Skip to main content

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. Adamatzky A (2012) Slime mold solves maze in one pass, assisted by gradient of chemo-attractants. IEEE Trans NanoBiosci 11(2):131–134

    Article  Google Scholar 

  2. Adamatzky A, Prokopenko M (2012) Slime mould evaluation of Australian motorways. Int J Parallel Emerg Distrib Syst 27(4):275–295

    Article  Google Scholar 

  3. Akiyama T, Yamanishi H (1993) Travel time information service device based on fuzzy sets theory. In: Proceedings., 2nd international symposium on uncertainty modeling and analysis, 1993. IEEE, pp 238–245

  4. Azadeh A, Moghaddam M, Khakzad M, Ebrahimipour V (2012) A flexible neural network-fuzzy mathematical programming algorithm for improvement of oil price estimation and forecasting. Comput Ind Eng 62(2):421–430

    Article  Google Scholar 

  5. Bar-Gera H (2010) Traffic assignment by paired alternative segments. Transp Res Part B Methodol 44(8):1022–1046

    Article  Google Scholar 

  6. Beckmann M, McGuire CB, Winsten CB (1956) In: Studies in the economics of transportation. p 226

  7. Burrell JE (1968) Multiple route assignment and its application to capacity restraint. In: Proceedings of 4th international symposium on the theory of traffic flow

  8. Chen A, Ryu S, Xu X, Choi K (2014) Computation and application of the paired combinatorial logit stochastic user equilibrium problem. Comput Oper Res 43:68–77

    MathSciNet  Article  MATH  Google Scholar 

  9. Chen D, Chen J, Jiang H, Zou F, Liu T (2014) An improved PSO algorithm based on particle exploration for function optimization and the modeling of chaotic systems. Soft Comput 19:3071–3081

    Article  Google Scholar 

  10. Deng Y (2015) Generalized evidence theory. Appl Intell 43(3):530–543

    Article  Google Scholar 

  11. Deng Y (2016) Deng entropy. Chaos Solitons Fractals 91:549–553

    Article  MATH  Google Scholar 

  12. Deng Y (2017) Fuzzy analytical hierarchy process based on canonical representation on fuzzy numbers. J Comput Anal Appl 22(2):201–228

    MathSciNet  Google Scholar 

  13. Deng Y, Chen Y, Zhang Y, Mahadevan S (2012) Fuzzy Dijkstra algorithm for shortest path problem under uncertain environment. Appl Soft Comput 12(3):1231–1237

    Article  Google Scholar 

  14. Deng Y, Liu Y, Zhou D (2015) An improved genetic algorithm with initial population strategy for symmetric TSP. Math Probl Eng 2015:212,794

    Google Scholar 

  15. Dial RB (1971) A probabilistic multipath traffic assignment model which obviates path enumeration. Transp Res 5(2):83–111

    Article  Google Scholar 

  16. Du W, Gao Y, Liu C, Zheng Z, Wang Z (2015) Adequate is better: particle swarm optimization with limited-information. Appl Math Comput 268:832–838

    MathSciNet  Google Scholar 

  17. Du WB, Ying W, Yan G, Zhu YB, Cao XB (2016) Heterogeneous strategy particle swarm optimization. IEEE Trans Circuits Syst II Expr Br PP(99):1–1. doi:10.1109/TCSII.2016.2595597

    Google Scholar 

  18. Du WB, Zhou XL, Lordan O, Wang Z, Zhao C, Zhu YB (2016) Analysis of the chinese airline network as multi-layer networks. Transp Res Part E Logist Transp Rev 89:108–116

    Article  Google Scholar 

  19. Dutta AJ, Tripathy BC (2012) On I-acceleration convergence of sequences of fuzzy real numbers. Math Model Anal 17(4):549–557

    MathSciNet  Article  MATH  Google Scholar 

  20. Et M (2012) On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers. Iran J Sci Technol (Sci) 36(2):147–155

    MathSciNet  MATH  Google Scholar 

  21. Ezzati R, Saneifard R (2010) A new approach for ranking of fuzzy numbers with continuous weighted quasi-arithmetic means. Math Sci 4(2):143–158

