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.
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