Route Choice Model Considering Generalized Travel Cost Based on Game Theory

Abstract

This paper aims at testing the influence of emission factors on travelers’ behavior of route choice. The generalized travel cost is defined as the linear weighted sum of emission factors, travel time, and travel time reliability. The relational model of exhaust volume and traffic volume is established using the BPR (Bureau of Public Road) function to calculate the cost of travel regarding emission. The BPR function is used to measure the road segment travel time, while the reliability is used to quantify the cost of travel time fluctuation. At last, the route choice model considering the generalized travel cost is established based on the game theory. The calculating and analyzing of results under a miniature road network show that the weight coefficient of travel cost influences the travelers’ behavior of route choice remarkably and the route choice model which takes emission into account can reduce the exhaust of road network effectively, approximately 11.4% in this case.

1. Introduction

How to reduce exhaust emission has become a very urgent subject as the number of automobile grows rapidly in China. At the moment, research on energy saving and emission reduction about the vehicle itself has been relatively mature, while few studies were conducted concerning this problem from the perspective of traffic management. The influence of emission factors on travelers’ behavior of route choice will be studied in this paper from the aspect of traffic management, which can provide information about vehicle emission for traffic managers in order to decrease energy consumption and emission to some extent.

Travel time cost and fuel consumption are usually considered in travel cost estimation [1–3]. However, in general, former route choice models only took travel time cost into consideration. The researches on the relationship between route choice model and exhaust emission are few [4, 5]. And there is only information about route length and travel time in the traffic guidance system and no other valuable information concerning emission. Travel cost was usually described using travel time which was presented by the BPR function.

There are some route choice models of travelers such as stochastic user equilibrium model and elastic demand equilibrium model [6]. As the prisoner’s dilemma games and its extensions have been studied frequently [7–10], the paly game is also used in the traffic distribution [11]. However, their BPR functions do not well reflect different demands of travelers in choosing routes, because emission has gradually become a factor affecting route choice. And trip demand objectives are not single and are often mutually contradictory, for example, shorter travel time, higher reliability, and security [12, 13]. Hence, generalized travel cost, which took emission into consideration, was studied in the paper based on the emission model for road segment. Then, the effect of route choice behavior and cost factors on route choice of travelers was discussed.

2. Generalized Travel Cost Considering Emission

Travelers are paying more attention to the emission problem in route choice besides the consideration of travel time and its fluctuation, because of the aggravation of the energy crisis and the pressing need of environmental protection. It is the ideal goal of travelers to reduce emission, decrease travel time, increase reliability, and so forth [14]. However, these goals often lead to contradiction and conflicts. Travelers are always seeking compromise in pursuing them. Therefore, this paper is based on the generalized travel cost of the above factors. The generalized travel cost is defined as the linear weighted sum of travelers’ exhaust emission factor, travel time, and travel time reliability. So the generalized travel cost function may be represented as 𝑐_𝑎=𝑤₁𝑙_𝑎𝑓_𝑎(⋅)+𝑤₂𝑇_𝑎+𝑤₃𝛾𝑇_𝑎, (1) where 𝑐_𝑎 is the generalized travel cost in road segment 𝑎;𝑓_𝑎(⋅) represents the emission factor on road segment 𝑎, (g/km); 𝑙_𝑎 is the length of road segment 𝑎 (km); 𝑇_𝑎 is the travel time of road segment 𝑎 (s); 𝛾(𝑇_𝑎) is the travel time fluctuation of road segment 𝑎; 𝑤₁, 𝑤₂, and 𝑤₃ are the weight of exhaust emission, travel time, and travel time reliability in the generalized travel cost, respectively. These three weight coefficients may reflect travelers’ attitude towards hazards. The larger 𝑤₁ is, the smaller 𝑤₂ and 𝑤₃ will be, which means travelers pay more attention to emission, tending to consider emission as a route choice criterion. Each weight can be determined on the basis of SP (stated preference) survey.

