Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T10:50:03.743Z Has data issue: false hasContentIssue false
  • English
  • Français

STOCHASTIC MULTI-ITEM INVENTORY SYSTEMS WITH MARKOV-MODULATED DEMANDS AND PRODUCTION QUANTITY REQUIREMENTS

Published online by Cambridge University Press:  27 April 2012

Aykut Atalı  and
Özalp Özer
Aykut Atalı
Affiliation:
McKinsey & Company, Chicago, IL 60603 E-mail: [email protected]
Özalp Özer
Affiliation:
School of Management, The University of Texas at Dallas, 800 W. Campbell Rd, Richardson, TX 75080 E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a multi-item two-stage production system subject to Markov-modulated demands and production quantity requirements. The demand distribution for each item in each period is governed by a discrete Markov chain. The products are manufactured in two stages. In the first stage, a common intermediate product is manufactured, followed by product differentiation in the second stage. Lower and upper production limits, also known as production smoothing constraints, are imposed on both stages for all items. We propose a close-to-optimal heuristic to manage this system. To do so, we develop a lower bound problem and show that a state-dependent, modified base-stock policy is optimal. We also show when and why the heuristic works well. In our numerical study, the average optimality gap was 4.34%. We also establish some monotonicity results for policy parameters with respect to the production environment. Using these results and our numerical observations, we investigate the joint effect of (i) the two-stage production process, (ii) the production flexibility, and (iii) the fluctuating demand environment on the system's performance. For example, we quantify the value of flexible production as well as the effect of smoothing constraints on the benefits of postponement. We show that a redesign of the production process to allow for delayed product differentiation is more effective and valuable when it is accompanied by an investment in production flexibility.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 2 , April 2012 , pp. 263 - 293
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Atalı, A. & Özer, Ö. (2011). An online addendum to: “Stochastic multi-item inventory systems with Markov-modulated demands and production quantity requirements.” Available from the authors.Google Scholar
2.Aviv, Y. & Federgruen, A. (1997). Stochastic inventory models with limited production capacity and periodically varying parameters. Probability in the Engineering and Informational Sciences 11: 107–135.CrossRefGoogle Scholar
3.Aviv, Y. & Federgruen, A. (2001). Capacitated multi-item inventory systems with random and seasonally fluctuating demands: Implication for postponement strategies. Management Science 47: 512–531.CrossRefGoogle Scholar
4.Aviv, Y. & Federgruen, A. (2001). Design for postponement: A comprehensive characterization of its benefits under unknown demand distributions. Operations Research 49: 578–598.CrossRefGoogle Scholar
5.Beyer, D. & Sethi, S.P. (1997). Average cost optimality in inventory models with markovian demands. J. Optimization Theory and Application 92: 497–526.CrossRefGoogle Scholar
6.Beyer, D., Cheng, F., Sethi, S.P., & Taksar, M.I. (2010). Markovian demand inventory models. International series in operations research and management science. New York, NY: Springer.Google Scholar
7.Chan, E.W. & Muckstadt, J.A. (1999). The Effects of load smoothing on inventory levels in a capacitated production and inventory system. School of Operations Research and Industrial Engineering, Technical Report No. 1251, Cornell University, Ithaca, NY.Google Scholar
8.Chen, F. & Song, J.S. (2001). Optimal policies for multiechelon inventory problems with markov-modulated demand. Operations Research 49: 226–234.CrossRefGoogle Scholar
9.Ciarallo, F.W., Akella, R. & Morton, T.E. (1994). A periodic review, production planning model with uncertain capacity and uncertain demand-optimality of extended myopic policies. Management Science 40: 320–332.CrossRefGoogle Scholar
10.Eppen, G. & Schrage, L. (1981). Centralized ordering policies in a multiwarehouse system with leadtimes and random demands. In Multi-level production/inventory control systems: theory and practice. Schwarz, L. (ed.), TIMS Studies in Management Sciences, North Holland, Amsterdam, The Netherlands, pp. 51–69.Google Scholar
