A Balanced Assignment Mechanism for Online Taxi Recommendation

Transcription

1 217 IEEE 18th International Conference on Mobile Data Management A Balanced Assignment Mechanism for Online Taxi Recommendation Guang Dai, Jianbin Huang, Stephen Manko Wambura, Heli Sun School of Computer Science and Technology, Xidian University, Xi an, China School of Software, Xidian University, Xi an, China Department of Computer Science and Technology, Xi an Jiaotong University, Xi an, China Abstract Majority of taxi recommender systems mainly focused on satisfaction of passengers without considering fairness in assignment of taxi drivers. In this paper we propose a balanced assignment mechanism for online taxi recommendation (BAMOTR). BAMOTR provides a mechanism for fair assignment of drivers at some locations with specific routes to pick up passengers and ensures a short waiting time for passengers. Fair assignment is intended to minimize the differences in income among the taxi drivers. Analysis shows out that fair assignment of drivers and shortening the time the passenger wait before pick up is a trade-off problem. In this paper, we set a regulatory factor that can adjust the trade-off between fair assignment of drivers and shortening of waiting time of passengers. We also propose an efficient range refinement algorithm to solve online taxi recommendation problem in BAMOTR. It is theoretically and experimentally proved that range refinement algorithm ensures the same recommendation result as brute-force algorithm, however it greatly reduces the time overhead. We validate the performances of BAMOTR with extensive evaluations. Experimental results show that BAMOTR achieve better recommendation fairness than compared approaches and guarantee a short waiting time for passengers to be picked up. I. INTRODUCTION Taxi recommender systems have emerged recently with the development of Global Positioning System (GPS), mobile communication technologies, sensor and wireless networking technologies. Taxi recommender systems are very important as they can contribute to several benefits such as suggesting right places for getting passengers, increasing revenue to taxi drivers, reducing waiting time, traffic jams as well as minimizing fuel consumption and hence increasing the number of trips the drivers can perform. Fundamentally, there are two suggested modes for taxi recommender systems. The first one is Grab Single Mode that a passenger request is sent to multiple taxis, and DiDi is a good example in this category. The second mode is Assignment Mode that a passenger request is assigned to a taxi, and Uber mode is a good example in this category. Tremendous evolution in technology has seen DiDi switching to Assignment Mode. This is attributed to the fact that in Grab Single Mode a request is sent to multiple taxis, as these taxis have likely equal chances to grab the passenger request, this might lead to unreasonable allocation of taxis. This situation might lead to several circumstances including the following. Firstly the assigned taxi might not be in an optimum position to pick up the passenger and hence increase waiting time for the passenger and driver time simultaneously. Secondly the other drivers who are in nearby positions might be deprived of the opportunity. The best solution to this problem is to apply Assignment Mode which assign a taxi to pick up a passenger according to some strategies, such as minimizing passenger waiting time. However, the question comes which taxi should be recommended to a passenger? Recommender systems focusing on public transportation service have been studied extensively [1] [6]. In reality, in order to reduce the waiting time of passengers, recommender systems usually recommend the taxi that is nearest to passengers. Liao and Lee [1], [2] aim to reduce the waiting time of passengers, but they have proposed systems which focus only on passengers, without taking into consideration the fairness in assignment of taxis, which may lead to big difference in income of drivers. The taxi-sharing systems proposed in [3], [4] are intended to maximize the profit of taxis by ridesharing, however, these systems can lead also to big difference in income of drivers, and a relatively increased waiting time for passengers. The works in [5], [6] do not also take into consideration fair assignment of drivers as they recommend some routes to taxi drivers, such that given routes might have potential passengers. In this paper we propose BAMOTR, a balanced assignment mechanism for online taxi recommendation, which can considerably bridge the gap in income between drivers and guarantee a short waiting time for passengers. We theoretically and experimentally prove that range refinement algorithm with the local shortest path algorithm (RRA-LSP) ensures the same recommendation result as brute-force algorithm, however it greatly reduces the time overhead. We evaluate the fairness in recommendation and shortening of waiting time by BAMOTR based on the real city network and historical taxi trajectories in Shanghai, China. Experimental results show that BAMOTR greatly reduce the differences in drivers income and guarantee a short waiting time for passengers when a proper regulatory factor is selected. The main contributions of the paper are as follows: We formulate an evaluation function which optimally and efficiently selects one taxi from a given group of competing taxis for each passenger request, and define an online taxi recommendation problem in BAMOTR. We propose two approaches to solve this problem. We analytically show that RRA-LSP enhances both great /17 $ IEEE DOI 1.119/MDM