The generalized travel cost of travelers can be represented as 𝑐^𝑤_𝑘=_𝑎∈𝐴𝑐_𝑎𝛿^𝑤_𝑎𝑘, (2) where 𝑐^𝑤_𝑘 is the generalized travel cost of route 𝑘 in OD pair 𝑤; 𝛿^𝑤_𝑎𝑘 is a 0-1 variable; if road segment 𝑎 lies in the route 𝑘 of the OD pair 𝑤, then 𝛿^𝑤_𝑎𝑘=1; otherwise 𝛿^𝑤_𝑎𝑘=0.

2.1. Quantification of Travel Cost Index

2.1.1. Travel Time

The BPR (Bureau of Public Roads) travel time function proposed by the United States Federal Highway Administration is the most representative achievement in the research of travel time model. The most significant influence of the model is on the field of traffic planning and it is most widely used in this field. It was initially used in the highway network planning, later in urban road network planning [15]. The mathematical expression is 𝑇_𝑎=𝑡_𝑎𝑥1+𝛽^𝑎𝐶_𝑎^𝑛, (3) where 𝑇_𝑎 is the travel time of road segment 𝑎, (s); 𝑡_𝑎 is the free flow travel time of road segment 𝑎, (s); 𝑥_𝑎 is the traffic flow of road segment 𝑎, (pcu/h); 𝐶_𝑎 is the traffic capacity of road segment 𝑎, (pcu/h); 𝛽, 𝑛 are the model parameters; the recommended values are 𝛽=0.15,𝑛=4 in highway network application.

2.1.2. Travel Time Fluctuation

In 1991, Asakura and Kashiwadani [16] presented the travel time reliability concept to reflect travel time fluctuation. Travel time reliability is defined as the probability that a trip may be completed within a specified time under a certain LOS (level of service) demand, which is a measure of travel time stability index and describes the flexibility of road network in stochastic traffic conditions. In this paper, the travel time reliability is defined as 𝑅𝑇_𝑎=𝑃𝑇_𝑎≤𝑡_𝑎+Δ^′, (4) where Δ^′ is the acceptable threshold of buffer time; it usually takes 5%, 10%, 15%, or 20% of the free travel time. For the simple reason that people have different tolerance degrees in various traffic congestions in different areas, Δ^′ may be determined through the SP investigation; other parameters have the same meanings as previously mentioned. Substitute (3) into (4) 𝑅_𝑎𝑇_𝑎=𝑃𝐶_𝑎≥𝑥_𝑎𝑡_𝑎𝛽Δ^′1/𝑛. (5) Due to the capacity 𝐶_𝑎 is a random variable, the distribution function 𝐹_𝐶𝑎(𝑥) can be obtained by field observation; it may be written as 𝑅_𝑎𝑇_𝑎=1−𝐹_𝐶𝑎𝑥_𝑎𝑡_𝑎𝛽Δ^′1/𝑛,𝑟𝑇_𝑎=1−𝑅_𝑎𝑇_𝑎=𝐹_𝐶𝑎𝑡_𝑎𝛽Δ^′1/𝑛𝑥_𝑎. (6) In the formula, 𝑟(𝑇_𝑎) is the probability of road segment 𝑎 not meeting the requirement of travel time reliability; it is deemed that 𝑟(𝑇_𝑎) is the negative utility of travel time fluctuation.

2.1.3. Exhaust Emission Model

Average speed of vehicles is one of the most important factors that affect the exhaust emission. Margiotta [17] believed that emission factors were sensitive to speed and put forward the relational model between them. Wang et al. [18] calculated emission factors of vehicles under different average speeds using the modified MOBILE model based on the field data in Nanjing and then carried out a curve fitting taking emission factors as the dependent variable and the average speed of motor vehicle as the independent variable. The polynomial computational model for traffic exhaust emission can be written as follows: 𝑓_𝑎=𝑏₀+𝑏₁𝑣_𝑎+𝑏₂𝑣²_𝑎+𝑏₃𝑣³_𝑎, (7) where 𝑓_𝑎 represents emission factor of passenger car on road segment 𝑎, (g/km); 𝑣_𝑎 is the average speed of vehicles on road segment 𝑎, (km/h),  0≤𝑣_𝑎≤90; 𝑏₀, 𝑏₁, 𝑏₂, and 𝑏₃ are regression coefficients.

