Optimal pricing in ride-share platforms

Abstract

Ride-share platforms are contemporary businesses that match passengers with drivers, unlike taxis that can be hailed from the street. In the literature, the problem of optimizing the operations of such companies is mostly considered in static settings. We use in this paper a dynamic model and propose differential equations to model the evolution of the system. The objective is to maximize the profit during the planning horizon. Using optimal control theory, we determine the optimal rate of change in the ride price rate. An illustrative example along with sensitivity analyses shows the effect of the system parameters on the optimal solution obtained.

Keywords:

• ride-sharing
• optimal control
• optimal pricing

1. Introduction

With the dramatic increase of smartphone sales around the world and the internet easy access, online service requesters are a major force in the consumption market. The economy of sharing is one of the booming concepts evolving rapidly to match supply represented by online and mainly through smartphone app requests and service providers. Services range from food delivery (e.g., Uber eats), grocery delivery (e.g., Instacart), housing accommodation (e.g., Airbnb) to transportation (Uber, Lyft).

For example, Airbnb that offers rental services (house, apartment, room or even a couch) business expanded rapidly over the years, where the number of bookings increased from 52 million in 2016 to 272 million in 2019 (Business of apps, 2012). The same expansion outlook is seen for the transportation or mobility apps. According to Uber (invester.Uber.com, 2021), the number of rides fulfilled almost doubled between the second quarter of 2017 (889 million) and the first quarter of 2020 (1658 million). Although Uber business was hardly hit by coronavirus pandemic in 2020, it did bounce back in Q4 2020. The annual number of Uber users in 2021 was 118 million, which represents a 6% increase on 2019 figures and 26% increase on 2020 figures (www.businessofapps.com, 2023).

Online service providers like Airbnb and Uber are called two-sided platforms, as they match service requests from customers to service providers. In the case of mobility or transportation service providers called ride-share platforms, like Uber (US), Didi (China), Careem (Middle East), pricing is one of the most challenging tasks as it depends on many variables that vary over time. In most platforms, the rider opens the app, requests a ride (pickup and destination locations) then the price will show up; then, the rider either confirms the ride request or leaves the ride-share platform. Leaving and not confirming the ride request can happen for a variety of reasons: a higher price than the rider expected, non-availability of drivers, a non-acceptable waiting time for pickup, etc. Other than ride requests, the ride-share platform operates under (but not limited to) the following variables: number of drivers, traffic, ride distance, ride timings, etc. Consequently, pricing in ride-share platform is challenging and requires models that capture all those interacting variables that change over time.

Ride-sharing platforms are considered successful when ride-sharing fleet companies are making a profit through assigning optimal pricing and are able to manage their resources efficiently. Many types of research have addressed the optimal pricing of ride-sharing platforms. For instance, in the work (Al-Abbasi et al., 2019), the authors propose DeepPool platform, which is a distributed free-model that uses Q-network techniques. Ride-sharing services were improved by incorporating a deep learning model and demand statistics. By interacting with the environment, the DeepPool platform was able to manage dispatching vehicles in different dynamic environments. Results reveal better performance when compared to other strategies.

In the work (Liang et al., 2020), the authors provide an integer non-linear programming model to optimize profit for automated taxi systems. The proposed model was hard to solve since it considers different constraints and decision variables such as traffic congestion, travel times, and dynamic travel demand. However, the authors solved the problem using a Lagrangian relaxation algorithm, which provides a near-optimal solution for the proposed problem. Results show the efficiency of the proposed model in terms of number of served requests and delay time. Moreover, ride-sharing service quality can be improved when a delay penalty is applied.

