User preferences in ride-sharing mathematical models for enhanced matching

Abstract

Ride-sharing services have attracted significant interest due to overcrowding, limited energy resources, and environmental concerns. This study proposes a mathematical programming model that integrates user preferences into the ride-sharing problem with transfer. Passenger transfer in ride-sharing problems addresses the limitations of the matching process, especially in less-populated areas. It allows passengers to be dropped off at meeting points and continue with another driver. Moreover, considering user preferences in ride-sharing systems is crucial for enhancing efficiency and user satisfaction. Accordingly, we propose a Preference-Driven Matching Algorithm for the matching process. Our proposed algorithm captures user preferences and provides potential matches. In addition, we introduce an Iterative Enhance-and-Optimize Algorithm capable of producing high-quality solutions within short computational times. We evaluate the efficiency of these approaches across various instances, focusing on real-scale scenarios. Based on the results, the model with preferences demonstrates effective performance compared to other methods. Our findings underscore the importance of user preferences in optimizing ride-sharing problems and highlight the trade-off between efficiency and user satisfaction. By incorporating user preferences, our approach results in higher user satisfaction, a more responsive and efficient system with reduced response times, and increased demand and revenue by servicing more users.

Introduction

In the rapidly evolving urban transportation landscape, ride-sharing is becoming a revolutionary force changing how people move around cities¹. Establishing ride-sharing is an important step considering obstacles such as traffic congestion, environmental issues², and the global demand for access to transportation services³. Facing this paradigm shift, more in-depth research on the dynamics of shared service management is needed. Based on the 2009 National Household Travel Survey (NHTS), the average seat utilization in a vehicle with a capacity of 4 seats is only 1.7. Additionally, according to the official statistics of the International Energy Agency⁴, each additional person added to transportation results in a 7.7% reduction in fuel consumption.

Ride-sharing refers to a system where passengers and drivers are matched by an automated system provided by a company in real-time. This system allows the matching of passengers with non-recurring, short-notice, and on-demand requests^5,6. The nature of this problem is different from the carpooling problem. The concept of carpooling has been introduced by large organizations to encourage employees to travel together to and from work. This concept aimed to reduce the number of cars utilized for daily commutes. It refers to a long-term, prearranged, demand-responsive trip between two or more people⁷. Once a carpooling team is formed, the driver must pick-up all members from their residences in a pre-determined sequence, then drop them off at their workplaces in a pre-determined order, ending at the driver’s workplace. This process must then be repeated in roughly the reverse order at the end of the workday⁶.

There are many challenges in ride-sharing services, including social, privacy, and security issues, and it is essential to know how to overcome these challenges in the long run. In addition, it is imperative to continue to develop solutions to optimize ride-sharing systems in the real world⁸. Research shows that time, cost, and convenience are the most significant factors when choosing a ride-share, and safety is also a top consideration⁹. Therefore, to solve this problem, it is essential to consider user preferences, rental benefits, travel costs, and safety issues in the ride competition^6,10. A major challenge when planning a ride-sharing service with user preferences is the complex interaction between user preferences. Current literature shows that psychological barriers are a significant barrier to user decision-making in transportation. In addition, in the traditional Ride-Sharing problem, matching is possible if the driver with the same route, preferences, and schedule picks up the passengers at the departure point¹¹. Therefore, this matching is more likely to occur in densely populated areas and densely lined roads. In other words, if a passenger requests a ride from a less populated area to downtown or shares an indirect and complex route with others, their request may not be answered. The passenger might have to endure an extended wait for the driver to accept it, resulting in a loss of demand for companies offering these services. Current advances allow passengers to be dropped off at meeting points and continue with another driver (transfer between drivers).

The research question focuses on determining how passengers can reach their destinations in one or more sequential trips, given the preferences of drivers and passengers. The importance of this research lies in its ability to tackle the gap between operational efficiency and user satisfaction.

Some platforms offer ride-sharing services, which have become an effective way to reduce traffic congestion and pollution. These services, particularly on well-known platforms like Uber and Lyft, not only help users save on costs but also enable them to share rides with other passengers heading to similar destinations. For instance, UberX Share allows riders to share their trip with others going in the same direction, providing discounts of up to 20%. This service adds an average of six minutes to the journey and aims to reduce car usage and emissions. Additionally, Uber’s multi-stop feature allows riders to add up to five extra stops during their trip, making it convenient for errands or picking up friends. Similarly, Lyft enables passengers to share their journey with up to two other riders, offering a more affordable and efficient option. The BlaBlaCar platform connects drivers with empty seats to passengers traveling in the same direction, helping to share travel costs. Furthermore, DiDi Global Inc. provides a range of services, including ride-hailing and shared mobility, leveraging AI and smart transportation to support sustainable urban mobility.

Despite these advancements, the ability to transfer passengers between different drivers and incorporate personalized preferences has largely been neglected in current ride-sharing platforms. This gap highlights the need for a more advanced system that addresses user satisfaction on a deeper level. However, discussions on transfers and user preferences have yet to be directly implemented in these applications. The objective of this study is to examine the implementation of user preferences and transfer options to enhance user satisfaction. In this context, there are three main stakeholders: passengers, drivers, and companies, each playing a crucial role in the overall satisfaction of the system. Current ride-sharing models, however, do not prioritize user preferences, providing only a minimal level of satisfaction, which can lead to user dissatisfaction and customer attrition in the long run.

To effectively implement the proposed models and algorithms, both hardware and software infrastructures are essential. This paper presents the mathematical models and algorithms aimed at improving user satisfaction by focusing on their preferences, thereby offering more effective solutions for ride-sharing systems. With these foundational elements established, companies can take the initiative to enhance their software components and leverage these insights to improve overall user satisfaction.

We believe that understanding user preferences is important for service providers, as well as city planners and policymakers, to shape the future of shared mobility. The practical applications of our findings can lead to more efficient and user-centric ride-sharing systems. By incorporating user preferences and passenger transfer into the ride-sharing model, this research not only improves operational efficiency but also enhances user satisfaction. This dual focus is essential for creating sustainable urban transport systems that can adapt to varying demands and preferences. The insights from this study can inform the development of policies and strategies that promote higher vehicle occupancy rates, reduce fuel consumption, and mitigate traffic congestion, ultimately contributing to environmental sustainability. Additionally, these improvements can lead to increased demand and revenue for ride-sharing services, making them more viable and attractive to a broader range of users.

