基于方格的道路网络中出租车乘客效应函数模型和优化策略

0 Introduction

Nowadays, taxi-hailing applications have been popular in China. It helps customers take a taxi quickly and comfortably. Taxi is an important transportation mode which always circles around the city to search for customers. The taxi occupying the road space will worsen air quality and aggravate the traffic congestion. In order to establish the relationship between taxi drivers and customers, Yang et al.[1] developed a model to determine the taxi movements on a given road network. On the basis of the assumption that each vacant taxi driver attempts to minimize his or her expected searching time to find a customer, the customer-search behavior of taxi services appeared in this model. Afterwards, the model was further improved by Wong et al.[2-5] to build a diversified model. Nevertheless, the high profit return vacant taxi drivers travelling to remote areas to pick up customers has been ignored.

Considering the profitability, Wong et al.[6] developed a taxi network model. They explicitly examined the effects of the perceived profitability on the customer-search behavior of vacant taxis, as well as the expected profit earned by the drivers picking up customers in particular zones. Yang et al.[7] further extended this concept considering the profit per unit time through incorporating the operational cost as well as the time consumption involved in taxi trips to maximize the profits from customer-search. However, the search behavior models in the preceding taxi network models are in logit form and have not been calibrated and validated. Wong’s logit model for calibration and verification based on the global positioning system (GPS) data was established from 460 urban taxis to predict the drivers’ partition of strategy selection in peak and off-peak periods [8]. Szeto et al.[9] extended the consideration into every hour in a day. They ignored the fact that vacant taxi drivers may meet a customer on the way to their destinations. To address the challenges, a game theoretic approach has been proposed to describe the taxi drivers’ strategic behaviors.

许多人认为“人不应该有情绪”，所以不肯承认自己有负面的情绪。要知道，人一定会有情绪的，压抑情绪反而带来更不好的结果，学会体察自己的情绪，是情绪管理的第一步。

Wong et al.[10] proposed a local customer-search model combining both the modeling principles of the logit-based search model and the intervening opportunity model. The logit-based modeling concept is used to handle the multi-directional cases. But the opportunity model did not consider the aspect of the customer utility. On the other hand, the equilibria of the bilateral taxi-customer searching and meeting on networks become an important issue. Yang et al.[11] proposed an equilibrium model to characterize such issues. The Nash Equilibrium implied the balance the utility functions between taxi drivers and customers. Unlike the previous researches, this paper considered the utility of the customers.

In the same issue of game strategy, Lee et al.[12] proposed a novel control strategy in order to facilitate the taxi-customer matching process in booking-free taxi service, and a limited information-sharing strategy (LISS) for the taxi-customer searching problem in booking-free taxi service, in which both the taxi and the customer are equipped with mobile devices to communicate with each other within limited searching ranges. The proposed LISS is based on the game theoretical formulation in which a learning algorithm is developed to find the pure Nash equilibrium. The strategy for taxi drivers is relatively complex, since the taxi drivers should do many calculations to decide how to work. Gan et al.[13] proposed a simplified and optimized strategy for taxi systems by reviewing the existing taxi system researches for modeling taxi system dynamics. The taxi system efficiency optimization problem was introduced, and a game theoretic approach for optimizing the efficiency of taxi systems was presented. But the model in [13] was too simple to calculate the taxi utility accurately and the utility of customers was not taken into account.

3.2.2 注射前再次核对注射剂量缺失 注射前核对的缺失容易造成注射过量或者不足，引起血糖控制不稳定，影响治疗效果。表3显示，300例患者中，注射前再次确认注射剂量者82例，只占27.3%。汤莉娜等［7］研究结果也反映了同样的问题。不核对剂量会引起胰岛素注射过量或不足，注射过量会引起低血糖，注射量不足会导致血糖控制不理想，从而增加了不必要的医疗开支。其中老年人的视力退化和记忆力不佳，导致注射剂量不准确，重复注射和遗漏注射的现象屡见不鲜。因此，医护人员应特别关注患者对剂量的把控，健康教育中要教会患者如何检查与核对，从而降低注射带来的风险。