In the work (Jacob & Roet-Green, 2021), the authors consider the pool option in a ride-sharing platform, where passengers get to share the ride with other passengers for a reduced fare. Using a queueing model, the optimal revenue is achieved for both pooled and solo rides. The result shows that maximum revenue is achieved under low congestion. On the other hand, when the number of passengers and drivers increases, driver revenue is decreased. In the work (Banerjee et al., 2015), the authors use a queueing theoretical approach to solve the problem of pricing for the ride-sharing platforms. They suggested a queuing model where drivers and passengers are the agents with Poisson distribution arrivals: passengers live in the system for one ride and depart after being served, and drivers can be either available (ready to serve a ride) or busy (serving a ride). In terms of pricing, they suggested a pricing scheme that varies with the number of available drivers and dynamically equal to a multiple of the base price (enough to cover the driver and platform expenses). The authors found out that throughput and revenue under dynamic pricing does not outperform the optimal static pricing policy (single price). In the work (Battifarano & Qian, 2019), the authors propose a log-linear regression with L1 regularization-based framework to predict surge pricing, which is the real-time change in prices during high ride demand to more efficiently allocate drivers. This prediction benefits both riders (time and cost) and drivers (profit). The authors applied the model to predict Uber and Lyft surge multipliers in the city of Pittsburgh and found out that the model outperforms other methods (overall mean, historical mean, and non-linear methods).

In another work (Zha et al., 2018), to tackle the temporal and spatial characteristics of the ride-share platforms, like Uber’s surge pricing and Lyft’s prime time pricing, the authors propose a geometric matching and spatial pricing framework (where a rider is matched with the closest driver within a matching radius) that can be used by ride-share platforms. The authors found that a revenue-maximizing platform will not only adopt a pricing scheme to balance demand and supply but also to increase revenue through higher prices. They also found that the platform and drivers benefit more under the revenue-maximizing scheme than the customers. To counter the pricing scheme negative effect on the customer welfare, the authors propose a commission rate regulation.

The study in (Sun et al., 2019) investigates the ride-share platforms pricing under two types of driver selection: first driver to respond and closest driver to customer. For the former pricing type, the authors found that the optimal price acceptable by both sides (driver and customer) consists of a base fare (a function of distance), a rush hour congestion fee, and an emergency fee. For the latter pricing type, the driver travelling distance to reach the customer as well as the customer waiting time is minimized, which will benefit the platform by charging the customer a high price.

The work (Yang et al., 2020) proposes a model that optimizes matching time interval (the time it takes the platform to accumulate drivers and riders before matching) and matching radius (maximum allowed distance between driver location and customer pickup location) in on-demand ride-sourcing platform. Numerical experiments were then conducted to investigate the effect of the optimized variables on passenger waiting time, vehicle utilization, and matching rate. Results show that the matching rate and expected pickup time increase with the matching radius till a certain level, then become independent of it. They also show that a global optimal matching time interval impact on system performance depends on supply and demand (maximized if supply is greater than demand and vice versa).

The demand for rides is associated with the quality of the service like the waiting time, availability of drivers at the pickup location, ease of use and reliability of the app, as well as the ride price. Like any other two-sided platform, the ride price set by the platform has to be chosen very carefully to entice more riders into the platform, which will increase ride requests. Traffic will increase if the price offered by the platform is lower than the customer reference price, which is the price the customer is willing to pay for a service after evaluation of the service quality and comparison between service providers (Niedrich et al., 2001; Xue et al., 2016).,

Concerning solution procedure, most papers available in the literature assume a static setting and use optimization techniques, statistical regressions, or a queueing theoretic approach in the steady state. In contrast, we assume a dynamic system and use an optimal control approach. To the best of our knowledge, this paper is the first to use such approach. Optimal control theory is a branch of mathematics. It aims at optimizing some dynamical system by finding a control that optimizes some objective function. It has successful applications in many areas of science such as biology (Lenhart & Workman, 2007), biomedicine (Swan, 1984), and engineering (Geering, 2007), industry (Rapoport & Pleshivtseva, 2007), management science and economics (Sethi, 2019; Tu, 1984), etc.

