Heavy Traffic Analysis of the Dynamic Stochastic Inventory Routing Problem

Cover Heavy Traffic Analysis of the Dynamic Stochastic Inventory Routing Problem
Heavy Traffic Analysis of the Dynamic Stochastic Inventory Routing Problem
Martin I Reiman
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Nonetheless, we will maintain equation (61) as is, and view the time-dependent coefficient as a refinement of the heavy traffic Hmit.
Let 3?(i) G {TSP, DS} denote the routing mode that is used at time t in the limiting control problem. We want to consider intervals of time during which one of the two controls is used exclusively, and approximate the resulting normalized netput process by a diffusion with control-dependent drift and variance. In order to do this we need to reconcile the differen
...t definitions of the netput process that are approximated in the two cases: the embedded netput process in the TSP case, and the averaged netput process in the DS case. The lack of a cyclic structure in the DS case forced us to use averaging there, while the choice of embedding in the TSP case was made for convenience. Thus we translate the embedded netput process from the TSP case, x{t)-> to an averaged netput process x[t). This translation consists of adding a constant, ?7, which needs to be determined: \(x(l + e^T{'2-zi)gUD(z3-:2) - e'^r{z2-2i)) 4- j)£, (e'^r(z2-zi) _ 1) j>^e''7-(22-Zl)gi>D(23-22) and A;.

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