A new two-stage heuristic for the recreational vehicle scheduling problem
Introduction
In this paper, we address the problem of scheduling vehicles in a recreational vehicle operation. Consider a recreational vehicle (RV) rental company that operates from different locations (or depots). At any point in time, the company’s inventory of RVs of different types is either with a customer (in other words, on a booking) or idle, at any of its depots across the geography in which the company operates. A booking is made against a customer request, which specifies a pickup location (a depot), a drop-off location (another depot which may or may not be the same as the pickup depot), a booking pickup time, the requested vehicle type and an anticipated drop-off date and drop-off time. The customer request may also specify the ranking (or preference) of other vehicle types in case of non-availability of the vehicle type that is requested. Sometimes, a customer may request for a specific vehicle, which is known as a special booking. Booking requests may be satisfied by using the available inventory of vehicles of the required type. In the case of non-availability, a vehicle may either be relocated from another location (termed relocation), or a vehicle of another type may be assigned to the booking (termed substitution). The cost that is associated with relocation is a function of the duration of the relocation.
Apart from relocation and substitution, other details to be considered are the vehicle turnaround time at the depot (between the conclusion of a booking and the commencement of the next booking) and any pre-planned fleeting and de-fleeting decisions. The turnaround time is the time that is required for cleaning or mechanical servicing of vehicles between successive bookings. Fleeting and de-fleeting operations refer, respectively, to the addition and disposal of vehicles from the service at various points in time in the planning horizon. These fleeting and de-fleeting decisions are assumed to be known at the commencement of the planning horizon. The schedule also needs to consider any planned maintenance activities for a vehicle during the planning horizon. Special bookings, if any, that demand a specific vehicle and any fixed maintenance schedule for the vehicle need to be prioritised over all other bookings. In the current study, the maintenance requirements, apart from the request for a specific vehicle, are also considered as the special bookings. All special bookings need to be satisfied by the schedule.
The RV business is quite different from (normal) car rental operations and requires significant care and attention when it comes to vehicle scheduling. The RV business caters predominantly to holidaymakers, while the car rental business caters mostly to business and commercial needs. As a result, most bookings in the RV business have long booking lead times (the duration between the actual pickup time and the time at which the booking request is made) and long booking horizons (the duration between the pickup time and the drop-off time). Furthermore, a large proportion of the bookings in the RV business tends to be point-to-point. In direct contrast, most car rental operations see short booking lead times, relatively short booking durations and booking patterns in which the pickup location and the drop-off location are the same. As a result, in the RV business, it is quite common for the vehicle inventory to be off the books (and unavailable to service any booking requests) for large durations, only to re-emerge at a depot that is many thousands of kilometres away from where they were off the company’s inventory. The long booking lead times in the RV business are mainly prevalent because holidaymakers tend to plan their RV holidays well in advance and make their RV requests when they plan their trips. This is both a boon and a disadvantage from the point of view of maximising the vehicle utilisation. Thus, specific vehicle scheduling models need to be developed for RVs: the right vehicle needs to be assigned to the right booking request at the right time at least cost.
Given all the information about accepted bookings, the availability of vehicles at different locations at the start of the planning horizon, the static recreational vehicle scheduling problem (hereafter, referred to as RVSP) determines a schedule for each vehicle over the planning horizon, by assigning accepted bookings to vehicles in such a way that the overall cost of assignment is minimized, while satisfying the following constraints:
- i
Each booking should be assigned to at most one vehicle.
- ii
In any given time period, a vehicle can serve at most one booking.
- iii
Every vehicle should abide by the maintenance or special bookings requirements, if any.
The RVSP makes the following assumptions.
- i
No pre-emption: A vehicle that currently serving a booking can not be stopped and assigned to another booking.
- ii
All booking information (start/end location/time) is known with certainty and is fixed for the planning duration.
- iii
Fleeting and de-fleeting decisions, and fixed and frozen maintenance schedules are assumed to be known at the commencement of the planning horizon.
- iv
No allowance is made for breakdowns.
The RVSP cost structure includes relocation, substitution and turnaround costs along with the penalty cost of not assigning a booking to a vehicle. The relocation cost is a function of the relocation time, while the substitution cost occurs due to the decision to assign a vehicle of a higher-value to a booking in the case of unavailability of the preferred vehicle type. The turnaround cost is the cost that is required to prepare the vehicle for the next booking.