In light of the research works discussed above, different quantitative techniques such as simulation or optimization have been employed by researchers. For example, simulation is more of a descriptive than a prescriptive technique. It allows decision-makers to observe the results of their experimentation and it is similar to the “what if?” approach of sensitivity analysis. However, optimization is a prescriptive technique where decision variables are not functions of time. Also, another technique that has been used is queueing theory. However, when it is used, researchers assume that a steady state has been reached so that variables are no longer functions of time. On the other hand, optimal control is well suited to our problem because it is dynamic. Furthermore, our variables (the number of ride requests and the price) are functions of time and the solutions obtained using optimal control are functions of time.

Application of optimal control theory to ride-sharing systems is sparse, and very few works came to our attention. Authors in Chen and Cassandras (2020) proposed a dynamic vehicle assignment strategy in a real-time ride-sharing system to minimize the system-wide waiting and traveling times of passengers. To deal with the “curse of dimensionality” in RSS optimization problem formulations, authors adopted an event-driven Receding Horizon Control (RHC) approach, which can react to random events in real time. Modelling the RSS as a discrete event system, both passenger waiting and traveling times were optimized. The paper mainly focused on reducing the complexity of the vehicle assignment problem. However, it has not considered how profit can be optimized through addressing the optimal rate of change in the ride price rate.

Sadeghi and Smith (2019) proposed an interactive bathtub model to describe the traffic dynamics of ride-sourcing vehicles in a city with undifferentiated streets solely served by ride-sourcing services. The model is simple and requires only basic input information. It considers three states of ride-sourcing vehicles: idling, picking-up/collecting, and delivering, along with factors such as travel time, waiting time, and service time to simulate vehicle movements accurately. Although the interactive bathtub model provides a framework for developing effective control strategies to manage traffic congestion in cities served by ride-sourcing vehicle, travelers choice models, and pricing instruments have not been introduced to minimize waiting and travelling times.

In Fan (2023) two indirect control methods to optimize the set of waiting locations of drivers were introduced to minimize the expected wait time of customers. Nonetheless, the algorithm proposed has only found a near-optimal controls for both the sharing information control method and the pay-to-control method. Yet, dynamic pricing or routing incentives have not been investigated to further optimize customer waiting time.

In Yengejeh and Smith (2021), authors likewise proposed two indirect control methods; sharing information to a subgroup of drivers on the location of other waiting drivers and paying drivers to relocate (pay-yo-control). The model is further modified to optimize waiting time; approximation and LP-rounding algorithms have been provided to manipulate drivers’ decision towards relocating to a desired waiting location, hence minimizing the maximum wait-time for both customers and drivers. However, this model still lacks the ability to deal with scalability issue, with the number of drivers and customers in the system due to the combinatorial nature of the proposed approach. The model also has not taken advantage of the real-life scenario where the number of available drivers and the number of ride requests change over time, which can further optimize the driver’s profit (through applying dynamic pricing).

In Luo and Saigal (2017) authors proposed a continuous-time continuous-space approach to dynamic pricing for on-demand ride-sharing, which can help to address the curse of dimensionality and improve the efficiency of pricing decisions. The authors argue that this approach is effective for providing efficient pricing decisions, which can ultimately lead to increased revenue for the ride-sharing platform. However, the computation effort of such approach requires much more computational resources than other methods, which might make hard to implement, particularly for large-scale systems.

The above literature review shows that the application of optimal control theory to ride-sharing systems has many gaps, and our goal is to address one of them. As stated earlier, the pricing dynamics of ride-share platforms are affected by positive (i.e., market growth) and negative factors (i.e., price sensitivity). For such platforms to continue to exist, a profit needs to be maximized. To this end, this paper proposes an optimal control model, with the main objective of maximizing the total profit of a ride-share platform. Using optimal control theory, the change rate in the ride price rate under different circumstances is analyzed to find the optimal ride price over a given planning horizon.

The problem is described in the next section, and the model along with the model assumptions and solution are presented in Section 3. Section 4 is devoted to an illustrative example with sensitivity analyses to assess the effect of the system parameter on the optimal solution. Section 5 concludes the paper.

