Innovative Applications of O.R.
A co-evolutionary matheuristic for the car rental capacity-pricing stochastic problem

https://doi.org/10.1016/j.ejor.2019.01.015Get rights and content

Highlights

  • New two-stage stochastic model including demand uncertainty and price-sensitivity.

  • Innovative matheuristic generates solutions to first-stage decisions and scenarios.

  • Genetic algorithm with co-evolution of solutions and scenarios.

  • Little information on random variables is needed to generate scenarios.

  • Method is adaptable to different decision-making risk profiles.

Abstract

When planning a selling season, a car rental company must decide on the number and type of vehicles in the fleet to meet demand. The demand for the rental products is uncertain and highly price-sensitive, and thus capacity and pricing decisions are interconnected. Moreover, since the products are rentals, capacity “returns”. This creates a link between capacity with fleet deployment and other tools that allow the company to meet demand, such as upgrades, transferring vehicles between locations or temporarily leasing additional vehicles.

We propose a methodology that aims to support decision-makers with different risk profiles plan a season, providing good solutions and outlining their ability to deal with uncertainty when little information about it is available. This matheuristic is based on a co-evolutionary genetic algorithm, where parallel populations of solutions and scenarios co-evolve. The fitness of a solution depends on the risk profile of the decision-maker and its performance against the scenarios, which is obtained by solving a mathematical programming model. The fitness of a scenario is based on its contribution in making the scenario population representative and diverse. This is measured by the impact the scenarios have on the solutions.

Computational experiments show the potential of this methodology regarding the quality of the solutions obtained and the diversity and representativeness of the set of scenarios generated. Its main advantages are that no information regarding probability distributions is required, it supports different decision-making risk profiles, and it provides a set of good solutions for an innovative complex application.

Introduction

When planning a selling season, a car rental company must decide on the fleet size and mix, i.e., the capacity it will have to meet demand throughout the season and rental locations. The demand is uncertain and highly price-sensitive. Therefore, the prices charged by a company are connected with and should influence the capacity decisions. Capacity decisions are also connected with other instruments that allow the company to “meet” its demand, which range from offering upgrades to transferring vehicles between locations or temporarily leasing additional vehicles.

The goal of this work is to provide decision-makers with profitable solutions to capacity and pricing decisions, assessing and increasing their ability to deal with the different realizations of uncertainty, represented by scenarios, when little information regarding those is available. The methodology developed is based on a co-evolutionary genetic algorithm, where parallel populations of solutions and scenarios co-evolve, depending on each other for the fitness evaluation of their individuals. On the one hand, this method aims at obtaining a representative and diverse population of scenarios, measured according to the impact they have on the population of solutions. On the other hand, the solutions evolve according to different decision-making risk profiles that assess its performance against the population of scenarios.

This work deals with the integration of capacity and pricing decisions under uncertainty within the context of the car rental business. In this section, the relevance of the application and methodological scope of the work will be discussed. Firstly, the recently growing body of research on car rental fleet management and pricing will be briefly reviewed. This is an innovative and different application because the capacity is rented rather than sold. However, previous works that tackled the integration of pricing and capacity, although not directly applicable, can bring relevant insights to this problem. A stochastic approach to the problem is considered, where the uncertainty is represented by scenarios. Stochastic problems with similar characteristics are briefly reviewed regarding methodological approaches. Moreover, previous fundamental works that laid the foundation for the methodological idea developed in this paper will be presented.

The car rental fleet management problem is initially structured in Pachon, Iakovou, Ip, and Aboudi (2003) and Pachon, Iakovou, and Ip (2006). Fink and Reiners (2006) extends the operational issues within fleet management and deployment, considering essential and realistic practical needs. In Oliveira, Carravilla, and Oliveira (2017c), the link with revenue management issues is introduced, and the body of research developed in this field is reviewed and structured. Existing gaps and relevant future research directions are discussed, including the integration of pricing and/or capacity allocation (revenue management issues) with operational decisions related to fleet size/mix and deployment. The need to consider uncertainty in demand in order to approximate the model to reality is also highlighted.