Average speed of vehicles on a road segment can be calculated by travel time as follows, for the average speed of the vehicle equals road length divided by average travel time: 𝑣_𝑎=𝑙_𝑎𝑇_𝑎=𝑙_𝑎𝑡_𝑎1+𝛽𝑥_𝑎/𝐶_𝑎^𝑛, (8) where 𝑙_𝑎 is the length of road segment, (km); other parameters are of the same meaning as mentioned previously. The volume of exhaust emission can be expressed as the function of the length of road segment, traffic volume, and its capacity.

Pluging (8) into (7), we can get the following model of the traffic volume and exhaust emission: 𝑓_𝑎=𝑏₀+𝑏₁𝑙_𝑎𝑡_𝑎1+𝛽𝑥_𝑎/𝐶_𝑎^𝑛+𝑏₂𝑙_𝑎𝑡_𝑎1+𝛽𝑥_𝑎/𝐶_𝑎^𝑛2+𝑏₃𝑙_𝑎𝑡_𝑎1+𝛽𝑥_𝑎/𝐶_𝑎^𝑛3. (9)

2.2. Weight of Cost Index

Weight of different cost index may be determined on the basis of SP data, using the relative comparison method. First, make all indexes a square matrix. Second, compare every two indexes and score them, using [0,1] scoring method. Third, sum up each index score and do the normalization. The weight of index 𝑖 may be expressed as 𝑤_𝑖=∑^𝑛_𝑗=1𝑎_𝑖𝑗∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑎_𝑖𝑗, (10) where 𝑎_𝑖𝑗 is the importance of index 𝑖 relative to index 𝑗. Other parameters have the same meanings as mentioned previously.

2.3. Nondimensionalization of Indexes

To make different dimensional evaluation index comparable and additive, the method of mathematical transformation was used to eliminate the influence of index dimension in generalized travel cost. This paper uses linear proportion method to process nondimensionalization. The formula is 𝑦_𝑖=𝑥_𝑖𝑥^′_𝑖, (11) where 𝑥_𝑖 is the sample value; 𝑦_𝑖 is the dimensionless results of 𝑥_𝑖; and 𝑥^′_𝑖 is the minimum, maximum, or average value of the sample value.

3. Game Theory Model

The theory of games is a study about the interaction between rational decision makers. Each player has a number of strategies (feasible actions), which determines the outcome of the game and the payment to each player. In this game, a network user looks for a route to minimize the expected travel cost while an “evil entity” imposes road segment costs on the user so as to maximize the expected trip cost. This is assumed to be a two-player, noncooperative, and zero-sum game. The user guesses what road segment costs will be imposed and the evil entity guesses which route will be chosen. The mixed strategy that the Nash equilibrium proposed offers a useful measure for network reliability, since it yields the expected trip cost when the user is extremely pessimistic about the state of the network [19, 20].

We assume that situation 𝑗 implies performance degradation, failure, or attack on road segment 𝑖. The problem is to solve the following maximization and minimization model: min_𝑝𝑖max_𝑞𝑗𝐶=_𝑖𝑗𝑝_𝑖𝑐_𝑖𝑗𝑞_𝑗=_𝑖𝑗𝑘𝛿_𝑖𝑘ℎ_𝑘𝑐_𝑖𝑗𝑞_𝑗=_𝑘ℎ_𝑘𝑗𝑔_𝑘𝑗𝑞_𝑗,St._𝑗𝑞_𝑗𝑞=1_𝑗≥0,_𝑘𝛿_𝑖𝑘ℎ_𝑘=𝑝_𝑖,_𝑘ℎ_𝑘ℎ=1,_𝑘≥0, (12) where 𝑝_𝑖 is the probability that road segment 𝑖 is chosen; 𝑞_𝑗 is the probability of situation 𝑗; 𝑐_𝑖𝑗 is the travel cost of road segment 𝑖 under situation 𝑗; 𝑔_𝑖𝑗 is the travel cost of path 𝑘 under situation 𝑗; ℎ_𝑘 is the probability path 𝑘 is chosen; 𝛿_𝑖𝑘 equals 1 if road segment 𝑖 is on path 𝑘, 0 otherwise.

To facilitate the application of the route choice model, the method of successive average (MSA) scheme is used and it can be described as follows.

Step 1. Initialize 𝑞_𝑗 for all situations 𝑗 and 𝑛←1.