Therefore, the primary aim of this paper is to present a mathematical model for the ride-sharing problem with transfer and user preferences. To achieve this, we will first examine the concept of transfer and user preferences in the ride-sharing problem. Following this, we propose a Preference-Driven Matching (P-DM) algorithm to facilitate the process of matching passengers and drivers considering user preferences. In the next stage, a non-linear mathematical model will be developed for the ride-sharing problem with transfer, afterward linearizing the model to facilitate analysis. This model emphasizes the importance of user preferences in passenger-driver matching. Then, we present an Iterative Enhance-and-Optimize (IE-O) algorithm that can identify high-quality solutions in short computational times, and make it suitable for long-term use. The results of the mathematical model and algorithm for different ride-sharing networks were compared, considering the effectiveness of different approaches.

The study is structured as follows: section “Literature review” describes the latest studies regarding ride-sharing. Section “Theoretical framework” explores the concept of transfer and user preferences within ride-sharing, introducing the P-DM algorithm for the matching process that considers user preferences. Section “The ride-sharing problem with transfer and preferences” introduces a mathematical model for the ride-sharing problem, incorporating transfer and user preferences. Section “Solution methodology and comparative analysis” outlines the solution methodology employed to obtain feasible solutions and presents the managerial recommendations. Finally, the last section offers conclusions and suggests avenues for future research.

Literature review

The growth of e-hailing services has made ride-sharing a popular and efficient mode of transportation. Addressing the complexities of ride-sharing problems requires an integrated approach that considers user preferences, passenger transfer, and dynamic route planning. A crucial aspect of this problem is optimizing vehicle-passenger matching, pricing, and dispatching, while also considering the impact of transfer, demand-aware route planning, and algorithmic decision-making¹².

The research literature will be examined from two distinct perspectives to achieve a comprehensive analysis. First, the integration of user preferences will be explored to understand how individual choices and preferences impact the effectiveness and efficiency of ride-sharing systems. This includes examining factors such as personal comfort, convenience, and social preferences in the matching process. Second, passenger transfers will be examined to assess the feasibility and benefits of allowing passengers to switch vehicles during their journey. This involves studying existing models and strategies for integrating ride-sharing with public transport systems and optimizing transfer points to enhance overall system efficiency and user satisfaction. By investigating these two perspectives, the study aims to provide a comprehensive understanding of ride-sharing services’ current state and future potential.

User preferences in ride-sharing

Understanding and integrating user preferences are crucial for improving the performance and reliability of ride-sharing systems. Thaithatkul et al.¹³ formulated a matching model between passengers by considering user preference and investigated how user preference affects the ride-sharing system’s performance. The findings demonstrate that user preference significantly affects the performance and reliability of ride-sharing systems. Additionally, the results of numerical experiments showed that the steadiness of user personality affected individual and system success rates.

König and Grippenkoven¹⁴ aimed to understand users’ preferences for ridepooling services by employing a Discrete Choice Experiment with 410 participants. Through a literature review and focus groups, they identified six key attributes—fare, walking distance, booking time, departure time shift, travel time, and information provision. The results indicated that these attributes significantly influence user choice behavior. Furthermore, Mitropoulos et al.¹⁵ conducted a systematic review of global ride-sharing studies, identifying key aspects such as online platforms, user preferences, and barriers. They categorized these barriers into economic, technological, business, behavioral, and regulatory dimensions, emphasizing the positive impact of customizable ride-sharing platforms on user satisfaction.

Cui et al.¹⁶ focused on modeling the social preferences of ride-share passengers, addressing challenges in partner selection for ride-sharing. They conducted an online survey in the United States to explore respondents’ ride-sharing preferences and identified 13 attitudinal dimensions representing social preferences. Through factor analysis, they condensed these dimensions into four variables: Organizational, Socioeconomic, Smoking/Drinking Habits, and Sociodemographic. The study revealed that passengers with higher scores in the Sociodemographic factor tended to prefer partners with similar characteristics. Moreover, their findings underscored the importance of considering diverse social preferences in designing effective ride-sharing systems.

Zhao et al.¹⁷ introduced the Online Stable Matching under Known Identical Independent Distributions model for task assignment in taxi dispatching, addressing the dual objectives of maximizing platform profit and accounting for user preferences. They also proposed a linear program-based online algorithm, LP-ALG, which optimizes profit while minimizing blocking pairs—representing misalignment between driver and passenger preferences. Experimental results from both synthetic and real-world datasets demonstrate that the proposed algorithm significantly outperforms baseline models, particularly in scenarios where the demand for rides exceeds the available driver supply.

Golpayegani et al.¹⁸ presented a novel approach to taxi-sharing, emphasizing its economic and environmental benefits. The study advocates intelligent taxi-dispatch systems that align taxi distribution with urban population density to address demand. Unlike traditional approaches focusing solely on matching passengers based on origin and destination, this research acknowledges the intricacy of individual preferences, such as convenience, time, cost, and environmental factors.

Ruch et al.¹⁹ explored the trade-offs between efficiency gains and service quality in ride-sharing mobility-on-demand systems across diverse transportation scenarios, including dense urban and line-shaped rural settings. Their findings highlighted that while ride-sharing can deliver moderate efficiency improvements, concerns about reduced convenience, privacy implications, and extended travel times raise questions about its overall viability. Alok et al.²⁰ investigated the impact of cultural norms, trust issues, accessibility, and affordability on favoring unconventional contractual rides over ride-sharing apps. The research sheds light on the intricate and context-specific nature of consumer preferences within the ride-sharing industry, including the crucial role of privacy concerns.