2. Problem description

In this work, the problem of obtaining the optimal ride price over time for ride-share platforms is sought. Ride-share platforms generate revenues from charging riders a ride price upon their request through an app. In the developed model, a decision-making horizon is defined, and decisions depend on each other over time. The developed model tries to catch the dynamics of a real-life scenario where the number of available drivers and the number of ride requests change over time. This makes the proposed model solutions applicable to many ride-share platforms.

There are many real-time factors that affect the attractiveness of the platform and acceptance of the ride price by the ride requester. A lower price (compared to the reference price) will attract more ride requests; however, the platform should satisfy these ride requests through an optimal supply of drivers to avoid disutility to riders.

In the developed model, the number of ride requests increases over time, which is represented by a growth rate. This factor is drawn from real life where the market growth (number of users and adopters) is very common in any new business. The ride-share platform needs to decide the optimal price in each time period, such that its total profit is maximized.

3. Model assumptions, formulation, and solution

The problem is dynamic, and an optimal control approach seems appropriate. This formulation has not been attempted before, and it may lead to some insights that other methods may not provide.

3.1. Model assumptions

In the following, we will use optimal control theory to find the best pricing policy in ride-sharing platforms. To describe the model, the following assumptions are made:

1. There are two actors in the system, the passengers that are represented by the number of ride requests and the drivers available to take the rides.

2. The platform charges the passenger a price $\mathit{p}\left(\mathit{t}\right)$ (depending on the number of drivers available, base cost, distance, trip duration).

3. At time $\mathit{t}$, the passenger will take the ride if $\mathit{p}\left(\mathit{t}\right)\le \mathit{R}$, where $\mathit{R}$ is the reference price, or leaves the system.

4. The utility to the passenger is $\mathit{R}-\mathit{p}$ if served and 0 if not.

5. The rate of ride requests depends on $\mathit{R}$ and $\mathit{p}$ and equals $\mathit{\mu }$.

6. A passenger takes a ride and leaves the system.

7. A passenger is negatively sensitive to the price (instantaneously for the ride).

For a ride-share platform, there is a cost associated with serving each customer denoted as $\in$ in our model. This cost covers the cost of developing, optimizing, and operating the application. It also can cover the cost of attracting and training drivers, etc. Another cost pertains to changing the ride price. This cost is captured by $\mathit{\beta }$ in the developed model and will be considered as quadratic, as the marginal impact of changing the ride price usually increases as the amount of change price increases (Kumar & Sethi, 2009). The notation used is summarized in Table 1 below:

Table 1. Notation

$\mathit{T}$	Planning horizon (day, week, month, etc.)
$\mathit{x}\left(\mathit{t}\right)$	Number of ride requests at time $\mathit{t}$ after checking the price (state variable)
$\mathit{p}\left(\mathit{t}\right)$	Ride price rate (per hour) at time $\mathit{t}$ (state variable)
$\mathit{u}\left(\mathit{t}\right)$	Rate of change in the ride price rate (decision variable)
$\mathrm{\Phi }$	Sensitivity of riders to the ride price rate
$\mathit{\mu }$	Market growth rate for ride requesters
$\mathit{\pi }$	Utility factor for passengers for difference between reference price and shown price
$\in$	Cost of serving riders
$\mathit{\beta }$	Cost of changing the price rate

3.2. Optimal control formulation

To formulate the model, we use two state variables (the number of ride requests and the ride price rate) and one control variable (the rate of change in the ride price rate). The dynamics of the system are modelled by the system of differential equations

$\frac{d}{\mathit{dt}}\mathit{x}\left(\mathit{t}\right)=\mathit{\mu }-\mathit{\varphi u}\left(\mathit{t}\right)+\mathit{\pi }\left[\mathit{R}-\mathit{p}\left(\mathit{t}\right)\right],\mathit{x}\left(0\right)=x_0,$