In this paper, authors will strengthen the road network model according to the cells and will consider the concept of the customer utility. The innovation customer utility function model will be proposed and three situations will be analyzed in cell-based road network. The customers will make the decisions by the utility functions and optimal strategies to gain the best interests. Two independent variables of the speed of taxi and the distance between customer and taxi are introduced according to what customers focus on and care about. The authors observe the changing trend of the data and will obtain the conclusion that whether the customers should walk to the destination for a while to gain more utility value.

This paper comprises four further sections. The system model will be introduced in detail in Section 1. Section 2 will present the models of customer utility functions, including the definitions of the utility functions and the comparisons of different customer movement situations. Section 3 will give the optimal strategies to the customer to make the decisions according to the best utility function. Simulations will be performed with the speed and the distance as independent variables in Section 4. And Section 5 will provide the conclusions and some expectations in the future.

1 System model

The research is according to the model in [10] where the taxi drivers can search the local customers using a cell-based approach. The research district is divided into identical squares to form a cell-based network. It is assumed that the network circumstance is unobstructed.

Fig.1 shows an illustration of the cell-based network for an urban area. The length of each square is constant. The size of each cell can be adjusted subject to the search level of modeling accuracy and searching range. Meanwhile, taxi and customer movements between these cells are represented by bi-directional links connected between each pair of the neighboring cells [10].

In Fig.1, the average number of cells travelled by vacant taxi drivers before picking up a customer is used to describe the number of search decisions. On account of that the assumption that each driver will make one search decision in each cell, i is expressed as the long-term average for an individual driver in the mind set based on the past experience. i = 1 means that the taxi driver is expected to search for and successfully meet a customer in one cell. The taxi drivers and customers may know the locations according to the hailing-taxi application and the application will provide the distance between the taxi driver and the customer by GPS. Meanwhile, the customer utility makes the model being established with the consideration of the customers interest function.

RCA水质模型中的水质变量包括：溶解氧（DO）、1种形态的碳 （LDOC）、3种形态的氮（LDON，NH4T，NO23）、2种形态的磷（LDOP，PO4T）。 本次采用离线方式对大伙房水库富营养化进行模拟，水质模型从2017年4月1日开始运行至当年的12月1日，水质时间步长为10min，水动力时间步长设定为30s。

  

Fig.1 Illustration of the cell-based network

In order to make a calculation easily, i is used to replace the distance in Matlab simulation which is an approximate distance value.The variable definitions of basic parameters are shown in Table 1.

 

Table 1 Variables definitions

  

ParametersExplanation iNumber of cells in the road networksPure strategy of the customerdtcDistance between the taxi and customerdtdDistance between the taxi and destinationdcdDistance between the customer and destinationvtConstant speed of the taxivcConstant speed of the customerL1Taxi travelling time with customer waiting in the situL2Taxi travelling time with dtc ≤ dcdL3Taxi travelling time with dtc≥ dcdWTime of customer waiting for traffic lights or taxiWiTime of customer waiting for traffic lights or taxi in cell iGTime of customer walking to the destinationGiTime of customer walking to the destination in cell i

A taxi-customer market is a dynamic time-varying system where the vehicles and people are always moving. In this paper, the consuming time is converted into the cells of the customers or the travelling taxis. We transform the concept of time into the path length and the customers will walk to the destination until the taxi arrives. In the meantime, customers have already walked in a distance and the total distance can be divided into some segments. According to the speed of taxi and customer, the total distance can be calculated. And if the length of each cell(square) is determined, the number of cells of different distance can be proposed, where i presents the number of cells, which the customer go through before he/she takes a taxi.