    MathSciNet  MATH  Google Scholar 

  22. Fan G, Zhong D, Yan F, Yue P (2016) A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers. Expert Syst Appl 44:289–303

    Article  Google Scholar 

  23. Farahani RZ, Miandoabchi E, Szeto W, Rashidi H (2013) A review of urban transportation network design problems. Eur J Oper Res 229(2):281–302

    MathSciNet  Article  MATH  Google Scholar 

  24. Friesz TL (1985) Transportation network equilibrium, design and aggregation: key developments and research opportunities. Transp Res Part A Gen 19(5):413–427

    Article  Google Scholar 

  25. Friesz TL, Kim T, Kwon C, Rigdon MA (2011) Approximate network loading and dual-time-scale dynamic user equilibrium. Transp Res Part B Methodol 45(1):176–207

    Article  Google Scholar 

  26. Frikha A, Moalla H (2015) Analytic hierarchy process for multi-sensor data fusion based on belief function theory. Eur J Oper Res 241(1):133–147

    MathSciNet  Article  MATH  Google Scholar 

  27. Fujii S, Kitamura R (2004) Drivers’ mental representation of travel time and departure time choice in uncertain traffic network conditions. Netw Spat Econ 4(3):243–256

    Article  MATH  Google Scholar 

  28. George B, Kim S (2013) Spatio-temporal networks: an introduction. In: Spatio-temporal networks. Springer, New York, pp 1–6

  29. Ghatee M, Hashemi SM (2009) Traffic assignment model with fuzzy level of travel demand: an efficient algorithm based on quasi-logit formulas. Eur J Oper Res 194(2):432–451

    MathSciNet  Article  MATH  Google Scholar 

  30. Ghezavati V, Nia NS (2014) Development of an optimization model for product returns using genetic algorithms and simulated annealing. Soft Comput 19:3055–3069

    Article  Google Scholar 

  31. Giachetti RE, Young RE (1997) A parametric representation of fuzzy numbers and their arithmetic operators. Fuzzy Sets Syst 91:185–202

    MathSciNet  Article  MATH  Google Scholar 

  32. Gildeh BS, Gien D (2001) La distance-dp, q et le cofficient de corrélation entre deux variables aléatoires floues. Actes de LFA 2001:97–102

    Google Scholar 

  33. Giri PK, Maiti MK, Maiti M (2014) Fuzzy stochastic solid transportation problem using fuzzy goal programming approach. Comput Ind Eng 72:160–168

    Article  Google Scholar 

  34. Guha D, Chakraborty D (2010) A new approach to fuzzy distance measure and similarity measure between two generalized fuzzy numbers. Appl Soft Comput 10(1):90–99

    Article  Google Scholar 

  35. Hassanzadeh R, Mahdavi I, Mahdavi-Amiri N, Tajdin A (2013) A genetic algorithm for solving fuzzy shortest path problems with mixed fuzzy arc lengths. Math Comput Model 57(1):84–99

    MathSciNet  Article  MATH  Google Scholar 

  36. Henn V (2000) Fuzzy route choice model for traffic assignment. Fuzzy Sets Syst 116(1):77–101

    Article  MATH  Google Scholar 

  37. Hu H, Li Z, Al-Ahmari A (2011) Reversed fuzzy petri nets and their application for fault diagnosis. Comput Ind Eng 60(4):505–510

    Article  Google Scholar 

  38. Jiang W, Luo Y, Qin X, Zhan J (2015) An improved method to rank generalized fuzzy numbers with different left heights and right heights. J Intell Fuzzy Syst 28:2343–2355

    MathSciNet  Article  MATH  Google Scholar 

  39. Jiang W, Wei B, Qin X, Zhan J, Tang Y (2016) Sensor data fusion based on a new conflict measure. Math Probl Eng. doi:10.1155/2016/5769061

