Physarum polycephalum assignment: a new attempt for fuzzy user equilibrium

Abstract

The fuzzy user equilibrium problem in urban traffic assignment has attracted much attention since its great theoretical significance and wide application. Based on the fact that travelers tend to choose the minimum-cost path between every origin–destination pair of the traffic network, an equilibrium is emerging over time. However, in the real world, travelers’ selection of paths is often fuzzy with the lack of global information. In this paper, by aid of the Physarum polycephalum algorithm, we propose a model for solving the fuzzy user equilibrium problem. P. polycephalum can build a bio-network and assign the flow according to the location and the size of the food source. Taking full advantage of this feature, the proposed model associates the traffic demand with the food source and unifies the bio-network and the traffic network. The solution of the fuzzy user equilibrium problem is the flow assignment in the bio-network. To test the performance of the proposed method, we conduct experiments on some traffic networks selected from recent related works. The results show that the proposed method is efficient.

Appendices

Appendix 1: Physarum-type algorithm

Appendix 2: Fuzzy set and fuzzy number

Definition 1

A fuzzy set $\widetilde{A}$ defined on a universe X may be expressed as:

$$\begin{aligned} \widetilde{A} = \left\{ {\left\langle {x,\mu _{\widetilde{A}} \left( x \right) } \right\rangle \left| {x \in X} \right. } \right\} \end{aligned}$$

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where $\mu _{\widetilde{A}} \rightarrow \left[ {0,1} \right]$ is the membership function of $\widetilde{A}$. The membership value $\mu _{\widetilde{A}} \left( x \right)$ describes the degree of $x \in X$ in $\widetilde{A}$.

Definition 2

A fuzzy set $\widetilde{A}$ of X is normal iff $\sup _{x \in X} \mu _{\widetilde{A}} \left( x \right) = 1$.

Definition 3

A fuzzy set $\widetilde{A}$ of X is convex iff $\mu _{\widetilde{A}} \left( {\lambda x + \left( {1 - \lambda } \right) y} \right) \ge \left( {\mu _{\widetilde{A}} \left( x \right) \wedge \mu _{\widetilde{A}} \left( y \right) } \right) ,\; \forall x,y \in X,\forall \lambda \in \left[ {0,1} \right]$, where $\wedge$ denotes the minimum operator.

Definition 4

A fuzzy set $\widetilde{A}$ is a fuzzy number iff $\widetilde{A}$ is normal and convex on X.

Definition 5

A triangular fuzzy number $\widetilde{A}$ is a fuzzy number with a piecewise linear membership function $\mu _{\widetilde{A}}$ defined by:

$$\begin{aligned} \mu _{\widetilde{A}} = \left\{ \begin{array}{l} 0,\quad \quad \quad \quad x \le a_1 \\ \frac{{x - a_1 }}{{a_2 - a_1 }},\quad \;\;a_1 \le x \le a_2 \\ \frac{{a_3 - x}}{{a_3 - a_2 }},\quad \;\;a_2 \le x \le a_3 \\ 0,\quad \quad \quad \quad a_3 \le x \\ \end{array} \right. \end{aligned}$$

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which can be denoted as a triplet $\left( a_1, a_2, a_3 \right)$. A triangular fuzzy number $\widetilde{A}$ in the universe set X conforms to this definition shown in Fig. 4.

Fig. 4

A triangular fuzzy number $\widetilde{A}$

Based on Giachetti and Young (1997), fuzzy arithmetic on triangular is shown as follows.

Definition 6

Assuming that both $\widetilde{A} = \left( a_1, a_2, a_3 \right)$ and $\widetilde{B} = \left( b_1, b_2, b_3 \right)$ are triangular numbers, then the basic fuzzy operations are:

$$\begin{aligned} \widetilde{A} \oplus \widetilde{B}= & {} \left( {a_1 + b_1 ,a_2 + b_2 ,a_3 + b_3 } \right) \quad \mathrm{for \; addition,} \end{aligned}$$

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$$\begin{aligned} \widetilde{A} \ominus \widetilde{B}= & {} \left( {a_1 - b_3 ,a_2 - b_2 ,a_3 - b_1 } \right) \nonumber \\&\quad \mathrm{for \; subtraction,} \end{aligned}$$

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$$\begin{aligned} \widetilde{A} \otimes \widetilde{B}= & {} \left( {a_1 \times b_1 ,a_2 \times b_2 ,a_3 \times b_3 } \right) \nonumber \\&\quad \mathrm{for \; multiplication,} \end{aligned}$$

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$$\begin{aligned} \widetilde{A} \oslash \widetilde{B}= & {} \left( {a_1 \; / \; b_3\, ,a_2 \; / \; b_2\, ,a_3 \; / \; b_1 \; } \right) \nonumber \\&\quad \mathrm{for \; division.} \end{aligned}$$

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For example, let $\widetilde{A} = \left( 8,10,12 \right)$ and $\widetilde{B} = \left( 4, 5, 6 \right)$ be two triangular fuzzy numbers. Based on Eq. (19), four basic operations can be derived as:

$$\begin{aligned} \widetilde{A} \oplus \widetilde{B}= & {} \left( {8,10,12} \right) \oplus \left( {4,5,6} \right) = \left( {12,15,18} \right) . \\ \widetilde{A} \ominus \widetilde{B}= & {} \left( {8,10,12} \right) \ominus \left( {4,5,6} \right) = \left( {2,5,8} \right) .\\ \widetilde{A} \otimes \widetilde{B}= & {} \left( {8,10,12} \right) \otimes \left( {4,5,6} \right) = \left( {32,50,72} \right) .\\ \widetilde{A} \oslash \widetilde{B}= & {} \left( {8,10,12} \right) \oslash \left( {4,5,6} \right) = \left( {8/6,2,3} \right) . \end{aligned}$$

The results of the above operations are depicted in Fig. 5.

