A resource sharing (sharing platform) scheme on online taxi services

. This paper will review the match between single driver and single rider in online taxi services through a resource sharing (sharing platform) for the operators with the objectives to maximize the profit for drivers (operators) and minimize waiting time for passengers so that the matching rate is higher. A low matching rate between rider and driver can cause the consumer to drop the services. The matching between single driver and single rider in online taxi services through a sharing platform scheme is formulated in maximum weighted bipartite matching problem. To solve the proposed model, we use Kuhn Munkres Algorithm, while to solve the shortest path for the driver to pick up the passenger and the shortest path of passenger's origin destination, modified Dijkstra with adaptive algorithm based on Wei Peng et.al (2012) is used. Based on illustrative example with several cases, we found a resource sharing scenario can optimize the matching between driver and rider and moreover can solve the surge pricing problem which is deemed as less transparant to customer


Introduction
Ride sourcing services, more widely known as online taxi, have become a new transportation alternative to people in big cities all over the world, including in Jakarta, Indonesia. The ride sourcing services use private vehicles as a means of public transportation for passengers similar to taxi services, which are regulated through an application utilizing technology of smartphone, internet and GPS [1]. The inadequacy of public transportation services, while the use of technology is very fast growing in the society for the past few years, has boosted the use of online taxi services.
In Indonesia, specifically in Jakarta, previously there were 3 big companies operating in online taxi services, which were Uber, Grab and GoCar [2], whereas in 2018 Uber was acquired by Grab. Each company has its own specific characteristics in their online taxi services for the consumers. These companies providing the online transportation services are not interlinked with one another, so that the route setting to match drivers and riders is also not interlinked, while a lot of the drivers are actually joining or working under 2 or 3 operators at the same time. On the other hand, from the operational point of view, the online taxi services are not yet optimal, since the waiting time for customers is still quite long or is not as per estimated waiting time stated in the app, and the estimated arrival time is not correct either. In addition, in certain hours and conditions, the fare charged to customers can drastically increase or fluctuate, which is called a surge pricing, which depends on the available vehicles and customer demand levels in the specific areas.
Based on operational issue of ride sourcing services, in this paper, we will evaluate the matching between single drivers and single riders in online taxi services, using a resource sharing (sharing platform/operator) schemes, in order to optimize the profit for operators by minimizing the rider's waiting time so that the matching rate can be higher. A low matching rate between drivers and riders can cause the customers to drop the service. The resource-sharing (sharing platform) scheme is interlinking drivers from different operators (companies) in providing services to the customers, where a driver from company X can serve customers from company Y and vice versa, with a predetermined profit sharing. In addition, the resource sharing scheme is expected to solve the surge pricing problem which is deemed as less transparent for the customers [3]. The matching between single driver and single rider in online taxi services through a sharing platform scheme is formulated in maximum weighted bipartite matching problem. To solve the proposed model, we use Kuhn Munkres Algorithm [4], while to solve the shortest path for the driver to pick up the passenger and the shortest path of passenger's origin destination, modified Dijkstra with adaptive algorithm based on Wei Peng et.al (2012) [5] is used.
The rest of the paper is organized as follows. Section 2 presents a literature review, while Section 3 describes the problem formulation. Section 4 on the other hand presents the solution method using modified Dijkstra with adaptive algorithm [5] and Kuhn Munkres Algorithm [4], and Section 5 presents illustrative example which describes a possible scenario that can occur. Finally, Section 6 concludes and illustrates future research directions.

