Impact of Dampening Demand Variability in a Production/Inventory System with Multiple Retailers

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 27)

Abstract

We study a supply chain consisting of a single manufacturer and two retailers. The manufacturer produces goods on a make-to-order basis, while both retailers maintain an inventory and use a periodic replenishment rule. As opposed to the traditional (r, S) policy, where a retailer at the end of each period orders the demand seen during the previous period, we assume that the retailers dampen their demand variability by smoothing the order size. More specifically, the order placed at the end of a period is equal to β times the demand seen during the last period plus (1 − β) times the previous order size, with β ∈ (0, 1] the smoothing parameter. We develop a GI/M/1-type Markov chain with only two nonzero blocks A0 and Ad to analyze this supply chain. The dimension of these blocks prohibits us from computing its rate matrix R in order to obtain the steady state probabilities. Instead we rely on fast numerical methods that exploit the structure of the matrices A0 and Ad, i.e., the power method, the Gauss–Seidel iteration, and GMRES, to approximate the steady state probabilities. Finally, we provide various numerical examples that indicate that the smoothing parameters can be set in such a manner that all the involved parties benefit from smoothing. We consider both homogeneous and heterogeneous settings for the smoothing parameters.

Key words

Structured Markov chains Supply chain Inventory MSC: Primary 60J22 Secondary 90B30 90B05 

References

  1. 1.
    Bini, D.A., Meini, B., Steffé, S., Van Houdt, B.: Structured Markov chains solver: algorithms. In: SMCtools Workshop. ACM Press, Pisa (2006)Google Scholar
  2. 2.
    Bobbio, A., Horváth, A., Telek, M.: The scale factor: a new degree of freedom in phase type approximation. Perform. Eval. 56, 121–144 (2004)CrossRefGoogle Scholar
  3. 3.
    Boute, R., Lambrecht, M., Van Houdt, B.: Performance evaluation of a production/inventory system with periodic review and endogeneous lead times. Nav. Res. Logist. 54, 462–473 (2007)MATHCrossRefGoogle Scholar
  4. 4.
    Boute, R.N., Disney, S.M., Lambrecht, M.R., Van Houdt, B.: An integrated production and inventory model to dampen upstream demand variability in the supply chain. Eur. J. Oper. Res. 178, 121–142 (2007)MATHCrossRefGoogle Scholar
  5. 5.
    Fernandes, P., Plateau, B., Stewart, W.: Efficient descriptor-vector multiplications in stochastic automata networks. J. ACM 45, 381–414 (1998)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Golub, G.H., Van Loan, C.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  7. 7.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability. SIAM, Philadelphia (1999)MATHCrossRefGoogle Scholar
  8. 8.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models. John Hopkins University Press, Baltimore (1981)MATHGoogle Scholar
  9. 9.
    Philippe, B., Saad, Y., Stewart, W.J.: Numerical methods in Markov chain modeling. Oper. Res. 40, 1156–1179 (1992)MATHCrossRefGoogle Scholar
  10. 10.
    Saad, Y., Schultz, M.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Stewart, W.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)MATHGoogle Scholar
  12. 12.
    Vázquez Cortizo, D., García, J., Blondia, C.: FS-ALOHA\(++\), a collision resolution algorithm with QoS support for the contention channel in multiservice wireless LANs. In: Proceedings of IEEE Globecom (1999)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Antwerp – IBBTAntwerpBelgium
  2. 2.Department of Electrical and Electronics EngineeringUniversidad de los AndesBogotáColombia

Personalised recommendations