Supply Streams

Abstract

A supply stream is a continuous version of a supply chain. It is like a series inventory system, but stock can be held at any point along a continuum, not just at discrete stages. We assume stationary parameters and aim to minimize the long-run average total cost. We show that a stationary continuous-stage echelon base-stock policy is optimal. That is, at each geographic point along the supply stream, there is a target echelon inventory level, and the optimal policy at all times is to order and dispatch material so as to move the echelon inventory position as close as possible to this target. We establish this result by showing that the solutions to certain discrete-stage systems converge monotonically to a limit, as the distances between the stages become small, and this limit solves the continuous-stage system. With demand approximated by a Brownian motion, we show that, in the continuous-stage limit, the supply stream model is equivalent to one describing first-passage times. This linkage leads to some interesting and useful results. Specifically, we obtain a partial differential equation that characterizes the optimal cost function, and we find a closed-form expression for the optimal echelon basestock levels in a certain special case, the first in the inventory literature. These expressions demonstrate that the well-known square-root law for safety stock does not apply in this context.

An approximation for the inverse first passage time problem

“…(This was shown independently by Cheng et al [2] and Song and Zipkin [12].) Since (1) implies that¯ (ẑ) < F (t), we haveb(t) > b − (t).…”

“…Thus, solving (1) can be interpreted as solving a version of the newsvendor problem. This interpretation is meaningful in the context of the inventory model of [12].…”

“…The IFPT problem has applications in biology, finance, and other areas. Recently, Song and Zipkin [12] formulated an inventory-control problem which, in certain cases, turns out to be equivalent to the IFPT problem. See the other references below for further discussion of applications and pointers to the literature on them.…”

“…Zucca and Sacerdote [14] described two solution methods, one employing a piecewiselinear approximation of the boundary, and the other based on discretizing a Volterra integral equation. Song and Zipkin [12] presented a different algorithmic approach. Other work on this and related problems include [1], [7], and [10].…”

Dynamic Speed Optimization in Supply Chains with Stochastic Demand

“…The three papers above focus on multi-echelon systems. Song and Zipkin (2013) use the traditional approach of cost accounting period by period and obtain a partial differential equation to characterize the optimal policy. In contrast, Axsäter and Lundell (1984) and Berling and Martínez-de-Albéniz (2011a) use the unit-by-unit decomposition approach, similar to this paper.…”