    MathSciNet  Google Scholar 

  40. Jiang W, Wei B, Xie C, Zhou D (2016) An evidential sensor fusion method in fault diagnosis. Adv Mech Eng 8(3):1–7

    Google Scholar 

  41. Jiang W, Xie C, Luo Y, Tang Y (2016) Ranking z-numbers with an improved ranking method for generalized fuzzy numbers. J Intell Fuzzy Syst

  42. Jiang W, Xie C, Wei B, Zhou D (2016) A modified method for risk evaluation in failure modes and effects analysis of aircraft turbine rotor blades. Adv Mech Eng 8(4):1–16. doi:10.1177/1687814016644579

    Google Scholar 

  43. Jiang W, Zhan J, Zhou D, Li XA (2016) method to determine generalized basic probability assignment in the open world. Math Probl Eng 2016:11 doi:10.1155/2016/3878634

  44. Jones J, Adamatzky A (2014) Material approximation of data smoothing and spline curves inspired by slime mould. Bioinspir Biomim 9(3):036016

  45. Kammoun HM, Kallel I, Casillas J, Abraham A, Alimi AM (2014) Adapt-traf: An adaptive multiagent road traffic management system based on hybrid ant-hierarchical fuzzy model. Transp Res Part C Emerg Technol 42:147–167

    Article  Google Scholar 

  46. Kauffman A, Gupta MM (1991) Introduction to fuzzy arithmetic: theory and application. Van Nostrand Reinhold, New York

    Google Scholar 

  47. Liang D, Pedrycz W, Liu D, Hu P (2015) Three-way decisions based on decision-theoretic rough sets under linguistic assessment with the aid of group decision making. Appl Soft Comput 29:256–269

    Article  Google Scholar 

  48. Liu HC, You JX, Fan XJ, Lin QL (2014) Failure mode and effects analysis using D numbers and grey relational projection method. Expert Syst Appl 41(10):4670–4679

    Article  Google Scholar 

  49. Liu HC, You JX, You XY, Shan MM (2015) A novel approach for failure mode and effects analysis using combination weighting and fuzzy vikor method. Appl Soft Comput 28:579–588

    Article  Google Scholar 

  50. Liu Y (2014) A method for 2-tuple linguistic dynamic multiple attribute decision making with entropy weight. J Intell Fuzzy Syst Appl Eng Technol 27(4):1803–1810

    MathSciNet  MATH  Google Scholar 

  51. Mahdavi I, Nourifar R, Heidarzade A, Amiri NM (2009) A dynamic programming approach for finding shortest chains in a fuzzy network. Appl Soft Comput 9:503–511

    Article  Google Scholar 

  52. Manimala K, David IG, Selvi K (2014) A novel data selection technique using fuzzy C-means clustering to enhance SVM-based power quality classification. Soft Comput 19:3123–3144

    Article  Google Scholar 

  53. Manual TA (1964) Bureau of public roads. US Department of Commerce, Washington, p 113

  54. Mardani A, Jusoh A, Zavadskas EK (2015) Fuzzy multiple criteria decision-making techniques and applications—two decades review from 1994 to 2014. Expert Syst Appl 42(8):4126–4148

    Article  Google Scholar 

  55. Nakagaki T, Iima M, Ueda T, Nishiura Y, Saigusa T, Tero A, Kobayashi R, Showalter K (2007) Minimum-risk path finding by an adaptive amoebal network. Phys Rev Lett 99(6):068104

    Article  Google Scholar 

  56. Nakagaki T, Yamada H, Tóth Á (2000) Intelligence: Maze-solving by an amoeboid organism. Nature 407(6803):470

    Article  Google Scholar 

  57. Nakagaki T, Yamada H, Toth A (2001) Path finding by tube morphogenesis in an amoeboid organism. Biophys Chem 92(1):47–52