本文主要使用来自中国政府发布统计的两套数据。其中一套来源于我国非国有企业的工业统计数据库，为2008—2015年国家统计局对全部国有、主营业务收入比500万元高的非国有企业的统计数据。对该数据库中的数据进行如下操作：

2 Customer utility function models

The cell-based taxi-customer road network and the customer strategy model have been built. In addition, the variables have been defined. In order to prove the proposed strategy and models more accurately, the following three situations should be considered: (1) the customer is in-situ waiting; (2) the meeting state between taxi and customer; and (3) the chasing state between taxi and customer. In the case of common sense, situation (2) with the customer walking to the destination will bring a greater utility which can be evaluated by the time consumption.

2.1 Customers waiting in-situ

The current researches usually adopt the game theoretic considerations to optimize the efficiency of Taxi Systems so as to maximize the efficiency of a taxi system through adjusting the taxi fare. To this end, the first step is to know how the system efficiency is affected by the fare price in taxi market [13]. In this section, a customer utility function model is established in cell-based road network based on an advanced transportation research [8,13], and will be formulated. Taxi service has long been an indispensable part of public transport in modern cities. Unlike other types of public transportation systems (e.g., bus and metro systems), the taxi system is a highly decentralized system operated by a large number of self-controlled taxi drivers who can freely decide their operation schedules and movements.

  

Fig.2 First situation of the customer

In Fig.2, a customer is waiting for the taxi after with a hailing taxi app usage. In this situation, no matter if the customer meets the taxi with a greeted way (walk to the taxi location) or the taxi has to catch up with the customer, the variations of the waiting time W and the taxi speed vt are the same. The speed of taxi is supposed to be constant. The customer will wait at the original location until taxi driver comes, then G=0 and W will satisfy:

 

so as to obtain the extremum U2.

U1=(W,vt)=(σ1·W)-1+(σ2·L1)-1

因此，在高校艺术类人才特别是设计类人才培养过程中，古代文明的文化借鉴、思考是重要的艺术修养来源，现代文明及当代设计根基都来源于此，当代设计风格中无不体现着古代文明的符号和元素。在培养学生知识扩展层面，必须引入中西方文化的融合学习，包含多国家的重要城市地标、历史、艺术风格、艺术家作品、文化产物、设计概念和理念等，要充分发挥互联网信息技术的应用性，拓宽视野，培养学生从中得到完整的、有效的解读，多维观察，提出见解，使得综合设计能力能够真正提升。

(1)

So, the utility function for the first situation shown in Eq.(1) can be calculated as follows:

 

(2)

The function can be obtained by:

1.1.1 水泥池 试验在武汉农业气象试验基地内的水泥池中进行。共用大小相等的18个2.0 m×3.0 m的水泥养殖池。每个池都设有进排水口，池内均养殖了虾藻，模拟虾苗生长的自然环境。

 

(3)

2.2 Customers walking to the destination

Next, two situations with the customers walking to the destination will be discussed, namely the meeting utility function of taxi and customer as well as the chase utility function of taxi and customer.

1.2.1 心脏支架。1）就目前支架而言，按来源分为国产及进口两种：①国产常见品牌：微创医疗（Firebird支架）、乐普医疗（同心Parter支架）、赛诺医疗（BUMA支架）、吉威医疗（爱克赛尔支架）等；②进口常见品牌：美敦力公司（Endeavor Resolute支架）、雅培公司（XIENCE XPedition支架）、波士顿科学（Promus支架）。

2.2.1 The meeting utility function of taxi and customer

When a customer wants to take a taxi with the preference to the better utility function, and he/she will take the front taxi between the customer and the taxis or the rear taxi between the customer and the destination. Now, it should testify the relationship between the utility function and the distance between the taxi and the customer.

The intersection point of U2 and U3 is also a positive number.

Fig.3 shows that a customer is waiting for the taxi after ordering on a taxi-hailing app. So there may be a kind of competition. The vehicles should choose the best or the optimal waiting customers according to the interests of the customers. In such situation, the distance between the taxi and the destination is less than that between the customers and the destination, i.e. there is dtd dcd.