$\frac{d}{\mathit{dt}}\mathit{p}\left(\mathit{t}\right)=\mathit{u}\left(\mathit{t}\right),\mathit{p}\left(0\right)=p_0,$

where $x_0$ and $p_0$ are known constants. The objective is to maximize the profit during the planning horizon $\left[0,\mathit{T}\right]$:

$\mathrm{Max}\mathit{J}=\underset{0}{\overset{T}{\int }}\left[\mathit{p}\left(\mathit{t}\right)\mathit{x}\left(\mathit{t}\right)-\in \mathit{x}\left(\mathit{t}\right)-\mathit{\beta u}{\left(\mathit{t}\right)}^2\right]\mathit{dt}.$

3.3. Model solution

The optimal solution of the above problem is obtained by applying the necessary optimality conditions in the form of the Pontryagin maximum principle (see, for example, Sethi, 2019). We introduce the Hamiltonian function:

$\mathit{H}=\left[\mathit{p}\left(\mathit{t}\right)\mathit{x}\left(\mathit{t}\right)-\in \mathit{x}\left(\mathit{t}\right)-\mathit{\beta u}{\left(\mathit{t}\right)}^2\right]+{\lambda }_1\left(\mathit{t}\right)\left\{\mathit{\mu }-\mathit{\varphi u}\left(\mathit{t}\right)+\mathit{\pi }\ast \left[\mathit{R}-\mathit{p}\left(\mathit{t}\right)\right]\right\}+{\lambda }_2\left(\mathit{t}\right)\mathit{u}\left(\mathit{t}\right),$

where ${\lambda }_1\left(\mathit{t}\right)$ and ${\lambda }_2\left(\mathit{t}\right)$ are the adjoint functions associated with the differential equation constraints, respectively.

The necessary condition

$\frac{\mathrm{\partial H}}{\mathrm{\partial u}\left(\mathit{t}\right)}=0,$

readily yields the optimal control variable:

$u^{\ast }\left(\mathit{t}\right)=-\frac{\varphi }{2\mathit{\beta }}{\lambda }_1\left(\mathit{t}\right)+\frac{1}{2\mathit{\beta }}{\lambda }_2\left(\mathit{t}\right).$

Note that the sufficient condition:

$\frac{{\mathrm{\partial }}^2\mathit{H}}{\mathrm{\partial u}{\left(\mathit{t}\right)}^2}=-2\mathit{\beta }\le 0,$

guarantees a unique global maximum. Substituting the optimal control variable into the state equations, and using the adjoint equations

$\frac{d}{\mathit{dt}}{\lambda }_1\left(\mathit{t}\right)=-\frac{\mathrm{\partial H}}{\mathrm{\partial x}\left(\mathit{t}\right)},\frac{d}{\mathit{dt}}{\lambda }_2\left(\mathit{t}\right)=-\frac{\mathrm{\partial H}}{\mathrm{\partial p}\left(\mathit{t}\right)},$

yields the following differential system:

$\frac{d}{\mathit{dt}}\mathit{x}\left(\mathit{t}\right)=-\mathit{\pi p}\left(\mathit{t}\right)+\frac{{\varphi }^2}{2\mathit{\beta }}{\lambda }_1\left(\mathit{t}\right)-\frac{\varphi }{2\mathit{\beta }}{\lambda }_2\left(\mathit{t}\right)+\mathit{\mu }+\mathit{\pi R}$

$\frac{d}{\mathit{dt}}\mathit{p}\left(\mathit{t}\right)=-\frac{\varphi }{2\mathit{\beta }}{\lambda }_1\left(\mathit{t}\right)+\frac{1}{2\mathit{\beta }}{\lambda }_2\left(\mathit{t}\right)$

$\frac{d}{\mathit{dt}}{\lambda }_1\left(\mathit{t}\right)=-\mathit{p}\left(\mathit{t}\right)+\in$

