Crowd-Starting a Shared (Shuttle) Service With Customer Suggestions

Abstract

Digital platforms have improved the efficiency and quality of smart city operations by soliciting more customer inputs, for example, in the form of suggestions. One innovative option in urban transportation is the shared shuttle service, which lies between traditional public transportation and ride-hailing services. Platforms that offer these services can gather customer suggestions in a “crowd-starting” manner, which provides valuable insights into customer needs. However, this also presents a challenge in balancing service coverage and quality to meet customer needs implied by their suggestions. To address this issue, we introduce an optimization framework designed to maximize expected profit by leveraging customer response models which characterize how customers will respond to different service attributes and how their suggestions inform these responses. When estimating these response models, we present methods involving isotonic penalty and shrinkage tailored for handling small datasets. To demonstrate the practical implications, we apply our model to a shared shuttle service case study and discuss practical considerations, such as the value of information, the effectiveness of our estimation approaches, and the benefits of involving customers in the service design process.

Keywords

Shuttle service design smart city operations data-driven optimization customer suggestions

1. Introduction

Smart cities are increasingly using customer/citizen data to improve their operations, particularly in the realm of urban mobility. With the growth of ride-sharing services, customers are now more actively involved in shaping the transportation options available to them. They share personal travel information with platforms that either provide services directly or work with suppliers to offer the service. This customer engagement allows the platform to improve service quality or even create new services. The exchange of data between customers and service providers is a key factor in this process, as it enables the platform to better understand the needs and preferences of its customers and to design services that meet their specific requirements.

The use of customer engagement in the design of smart city services has been demonstrated through the case of Beeline, a crowd-starting transportation platform that offers shared shuttle services in Singapore (Hasija et al., 2020; Mak, 2022). Beeline involves customers in the design process by allowing them to suggest new routes, which are then leveraged by the platform to propose new shuttle services. If a new proposal receives sufficient customer adoption within a specific timeframe, it will be implemented for long-term operations. This approach, known as crowd-starting, empowers customers to play a role in the design of the service and allows the platform to create services that meet the needs and preferences of its customer base.

Shared shuttle services, such as LyftShuttle in San Francisco (Figure 1), have been around for some time. These services, which operate similarly to public transportation with many stops, offer a convenient and affordable transportation option. Beeline, on the other hand, is a region-to-region service (Figure 2) that only has stops in the pickup and dropoff regions, targeting daily commuters between residential and commercial areas. This allows Beeline to provide competitive service quality at an affordable price compared to ride-sharing services. It is worth noting that LyftShuttle designs its routes purely leveraging its enormous amount of historical ride-sharing data, while Beeline discovers new demand with individual preferences through customer suggestions. This highlights the importance of customer engagement in the design of shared shuttle services.

To design a new shuttle service that optimizes customer adoption, it is imperative to strike a balance between two critical factors: service coverage, which represents the potential customer base interested in the new route, and service quality, denoting customer satisfaction upon using the service. A lengthy route with substantial detours has the potential to reach more locations, thereby reducing the walking distances for customers; however, it may also result in increased travel time for all passengers on board. To effectively manage this trade-off and craft efficient routes, the platform solicits customer inputs as a means of gathering additional insights for the design process. This initiative serves to enhance the platform’s comprehension of the distinctive needs and preferences of its customers, thereby enabling the design of a service that not only caters to a broader customer base but also elevates overall satisfaction levels among customers.

1.1. Research Question and Main Results

Motivated by the business model of Beeline, this article aims to study how customer engagement should be leveraged to design new shared shuttle services that attract more customer adoption. Following the chart in Figure 3, we illustrate the flow of operations and activities. A customer enters the platform and first browses routes by providing their travel information (e.g., home location and office location). The existing routes (can be none) that match the travel information are presented to the customer (Figure 4). If satisfied with any route, the customer joins the route and adopts the service (Figure 5). If there is no desirable route, the customer either leaves or suggests a route. When suggesting, the customer is first prompted to choose feasible pickup and dropoff locations. These locations may differ from the original travel information because door-to-door services are usually not efficient for shared shuttles, and only specific locations are designated as shuttle stops. In the case of Beeline, only nearby bus stations (e.g., Punggol Town Center in Figure 4) can be candidate pickup and dropoff locations. Moreover, the suggestion also includes the departure and arrival times at pickup and dropoff locations, respectively. By collecting sufficient suggestions, the platform may be motivated to design a new route (e.g., a full schedule of pickup and dropoff stops) and make it public on the platform. The customer will be notified if a new route is designed related to the suggestion and make an adoption decision regarding this new route.

Integrating customer feedback into the service design process presents inherent challenges. As previously mentioned, each customer’s input essentially equates to a highly personalized route proposal. Given the inevitable diversity in these suggestions, it becomes a challenging task to seamlessly align them with the various requirements while creating a new shared route. Thus, understanding the uncertain adoption behavior, stemming from the disparities between suggested routes and designed routes, becomes imperative.

In addressing this challenge, Beeline initially adopted a somewhat simplistic approach, presuming that customers would readily adopt the service as long as the deviations from their suggestions were not overly significant, as documented by Sim (2017). Unfortunately, this approach failed to garner substantial adoption for many of the newly proposed routes.¹ The inefficiency of the crowd-starting process in the absence of robust demand was a significant factor contributing to Beeline’s decision to cease operations in January 2020, after nearly 5 years of service. In accordance with a report by CNA (2019), in 2017, the service had 130 routes and 19,000 monthly bookings, that was less than five bookings per route per day. To lay the foundation for the viability of future crowd-started shared shuttle services, this paper delves into the challenge of shuttle service design involving customer suggestions, incorporating models of uncertain adoption with the ultimate goal of maximizing service profitability. Our main results are summarized as follows:

Our proposed model synthesizes individual customer preferences implied from their suggestions with communal elements tailored for the local market. Through a comparison with two benchmarks, we delineate the value of information (VOI) from customer input in two key aspects: the value obtained from gathering travel information of potential customers (e.g., from their browsing history), and the value obtained from understanding individual route preferences (e.g., from customer suggestions). This decomposition of VOI allows us to explore the ways in which service design can leverage customer inputs.

We employ the random utility framework and adopt a non-parametric approach to characterize customer response models for service design optimization, providing model flexibility as well as computational tractability. This approach accommodates three scenarios: the straightforward case where customers share the same preferences and truthfully express their preferences in suggestions, as well as the intricate cases where customers possess varying preferences and encounter additional randomness in their suggestions, making it less informative for their later adoption decisions.

Inspired by non-parametric regression literature, we have developed efficient techniques to estimate these response models. These methods incorporate two types of regularization: isotonic penalty and shrinkage. Both approaches embrace the concept of data pooling, leading to more reliable estimates. Moreover, we address concerns related to endogeneity by providing a solution for de-biasing, ensuring the robustness of our analyses.

In our case study on shared shuttle service design, we conducted extensive simulations and summarize valuable insights into the crowd-starting process: First, our proposed model designs shuttle routes that cautiously tailor to customer suggestions without adversely affecting others, leading to higher expected profits. Second, the VOI becomes more pronounced when demand is widely distributed across the region. In particular, the value of understanding individual preferences is pertinent when customers have heterogeneous preferences. Third, the power of pricing decisions brings an additional amount of VOI that is indirect: more personal information enables higher personalized services, allowing for increased prices and thus profits. Finally, while isotonic penalty and shrinkage serve distinct roles as regularization methods, both offer substantial benefits in various scenarios, particularly when dealing with limited sample sizes.

1.2. Related Works

This work is related to various topics in operations management, transportation, marketing, and economics. Smart city initiatives aim to improve the quality of life and increase sustainability for citizens through data-driven, technology-focused initiatives. This has led to a proliferation of research topics in operations management related to these efforts, including electric vehicles (Mak et al., 2013; Avci et al., 2015), autonomous vehicles (Mirzaeian et al., 2021; Reed et al., 2022; Siddiq and Taylor, 2022), last-mile services (Qi et al., 2018; Fatehi and Wagner, 2022; Mao et al., 2022), ride-sharing (Feng et al., 2021; Benjaafar et al., 2022; Cohen et al., 2022), carpooling (Cohen et al., 2023; Wang and Zhang, 2022), among others.