    Article  Google Scholar 

  58. Nikolova E, Stier-Moses NE (2014) A mean-risk model for the traffic assignment problem with stochastic travel times. Oper Res 62(2):366–382

    MathSciNet  Article  MATH  Google Scholar 

  59. Ning X, Yuan J, Yue X (2016) Uncertainty-based optimization algorithms in designing fractionated spacecraft. Sci Rep 6:22979

    Article  Google Scholar 

  60. Ning X, Yuan J, Yue X, Ramirez-Serrano A (2014) Induced generalized choquet aggregating operators with linguistic information and their application to multiple attribute decision making based on the intelligent computing. J Intell Fuzzy Syst 27(3):1077–1085

    MathSciNet  MATH  Google Scholar 

  61. Ning X, Zhang T, Wu Y, Zhang P, Zhang J, Li S, Yue X, Yuan J (2016) Coordinated parameter identification technique for the inertial parameters of non-cooperative target. PloS ONE 11(4):e0153,604

    Article  Google Scholar 

  62. Ocalir EV, Ercoskun OY, Tur R (2010) An integrated model of GIS and fuzzy logic (FMOTS) for location decisions of taxicab stands. Expert Syst Appl 37(7):4892–4901

    Article  Google Scholar 

  63. Pedrycz W, Al-Hmouz R, Morfeq A, Balamash AS (2014) Building granular fuzzy decision support systems. Knowl Based Syst 58:3–10

    Article  Google Scholar 

  64. Ramazani H, Shafahi Y, Seyedabrishami S (2011) A fuzzy traffic assignment algorithm based on driver perceived travel time of network links. Sci Iran 18(2):190–197

    Article  MATH  Google Scholar 

  65. Ridwan M (2004) Fuzzy preference based traffic assignment problem. Transp Res Part C Emerg Technol 12(3):209–233

    Article  Google Scholar 

  66. Sadi-Nezhad S, Khalili Damghani K (2010) Application of a fuzzy topsis method base on modified preference ratio and fuzzy distance measurement in assessment of traffic police centers performance. Appl Soft Comput 10(4):1028–1039

    Article  Google Scholar 

  67. Smith M (1979) The existence, uniqueness and stability of traffic equilibria. Transp Res Part B Methodol 13(4):295–304

    MathSciNet  Article  Google Scholar 

  68. Smith M, Hazelton ML, Lo HK, Cantarella GE, Watling DP (2014) The long term behaviour of day-to-day traffic assignment models. Transp A Transp Sci 10(7):647–660

    Google Scholar 

  69. Talebian A, Shafahi Y (2015) The treatment of uncertainty in the dynamic origin–destination estimation problem using a fuzzy approach. Transp Plan Technol 38(7):795–815

    Article  Google Scholar 

  70. Tero A, Kobayashi R, Nakagaki T (2007) A mathematical model for adaptive transport network in path finding by true slime mold. J Theor Biol 244(4):553–564

    MathSciNet  Article  Google Scholar 

  71. Tero A, Takagi S, Saigusa T, Ito K, Bebber DP, Fricker MD, Yumiki K, Kobayashi R, Nakagaki T (2010) Rules for biologically inspired adaptive network design. Science 327(5964):439–442

    MathSciNet  Article  MATH  Google Scholar 

  72. Tripathy BC, Borgohain S (2011) Some classes of difference sequence spaces of fuzzy real numbers defined by orlicz function. Adv Fuzzy Syst 2011:6. doi:10.1155/2011/216414

  73. Tripathy BC, Das PC (2012) On convergence of series of fuzzy real numbers. Kuwait J Sci Eng 39.1A:57–70

  74. Tripathy BC, Debnath S (2013) \(\gamma \)-open sets and \(\gamma \)-continuous mappings in fuzzy bitopological spaces. J Intell Fuzzy Syst Appl Eng Technol 24(3):631–635

    MathSciNet  MATH  Google Scholar 

  75. Tripathy BC, Ray GC (2012) On mixed fuzzy topological spaces and countability. Soft Comput Fusion Found Methodol Appl 16(10):1691–1695