In a previous paper – Oliveira, Carravilla, and Oliveira (2018) – we tackled the first research direction. A mathematical model for the deterministic integration of dynamic pricing and capacity decisions was proposed. Due to the complexity of the problem, a matheuristic was proposed. This matheuristic is based on a decomposition of the problem, where the price decisions are directly encoded in the chromosomes, and the remaining decisions and the fitness of the full solution are obtained by solving a mathematical programming model. Moreover, some performance-boosting initial population generation procedures were proposed.

In this work, we propose to tackle the even more complex problem that arises when uncertainty is incorporated. Moreover, additional realistic requirements (such as price hierarchy) are included, and demand is modeled considering its relationship with competitor prices.

Pricing decisions have often been tackled independently of capacity and inventory decisions. A recent and growing body of research on the integration of these topics has been arising.

Den Boer (2015) presents an interesting and thorough literature review on the topic of dynamic pricing, primarily focused on learning processes. Following the structure proposed by the author, the car rental pricing problem herein considered can be seen as a dynamic pricing problem with inventory effects, more specifically “jointly determining selling prices and inventory–procurement”. In Gallego and van Ryzin (1994), the dynamic pricing problem for inventories with price-sensitive and stochastic demand is tackled, including an extension where the initial stock is considered as a decision variable. The rental facet of the problem at hand hinders the direct application of the insights drawn. Focusing on perishable assets, a dynamic pricing problem under competition is studied in Gallego and Hu (2014). Here, the dynamics of an oligopoly are considered, dealing with substitutability among assets. These characteristics are more similar to the car rental market, where vehicles that are available at a particular day (or the corresponding available days-of-use) “expire” since they cannot be used in a future time period. Relevant results are obtained regarding dynamic pricing strategies. As this, other important works have dealt with similar environments with insightful outcomes. Adida and Perakis (2010) present an interesting work, where different joint dynamic pricing and inventory control models that deal with demand uncertainty (which depends linearly on price) are considered, within a make-to-stock manufacturing context. This work compares stochastic and robust optimization approaches, introduces different formulations and compares their computational performance.

Nevertheless, the car rental business is characterized by the return of its “sold inventory” in a pre-determined future time period and location. This causes significant changes to the problem structure and renders the problem even more complex to solve. In Oliveira, Carravilla, and Oliveira (2017a), a dynamic programming approach is developed for a deterministic and simpler version of this problem and this question is further discussed.

In innovative transportation systems based on the sharing paradigm, this issue is also present. Bike-sharing, for example, has been a key driver of research on managing capacity to meet demand better, considering variations throughout time and space. Relevant works have focused on capacity reallocation, such as Freund, Henderson, and Shmoys (2017), yet only a few works have focused on the role of pricing in influencing demand, like Chemla, Meunier, Pradeau, Calvo, and Yahiaoui (2013). Nevertheless, this mobility system shows some differences to the car rental business, which hinder the direct application of the developed techniques, such as the homogeneity of the fleet, the design of the repositioning schemes and the motivations (and consequent distribution) of demand, among others. Other innovative businesses are also driving research in this field, such as ride-sharing or e-hailing platforms such as Uber. For example, in Bimpikis, Candogan, and Saban (2016), pricing decisions for a ride-sharing system are introduced to manage the supply-demand balance considering not only variations in time but also the geographical distribution of demand and supply. As the car rental business, the vehicles are shared and thus capacity “returns” to be used by another client. However, the capacity decisions are not centralized in the same decision-maker, as the pricing itself may induce more or less supply of drivers, depending on the business format of the e-hailing platform.

Additionally, the relationship between demand and price in the context of car rental is distinctive and challenging to determine due to the effect of competition and to the myriad of products offered (rental types) that share the same resources (vehicle fleet). Therefore, new approaches are needed to tackle this problem.

Scenarios can be important tools for companies dealing with relevant uncertainties. Moreover, the process of scenario generation is critical for the practical relevance of the results obtained.