In this article, we focus on the business models in shared mobility. Modern ride-sharing platforms normally provide carpooling services with a small capacity (e.g., up to four). Santi et al. (2014) and Alonso-Mora et al. (2017) investigated the opportunity of ride-sharing with a larger capacity by analyzing the trade-off between the collective benefit of sharing and the personal discomfort of passengers. They show that within moderate waiting time or delay for customers, most demand can be well covered by much fewer vehicles with a capacity of more than four. Further, Vazifeh et al. (2018) addressed the minimum fleet problem for the high-capacity shared mobility system. These works justify the practical motivation of Beeline and also highlight the subtlety in balancing personalization and sharing in our model. There are some attempts to combine user data and shared shuttle design. Tong et al. (2017) optimized the assignment and routing decisions in the customized bus service while satisfying individual demand requests. Instead of proactively eliciting the demand information from customers, Qiu et al. (2018) extracted demand patterns from historical passenger trip data and then designs new customized routes. These works assume demand is deterministic from the source data and focus on solution methodologies, especially routing algorithms. However, the service quality of each customer depends on how the service/route is designed, which further affects the willingness to adopt. Therefore, it is crucial to carefully characterize the uncertain adoption behavior.

Motivated by the stream of literature in marketing and economics studies on how true (revealed) preferences can be better understood when customers state their preferences, we employ two related random utility models to study the connection between adoption and suggestion. Wardman (1988) uncovered a strong correlation between the revealed preference and the stated preference in modeling travel behavior. Adamowicz et al. (1994) and Ben-Akiva et al. (1994) managed to combine the revealed preference and the stated preference in different ways for better choice modeling. Moreover, Parker et al. (2018) studied how citizens respond to an economic policy by showing that the stated preferences are valuable in predicting behavior and in estimating population aggregates. Different from this stream of literature that focuses on estimating customer preferences at the population level through structural models, we adopt a non-parametric approach. This approach considers suggestions as an extra lever to leverage the diversity in customer preferences for optimizing service design. Furthermore, this non-parametric method allows for varied preference lists among customers, which can be regularized using an isotonic penalty as a soft constraint.

The shared service design that optimizes customer adoption is, in spirit, close to the single product design that aims to maximize the total market share. By parameterizing the utility values for the separate parts of the product (assigned to the multiple attributes) to model the choice decisions, many optimization algorithms have been developed for solving the single product design problem. Early works like Kohli and Krishnamurti (1987) and Balakrishnan and Jacob (1996) proposed heuristic methods and later Camm et al. (2006) and Shi et al. (2001) studied exact but more sophisticated methods. Recently, Akçakuş and Mišić (2021) derived an exact approach for standard logit-based share-of-choice product design by reformulating the problem into mixed-integer convex programming. Our paper utilizes non-parametric modeling for adoption decisions and also takes suggestions into consideration. There are various extensions to the single product design, such as dynamic models in new product development (e.g., Zirger and Maidique, 1990) and mechanism design models in product line design (e.g., Moorthy, 1984) which are beyond the current scope of our paper. However, these topics represent interesting areas for future research.

Our estimation framework builds on the non-parametric regression. Brunk (1955) and Brunk et al. (1972) initiated the study of non-parametric regression with monotone constraints by providing necessary statistical properties. Mukerjee (1988) later proposes a new procedure that leads to monotone estimators with better asymptotic properties. Chatterjee et al. (2018) studied the risk bound of least squares estimators for an unknown matrix from noisy observations under coordinate-wise monotone constraints. Han et al. (2019) extended the analysis to the case with higher dimension coordinates. Beyond point estimates, Deng et al. (2021) studied the problem of constructing confidence intervals to guide inference. Besides, Naeini and Cooper (2016) used nearly isotonic regression to calibrate probabilistic models where monotone constraints are relaxed such that the loss function only penalizes adjacent pairs that violate the monotonicity. By mapping the service design into customer’s service attributes, our non-parametric approach allows for heterogeneous preference lists among customers. Therefore, in our model calibration of customer response models, we relax the monotone constraint by imposing an isotonic penalty as a soft constraint.

Lastly, there has been significant recent progress in efficiently estimating multiple models with limited data by balancing personalization and pooling/sharing. This progress motivates our estimation procedure to include a shrinkage term in addition to the abovementioned isotonic penalty. For example, Aouad et al. (2023) proposed market segmentation trees for learning personalized response models, using a similar characterization of conditional response models. Their method adaptively generates the segmentation based on customer covariates and estimates corresponding response models to maximize prediction accuracy. In the context of demand prediction, Cohen et al. (2022) considered the problem of estimating predictive models for multiple products and proposes a data aggregation approach to enhance sample efficiency. Their method manages to identify whether a linear coefficient in the predictive model should be a customized parameter for each product, a partially shared parameter for several products, or a globally shared parameter for all products. When estimation is further embedded in stochastic optimization problems, Gupta and Kallus (2022) showed that it can be beneficial to couple independent stochastic optimization problems via a shrunken approach, highlighting the value of data pooling in a small-data regime. Lei et al. (2022) examined the benefit of data pooling in the practice of online retailing by treating the top-level sales information as a regularization for fitting the bottom-level prediction model.

The remainder of this article is organized as follows. We first discuss the platform’s problem in Section 2 by introducing the essential components for a service design. We then characterize the customers’ problem in Section 3. In Section 4, we discuss how to calibrate the models using data. After investigating a case study of shared shuttle service in Section 5, we conclude the article and discuss potential future improvements in Section 6.

2. Crowd-Starting Service Design

We study a platform that crowd-starts a new shared shuttle service for customers commuting from a residential area to a business area where a similar shuttle service is not yet available. As described in Section 1, customers browsing the platform provide their travel information that contains the origin and destination locations denoted by $w$ . A customer not seeing a suitable route may also provide a suggestion denoted by $s$ , for example, a suggested route including pickup and dropoff locations and times, which essentially conveys the customer’s individual service preferences. The platform plans a fixed service $θ$ that is offered to all customers as a universal service, for example, a shuttle route with designated stops and a specific timetable.

Although everyone sees the same shuttle route, customers may have different evaluations of $θ$ depending on their $w$ and $s$ . If the service design fits the suggestion very well (e.g., $θ$ contains the pickup and dropoff locations in $s$ with similar times), then a customer is likely to be satisfied; If the service design deviates from the suggestion (e.g., $θ$ leads to a significant detour compared to $s$ or there is a long walking distance from the origin location in $w$ to the stops in $θ$ ), then a customer may not be pleased. We use $Y \in {0, 1}$ to indicate whether a customer will adopt the service or not. Note that this response $Y$ is random to the platform. We define $P (Y = 1 | s, w; θ)$ as the conditional response model—the adoption probability given some designed route $θ$ and the suggested route $s$ for those with OD information $w$ . Besides proactively contributing suggestions to the platform, customers may also browse and reactively adopt the existing services. We define $P (Y = 1 | w; θ)$ to be the marginal response model without conditioning on one’s suggestion.

2.1. Platform’s Objective

Suppose there are $K$ customers who contribute their suggestions. Each customer $k \in [K]$ reveals their own OD information $w_{k}$ and provides their suggestion $s_{k}$ . Following the sequence of events in Figure 6, the platform designs the service taking into consideration these $K$ customer suggestions as well as being prepared for potential future customers. After the new shuttle route is designed, in addition to the $K$ customers who suggested, there will be $K^{'}$ future customers who would consider the new shuttle service but did not suggest. Because the platform may not precisely know a future customer’s travel information, we assume it possesses knowledge about the distribution denoted by $P (W = w)$ . Thus, the adoption probability of a future customer in this group can be characterized by $E_{W} [P (Y = 1 | W; θ)]$ using the marginal response model.

Let $r$ be the fixed price charged to all customers for the shuttle service (e.g., Beeline charged $5.6 per trip in Figure 5) and $c (θ)$ be the cost of operating the service $θ$ (e.g., the travel distance to cover and the fixed costs for the vehicle and the driver). The platform designs a shuttle route by the following optimization problem to maximize its expected profit:

\begin{aligned} \begin{aligned} max r [\sum_{k \in [K]} P (Y_{k} = 1 | s_{k}, w_{k}; θ) + K^{'} \cdot E_{W} [P (Y = 1 | W; θ)]] \\ - c (θ), \\ s.t . θ \in Θ, \end{aligned} \end{aligned}

(1)

where

Θ

is the feasible region for the service design. In the objective, the first term

\sum_{k \in [K]} P (Y_{k} = 1 | s_{k}, w_{k}; θ)

is the expected adoptions from the

K

customers who suggested, and the second term

K^{'} \cdot E_{W} [P (Y = 1 | W; θ)

accounts for the expected future adoptions once the new service is proposed. This objective balances the service to the

K

customer with suggestions and the broader prospective customer base. When

K^{'}

is large, the platform’s route design tends to resemble that of a public service, with a primary focus on meeting the needs of the general population. When

K^{'}

is small, the platform places greater emphasis on serving the

K

customers who provide suggestions. In this scenario, it is still important to note that the platform conscientiously balances the individualized and communal elements of the shuttle service while accommodating these

K

customers. This is crucial because tailoring the route to a specific group of customers may impact others adversely.