  76. Tripathy BC, Sarma B (2012) On I-convergent double sequences of fuzzy real numbers. Kyungpook Math J 52(2):189–200

    MathSciNet  Article  MATH  Google Scholar 

  77. Tsai SB, Chien MF, Xue Y, Li L, Jiang X, Chen Q, Zhou J, Wang L (2015) Using the fuzzy DEMATEL to determine environmental performance: a case of printed circuit board industry in taiwan. PloS ONE 10(6):e0129,153

    Article  Google Scholar 

  78. Wang HF, Liao HL (1999) User equilibrium in traffic assignment problem with fuzzy N–A incidence matrix. Fuzzy Sets Syst 107(3):245–253

    MathSciNet  Article  MATH  Google Scholar 

  79. Wardrop JG (1952) Road paper. Some theoretical aspects of road traffic research. In: ICE Proceedings: engineering divisions, vol 1. Thomas Telford, pp 325–362

  80. Yang C, Bruzzone L, Sun F, Lu L, Guan R, Liang Y (2010) A fuzzy-statistics-based affinity propagation technique for clustering in multispectral images. IEEE Trans Geosci Remote Sens 48(6):2647–2659

    Article  Google Scholar 

  81. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  82. Zavadskas EK, Antuchevicience J, Hajiagha SHR (2015) The interval-valued intuitionistic fuzzy MULTIMOORA method for group decision making in engineering. Math Probl Eng 2015:560,690

    Article  Google Scholar 

  83. Zhang B, Dong Y, Xu Y (2013) Maximum expert consensus models with linear cost function and aggregation operators. Comput Ind Eng 66(1):147–157

    Article  Google Scholar 

  84. Zhang B, Dong Y, Xu Y (2014) Multiple attribute consensus rules with minimum adjustments to support consensus reaching. Knowl Based Syst 67:35–48

    Article  Google Scholar 

  85. Zhang Y, Zhang Z, Deng Y, Mahadevan S (2013) A biologically inspired solution for fuzzy shortest path problems. Appl Soft Comput 13(5):2356–2363

    Article  Google Scholar 

  86. Zheng H, Chiu YC, Mirchandani PB (2013) On the system optimum dynamic traffic assignment and earliest arrival flow problems. Transp Sci 49(1):13–27

  87. Zhou D, Zhou H (2014) A modified strategy of fuzzy clustering algorithm for image segmentation. Soft Comput 19:3261–3272

    Article  Google Scholar 

  88. Zhou Z, Chen A, Bekhor S (2012) C-logit stochastic user equilibrium model: formulations and solution algorithm. Transportmetrica 8(1):17–41

    Article  Google Scholar 

Download references

Acknowledgements

The authors greatly appreciate the reviewer’s suggestions and the editor’s encouragement. The work is partially supported by National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA013801), National Natural Science Foundation of China (Grant Nos. 61174022, 61573290, 61503237), China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No.BUAA-VR-14KF-02).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yong Deng.

Ethics declarations

Conflicts of interest

Yang Liu declares that he has no conflict of interest. Yong Hu declares that he has no conflict of interest. Felix T. S. Chan declares that he has no conflict of interest. Xiaoge Zhang declares that he has no conflict of interest. Yong Deng declares that he has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by V. Loia.

Appendices

Appendix 1: Physarum-type algorithm

figurea

Appendix 2: Fuzzy set and fuzzy number

Definition 1

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

$$\begin{aligned} \widetilde{A} = \left\{ {\left\langle {x,\mu _{\widetilde{A}} \left( x \right) } \right\rangle \left| {x \in X} \right. } \right\} \end{aligned}$$
(17)

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:

$$\begin{aligned} \mu _{\widetilde{A}} = \left\{ \begin{array}{l} 0,\quad \quad \quad \quad x \le a_1 \\ \frac{{x - a_1 }}{{a_2 - a_1 }},\quad \;\;a_1 \le x \le a_2 \\ \frac{{a_3 - x}}{{a_3 - a_2 }},\quad \;\;a_2 \le x \le a_3 \\ 0,\quad \quad \quad \quad a_3 \le x \\ \end{array} \right. \end{aligned}$$
(18)