$\frac{d}{\mathit{dt}}{\lambda }_1\left(\mathit{t}\right)=-\mathit{x}\left(\mathit{t}\right)+\mathit{\pi }{\lambda }_1\left(\mathit{t}\right)$

This system of differential equations can be rewritten as

$\frac{d}{\mathit{dt}}\mathit{z}\left(\mathit{t}\right)=\mathit{Az}\left(\mathit{t}\right)+\mathit{b},$

where

$\mathit{z}\left(\mathit{t}\right)=\left[\begin{array}{c}\mathit{x}\left(\mathit{t}\right)\\ \mathit{p}\left(\mathit{t}\right)\\ {\lambda }_1\left(\mathit{t}\right)\\ {\lambda }_2\left(\mathit{t}\right)\end{array}\right],\mathit{A}=\left[\begin{array}{cccc}0& -\mathit{\pi }& \frac{{\varphi }^2}{2\mathit{\beta }}& -\frac{\varphi }{2\mathit{\beta }}\\ 0& 0& -\frac{\varphi }{2\mathit{\beta }}& \frac{1}{2\mathit{\beta }}\\ 0& -1& 0& 0\\ -1& 0& \mathit{\pi }& 0\end{array}\right],\mathit{b}=\left[\begin{array}{c}\mathit{\mu }+\mathit{\pi R}\\ 0\\ \in \\ 0\end{array}\right].$

Using standard methods, we obtain the optimal state variables:

$\begin{array}{rl}& x^{\ast }(\mathit{t})=\frac{(C_2{\varphi }^2+C_2\mathit{\pi }\sqrt{\mathit{\beta \varphi }}-2C_1\mathit{\varphi \pi t})e^{\sqrt{\frac{\varphi }{\beta }\mathit{t}}}+(C_3{\varphi }^2-C_3\mathit{\pi }\sqrt{\mathit{\beta \varphi }})e^{-\sqrt{\frac{\varphi }{\beta }\mathit{t}}}}{4{\varphi }^2}\\ & +\frac{\varphi ({\pi }^2-\mathit{R}{\pi }^2-\mathit{\mu \pi })t^2+2{\varphi }^2(\mathit{\mu }+\mathit{R\pi })\mathit{t}+2(C_1{\varphi }^2-{\varphi }^3-C_4\mathit{\varphi \pi }-\mathit{\beta \mu \pi }-\mathit{R\beta }{\pi }^2+\mathit{\beta }{\pi }^2)}{4{\varphi }^2},\\ & p^{\ast }(\mathit{t})=\frac{-C_2{e}^{\sqrt{\frac{\varphi }{\beta }\mathit{t}}}-C_3{e}^{\sqrt{{ }^{-}\frac{\varphi }{\beta }\mathit{t}}}+2(\mathit{\mu }+\mathit{R\pi }-\mathit{\pi })\mathit{t}+2(C_1+\mathit{\varphi })}{4\mathit{\varphi }}\end{array}$

and the optimal adjoint variables

${\lambda }_1\left(\mathit{t}\right)=\frac{C_2\sqrt{\mathit{\beta \varphi }}{e}^{\sqrt{\frac{\varphi }{\beta }}\mathit{t}}-C_3\sqrt{\mathit{\beta \varphi }}{e}^{-\sqrt{\frac{\varphi }{\beta }}\mathit{t}}-\mathit{\varphi }\left(\mathit{\pi R}-\mathit{\pi }\in +\mathit{\mu }\right)t^2-2\left(C_1\mathit{\varphi }-\in {\varphi }^2\right)\mathit{t}-2\left(C_4\mathit{\varphi }+\mathit{\beta \mu }-\mathit{\beta }\in \mathit{\pi }+\mathit{R\beta \pi }\right)}{4{\varphi }^2},$