Throughout the article, we posit that the platform has the knowledge of $K^{'}$ and $P (W = w)$ , which collectively represent market-level information relevant to the new shuttle service. For practice, these values could be estimated through market research or inferred from browsing history. While a thorough examination of these is outside the scope of this work, in the case study (as detailed in Section 5), we will discuss the characterization of the travel information distribution and assess how the model performs when the actual value of $K^{'}$ differs from the estimate.

So far, we have taken all the response models being given in problem (1). More details about the model characterization and calibration will be provided in Sections 3 and 4 after we quantify the VOI from customer suggestions in the following subsection.

2.2. Value of Information

One of our research questions is how the crowd-starting process brings value to the service design, in particular, through customer suggestions. To this end, we introduce another two benchmarks by modifying the optimization problem (1).

In the first benchmark, we consider a hypothetical setting where the $K$ customers only reveal their travel information without suggesting routes. Here, the platform only needs to modify problem (1) by replacing the conditional response model of $K$ customers with the marginal response (MR) model and obtains the service design $θ_{M R}^{*}$ as below:

\begin{aligned} θ_{M R}^{*} & = \underset{θ \in Θ}{\arg \max} r [\sum_{k \in [K]} P (Y_{k} = 1 | w_{k}; θ) + K^{'} \cdot E_{W} P (Y = 1 | W; θ)] \\ - c (θ) . \end{aligned}

(2)

The second benchmark considers even less information available. That is, suppose there is no customer input from the

K

customers, the platform determines the service

θ_{0}^{*}

by:

θ_{0}^{*} = \underset{θ \in Θ}{\arg \max} r [(K + K^{'}) \cdot E_{W} P (Y = 1 | W; θ)] - c (θ) .

(3)

This serves as a practical benchmark when the platform is developing the service for a public bus service.

For any service design $θ$ , we evaluate its true expected profit with

\begin{aligned} Z (θ) & = r [\sum_{k \in [K]} P (Y_{k} = 1 | s_{k}, w_{k}; θ) + K^{'} \cdot E_{W} [P (Y = 1 | W; θ)]] \\ - c (θ), \end{aligned}

whose realization depends on the number of customers, their suggestions if any, and their adoption decisions.

Let $θ_{C R}^{*}$ be the optimal solution of (1) with the true expected profit $Z (θ_{C R}^{*})$ . We quantify the VOI as the profit increase achieved by $θ_{C R}^{*}$ over $θ_{0}^{*}$ :

\begin{aligned} VOI & = Z (θ_{C R}^{*}) - Z (θ_{0}^{*}) \\ = Z (θ_{M R}^{*}) - Z (θ_{0}^{*}) + Z (θ_{C R}^{*}) - Z (θ_{M R}^{*}) . \end{aligned}

(4)

Here, VOI can be interpreted in two parts:

Z (θ_{M R}^{*}) - Z (θ_{0}^{*})

is the value of collecting travel information of particular customers (i.e.,

w_{k}

) as opposed to having distributional information only about the overall population (i.e., the distribution of

w

), and

Z (θ_{C R}^{*}) - Z (θ_{M R}^{*})

is the value of understanding individual preferences for the service (i.e., implied by

s_{k}

). We will investigate these in more detail in our case study (Section 5).

3. Customer Adoption and Suggestion

In this section, we model how customers adopt service as well as how they provide suggestions so as to characterize the response models.

3.1. Customer Utility With Service Attributes

The adoption of a service depends on how well the designed service fits an individual’s travel needs and preferences. That is, we need to measure the individualized service quality from the universal service (e.g., a given shuttle route) and its impact on the willingness to adopt. Thus, we introduce customer $k$ ’s designed attributes $x_{k}^{θ} = g (w_{k}, θ)$ , where $g (w_{k}, θ)$ is a mapping that transforms a universal service $θ$ into different dimensions of service quality perceived by the customer according to her contextual travel information $w_{k}$ . For example, as we detailed in Section 5, a customer evaluates a shuttle route by the walking distance from home to the nearest stop, and the additional time spent en route compared to a typical ride-hailing service. This mapping allows we to study the response models based on designed attributes $x_{k}^{θ}$ for the individual customer $k$ .

We employ the random utility model for service adoption. When considering the shared shuttle service $θ$ , customer $k$ perceives the designed attributes $x_{k}^{θ}$ . The customer’s random utilities ${\tilde{u}}_{1, k} (x_{k}^{θ})$ and ${\tilde{u}}_{0, k} (z_{0})$ for adopting the shared service $θ$ and the outside option $z_{0}$ , respectively, are given by

{\tilde{u}}_{1, k} (x_{k}^{θ}) = μ_{k} (x_{k}^{θ}) + {\tilde{ϵ}}_{1, k}, {\tilde{u}}_{0, k} (z_{0}) = μ_{k} (z_{0}) + {\tilde{ϵ}}_{0, k},

(5)

where

μ_{k} (x)

captures the intrinsic utility of a service with designed attributes

x

. The intrinsic utility for the outside option is typically normalized to

μ_{k} (z_{0}) = 0

. Additionally,

{\tilde{ϵ}}_{0, k}

and

{\tilde{ϵ}}_{1, k}

are random noise terms that capture the idiosyncrasy among customers, and i.i.d. for individuals. With (5), the customer will adopt the shared service (i.e.,

Y_{k} = 1

) if

{\tilde{u}}_{1, k} (x_{k}^{θ}) \geq {\tilde{u}}_{0, k} (z_{0})

. This adoption behavior is also related to the concept of satisficing theory (Simon, 1956), in which a customer adopts a service as long as it meets their needs (e.g., a threshold in utility).

One key aspect of crowd-started services like Beeline is that customers can provide suggestions for the service design proactively. Ideally, these suggestions should enable the platform to create a shuttle service customized to individual needs. Yet, customers often suggest routes based solely on their preferences, overlooking the communal nature of the shuttle service. Consequently, it is challenging for a shared service to accommodate all these suggestions. Thus, it is important to examine how the disparity between a design denoted as $θ$ and an individual’s suggestion represented by $s_{k}$ affects the decision to adopt.² Similar to the designed attributes introduced above, we define customer $k$ ’s suggested attributes as $x_{k}^{s} = g (w_{k}, s_{k})$ as a mapping from the suggested route to the same dimensions of service quality.

To build up the relation between designed attributes and suggested attributes, we introduce the utility model for customer suggestion, inspired by the comparison between “revealed preference” and “stated intention” by Ben-Akiva and Morikawa (1990). This article conducts a travel demand analysis with stated preference data and shows that “a survey of stated intentions may yield responses with significant biases and large random errors.” It also refers to the statistics from Suzuki et al. (1986) that “42.9% of respondents expressed intention to use a new subway line while only 34.5% of the respondents were actually using it.” Suppose the customer $k$ provides suggested attributes $x_{k}^{s}$ . At the time of suggestion, the customer considers the utility from her suggested service ${\hat{u}}_{1, k} (x_{k}^{s})$ and that from the outside option ${\hat{u}}_{0, k} (z^{0})$ as follows:

{\hat{u}}_{1, k} (x_{k}^{s}) = μ_{k} (x_{k}^{s}) + {\hat{ϵ}}_{1, k}, {\hat{u}}_{0, k} (z^{0}) = μ_{k} (z^{0}) + {\hat{ϵ}}_{0, k} .

(6)

Ben-Akiva and Morikawa (1990) statistically showed that the same intrinsic utility is retained in both the “revealed preference” and “stated intention” models, so we assume utility functions in (5) and (6) share a common intrinsic utility. However, for each customer, the noise term

{\hat{ϵ}}_{1, k} ({\hat{ϵ}}_{0, k})

can be different from

{\tilde{ϵ}}_{1, k} ({\tilde{ϵ}}_{0, k})

in (5). This potential discrepancy reflects how individual tastes change between suggestion and adoption, especially when the duration is long (e.g., it can be as long as one month in the case of Beeline). This phenomenon also has been observed in studies on stated intention and revealed preference (Ben-Akiva and Morikawa, 1990; Swait and Louviere, 1993). With (6), we have

{\hat{u}}_{1, k} (x_{k}^{s}) \geq {\hat{u}}_{0, k} (z_{0})

when

s_{k}

is suggested. Although the values of the utilities nor their difference are precisely known, the suggested attributes still provide a reference point (e.g., via this inequality) for characterizing the response models.