  

Fig.3 Second situation of the customer

For balancing the interests of the customers, the utility function should be taken into account. Therefore, some other values will be calculated. The comprehensive cost of each node in the network should be considered as well. The difference of i depends on the length of the cell. According to the discussion above, i represents the number of cells that the customers will go through. Assumed that the customers move at a fixed speed vc. In each cell being passed to the distance, Gi can be calculated as:

 

and the whole waiting time consists of each cell distance i ( G=∑Gi) and the time of waiting for the traffic lights (W=∑Wi). Therefore, the overall utility function consists of each distance i calculated by all the waiting time (for traffic lights) plus the walking time. L2 represents the travelling time to the destination in this situation. Obviously, the more time it takes the customer to go to the destination, the lower value will the utility function obtain.

where σ1, σ2are parameters used for converting the time cost into monetary cost. According to Fig. 1, the travelling time of this situation with the customer waiting in-situ equals:

 

(4)

The subfunctions can be calculated as follows:

 

(5)

 

(6)

According to Fig.2, L2 can be calculated as:

 

(7)

and

 

(8)

Combining with Eq.(5), Eq.(6), Eq.(7), the utility function of Eq.(4) in the second situation will be:

 

(9)

where σ1,σ2,σ3 are the parameters used for converting the time cost into the monetary cost with the same effect in the utility functions of all the three situations.

Theorem 1 When the distance between the customer and taxi is close enough, there exists a largish utility function U2. When the distance between the customer and taxi increases gradually, there exists a minimum of the utility function. And after that, the utility function will be increasing with the increase of the distance.

Proof The utility function U2 in the second situation can be calculated and analyzed by Eq.(9), especially the extremum of the U2 and the partial derivative of dtc. By simplifying the utility function U2, there is:

 

(10)

where vt+vc is defined as X and this partial formula is equal to 0:

 

(11)

Then the former formula can be further calculated as:

σ2·(dcd·X-vc·dtc)2=vt·vc·σ3·(dtc)2

(12)

And there is:

 

(13)

where dtc represents the distance between the taxi driver and the customer, l is adopted to replace dtc. Specially, if there is a congestion or a red light, W function will be larger than that on an unobstructed road. The customer will get into the taxi until taxi comes. Then the customer does not need to walk anymore. This standard can be measured by the time consumption. W presents the customer’s waiting time and L1 presents the travelling time. The more time it takes the customer to wait, the lower value of utility function it will be. Therefore, the waiting customer’s utility function can be defined as follows:

Now, the minimum can be proved by substituting into Eq.(10) and calculating .

Since there is can be deduced obviously.

If the left hand side of the derivative value is greater than zero and the right hand side of the derivative value is less than zero, it will first increase then decrease so as to get the minimum.

2.2.2 Chase utility function of the taxi and customer

由表5可知，针对无压W1胎体配方，H304和T304焊料的焊接强度均值分别是125 MPa和126 MPa，基本一致，但H304焊料的焊接强度最高值为152 MPa，且其他焊料强度均不足100 MPa，可见最适合的焊接材料是H304和T304焊料。

There may be the condition that the customers walking to the destination in some period i to wait for the taxi at the same time. In Fig.4, there is a customer waiting for the taxi after ordering on a hailing taxi app. In such situation, it is assumed that the destination lies further away from the taxi than from the customer, i.e.dtd≥dcd. In this situation, however, the customer would not meet the taxi with a greeting way. On the contrary, the customer may walk towards the destination and the taxi driver will have to catch the customer with help of GPS.