There are static and dynamic versions of the RVSP. In the dynamic version, given an existing schedule, decisions need to be made on, for example, (a) whether to accept a specific rental request in the future; (b) the changes that are to be made to the schedule due to reasons such as a booking no-show or a sudden change in the drop-off time/date for a specific booking. Accordingly, the dynamic RVSP (DRVSP) alters the planned schedule with a strong emphasis on maintaining feasibility (rather than optimality). The RVSP, on the other hand, deals with obtaining an optimal schedule for the available vehicles by assigning the vehicles to the accepted bookings, considering various operational costs such as relocation, substitution, and turnaround. The static RVSP tries to optimise allocations so as to reduce the cost of allocation and operation. In practice, the RVSP will be required to be solved at least once in a day, mostly at times when there is significantly low volume of booking requests transactions. Since the booking system will be offline during this period, it is expected that the RVSP will provide solutions within an hour (see Ernst et al. (2011)). The RVSP schedule is then used at the various depots to service the various requests that are due to be allocated on the next day. Another aim of the RVSP is to maintain a near optimal schedule that can be used as inputs to the DRVSP.
Ernst et al. (2011) reported that, for large-sized instances, the CPLEX solver with the formulation provided in the paper could not find feasible solutions in the allowed time of one hour. The heuristic that is proposed in the above paper could also not consistently find near optimal solutions in the one-hour time limit. Hence, it is necessary to design a method that consistently produces good-quality solutions within the required maximum solution time for realistic instances. With this as a motivation, the focus of the current study is to provide efficient mathematical formulations and solution procedures for the RVSP, which can consistently provide good-quality solutions in a reasonable amount of time. In this paper, we begin by suggesting simple modifications to the existing formulation. We then suggest a novel, stronger formulation and also devise a two-stage heuristic to quickly solve the problem.
The rest of the paper is organised as follows. We discuss the vehicle scheduling literature that is specific to the RVSP in Section 2. We present our new mathematical formulation for the RVSP (and also two existing formulations from the literature) in Section 3. In Section 4, we discuss the equivalence between all the three formulations. The solution methodology is discussed in Section 5 followed by computational results in Section 6. The conclusions and pointers for further research are discussed in Section 7.
Section snippets
Literature survey
The use of decision support systems in the car rental industry can be seen even in the 1970s. Edelstein and Melnyk (1977) designed a pool control system to make fleet size, vehicle transfer, and booking request acceptance decisions. Since then a fair amount of work on pool control system and revenue management has been reported (Carroll, Grimes, 1995, Geraghty, Johnson, 1997, Pachon, Iakovou, Ip, Aboudi, 2003, Yang, Jin, Hao, 2008, You, Hsieh, 2014). This stream of research dealt with the
Mathematical formulations
There are two different formulations for the RVSP in the literature: a network-flow-based formulation and an assignment-based formulation. The network-flow-based formulation models the RVSP as a minimum cost network flow problem, on a time-expanded network of locations, whereas the assignment formulation models the RVSP as an assignment problem that assigns available vehicles to bookings. We propose a new inventory formulation, which models the RVSP as a multiperiod multi-product inventory
Equivalence of formulations
In this section, we provide proofs that the formulations that we presented in the previous section are all equivalent. By equivalence, we mean that the linear programming (LP) solution of all the formulations is the same. We will transform the network formulation into the inventory and the assignment formulation. All feasible flows that are candidates for the optimal solution are retained during the transformation. Hence, the optimal solution will be the same for all the formulations. A
Heuristic solution methodology
Previous studies on the RVSP (see Section 2) suggested that it was essential to design a method that could consistently produce good solutions within the required maximum solution time. With this as a motivation, Ernst et al. (2011) proposed a heuristic that produces feasible solutions in a reasonable amount of computational time. This uses parallel computing in order to reduce the overall solution time. They reported that the parallelised heuristic was able to find feasible solutions (even for
Computational results
We used a dataset with sixty real-life problem instances. The description of the dataset is presented in Table 6. The instances are arranged in the increasing order of the number of variables in the assignment formulation. Due to data confidentiality, we received the data in a format that is suitable for the assignment formulation. An instance has been divided into different assignment components. Hence, the number of vehicle types in Table 6 refers to the number of assignment components in a
Conclusions and future work
In this article, we have proposed a new formulation for the RVSP. The formulation can be seen as an improvement to the existing network flow formulation under the assumption of infinite capacity of relocation arcs. The detailed explanation of all the formulations is presented. The existing assignment formulation is modified using the node and arc aggregation to improve the solution time when solved using the CPLEX solver. The performance of the CPLEX solver with the inventory formulation is
Acknowledgments
We thank the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Australia for providing the data and the authors of Ernst et al. [2011] for providing the C++ code of M-EHeur, for the experimentation. We extend our gratitude also to the two anonymous referees and the associate editor for their suggestions, which significantly improved the quality of the paper.
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