${\lambda }_2\left(\mathit{t}\right)=-\frac{C_2\sqrt{\mathit{\beta \varphi }}{e}^{\sqrt{\frac{\varphi }{\beta }}\mathit{t}}-C_3\sqrt{\mathit{\beta \varphi }}{e}^{-\sqrt{\frac{\varphi }{\beta }}\mathit{t}}+\mathit{\varphi }\left(\mathit{\mu }+\mathit{R\pi }-\in \mathit{\pi }\right)t^2+2\left(C_1\mathit{\varphi }-\in {\varphi }^2\right)\mathit{t}+2\left(C_4\mathit{\varphi }-\mathit{\beta \mu }-\mathit{R\beta \pi }+\mathit{\beta }\in \mathit{\pi }\right)}{4\mathit{\varphi }}.$

The optimal control variable is obtained as the derivative of the optimal price $p^{\ast }\left(\mathit{t}\right)$. Finally, the unknown constants $C_1,C_2,C_3,C_4$ are derived using the initial and transversality conditions:

$\mathit{x}\left(0\right)=x_0,\mathit{p}\left(0\right)=p_0,{\lambda }_1\left(\mathit{T}\right)=0,{\lambda }_2\left(\mathit{T}\right)=0.$

4. Numerical example

Consider a ride-share platform that is interested in the optimal rate of change in the ride price rate and in the optimal number of ride requests and ride price rate. In this numerical example, the base parameter values are as follows:

$\mathit{T}=20,x_0=2,p_0=1,\in =0.01,\mathit{\mu }=10,\mathit{\beta }=\mathrm{0.3.}\mathit{\pi }=3.9,\mathit{\varphi }=5,\mathit{R}=1.$

Implementing the results of the previous section we obtain the optimal state and control variables as shown in Figure 1. We also obtain the optimal objective function value $J^{\ast }=4.0629\times {10}^5.$

Figure 1. Optimal state and control variables in the base case.

To assess the effect of the different parameters on the state variables (number of ride requests and ride price) and the optimal objective function value, in each numerical experiment, we vary one parameter, and the rest of parameters will be set at their basic values.

4.1. Impact of ϕ, the riders’ sensitivity to the ride price rate

Figure 2 presents the impact of the rider sensitivity $\mathit{\varphi }$ to ride price rate on the optimal price rate trajectory (right) and the number of ride requests (left). As expected, the number of ride requests decreases as the sensitivity to ride price rate increases from $\mathit{\varphi }=3$ to $\mathit{\varphi }=7$. As the number of subscribers increases over time, the optimal price rate increases, as shown in Figure 2. This reflects the fact that the riding platform will always capitalize on the increase in ride requests to increase ride rates.

Figure 2. Effect of ϕ on the optimal state variables.

In Table 2, the objective function decreases with $\mathit{\varphi }$, as the number of ride requests decreases, and even the increase of the ride price rate does not compensate the decrease of profit because of the decrease of ride requests. For instance, the profit is equal to $5.46\times {10}^5$ and $3.42\times {10}^5$, for $\mathit{\varphi }=3$ and $7$, respectively.

Table 2. Effect of ϕ on the optimal objective function value

$\mathrm{\Phi }$	1	3	5	7	9	11	13	15
$J^{\ast }\left(\times {10}^6\right)$	1.1499	0.5464	0.4063	0.3427	0.3052	0.2794	0.2598	0.2439

4.2. Impact of the natural growth rate $\mathit{\mu }$

The impact of the natural growth rate $\mathit{\mu }$ on the optimal ride price rate and the number of ride request trajectories in Figure 3, right and left, respectively. Actually, an increase in the growth rate increases the number of ride requests. Along with the increase of number of ride requests, the ride price is also increasing with $\mathit{\mu }$. This is due to the fact that the platform will capitalize on the increasing demand to increase prices, which is a natural behavior for any business to increase prices once demand picks up.

Figure 3. Effect of μ on the optimal state variables.

The effect of $\mathit{\mu }$ is reflected on the profit objective function which increases with $\mathit{\mu }$. According to Table 3, the profit is equal to $0.4\times {10}^6$ and $1.2\times {10}^6$, for $\mathit{\mu }=10$ and $20$, respectively.