  

Fig.4 Third situation of the customer

The percentage pi of the customer walking into any cell has the same meaning with the previous subsection and the whole waiting time consists of the cell distance i and the time of waiting for traffic lights: W=∑Wi. Analogously, the whole walking time consists of each segment of distance i: G=∑Gi, which can be obtained by:

 

where dtc represents the distance between the taxi and the customer, l can be used to replace dtc in simulating based on Matlab. The waiting time mainly caused by the red-light. And the utility function of Ui will be:

 

(14)

and there is:

 

(15)

L3 in Fig.3 can be calculated by:

2.2.7 休息与睡眠 要保证患者在手术前得到充分休息，手术前晚家属陪伴，减轻焦虑。完成手术前治疗后，可给患者适量镇静剂，如艾司唑仑片，但用药应在手术前用药4 h以上，减少药物协同作用，防止出现呼吸抑制状况。

 

(16)

So, the utility function U3 in the third situation will be obtained as follows:

 

(17)

Combining Eq.(10), Eq.(11), Eq.(12) with Eq.(13), the final utility function U3 can be presented as:

 

(18)

Finally, the former function in Eq.14 can be presented as:

“司米模式”，创造体验性营销模式，具有重要意义。除了销售，门店还要实现品牌的“表达”，并重视顾客消费过程中的体验感受，顾客的体验来自于某种经历对感觉、心灵和思想的触动，它把企业、品牌与顾客的生活方式联系起来，赋予顾客个体行动和购买时机更广泛的心理感受和社会意义，创造无数营销的成功案例。

 

(19)

Theorem 2 When the distance between the customer and taxi is close enough, there exists a largish utility function U3. When the distance between the customer and taxi increases, there will exist a minimum of the utility function. After that, the utility function value will increase with the distance increasing.

Proof The utility function U3 in the second situation can be calculated and analyzed by Eq.(15) to consider the extremum of the U3 and the partial derivative for dtc. By simplifying the utility function U3, there will be:

 

(20)

This partial formula equals to 0 and vt -vc is defined as Y, so there will be:

 

(21)

Then the former formula can be further calculated:

 

(22)

The extremum U3 can be obtained according to Eq.(22).

Let us prove that it is the minimum value, and is put into Eq.(20).

Proof The intersection point of U2 and U3 can be calculated in Eq.(9) and Eq.(19) as:

According to it can be deduced that

It is the same as the former concept. If the left hand side of the derivative value is greater than zero and the right hand side of the derivative value is less than zero, it will first increase and then decrease so as to obtain the minimum value.

Thus it can be seen that both the two functions have the minimum values.

3 Optimal strategies

According to sections above, the conclusion can be drawn that the customers walking to the destination can get more interests (utility functions) than those waiting in-situ. Now the relationship between the speed of taxi and the utility function should be verified.

Theorem 3 When vt is proportional to U, the utility function of the first situation will depend on the speed of the taxi. The higher speed a taxi is moving at, the greater value the utility function U will be. When the speed of a taxi vt is 0, the value of U will achieve the minimum value (the minimum utility function).

Proof As it is known, the first situation illustrates that the customer will wait in-situ. According to the utility function U1 of the first situation, Eq.(2) is calculated and analyzed. Eq.(3) is put into Eq.(2) to obtain the result as:

 

It can be seen that vt is proportional to U1. Therefore, the speed of the taxi determines the value of the utility function. The same results are got about utility function U2 and U3 according to the demonstrations.

Remark 1 Obviously, the faster is the taxi travelling, the less time it will cost the customer to wait for the coming taxi and the more benefits it will gain. The speed of the taxi, however, must be controlled in the limited range in order to ensure the safety of the transportation.

供应商与物资供应段、物资供应段与生产站段之间库存信息、生产信息、资金信息、运输信息高频率地精确交换是VMI模式有效实施的基础。为保证信息在整个供应链体系中高效运转，需借助EDI或Internet网络信息技术在供应商、物资供应段与生产站段之间构建起供应链管理系统。

Theorem 4 When the distance between the customer and taxi ranges from 0 to the customer will select the better utility function of U with the distance between the taxi and the destination being smaller than that between the taxi and the customer, namely the front taxi, and will walk to the destination.In contrast, when the distance between customer and taxi ranges from to the further distance, the customers will select the better utility function of U with the distance between the taxi and the destination further than that between the taxi and the customer, namely the rear taxis, and will walk to the destination.

And the formula can be obtained.