Fig. 5

An example of fuzzy arithmetic operations on triangular fuzzy numbers $\widetilde{A} = \left( 8,10,12 \right)$ and $\widetilde{B} = \left( 4, 5, 6 \right)$

Recently, fuzzy distance, as a measure of distance between two fuzzy numbers, has gained much attention from researchers and been widely applied in data analysis, classification and so on (Guha and Chakraborty 2010; Sadi-Nezhad and Khalili Damghani 2010). In this paper, the ${\hbox {Dis}}_{p,q}$-distance proposed in Gildeh and Gien (2001) is adopted to measure the difference between two fuzzy numbers.

Definition 7

The ${\hbox {Dis}}_{p,q}$-distance, indexed by parameters $1< p < \infty$ and $0< q < 1$, between two fuzzy numbers $\widetilde{A}$ and $\widetilde{B}$ is a nonnegative function given by Gildeh and Gien (2001) and Mahdavi et al. (2009):

$$\begin{aligned}&{\hbox {Dis}}_{p,q} \left( {\widetilde{A}\mathrm{{,}}\widetilde{B}} \right) \nonumber \\&\quad = \left\{ \begin{array}{l} \left[ \left( {1 - q} \right) \int _0^1 \left| {A_\alpha ^ - - B_\alpha ^ - } \right| ^p \;\hbox {d}\alpha + q\int _0^1 \left| {A_\alpha ^ + - B_\alpha ^ + } \right| ^p\; \hbox {d}\alpha \right] ^{{1/p}} ,\quad p< \infty , \\ \left( {1 - q} \right) \mathop {\sup }\limits _{0< \alpha \le 1} \left( {\left| {A_\alpha ^ - - B_\alpha ^ - } \right| } \right) + q\mathop {\inf }\limits _{0 < \alpha \le 1} \left( {\left| {A_\alpha ^ + - B_\alpha ^ + } \right| } \right) ,\quad \quad p = \infty . \\ \end{array} \right. \nonumber \\ \end{aligned}$$

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The analytical properties of ${\hbox {Dis}}_{p,q}$ depend on the first parameter p, while the second parameter q of ${\hbox {Dis}}_{p,q}$ characterizes the subjective weight attributed to the end points of the support. Having q close to 1 results in considering the right side of the support of the fuzzy numbers more favorably. Since the significance of the end points of the support of the fuzzy numbers is assumed to be same, the $q = (1/2)$ is adopted in this paper.

According to studies by Mahdavi et al. (2009) and Hassanzadeh et al. (2013), with $p=2$ and $q=(1/2)$, the general form of fuzzy distance ${\hbox {Dis}}_{p,q}$ can be converted into different forms, as two fuzzy numbers $\widetilde{A}$ and $\widetilde{B}$ take different types.

For triangular fuzzy numbers $\widetilde{A} = \left( a_1, a_2, a_3 \right)$ and $\widetilde{B} = \left( b_1, b_2, b_3 \right)$, the fuzzy distance between them can be represented as:

$$\begin{aligned}&{\hbox {Dis}} \left( {\widetilde{A}\mathrm{{,}}\widetilde{B}} \right) \nonumber \\&\quad = \sqrt{\frac{1}{6}\left[ {\sum \limits _{i = 1}^3 {\left( {b_i - a_i } \right) ^2 } + \left( {b_2 - a_2 } \right) ^2 + \sum \limits _{i \in \left\{ {1,2} \right\} } {\left( {b_i - a_i } \right) \left( {b_{i + 1} - a_{i + 1} } \right) } } \right] }\nonumber \\ \end{aligned}$$

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Appendix 3: Deng and Hassanzadeh’s method (Deng et al. 2012; Hassanzadeh et al. 2013)

Given a triangular fuzzy number $\widetilde{A} = (a_1, a_2, a_3)$, Deng et al. obtain the crisp number $P(\widetilde{A})$ of $\widetilde{A}$ according to the following Eq. (25):

$$\begin{aligned} P(\widetilde{A})=\frac{1}{6}(a_1+4\times a_2+a_3) \end{aligned}$$

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Hassanzadeh et al. use Eq. (26) to transform the fuzzy number $\widetilde{A}$ to the crisp number $P(\widetilde{A})$.

$$\begin{aligned} P(\widetilde{A})=\sqrt{\frac{1}{2}\sum _{i=1}^n{(a_1^i)^2}+\frac{1}{2}\sum _{i=1}^n{(a_3^i)^2}} \end{aligned}$$

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where n is the upper index of the cuts. For example, we here consider $\widetilde{A}=(9, 15, 20)$ with $n=10$. Then, we have:

$$\begin{aligned} a_1^i= & {} \{19.5, 19, 18.5, 18, 17.5, 17, 16.5, 16, 15.5, 15\}\\ a_3^i= & {} \{15, 14.4, 13.8, 13.2, 12.6, 12, 11.4, 10.8, 10.2, 9.6\} \end{aligned}$$

in which $a_1^i-a_1^{i+1}=(a_2-a_1)/n$ and $a_3^i-a_3^{i+1}=(a_3-a_2)/n$. As a result, the crisp number of this fuzzy number is:

$$\begin{aligned} P(\widetilde{A})=47.64 \end{aligned}$$

comparing the Deng’s result:

$$\begin{aligned} P(\widetilde{A})=14.83. \end{aligned}$$