3.2. Characterizing Response Models

Based on the two utility models (5) and (6), we can now characterize the response models introduced in Section 2. There are various approaches in this regard, for example, by parameterizing the utility functions. For example, one common method is the multinomial logit (MNL) model, which assumes a linear intrinsic utility and an i.i.d. Gumbel random noise term. This model has closed-form expressions for adoption probabilities, making it easy to estimate the coefficients in the linear model. Without the linear assumption, in this article, we follow a non-parametric approach that bridges the utility functions and the response models. We do not specify any functional form for the intrinsic utility or the random noises, providing flexibility for capturing the relation between suggestion and adoption in various scenarios.

To begin with, we discretize each dimension of the attributes into several levels. As an example, we consider two service attributes: walking distance and detour time, which are divided into discrete levels of 100 m and 5 minutes, respectively. Thus, we define the set of the discretized service attributes: $Z = {z_{1}, z_{2}, \dots, z_{M}} \subset R^{2}$ where each $z_{i}$ is a combination of discretized levels, for example, $z_{i} = (100 m, 20 m i n u t e s)$ . Both designed attributes and suggested attributes belong to the set $Z$ . For any $z_{i}, z_{j} \in Z$ , if the adoption utilities from (5) satisfy ${\tilde{u}}_{1, k} (z_{i}) > {\tilde{u}}_{1, k} (z_{j})$ , or equivalently $μ_{k} (z_{i}) > μ_{k} (z_{j})$ , we say $z_{i}$ is preferred over $z_{j}$ by the customer $k$ , which denoted by $z_{i} ≻ z_{j}$ . Similarly, for some designed attributes $z_{i} \in Z$ and the outside option $z_{0}$ , if the adoption utilities satisfy ${\tilde{u}}_{1, k} (z_{i}) > {\tilde{u}}_{0, k} (z_{0})$ , we say the customer $k$ prefers the new service to the outside option, that is, $z_{i} ≻ z_{0}$ . Furthermore, we define coordinate-wise monotone pairs as follows: if attributes $z_{i}$ dominate $z_{j}$ in all dimensions, then $μ_{k} (z_{i}) \geq μ_{k} (z_{j})$ holds for every customer $k$ and thus $z_{i} ≻ z_{j}$ , for example, $z_{1} = (200 m, 15 m i n u t e s) ≻ z_{2} = (300 m, 20 m i n u t e s)$ . These are known as shape constraints (Chatterjee et al., 2018; Han et al., 2019; Deng et al., 2021). We collect all these coordinate-wise monotone pairs in the set $M$ .

Recall the response models introduced at the beginning of Section 2. With the mapping $x_{k}^{θ} = g (w_{k}, θ)$ , the marginal response model $P (Y_{k} = 1 | w_{k}; θ)$ can be written as $P (Y_{k} = 1 | x_{k}^{θ})$ . Let $p_{i} := P (Y_{k} = 1 | x_{k}^{θ} = z_{i})$ be the probability that $z_{i}$ is preferred over $z_{0}$ . Thus, if $z_{2} ≻ z_{1}$ , then $p_{2} \geq p_{1}$ . Similarly, we define $p_{i | j} := P (Y_{k} = 1 | x_{k}^{θ} = z_{i}, x_{k}^{s} = z_{j})$ as the probability that $z_{i}$ is preferred over $z_{0}$ given the suggested attributes $z_{j}$ .

Below, we demonstrate the impact of suggestions in three scenarios, each distinct by the structural assumptions applied to utility models. Using a simple example with two instances of suggested attributes for walking distance and detour time $(200 m, 25 m i n u t e s)$ and $(400 m, 20 m i n u t e s)$ , we can visualize the adoption probabilities in conditional response models for each scenario.

1.
Suppose customers share the same intrinsic utility functions: $μ_{k} (\cdot) = \bar{μ} (\cdot), \forall k$ . In this case, customers share the same preference list over $Z$ , and their heterogeneity is only characterized by the outside option $z_{0}$ . If the noise terms in the adoption utilities (5) and suggestion utilities (6) coincide: $P ({\tilde{ϵ}}_{1, k} = {\hat{ϵ}}_{1, k}) = P ({\tilde{ϵ}}_{0, k} = {\hat{ϵ}}_{0, k}) = 1$ and the gap in suggestion ${\hat{u}}_{1, k} (x_{k}^{s}) - {\hat{u}}_{0, k} (z_{0}) = C$ is a constant, the conditional adoption probabilities will be either 0 or 1, as illustrated in Figure 7(a) and (d). This is an ideal scenario where customers are thought to be willing to adopt the service if their suggestions are met.
2.
Suppose customers share only the same intrinsic utility functions: $μ_{k} (\cdot) = \bar{μ} (\cdot), \forall k$ , but the correlation between noise terms can be arbitrary. In this scenario, customers still share the same preference list over $Z$ but the suggestion may not be very informative as shown in Figure 7(b) and (e).
3.
Figure 7(c) and (f) presents the third scenario that allows customers to have different intrinsic utility functions $μ_{k} (\cdot)$ . It is possible that different suggested attributes imply heterogeneous preferences over $Z$ .

Figure 7.
Conditional adoption probability with two instances of suggested attributes $(200 m, 25 m i n u t e s)$ and $(400 m, 20 m i n u t e s)$ under three scenarios.
4. Response Model Calibration

Building upon the customer response models detailed in the previous section, this section delves into the process of calibrating these models using historical data for service design. Given a proactive customer $n \in [N^{†}]$ who provided suggestions in the design of historical routes, we first collect the suggestion $s_{n}$ and travel information $w_{n}$ . According to the service $θ_{n}$ offered to this customer, we observe the subsequent adoption decision $y_{n}$ . Following the mapping of service attributes, that is, $x_{n}^{θ} = g (w_{n}, θ_{n})$ and $x_{n}^{s} = g (w_{n}, s_{n})$ , the collection of data is then represented by $D^{†} = {(x_{n}^{s}, x_{n}^{θ}, y_{n}), \forall n \in [N^{†}]}$ . Besides, a reactive customer $n \in [N^{‡}]$ with travel information $w_{n}$ browsed an existing route $θ_{n}$ and made an adoption decision $y_{n}$ . We denote this collection of data by $D^{‡} = {(x_{n}^{θ}, y_{n}), \forall n \in [N^{‡}]}$ .

Following the notations in Section 3, we first estimate the marginal adoption probability $p_{i}$ for $z_{i} \in Z$ by the minimization of mean squared error (MSE):

\begin{aligned} min_{p} \sum_{i \in [M]} \sum_{n \in D_{i}} (y_{n} - p_{i})^{2}, \\ s.t. p_{i_{1}} \geq p_{i_{2}}, \forall (i_{1}, i_{2}) \in M, \end{aligned}

(7)

where

D_{i}

is the subset of

D^{†} \cup D^{‡}

, where

x_{n}^{θ} = z_{i}

. The constraints are specified for the coordinate-wise monotone pairs in the set

M

. This optimization problem is well studied (Chatterjee et al., 2018; Han et al., 2019; Deng et al., 2021) with sound asymptotic properties. We use

\hat{p}

to denote the optimal solution to (7). Note that the estimate of the marginal response model

\hat{p}

entails an empirical preference list over

Z

. W.L.O.G., suppose that

{\hat{p}}_{M} \geq {\hat{p}}_{M - 1} \geq \dots \geq {\hat{p}}_{2} \geq {\hat{p}}_{1}

. This preference list extends the monotone property in (7) by incorporating the observed data. That being said, it may not be consistent with the true rank of service attributes in

Z

with a finite sample.