2 reduction of the time overhead and provision of the same recommendation results as brute-force algorithm. We prove some theorems pertaining to RRA-LSP, these theorems guarantee that RRA-LSP have the same the recommendation results as brute-force algorithm. We design a balanced assignment mechanism to recommend taxis for a group of passengers while taking care of conflict resolution from multiple requests, which finds tradeoff between the fairness in assignment of drivers and the shortening of the waiting time of passengers. We provide evaluation results and the recommendation on fairness provided by BAMOTR and the efficiency produced by RRA-LSP based on extensive experiments. The rest of the paper is organized as follows. Section 2 discusses related work, highlighting the difference between BAMOTR and the existing ones. Section 3 formulates the online taxi recommendation problem in BAMOTR. Section 4 propose two approach to solve this problem. Section 5 presents a balanced assignment mechanism. Section 6 illustrates experimental results. We final conclude the paper in Section 7. II. RELATED WORK Recommender systems focused on public transportation service have been studied extensively [1] [12]. Generally, these recommender systems can be divided roughly into two categories. The first category recommend taxis according to dynamic requests. The second category recommend taxis according to historic trajectories from which mobility of patterns of passengers and pick-up behaviors of drivers can be mined. For the first category of recommender systems, authors in [1], [2], [7] only focus on passenger waiting time when recommending taxis. when a new request is received, Liao [1] recommends the nearest taxi to the pick up passenger. Lee et al. [2] considers factors such as traffic situation in addition to various traffic related factors, and try to minimize passenger waiting time. Seow et al. [7] focus on minimizing total passenger waiting time by concurrent dispatching multiple taxis and allowing taxis to exchange their booking assignments. The taxi-sharing systems proposed in [3], [4], [8], [13], which allow occupied taxis to pick up new passengers on the fly, promising that increase the income of these taxis by ridesharing. Several studies in the second category have similarly be conducted extensively. Sequence of pick-up points learned from the historic trajectories are recommended to drivers in [9] [11]. Systems proposed in [5], [6] provide taxi drivers with actual driving routes connecting these points. Qian et al. [12] recommend driving routes construction with connected road sections to drivers according to a route assignment mechanism. Our problem is closely related to the first category where we recommend taxis according to dynamic requests. Solutions proposed in [3], [4], [8] are not favorable and optimally applicable to our problem as they do not take into consideration on fair and balanced assignment of taxis as well differences in income of the drivers. Similarly existing taxi recommender systems [1], [2], [7] do not put consideration into factors we have outlined above. Therefore, we propose BAMOTR considering fairness in taxi recommendation. Qian et al. [12] propose fair assignment mechanism on recommending routes to drivers, hence making their work completely different from ours as we are focusing on fair assignment mechanism on requests by passengers. To the best of our knowledge, the problem of recommending taxis for a group of passengers considering fair assignment of taxis has seldom been studied in the existing taxi recommender systems. There are challenges that we have taken into consideration in finding tradeoff, for instance ensuring fair assignment of drivers while minimizing differences in their income and at the same time shortening waiting time of the passenger. If we recommend taxis by only considering the waiting time, then we can recommend the earliest arriving taxi to passengers, but the question is can we guarantee the fairness in assignment of drivers? On the other hand, if we recommend taxis by only considering the fairness in assignment of drivers, again comes a question can we recommend the taxi with the minimum travel time to passengers, and guarantee the shortening waiting time? In this paper, we propose a balanced assignment mechanism to address the above challenges. III. PROBLEM FORMULATION A. Basic Concepts Definition 1. A road network is a directed graph G =(I,R), where I is a set of intersections and R I I is a set of road sections. An intersection I i denotes a road junction or road end. An road section R k =(I i,i j ) R represents from its initial intersection I i to its final intersection I j. To ease the following discussions, we denote the initial and final intersections of road section R k as R k.s and R k.e, respectively. Definition 2. A path P =< R 1,R 2...R k > is a sequence of k road sections, where R i R and R i R j if i j. The consecutive road section must share an intersection, i.e., R i.e = R i+1.s where 1 i k. Definition 3. A request set Req = {req 1,req 2...req m } is a group of m passenger requests. A request req i is associated with four main properties, the departure location req i.s, the destination req i.d, the request time req i.t and the travel cost TC i generated by the request, which is calculated according to the shortest path from req i.s to req i.d. Taking Shanghai as an example, if the shortest path from req i.s to req i.d is 5 km, The starting taxi billing price within 3 kilometers is 14 yuan and after that the transportation expenses per kilometer is 2.5 yuan, total travel cost TC i =14+(5-3)*2.5=19 yuan. Definition 4. A taxi driver set D = {D 1,D 2...D n } is a group of n competing taxi drivers. A taxi driver D j is associated with two main properties, The current location D j.loc and the accumulated income AI j of the taxi driver. 13