Given that online matching often involves sharing the locations of passengers or drivers, privacy protection has emerged as a major concern in the mobile internet era, with vast amounts of personal information and sensitive data facing increasing exposure risks daily²¹. To address this issue, Feng et al.²² proposed PBAG, a blockchain-based authentication protocol with global-updated commitment for enhanced privacy protection in the Internet of Vehicles. The PBAG protocol enhances privacy-preserving authentication in the Internet of Vehicles by using blockchain technology to securely connect vehicles and infrastructures. It also addresses privacy concerns by ensuring anonymity, while maintaining the ability to trace identities if needed. Additionally, Cheng et al.²³ introduced a lightweight privacy preservation scheme with efficient reputation management for mobile crowdsensing in vehicular networks, which includes algorithms for verifying reputation values and selecting reliable sensing vehicles to prevent forgery while protecting location and identity privacy and reducing overhead. Liu et al.²⁴ further developed the Privacy-Preserving Reputation Updating scheme, which focuses on privacy-preserving reputation updating in cloud-assisted vehicular networks. The scheme enhances reputation management by enabling a Cloud Service Provider to handle reputation feedback with strong privacy and security measures. These advancements play a critical role in ensuring secure and private ride-sharing systems.

In the following, and emphasizing the importance of privacy protection, Miao et al.²⁵ proposed the Task Assignment with Federated Preference Learning framework for task assignment, incorporating worker preferences while preserving data privacy. This framework develops local preference models through federated learning and allocates tasks based on these preferences.

The implications of these findings extend to transportation companies and technology developers, emphasizing the necessity to consider various factors in designing and implementing ride-sharing services. These insights align with the growing emphasis on sustainability and efficient resource utilization in urban transportation systems, substantially contributing to broader discussions on sustainable development goals.

The literature review on user preferences indicates that integrating users’ preferences improves the performance and user satisfaction of ride-sharing systems by enhancing system efficiency and user experience. However, a notable research gap exists due to the lack of practical implementation of these preferences in mathematical models. This highlights the need for further research to operationalize these considerations effectively.

Passenger transfer in ride-sharing

Passenger transfers play a crucial role in ride-sharing systems, extensively studied in the current literature. Recognizing the significance of transfers in ride-sharing, Agatz et al.²⁶ introduced a comprehensive classification of ride-sharing, with four distinct types: single driver and single rider, single driver and multiple riders, multiple drivers and single rider, and multiple drivers and multiple riders. These classifications encompass a range of scenarios wherein passengers seamlessly transfer between drivers based on preferences and the availability of suitable matches. This type of system has already been described by Gruebele²⁷, who shows that passenger transfer locations can be public transportation stations, shopping malls, etc.

Stiglic et al.²⁸ examined the advantages of integrating ride-sharing with public transit, specifically addressing challenges related to downtown accessibility. This integration proves particularly advantageous in suburban areas with limited public transportation options. Moreover, it mirrors situations where passengers can transition between two or three drivers, providing flexible routes to reach their destinations. Additionally, Liu et al.²⁹ investigated integrating ride-hailing services with public transport using a Stochastic User Equilibrium model for multimodal transport systems. This study demonstrated that integrating public transport systems with ride-hailing services can optimize traffic flow distribution.

Ma et al.³⁰ proposed an innovative ride-sharing strategy that impeccably integrates transit services. This strategy offers diverse options, including door-to-door drop-offs, transit station commitments, and mixed pickups and drop-offs using different vehicles. The study implemented tailored-online-algorithms, resulting in substantial reductions in ride-share vehicle travel time (40–60%) and passenger journey times (50–60%) during periods of high demand.

Chen et al.³¹ developed a model addressing the dynamic ride-sharing problem for company employees with predetermined schedules and travel times. The ILP model in their study incorporates return restrictions to satisfy business needs, including meeting points and the option for riders to transfer between drivers, reducing employee travel costs by integrating precise travel information, feasible deviations, and vehicle capacity within specific time intervals. The results show that ride-sharing can create up to 31.3% mileage savings and up to 28.7% reduction in the number of cars needed to fulfil employees’ travel schedules.

Recently, Wang et al.³² extended the traditional ride-sharing problem by allowing passenger transfers between vehicles at transfer stations. Using a collaborative decision framework that integrates deep reinforcement learning and integer programming techniques, they were able to overcome this problem. They used ILP to obtain the optimal online dispatching and matching strategy in each decision stage, and deep reinforcement learning to learn the approximate state value of each vehicle that incorporates some strategies to limit the state space and reduce the computational complexity. Performing numerical studies on the real-world trip dataset in Chengdu, they demonstrated that the proposed method outperforms several state-of-the-art methods and that ride-sharing with passenger transfer is more beneficial than traditional ride-sharing. This approach represents a forward-looking solution to enhance the efficiency and flexibility of ride-sharing systems.

The literature on passenger transfers highlights that integrating transfers within ride-sharing systems can enhance system efficiency. Optimizing transfer points and utilizing novel strategies can improve user experience while reducing costs and travel time. However, the primary research gap in this domain is the lack of practical and applied studies that examine passenger transfer between drivers, necessitating the development of mathematical models and algorithms to address this problem.

Research gap

The literature has provided valuable insights into the evolving landscape of ride-sharing services, encompassing technological, economic, cultural, and user satisfaction aspects. It has addressed key areas such as the integration of ride-sharing with public transit, vehicle transfers, understanding user preferences and satisfaction, and proposing mathematical models to optimize ride-sharing services and enhance user satisfaction.

Despite these advancements, existing literature has limitations regarding the practical application of passenger transfers and user preferences. Most studies have discussed these aspects theoretically without implementing them in a mathematical model. Specifically, while theoretical frameworks and conceptual discussions provide valuable insights, they often fail to translate into actionable strategies or algorithms that can be applied in real-world scenarios. This gap underscores the need for further research to develop and validate mathematical models that effectively integrate passenger transfers and user preferences, thereby enhancing the practical utility and operational efficiency of ride-sharing systems. Additionally, the lack of empirical studies and real-world applications highlights the necessity for case studies that can bridge the gap between theory and practice.

To tackle these research gaps, this study offers three main contributions: the introduction of the P-DM algorithm, which ensures that passengers and drivers with compatible preferences, such as smoking, music, and other criteria, are paired together. This approach not only increases user satisfaction but also improves the overall efficiency of the ride-sharing system by minimizing mismatches and optimizing the use of available vehicles; the development of a mathematical model that incorporates user preferences and passenger transfers. The model unequivocally ensures that the ride-sharing system can effectively accommodate users’ individual preferences and facilitate seamless passenger transfers between different drivers. By including these factors in the model, the study aims to enhance the practicality and operational efficiency of ride-sharing services; and the implementation of the IE-O algorithm, developed to generate high-quality solutions for real-scale examples in real-time. This algorithm is designed to handle the complexities and scale of real-world ride-sharing scenarios, providing robust and efficient solutions that can be implemented practically.