Then we estimate the conditional adoption probability $p_{i | j}$ for $z_{i}, z_{j} \in Z$ by conditioning on each suggestion $z_{j}$ :

\begin{aligned} \begin{array}{cc} min_{p_{\cdot | j}} \sum_{i \in [M]} \sum_{n \in D_{i | j}^{†}} (y_{n} - p_{i | j})^{2} + ζ \sum_{i \in [M - 1]} (p_{i | j} - p_{i + 1 | j}) 1 {p_{i | j} > p_{i + 1 | j}}, \\ s.t. p_{i_{1} | j} \geq p_{i_{2} | j}, \forall (i_{1}, i_{2}) \in M, \end{array} \end{aligned}

(8)

where

D_{i | j}^{†}

is the subset of

D^{†}

, where

x_{n}^{θ} = z_{i}

and

x_{n}^{s} = z_{j}

. Compared to (7), we incorporate the preference list from the estimated marginal response model as an isotonic penalty in the objective function. This is motivated by Naeini and Cooper (2016) where a similar penalty is utilized to calibrate multiple probabilistic classifiers. When

ζ = \infty

, (8) becomes the classical isotonic regression where the preference list from the estimated marginal response model is imposed as a hard constraint. Under the first and second scenarios outlined in Section 3.2 where there exists a shared preference list across

Z

, opting for an exceedingly large

ζ

may not be warranted, as the monotone structure derived from (7) may not be accurate. In contrast, in the third scenario, where no unique preference list exists, setting

ζ = 0

may not be a straightforward solution, as the isotonic penalty can still serve as a regularization method to mitigate overfitting. Therefore, we guide the practical selection of

ζ

through cross-validation.

4.1. MSE With Shrinkage Under Limited Data

If the platform only has limited historical data, the estimation of the conditional response models in (8) may not be reliable. In particular, for a large set $Z$ , the sample size $N_{i | j}^{†} = | D_{i | j}^{†} |$ for estimating the conditional adoption probability $p_{i | j}$ can be insufficient, which leads to significant estimation error due to overfitting. Therefore, we propose the following MSE minimization with shrinkage:

\begin{aligned} \begin{array}{cc} min_{p_{\cdot | j}} \sum_{i \in [M]} \sum_{n \in D_{i | j}^{†}} (y_{n} - p_{i | j})^{2} + \sum_{i \in [M - 1]} ζ (p_{i | j} - p_{i + 1 | j}) 1 {p_{i | j} > p_{i + 1 | j}} \\ + \sum_{i \in [M]} α_{i} E_{Y \sim p_{i}^{0}} [(Y - p_{i | j})^{2}], \\ s.t. p_{i_{1} | j} \geq p_{i_{2} | j}, \forall (i_{1}, i_{2}) \in M . \end{array} \end{aligned}

(9)

Compared to (8), we add a regularization that computes the expected loss when

Y

follows a Bernoulli distribution parametrized by a given

p_{i}^{0}

. The value of

p_{i}^{0}

is specified beforehand and is independent of

j

. Essentially, the idea is to shrink the estimation towards some anchor distribution

p^{0}

, hedging against the risk of estimation error from sampling.

Remark

When $M = | Z | = 1$ , the problem reduces to the estimation of a single Binomial without the monotonic constraint. By setting $p^{0} = 0.5$ , the optimal solution, in this case, coincides with the center of Wilson score, which is a popular tool in statistics for constructing confidence interval of Binomials (Brown et al., 2001).

The modified formulation (9) is also highly related to Gupta and Kallus (2022). They demonstrate the benefit of data pooling, which shrinks the solutions of multiple independent stochastic optimization problems towards some anchor distribution. The approach is constructive in high-dimensional problems with quadratic loss because of a notable effect in statistics often referred to as Stein’s paradox. In our problem, the pooling effect actually has two ways. The first one is achieved by the isotonic penalty, which pools the data within the same suggestion by connecting the losses of different service attributes in accordance with the monotone structure. The second one is integrated by fixing a common anchor point $p_{i}^{0}$ in the loss function, which pools the data across different suggestions. Specifically, we extend Example 2.1 in Gupta and Kallus (2022) to show that MSE minimization with shrinkage can improve the estimation in an asymptotic view.

Proposition 1

Given any $p^{0}$ , let ${\hat{p}}_{\cdot | j}$ and ${\hat{p}}_{\cdot | j}^{'}$ denote the optimal solutions to (8) and (9), respectively, for each $j$ with $ζ = \infty$ . The true loss function is defined as $L ({\hat{p}}_{\cdot | j}) = \sum_{i \in [M]} E_{Y \sim p_{i | j}} [(Y - {\hat{p}}_{i | j})^{2}]$ . Suppose that the sample sizes satisfy $N_{i | j}^{†} = \bar{N}, \forall i, j$ , then there exists a uniform $α^{*} = α_{1} = \dots = α_{M}$ such that as $M \to \infty$ ,

\sum_{j \in [M]} L ({\hat{p}}_{\cdot | j}^{'}) \leq \sum_{j \in [M]} L ({\hat{p}}_{\cdot | j}) .

The inequality is strict with

α^{*} > 0

if the true probability

p \neq p^{0}

and

p \neq \bar{p}

, where

\bar{p}

refers to the sample mean.

Proof Proof

The value of optimal $α^{*}$ is typically unknown. In practice, we can conduct cross-validation for $α$ , as discussed by Gupta and Kallus (2022). The detailed steps are provided in Supplemental Appendix C. Lastly, we discuss the choice of anchor distribution $p^{0}$ . One naive choice is $p_{i}^{0} = \frac{1}{2}, \forall i$ which forms a non-informative prior. This setup is a so-called a fixed anchor, under which the relationship among Binomials is omitted. Alternatively, let us consider another anchor that leverages our problem structure. We set $p^{0} = \hat{p}$ which is exactly the estimated marginal response model. Our simulation also shows that this anchor outperforms the fixed anchor, and thus we will employ it for further numerical experiments.

Although the estimation problems can be solved efficiently, we may encounter the missing data issue when estimating the response models. That is, if $D_{i | j}^{†}$ is empty for some pair $(i, j)$ , then the estimation for $p_{i | j}$ is not well defined. Fortunately, the missing data issue, in this case, could be naturally resolved by our modified MSE minimization with shrinkage. Suppose $D_{i - 1 | j}^{†}$ and $D_{i + 1 | j}^{†}$ are non-empty, and ${\hat{p}}_{i - 1 | j}^{'}$ and ${\hat{p}}_{i + 1 | j}^{'}$ are the optimal solutions in (9). Then it turns out that

{\hat{p}}_{i | j}^{'} = \underset{{\hat{p}}_{i + 1 | j} \geq p_{i | j} \geq {\hat{p}}_{i - 1 | j}}{\arg \min} | p_{i | j} - p_{i}^{0} | .

This is well-defined and determined by the anchor distribution and the monotone structure. If the probability

p_{i}^{0}

is between

{\hat{p}}_{i - 1 | j}^{'}

and

{\hat{p}}_{i + 1 | j}^{'}

, then the estimation

{\hat{p}}_{i | j}^{'}

reduces to the anchor. Otherwise, either

{\hat{p}}_{i - 1 | j}^{'}

{\hat{p}}_{i + 1 | j}^{'}

will be the optimal solution, depending on their distance to the anchor. In terms of estimating the marginal response model in (7), the missing data issue could be much less likely since the model complexity is smaller and the sample size is relatively larger. Even if it occurs, we can mitigate the situation by incorporating a non-informative anchor

p^{0} = 0.5

to (7). See Supplemental Appendix B.

4.2. Endogeneity

When developing our estimation methods, we build on the existence of historical data but are silent on the underlying data generation process. If the service design $θ$ was randomly constructed, then a customer $k$ would perceive designed attributes $x_{k}^{θ}$ as randomly assigned, which leads to consistent estimators. Unfortunately, the quality of estimation may deteriorate in the presence of endogeneity when the designed attributes $x_{k}^{θ}$ are correlated with the noise terms $ϵ_{k} = ({\tilde{ϵ}}_{1, k}, {\tilde{ϵ}}_{0, k})$ in utility models.

Consider the following example for estimating the marginal response model. In a given service area, one group of customers resides at a considerable distance from the subway station, while another group lives in proximity to it. The latter group, owing to their geographical advantage, is more likely to suggest routes with higher service quality, as they have the subway station as a convenient outside option. This discrepancy can significantly influence how the platform designs routes to cater to both customer groups and maximize profits. As a result, the designed attributes customers receive in the shuttle routes may depend on their random utilities.

Addressing the issue of endogeneity in econometrics typically involves the identification of a suitable instrumental variable. However, finding such an instrumental variable can be a complex task and often relies on robust data availability, which falls beyond the scope of this article. Nevertheless, we briefly discuss one remedy for conducting estimation under endogeneity in our problem, which hinges on the conditional independence assumption: $ϵ_{k} ⊥ x_{k}^{θ} | s_{k} .$ This assumption posits that, when conditioning on the customer’s suggestion $s_{k}$ , the designed attributes $x_{k}^{θ}$ for this customer become independent of the error terms $ϵ_{k}$ in this customer’s utility models. This assumption is mild and generally hold in our context based on the shuttle route design optimization (1). We note that the route design $θ$ depends on the customer $k$ ’s suggestion and the information from other customers. Once we integrate the suggestion, no further information pertaining to this customer’s random utility is utilized, and the designed route is only influenced by the random utilities of other customers. As we assume i.i.d. idiocratic noise $ϵ_{k}$ among customers, the assumption is well established.