3 Variable TABLE I. DESCRIPTION OF MAJOR NOTATIONS Description m, n the numbers of requests and taxi drivers respectively. req i,d j the i-th request and the j-th taxi driver respectively. a regulatory factor, [, 1]. t i,j the travel time from D j to req i, t i,j =SP i,j /v. D ear, the earliest taxi to arrive req i. D near the nearest taxi from req i. TC i the travel cost generated by req i AI j the accumulated income of D j. EV A i,j the EVA value of D j for req i SP i,j,d i,j, the shortest path, the Euclidean distance and the LSP i,j local shortest path from D j to req i respectively. the i-th narrowed subrange in RRA that a circle with G i,g final center at req i and a radius of r i, and the final narrowed subrange in RRA. i=1,2... AImin i the minimum accumulated income of dirvers in G i, i=,1,2... For example, if a taxi driver D j have picked up two passenger request req 1 and req 2 so far, the accumulated income AI j =TC 1 + TC 2. For the ease of reference, we summarize the commonly used notations throughout the paper in Table I. B. Evaluation Function With respect to the above discussion, we know that we need to find the tradeoff between the fair assignment of drivers and the shortening of waiting time of passenger by assigning a right driver to a passenger request. Therefore, we propose an evaluation function which takes into account two evaluation principles. 1. Trade-off Principle. The fair assignment of drivers which means that minimize the differences of drivers income, and the shortening of waiting time of passengers such that conditions remains favorable to both passengers and drivers. We should tradeoff between them according to the previous discussion. 2. Priority Principle. Business is customer oriented so we need to develop a strategy such that we can ensure the satisfaction of passengers. Otherwise, customers may be lost. Naturally, the priority principle means that the shortening of waiting time of the passengers have higher priority than the fair assignment of drivers. Therefore, for a given passenger request req i, we need to evaluate which driver D j (j (1...n) ) is best suited to be recommended to pick up the passenger. According to the above two principles, we define the EVA function as: EV A i,j =(1 )AI j + e Δti,j (1) where [, 1] and Δt i,j = t i,j t i,ear j (1...n). In Eq. 1, AI j represents the accumulated income of D j so far. t i,j represents the travel time from D j to req i.itis obvious that t i,ear = min(t i,j ). Thus, Δt i,j is an extra time caused by recommending D j rather than D ear. Apparently, Δt i,j, holds true when D j is the earliest arriving driver. Quite a few advanced travel time prediction techniques [14] (e.g., incorporating real time traffic conditions) can be applied to estimate the travel time. However, since the traffic prediction TABLE II. AN EXAMPLE OF ONLINE TAXI RECOMMENDATION PROBLEM IN BAMOTR AI j t,j Δt,j EV A,j D D * D D D is not a focus of this paper, we calculate travel time t i,j = SP i,j /v for the sake of simplicity. The exponential processing of Δt i,j realizes the priority principle where is a regulatory factor. When is chosen such that its value approaches to 1, it means that we more focus on waiting time. On the other hand, when its value approaches to, it means that we care more on the difference of drivers income. C. Problem Formulation Definition 5. Online taxi recommendation problem in BAMOTR is defined as following. For a given request req i, we aim to recommend the optimal taxi with the minimum EVA value to pick up the passenger. Therefore, For a request req i, our objective is: min : EV A i,j (2) D j,j [1,n] To simplify the discussion, the problem is illustrated via an example. We assume that there are five drivers and a request req and set =.7. The AI j and t,j of each driver is described in Table II. Thus we can calculate Δt,j = t,j t,ear where t,ear =3, and calculate the EVA value of each driver according to Eq. 1. We will recommend D 2 to pick up req according to Eq. 2. Although D 1 is the earliest arriving driver, it has a large accumulated income. For D 4, although it possesses the minimum accumulated income, it still has the maximum EVA value because it will let the passenger to wait extra 5 minutes. These fulfil the trade-off principle and priority principle. By analyzing the taxi trajectories collected in Shanghai, China, we have the following statistical results. The distribution and cumulative distribution function (CDF) of the time of passengers on a taxi are plotted in Fig. 1. From the figure, it can be seen that about 8% of passengers time on a taxi is less than 2 minutes. This means that most of passengers destinations are not far from their departure. The distribution and CDF of travel cost generated by requests are depicted in Fig. 2. We calculate the travel cost according to the Shanghai taxi billing. It is observed that almost 6% of passengers travel cost is less than 2 Yuan. The waiting time of passengers has been analyzed in [12], it shows that almost 4% of taxis pick up a passenger in less than 5 minutes. The standard deviation of about 25 driver s accumulated income per day are plotted in Fig. 3, from the figure, it can be clearly observed that as the number of requests increases, the difference in income of the drivers also increases. 14

4 (a) (b) Fig. 1. Distribution and CDF of the time of passengers on a taxi. (a) (b) Fig. 2. Distribution and CDF of the travel cost generated by passengers. A. Brute-force Algorithm IV. METHODOLOGY For a given request, our goal is to find a taxi with the minimum EVA value, and recommend it to pick up the passenger. For obtaining the taxi, intuitively, we should calculate the EVA value of all taxis. Thus we must find the shortest path from each taxi to the passenger according to Eq 1. Obviously, this is a single source shortest path problem. Therefore, brute-force algorithm initially find the shortest path from each taxi to the passenger by Dijkstra s algorithm [15], then calculate the EVA value of all taxis, and recommend the optimal taxi to pick up the passenger. With brute-force algorithm, we can get the optimal recommendation results, because we traverse all taxis to find the optimal taxi. However, brute-force algorithm have a great computational overhead, for each passenger request req i,in order to calculate the EVA value of taxis, it uses Dijkstra s algorithm on the entire road network. Intuitively, for req i, if D j is far away from it, D j is almost impossible to be recommended, because the EVA value of D j increases exponentially as Δt i,j. When t i,ear is fixed, a big t i,j will cause a big Δt i,j, and a small increase