These innovations highlight the practicality and superiority of our approach compared to existing methodologies, offering a comprehensive and sustainable solution for ride-sharing systems. By incorporating user preferences and facilitating passenger transfers, the proposed model effectively bridges existing research gaps. The practical application of these algorithms in real-scale scenarios demonstrates their potential to revolutionize the ride-sharing industry, making it more user-centric and operationally efficient.

Theoretical framework

In this section, we explore two fundamental aspects that impact the efficacy and functionality of ride-sharing systems: passenger transfer and user preferences. Ride-sharing has revolutionized urban mobility by offering cost-effective and convenient travel options. However, we must examine passenger transfer and user preferences within this framework to maximize its benefits and address inherent challenges. We aim to elucidate these factors’ critical role in shaping ride-sharing systems. For this purpose, we will explore these concepts in detail in sections “Passenger transfer” and “User preferences”.

Passenger transfer

Consider the unweighted graph depicted in Fig. 1. The origin and destination of passengers A, B, and C are specified. Based on the positions of the vehicles in the network, passengers A and C can match with vehicles I and II, respectively. However, passenger B cannot travel due to a shortage of available vehicles. In this situation, ride-sharing with transfer can be a solution. Driver II can deviate slightly from the route to pick-up passenger B from their origin and drop-off passenger B at the transfer point before proceeding to the destination of passenger C. Subsequently, passenger B can continue their journey with driver I, who is already carrying passenger A. In this manner, passenger B can reach their final destination by transferring between drivers II and I.

Fig. 1.

Graph network depiction and vehicle route I and II.

The advantage of transfer in ride-sharing lies in its ability to efficiently and conveniently transport passengers to their final destinations, even when their routes do not perfectly align with a single driver. This provides a flexible transportation option for passengers with varying origins and destinations. Additionally, it offers a cost-effective and convenient mode of travel that contributes to traffic congestion reduction.

User preferences

The efficient matching of passengers to drivers presents a critical challenge, demanding meticulous consideration of diverse user preferences. The matching process entails a binary variable indicating whether a passenger is paired with a driver, taking the value of 1 if matched and 0 otherwise. The success of this matching process depends on satisfying the preferences of users. For instance, imagine a situation where one user is a smoker while the other prefers a non-smoking environment. Similarly, consider a scenario where one user enjoys alcohol while the other leads a sober lifestyle. In such cases, differing preferences can complicate the search for a compatible match.

The complexity of these challenges highlights the importance of understanding and incorporating user preferences into the ride-sharing framework. This requires considering various preferences and developing innovative solutions to address individual needs. Such an approach is crucial for enhancing modern ride-sharing systems’ overall efficiency and satisfaction. The classification of user attributes is depicted in Fig. 2.

Fig. 2.

Categorization of user attributes.

Given these intricacies, we propose an algorithm for efficient and personalized matching. Our proposed algorithm considers user preferences in the matching process to increase the overall service quality and satisfaction for passengers and drivers. The algorithm captures user preferences and attributes comprehensively, providing potential matches.

Algorithm 1 provides a detailed description and pseudocode of the P-DM algorithm tailored for the ride-sharing problem. Let represent the number of users an d represent the number of attributes.

Algorithm 1.

P-DM algorithm.

The P-DM algorithm is designed to match users by considering both their attributes and individual preferences. Its primary goal is to identify mutually compatible users based on multiple predefined attributes. The process begins by collecting inputs, including user attributes (such as gender, age, music preferences, etc.), along with their preferences for these attributes, as well as the total number of users () and the number of attributes () involved in the matching process.

The algorithm initializes an empty set of potential matches, denoted as , which will store the identified matches. Furthermore, for each user i, a corresponding set is created, initially including all other users as candidates for potential matches with user i. The algorithm then evaluates the preferences of user i for each attribute, one by one. If user i is indifferent to an attribute, no changes are made to . Otherwise, the algorithm deletes users who do not align with that preference, gradually refining for user i.

As the algorithm processes all users in this manner, each user’s is continuously updated to reflect only those users who satisfy all of their preferences. Once these sets are finalized, the algorithm performs a final check to ensure mutual compatibility. In other words, a user is only considered a potential match for another if the preference is reciprocated (both users must have identified each other as compatible based on their preferences). If this mutual agreement is found, the pair is added to the final set .

In summary, the P-DM algorithm functions by iteratively narrowing down the pool of potential matches for each user (), based on their individual preferences, while ensuring mutual compatibility before finalizing any pairing. Through this dynamic and preference-driven process, the algorithm efficiently identifies matches that are well-aligned with users’ stated preferences. An example is provided below for further clarification.

Consider the scenario involving passengers P1 to P4 and drivers D1, D2, and D3. Assume that information about these users in terms of five attributes (Smoking habit, Listening to music, Age, Favorite vehicle type, and Gender) is recorded in the system. The attributes of the users are listed in Table 1.

Table 1.

The attributes of users.

User	Attribute
	Smoking habit	Listening to music	Age	Vehicle type	Gender
P1	Smoking	No	22	–	Female
P2	Non-smoking	No	19	–	Male
P3	Non-smoking	Yes	36	–	Female
P4	Non-smoking	No	43	–	Male
D1	Non-smoking	Yes	26	Luxury	Male
D2	Non-smoking	Yes	51	Basic comfort	Male
D3	Non-smoking	Yes	30	Luxury	Female

Now, decisions must be made concerning potential ride-sharing partners based on each user’s preferences. The list of preferences for each user is provided in Table 2.

Table 2.

The preferences of users.

