Keywords

1 Introduction

With the development of the air passenger transport market, the capacity of diversified transportation modes at airports is facing more challenges. Among them, the efficient operation of airport taxi pick-up points is a key issue, but scholars are more inclined to study service level in ride area. Few people have paid attention to the driver’s decision-making before picking up passengers, and the issue of the income balance of different operating distances. Therefore, the correct establishment of a mathematical model, a selection strategy for drivers, and a reasonable arrangement of planning pick-up points are significant for improving the efficiency of taxi operations and maintaining the benefits of taxi drivers.

In terms of taxi decision and profit maximization, Zhang et al. established driver’s choice decision model based on time periods and multi-objective programming model based on queuing theory [1]. Wang established a judgment formula based on the comprehensive supply and demand relationship and profit relationship and achieved the long-distance and short-distance taxi driver income balance by dividing the level [2]. Lv et al. established a multi-objective programming model, which was solved using genetic algorithms to obtain a reasonable distribution scheme in airport with the highest riding efficiency [3]. Zheng established a fitting model by collecting relevant airport data and analysis and calculation methods, and a reasonable scheme is designed for the allocation of taxi resources [4].

The revenue of airport taxis is related to both whether the taxi carrying a long-distance or a short-distance passenger and whether a return taxi carrying a long-distance or a short-distance passenger. When the airports set the pick-up points, they should consider giving a “priority” to those drivers whose latest trip was a short distance, so that the revenue of these taxis is as balanced as possible. Secondly, the existing multi-point side-by-side taxi queuing service system [5] has not maximized the riding efficiency, so taxi boarding points should be set more reasonably and efficiently.

The structure of this paper is as follows. Section 2 introduces the establishment of hybrid strategy model, which simplifies the problems to the game process of waiting or not among driver groups. Section 3 gives the scheme to make short-distance passenger-carrying drivers get priority to make their income equal, and correspondingly gives the scheme of a pick-up point, which has two parallel lanes, to improve the efficiency of the pick-up point, and shows the effectiveness of this scheme.

2 Hybrid Strategy Model

2.1 Hybrid Strategy

The research of traffic psychology shows that factors such as the driver’s driving age, gender, risk perception ability, emotion, and decision-making style will have an impact on the driver’s driving decision [6]. Combining with the taxi drivers in the target airport of this article, we observed that the changes in the number of passengers at the airport and the driver’s expected income will also affect the driver’s decision.

To facilitate the following description, the following concepts are introduced here:

  • Income (\( W \)): Refers to the driver’s final income under the different options.

  • Estimated income (\( I \)): Refers to the income that the driver may obtain from entering the waiting area and successfully carrying passengers.

  • Time cost (\( C_{1} \)): Refers to the revenue lost during the waiting period from when the driver enters the waiting area to successfully carry passengers.

  • No-load cost (\( C_{2} \)): Refers to the no-load cost (gas fee) paid by the driver when he chooses an empty vehicle to return to the urban area and the possible loss of passenger income.

  • Other cost (\( C_{0} \)): additional costs incurred in other time periods.

For the taxi drivers, they have only two decision-making schemes. One is waiting in line for passengers (hereinafter referred to as Scheme A), and the other is to empty the taxi and return to the city to carry passengers (hereinafter referred to as Scheme B).

Therefore, from the standpoint of the taxi driver (self), maximizing revenue is the key to the decision. The driver’s competitors are the biggest distractions affecting the driver’s earnings. Thus, the problem can be simplified as a game process of choice between Scheme A and Scheme B among taxi drivers.

Figure 1 reflects the income calculation principle of different decision-making schemes [7].

Fig. 1
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Principle of income calculation

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The qualitative relationship between the income and cost is as follows:

$$ {\text{Scheme}}\,{\text{A}}:W = I - C_{1} - C_{2} - C_{0} $$
(1)
$$ {\text{Scheme}}\;{\text{B}}:W = C_{2} - C_{0} $$
(2)

Also, due to the objective existence of tourist peak-season and tourist off-season, the income and cost of drivers are not fixed, but fluctuate periodically with time.

In this step, we supposed that the income of “oneself” and the “competitor” after the game are Wxy, Wxy, where “x” represents “oneself” and “y” represents the “competitor”. Then, we supposed that the probability of “oneself” choosing scheme A is \( p\text{ }(0 < p < 1) \), so the probability of choosing scheme B is \( 1 - p \). Similarly, a “competitor” chooses scheme A with probability \( q\text{ }(0 < q < 1) \), so the probability of choosing scheme B is \( 1 - q \). Thus, the hybrid game matrix of “self” and “competitor” is shown in Table 1 (the content of the matrix is represented in brackets).

Table 1 Hybrid game matrix between “self” and “competitor”
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2.2 Utility Analysis

We assumed that the probability of the “competitor” choosing Scheme A is \( q \), then the utility function \( U(x,y) \) of “self” Scheme A and Scheme B are chosen (3) and (4) as follows, respectively:

$$ U(1,q) = W_{11} q + W_{12} (1 - q) $$
(3)
$$ U(0,q) = W_{21} q + W_{22} (1 - q) $$
(4)

Let \( U(1,q) = U(0,q) \), get the probability:

$$ q = \frac{{W_{22} - W_{12} }}{{W_{11} - W_{12} - W_{21} + W_{22} }} $$
(5)

So as to get the income expectation of “self” as:

$$ E = W_{11} q + W_{12} (1 - q) $$
(6)

In the same way, we assumed that the probability of “self” choosing Scheme A is P. Let \( U(p,1) = U(p,0) \), then the probability can be obtained:

$$ p = \frac{{W_{22}^{\prime } - W_{12}^{\prime } }}{{W_{11}^{\prime } - W_{12}^{\prime } - W_{21}^{\prime } + W_{22}^{\prime } }} $$
(7)

Decision-making suggestions are provided according to the above processes:

  • When the probability of the “competitor” choosing Scheme A is equal to \( q \), “self” can choose Scheme A or B. When the probability of the “competitor” choosing Scheme A is greater than \( q \), the “own” Scheme A is more dominant. On the contrary, the “self” Scheme B is more dominant.