in Δt i,j will result in rapidly increasing in EVA value. Since these taxis that are far away from req i are almost impossible to be recommended, how do we avoid searching for these taxis? A simple idea is that we only consider taxis within a local range. B. Range Refinement Algorithm According to the above discussion, we can only consider taxis within a local range to avoid searching for taxis that are far away from req i. However, how do we determine the extent of the local range? Suppose we randomly select a local range, such as a local range that a circle with center at req i and a radius of 3 km, but the optimal recommendation results will not be guaranteed. Therefore, there remain one question in this case, whether there is a way to find a local range Fig. 3. Standard Deviation of driver s accumulated income. which guarantee the same recommendation results as bruteforce algorithm, the solution of which is provided by range refinement algorithm (RRA). RRA firstly narrows the range by iterating and gets a subrange, in which RRA recommend a taxi to the passenger. Specifically, The subrange narrowed by RRA include the optimal taxi, which will be recommended to pick up the passenger (it can be proved by Theorem 1 and Properties 1 and 2). Not only does RRA narrow the scope, but also ensures the same recommendation results as brute-force algorithm. Some theorems are introduced first to describe how to narrow the scope in RRA. Before introducing these theorems, we introduce some variables. For a given request req i,we assume G is the initial range (G is the original road network), AImin is the minimum accumulated income of all drivers within G, and AI ear is the accumulated income of D ear. We also assume in RRA. In fact, if =,it means that we only focus on the difference of drivers income. Thus, we just need to recommend a taxi with the minimum accumulated income each allocation, which can be found by a simple traversal. Theorem 1. In RRA, for a given request req i,wecangeta subrange G that a circle with center at req i and a radius of r. The taxi, only within the range G, may be the optimal taxi. r is defined as r = SP i,ear + v ln (AIear AI 1+ min ) (1 ) (3) Proof: We first calculate the EVA value of the earliest arriving taxi D ear EV A i,ear =(1 )AI ear +e (ti,ear ti,ear) =(1 )AI ear + we also calculate the EVA value of other taxi D j EV A i,j =(1 )AI j + e (ti,j ti,ear) (SP =(1 )AI j + e i,j SPi,ear)/v Generally, it is not easy to get SP i,j for each driver D j (j (1...n)) unless use Dijkstra s algorithm on the entire road network. We assume d i,j is a Euclidean Distance between D j and req i. Apparently, SP i,j d i,j. we can calculate the EVA value of D j with d i,j EV A i,j =(1 )AI j + e (di,j SPi,ear)/v Thus, EV A i,j EV A i,j. Equality holds if SP i,j = d i,j.if EV A i,j >EVA i,ear, wehaveev A i,j >EVA i,ear, and D j 15

5 is not the optimal taxi. We assume a taxi with AI min is D s, and assume d i,s is the Euclidean Distance between D s and req i. We can calculate the EVA value of D s with d i,s EV A i,s =(1 )AI min + e (di,s SPi,ear)/v we know that there is possible for EV A i,s EV A i,ear only when EV A i,s EV A i,ear, otherwise EV A i,s >EVA i,ear because EV A i,s EV A i,s and EV A i,s >EVA i,ear. When EV A i,s EV A i,ear, then (1 )AI min + e (di,s SPi,ear)/v (1 )AI ear + e (di,s SPi,ear)/v (1 )(AI ear AI min ) d i,s SP i,ear + v ln (1 )(AIear AI min ) Equality holds only when EV A i,s = EV A i,ear, and we assume (1 )(AIear AI min d = SP i,ear + v ln ) +1 If d i,s >d, then EV A i,s >EVA i,ear, and we can confirm that D s is definitely not the optimal taxi. Now we will prove that for any taxi D j, the following proposition is true. If d i,j > d, then EV A i,j > EVA i,ear for each D j (j (1...n)). For D j, we known AI j AI min, and if d i,j >d,wehave EV A i,j =(1 )AI j + e (di,j SPi,ear)/v (1 )AI min + e (di,j SPi,ear)/v > (1 )AI min + e (d SPi,ear)/v = EV A i,ear Due to EV A i,j EV A i,j, then EV A i,j >EVA i,ear Consequently, for any taxi D j (j (1...n)), if the Euclidean Distance d i,j between D j and req i is greater than d, D j is definitely not the optimal taxi. We set r = d = SP i,ear + v ln (AIear AI 1+ min ) (1 ) Therefore, the optimal taxi must be within this range G that a circle with center at req i and a radius of r. Through Theorem 1, we can get a subrange which include the optimal taxi. Move over, there are two properties in RRA. Property 1. The narrowed subrange G can be continuously narrowed iteration until AI i 1 min is equal to AIi min Proof: Through Theorem 1, we can get a subrange G from G. Generally, we can get G i+1 from G i, let AImin i is the minimum accumulated income of all drivers within G i, and assume the radius of G i is r i, then the radius of G i+1 is (1 )(AIear AI i min r i+1 = SP i,ear + v ln ) +1 Due to G i G i 1, therefore AImin i AIi 1 min. Then r i+1 r i = v (ln <= (1 )(AIear AI i min ) +1 (1 )(AIear AIi 1 min ln ) +1 ) Therefor r i+1 r i. Equality holds if and only if AImin i = AI i 1 min, otherwise, r i+1 < r i, it means that the subrange G i can be further narrowed to be G i+1. Therefore, when AImin i = AIi 1 min, G i+1 is the same as G i, and the subrange will converge. We will get the final narrowed subrange G final. Property 2. The shortest path SP i,opt from the optimal taxi D opt to req i must be within G final. Proof: In Theorem 1, we known that if d i,j > r,we have EV A i,j > EVA i,ear, and D j is definitely not the optimal taxi. In fact, we can prove that if SP i,j > r,we have EV A i,j >EVA i,ear. Similarly to the proof in Theorem 1, For each D j, we known AI j AImin, and if SP i,j >r, we have (SP EV A i,j =(1 )AI j + e i,j SPi,ear)/v (1 )AImin (SP + e i,j SPi,ear)/v > (1 )AImin + e (r SPi,ear)/v = EV A i,ear Thus EV A i,j > EVA i,ear, and D j is definitely not the optimal taxi. if combining with the Property 1, we known that if SP i,j > r final, we have EV A i,j > EVA i,ear. Therefore, if a taxi is the optimal taxi D opt, The shortest path SP i,opt r final, which means SP i,opt must be within G final. According to