User	Preferences
P1	(Indifference, Indifference, 18–30, Luxury, Female)
P2	(Indifference, Indifference, 18–45, Indifference, Indifference)
P3	(Non-smoking, Yes, 31–70, Indifference, Female)
P4	(Indifference, Indifference, Indifference, Indifference, Indifference)
D1	(Non-smoking, No, Indifference, –, Male)
D2	(Non-smoking, No, 31–45, –, Male)
D3	(Indifference, Indifference, Indifference, –, Indifference)

Considering these preferences and user attributes, users capable of sharing a ride and non-feasible matches are determined in the preliminary phase before solving the mathematical model. For instance, P1 is indifferent to the ‘Smoking habit’ attribute; therefore, P1 is compatible with all passengers and drivers (P2, P3, P4, D1, D2, D3). However, on the condition that other users do not have a problem with P1 being a smoker.

Subsequently, users compatible with P1 based on the 'Listening to music’ attribute are considered. As P1 is indifferent to this attribute, P1 can share a ride with all users. The next step involves examining the ‘Age’ attribute; P1 prefers to share a ride with individuals aged 18–30. Therefore, based on the Age attribute stated in Table 1, P1 can ride with users (P2, D1, D3).

Continuing with the ‘Favorite vehicle type’ attribute, P1 prefers a luxury vehicle. Since both D1 and D3 have this attribute, as indicated in Table 1, P1 can still share a ride with users (P2, D1, D3). Finally, considering the ‘Gender’ attribute and P1’s preference for a female user, P1 can share a ride with D3.

The final row has been arranged to verify the compatibility between P1 and D3 and, in turn, whether D3 is mutually compatible with P1 in terms of the preferences. If both users align with each other based on their preferences, then P1 can match with D3 according to the mathematical model. A list of potential users who may share a ride is provided in Table 3.

Table 3.

List of potential users who could share a ride.

Attribute	Passengers				Drivers
	P1	P2	P3	P4	D1	D2	D3
Smoking habit	P2, P3, P4, D1, D2, D3	P1, P3, P4, D1, D2, D3	P2, P4, D1, D2, D3	P1, P2, P3, D1, D2, D3	P2, P3, P4	P2, P3, P4	P1, P2, P3, P4
Listening to music	P2, P3, P4, D1, D2, D3	P1, P3, P4, D1, D2, D3	D1, D2, D3	P1, P2, P3, D1, D2, D3	P2, P4	P2, P4	P1, P2, P3, P4
Age	P2, D1, D3	P1, P3, P4, D1, D3	D2	P1, P2, P3, D1, D2, D3	P2, P4	P4	P1, P2, P3, P4
Vehicle type	P2, D1, D3	P1, P3, P4, D1, D3	D2	P1, P2, P3, D1, D2, D3	Undefined	Undefined	Undefined
Gender	D3	P1, P3, P4, D1, D3	None	P1, P2, P3, D1, D2, D3	P2, P4	P4	P1, P3
Potential matches	D3	P4, D1	None	P2, D1, D2	P2, P4	P4	P1

Table 3 presents a list of potential users who could share a ride, with the last row indicating potential matches. Based on this information, constraints are established for preprocessing and can be input into the mathematical model. As shown in Table 3, P1 can only travel with D3, and P1 cannot be matched with any passengers. Consequently, the following restrictions can be entered into the mathematical model:

1

2

3

4

5

If equals 1 when passenger r is matched to driver k, otherwise, it equals 0; and equals 1 when passenger r is transferred, otherwise, it equals 0. Therefore, since P1 can only travel with D3, , and P1 cannot be transferred between drivers, hence .

Similarly, preprocessing constraints can be applied for P2 as follows:

6

7

8

9

Similar constraints are formulated for the remaining passengers, guaranteeing feasible matches within the ride-sharing context.

10

11

12

13

14

15

16

The goal is to maximize users’ overall satisfaction by matching them appropriately with passengers and drivers based on their individual preferences.

The ride-sharing problem with transfer and preferences

This section provides a mixed-integer nonlinear programming model to solve a ride-sharing problem with transfers optimally. It is assumed that V vehicles serve P passengers, and each passenger has an origin and destination point, denoted by o(r) and d(r), both of which are associated with time windows. Passengers can transfer between drivers, and the maximum number of trips a passenger can take to the destination is two. Trip costs and distances are calculated based on the network topology.

In addition, both drivers and passengers are required to provide information, such as their preferences and attributes, before participating in the service. This ensures that the system has a comprehensive understanding of their characteristics. Moreover, the model is executed at regular intervals, such as every 30 s. During each interval, new passengers are dynamically added to the system, and existing matches are updated accordingly. This dynamic approach reflects the real-time behavior of ride-sharing systems, enabling the system to respond effectively to new ride requests while maintaining operational efficiency.

Given a directed graph G = (Ns, E), where Ns denotes the set of nodes, E represents the set of edges, O signifies the set of origin points of vehicles, and . Let represent the number of seats demanded by passenger r, and Cap denotes the vehicle capacity. Additionally, denotes the trip costs from point i to j, whereas indicates the trip distance from point i to j. Each passenger is associated with the earliest pick-up time and the latest drop-off time . Furthermore, the transport income from passengers r is represented by , and transfer time by Tt. Finally, the decision variables include the following.

	1 if passenger r is matched to driver k; otherwise, 0
	1 if driver k travels from point i to j; otherwise, 0
	1 if passenger r is transferred; otherwise, 0
	1 if driver k takes passenger r from point i to j; otherwise, 0
	Non-negative variable indicating the time at which vehicle k reaches node i

The problem can be formulated as the following model.

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30

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33

Building upon the studies, the main focus of research in the field of ride-sharing problems has been the optimization of driver-passenger matching. As a result, the objective function (17) in this model is formulated to maximize the number of matching instances and, subsequently, enhance overall profit.

To ensure efficient service, constraints (18) guarantee that a passenger can reach their destination with at most two drivers. Vehicles cannot travel simultaneously to two nodes or reach a single node from two different edges due to constraints (19) and (20). Equations (21) and (22) represent the limits for flow conservation and path continuity for the vehicle. Specifically, constraint (22) indicates that drivers can exit a node only if they have previously entered it. Under constraint (23), a vehicle is allowed to depart from the origin only if it has been matched with a passenger. Inequalities (24) state that the vehicle’s capacity is not exceeded.

