Research 58(2) 486-491.
  • [Zipkin] 8.1-8.3.3

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Class 6: Serial Systems: Infinite-Horizon Models¶

  • Echelon and local base-stock policies
  • Echelon and local (r,Q) policies
  • Echelon and local (s,T) policies
  • Lower bounds, heuristics, and approximations

Readings:

  • Chen, F. 1999. Decentralized supply chains subject to information delays. Management Science 45 1076-1090.
  • Shang, K., J.-S. Song. 2007. Supply chains with economies of scale: Bounds and approximations. Operations Research 55 843 -853.
  • Shang, K., S. Zhou. 2010. Optimal and heuristic echelon (r,nQ,T) policies in serial inventory systems with fixed costs. Operations Research 58 414-427.
  • Shang, K., J.-S. Song. 2003. Newsvendor bounds and heuristic for optimal inventory policies in serial supply chains. Management Science 49 618-638.
  • Shang, K. 2008. Note: A simple heuristic for serial inventory systems with fixed order costs. Operations Research 56 1039-1043.
  • Shang, K., S. Zhou. 2010. Optimal and heuristic echelon (r,nQ,T) policies in serial inventory systems with fixed costs. Operations Research 58 414-427.
  • Chen, F., Y.-S. Zheng. 1994. Lower bounds for multi-echelon stochastic inventory systems. Operations Research 40 1426-1443.
  • Shang, K., S. Zhou. 2009. A simple heuristic for echelon (r,nQ,T) policies in serial supply chains. Operations Research Letters 37 433-437.
  • Shang, K., S. Zhou, G.-J. van Houtum. 2010. Improving supply chain performance: real-time demand information and flexible deliveries. Manufacturing & Service Operations Management 12 430-448.

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Class 7: Distrbution Systems¶

  • Infinite-horizon model with base-stock policies
  • Balance assumption and myopic allocation
  • FCFS allocation
  • Lower bounds and approximations
  • Closed-form solution: Taylor approximation

Readings:

  • [Zipkin] 8.5-8.6
  • Chen, F., Y.-S. Zheng. 1994. Lower bounds for multi-echelon stochastic inventory systems. Operations Research 40 1426-1443.
  • Gallego, et al. 2007. Bounds, heuristics, and approximations for distribution systems. Operations Research 55 503-517.
  • Shang, K., Z. Tao, S. Zhou. 2015. Optimizing reorder intervals for two-echelon distribution systems with stochastic demand. Operations Research 63(2) 458-475.

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Class 8: Inventory Systems with Financial Flows¶

  • Single-stage models with financial flows
  • Serial systems with financial flows

Readings:

  • Modigliani, F., M. H. Miller. 1958. The cost of capital, corporate finance and the theory of investment. The American Economic Review 48(3) 261-297.
  • Shang, K. 2019. Cash Beer Game. Foundations and Trends® in Technology, Information and Operations Management 12(2-3) 173-188.
  • Gaur, V., S. Seshadri. 2005. Hedging inventory risk through market instruments. Manufacturing and Service Operations Management 7(2) 103-120.
  • Chen, X., M. Sim, D. Simchi-Levi, P. Sun. Risk aversion in inventory management. Operations Research 55(5) 828-842.
  • Luo, W., K. Shang. 2019. Technical Note: Managing inventory for firms with trade credit and deficit penalty. Operations Research 67(2) 468-478.
  • Luo, W., K. Shang. 2015. Joint inventory and cash management for multi-divisional supply chains. Operations Research 63(5) 1098-1116.
  • Kouvelis, P., Li, R. 2020. Integrated risk management for newsvendors with VaR constraints. Manufacturing and Service Operations Management 21(4) 816-832.

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Class 9: Data-Driven Inventory Models: Offline (Supervised) Learning¶

  • Single-stage systems
  • Distribution systems
  • Concentration inequalities and bounds

Readings:

  • Chapter 2 Basic tail and concentration bounds. Lecture notes by Martin Wainwright, Dept. of Statistics, UC Berkeley.
  • Levi, R., R. Roundy, D. Shmoys. 2007. Provably Near-Optimal Sampling-Based Policies for Stochastic Inventory Control Models. Mathematics of Operations Research 32 821-839.
  • Lin, M, Huh, W.T. Krishnan, H., Uichanco, J. 2022. Data-driven Newsvendor problem: Performance of the sample average approximation. Operations Research 70(4) 1996–2012.
  • Bertsimas, D., N. Kallus. 2020. From predictive to prescriptive analytics. Management Science 66(3) 1025-1044.
  • Levi, R., G. Perrakis, J. Uichanco. 2015. The data-driven newsvendor problem: New bounds and insights. Operations Research 63(6) 1294-1306.
  • Ban, G.-Y., C. Rudin. 2019. The big data newsvendor: practical insights from machine learning. Operations Research 67(1) 90-108. (Please download the online appendix and go through the proofs.)
  • Huang, J.-K., K. Shang, Y. Yang, L. Zhou, D. Li. 2023. Taylor approximation of inventory policies for one-warehouse multi-retailer systems with demand feature information. Management Science.