Theorem 1 and Properties 1 and 2, RRA can provide the shortest path for the optimal taxi to pick up the passenger. This means that RRA have the same recommendation results as brute-force algorithm. However, RRA has a premise that the value of SP i,ear and AI ear is known. To calculate these values, we have to find D ear firstly. Bruteforce algorithm can find D ear by using Dijkstra s algorithm on the entire road network. But it spends a lot of time for finding D ear. To avoid the time cost for calculating the value of SP i,ear and AI ear, we propose the following algorithm, which get a subrange that is larger than subrange narrowed by RRA, and also guarantees the same recommendation result as brute-force algorithm. Algorithm 1 LSP. Input: Request req i, Driver D i, Road Network G Output: LSP i,j 1: Get the x-axis x 1 and y-axis y 1 of req i and the x-axis x 2 and y-axis y 2 of D j ; 2: LSP i,j ; 3: x min min(x 1,x 2 ); 4: x max max(x 1,x 2 ); 5: y min min(y 1,y 2 ); 6: y max max(y 1,y 2 ); 7: while LSP i,j = do 8: x min x min C; //C is a constant. 9: x max x max + C; 1: y min y min C; 11: y max y max + C; 12: G l a local rectangular range surrounded by (x min,y min ), (x min,y max), (x max,y min ) and (x max,y max); 13: LSP i,j the local shortest path from D j to req i in G l ; 14: end while 15: return LSP i,j ; 16

6 let r r = LSP i,near + v ln 1+ (AI lmax AI (SP i,ear + v ln (AIear AI 1+ min ) (1 ) ) = min ) (1 ) Fig. 4. An example of LSP. 1) Range Refinement Algorithm with Local Shortest Path Algorithm (RRA-LSP): To simplify the discussion, we first introduce the local shortest path algorithm (LSP) via an example and Algorithm 1 outlines LSP. Fig. 4 shows that there is a request req i and a driver D j, our goal is to quickly find a connectivity path from D j to req i. In reality, urban road network generally has relatively strong connectivity. Thus, we can quickly find a connectivity path from D j to req i with a relatively large probability in a local road network, which is main idea of LSP. Therefore, we first get a local range G 1 (the red range) according to Algorithm 1 such that if there is connectivity path from D j to req i, then stop. But actually we do not get a connectivity path in G 1. Thus, we re-calculate a local range G 2 (the blue range) according to Algorithm 1. Fortunately, we can get a connectivity path from req i to D j in G 2, which is the local shortest path LSP i,j from D j to req i. RRA-LSP initially finds a local shortest path LSP i,near from D near to req i by using LSP, and get a local range that a circle with center at req i and a radius of LSP i,near. In fact, we can directly find the shortest path SP i,near from D near to req i by using some shortest path algorithm, such as Dijkstra s algorithm [15] and A* algorithm [16], but LSP is more efficiency than these algorithm in this problem because LSP can quickly find a local shortest path. We assume AI lmax is the maximum accumulated income of drivers in this local range. Then we have the following Theorem 2. Theorem 2. In RRA-LSP, for a given request req i, we can get a subrange G that a circle with center at req i and a radius of r. The taxi, only within the range G, may be the optimal taxi. r is defined as r = LSP i,near + v ln 1+ (AI lmax AI min ) (1 ) (4) Proof: Due to LSP i,near is a local shortest path from D near to req i and D near is not certainly the earliest arriving taxi D ear,wehavelsp i,near SP i,near and SP i,near SP i,ear respectively. Therefore, LSP i,near SP i,ear. Move over, AI lmax is the maximum accumulated income of drivers in a range that a circle with center at req i and a radius of LSP i,near. The range include D ear. Otherwise, SP i,ear > LSP i,near. It means D ear is not the earliest arriving taxi. Therefore, D ear is within the range, and we have AI lmax AI ear. In Eq 2, we replace SP i,ear and AI ear by LSP i,near and AI lmax, respectively. then r = LSP i,near + v ln 1+ (AI lmax AI min ) (1 ) Thus, r r. It means we expand the narrowed subrange. Therefore, the optimal taxi must be within this range G that a circle with center at req i and a radius of r. From Theorem 2, we know that RRA-LSP expand the final narrowed range, which may bring some extra time cost when using Dijkstra algorithm in the final narrowed range. But it avoids the time cost for finding D ear. The experimental results show that the time cost of RRA-LSP is almost 24% of that of brute-force algorithm, and get the same recommendation result as brute-force algorithm. In addition, it is seem obvious that Properties 1 and 2 is still established in RRA-LSP. Algorithm 2 outlines RRA-LSP. Algorithm 2 RRA-LSP. Input: A Request req i, Driver Set D={D i }, Road Network G Output: Result=(D opt,req i ) 1: Result null; 2: D near the taxi nearest to req i ; 3: AImin the minimum accumulated income of drivers in G; 4: LSP i,near calculate the local shortest path from D near to req i by LSP; 5: G local a circle with center at req i and a radius of LSP i,near ; 6: AI lmax the maximum accumulated income of drivers in G local ; 7: k 1; 8: AImin k -1; 9: while AImin k AIk 1 min do 1: if AImin k then 11: k k+1; 12: end if 13: r k calculate radius according to Theorem 2 and Property 1; 14: G k a circle with center at req i and a radius of r k ; 15: AImin k the minimum accumulated income of drivers in G k; 16: end while 17: G final G k ; 18: D sub D j within G final ;//D j D; 19: calculate the path P i,j from D j to req i by Dijkstra s algorithm on G final ;//D j D sub ; 2: D sub filt(d sub); // removing disconnected drivers from D sub ; 21: calculate the EV A i,j of D j according to Eq. 1; // D j D sub ; 22: D opt find the optimal taxi from D sub ; 23: Result (D opt,req i ); 24: AI opt AI opt + TC i ; // update the accumulated income of D opt. 25: D D \ D opt; 26: return Result; 2) Example: To better understand Algorithm 2, we use an example shown in Fig. 5 to illustrate the process of RRA-LSP. In the example, we assume =.7 and v = 5m/min. As shown in Fig. 5(a), there is a request and lots of taxis in G,we first calculate the local shortest path LSP near from the nearest taxi to the request by Algorithm 1. Then we can find AI lmax and AImin. We assume that the values of LSP near, AI lmax and AImin are 12, 32 and 6, respectively. Thus, we can calculate r 1 = 3561 according to Eq. 4, and get a subrange G 1 that a circle with center at the request and a radius of r 1 as shown in Fig. 5(b). We then find AImin 1, and assume AImin 1 = 2. Now we should find whether AI1 min and AI min 17