Literature review
Ride-sourcing service was started in San Fransisco by Uber in 2009 [6], and it was fast spreading to other big cities all over the world, as other companies providing similar services were also popping up. The ride-sourcing service itself was adopting the previously existing ridesharing concept, so that the ride-sourcing service was often considered as ride-sharing services, but with a profit-oriented concept [1,6].
Different from the common public transportation services, a ride-sourcing service is using a dynamic pricing system, which is taking into consideration the real supply and demand condition in each specific area [7]. For example, when the demand for service is high, while the vehicle availability (supply) is low, the fare will be higher than the regular fare, known as surge pricing. But the formulation for the surge pricing is not transparent to the consumers so that they can feel cheated and drop the service [3]. On the other hand, drivers are taking advantage during the surge pricing, since they will earn more money, so that more drivers opt to drive during the surge pricing time [8]. If this situation continues, in the long run customers will leave the service.
The application of surge pricing is not always harmful, because a flat fare can be unfair for a specific zone [9]. However, since this ride-sourcing service is relatively new, the research reviewing the services, specifically on service fares is still very limited. Galichon and Hsieh [10] developed revised surge pricing algorithm to minimize the total inefficiency resulting from time waited in line considering demand and supply uncertainties, while Zha et.al [9] developed the commission rate cap regulation that reaps the flexibility of spatial pricing to solve the surge pricing problem.
In this paper, we will develop a resource-sharing (sharing platform) scheme, which is interlinking drivers from different operators (companies) in providing services to the customers, where a driver from company X can serve customers from company Y, and vice versa, with a predetermined profit sharing. This scheme is expected to benefit the customers with shorter waiting time, while the drivers can optimize their income, and in addition will make the surge pricing calculation, which in this case is a predetermined profit, more transparent so that no parties will feel cheated or harmed.
According to Agatz [11], the problem in optimizing the ride-sharing services for single driver-single rider case can be represented as bipartite matching problem. In this study, the ride-sourcing services for the single driversingle rider will be formulated in maximum weighted bipartite matching.

Problem formulation
The whole process of the assignment of a request to a vehicle with resource sharing (sharing platform) scenario can be described in figure (1)-(3), assuming there are two different operators (platforms) providing ride sourcing services, where it can be enlarged to n platforms.   Initially, customer will announce the request to platform 1 (Figure 1), and then platform 1 will check the availability and feasibility of the vehicle that can serve the customer (2a), while announcing to platform 2 about the request (2b) as described in Figure 2. When the platform 1 announces the request to platform 2, it also offers the trip fare that should be paid by the customer and profit that can be shared if a vehicle on platform 2 serve the request. Next, platform 2 will check the availability and feasibility of its vehicle (2c) and then responds to the platform 1, which available vehicle on it that feasible (Figure 2).
At the final stage (Figure 3), platform 1 will announce to customer (3a) which vehicles are available, from platform 1 and platform 2, each having different waiting time and trip fare that should be paid by the customer. Then the customer will choose which vehicle will serve him/her (3b). Next, platform 1 will assign the chosen vehicle to customer (3c).
Based on the assignment process as described above, the resource sharing scheme to optimize ride sourcing services for single driver-single rider problem considered in this paper is as follow. Let the number of operators From the passenger's point a view, they want to be picked up as soon as possible, so that the waiting time to be picked up is crucial. However, when the available vehicle from the request platform is limited be the set of all passengers of platform ; = 1,2.
Given a complete bipartite graph = ( , ), where = ∪ be the set of nodes, is associated with vehicle nodes and T is associated with passenger nodes and be the set of edges, where = {( , )| ∈ , ∈ } Let F be the set of feasible matches, the binary decision variable indicate whether the edge is in an optimal matching ( = 1) or not ( = 0), then the single driver (vehicle) -single rider matching problem that maximizes the number of matches in order to maximize the profit for drivers (operators) and minimize waiting time for passengers with sharing platform scenario can be formulated as follows: The objective function (1) maximizes the number of matches. Constraints (2) and (3) assure that each vehicle and each rider is only included in at most one match in an optimal matching.To determine the set of feasible matches F, we impose time feasibility constraints, as detailed below. Each vehicle ∈ has a maximum vehicle capacity and associated with , time at which it leaves from its origin .

Time feasibility
If vehicle ∈ serves passenger ∈ , let be the distance from vehicle v's current location (its origin) to passenger's pick up location , be travel time of vehicle to reach passenger's pick up location from vehicle v's current location (its origin) and be waiting time of passenger to be picked up by vehicle .
For a given passenger ∈ and vehicle ∈ , we can determine the time flexibility that is required to make a feasible match between them, as follows. The vehicle leaves its current location at time , and should arrive at passenger's pick up location between passenger' s earliest pick up time and latest pick up time , as described in condition (5).
Next, the vehicle which serves passenger , should arrive at passenger's drop off location between passenger's earliest drop off time and latest drop off time , as described in condition (6).
After time feasibility constraints checked, then we determine trip fare and operational cost for each feasible match, as detailed in the next subsection.