  • When the ideal income of “self” is less than the income expectation \( E \), Scheme A should be selected, that is, queuing up passengers to obtain greater income. When the ideal income is greater than the income expectation \( E \), Scheme B should be selected, that is, empty the car without carrying passengers. When the two are equal, either Scheme A or B will work.

3 Income Equilibrium

3.1 Principle of Income Equilibrium

The types of passenger-carrying drivers can be divided into long-distance passenger-carrying and short-distance passenger-carrying. Among them, the revenue and cost of the two types of drivers over time are shown in Figs. 2 and 3.

Fig. 2
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Revenue of the two types of drivers change over time

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Fig. 3
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Cost of the two types of drivers change over time

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It can be seen from Fig. 3 that the profit level of long-distance passenger-carrying drivers is much higher than that of the short-distance passenger-carrying drivers. Besides, since the no-load cost and time cost of the long-distance passenger-carrying driver is lower than the short-distance passenger-carrying driver, the cost level is also lower than that of the short-distance passenger transportation.

As time goes by, the polar differences between the two sides will become larger and larger, which could disrupt the stability of the taxi economy market. At this time, the airport management department must provide a certain “priority” to the drivers whose latest trip is a short distance to ensure that the benefits of both parties are balanced [8], this is the income equilibrium.

The two sides of the game in this problem are the short-distance passenger driver and the long-distance passenger driver. To analyze this problem, the hybrid decision-making model established in Sect. 2 can be obtained. The mathematical expression of the equilibrium of the two parties’ income [7] is:

$$ W_{11} q + W_{12} (1 - q) = W_{11}^{\prime } p + W_{21}^{\prime } (1 - p) $$
(8)

It means that the income expectation of “self” is equal to the income expectation of the “competitor”, namely:

$$ E = E^{\prime } $$
(9)

3.2 “Priority” Scheme Design

We analyzed a model, which is a double-sided multi-point cross-tandem queue service system. In the scenario, we set there are two parallel lanes in the taxi ride area with this model. The model is a double-sided queuing system, and the pick-up points are cross-distributed, providing detour space for the vehicles from the rear. After entering the riding area, taxis could still enter any pick-up point and wait in line. When picking up passengers, they could choose to leave the riding area on the original road or take a detour. Passengers are diverted to the two sides of the two parallel lanes in the ride area through the dedicated passage, forming a line respectively, and the first passenger in the line could choose different pick-up points. The model diagram is shown in Fig. 4.

Fig. 4
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Double-sided multi-point cross-tandem queue service system

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To give a priority for short-distance passenger-carrying drivers, before the vehicles enter the pick-up points, they should be classified into four kinds: ordinary and priority with a long and short distance, then be arranged on both sides of the ride area respectively, as shown in Figs. 5 and 6.

Fig. 5
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Classification

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Fig. 6
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Priority queuing scheme for short-distance passenger transportation-ride area

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When a taxi arrived at the airport, the driver should select a long or short distance to go for the next trip. Then, if the latest trip is short, the driver could enter the priority lane. If not or this is the first trip from the airport today, the driver should enter the ordinary lane. Therefore, all vehicles can be divided into four categories: ordinary vehicles A that will carry long-distance passengers, priority vehicles B that will carry long-distance passengers, ordinary vehicles C that will carry short-distance passengers, and priority vehicles D that will carry short-distance passengers.

The priority of short-distance passenger-carrying return vehicles is reflected in the separate queuing channel.

Passengers, before entering the ride area, are divided into two types: long-distance passengers and short-distance passengers. The two types of passengers enter the corresponding ride area in a line and follow the instructions of the administrator to enter the pick-up point and wait for taxis.

3.3 Effect of Hybrid Strategy Model with “Priority”

Based on the priority queuing schemes in Figs. 5 and 6, the hybrid strategy model in Sect. 2.1, the hybrid game matrix of “self” and “competitor” is re-established [9] as Table 2.

Table 2 Hybrid game matrix between yourself and the vehicle in front (with priority)
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Obtaining \( E = E^{\prime} = \frac{7}{3} \) from Eq. (9) shows that the income expectation of “self” is equal to the income expectation of “competitor”, which means the income of short-distance passenger-carrying drivers and long-distance passenger-carrying drivers are balanced. It proves the scheme of priority queuing designed in this paper is effective.

4 Conclusions

This paper takes airport taxis as the research object and establishes a hybrid strategy model to be used into two problems. First, simplify the problem to the game process of whether the driver is waiting. Second, calculate the effectiveness of a scheme to balance the income of short-distance and long-distance passenger-carrying drivers. By analyzing above, the following conclusions can be drawn: through the hybrid strategy model, drivers can intuitively compare the final revenue expectation with their own income expectation, so as to choose the most favorable decision. The hybrid strategy model has good portability, it can adapt to different conditions by changing the main body of the game. Meanwhile, changing the scheme of the ride area, distinguishing taxi types, and giving priority to short-distance passenger-carrying drivers on an existing basis can effectively balance the benefits of the driver group.