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Class 10: Data-Driven Inventory Models: Online Learning¶

  • Single-stage systems
  • Lost sales model with censored demand
    • Parametric learning: Bayesian approach
    • Nonparametric learning: Adaptive SGD-based algorithms

Readings:

  • Azoury, K. S. 1985. Bayes solution to dynamic inventory models under unknown demand distribution. Management Science 31(9) 1150-1160.
  • Huh, W. T., P. Rusmevichientong. 2009. A nonparametric asymptotic analysis of inventory planning with censored demand. Mathematics of Operations Research 34(1) 103-123.
  • Huh, W.T., G. Janakiraman, J. Muckstadt, P. Rusmevichientong. 2009. Asymptotic optimality of order-up-to policies in lost sales inventory systems. Management Science 55(3) 404-420.
  • Ding, J., Rong, Y, Huh, W.T. 2024. Feature-based inventory control with censored demand. Manufacturing & Service Operations Management 26(3) 1157-1172.
  • Yang, C, Huh, W.T. 2024. A non-parametric learning algorithm for a stochastic multi-echelon inventory problem. Production and Operations Management 33(3) 701-720.
  • Huh, W.T., Janakiraman, G., Muckstadt, J., Rusmevichientong, P. 2009. An adaptive algorithm for finding the optimal base-stock policy in lost sales inventory systems with censored demand. Mathematics of Operations Research 34(2) 397-416.
  • Huh, W.T., Levi, R., Rusmevichientong, P., Orlin, J. 2011. Adaptive data-driven inventory control with censored demand based on Kaplan-Meier estimator. Operations Research 59(4) 929-941.

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Class 11: Reinforcement Learning on Inventory Applications¶

  • Approximate dynamic programming
  • Monte-Carlo evaluation and policy improvement
  • Temporal difference learning
  • Q-learning
  • Actor-critic method
  • Deep neural net

Readings:

  • Ch. 5, 6, 15.8 of Reinforcement Learning: An Introduction by R. Sutton and A. Barto.
  • Gijsbrechts, J., R. Boute, J. Van Mieghem, and D. Zhang. 2022. Can deep reinforcement learning improve inventory management? performance on lost sales, dual-sourcing, and multi-echelon problems. Manufacturing and Service Operations Management 24(3) 1349-1368.
  • Mao, W., K. Zhang, R. Zhu, D. Simchi-Levi, T. Basar. 2025. Model-free nonstationary reinforcement learning: Near-optimal regret and applications in multiagent reinforcement learning and inventory control. Management Science 71(2) 1564-1580.

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Class 12: Final Project Presentation¶

  • Students present the final project following the guidance in the course policy section.

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Open Research Questions¶

  • Based on Shang and Song (2003) and Levi. et al. (2007), is it possible to develop $\alpha$-optimal echelon base-stock policies?
  • Based on Gaur and Seshadri (2005) and Luo and Shang (2019), one can conclude a 2 by 2 table where the horizontal axis represents whether financial markets are perfect and the vertical axis represents the risk attitudes (i.e., either risk neutral or risk averse). A standard newsvendor model is dervied from the assumption that the financial market is perfect and the firm is risk neutral. Luo and Shang (2019) study a risk-neural firm facing imperfect financial markets. Gaur and Seshadri (2005) study a risk-averse firm facing perfect financial markets. It would be interesting to explore the fourth quadrant.
  • Using Q-learning or other learning algorithms to study a lost-sales inventory model with lead times. Can you derive analytical regret bounds for your learning algorithm and compare the resulting policy with existing heuristics?
  • Using a finite-horizon, single-stage model as a basis to compare and contrast learning algorithms, including offline and online approaches. This can be a literature review paper that includes perishable (newsvendor-like) and nonperishable inventory systems.
  • You are encouraged to explore topics related to transfer learning, offline-data-online-learning, and multi-agent learning problems.

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