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
figure4

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:

$$\begin{aligned} \widetilde{A} \oplus \widetilde{B}= & {} \left( {a_1 + b_1 ,a_2 + b_2 ,a_3 + b_3 } \right) \quad \mathrm{for \; addition,} \end{aligned}$$
(19)
$$\begin{aligned} \widetilde{A} \ominus \widetilde{B}= & {} \left( {a_1 - b_3 ,a_2 - b_2 ,a_3 - b_1 } \right) \nonumber \\&\quad \mathrm{for \; subtraction,} \end{aligned}$$
(20)
$$\begin{aligned} \widetilde{A} \otimes \widetilde{B}= & {} \left( {a_1 \times b_1 ,a_2 \times b_2 ,a_3 \times b_3 } \right) \nonumber \\&\quad \mathrm{for \; multiplication,} \end{aligned}$$
(21)
$$\begin{aligned} \widetilde{A} \oslash \widetilde{B}= & {} \left( {a_1 \; / \; b_3\, ,a_2 \; / \; b_2\, ,a_3 \; / \; b_1 \; } \right) \nonumber \\&\quad \mathrm{for \; division.} \end{aligned}$$
(22)

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
figure5

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):

$$\begin{aligned}&{\hbox {Dis}}_{p,q} \left( {\widetilde{A}\mathrm{{,}}\widetilde{B}} \right) \nonumber \\&\quad = \left\{ \begin{array}{l} \left[ \left( {1 - q} \right) \int _0^1 \left| {A_\alpha ^ - - B_\alpha ^ - } \right| ^p \;\hbox {d}\alpha + q\int _0^1 \left| {A_\alpha ^ + - B_\alpha ^ + } \right| ^p\; \hbox {d}\alpha \right] ^{{1/p}} ,\quad p< \infty , \\ \left( {1 - q} \right) \mathop {\sup }\limits _{0< \alpha \le 1} \left( {\left| {A_\alpha ^ - - B_\alpha ^ - } \right| } \right) + q\mathop {\inf }\limits _{0 < \alpha \le 1} \left( {\left| {A_\alpha ^ + - B_\alpha ^ + } \right| } \right) ,\quad \quad p = \infty . \\ \end{array} \right. \nonumber \\ \end{aligned}$$
(23)

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:

$$\begin{aligned}&{\hbox {Dis}} \left( {\widetilde{A}\mathrm{{,}}\widetilde{B}} \right) \nonumber \\&\quad = \sqrt{\frac{1}{6}\left[ {\sum \limits _{i = 1}^3 {\left( {b_i - a_i } \right) ^2 } + \left( {b_2 - a_2 } \right) ^2 + \sum \limits _{i \in \left\{ {1,2} \right\} } {\left( {b_i - a_i } \right) \left( {b_{i + 1} - a_{i + 1} } \right) } } \right] }\nonumber \\ \end{aligned}$$
(24)

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):

$$\begin{aligned} P(\widetilde{A})=\frac{1}{6}(a_1+4\times a_2+a_3) \end{aligned}$$
(25)

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

$$\begin{aligned} P(\widetilde{A})=\sqrt{\frac{1}{2}\sum _{i=1}^n{(a_1^i)^2}+\frac{1}{2}\sum _{i=1}^n{(a_3^i)^2}} \end{aligned}$$
(26)

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:

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, Y., Hu, Y., Chan, F.T.S. et al. Physarum polycephalum assignment: a new attempt for fuzzy user equilibrium. Soft Comput 22, 3711–3720 (2018). https://doi.org/10.1007/s00500-017-2592-9

Download citation

Keywords

  • Traffic assignment
  • Fuzzy user equilibrium
  • Physarum polycephalum
  • Bio-network