我一听，手上的多米诺骨牌抖了抖，差点把我三天的辛苦化为泡影。我忍不住低声咒骂，这个嘴上没把门的兔崽子，他这么一嗓子嚷下去，我妈铁定知道了。

 

(23)

Then there will be:

 

 

(24)

In order to simply the operation, vt+vc is defined as X and vt -vc as Y, σ2·vc+σ3·vt as B, σ2·σ3·dcd as C, σ2·σ3·vc as D. So, there is:

 

(25)

In Eq.(25), dtc is a independent variable. As is known to all, dtc>0, so the solution is got by calculating:

 

(26)

实验组糖尿病子宫肌瘤伴不孕患者的手术用时、手术出血量、住院时间均明显低于对照组，差异有统计学意义（P＜0.05）。 如表 1。

Since dtc is the positive number, the functional image can be outlined lying in the first quadrant. In the section above, the two function images are adopted to obtain the minimum values. Now we put the two minimum values into their original functions.

By calculating, there is Because of an inequality can be worked out as The function images can also be determined.

Remark 2 When a customer wants to take a taxi, he/she may select the front taxi between the customer and the destination or the rear one. In either case, the utility function of U2 and U3 are always bigger than the utility function of U1, which means waiting in-situ. Accordingly, when the distance of taxi and the customer is getting closer, it will cost the customer less time to wait for the coming taxi and gain more benefits. When the distance is larger than certain value, the customer may gain less benefits. Customer can make a decision to select a taxi with preference to the better utility function according to this point.

4 Simulations

Based on the three situations discussed above, different utility functions in different situations are obtained. But there are many same elements in the three utility functions. That is to say, every utility function consists of three parts, representing the time of waiting for traffic lights, the time of walking to the destination and the time of travelling to the destination by taxi, respectively.

Now, the two utility functions are compared based on Matlab to analyze which function of customers’ walking is the better way. Firstly, the utility functions of Eq.(2) and Eq.(9) are compared, which are essentially the comparisons of the customers waiting in-situ with the customers walking to the destination (dtc dcd).

In order to simplify the calculation, several variables are adopted and defined, which are assumed to be constant on the basis of the common sense and ideal situations. Since the parameters σ1, σ2, σ3 are used for converting the time costs into the monetary costs, the parameters equal to the same value: σ1=σ2=σ3=0.5.

Assumed that the time of waiting for the traffic lights is idealized, W in period i is 60 s and the international standard is unified as W=60 s/3 600 s·h-1=1/60·h-1. In order to unify the distance value and simplify the calculation, the concept of the number of cells is introduced. In this cell-based network of an urban area, the length of each square equals to the constant. The size of each cell can be adjusted according to the searching level of the modeling accuracy and searching range, and the length of each square is set to be 100 m. Then the initial average distance between the destination and customer is set to be 10 km. In order to conform to the concept of the segment of distance, there will be: dcd=10/0.1=100 and L1= dcd /vt =100/vt. It indicates that the distance between the customer and the taxi should be no more than 2 km according to the hailing-taxi applications, i.e. the taxi drivers can search the customer willing to hail a taxi in such scope. So, there will be dtc=2/0.1=20. Now, the meeting situation of the taxi and customer is considered with the cell of distance i being taken into account, where the variable i can be calculated from Fig.3 and there will be:

 

(27)

According to Eq.(27), the constant dtc and vt can be substituted to make the number of cell of distance i equal to 100/(vt -5). Thus, it can be seen that the speed of the taxi must more than 5 km·h-1. Otherwise, i will be negative. Therefore, the range of the speed in the city road is from 5 to 80 km·h-1, which means the taxi will not chase the customer at all.

4.1 Speed of the taxi as an independent variable

According to the driving experience, the range of the speed in the city road is from 0 to 80 km·h-1. The general walking speed is about 5 km·h-1. The function about U and vt can be established, where vt is defined as an independent variable and U as a dependent variable.Now the conclusion can be validated from Theorem 3.

