Production, Manufacturing and Logistics
Fluid dynamics models and their applications in transportation and pricing

https://doi.org/10.1016/j.ejor.2004.07.047Get rights and content

Abstract

Fluid dynamics models provide a powerful deterministic technique to approximate stochasticity in a variety of application areas. In this paper, we study two classes of fluid models, investigate their relationship as well as some of their applications. This analysis allows us to provide analytical models of travel times as they arise in dynamically evolving environments, such as transportation networks as well as supply chains. In particular, using the laws of hydrodynamic theory, we first propose and examine a general second-order fluid model. We consider a first-order approximation of this model and show how it is helpful in analyzing the dynamic traffic equilibrium problem. Furthermore, we present an alternate class of fluid models that are traditionally used in the context of dynamic traffic assignment. By interpreting travel times as price/inventory–sojourn-time relationships, we are also able to connect this approach with a tractable fluid model in the context of dynamic pricing and inventory management.

Introduction

In this paper, we study a variety of fluid dynamics models as they arise in diverse application domains such as transportation and dynamic pricing. This analysis allows us to investigate and provide insights to dynamic phenomena that arise in a variety of systems that share similar characteristics. Such systems include transportation networks as well as supply chain and inventory management systems. An underlying common characteristic in these systems is some form of travel time. These systems are dynamic in nature. In particular, due to their service time and the inherent disequilibrium between demand and supply, these systems give rise to dynamic delays. For example, in ground transportation, poor-quality roads and congested traffic conditions cause travellers to experience delays in traversing a network’s path. In inventory management systems, a high unit price and a high level of inventory of a good may cause a newly produced unit of that good to incur a delay before it is sold.

As a result, it is important for traffic planners to understand and manage the nature of travellers’ delays (costs) in urban and highway transportation systems, and for the supply chain industry to design optimal pricing and inventory management strategies that maximize profits, reduce inventory levels, and effectively manage the delays that goods incur before being sold.

Therefore, understanding the nature of the dynamic phenomena arising in these systems, exploring their common characteristics, and designing mechanisms to manage them effectively, have potential for tremendous economic, social and political impacts. In this paper, we will explore the relationship between these systems.

The literature in transportation takes two approaches when modeling travel times. The first and more traditional approach assumes a predetermined functional form that describes the relationship between travel times and flow rates. This is typically determined through a statistical analysis. Practitioners in the transportation community have been using several functional forms to describe travel times. These include the BPR function [39] which is used to estimate travel times at priority intersections, and is a polynomial function. Meneguzzer et al. [35] proposed an exponential travel time function for all-way-stop intersections. Akcelik [1], [2] also proposed a polynomial-type travel time function for links at signalized intersections. Nevertheless, considering a specific link travel time function in advance has some drawbacks. These travel time functions may not describe accurately peak period traffic dynamics especially since there are dramatic changes in traffic conditions in a short period of time. Travel times in a dynamic transportation network depend both on prevailing traffic conditions and future traffic conditions relative to the departure time. As a result, this approach can lead to controversial results (see [13]). A second approach for modeling travel times considers travel time functions as an output rather than an input in the model. One can achieve this by determining the functional forms for travel times through an analytical method. Perakis [45] and Kachani and Perakis [23] provide a first order model through the hydrodynamic model of Lighthill and Witham [28]. Using the model by Newell [38], Kuwahara and Akamatsu [27] also discuss an analytical instantaneous travel time function. Nevertheless, the travel time they derive does not represent the actual (experienced) travel time, unless traffic conditions remain constant over time. This latter stream of research does not assume in advance specific link travel time functions. It is attractive due to the fact that it takes into account the traffic flow dynamics but also provides analytical solutions. Our goal in this paper is to follow this approach by proposing a second-order model that incorporates additional phenomena such as effects from neighboring links. Furthermore, we wish to examine its relation to alternate fluid formulations in the literature. Finally, our goal is to illustrate how fluid models are useful in a variety of application areas that we discuss in this paper. Examples include transportation as well as dynamic pricing and supply chains. Friesz et al. [5], [15], [16], [17], Ran and Boyce [48], Ran et al. [49], Wu et al. [54], and Xu et al. [55] have developed and studied fluid models for the Dynamic User-Equilibrium problem that rely on the use of an analytical travel time function, given to the model as an input.

