A study of replacement rules for a parallel fleet replacement problem based on user preference utilization pattern and alternative fuel considerations

Abstract

Parallel fleet replacement problems deal with determining an optimal replacement schedule that results in a minimal total cost of owning and operating a fleet within a finite planning horizon. In this paper, the fleet consists of service vehicles, varying in age and cumulative mileage, commonly owned by government agencies or private business organizations. The paper focuses on three main aspects of the problem, including the effects of some widely used replacement rules, consideration of alternative fuel vehicles for replacement, and a new “user preference” utilization pattern. The vehicle replacement rules under study include purchase-new-vehicles-only, no-splitting-in-selling, one-purchase-choice, older-vehicles-selling, and all-or-none rules. The initial fleet consists of gasoline vehicles, while alternative-fuel vehicles using either compressed natural gas (CNG) or liquefied petroleum gas (LPG) are included as challengers in every replacement decision period. The user preference utilization pattern denotes a pattern in which yearly usage of vehicles declines as the vehicles grow older or have higher cumulative mileage. Integer programming formulation of the defined problem that incorporates these new factors is provided. Small numerical examples under various scenarios and a large problem instance with model parameters estimated from actual data are solved to optimality to gain some insights about the replacement rules and other considered factors.

Introduction

Parallel fleet replacement problems (PFRPs) involve determining an optimal replacement schedule that results in a minimum total cost of owning and operating a fleet of vehicles over a finite or infinite planning horizon. In general, a vehicle’s performance or efficiency deteriorates over its useful life due to usage and age. In other words, when vehicles become older or have higher cumulative mileages, they incur higher costs to own and operate. These costs include operating cost, maintenance cost, repair cost, fixed cost, and other overhead costs. Such higher costs combined with less efficiency call for vehicle replacement.

From a real application standpoint, fleet replacement problems occur in many situations. Examples are government agencies and private business organizations that maintain fleets of vehicles to satisfy public service or user demands for transportation. Vehicles in the fleet can be organized into classes categorized by size and/or function. Within a class, vehicles usually vary in their ages and cumulative mileages. This variety directly affects user preferences for using newer vehicles, given that various vehicles are available to provide the same service or function. This utilization pattern is referred to as “user preference” in this paper. When replacement decisions are made, the effect of this utilization pattern should be considered. This is because as older vehicles are replaced by new vehicles, the new ones become the most highly utilized, which makes their operational costs higher than those of the older vehicles. The pattern of user preference utilization is inconsistent with an often-made assumption in many previous studies that the operating and maintenance (O&M) cost is a non-decreasing function of age, which is a result of implicitly assuming constant annual utilization for all vehicles (or assets in other applications). Thus, this utilization pattern is studied in this paper.

In addition to the user preference utilization pattern, two other real-world aspects of the problem are considered. These include the current situation of high and fluctuating gasoline prices and the availability of other cheaper fuel alternatives, and some often-used replacement rules in making replacement decisions. The objectives are to investigate the effects of these real-world aspects, especially the replacement rules, on the optimal fleet replacement schedule. The problem is formulated as an integer program, and a set of problem instances is solved to optimality under various scenarios to examine the effect of these factors. The fleet data used in the computational study are taken from an actual fleet, gathered from several reliable sources, or estimated from real data. The findings from solving the large problem instance then lead to suggestions for fleet replacement decisions. Although the focus is particularly on the replacement of a vehicle fleet, the findings should be applicable to other replacement problems with similar characteristics.

There are a large number of relevant research studies on replacement problems in the literature, but to the best of our knowledge no studies have explicitly taken user preference utilization patterns or alternative fuel vehicles into account. The serial replacement problem of a single asset is well studied; see, for example, Fraser and Posey, 1989, Hopp and Nair, 1991, and Bean, Lohmann, and Smith (1994). In multiple-asset cases, replacement decisions are considered in parallel, and thus are called parallel replacement problems because assets are interdependent for one or more of the following reasons: (1) they exist to satisfy a prespecified level of demand (Hartman, 1999); (2) their replacements compete for a common limited budget pool (Karabakal, Lohmann, & Bean, 1994); (3) an economy of scale is associated with asset investment costs (Chen, 1998, Hopp et al., 1993, Jones et al., 1991, Tang and Tang, 1993). In other words, the parallel replacement problem involves tradeoffs between the capital expenses of acquiring new assets, capital gains from salvage values of old assets, and the operational costs of new versus old assets. Recent studies on the parallel replacement problem with various problem characteristics and settings can be found in Hartman, 2000, Hartman, 2001, Hartman, 2004, Hartman, 2006, Hartman and Ban, 2002, and Chang (2005), among others.

Hartman (2000) considers a generalized parallel replacement problem with replacement costs, budget, and demand constraints. An integer programming formulation for the problem is presented, as well as the linear programming relaxation, which is shown to have integer extreme points if the binary variables for economies of scale are fixed. Hartman, 2001, Hartman, 2004 studies the parallel replacement problem that explicitly considers the deterioration effects of age and cumulative utilization factors. The studies assume that the decision maker has some control over asset utilization patterns, and that workload allocation is treated as part of the decision. The effects of probabilistic asset utilization on replacement decisions are reported in Hartman (2001), and an efficient optimal solution procedure based on a stochastic dynamic programming approach is provided for the two-asset problem in Hartman (2004).

Hartman and Ban (2002) consider the series–parallel replacement problem, where assets are multiple heterogeneous machines in a parallel flow shop environment. They formulate an integer program for determining the optimal purchase, salvage, utilization, and storage decisions for each asset over a finite horizon. The model is found to be difficult to solve; therefore they provide the valid inequalities to improve the lower bound by LP relaxation, and the initial upper bound by a dynamic programming approach. Chang (2005) presents a fuzzy strategic analysis for equipment replacement that models existing equipment cost and market obsolescence fuzzily, in addition to equipment deterioration. Two numerical examples are also given. Hartman (2006) investigates the replacement rule, in which an asset is replaced at the end of its economic life, to see whether this is a good rule for the finite-horizon problem. The cases when the replacement rule is good and when it is significantly off-optimal are identified.