Table 3. Effect of $\mathit{\mu }$ on the optimal objective function value

$\mathit{\mu }$	5	7	10	12	15	17	20	22
$J^{\ast }\left(\times {10}^6\right)$	0.1655	0.2491	0.4063	0.5324	0.7534	0.9221	1.2069	1.4181

4.3. Impact of the cost of changing the price rate $\mathit{\beta }$

In Figure 4, the effect of the cost of changing the price ride is investigated. Clearly, $\mathit{\beta }$ has a negative effect on the ride price and the number of subscribers as the changing prices becomes more expensive.

Figure 4. Effect of β on the optimal state variables.

If $\mathit{\beta }$ keeps increasing, the profit might become negative as it can be seen from Table 4. For example, when $\mathit{\beta }$ is equal to 5, the revenue cannot compensate the high changing price cost, which will result in a negative revenue.

Table 4. Effect of $\mathit{\beta }$ on the optimal objective function value

$\mathit{\beta }$	0.3	0.7	1	3	5	7	9	11
$J^{\ast }\left(\times {10}^8\right)$	0.0041	0.0030	0.0010	−0.0717	−0.3271	−0.8613	−1.7558	−3.0797

4.4. Impact of epsilon $\mathit{\epsilon }$

In Figure 5, the effect of the cost of serving each ride is investigated. Clearly, has a negative effect on the ride price and the number of subscribers as serving customers becomes more expensive.

Figure 5. Effect of ϵ on the optimal state variables.

This effect is reflected in profit that decreases while epsilon increases (Table 5).

Table 5. Effect of $\in$ on the optimal objective function value

$\in$	0.01	0.1	0.5	1	2	3	4	5
$J^{\ast }\left(\times {10}^5\right)$	4.0629	3.9123	3.2796	2.5724	1.4377	0.6755	0.2860	0.2690]

4.5. Effect of $\mathit{\pi }$ utility factor for passengers for difference between reference price and shown price

The results of this experiment are shown in Figure 6. As expected, the number of ride requests increases when the utility factor for passengers becomes higher. In addition, Figure 6 indicates that the platform can actually capitalize on the increase of the utility factor and ride requests to increase prices.

Figure 6. Effect of π on the optimal state variables.

The impact of the utility factor on the objective function is detailed in Table 6. For example, the profit is equal to $4.06\times {10}^5$ and $4.67\times {10}^5$ for $\mathit{\pi }=3.9$ and $7$, respectively.

Table 6. Effect of $\mathit{\pi }$ on the optimal objective function value

$\mathit{\pi }$	1	3	5	7	9	11	13	15
$J^{\ast }\left(\times {10}^5\right)$	5.8280	4.0629	4.2174	4.6773	5.2082	5.7136	6.1234	6.3718

5. Conclusion

We have developed in this paper a dynamic model for a ride-share platform. Optimal control is achieved through the rate of change in the ride price rate. The results show that the number of ride requests is negatively affected by the ride price rate, which reflect the fact that the riding platform will always capitalize on the increase in ride requests to increase ride rates. Moreover, when growth rate increases, the number of ride requests increases, which in turn increase the ride price as well. Result reveals that high changing price cost generates a negative revenue. Also, the cost of serving each ride is found to have a negative effect on the ride price and the number of subscribers. Finally, result indicates that the proposed platform can capitalize on the increase of the utility factor and ride requests to increase prices. While this work tries to investigate the ride share pricing using an optimal control approach, it still lacks to capture all the dynamics of the problem like controlling the number of drivers joining the platform, raising prices during certain times (i.e., during night), constraints for the maximum number of drivers and the price … etc.

For future work, it is recommended to expand the problem to cover the dynamics mentioned earlier.

Acknowledgments

The authors would like to thank the reviewers for carefully reading the paper and making numerous suggestions for its improvement.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.