The key to the approach we take in this paper, is the use of fluid models in two main application areas. A further motivation for using fluid models is that they provide good policies in a variety of settings. Furthermore, these models have been shown to approximate well the underlying stochasticity of problems in a deterministic way (see [3], [6], [21], [36]).

The contributions in this paper are the following:

  • We develop a general second-order model for travel times in a dynamic transportation network.

  • Using this model, we derive closed-form solutions for travel times. These travel times correspond to the ones used in practice. We illustrate and connect these travel time functions through some numerical examples.

  • We consider some applications of the fluid models in this paper. In particular, we connect our model with the dynamic user-equilibrium problem as well consider its application to dynamic pricing and inventory management of non-perishable products.

The paper by McGill and van Ryzin [34], Bitran and Caldentey [7] and the references therein, provide a thorough review of revenue management. Elmaghraby and Keskinocak [14] review the literature and current practices in dynamic pricing while Yano and Gilbert [56] review models for joint pricing and production under a monopolistic setup.

The structure of this paper is as follows. In Section 2, we consider a general second-order fluid dynamics model for determining travel times. We study this model and some of its simplifications that allow us to propose closed form solutions for travel times. Furthermore, to illustrate these results, we consider some numerical examples. In Section 3, we illustrate how our results can be useful in the context of dynamic user-equilibrium. In Section 4, we review an alternate fluid dynamics model for studying the dynamic user-equilibrium model (the Dynamic Network Loading Model). In Section 5, we also demonstrate how we can use our results in the context of dynamic pricing and inventory management. Finally, in Section 6, we discuss our conclusions.

Section snippets

A second-order fluid model

In this section we present a second-order fluid model for determining travel times in dynamic transportation networks. In Section 2.1, we present the necessary notation, while in Section 2.2, we present the general model. In Section 2.3, we consider the case of separable velocity functions. These give rise to two simplified second-order models. In Section 2.4, using piecewise linear and piecewise quadratic approximations of the flow rate, we derive closed form solutions for travel times.

Application to the dynamic user-equilibrium problem

In this section we illustrate how the second-order fluid model we examined above for determining travel times, connects with the dynamic user-equilibrium problem. In particular, we propose a model for determining dynamic user-equilibrium flow patterns that does not consider travel times functions as an input to the model but rather incorporates the second-order travel time model as part of the feasibility constraints. The model we propose will assume that drivers make their route choices under

An alternate fluid model for the dynamic user-equilibrium problem

In this section we review a class of fluid models from the literature that provide an alternate way to describe the evolution of traffic in a dynamic transportation network. Furthermore, this model can be combined with the dynamic user-equilibrium conditions. This class of models is also referred to as the Dynamic Network Loading Model.

In particular, in Section 4.1 we review the essential notation we will need to formulate the problem. In Section 4.2, we review the Dynamic Network Loading

An application to dynamic pricing and inventory control

In what follows, we briefly review the continuous-time model for the dynamic pricing problem using a delay approach as introduced in Kachani and Perakis [24]. We consider a make-to-stock production environment of a single manufacturer producing several products with dynamic shared production capacity. This model aims to address how to simultaneously determine optimal dynamic production, inventory and pricing policies.

Similarly to the previous section, we build a model by taking a fluid dynamics

Conclusions

In this paper, we examined two classes of fluid models. The first class was based on the laws of hydrodynamic theory while the second was based on the Dynamic Network Loading Model. The first class of models allowed us to propose several families of travel time functions for dynamic transportation networks. These travel times correspond to what practitioners are using. Furthermore, through some numerical examples we established some insights on these functions.

The second fluid modeling approach

Acknowledgements

Preparation of this paper was supported, in part, by the MIT-Singapore Alliance, the PECASE Award DMI-9984339 from the National Science Foundation, the Charles Reed Faculty Initiative Fund and the New England University Transportation Research Award. We are grateful to the referees for their insightful comments and suggestions that have helped us improve both the exposition and the content of this paper.

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