Approximate Approach to Calculating the Order Fill Rate under Purchase Dependence

- Journal title : Journal of the Korean Operations Research and Management Science Society
- Volume 41, Issue 2, 2016, pp.35-51
- Publisher : The Korean Operations and Management Science Society
- DOI : 10.7737/JKORMS.2016.41.2.035

Title & Authors

Approximate Approach to Calculating the Order Fill Rate under Purchase Dependence

Park, Changkyu; Seo, Junyong;

Park, Changkyu; Seo, Junyong;

Abstract

This paper proposes a new approximate approach to calculate the order fill rate and the probability of filling an entire customer order immediately from the shelf in a business environment under purchase dependence characterized by customer purchase patterns observed in such areas as marketing, manufacturing systems, and distribution systems. The new approximate approach divides customer orders into item orders and calculates fill rates of all order types to approximate the order fill rate. We develop a greed iterative search algorithm (GISA) based on the Gauss-Seidel method to avoid dimensionality and prevent the solution divergence for larger instances. Through the computational analysis that compares the GISA with the simulation, we demonstrate that the GISA is a dependable algorithm for deriving the stationary joint distribution of on-hand inventories in the type-K pure system. We also present some managerial insights.

Keywords

Order Fill Rate;Purchase Dependence;Iterative Approach;Quasi-Birth-and-Death Process;

Language

Korean

References

1.

박창규, 서준용, "구매종속성이 재고관리에 미치는 영향", 경영과학, 제30권, 제3호(2013), pp.17-31.

2.

박창규, "구매종속성을 고려한 부분 부재고 재고모형", 한국경영과학회지, 제39권, 제1호(2014), pp.83-100.

3.

박창규, "구매종속성이 존재하는 상황에서 부분 부재고 EOQ 모형에 대한 고찰", 경영과학, 제32권, 제1호(2015), pp.65-82.

4.

Athappilly, K., M.A. Razi, and J.M. Tarn, "A multi-technique data mining approach to exploring consumer behaviors," Human Systems Management, Vol.29, No.3(2010), pp.153-163.

5.

Benjaafar, S. and M. EI Hafsi, "Production and inventory control of a single product assemble-to-order system with multiple customer classes," Management Science, Vol.52, No.12(2006), pp.1896-1912.

6.

Dayanik, S., J.S. Song, and S.H. Xu, "The effectiveness of several performance bounds for capacitated production, partial-order-service, assemble-to-order systems," Manufacturing and Service Operations Management, Vol.5, No.3(2003), pp.230-251.

7.

Gao, C., H. Shen, and T.C.E. Cheng, "Orderfulfillment performance analysis of an assemble-to-order system with unreliable machines," International Journal of Production Economics, Vol.126, No.2(2010), pp.341-349.

8.

Gunawardena, A.D., S.K. Jain, and L. Snyder, "Modified iterative methods for consistent linear systems," Linear Algebra and its Applications, Vol.154-156(1991), pp.123-143.

9.

Hadjidimos, A., D. Noutsos, and M. Tzoumas, "More on modifications and improvements of classical iterative schemes for M-matrices," Linear Algebra and its Applications, Vol.364 (2003), pp.253-279.

10.

Hoen, K.M.R., R. Gullu, G.J. van Houtum, and I.M.H. Vliegen, "A simple and accurate approximation for the order fill rates in lostsales assemble-to-order systems," International Journal of Production Economics, Vol. 133, No.1(2011), pp.95-104.

11.

Iravani, S., L. Luangkesorn, and D. Simchi-Levi, "On assemble-to-order systems with flexible customers," IIE Transactions, Vol.35, No.5(2003), pp.389-403.

12.

Iravani, S., L. Luangkesorn, and D. Simchi-Levi, "A general decomposition algorithm for parallel queues with correlated arrivals," Queueing Systems, Vol.47, No.4(2004), pp.313-344.

13.

Kohno, T., H. Kotakemori, and H. Niki, "Improving the modified Gauss-Seidel method for Z-matrices," Linear Algebra and its Application, Vol.267(1997), pp.113-123.

14.

Kotakemori, H., K. Harada, M. Morimoto, and H. Niki, "A comparison theorem for the iterative method with the preconditioner(I+Smax)," Journal of Computational and Applied Mathematics, Vol.145, No.2(2002), pp.373-378.

15.

Kotakemori, H., H. Niki, and N. Okamoto, "Accelerated iterative method for Z-matrices," Journal of Computational and Applied Mathematics, Vol.75, No.1(1996), pp.87-97.

16.

Lu, Y., J.S. Song, and D.D. Yao, "Order fill rate, lead time variability, and advance demand information in an assemble-to-order system," Operations Research, Vol.51, No.2 (2003), pp.292-308.

17.

Milaszewicz, J.P., "Improving Jacobi and Gauss-Seidel iterations," Linear Algebra and its Applications, Vol.93(1987), pp.161-170.

18.

Morimoto, M., K. Harada, M. Sakakihara, and H. Sawami, "The Gauss-Seidel iterative method with the preconditioning matrix(I+S +Sm)," Japan Journal of Industrial and Applied Mathematics, Vol.21, No.1(2004), pp.25-34.

19.

Niki, H., T. Kohno, and M. Morimoto, "The preconditioned Gauss-Seidel method faster than the SOR method," Journal of Computational and Applied Mathematics, Vol.219, No.1(2008), pp.59-71.

20.

Noutsos, D. and M. Tzoumas, "On optimal improvements of classical iterative schemes for Z-matrices," Journal of Computational and Applied Mathematics, Vol.188, No.1(2006), pp.89-106.

21.

Park, C. and J. Seo, "Consideration of purchase dependence in inventory management," Computers and Industrial Engineering, Vol.66, No.2(2013), pp.274-285.

22.

Song, J.S., "On the order fill rate in a multiitem, base-stock inventory system," Operations Research, Vol.46, No.6(1998), pp.831-845.

23.

Song, J.S., "Order-based backorders and their implications in multi-item inventory systems," Management Science, Vol.48, No.4(2002), pp. 499-516.

24.

Song, J.S., S. Xu, and B. Liu, "Order-fulfillment performance measures in an assembleto-order system with stochastic lead times," Operations Research, Vol.47, No.1(1999), pp. 131-149.

25.

Song, J.S. and D.D. Yao, "Performance analysis and optimization of assemble-to-order systems with random lead times," Operations Research, Vol.50, No.5(2002), pp.889-903.

26.

Song, J.S. and P. Zipkin, "Supply chain operations: assemble-to-order system," In:de Kok, T. and S. Graves, (Eds.) Handbooks in Operations Research and Management Science, North-Holland, Amsterdam, (2003), pp. 561-596.

27.

Usui, M., H, Niki, and T. Kohno, "Adaptive Gauss-Seidel method for linear systems," International Journal of Computer Mathematics, Vol.51, No.1/2(1994), pp.119-125.

28.

Vliegen, I.M.H. and G.J. van Houtum, "Approximate evaluation of order fill rates for an inventory system of service tools," International Journal of Production Economics, Vol. 118, No.1(2009), pp.339-351.

29.

Yang, Y., H. Liu, and Y. Cai, "Discovery of online shopping patterns across websites," INFORMS Journal on Computing, Vol.25, No.1(2013), pp.161-176.

30.

Ye, J. and S. Li, "Folding algorithm:a computational method for finite QBD processes with level-dependent transitions," IEEE Transactions on Communications, Vol.42, No.234 (1994), pp.625-639.