7 Fig. 6. An example of handling Conflict. are equal according to Property 1. if AImin 1 = AI min,we can not continue to narrow the scope, and G 1 is the final narrowed subrange, otherwise there is AImin 1 >AI min,we can get a smaller subrange. In this example, it seems obviously that AImin 1 = 2 >AI min =6, thus, we can continue to narrow G 1, and calculate r 2 = 318. Thus, we get a smaller subrange G 2 shown in Fig. 5(c). We then find AImin 2, and assume AImin 2 = 2. In this iteration, AI2 min = AI1 min = 2. Thus, we can not continue to narrow the scope, and G 2 is the final narrowed subrange G final. We final recommend a taxi with a specific route to pick up the request in G final. The taxi must be the optimal taxi D opt, and the specific route must be the shortest path SP opt, which is proved in Theorems 1 and 2 and Properties 1 and 2. However, for brute-force algorithm, it will search the entire road network G to find D opt. Therefore, Not only does RRA-LSP greatly reduces the time overhead, but also ensures the same recommendation results as bruteforce algorithm. V. TAXI ASSIGNMENT MECHANISM For a given request req i, we aim to recommend the optimal taxi to pick up the passenger. However, the reality is always that there are multiple requests which need to be assigned at the same time. When we recommend taxis to multiple requests at the same time, there may be a conflict that a taxi is recommended to multiple requests, provided that the taxi have the minimum EVA for these requests. As shown in Fig. 6(a), D 1 is recommended req 1 and req 2 at the same time. Taking into consideration the stated conflict, we propose a realistic rule to solve it. Realistic Rule for Handling Conflict. When a taxi is recommended to multiple requests, the rule find one with the maximum travel cost from these requests, and recommend the taxi to pick up this request. The reason is that if a taxi is Fig. 5. An example of RRA-LSP. recommended to multiple requests, then the taxi has the right to choose, therefore the driver will choose a request with the maximum travel cost because he or she can earn more money. Under this principle, as shown in Fig. 6(b), req 2 will be recommended to D 1. Generally speaking, each time the conflict occurs, we give priority in allocating the request with the maximum travel cost. This means that we can sort the multiple requests according to the travel cost in descending order, and then assign these requests in turn. Therefore, we design a taxi assignment mechanism, considering the conflict, which recommend a taxi for each request. The assignment problem has been addressed largely by mechanism designers [17], [18], and authors of [12] design a route assignment mechanism to assign routes to taxis. As discussed in [12], a fair assignment mechanism for drivers should meet some conditions. And here, our taxi assignment mechanism should meet the following conditions. Firstly we need to consider that there is a fairness in assignment of drivers to requests. Secondly we should try to minimum waiting time for passenger as already discussed earlier. Incorporating these conditions we design a balanced assignment mechanism to taxis through the following steps: 1) At the beginning of the assignment, the accumulated income of each driver D j D is set to AI j =. 2) At stage k, the requests are sorted on their travel cost in descending order. We then recommend taxis for these requests in turn by RRA-LSP. It should be noted that a taxi which have been assigned, will no longer participate in the next allocation. 3) After the assignment at stage k, the accumulated income of driver D j is updated as AI j = AI j + CT i when the request req i is picked up by the driver D j. If a driver not pick up any request, the accumulated income of the driver will not be updated. The mechanism can reduce the difference in drivers income and guarantee the shortening of waiting time of passengers, which is also verified by experimental results. VI. EXPERIMENTAL EVALUATION To verify that BAMOTR reduces the difference of drivers income, and guarantee the shortening of waiting time of passengers, we conduct comprehensive experiments. Each experiment randomly generated passenger requests and taxi 18

8 Fig. 7. SDI: =.1. Fig. 8. SDI: =.1. Fig. 9. SDI: =.1. Fig. 1. SDI: =.4. Fig. 11. SDI: =.7. Fig. 12. SDI: =.9. Fig. 13. SDI: =.99. Fig. 14. SDI: =.999. passenger satisfaction, less waiting time will lead to higher passenger satisfaction. B. Compared Methods Fig. 15. SDI: number of assignment: 1. TABLE III. EVALUATION CONFIGURATIONS The number of drivers 5 The number of requests 15 The regulatory factor.1,.1,.1,.4,.7,.9,.99,.999 The number of assignments 1, , 1 drivers in the intersection based on historical taxi trajectory dataset. We use 3km/h as the driving speed in the experiments and set the constant C in LSP as 5 miles because most of road section is shorter than 5m. A. Setting Road Network. The simulation is based on the road network of Shanghai, which contains about 2242 road intersections and road sections. Trajectories. The trajectories were collected in Shanghai, China from approximately 4 probe-taxis operating over a period of 126 days [19]. A trajectory consists of a sequence of points. Each point contains seven fields: ID, timestamp, longitude, latitude, speed, angle, and status. The meaning of the first six fields is well understood. The last field is the current status of a taxi, indicating vacant and 1 for occupied. Standard Deviation of Driver s Income (SDI). We compute (AIj AI) SDI by SDI = 2 n (j (1, 2...n)). SDI can be used to estimate