Toward a Possibilistic Swarm Multi-robot Task Allocation: Theoretical and Experimental Results


Selecting the best task to execute (task allocation problem) is one of the main problems in multi-robot systems. Typical ways to address this problem are based on swarm intelligence and very especially using the so-called response threshold method. In the aforementioned method a robot has a certain probability of executing a task which depends on a task threshold response and a task stimulus. Nevertheless, response threshold method cannot be extended in a natural way to allocate more than two tasks when the theoretical basis is provided by probability theory. Motivated by this fact, this paper leaves the probabilistic approach to the problem and provides a first theoretical framework towards a possibilistic approach. Thus, task allocation problem is addressed using fuzzy Markov chains instead of probabilistic processes. This paper demonstrates that fuzzy Markov chains associated to a task allocation problem can converge to a stationary stage in a finite number of steps. In contrast, the probabilistic processes only can converge asymptotically, i.e. the number of steps is not bounded in general. Moreover, fuzzy Markov chains predicts in a better way the future behavior of the system in the presence of vagueness when measuring distances. The simulations performed confirm the theoretical results and show how the number of steps needed to get a stable state with fuzzy Markov chains is reduced more than 10 times and the system’s behavior prediction can be improved more than a 60% compared to probabilistic approaches.

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  1. 1.

    Agassounon W, Martinoli A (2002) Efficiency and robustness of threshold-based distributed allocation algorithms in multi-agent systems. In: 1st international joint conference on autonomous agents and multi-agents systems, Bolonia, Italy, July 2002, pp 1090–1097

  2. 2.

    Avrachenkov K, Sanchez E (2002) Fuzzy markov chains and decision making. Fuzzy Optim Decis Mak 1:143–159

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bonabeau E, Sobkowski A, Theraulaz G, Deneubourg J-L (1997) Adaptive task allocation inspired by a model of division of labor in social insects. In: Lundh D, Olsson B, Narayanan A (eds) Bio computation and emergent computing. World Scientific, Singapore, pp 36–45

    Google Scholar 

  4. 4.

    Brash I, Gerchak Y (1978) Markov chains with finite convergence time. Stoch Process Appl 7:247–253

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Dubois HPD (1980) Fuzzy sets and systems: theory and applications. Academic Press, Cambridge

    Google Scholar 

  6. 6.

    Duan J (2004) The transitive clousure, convegence of powers and adjoint of generalized fuzzy matrices. Fuzzy Sets Syst 145:301–311

    Article  Google Scholar 

  7. 7.

    Dubois D, Prade H, Sandri S (1993) On possibility/probability transformations. In: Lowen R, Roubens M (eds) Fuzzy logic. Theory and decision library, vol 12. Springer, Netherlands, pp 103–112

  8. 8.

    Gerkey BP (2003) On multi-robot task allocation. PhD thesis, Center of robotics and embedded systems, University of Southern California, Los Angeles, USA, August 2003

  9. 9.

    Gerkey BP, Mataric M (2002) Sold!: auction methods for multi-robot coordination. IEEE Trans Robot Autom 18(5):758–768 (Special Issue on Multi-robot Systems)

    Article  Google Scholar 

  10. 10.

    Gerkey BP, Mataric M (2004) A formal analysis and taxonomy of task allocation in multi-robot systems. Int J Robot Res 23(9):939–954

    Article  Google Scholar 

  11. 11.

    Guerrero J, Oliver G (2012) Swarm-like methodologies for executing tasks with deadlines. J Intell Robot Syst 68(1):3–19

    Article  Google Scholar 

  12. 12.

    Heap B, Pagnucco M (2013) Repeated sequential single-cluster auctions with dynamic tasks for multi-robot task allocation with pickup and delivery. Lect Notes Comput Sci Multiagent Syst Technol 8076:87–100

    Article  Google Scholar 

  13. 13.

    Kalra N, Martinoli A (2006) A comparative study of market-based and threshold-based task allocation. In: 8th international symposium on distributed autonomous robotic systems, Minneapolis, USA, July 2006, pp 91–102

  14. 14.

    Kemeny J, Snell J (1960) Finite Markov chains. Springer, Berlin

    Google Scholar 

  15. 15.

    Kim M, Baik H, Lee S (2014) Response threshold model based uav search planning and task allocation. J Intell Robot Syst 75:625–640

    Article  Google Scholar 

  16. 16.

    Kuhn HW (1955) The hungarian method for the assignment problem. Nav Res Logist Q 2(1):83–97

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Lindqvist B (1981) Ergodic Markov chains with finite convergence time. Stoch Process Appl 11(1):91–99

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Navarro I, Matía F (2013) An introduction to swarm robotics. ISRN Robot 2013:1–10

  19. 19.

    Ranjbar-Sahraei B, Weiss G, Tuyls K (2013) A macroscopic model for multi-robot stigmergic coverage. In: Proceedings of the 12th international conference on autonomous agents and multiagent systems (AAMAS 2013), pp 1233–1234

  20. 20.

    Rubinstein RY, Kroese DP (2008) Simulation and the Monte Carlo method, 2nd edn. Wiley, Hoboken

    Google Scholar 

  21. 21.

    Vajargah BF, Gharehdaghi M (2014) Ergodicity of fuzzy Markov chains based on simulation using sequences. Int J Appl Math Comput Sci 11(2):159–165

    Google Scholar 

  22. 22.

    Valentini G, Ferrante E, Hamann H, Dorigo M (2016) Collective decision with 100 kilobots: speed versus accuracy in binary discrimination problems. Auton Agents Multiagents Syst 30(3):553–580

    Article  Google Scholar 

  23. 23.

    Yang Y, Zhou C, Tin Y (2009) Swarm robots task allocation based on response threshold model. In: 4th international conference on autonomous robots and agents, Willengton, New Zeland, February 2009, pp 171–176

  24. 24.

    Zadeh L (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28

    MathSciNet  Article  MATH  Google Scholar 

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This research was funded by the Spanish Ministry of Economy and Competitiveness under Grants DPI2014-57746-C03-2-R, TIN2014-53772-R, TIN2014-56381-REDT (LODISCO), TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds.

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Correspondence to José Guerrero.

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Guerrero, J., Valero, Ó. & Oliver, G. Toward a Possibilistic Swarm Multi-robot Task Allocation: Theoretical and Experimental Results. Neural Process Lett 46, 881–897 (2017).

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  • Multi-robot
  • Possibility theory
  • Swarm intelligence
  • Task allocation