Scenario generation consists of defining discrete outcomes (realizations) for all random variables and time periods (Høyland & Wallace, 2001), especially useful for stochastic problems. Mitra and Di Domenica (2010) review the scenario generation methods applied in the literature for stochastic programming models, including sampling-based generation (e.g., Monte Carlo, bootstrap or conditional sampling methods), statistical methods (e.g., property matching or regressions) and simulation-based generation (e.g., Vector Auto Regressive methods), as well as other less used methods (e.g., hybrid methods). The authors discuss relevant, desirable characteristics that all scenario generation methods should incorporate: including a variety of factors and existing correlations, considering the purpose of the model (to understand e.g., if it is more relevant to capture variance or higher moments), being consistent with any theory and with empirical data observations. Kaut and Wallace (2003) evaluate different scenario generation methods and propose two properties (and corresponding methodologies to test them) that a method should satisfy to be applicable and relevant to a given problem. Most of these techniques involve a considerable amount of knowledge about the uncertainty and random variables, e.g., their probability distribution.

Solving integer stochastic mathematical programming models is becoming a promising approach to obtain good and accurate solutions for complex real-world situations, such as hazard management of post-fire debris flows or transportation network protection against extreme events such as earth-quakes (Krasko, Rebennack, 2017, Lu, Gupte, Huang, 2018). Often, solution approaches are required to deal with the inherent complexity, such as decomposition or (meta)heuristics (Özcan, 2010, Puga, Tancrez, 2017, Yan, Tang, Fu, 2008).

Genetic algorithms have been proposed to tackle complex stochastic problems (Gu, Gu, Cao, Gu, 2010, Wang, Makond, Liu, 2011). In these works, random variables are often associated with probability distributions; thus scenarios are generated by random sampling or simulation. Furthermore, the hybridization of genetic algorithms and linear programming has been successfully used to develop alternative stochastic methodologies (Reis, Walters, Savic, & Chaudhry, 2005).

In this field, scenario generation is heavily dependent on the knowledge of probability distributions for the random variables and consists on selecting a small set of scenarios that represent it well, which is highly complicated in the multivariate case (Löhndorf, 2016). The author presents an empirical analysis of popular scenario generation methods for stochastic optimization. State-of-the-art methods are compared regarding solution quality, using a problem where analytical solutions are available. Their adequacy is dependent on the problem characteristics and probability distributions. Guastaroba, Mansini, and Speranza (2009) focus on optimal portfolio selection problem and compare scenario generation techniques for this problem. One of the conclusions is that the adequacy of the method depends on the risk profile of the decision-maker.

For this problem, using scenarios to represent uncertainty has a practical interest concerning the application of the method, since scenarios can be useful to help decision-makers understand and act upon the outputs. Nevertheless, the only information regarding the uncertain parameters available for this problem is the bounds on the values they can take. Therefore, a methodology that tackles this lack of information is needed.

Herrmann (1999) proposes a metaheuristic based on genetic algorithms that is especially adequate for problems where the set of scenarios is too large for each element to be evaluated individually, or even known. In this work, the author proposes the co-evolution of solutions and scenarios in two parallel spaces, as follows.

Considering that SO is the set of all solutions and SC the set of all possible scenarios, the value obtained by a solution iSO when scenario jSC occurs is given by F(i, j). In this work, the goal was to find the solution that performs best for the worst-case, which is translated (in a minimization problem) to: miniSOmaxjSCF(i,j). The author thus proposes a two-space genetic algorithm where scenarios (SC) and solutions (SO) co-evolve in different populations (PSO and PSC) composed of individuals whose fitness depends not only on its characteristics but also on the characteristics of the other population. This genetic algorithm favors solutions with better worst-case performances and scenarios with worse “best solutions”. The fitness of a solution i0 is evaluated as maxjPSCF(i0,j) (worst scenario for this solution), while the fitness of a scenario j0 is evaluated as miniPSOF(i,j0) (best solution for this scenario). The groundbreaking idea in this work is that using efficient genetic algorithms to evolve populations of scenarios requires only an initial sample that will evolve and is thus expected to adequately represent the full set, which would otherwise take significantly more effort to explore. Simultaneously, the solutions evolve to perform increasingly better. This work is continued by other authors, namely Jensen (2001) who proposes a ranking-based evaluation for scenario fitness that performs better and fixes symmetry and bias issues of the original approach.