how much difference in drivers income, which can reflect the recommendation fairness. Passenger Waiting Time. Waiting time is the duration of time from the moment a passenger makes a request to the time when the passenger gets a taxi. It reflects the degree of In this paper, The proposed method RRA-LSP is compared with three approaches. The first one is the brute-force algorithm, which have been discussed previously in the paper. The second approach is greedy algorithm, which recommend a taxi to each request in a local range considered as a circle with center at the request and a radius of 3 miles according to the balanced assignment mechanism proposed in BAMOTR. If there are not taxis within this local range or the path from each taxi within this local range to the request is disconnect, greedy algorithm will recommend the nearest (Euclidean distance) taxi to the request. It should be noted that when a taxi is recommended to a request, the path from the recommended taxi to the request should be provided to calculate passenger waiting time. The last approach is NTR (Nearest Taxi Recommendation method), which is proposed in literature [1]. NTR directly recommend the nearest vacant taxi to pick up a passenger, and is one of the classic representative of existing methods, such as [1], [2], [7], which focusing on reducing waiting time consequently resulting in higher passenger satisfaction, however it doesn t guarantee for the fairness in assignment of drivers. As we known, NTR is also widely used in reality, for example Didi and Uber. By comparing RRA-LSP with NTR, we can verify that BAMOTR achieve fairer recommendation results than these existing methods. There are other taxi recommendation methods that can not be compared with BAMOTR. Such as [3], [4], [8], as they allow occupied taxis to pick up new passengers on the fly as opposed to our solution. The method in [5], [9], [12] recommend a path to taxi by mining passenger mobility patterns from the history trajectories. They do not recommend a taxi according to a dynamic request, but recommend a path to a taxi for reducing taxi idle driving time when there is no request. Therefore, the focus of these methods is different from BAMOTR. 19

9 C. Evaluation Results 1) Evaluation on SDI: In the experiment, the parameter settings are shown in Table III. The results are depicted in Figs There are several observations that can be made from the figures. Firstly, Brute-force and RRA-LSP have the same SDI no matter how much is. This is attributable to the fact that RRA-LSP has the same recommendation results as Bruteforce, which is consistent with the proof of Theorems 1 and 2 and Properties 1 and 2. Secondly, the SDI of NTR increases rapidly with the number of assignments, and independent of the value of. More over, the SDI of RRA-LSP, Brutefroce and Greedy is smaller and smaller than that of NTR with the number of assignments. It is becuase that RRA-LSP, Brute-froce and Greedy recommend taxis according to the balanced assignment mechanism in BAMOTR, but NTR just focus on waiting time, without considering the fair assignment of drivers. Thirdly, when is closer to, the SDI of RRA- LSP is smaller and increases more slowly with the number of assignments, and when is closer to 1, the SDI of RRA-LSP is more closer to NTR. This is because we care much about the difference in drivers income when is closer to, and we focus much on waiting time when is closer to 1. Finally, the SDI of these methods is plotted in Fig. 15 when the number of assignments is 1, it is clearly observed that except NTR, the SDI of other methods increases with. Fig. 16. Average Waiting Time. Fig. 17. Distribution of waiting time of NTR. Fig. 18. Average Running Time. Fig. 19. Running Time: =.1. 2) Evaluation on Waiting Time: The experimental set up is the same and the number of assignments is set to 1. Thus, the waiting time is an average of 1 allocations. The results are plotted in Fig. 16. As in the previous experiment, Brute-force and RRA-LSP have the same result in waiting time. The reason is the same as what is described in the previous experiment. We observe that when is smaller, the average waiting time of RRA-LSP is longer than NTR, this is because we are concerned much on the difference of drivers income when is smaller. We also observe that the average waiting time of RRA-LSP and Greedy decreases with, and the average waiting time of RRA-LSP is smallest when is bigger than.6. This means that in order to obtain a shorter waiting time, we need to increasing, however from the last experiment, we know that SDI of RRA-LSP increases as increases. Thus, we should select a proper to find tradeoff between SDI and waiting time, from the expermental data it can be seen that the value of for this case is equal to.7. The distribution of the waiting time of NTR is plotted in Fig. 17. It is similar as the real distribution of waiting time encountered in [12]. 3) Evaluation on Running Time: In this experiment, we explore the average running time of 1 allocations of each method which is depicted in Fig. 18. We observe that RRA- LSP has the least running time when larger than.2. On average, RRA-LSP incurs 4%, 48% and 286% more efficiency than NTR, Greedy and Brute-force, respectively. It means that RRA-LSP greatly reduces the time overhead relative to Brute- Force. In more detail, we explore the running time varies with the number of assignment when =.7. The results are plotted in Fig. 19. We observe that as the number of assignment increases, the running time of brute-force algorithm increases quickly, unlike methods whose running times increase slowly. D. Scalability More insights of scalability are presented for four approaches in this section. SDI, waiting time and running time are evaluated by changing the number of taxi drivers and the number of passenger requests. 