Unfortunately, the conditional independence assumption may not directly lead to a practical solution because there will be very few customers sharing the exact same suggestion $s_{k}$ . Therefore, we employ the relaxed assumption: $ϵ_{k} ⊥ x_{k}^{θ} | x_{k}^{s},$ by replacing $s_{k}$ with the suggested attributes $x_{k}^{s}$ , and we develop the following three-step estimation procedure:

1.
Estimate the conditional response models $P (Y = 1 | x^{θ}, x^{s})$ for all $x^{s} \in Z$ .
2.
Estimate the distribution of suggested attributes $P (X^{s} = x^{s})$ .
3.
Construct the marginal response model using
$P (Y = 1 | x^{θ}) = \sum_{x^{s} \in Z} P (X^{s} = x^{s}) \cdot P (Y = 1 | x^{θ}, x^{s}) .$

Remark
This approach is inspired by the stratification/blocking idea from the causal inference literature (Imbens, 2004), where the treatment assignment is not completely randomized but conditioning on observable confounders, is independent of potential outcomes. Since the suggestion is naturally involved in our problem, we can take it as an observable confounder. We also highlight the difference from the literature on causal inference with observational study. Typically, the endogeneity comes from the fact that a person’s behavior is associated with many confounding factors. For example, how often one smokes can never be randomized, and in reality, it could depend on age, gender, etc. Oftentimes, although some covariates information is observed, it remains challenging to deal with unobserved confounders. In our problem, the endogeneity of the designed attributes $x^{θ}$ comes from the platform’s design process, but not the customers. Consequently, as long as we control the information used for service design, the estimation is conditionally unconfounded.

5. Case Study

We conduct a case study on the shared shuttle service design, enabling us to explore operational insights and practical considerations in detail. Consider many customers who have similar origin and destination regions for a new shared shuttle service. A toy example of this is shown in Figure 8: the shuttle first picks up customer 1 near her home and then moves to the second stop, where customers 2 and 3 meet the shuttle after some walk. Finally, the shuttle picks up customer 4 and travels to the destination with all four customers on board.

We use the dataset from a taxi company in Singapore to generate OD information on traveling demand. The dataset contains the details of all booking jobs, including pickup location and time, dropoff location and time, travel distance and time, fare, etc. We aggregate three-month booking data from 7 a.m. to 9 a.m. on each day to create our demand pool. In particular, we focus on the historical data whose pickup locations belong to three residential districts, and the dropoff location is Downtown. The three districts have distinct demand distributions and geographical features as shown in Figure 9. Subsequently, we will replicate the simulation and experiment for each district.

5.1. Simulation Setup

5.1.1. Service Attributes

We begin by outlining how we map travel information and a new shuttle route to attributes, denoted as $x_{k}^{θ} = g (w_{k}, θ)$ . These attributes encompass walking distance and detour time compared to a taxi service. When comparing shared shuttle service to a taxi service, customers are often willing to accept some walking distance and detour time in exchange for a more cost-effective option. For our simulations, we derive the two-dimensional attributes for each customer using travel information and Google Maps. Specifically, we establish five levels of walking distance in meters, represented as ${500, 400, 300, 200, 100}$ . It is assumed that customers will not opt for the shuttle and will instead choose the outside option if the walking distance exceeds 500 m. Recent research in urban transportation has examined the impact of walking distance on service adoption. For example, Kabra et al. (2020) found that in a bicycle-sharing system, each additional 100 m of walking to the station reduced the likelihood of usage by nearly 20%, and most usage occurred within 500 m (roughly 5 minutes). Similarly, Lyu and Teo (2022) suggested that in a locker-distribution system, customers walking to the locker within 5 minutes was crucial for service adoption. Concerning detour time, we establish three levels, namely ${15, 10, 5}$ minutes. Along with walking distance, these attributes contribute to the creation of the set $Z$ containing 15 alternatives.

5.1.2. Utility Models

We then define the customer utility functions that govern the response models. Following Ben-Akiva and Morikawa (1990), we consider a linear function for the intrinsic utility $μ_{k} (\cdot)$ and normalize the outside option with $μ_{k} (z^{0}) = 0$ . Then from (5), the random utilities of designed attributes $x_{k}^{θ}$ and the outside option are represented by

{\tilde{u}}_{1, k} (x_{k}^{θ}) = β_{0} + β_{k}^{'} x_{k}^{θ} + {\tilde{ϵ}}_{1}, {\tilde{u}}_{0, k} (z^{0}) = {\tilde{ϵ}}_{0} .

(10)

The adoption rule is equivalently

β_{0} + β_{k}^{'} x_{k}^{θ} + {\tilde{ϵ}}_{1} - {\tilde{ϵ}}_{0} > 0

. In particular, we let

β_{0} = 4, β_{k, 1} = - 0.0075, β_{k, 2} = - 0.2

, and

{\tilde{ϵ}}_{1}, {\tilde{ϵ}}_{0} \sim N (0, 1)

. Then we elaborate on how this utility model is applied in the three scenarios we discussed in Section 3.2 for the following numerical experiments. For the first scenario, the suggestion is simply generated with

x_{k}^{s} = min_{z \in Z} {β_{0} + β_{k}^{'} z + {\tilde{ϵ}}_{1} - {\tilde{ϵ}}_{0} > 0}

. For the second scenario, we introduce perturbations

τ_{1}, τ_{0} \sim N (0, 0.16)

in addition to the noise terms such that

{\hat{ϵ}}_{1} = {\tilde{ϵ}}_{1} + τ_{1}

and

{\hat{ϵ}}_{0} = {\tilde{ϵ}}_{0} + τ_{0}

, and then the suggestion follows

x_{k}^{s} = min_{z \in Z} {β_{0} + β_{k}^{'} z + {\hat{ϵ}}_{1} - {\hat{ϵ}}_{0} > 0}

. For the third scenario, besides the perturbations in suggestion, we incorporate the heterogeneous intrinsic utilities by letting

β_{k, 1} \sim U (- 0.012, - 0.003)

and

β_{k, 0} \sim U (- 0.35, - 0.05)

. That is, customers may have their own preferences in evaluating walking distance and detour time, while the market-level average coefficients are consistent with those in the first two scenarios (i.e.,

E [β_{k, 1}] = - 0.0075, E [β_{k, 2}] = - 0.2

5.2. Numerical Results

We unfold the numerical experiments in two parts. In the first part, we focus on the service design strategies under known response models. That is, we aim to investigate how different design strategies impact the shuttle route and the expected profit, understanding the VOI. In the second part, we incorporate data and estimation techniques, examining not only the effectiveness of our estimation methods but also addressing issues such as endogeneity and robustness.

5.2.1. Route Design and Performance

For the shuttle route optimization, we cluster origins into hexagon areas whose centers are designated as candidate pickup points. As in the case of Beeline, a shuttle route may only stop at selected bus stops in a neighborhood. Furthermore, this significantly reduces the computational burden for routing. In Figure 10, the heatmaps visualize the demand distributions after clustering in each district.

Based on the characterization of pickup points above, we derive the explicit MIP formulation for the service design problems in Supplemental Appendix D, and here we sketch the key steps. First, we model the routing and scheduling decisions using multi-commodity flows over a spatial–temporal network. Second, for the adoption probability $P (Y_{k} = 1 | x_{k}^{θ}, x_{k}^{s})$ , we introduce auxiliary binary variables to indicate what designed attributes in the discrete set of attributes $Z$ are received by customers. Third, for $E_{W} [P (Y = 1 | W; θ)$ , we approximate it by the weighted average of the demand distribution over clusters. Lastly, we specify a unit price of $r$ for each adoption and a unit cost of $c$ per kilometers for the length of the shuttle route.

For each instance, we independently sample $K$ customers, of which their location information is from the historical orders and their suggestions are from the utility models introduced in Section 5.1. Then we solve the route optimization problems following three different strategies in (1), (2), and (3) denoted by $Π_{C R}$ , $Π_{M R}$ , and $Π_{0}$ , respectively, to obtain the designed routes. Upon that, together with the initial $K$ customers, we independently sample another $K^{'}$ customers to evaluate the total adoption and profit following the true response models. By generating 10,000 instances, we evaluate the average out-of-sample profit as the performance metric for each design strategy.