Fig.5 shows the increasing trend, in which the value of utility function will increase with the speed of the taxi and the value of utility function U1 ranges from 0.07 to 48.2. The value of utility function U2 ranges from 239.7 to 319.8. In addition, the value of utility function U3 ranges from 249.6 to 322.1, and the taxi travels so fast that the customer can achieve satisfied utility.

According to Fig.5, the values of utility functions U2 and U3 are also greater than the value of utility function U1, no matter how fast the taxi is travelling. Then, it can be concluded that whether the distance between the taxi and the destination is smaller than that between the customer and the destination or not, the utility function will be greater with the customer walking to the destination.

The customers should walk towards the destination for a distance after ordering a taxi. Meanwhile, the taxi driver will communicate with the customer to confirm.

4.2 Distance between the taxi and customer as an independent variable

In this subsection, distance between the taxi and customer dtc is considered as the other independent variable. In this condition, the range of distance between the taxi and customer dtc is from 100 m to 2 000 m (0.1 km to 2 km). As before, the size of each cell can be adjusted according to the searching level of modeling accuracy and the searching range, and the length of each square is set to be 100 m. That is to say, the range of i is from 1 to 20.

The key point is that the speed of taxi is defined as 50 km·h-1. Although the definition is idealistic, this is to find another relationship between the independent variable and the dependent variable. All the necessary parameters and independent variables are substituted into Eq.(9). From the simulation and comparison, the conclusion will be drawn from Theorem 4 and Fig.6.

In Fig.6, the utility function U1 is diminishing according to the trend of curves. The utility function is a decreasing function where the value is gradually decreasing along with the number of cells (distance between taxi and customer at the original position). The range of the utility function U1 is from 155.9 to 60.2. So it can be deduced that the farther the taxi lies away from the customer, the smaller of the utility function of the customer will be.

The value of utility function U2 will range from 345.6 to 385.5 and the value of utility function U3 will range from 380.6 to 353.7. When the number of cells changes from 0 to 10, the value of utility function U2 is higher than value of utility function U3; when the number of cells changes from 10 to 20, the value of utility function U3 is higher than value of utility function U2. The intersection point between these two utility functions is from 10 to 11. The number of cells can be transformed into the distance between the taxi and customer with the unit of km.

The utility functions of U2 and U3 also have different monotonicity features. The utility functions of U2 and U3 are always higher than the utility function of U1 no matter how far the taxi is from the customer. And the utility functions of U2 and U3 have the same minimum. By means of the distance between the taxi and customer, the viewpoint can also be verified that the customer walking to the destination can gain much utility and achieve many benefits.

If the initial distance between the taxi and customer is from 0 to dt0c0, the customer can select the front taxi to gain much utility. Assumed that the initial distance between the taxi and customer is dt0c0, the customer may choose the rear taxi to the destination to obtain much utility.

  

Fig.5 Three utility functions of U (by speed of taxi)

 

 

Fig.6 Three utility functions of U (by distance)

5 Conclusions

The customer utility function models and relative optimal strategies in cell-based road network are proposed. Taxi drivers and customers have known the locations of each other based on the mobile phone network. Three situations in the cell-based network are analyzed and the utility functions are proposed. The customers will choose one of three situations adaptively. It is found that if a customer walks to the destination for a while, he/she will obtain more utility and benefits. If the taxi speed is fast enough, he/she may gain the best utility and benefits. Moreover, for different initial relative positions of the taxi and customer, there will be different utility functions (the other parameters are fixed). If the initial distance between the taxi and customer is from 0 to dt0c0(consisting of vt, vc, dcd ), the customer can choose the front taxi to save time and gain utility. If the initial distance between the taxi and customer is dt0c0 and more, the customer may choose the rear taxi to obtain more utility.

The customer utility function model proposed in this paper is idealized. In the actual city road, the traffic congestion and the network delay should be considered. In the future, the existed research will be extended to improve the customer utility function model in practical road network and to calculate the analytical solution.

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