1) The number of taxi drivers: In this experiment, we fix the number of assignment is 1, the number of passenger requests is 15 and =.7, and change the number of taxi drivers. As shown in Fig. 2, SDI decreases with the number of taxi drivers for four approaches. For NTR, SDI decreases obviously with the number of taxi drivers. For other methods, there is a slight decrease. However, even when the number of taxi drivers is equal to 1, RRA-LSP and Brute-force still achieve 51% and 456% better recommendation fairness than Greedy and NTR, respectively. We observe that the average waiting time of these approaches decreases as the number of taxi drivers increases, and RRA-LSP and Brute-force has the least waiting time. We also observe that except NTR, the average running time of these methods decreases with the number of taxi drivers. In addition, when the number of taxi drivers is 1, RRA-LSP is 12%, 25% and 33% more efficiency than NTR, Greedy and Brute-force, respectively. 2) The number of passenger requests: The effect of the number of passenger requests is depicted in Fig. 21, as the number of passenger requests increases, the SDI of NTR increases quickly, but the SDI of other methods just have a little increases. When the number of passenger requests is 3, RRA-LSP and Brute-force achieve 31% and 617% better recommendation fairness than Greedy and NTR, respectively. We observe that the waiting time of all methods has a little increase with the number of passenger requests. We also observe that the average running time of these methods increases with the number of passenger requests. Moreover, when the number of passenger requests is 3, RRA-LSP is 11

10 (a) SDI (b) Average waiting time (c) Average running time Fig. 2. Evaluation on the number of drivers: =.7, number of assignment: 1, number of requests: 5. (a) SDI (b) Average waiting time (c) Average running time Fig. 21. Evaluation on the number of requests: =.7, number of assignment: 1, number of drivers: 5. -2%, 79% and 231% more efficiency than NTR, Greedy and Brute-force, respectively. VII. CONCLUSION In this paper, we propose BAMOTR, a balanced assignment mechanism for online taxi recommendation, which is motivated by majority of existing taxi recommender systems that mainly focused on satisfaction of passengers without considering fairness in assignment of taxi drivers. In BAMOTR, a balanced assignment mechanism is proposed to solve a balance problem on how to find tradeoff fair assignment of drivers and shortening of waiting time of passengers? In addition, BAMOTR propose RRA-LSP, an efficient method, which ensures the same recommendation result as brute-force algorithm, however it is much more efficiency than bruteforce algorithm. Analysis of experimental results shows that BAMOTR achieve better recommendation fairness for taxi drivers than compared methods, which is verified by SDI. In future work, we look forward to consider the real-time traffic conditions so that we can fine-tune our method to fit in dynamic real situations. REFERENCES [1] Z. Liao, Real-time taxi dispatching using global positioning systems, Communications of the ACM, vol. 46, no. 5, pp , 23. [2] D.-H. Lee, H. Wang, R. Cheu, and S. Teo, Taxi dispatch system based on current demands and real-time traffic conditions, Transportation Research Record: Journal of the Transportation Research Board, no. 1882, pp , 24. [3] S. Ma, Y. Zheng, and O. Wolfson, T-share: A large-scale dynamic taxi ridesharing service, in Data Engineering (ICDE), 213 IEEE 29th International Conference on. IEEE, 213, pp [4] C. Tian, Y. Huang, Z. Liu, F. Bastani, and R. Jin, Noah: a dynamic ridesharing system, in Proceedings of the 213 ACM SIGMOD International Conference on Management of Data. ACM, 213, pp [5] N. J. Yuan, Y. Zheng, L. Zhang, and X. Xie, T-finder: A recommender system for finding passengers and vacant taxis, IEEE Transactions on Knowledge and Data Engineering, vol. 25, no. 1, pp , 213. [6] M. Qu, H. Zhu, J. Liu, G. Liu, and H. Xiong, A cost-effective recommender system for taxi drivers, in Proceedings of the 2th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 214, pp [7] K. T. Seow, N. H. Dang, and D.-H. Lee, A collaborative multiagent taxi-dispatch system, IEEE Transactions on Automation Science and Engineering, vol. 7, no. 3, pp , 21. [8] B. Cao, L. Alarabi, M. F. Mokbel, and A. Basalamah, Sharek: a scalable dynamic ride sharing system, in th IEEE International Conference on Mobile Data Management, vol. 1. IEEE, 215, pp [9] J. Huang, X. Huangfu, H. Sun, H. Li, P. Zhao, H. Cheng, and Q. Song, Backward path growth for efficient mobile sequential recommendation, IEEE Transactions on Knowledge and Data Engineering, vol. 27, no. 1, pp. 46 6, 215. [1] Y. Ge, H. Xiong, A. Tuzhilin, K. Xiao, M. Gruteser, and M. Pazzani, An energy-efficient mobile recommender system, in Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 21, pp [11] M. Zhang, J. Liu, Y. Liu, Z. Hu, and L. Yi, Recommending pick-up points for taxi-drivers based on spatio-temporal clustering, in Cloud and Green Computing (CGC), 212 Second International Conference on. IEEE, 212, pp [12] S. Qian, J. Cao, F. L. Mouël, I. Sahel, and M. Li, Scram: A sharing considered route assignment mechanism for fair taxi route recommendations, in Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 215, pp [13] Y. Huang, F. Bastani, R. Jin, and X. S. Wang, Large scale real-time ridesharing with service guarantee on road networks, Proceedings of the VLDB Endowment, vol. 7, no. 14, pp , 214. [14] J. Yuan, Y. Zheng, X. Xie, and G. Sun, Driving with knowledge from the physical world, in Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 211, pp [15] E. W. Dijkstra, A note on two problems in connexion with graphs, Numerische mathematik, vol. 1, no. 1, pp , [16] W. Zeng and R. Church, Finding shortest paths on real road networks: the case for a*, International journal of geographical information science, vol. 23, no. 4, pp , 29. [17] N. Nisan and A. Ronen, Algorithmic mechanism design, in Proceedings of the thirty-first annual ACM symposium on Theory of computing. ACM, 1999, pp [18] A. Abdulkadiroglu and T. Sönmez, School choice: A mechanism design approach, The American Economic Review, vol. 93, no. 3, pp , 23. [19] Suvnet-trace data, 111