Restrictions (25) and (26) specify that passengers must be picked up at their origin and dropped off at their destination. Inequalities (27) and (28) indicate that a vehicle can take a path if a passenger is matched with it. Constraints (29) and (30) define time windows for pick-up and drop-off. According to constraint (31), drop-off points should be reached after pick-up points. Restriction (32) is associated with passenger transfer time. Furthermore, constraint (33) imposes valid values for binary and non-negative variables.

This mathematical model is not linear due to constraints (30), (31), and (32). To linearize these constraints, we can introduce auxiliary variables and additional linear equations that represent the relationships between these variables. This allows us to transform the original nonlinear constraints into a linear form that can be easily solved using linear programming techniques.

The problem in using (30) is that the product of and causes the constraints to be non-linear. While this is not generally true for any product of variables, a product involving a binary variable and a bounded continuous variable can always be linearized³³. Note that is binary and , hence linearization is possible.

Let us introduce the continuous variable which will be equal to the product of and for every integer solution.

30a

30b

30c

We can now replace (30) by its linear counterpart:

30new

Note that the W-variables only occur in (30a)–(30c). It follows that in the LP relaxation, it is optimal to pick as small as possible, to maximizes the slack of (30); picking a larger value for the W-variable only restricts the range of . Hence, it is not necessary to bound from above (30c). The full linearization of (30) is then given by:

30new

30a

30b

The problem in using (31) is that the product of and causes the constraints to be non-linear. Note that is binary and , hence linearization is possible (Let and ).

Let us introduce the continuous variable which will be equal to the product of and and which will be equal to the product of and for every integer solution. The full linearization of (31) is then given by:

31new

31a

31b

31c

31d

31e

31f

31g

31h

Constraint (32) is also nonlinear and can be linearized, the problem in using (32) is that the product of , and causes the constraints to be non-linear. Additionally, the multiplication of by constitutes the second part of the constraint, rendering the constraint nonlinear.

Let us introduce the continuous variable which will be equal to for every integer solution. The full linearization of is then given by:

32a

32b

Let us introduce the continuous variable which will be equal to the product of and and which will be equal to the product of and for every integer solution. The full linearization of (32) is then given by:

32new

32a

32b

32c

32d

32e

32f

32g

32h

32i

32k

With the linearization of the mathematical model complete, extensive tests were conducted across various scenarios to validate its efficacy. The next section analyzes the model’s performance with and without preferences, and then introduces an algorithm as the solution method.

Solution methodology and comparative analysis

To evaluate the effectiveness of the proposed mathematical model, we conducted extensive tests across multiple scenarios. we analyzed 10 instances featuring diverse user sizes and graph network configurations. The findings have been summarized and are presented in Table 4. The table outlines various parameters for each instance, including the number of passengers, vehicles, and edges in the network.

Table 4.

Evaluation results of the mathematical model without preferences.

Instance	P	V	Edges	Objective	CPU (s)
1	6	4	30	7.8	54.78
2	7	4	30	10.4	90.02
3	12	4	30	21.8	153
4	7	4	31	13.1	44.66
5	8	5	60	3.35	36.31
6	12	8	60	8.8	73.09
7	15	5	60	14.75	13.75
8	7	5	70	5.2	60.95
9	11	5	76	10.2	45.73
10	4	5	88	2.6	273.4

A PC with a Core i5 processor running at 2.5 GHz and 8 GB of RAM runs the instances. The optimization problems have been encoded in OPL and solved using CPLEX version 12.1.00.

According to Table 4, the computational time of the mathematical model increases significantly with an increase in edges. For example, in instances 10 and 4, with 88 and 31 edges respectively, the model’s computation times are 273.4 and 44.66 s, demonstrating a clear increase with higher edge counts. Generally, a higher number of edges indicates a more complex problem, which challenges the model with more difficult scenarios and adversely affects execution time. Therefore, to address this issue, we will proceed to compare the model, considering preferences.

Impact of preferences on mathematical model performance

This section provides a detailed comparative analysis of our mathematical model’s performance under two conditions: with and without considering preferences. The attributes considered for users include gender, vehicle type, age, music preferences, and smoking habits. Therefore, based on each user’s preferences, the constraints related to user preferences are created according to the output of the P-DM algorithm and have been added to the mathematical model.

Table 5 shows results for both the model without preferences and the model with preferences, covering objective values, computational time, transfer status, and user satisfaction levels. The transfer column in each example’s optimal solution indicates whether a passenger reached the destination via a transfer between drivers, denoting if a transfer occurred.

Table 5.

Comparative analysis of mathematical model performance with and without preferences.

Instance	P	V	Edges	Model				Model with preferences
				Objective	CPU (s)	Transfer occurred	Average satisfaction (%)	Objective	CPU (s)	Transfer occurred	Average satisfaction (%)
1	6	4	30	7.8	54.78	Yes	95	7.7	1.25	Yes	100
2	7	4	30	10.4	90.02	Yes	93.3	7.7	16.59	Yes	100
3	12	4	30	21.8	153	Yes	94	21.7	0.89	Yes	100
4	7	4	31	13.1	44.66	Yes	90.7	7.3	19.8	Yes	100
5	8	5	60	3.35	36.31	Yes	91.25	2.85	1.22	Yes	100
6	12	8	60	8.8	73.09	Yes	96.6	8.8	31.73	Yes	100
7	15	5	60	14.75	13.75	Yes	93.75	12.8	0.77	No	100
8	7	5	70	5.2	60.95	Yes	88.75	3.95	1.84	No	100
9	11	5	76	10.2	45.73	Yes	90	8.95	1.22	Yes	100
10	4	5	88	2.6	273.4	No	98.3	1.75	0.78	No	100

User satisfaction is contingent upon the outcome of the matching process. Specifically, if a user is matched with another user whose preferences differ, their satisfaction decreases by 5% for each incompatibility (take this as an assumption). For example, let’s consider passenger X, a non-smoker and non-drinker, who prefers to share a ride with someone with similar attributes. If X is matched with passenger Y, who is a smoker and alcohol consumer, X's satisfaction with the trip decreases by 10%. Notably, the satisfaction level in these tables represents the average users’ satisfaction for each instance.

The results of the observations show that considering user preferences significantly influences objective values and satisfaction levels. The without-preference model generally achieves better objective values, while the preference model consistently attains higher user satisfaction rates, underscoring the importance of user preferences in ride-sharing network optimization.