Fare determination and operational cost
Let be -th passenger of platform and be -th vehicle of platform . For a given passenger and vehicle , the trip fare which should be paid by passenger to vehicle is: 1. If = , that is and from the same platform, then where 1 < ≤ 1 + ∆, is the "Sharing" multiplier factor, and the value of ∆ is determined based on the converting money value of waiting time per unit time, usually the value of ∆ less than 50% of initial trip fare.
The operational cost to be incurred by vehicle to serve passenger : Then the revenue (profit) which could be gained by vehicle to serve passenger is: 1. If = , that is and from the same platform, then where 0.05 ≤ (1 − ) ≤ 0.1, with (1 − ) is the sharing revenue factor taken by initial platform Next, weight of feasible match is determined based on waiting time of passenger and revenue gained by driver (vehicle), as detailed in subsection 3.3.

Weight of feasible match
Let be the converting money value of per unit waiting time of passenger to be picked up by vehicle , The weight of each feasible match ( , ), where = ∈ and = ∈ , is 1. If = , that is and from the same platform, then ) > ( , ). Therefore we add parameter to the weight so the vehicle will prioritize choosing passenger from the same platform, as in (13.a).

Solution method
To determine the shortest path, which is the distance and the travel time from vehicle current location (origin) to passenger's pick up location, and passenger's OD, we use modified Dijkstra with adaptive algorithm based on Wei peng et.al (2012) [5] as described in figure 4 and figure 5. The following data structured which will be used in modified Dijkstra with adaptive algorithm [5] : : the matrix containing the edge weights, where : the min-priority queue containing the vertices to be visited. It is the same queue as that used in the classic Dijkstra's algorithm. Deg : the vector containing the degree of vertices, deg [i] is the degree of the i-th vertex; Order : the vector containing the indices of vertices to be used as sources. Order[i] is the index of i-th source vertex. The procedure Enqueue (Q, v) adds a vertex v in the minpriority queue Q. The procedure DeQueue (Q) gets a vertex from the queue Q which has the smallest shortestpath starting from s. To solve the maximum weighted bipartite matching problem as in (1)-(4), we will use Kuhn Munkres Algorithm (Hungarian Algoritm) [4].

Ilustrative example
As an illustrative example, which describes a possible scenario that can occur, suppose there are 4 passengers and 4 vehicles in total, where each operator having 2 passenger's request and 2 available vehicle as described in figure 6. Let 11 and 12 be two passengers who request ridesourcing services on operator 1, and 21 and 22 be two passengers who request ride sourcing services on operator 2. Let 11 and 12 be two available vehicles on operator 1, and 21 and 22 be two available vehicles on operator 2.
Assume all passengers request vehicle at the same time, let 11 pick up location at node 3, and drop off location at node 9; 12 pick up location at node 10, and drop off location at node 2; 21 pick up location at node 8, and drop off location at node 10; and 22 pick up location at node 6, and drop off location at node 1.
By using Modified Dijkstra and Adaptive Algorithm [10] as described on section 4, then we have the following passenger's waiting time for vehicle described in table 1.  As we can see from table 1, 22 's minimum waiting time is 4 unit of time, which picked up by 11 (different platform) or 22 (same platform). The minimum waiting for passenger 12 is also 4 unit of time if he/she picked up by vehicle 22 .
Let 1 and 1 ′ be the trip fare of platform 1 per unit distance of distance, where 1 = 3/unit of distance for the same platform and 1 ′ = 3,25/unit of distance for different platform. Here we have the value of sharing multiplier factor is 1 = 1,08.
Let 2 and 2 ′ be the trip fare of platform 2 per unit distance of distance, where 2 = 3,5/unit of distance for the same platform and 2 ′ = 3,6/unit of distance for different platform, with sharing multiplier factor 2 = 1,029.
Then we have the following trip fare should be paid by the passenger described in table 2.

Conclusion and future research
Based on our study it can be concluded resource sharing (sharing platform) scheme can optimize the matching between driver and rider, which will give benefit for the customers in shorter waiting time, while the benefit for the drivers is optimum income. Furthermore, the weight defined in this study, makes the vehicle (operator) to continue to prioritize providing services to passengers from the same platform.
We hope that the insights generated by our study can be used by ride-sourcing system providers to solve the surge pricing problem which is deemed as less transparent for the customers.
In this paper, the proposed model is only tested using a small example to illustrate a possible scenario that can occur. In future research work, we aim to use larger data to test the proposed model and using appropriate heuristic method to solve the problem.