Dynamic-Programming Approximations for Stochastic Time-Staged Integer Multicommodity-Flow Problems

Abstract

In this paper, we consider a stochastic and time dependent version of the min-cost integer multicommodity flow problem that arises in the dynamic resource allocation context. In this problem class, tasks arriving over time have to be covered by a set of indivisible and reusable resources of different types. The assignment of a resource to a task removes the task from the system, modifies the resource, and generates a profit. When serving a task, resources of different types can serve as substitutes of each other, possibly yielding different revenues. We propose an iterative, adaptive dynamic programming-based methodology that makes use of linear and/or nonlinear approximations of the value function. Our numerical work shows that the proposed method provides high quality solutions and is computationally attractive for large problems.

Sensitivity Analysis of a Dynamic Fleet Management Model Using Approximate Dynamic Programming

“…For notational brevity, we assume that it takes one time period to move between any pair of locations. It is straightforward to extend our analysis to the case where there are multiperiod travel times by using the approach described in Topaloglu and Powell (2006). We define the following:…”

“…The question of whether these suboptimal policies yield high-quality solutions is outside the scope of this paper. We refer the reader to Godfrey and Powell (2002a, b) and Topaloglu and Powell (2006), where the experimental work indicates that this class of policies beat standard benchmarks by significant margins. Here, we assume that we already have a "good" policy , and we are interested in computing the change in F 1 r 1 d 1 d T induced by changing an element of the state vector r 1 or the load availability vector d 1 .…”

“…Therefore, most of the stochastic fleet management models revolve around the idea of approximating the value function in a tractable manner. For stochastic fleet management models other than the one in Godfrey and Powell (2002a, b), we refer the reader to Frantzeskakis and Powell (1990), Crainic et al (1993), Carvalho and Powell (2000), and Topaloglu and Powell (2006).…”

“…Therefore, most of the stochastic fleet management models revolve around the idea of approximating the value function in a tractable manner. For stochastic fleet management models other than the one in Godfrey and Powell (2002a, b), we refer the reader to Frantzeskakis and Powell (1990), Crainic et al (1993), Carvalho and Powell (2000), and Topaloglu and Powell (2006).Given a set of value function approximations, these models behave just like simulation models, generating different trajectories of vehicle allocation decisions for different trajectories of load realizations. Therefore, a key question for their sensitivity analysis is to be able to assess how the decision trajectories change when certain model parameters are perturbed.…”

Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

“…Thus, they developed an approximation scheme that is fully polynomial in T and the error term for the problem. Other dynamic programming and approximate dynamic programming approaches for related problems are explored in Powell (2002, 2003) and Topaloglu and Powell (2006) (among others).…”