Additionally, our model revealed a substantial decrease in computational time when user preferences were considered. This reduction highlights the efficiency improvements obtained by integrating user preferences into the mathematical model. This finding underscores the practical advantages of incorporating user preferences in ride-sharing optimization, demonstrating its potential for enhancing computational efficiency and overall model performance.

In the following, upon scrutinizing the results of instances in both scenarios, with and without user preferences, it was discerned that certain edges were recurrently utilized in the optimal routes of the vehicles. This phenomenon emphasizes the inherent efficiency of ride-sharing, as the repetition of edges facilitates the optimization of travel paths, thereby reducing overall travel time and resource utilization. Additionally, it is noteworthy that these repeated edges are part of the minimum spanning tree, further emphasizing their significance in the optimization process. Furthermore, it was consistently observed that some passengers were served repeatedly within the system in all instances, indicative of the effectiveness and reliability of the implemented ride-sharing framework.

To further analyze the results, we will examine three indicators: User Satisfaction Improvement (USI), Computational Time Improvement (CTI), and Income Deviation (ID). These indicators will illustrate the superiority of the preference-based model compared to the model without preference.

USI measures the percentage increase in user satisfaction achieved by employing the model with user preferences compared to the model without preferences. This indicator provides crucial insights for managerial assessment, elucidating the efficacy of integrating user preferences and its impact on customer satisfaction and retention strategies.

CTI quantifies the CPU time ratio between models with and without preferences, guiding managers in assessing the computational efficiency consequence of user preferences for resource allocation and performance optimization.

Additionally, ID delineates the percentage decrease in objective value in the model with preferences, further elucidating the financial implications of integrating user preferences and its effect on customer retention strategies.

Here, Fig. 3 shows the values of the indicators, including USI, CTI, and ID. This scatter plot illustrates the comparative performance of the preference-based model and the model without preference across 10 instances. The y-axis shows the CTI in percentage. Each point on the plot corresponds to an instance and is sized by the USI percentage and colored by the ID percentage.

Fig. 3.

The impact of integrating preferences on the efficiency and income.

Larger circles indicate higher user satisfaction improvements, demonstrating the significant impact of considering user preferences. Points at the top of the y-axis indicate higher computational time improvements, highlighting the efficiency gained by incorporating user preferences into the model. The color gradient from light gray to black shows the range of income deviation, where darker colors represent lower deviations from the optimal solution.

As shown in Fig. 3, integrating user preferences into the mathematical model can offset the decrease in income with improvements in computational time and user satisfaction. This insight empowers managers to devise informed strategies to improve operational efficiency and achieve customer-centric outcomes. Finding a trade-off between these objectives is a fundamental challenge in optimizing ride-sharing problems.

IE-O algorithm

After presenting the mathematical model, the IE-O algorithm is discussed. The aim is to assess whether this algorithm can offer more effective solutions and improve results. The algorithm is tested on various problems, and the outcomes are compared with those of the mathematical model. If the results demonstrate improvement, it can be concluded that the algorithm is effective for these problems.

The IE-O algorithm is designed to iteratively enhance the solution of the mathematical model by integrating passengers one at a time. The algorithm starts by solving the model for the first passenger, adjusting variables, and then proceeds by solving the model for each new passenger. In each iteration, the current solution is evaluated and compared to the best objective value Z obtained so far, allowing for continuous improvement as the algorithm progresses.

In each iteration, the binary decision variables X, U, and Y are obtained from the model and fixed for the subsequent iterations. This procedure ensures that decisions made in earlier stages remain consistent as new passengers are introduced into the system. The iterative nature of the algorithm allows for incremental improvement. The IE-O Algorithm, along with its corresponding pseudocode, is detailed in Algorithm 2, where p represents the number of passengers.

Algorithm 2.

IE-O algorithm.

In the following, the efficacy of the IE-O algorithm is assessed through 10 examples, with comprehensive results documented in Table 6. These results demonstrate its performance across diverse scenarios.

Table 6.

Comparative analysis of IE-O algorithm performance with and without preferences.

P	V	Edges	IE-O algorithm					IE-O algorithm with preferences
			Objective	Gap (%)	CPU (s)	Transfer occurred	Average satisfaction (%)	Objective	Gap (%)	CPU (s)	Transfer occurred	Average satisfaction (%)
6	4	30	7.8	0	3.23	Yes	93	7.7	0	3.07	No	100
7	4	30	9	13	4.54	Yes	94	7.6	1.3	4.3	No	100
12	4	30	21.8	0	4.55	Yes	92	21.7	0	3.09	Yes	100
7	4	31	12.7	3	4.96	Yes	94.3	5.7	21.9	4.33	Yes	100
8	5	60	3.3	1.5	16.25	Yes	96.25	2.85	0	13.45	Yes	100
12	8	60	8.8	0	70.6	Yes	93.3	8.8	0	41.34	No	100
15	5	60	14.7	0.3	14.83	No	96.25	12.8	0	13.66	No	100
7	5	70	5.2	0	14.89	Yes	90	3.4	13.9	13.16	No	100
11	5	76	9.7	4.9	17.28	Yes	92.5	8.4	6.1	16.98	No	100
4	5	88	2.6	0	8.64	No	96.6	1.75	0	7.94	No	100

Table 6 provides significant insights into the comparative performance of the IE-O algorithm under various user preference conditions across different instances. The results suggest that the algorithm without preferences generally achieves higher objective function values than the algorithm with preferences. Additionally, the computational times for both modes are almost identical, indicating that considering user preferences does not significantly impact the algorithm’s complexity.

Furthermore, the gap between the objective function values obtained by the algorithm and the optimal solution is negligible in most instances in both modes. This implies that while the algorithm may not consistently achieve the optimal solution, the difference between its solution and the optimal solution is minimal, reinforcing the practical effectiveness of both approaches.

Figure 4 illustrates the percentage of CTI, USI, and ID within the IE-O algorithm with preferences, compared to the IE-O algorithm without preferences.

Fig. 4.

The impact of integrating preferences on the efficiency and income of the IE-O algorithm.

As shown in Fig. 4, considering preferences has led to increased user satisfaction (larger circles) in most cases. Additionally, in many instances, such as instances 1, 3, 5, and 6, incorporating preferences has reduced computational time (points higher on the y-axis). The darker circles in the chart also indicate lower deviations from the optimal solution.