5.2.1. Comparison of Routes

We first consider the baseline setup: $K = K^{'} = 10$ , $r = 4$ , $c = 1$ , and suggestions follow the third scenario. Using one problem instance from postal district 82, we showcase the routes designed with different strategies in Figure 11.

In Figure 11(a), we draw the route following the strategy $Π_{0}$ , where customers are not engaged. The route balances the trade-off between covering more demand and incurring fewer detours by only visiting areas with medium to high densities.

In Figure 11(b), we draw the route following the strategy $Π_{M R}$ , where some customer locations are revealed (shown as circles). Compared to Figure 11(a), the route is adjusted by removing some areas with medium densities and adding some areas with known demand.

In Figure 11(c), we draw the route following the strategy $Π_{C R}$ , where customers at the same locations as shown in Figure 11(b) provide their suggestions. Unlike case (b), customers eight and nine are deliberately overlooked due to their preference for high quality in the suggested attributes mapped from their suggestions, as shown in Figure 11(d). As a result, in the route from $Π_{C R}$ , customer 0 can now receive a reduced detour time, while customer 1 benefits from a shorter walking distance, aligning better with their respective suggestions.

Overall, the route by

Π_{M R}

deviates far from

Π_{0}

while the route by

Π_{C R}

tends to be less aggressive and shares more overlap with

Π_{0}

, accounting for suggestions. In Figure 12, we compare the adoption rates of the same

K = 10

customers together with the average adoption of the other

K^{'}

future customers, from different strategies.

5.2.1. Value of Information

Here we analyze the VOI defined in (4) by investigating the performance gap between strategies. In addition to the baseline setup, we conduct the experiments by varying the relative sizes between $K$ and $K^{'}$ while keeping the total $K + K^{'}$ constant. We also consider three scenarios of utility models (indexed as S1, S2, and S3) outlined in Section 3.2 with setup in Section 5.1. We summarize the results in Table 1 with the following observations:

All three districts share the same structural property of the expected profit. As $K$ increases, both the value of collecting individual travel information ( $Π_{M R} - Π_{0}$ ) and the value of understanding their preferences ( $Π_{C R} - Π_{M R}$ ) are enhanced, contributing to more VOI. From S1 to S2, as more uncertainty in suggestion is introduced, the gain from customer suggestions ( $Π_{C R} - Π_{M R}$ ) reduces. From S2 to S3, as more diversity in utility coefficients is introduced, the VOI decreases. However, the ratio $\frac{Π_{C R} - Π_{M R}}{Π_{C R} - Π_{0}}$ increases. The reason is that individual preference becomes relatively more important than travel information when customer utilities in service attributes are more heterogeneous.

Three postal districts differ due to their demand distributions illustrated in Figure 10. Postal district 55 has a very concentrated demand distribution such that the design of a shuttle route is relatively easier. This leads to a high profit even with the baseline strategy $Π_{0}$ and a smaller VOI compared to the other two districts. Postal districts 54 and 82 have more dispersed demand distributions, and, therefore, the baseline strategy fails to earn sufficient profit. Yet, this opens room for improvement when customer inputs are leveraged. We indeed see much larger gains in the expected profits, especially the value of understanding individual preferences ( $Π_{C R} - Π_{M R}$ ) from suggestions.

Table 1.
Average expected profits in different districts under different scenarios of utility models.

Postal district 55 Postal district 54 Postal district 82

$K = 5$ $K = 10$ $K = 15$ $K = 20$ $K = 5$ $K = 10$ $K = 15$ $K = 20$ $K = 5$ $K = 10$ $K = 15$ $K = 20$

S1 $Π_{0}$ 40.05 30.70 31.97

$Π_{M R} - Π_{0}$ 2.02 3.47 5.49 7.50 1.87 4.80 6.06 8.70 2.94 4.94 6.36 8.33

$Π_{C R} - Π_{M R}$ 1.12 1.95 2.29 3.56 1.47 2.30 4.86 6.56 2.07 3.18 5.07 6.44

S2 $Π_{0}$ 40.05 30.70 31.97

$Π_{M R} - Π_{0}$ 2.02 3.47 5.49 7.50 1.87 4.80 6.06 8.70 2.94 4.94 6.36 8.33

$Π_{C R} - Π_{M R}$ 0.84 1.08 1.13 1.60 1.30 1.80 2.31 3.91 1.11 1.54 2.51 3.34

S3 $Π_{0}$ 39.01 30.36 31.32

$Π_{M R} - Π_{0}$ 1.8 2.95 4.76 6.63 1.49 3.58 5.03 7.24 2.61 4.47 5.82 7.78

$Π_{C R} - Π_{M R}$ 0.64 0.94 1.05 1.55 1.05 1.56 2.19 3.64 0.73 1.34 2.01 3.22

		Postal district 55	Postal district 54	Postal district 82
S1	$Π_{0}$	40.05	30.70	31.97
	$Π_{M R} - Π_{0}$	2.02	3.47	5.49	7.50	1.87	4.80	6.06	8.70	2.94	4.94	6.36	8.33
	$Π_{C R} - Π_{M R}$	1.12	1.95	2.29	3.56	1.47	2.30	4.86	6.56	2.07	3.18	5.07	6.44
S2	$Π_{0}$	40.05	30.70	31.97
	$Π_{M R} - Π_{0}$	2.02	3.47	5.49	7.50	1.87	4.80	6.06	8.70	2.94	4.94	6.36	8.33
	$Π_{C R} - Π_{M R}$	0.84	1.08	1.13	1.60	1.30	1.80	2.31	3.91	1.11	1.54	2.51	3.34
S3	$Π_{0}$	39.01	30.36	31.32
	$Π_{M R} - Π_{0}$	1.8	2.95	4.76	6.63	1.49	3.58	5.03	7.24	2.61	4.47	5.82	7.78
	$Π_{C R} - Π_{M R}$	0.64	0.94	1.05	1.55	1.05	1.56	2.19	3.64	0.73	1.34	2.01	3.22

Furthermore, we introduce another benchmark strategy (11) motivated by the practice of Beeline’s algorithm Sim (2017)—as long as the route is within a maximal walking distance and detour time, the demand is captured. We define this strategy $Π_{B L}$ as follows:

\begin{aligned} (Π_{B L}) : θ_{B L}^{*} & = \underset{θ \in Θ}{a r g m a x} r [\sum_{k \in [K]} 1 {x_{k, 1}^{θ} \leq 500, x_{k, 2}^{θ} \leq 15} \\ + K^{'} \cdot E_{W} P (Y = 1 | W; θ)] - c (θ) . \end{aligned}

(11)

Taking the postal district 82 as an example, we compare the performance of

Π_{B L}

with the above three strategies in Figure 13. We find that

Π_{B L}

follows the same increasing trend as

K

grows initially as a result of the VOI from customer inputs. Unfortunately,

Π_{B L}

becomes worse as

K

grows further, which reveals the price of being over-optimistic regarding the consistency between customers’ suggestions and their adoption. In contrast, the strategy

Π_{C R}

addresses this issue by taking a probabilistic view of customer adoptions and utilizing customer suggestions via the conditional response model.

Lastly, we investigate the model performance when $K^{'}$ is misspecified. In particular, let $\tilde{K^{'}}$ be the model parameter in the approximation of the true $K^{'} = 10$ in all strategies. We conduct a sensitivity analysis of the misspecified $\tilde{K^{'}}$ in Figure 14 by varying it from 2 to 20. Given that the deviation is modest in scale, the performances of $Π_{C R}$ and $Π_{M R}$ are fairly robust and lead to consistently large benefits against $Π_{0}$ .

5.2.1. Impact of Price

We have assumed a fixed price as a parameter in our service design optimization. In practice, the platform can further adjust the unit price charged to customers. To this end, we first modify the utility functions to include the price:

{\tilde{u}}_{1, k} (x_{k}^{θ}) = β_{0} + β_{k, 1} x_{k, 1}^{θ} + β_{k, 2} x_{k, 2}^{θ} + β_{k, 3} r + {\tilde{ϵ}}_{1}, {\tilde{u}}_{0, k} (z^{0}) = {\tilde{ϵ}}_{0} .

where

β_{k, 3} \sim U (- 1.5, - 1)

refers to the price sensitivity and we adjust the intercept with

β_{0} = 9

In the experiment, we enumerate discretized prices by varying $r \in [3, 5]$ and solve the corresponding service design optimization under each strategy respectively. In Figure 15, we highlight the price that maximizes the average profit for each strategy using a dashed line. It is not surprising to see that extremely low and high prices worsen the profit. We observe that, in the optimum, $Π_{C R}$ charges the highest price, while $Π_{0}$ charges the lowest. This suggests an indirect VOI in our service design: when more personal information is provided, it enables the delivery of highly personalized services, allowing for the possibility of charging higher prices and, consequently, generating increased profit.

