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Quantifying the bullwhip effect in a supply chain with stochastic lead time. (English) Zbl 1125.90363
Summary: In a recent paper, J. Dejonckheere, S. M. Disney, M. R. Lambrecht and D. R. Towill, Eur. J. Oper. Res. 147, No. 3, 567–590 (2003; Zbl 1026.90030)] used control systems engineering (transfer functions, frequency response, spectral analysis) to quantify the bullwhip effect. In the present paper, we, like F. Chen, J. K. Ryan, Z. Drezner and D. Simchi-Levi [Manage. Sci. 46, No. 3, 436–443 (2000)], use the statistical method. But our method extends Dejonckheere et al. and Chen et al. in that we include stochastic lead time and provide expressions for quantifying the bullwhip effect, both with information sharing and without information sharing. We use iid demands in a $$k$$-stage supply chain for both. By contrast, Chen et al. provide lower bounds using autoregressive demand for information sharing and for information not sharing (with zero safety factor for stocks). Dejonckheere et al. validate Chen et al.’s results for a 2-stage supply chain without information sharing, using both autoregressive and iid normally distributed demands. We estimate the mean and variance of lead-time demand (LTD) from historical LTD data, rather than from the component period demands and lead time. Nevertheless, we also calculate the variance amplification like Chen et al., but with gamma lead times. With constant lead times, which Chen et al. used, our method yields lower variance amplification. As for the effect of information, we find that the variance increases nearly linearly in echelon stage with information sharing but exponentially in echelon stage without information sharing.

##### MSC:
 90B50 Management decision making, including multiple objectives 90B22 Queues and service in operations research
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##### References:
 [1] Bagchi, U.; Hayya, J.C.; Ord, J.K., Modeling demand during lead time, Decision sciences, 15, 2, 157-176, (1984) [2] Bagchi, U.; Hayya, J.C.; Chu, C.-H., The effect of lead-time variability: the case of independent demand, Journal of operations management, 6, 2, 159-177, (1986) [3] Cachon, G.P.; Fisher, M., Supply chain inventory management and the value of shared information, Management science, 46, 8, 1032-1048, (2000) · Zbl 1232.90028 [4] Chatfield, D.; Kim, J.; Harrison, T.; Hayya, J., The bullwhip effect in supply chains—impact of stochastic lead times, information quality, and information sharing: A simulation study, Production and operations management, 13, 4, 340-353, (2004) [5] Chen, F.; Drezner, Z.; Ryan, J.; Simchi-Levi, D., Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times, and information, Management science, 46, 3, 436-443, (2000) · Zbl 1231.90019 [6] Dejonckheere, J.; Disney, S.M.; Lambrecht, M.R.; Towill, D.R., Measuring and avoiding the bullwhip effect: A control theoretic approach, European journal of operational research, 147, 3, 567-590, (2003) · Zbl 1026.90030 [7] Dejonckheere, J.; Disney, S.M.; Lambrecht, M.R.; Towill, D.R., The impact of information enrichment on the bullwhip effect in supply chains: A control theoretic approach, European journal of operational research, 153, 3, 727-750, (2004) · Zbl 1099.90503 [8] Güllü, A.R., A two-echelon allocation model and the value of information under correlated forecasts and demands, European journal of operational research, 99, 2, 386-400, (1997) · Zbl 0930.90004 [9] Kendall, M., Stuart, A., 1977. The Advanced Theory of Statistics, vol. 1. MacMillan Publishing Co., Inc., London. · Zbl 0353.62013 [10] Lee, H.; So, K.C.; Tang, C.S., The value of information sharing in a two level supply chain, Management science, 46, 5, 628-643, (2000) · Zbl 1231.90044 [11] Mason-Jones, R.; Towill, D.R., Information enrichment: designing the supply chain for competitive advantage, Supply chain management, 2, 4, 137-149, (1997) [12] Merkuryev, Y., Pethova, J., Buikis, M., 2003. Simulation-based analysis of the bullwhip effect in supply chains. Paper presented at the EURO/INFORMS meeting, Istanbul Turkey, July 6-10, 2003. [13] Mitra, S.; Chatterjee, A.K., Leveraging information in multi-echelon inventory systems, European journal of operational research, 152, 1, 263-280, (2004) · Zbl 1045.90002 [14] Mood, A.M.; Graybill, F.A.; Boes, D.C., Introduction to the theory of statistics, (1974), McGraw-Hill New York · Zbl 0277.62002 [15] Ord, J.K., Families of frequency distributions, (1972), Griffin London · Zbl 0249.62005
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