According to these findings, considering preferences may result in lower income, but could enhance system performance in some ways. Therefore, while maximizing income is important, risk management is crucial to ensuring long-term success and operational effectiveness.

Integrated analysis of model and algorithm results

Following examining the mathematical model and IE-O algorithm, it becomes evident how integrating user preferences into ride-sharing problems with transfer impacts the solution. In this section, an extensive comparison is conducted of all methods: the model without preferences, the model with preferences, the IE-O algorithm, and the IE-O algorithm with preferences. For this purpose, consider Fig. 5, which demonstrates the superiority of the CPU time of the model with preferences in comparison to other methods (the model without preferences, the IE-O algorithm, and the IE-O algorithm with preferences).

Fig. 5.

Comparing computational times: model and the IE-O algorithm with and without preferences.

Upon examination of Fig. 5, a notable trend emerges: the model with preferences demonstrates a substantial reduction in CPU time and attains the 1st rank compared to the other methods in 8 out of 10 instances. Blue denotes the CPU time of the model with preferences, whereas gray denotes the CPU time of the model without preferences, the IE-O algorithm, and the IE-O algorithm with preferences.

One of the key advantages of the model with preferences is the reduction in computational time. By incorporating preferences, the feasible region is reduced, which lowers the model’s computational complexity and, consequently, the solution time. Furthermore, increasing user satisfaction is another strength of our model. By providing a customized experience for users, our model helps attract and retain long-term customers.

In traditional ride-sharing problems, matching is feasible only if the driver picks up passengers at the departure point with the same route, preferences, and schedule. Our model addresses this limitation by enabling the transfer of passengers, which can be particularly beneficial in complex routing scenarios. This flexibility enhances the user experience and overall satisfaction. Moreover, the proposed model can be implemented on simple servers. Additionally, its scalability allows for more efficient use of computational resources when additional cores are available, further enhancing performance and applicability.

In summary, our model offers shorter computational times, reduced complexity, and increased user satisfaction compared to the model without preferences and the IE-O algorithm, making it highly feasible for practical implementation across various scales. This information provides transportation managers with invaluable guidance in decision-making processes, enabling them to make informed choices that optimize operational effectiveness and resource allocation strategies.

Managerial recommendations

Initially, the advantages of increasing user satisfaction are evident as the model with preferences prioritizes user attributes, resulting in improved satisfaction levels. Subsequently, the reduction in response time to users is noteworthy, indicating a more responsive and efficient system that caters to user needs promptly. Finally, the increased demands and income stem from a more user-centric approach, attracting more users to the ride-sharing platform and ultimately boosting revenue.

Furthermore, although the direct environmental benefits were not measured in this study, potential improvements can be inferred. With the transfer conditions, the system operates more efficiently, making better use of vehicle capacity and transporting more passengers per trip compared to standard conditions. This efficiency can reduce emissions and fuel consumption, which shows a positive impact on environmental sustainability.

In addition, it is important to address the limitations and practical challenges associated with implementing ride-sharing systems in densely populated urban environments. One of the main challenges is managing large-scale data and its impact on response times for each request. Rapid responses are critical for ensuring system efficiency. In this study, data is updated at short intervals to allow the system to optimize automatically and respond promptly to new requests.

To further improve response times during periods of congestion, optimization methods such as Best Known Solution are recommended. These techniques enhance system performance by managing resources and time more efficiently, especially during peak demand conditions. Additionally, geographical segmentation offers an effective solution for handling large-scale data. By processing requests independently in each segment, the system can reduce the load and provide faster responses across different geographical areas. This approach improves scalability, particularly when the volume of requests increases significantly.

Another key challenge in implementing ride-sharing systems is the collection of user data. To ensure optimal matching, it is essential to gather data on user preferences during the registration phase. The platform must be designed to capture and store this information accurately in user profiles, ensuring the system can deliver personalized services. Moreover, ensuring user privacy is a key challenge in ride-sharing systems, as they involve the sharing of sensitive information, such as user locations and personal data. Protecting privacy and securing data are essential. Blockchain technology has emerged as a potential solution to these issues, playing a crucial role in safeguarding sensitive information and fostering trust in ride-sharing systems.

These insights are invaluable for transportation managers, providing them with a holistic understanding of the benefits of incorporating user preferences into ride-sharing systems. With this knowledge, managers can make informed decisions to maximize operational efficiency and refine resource allocation. Furthermore, they can use these insights to develop methodologies that enhance ridesharing systems’ computational efficiency while improving user satisfaction.

Conclusion

In this research, we examined the ride-sharing problem with transfer, emphasizing the significance of user preferences. Our study explored the implications of incorporating user preferences into ride-sharing problems and for this purpose, the P-DM algorithm is presented.

Based on the observations, user preferences significantly influence objective values and satisfaction levels. This highlights the need for a trade-off between efficiency and user-friendliness in ride-sharing systems. Moreover, our investigation showed a notable decrease in computational time when preferences were considered in the model. In addition to improving user satisfaction, this efficiency improvement simplifies computational processes, increasing model performance. Furthermore, our examination of the IE-O algorithm across various scenarios illuminates the interplay between computational efficiency and user-centric optimization. Based on the results, the model with preferences achieved the first rank in 8 out of 10 examples, compared with the other methods, and it could increase user satisfaction by 100%.

These findings have implications for transportation planners and guide them in selecting methodologies that effectively balance computational time with user satisfaction, thereby paving the way for more responsive and sustainable urban transport frameworks in the future. These results are based on observations and may change with new data. In the future, it is essential to prioritize user preferences. This might mean organizing preferences into required and optional categories and incorporating them into matching procedures. Such an approach holds the potential to enhance and optimize ride-sharing systems, ensuring they remain attuned to the evolving needs of users while maintaining computational efficacy. In addition, ride-sharing systems can better tailor their services to meet users’ needs, resulting in greater loyalty. Additionally, it could lead to greater competition in the industry, as companies strive to offer better services and more personalized experiences.

Funding

There is no funding for this research work.

Data availability

The data that support the findings of this study are available from the corresponding author, H. K, upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.