Queuing Models for Sizing and Structuring Rental Fleets

Abstract

T his paper has been motivated by a fleet optimization problem faced by one of the leading European cargo rail companies. The company operates a fleet of more than 100,000 rail cars and annually invests significant sums of money into new cars. Because the price tag of a new car is over 50,000 euros, planning such a fleet is an important activity at the company. In this paper, we develop and solve analytical models for fleet planning. We first describe the rental process and show how it can be modeled as a queuing loss system. We then develop a profit function and derive several structural results, such as the concavity of the profit function in the fleet size. Building on these structural results, we show how the fleet size can be optimized, how the fleet structure (i.e., the types of cars being used) can be optimized, and how a joint fleet of owned and leased cars can be optimized. Because some of the optimal methods are difficult to implement, we also develop and test an approximation that is easy to implement. To illustrate our findings and to validate our approach, we provide numerical results that are based on data of the company that motivated our research.

Modelling and analysis of a strategic fleet sizing problem for a furniture distributor

“…Topaloglu and Powell (2007) develop efficient sensitivity analysis methods for a stochastic fleet management model by computing the change in the objective function in response to an additional vehicle or load in the system. Papier and Thonemann (2008) propose a model to determine the optimal fleet size for a leading European cargo rail company.…”

Managing Rentals with Usage-Based Loss

“…Table 1 compares our rental inventory model to the other rental inventory models that also make the assumption of lost sales. In addition to the continuous-time rental inventory models of Tainiter (1964) and Whisler (1967) tabulated here, Papier and Thonemann (2008) build on the M/M/c/c queueing model in Harel (1988), where approximations, as well as lower and upper bounds, are developed for the lost sales rate as a function of the system capacity. Extending this model to account for a compound Poisson arrival process, Papier and Thonemann (2008) conduct a stationary queueing analysis to obtain structural results for a fleet sizing problem and provide an approximation suitable for implementation.…”

“…In addition to the continuous-time rental inventory models of Tainiter (1964) and Whisler (1967) tabulated here, Papier and Thonemann (2008) build on the M/M/c/c queueing model in Harel (1988), where approximations, as well as lower and upper bounds, are developed for the lost sales rate as a function of the system capacity. Extending this model to account for a compound Poisson arrival process, Papier and Thonemann (2008) conduct a stationary queueing analysis to obtain structural results for a fleet sizing problem and provide an approximation suitable for implementation. The use of the M/M/c/c or M/G/c/c queueing model as a basis for studying capacity management for rental systems further follows in Savin et al (2005), Gans andSavin (2007), Adelman (2008), Hampshire et al (2009), and Levi and Shi (2011).…”

Optimizing fleet size and delivery scheduling for multi-temperature food distribution

“…Following the formulation of a fleet-size optimization model by Papier and Thonemann [6], this study constructs an analytical model to determine the optimal fleet size for carriers providing multi-temperature food delivery services. This study focuses on the delivery scheduling of a single distribution center.…”

“…The expected profit function is concave in the fleet size because b m (X) enters the expected profit function with a negative sign. Even the optimal solution might not be unique; this study follows Papier and Thonemann [6] to choose the smallest optimal fleet size, X ⁄ , as the solution which yields min [p(X + 1) À p(X) < 0] .…”