Approximate Calculation of Order Fill Rate under Purchase Dependence
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
For the competitive business environment under purchase dependence, this paper proposes a new approximate calculation of order fill rate which is a probability of satisfying a customer order immediately using the existing inventory. Purchase dependence is different to demand dependence. Purchase dependence treats the purchase behavior of customers, while demand dependence considers demand correlation between items, between regions, or over time. Purchase dependence can be observed in such areas as marketing, manufacturing systems, and distribution systems. Traditional computational methods have a difficulty of the curse of dimensionality for the large cases, when deriving the stationary joint distribution which is utilized to calculate the order fill rate. In order to escape the curse of dimensionality and protect the solution from diverging for the large cases, we develop a greedy iterative search algorithm based on the Gauss-Seidel method. We show that the greedy iterative search algorithm is a dependable algorithm to derive the stationary joint distribution of on-hand inventories in the retailer system by conducting a comparison analysis of a greedy iterative search algorithm with the simulation. In addition, we present some managerial insights such as : (1) The upper bound of order fill rate can be calculated by the one-item pure system, while the lower bound can be provided by the pure system that consists of all items; (2) As the degree of purchase dependence declines while other conditions remain same, it is observed that the difference between the lower and upper bounds reduces, the order fill rate increases, and the order fill rate gets closer to the upper bound.
Abstract
For the competitive business environment under purchase dependence, this paper proposes a new approximate calculation of order fill rate which is a probability of satisfying a customer order immediately using the existing inventory. Purchase dependence is different to demand dependence. Purchase dependence treats the purchase behavior of customers, while demand dependence considers demand correlation between items, between regions, or over time. Purchase dependence can be observed in such areas as marketing, manufacturing systems, and distribution systems. Traditional computational methods have a difficulty of the curse of dimensionality for the large cases, when deriving the stationary joint distribution which is utilized to calculate the order fill rate. In order to escape the curse of dimensionality and protect the solution from diverging for the large cases, we develop a greedy iterative search algorithm based on the Gauss-Seidel method. We show that the greedy iterative search algorithm is a dependable algorithm to derive the stationary joint distribution of on-hand inventories in the retailer system by conducting a comparison analysis of a greedy iterative search algorithm with the simulation. In addition, we present some managerial insights such as : (1) The upper bound of order fill rate can be calculated by the one-item pure system, while the lower bound can be provided by the pure system that consists of all items; (2) As the degree of purchase dependence declines while other conditions remain same, it is observed that the difference between the lower and upper bounds reduces, the order fill rate increases, and the order fill rate gets closer to the upper bound.
참고문헌 (21)
- Gao, C., "Order-fulfillment performance analysis of an assemble-to-order system with unreliable machines", International Journal of Production Economics, 126(2), pp.341~349 (2010)
- Gunawardena, A. D., "Modified iterative methods for consistent linear systems", Linear Algebra and its Applications, 154-156, pp.123~143 (1991)
- Hadjidimos, A., "More on modifications and improvements of classical iterative schemes for M-matrices", Linear Algebra and its Applications, 364, pp.253~279 (2003)
- Iravani, S., "On assemble-to-order systems with flexible customers", IIE Transactions, 35(5), pp.389~403 (2003)
- Kohno, T., "Improving the modified Gauss-Seidel method for Z-matrices", Linear Algebra and its Application, 267, pp.113~123 (1997)
- Kotakemori, H., "A comparison theorem for the iterative method with the preconditioner (I+Smax)", Journal of Computational and Applied Mathematics, 145(2), pp.373~378 (2002)
- Kotakemori, H., "Accelerated iterative method for Z-matrices", Journal of Computational and Applied Mathematics, 75(1), pp.87~97 (1996)
- Milaszewicz, J. P., "Improving Jacobi and Gauss-Seidel iterations", Linear Algebra and its Applications, 93, pp.161~170 (1987)
- Morimoto, M., "The Gauss-Seidel iterative method with the preconditioning matrix (I+S+Sm)", Japan Journal of Industrial and Applied Mathematics, 21(1), pp.25~34 (2004)
- Niethammer, W., "The successive order relaxation method (SOR) and Markov chains", Annals of Operations Research, 103(1), pp.351~358 (2001)
- Niki, H., "The preconditioned Gauss-Seidel method faster than the SOR method", Journal of Computational and Applied Mathematics, 219(1), pp.59~71 (2008)
- Noutsos, D., "On optimal improvements of classical iterative schemes for Z-matrices", Journal of Computational and Applied Mathematics, 188(1), pp.89~106 (2006)
- Park, C., "Consideration of purchase dependence in inventory management", Computers & Industrial Engineering, 66(2), pp.274~285 (2013)
- 박경종, "고객 수요가 공급 사슬의 총재고비용과 주문만족율에 미치는 영향", 산업경영시스템학회지, 32(3), pp.93~98 (2009), 한국산업경영시스템학회
- Philippe, B., "Numerical methods in Markov chain modeling", Operations Research, 40(6), pp.1156~1179 (1992)
- Song, J. S., "On the order fill rate in a multi-item, basestock inventory system", Operations Research, 46(6), pp.831~845 (1998)
- Song, J. S., "Order-fulfillment performance measures in an assemble-to-order system with stochastic lead times", Operations Research, 47(1), pp.131~149 (1999)
- Usui, M., "Adaptive Gauss-Seidel method for linear systems", International Journal of Computer Mathematics, 51(1-2), pp.119~125 (1994)
- Ye, J., "Folding algorithm : a computational method for finite QBD processes with level-dependent transitions", IEEE Transactions on Communications, 42(234), pp.625~639 (1994)
- 윤승철, "재고보유단위로 관리되는 제품의 재고관리 방법에 관한 연구", 산업경영시스템학회지, 38(1), pp.124~130 (2015), 한국산업경영시스템학회
- Zheng, B., "Two new modified Gauss-Seidel methods for linear system with M-matrices", Journal of Computational and Applied Mathematics, 233(4), pp.922~930 (2009)