We aim to extend the idea of a two-space genetic algorithm to evolve solutions and scenarios to other decision-making risk profiles beyond the limitation of the worst-case perspective in Herrmann (1999). Considering the expected value as the goal to evaluate solutions (stochastic approach) rather than the worst-case value significantly impacts the evolution of the scenario population. This focuses the evolutionary drive in obtaining a representative population, rather than converging to the worst-case scenario. To achieve this, recent developments on the field of instance generation were considered. In Gao, Nallaperuma, and Neumann (2016), an evolutionary algorithm is proposed for generating instances that are diverse with respect to different features of the problem. It aims to “diversify” points in N-dimensions by ranking candidates based on distance to nearest neighbors in each axis. Using this technique with elitism leads to new children being added to the population only if they extend the extreme values or lie in a large gap between existing points. Also in Deb, Agrawal, Pratap, and Meyarivan (2002), the concept of crowding distance is used to estimate the density of solutions surrounding a particular point in a population. It compares to the largest cuboid enclosing the point without enclosing any other points, with a similar reference to nearest neighbors in each axis.

The main contributions of this paper are related to the mathematical model and the solution methodology proposed.

  • We propose a new two-stage stochastic model, extending the deterministic model proposed in Oliveira et al. (2018):

    • Its main innovative feature is that the stochastic capacity-pricing problem for car rentals is modeled. Few papers focus on the integration of pricing with capacity decisions, using tactical information and uncertainty to deal with strategic decisions, especially in the complex rental context, where inventory is not depleted but only temporarily unavailable.

    • The issue of vehicle group price hierarchy is included, on a more realistic approach to the problem.

    • Demand uncertainty and price-sensitivity are modeled in an innovative and efficient way, with a significant fit with the problem at hand and its strategic scope. The model is adaptable to different shapes of the demand-price function, considering the effect of competition.

  • We propose an innovative solution method to tackle the problem, based on the decomposition of the stochastic model in first-stage and second-stage decisions:

    • Solutions to the first-stage decisions and scenarios are generated in parallel with mutual impact on fitness evaluation, requiring little information on random variables to do so.

    • The fitness depends on the profit obtained by each pair (solution, scenario), which is calculated using a mathematical programming model.

    • The methodology is easily adaptable to different decision-making risk profiles.

    • Specific problem know-how can be used in the initial populations to boost the evolutionary procedure (e.g., providing extreme scenarios).

    • It can be implemented and run in a reasonable time in a decision-support system.

Overall, this methodology has a proper fit with the problem at hand, making it useful in real-world applications. Moreover, it is a methodology that can be easily extended to other problems where information regarding uncertainty is scarce.

This paper is structured as follow. Firstly, the problem will be stated and the mathematical model presented (Section 2). Then, Section 3 presents the co-evolutionary matheuristic developed and in Section 4 the results of the computational tests are discussed. Finally, conclusions are drawn and future work and promising research directions are discussed (Section 5).

Section snippets

Problem definition

Car rental companies preparing a season must decide on the size and mix of their fleet, i.e., the capacity they will have to face demand in that season. In order for this capacity to be used efficiently, some operational issues that will take place during the season must be considered, as well as the uncertain demand.

A car rental company has several rental stations that share the same fleet. Within the scope of this problem, these stations are often aggregated in regions or locations such that

Solution method

Two main issues underline the need for a specific solution method to solve the mathematical model presented in the previous section. First, the number of decision variables in a real-world-sized instance is significantly large in this problem. Even in a small case study, a significant amount of rental types (different combinations of starting/ending times and locations) leads to an even higher number of (binary) decision variables, which makes it difficult to solve this model to optimality.

Computational experiments, results and discussion

The goal of the computational experiments discussed in this section is to validate the value of the methodology proposed in terms of: (i) the quality of the solutions proposed, (ii) the diversity and representativeness of the generated set of scenarios, which support the “robustness” of the solutions, and (iii) the applicability and utility of the method when integrated in a decision-support system. Additionally, to validate the simple adaptation needed to use a different shape for the demand

Conclusions

This study presents not only a new approach to deal with an innovative application, but also methodological contributions that can be applied beyond this scope. This methodology can be adapted to provide good solutions to complex two-stage stochastic problems where the information on uncertainty is scarce. It does not require the decision-maker to define the scenarios or probabilities associated with them, but only to establish upper and lower bounds for the uncertain parameters. The scenarios

Acknowledgments

This work is partially financed by the ERDF European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation – COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT – Fundação para a Ciência e a Tecnologia within project “POCI-01-0145-FEDER-029279”.

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