5.2.2. Estimation Performance

Now we consider the estimation of response models with data and investigate its impacts on the service design.

5.2.2. Impact on Expected Loss

We first compare the prediction performance of different estimation approaches discussed in Section 4. We use the excess risk, defined by the difference between the expected loss of an estimated model and that of the true model, to directly measure the estimation quality.³ We let CR be the shorthand for the conditional response model estimated from (9) with isotonic penalty and shrinkage. We compare it with the other two approaches: the one without any regularization ( $ζ = α_{i} = 0$ ), and the one with only isotonic penalty ( $α_{i} = 0$ ). Moreover, we set another benchmark using a parametric approach. Since a linear utility model is configured in (10), we consider the logit model here. In order to make a more fair comparison, we also develop a regularized version of the logit model which shares the same spirit of our shrinkage approach. The definition of excess risk and the details for the parametric approach are deferred to Supplemental Appendix E.1.

Here we use 1,000 replicates of the training set in which the service attributes are randomly selected with the sample size $N$ and evaluate the average performance. The choices of $ζ$ and $α_{i}$ are given by cross-validation. Since the first scenario leads to deterministic models which are easy to estimate, we illustrate the performance in the second scenario and leave the simulation in the third scenario to Supplemental Appendix E.1. In Figure 16(a), we observe that the performance of CR is improved by adding isotonic penalty and shrinkage, particularly when dealing with limited data. Despite that the naive logit model performs worse than all non-parametric approaches with small data, the regularized version provides a significant improvement and dominates CR. This is not surprising as the logit model fits the true utility structure very well. To further explore the limitations and advantages, we introduce another two utility structures as the true model: the quadratic utility and the step-wise utility (see Supplemental Appendix E.1 for the explicit forms). Basically, the quadratic function implies that the utility decreases slowly as the walking distance increases from 100 to 200 m and decreases sharply as the distance increases further; the step-wise function implies that the utility does not change within 200 m and drops to another level between 200 and 400 m. In Figure 16(b), we observe that the excess risks of CR and regularized logit model are very close. While using linear utility for estimation remains a good approximation for the true quadratic one, the cost of misspecification nearly offsets the efficiency gains from parametric estimation. In Figure 16(c), we observe the negative impact of significant model misspecification, wherein logit models diminish sample efficiency.

5.2.2. Impact on Expected Profit

We then plug in the estimated response models and solve the optimization problems under different strategies. We investigate how estimation affects the out-of-sample expected profit in Figure 17, taking the postal district 82 as an example. Moreover, we include the performance of the route design using the true response models without estimation, defined as oracle solution. We observe similar relations among three estimation approaches for $Π_{C R}$ as implied by Figure 16—isotonic penalty adds more value in Scenario 2 while shrinkage is more substantial in Scenario 3. Additionally, the utility model in Scenario 2 is simpler than that in Scenario 3 and the estimation of the marginal response model is easier as well. As a result, the performance of $Π_{M R}$ is even better than $Π_{C R}$ without regularization in Scenario 2 with small data. This underscores the importance of employing more sophisticated estimation techniques to effectively harness the VOI from customer suggestions.

5.2.2. Endogeneity

As mentioned in Section 4.2, endogeneity could be an issue that leads to biased model estimation. We construct the following data generation process to examine the potential consequence of endogeneity and how our three-step approach alleviates the problem. First, taking postal district 82 as an example, we notice that there is one subway station in the local region (denoted by the star in Figure 18). We divide customers into two segments, denoted by blue and green, depending on whether their walking distance to the subway station is < 1 km or not. For the blue segment, we introduce an offset $δ > 0$ for the outside option’s utility ${\tilde{u}}_{0, k} (z_{0}) = δ + {\tilde{ϵ}}_{0}$ because these customers have higher valuations of the subway as the outside option. In contrast, for the green segment, we have ${\tilde{u}}_{0, k} (z_{0}) = - δ + {\tilde{ϵ}}_{0}$ . For each problem instance with 20 customers, their suggestions are generated based on the modified utility models following Scenario 3, and then we solve the service design optimization (1) accordingly, from which the designed route is offered to customers and their responses are realized. By repeating this procedure, we are able to estimate the marginal response model while collecting more samples. Note that the mechanism of endogeneity behind is not explicitly specified following any structural form, but is implicitly induced by the underlying utility models and the service design process.

We consider two cases for the offset parameter: $δ = 0.5$ and $δ = 1$ . In Figure 19, we first plot the excess risk using the standard approach (MR) given by (7). When the sample size grows, the excess risk does not diminish as the consequence of biased estimation, and the case with the larger offset leads to more bias. We then use our three-step approach (MR-D) to de-bias the estimation—the excess risk gradually decays. This simulation study shows that endogeneity could indeed be critical and can be properly overcome by our three-step approach.

6. Concluding Remarks

In this article, we explore the process of launching a shared shuttle service with customer engagement. We propose an optimization framework to maximize expected profit based on the characterization of customer response models. We also discuss the challenges of estimating these models and propose approaches with isotonic penalty and shrinkage for dealing with small data sets. We finally conduct a case study to illustrate the practical implications, in terms of the routing strategies, the VOI, and the performance of estimation approaches.

There are two limitations to this work that could be potentially addressed in future research. First, we formulate the service design problem in a static manner. That is, the platform has historical customer inputs and needs to design a new shuttle service. In practice, the timing of proposing a new service can also be optimized. For example, how to dynamically engage customers in the service design process is an interesting but challenging extension. This relates to topics like adaptive preference elicitation in marketing literature and active learning in machine learning literature.

Second, our paper follows the predict-then-optimize framework with a plug-in strategy where we treat the response models and the service design as separate tasks. However, the ultimate goal of estimation is not simply to calibrate the response models, but also to solve a downstream service design optimization problem. One possibility is to incorporate the impact of estimation errors when optimizing the service design. Given that our estimation of response models is essentially a variation of the sample average for multiple Bernoulli random variables, it may be beneficial to optimize for a more robust service design while taking into account the statistical properties of the estimate.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478241256383 - Supplemental material for Crowd-Starting a Shared (Shuttle) Service With Customer Suggestions

Supplemental material, sj-pdf-1-pao-10.1177_10591478241256383 for Crowd-Starting a Shared (Shuttle) Service With Customer Suggestions by Long He and Tu Ni in Production and Operations Management

Footnotes

Acknowledgements

The authors gratefully acknowledge the departmental editor, the senior editor, and three anonymous referees for constructive comments.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Long He

Supplemental Material

Supplemental material for this article is available online ().

Notes

How to cite this article

He L and Ni T (2024) Crowd-Starting a Shared (Shuttle) Service With Customer Suggestions. Production and Operations Management 33(8): 1739–1758.

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Supplementary Material

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		Postal district 55				Postal district 54				Postal district 82
		$K = 5$	$K = 10$	$K = 15$	$K = 20$	$K = 5$	$K = 10$	$K = 15$	$K = 20$	$K = 5$	$K = 10$	$K = 15$	$K = 20$
S1	$Π_{0}$	40.05				30.70				31.97
	$Π_{M R} - Π_{0}$	2.02	3.47	5.49	7.50	1.87	4.80	6.06	8.70	2.94	4.94	6.36	8.33
	$Π_{C R} - Π_{M R}$	1.12	1.95	2.29	3.56	1.47	2.30	4.86	6.56	2.07	3.18	5.07	6.44
S2	$Π_{0}$	40.05				30.70				31.97
	$Π_{M R} - Π_{0}$	2.02	3.47	5.49	7.50	1.87	4.80	6.06	8.70	2.94	4.94	6.36	8.33
	$Π_{C R} - Π_{M R}$	0.84	1.08	1.13	1.60	1.30	1.80	2.31	3.91	1.11	1.54	2.51	3.34
S3	$Π_{0}$	39.01				30.36				31.32
	$Π_{M R} - Π_{0}$	1.8	2.95	4.76	6.63	1.49	3.58	5.03	7.24	2.61	4.47	5.82	7.78
	$Π_{C R} - Π_{M R}$	0.64	0.94	1.05	1.55	1.05	1.56	2.19	3.64	0.73	1.34	2.01	3.22