Step 2. Set expected road segment costs to ∑_𝑗𝑐_𝑖𝑗𝑝_𝑗 for all road segments 𝑖.

Step 3. Build the least expected cost path; 𝑥_𝑖←1 if road segment 𝑖 is on this path, and 0 otherwise.

Step 4. 𝑝_𝑖←(1/𝑛)𝑥_𝑖+(1−1/𝑛)𝑝_𝑖 for all road segments 𝑖.

Step 5. Find the 𝑗 which maximizes ∑_𝑖𝑐_𝑖𝑗𝑝_𝑖; 𝑦_𝑗←1; for all situations 𝑘≠𝑗𝑦_𝑘←0.

Step 6. 𝑞_𝑗←(1/𝑛)𝑦_𝑖+(1−1/𝑛)𝑞_𝑗 for all situations 𝑗.

Step 7. 𝑛←𝑛+1 and return to Step 2 until a satisfactory convergence is achieved.

4. Numerical Example

A numerical example was chosen to verify the applicability of the model proposed in this paper. As shown in Figure 1, it includes 4 nodes, 5 road segments, and 1 OD pair (1→4). The value of OD pair is 2600 pcu/h. In BPR function, parameters 𝛽=0.15,𝑛=4, road segments length, free travel time, capacity and emission parameters, and so forth are shown in Table 1. In the numerical example, Δ^′ takes 15% of the free travel time 𝑡_𝑎. Matlab 7.0 was used in the calculations.

Table 1 

Attribute data of road network unit.

Figure 1 

Test road network.

The results show that the different attitudes of travelers towards travel cost have obvious effects on road network traffic assignment. From the results of Table 2, it can be found that when 𝑤₁ is gradually increasing until it approaches 1, which means gradually increase the weight of emission factor in travel cost, the travelers tend to select emission as a criterion of route choice. Meanwhile, the travelers will select the low emission route as their priority selection. Because route 1–4 is the longest, it is 1.68 times as long as route 2 and 1.33 times as long as route 3–5, and it has the largest emission in terms of unit distance, so the emission cost of route 1–4 is the highest. When 𝑤₁ is 1, the distributed traffic capacity on route 1–4 is the lowest; it is only 653 pcu/h, but the traffic assignments on other routes are 1043 pcu/h and 904 pcu/h. As a result, the emission of the whole local road network can be reduced by 11.4%. When 𝑤₂ increases and approaches 1, it indicates that travelers tend to select travel time as a criterion of route choice, and travelers tend to select the lower time-consuming route. Due to the fact that route 2 takes the shortest travel time, route 3–5 the next, and then route 1–4, when 𝑤₂ is 1, the distribution of traffic on route 2 is the highest, which is 1062 pcu/h, route 3–5 is 906 pcu/h, and route 1–4 is the lowest, which is only 662 pcu/h. When 𝑤₃ gradually approaches 1, it indicates that the travelers tend to treat reliability as a criterion. They will select the high-capacity and low-fluctuation route, and route 3–5 has the lowest capacity and highest fluctuation. Therefore, the assignment on route 3–5 is the smallest, which is only 795 pcu/h.

Table 2 

Volume under different weight coefficient (pcu/h).

5. Conclusions

On the basis of the research carried out on emission model, road segment travel time, travel reliability, and generalized travel cost are defined and quantified. Meanwhile, the traffic distribution model that considers emission is proposed based on game theory. By means of the calculation examples, it is proved that travelers’ different attitudes towards emission have obvious effects on route choice. The route choice model considering emission can distinctly reduce road network emission; in the example, it reduced by 11.4%. With the popularized application of ITS, further researches about the effects of information and other factors on route choice behavior should be carried out.

Acknowledgments

This research was supported by China Postdoctoral Science Foundation 2011M500676, National Education Ministry Humanities and Social Sciences Foundation 12YJCZH097, Heilongjiang Institute of Technology Doctoral Science Foundation 2011BJ05, Technology Research and Development Program of Shandong 2012G0020129, and Natural Science Foundation of Heilongjiang Province QC2011C060. The authors are also grateful to the anonymous referees for their helpful comments and constructive suggestions on an earlier version of the paper.