11.Evans, R. (1967). Inventory control of a multiproduct system with a limited production resource. Naval Research Logistics Quarterly 14: 173–184.CrossRefGoogle Scholar
12.Federgruen, A. & Zipkin, P. (1984). Approximations of dynamic, multi location production and inventory problems. Management Science 30: 69–84.CrossRefGoogle Scholar
13.Federgruen, A. & Zipkin, P. (1984). Allocation policies and cost approximations for multi-location inventory systems. Naval Research Logistics Quarterly 31: 97–129.CrossRefGoogle Scholar
14.Federgruen, A. & Zipkin, P. (1986). An inventory model with limited production capacity and uncertain demands II. The discounted-cost criterion. Mathematics of Operations Research 11: 208–215.CrossRefGoogle Scholar
15.Gaddum, J.W., Hoffman, A.J. & Sokolowsky, D. (1954). On the solution of the caterer problem. Naval Research Logistics Quarterly 1: 223–229.CrossRefGoogle Scholar
16.Gallego, G., Özer, Ö. & Zipkin, P. (2007). Bounds, heuristics and approximations for distribution systems. Operations Research 55: 503–517.CrossRefGoogle Scholar
17.Glasserman, P. & Tayur, S. (1994). The stability of a capacitated, multi-echelon production-inventory system under a base-stock policy. Operations Research 42: 913–925.CrossRefGoogle Scholar
18.Iglehart, D. & Karlin, S. (1962). Optimal policy for dynamic inventory process with nonstationary stochastic demands. In Studies in applied probability and management science, Arrow, K., Karlin, S., & Scarf, H. (eds.) Chapter 8. Stanford, CA: Stanford University Press.Google Scholar
19.Janakiraman, G. & Muckstadt, J.A. (2004). Inventory control in directed networks: A note on linear costs. Operations Research 52: 491–495.CrossRefGoogle Scholar
20.Janakiraman, G. & Muckstadt, J.A. (2009). A decomposition approach for a class of capacitated serial systems. Operations Research 57: 1384–1393.CrossRefGoogle Scholar
21.Kapuscinski, R. & Tayur, S. (1998). A Capacitated production-inventory model with periodic demand. Operations Research 46: 899–911.CrossRefGoogle Scholar
22.Karlin, S. (1960). Dynamic inventory policy with varying stochastic demands. Management Science 6: 231–258.CrossRefGoogle Scholar
23.Lee, H.L., Billington, C. & Carter, B. (1993). Hewlett-Packard gains control of inventory and service through design for localization. Interfaces 23: 1–11.CrossRefGoogle Scholar
24.Özer, Ö. 2003. Replenishment strategies for distribution systems under advance demand information. Management Science 49: 255–272.CrossRefGoogle Scholar
25.Özer, Ö. & Wei, W. (2004). Inventory control with limited capacity and advance demand information. Operations Research 52: 988–1000.CrossRefGoogle Scholar
26.Parker, R.P. & Kapuscinski, R. (2004). Optimal policies for a capacitated two-echelon inventory system. Operations Research 52: 739–755.CrossRefGoogle Scholar
27.Rappold, J.A. & Muckstadt, J.A. (2000). A computationally efficient approach for determining inventory levels in a capacitated multiechelon production-distribution system. Naval Research Logistics 47: 377–398.3.0.CO;2-K>CrossRefGoogle Scholar
28.Rappold, J.A. & Yoho, K.D. (2008). A model for level-loading production in the process industries when demand is stochastic. Production Planning and Control 19: 686–701.CrossRefGoogle Scholar
29.Sethi, S.P. & Cheng, F. (1997). Optimality of (s,S) policies in inventory models with markovian demand. Operations Research 45: 931–939.CrossRefGoogle Scholar
30.Simchi-Levi, D., Kaminsky, P. & Simchi-Levi, E. (2000). Designing and managing the supply chain. New York: McGraw-Hill.Google Scholar
31.Song, J. & Zipkin, P. (1993). Inventory control in fluctuating demand environment. Operations Research 41: 351–370.CrossRefGoogle Scholar
32.Swaminathan, J. & Tayur, S. (1998). Managing broader product lines through delayed differentiation using vanilla boxes. Management Science 44: S161–S172.CrossRefGoogle Scholar
33.Topkis, D.M. (1978). Minimizing submodular function on a lattice. Operations Research 26: 305–321.CrossRefGoogle Scholar
34.Topkis, D.M. (1998). Supermodularity and complementarity. Princeton, NJ: Princeton University Press.Google Scholar
35.Veinott, A. (1965). Optimal policy for a multi-product, dynamic, nonstationary inventory problem. Management Science 12: 206–222.CrossRefGoogle Scholar
36.Veinott, A. (1998). Lecture notes on supply chain optimization. CA: Stanford University.Google Scholar
37.Zipkin, P. (1989). Critical number policies for inventory models with periodic data. Management Science 35: 71–80.CrossRefGoogle Scholar