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On the stability of supply chains. (English) Zbl 1165.90346
Summary: This paper examines the stability of decentralized, multistage supply chains under arbitrary demand conditions. It looks for intrinsic properties of the inventory replenishment policies that hold for all customer demand processes and for policies with desirable properties. It is found that the overall conditions experienced by suppliers several stages removed from the final customer, e.g., the variances of the orders they receive and the inventories they keep, depend on the policy much more than on the demand process. A policy-specific but demand-independent upper bound for the order variance amplification factor of any decentralized policy is shown to exist, and its formula is presented. The bound is always tight for the suppliers at the end of a long chain so that a policy exhibits the “bullwhip effect” if and only if its bound is greater than $1$. A simple necessary condition for bullwhip avoidance is also identified in terms of a policy’s “gain.” Gain is the marginal change in average inventory induced by a policy when there is a small but sustained change in the demand rate. It is shown that all policies with positive gain produce the bullwhip effect if they do not use future order commitments. Because manufacturers can reduce costs by operating with positive gain, this explains the prevalence of the bullwhip effect. A family of commitment-based policies that can dynamically maintain any desired inventory level for any demand rate (e.g., achieve positive gain) without the bullwhip effect is also presented. The family includes just-in-time strategies as a special case. Simulation results are used as an illustration.

##### MSC:
 90B06 Transportation, logistics 90B30 Production models 91B42 Consumer behavior, demand theory
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