Patrick Jaillet

Variational inequalities and the pricing of American options

This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called Brennan-Schwartz algorithm for the valuation of American put options.

A priori solution of a travelling salesman problem in which a random subset of the customers are visited

Consider a problem of routing through a set of n points. On any given instance of the problem, only a subset consisting of k out of n points (0 ≤ k ≤ n) has to be visited, with the number k random with known probability distribution. We wish to find a priori a tour through all n points. On any given instance, the k points present will then be visited in the same order as they appear in the a priori tour. The problem of finding such a tour of minimum length in the expected value sense is defined as a Probabilistic Traveling Salesman Problem (PTSP). What distinguishes one PTSP from another is the probability distribution (or more generally, the probability “law”) that specifies the number k and the identity of the points that need to be visited on any given instance of the problem. After motivating the problem by applications, we first derive closed form expressions for computing efficiently the expected length of any given tour under very general probabilistic assumptions. We then provide, in a unified way...

A Decomposition Approach to the Inventory Routing Problem with Satellite Facilities

This paper presents a comprehensive decomposition scheme for solving the inventory routing problem in which a central supplier must restock a subset of customers on an intermittent basis. In this setting, the customer demand is not known with certainty and routing decisions taken over the short run might conflict with the long-run goal of minimizing annual operating costs. A unique aspect of the short-run subproblem is the presence of satellite facilities where vehicles can be reloaded and customer deliveries continued until the closing time is reached. Three heuristics have been developed to solve the vehicle routing problem with satellite facilities (randomized Clarke-Wright, GRASP, modified sweep). After the daily tours are derived, a parametric analysis is conducted to investigate the tradeoff between distance and annual costs. This leads to the development of the efficient frontier from which the decision maker is free to choose the most attractive alternative. The proposed procedures are tested on data sets generated from field experience with a national liquid propane distributor.

AIRLINE NETWORK DESIGN AND HUB LOCATION PROBLEMS

Abstract Due to the popularity of hub-and-spoke networks in the airline and telecommunication industries, there has been a growing interest in hub location problems and related routing policies. In this paper, we introduce flow-based models for designing capacitated networks and routing policies. No a priori hub-and-spoke structure is assumed. The resulting networks may suggest the presence of “hubs”, if cost efficient. The network design problem is concerned with the operation of a single airline with a fixed share of the market. We present three basic integer linear programming models, each corresponding to a different service policy. Due to the difficulty of solving (even small) instances of these problems to optimality, we propose heuristic schemes based on mathematical programming. The procedure is applied and analyzed on several test problems consisting of up to 39 U.S. cities. We provide comments and partial recommendations on the use of hubs in the resulting network structures.

Delivery Cost Approximations for Inventory Routing Problems in a Rolling Horizon Framework

The inventory routing problem considered in this paper is concerned with the repeated distribution of a commodity, such as heating oil, over a long period of time to a large number of customers. The problem involves a central depot as well as various satellite facilities which the drivers can visit during their shift to refill their vehicles. The customers maintain a local inventory of the commodity. Their consumption varies daily and cannot be predicted deterministically. In case of a stockout, a direct delivery is made and a penalty cost is incurred. In this paper, we present incremental cost approximations to be used in a rolling horizon framework for the problem of minimizing the total expected annual delivery costs.

Chapter 3 Stochastic and dynamic networks and routing

Publisher Summary This chapter discusses stochastic and dynamic networks and routing. The chapter discusses priori optimization in routing, shortest paths, traveling salesman-type problems and vehicle routing. These problems arise when decisions must be made before random outcomes (typically customer demands) are known. The chapter covers dynamic models of problems arising in transportation and logistics, and includes a discussion of important modeling issues, as well as a summary of dynamic models for a number of key problem areas. Dynamic networks provide an important foundation for addressing many problems in logistics planning. Algorithms that have been specialized for dynamic networks are presented. The results for solving infinite networks, including both exact results for stationary infinite networks, and model truncation techniques are briefly discussed. The chapter presents basic results and concepts from the field of stochastic programming, oriented toward their application to network problems. This discussion provides a general framework for formulating and solving stochastic, dynamic network problems. That framework is used to present two stochastic programming models.

Dynamic decision making for commercial fleet operations using real-time information

The application of intelligent transportation system technologies to freight mobility requires dynamic decision-making techniques for commercial fleet operations, using real-time information. Recognizing the productivity-enhancing operational changes possible using real-time information about vehicle locations and demands coupled with constant communication between dispatchers and drivers, a general carrier fleet management system is described. The system features dynamic dispatching, load acceptance, and pricing strategies. A simulation framework is developed to evaluate the performance of alternative load acceptance and assignment strategies using real-time information. Real-time decision making for fleet operations involves balancing a complicated set of often conflicting objectives. The simulation framework provides a means for exploring the trade-offs between these objectives. Results suggest that reductions in cost and improvements in service quality should result from the use of dynamic dispatching...

Online map-matching based on Hidden Markov model for real-time traffic sensing applications

In many Intelligent Transportation System (ITS) applications that crowd-source data from probe vehicles, a crucial step is to accurately map the GPS trajectories to the road network in real time. This process, known as map-matching, often needs to account for noise and sparseness of the data because (1) highly precise GPS traces are rarely available, and (2) dense trajectories are costly for live transmission and storage. We propose an online map-matching algorithm based on the Hidden Markov Model (HMM) that is robust to noise and sparseness. We focused on two improvements over existing HMM-based algorithms: (1) the use of an optimal localizing strategy, the variable sliding window (VSW) method, that guarantees the online solution quality under uncertain future inputs, and (2) the novel combination of spatial, temporal and topological information using machine learning. We evaluated the accuracy of our algorithm using field test data collected on bus routes covering urban and rural areas. Furthermore, we also investigated the relationships between accuracy and output delays in processing live input streams. In our tests on field test data, VSW outperformed the traditional localizing method in terms of both accuracy and output delay. Our results suggest that it is viable for low latency applications such as traffic sensing.

Online Routing Problems: Value of Advanced Information as Improved Competitive Ratios

We consider online versions of the traveling salesman problem (TSP) and traveling repairman problem (TRP) where instances are not known in advance. Cities (points) to be visited are revealed over time, while the server is en route serving previously released requests. These problems are known in the literature as the online TSP (TRP, respectively). The corresponding offline problems are the TSP (TRP) with release dates, problems where each point has to be visited at or after a given release date. In the current literature, the assumption is that a request becomes known at the time of its release date. In this paper we introduce the notion of a requests disclosure date, the time when a citys location and release date are revealed to the server. In a variety of disclosure date scenarios and metric spaces, we give new online algorithms and quantify the value of this advanced information in the form of improved competitive ratios. We also provide a general result on polynomial-time online algorithms for the online TSP.

Evaluation of Dynamic Fleet Management Systems: Simulation Framework

The problem of dynamic fleet management for truckload carrier fleet operations is introduced, and the principal elements of a simulation framework for the evaluation of dynamic fleet management systems are described. The application of the simulated framework to the investigation of the performance of a family of real-time fleet operational strategies, which include load acceptance, assignment, and reassignment strategies, also is described. The simulation framework described is an example of a first-generation tool for the evaluation of dynamic fleet management systems. Selected experimental results are highlighted. These are intended to illustrate some of the issues encountered in real-time fleet management and the role of the simulation modeling environment in investigating them.