NSGA-II based multi-objective pollution routing problem with higher order uncertainty

Pollution routing problem (PRP) is an NP-hard multi-objective optimization problem. The main goal is pollution reduction and secondary goals are cost/distance minimization, profit maximization etc. We have considered two unique models with two different set of objectives viz. (i) distance and fuel consumption, and (ii) weighted load and fuel consumption. Here, system parameters like demand, driver wages, timing constraints etc. can't be predicted a-priori and involve multiple opinions from the designers. Thus, such uncertain system parameters can be modelled using fuzzy sets. As type-1 fuzzy sets (T1 FSs) has limitations in modelling higher order uncertainty, this paper models these uncertain parameters with interval type-2 fuzzy sets (IT2 FSs). We have solved the problem by an efficient multi-objective evolutionary algorithm viz. NSGA-II (non-dominated sorting genetic algorithm-II). Numerical examples demonstrate the efficiency of the proposed technique over existing (crisp and type-1 fuzzy set based) approaches.

[1]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[2]  N. N. Karnik,et al.  Introduction to type-2 fuzzy logic systems , 1998, 1998 IEEE International Conference on Fuzzy Systems Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98CH36228).

[3]  Gilbert Laporte,et al.  An adaptive large neighborhood search heuristic for the Pollution-Routing Problem , 2012, Eur. J. Oper. Res..

[4]  Héctor Pomares,et al.  Multiobjective Optimization and Comparison of Nonsingleton Type-1 and Singleton Interval Type-2 Fuzzy Logic Systems , 2013, IEEE Transactions on Fuzzy Systems.

[5]  Frank Chung-Hoon Rhee,et al.  Uncertain Fuzzy Clustering: Interval Type-2 Fuzzy Approach to $C$-Means , 2007, IEEE Transactions on Fuzzy Systems.

[6]  Jerry M. Mendel,et al.  Interval Type-2 Fuzzy Logic Systems Made Simple , 2006, IEEE Transactions on Fuzzy Systems.

[7]  Jerry M. Mendel,et al.  Applications of Type-2 Fuzzy Logic Systems to Forecasting of Time-series , 1999, Inf. Sci..

[8]  Jerry M. Mendel,et al.  On clarifying some definitions and notations used for type-2 fuzzy sets as well as some recommended changes , 2016, Inf. Sci..

[9]  Gilbert Laporte,et al.  The time-dependent pollution-routing problem , 2013 .

[10]  Jerry M. Mendel,et al.  Type-2 fuzzy sets made simple , 2002, IEEE Trans. Fuzzy Syst..

[11]  Michel Gendreau,et al.  Heuristics for multi-attribute vehicle routing problems: A survey and synthesis , 2013, Eur. J. Oper. Res..

[12]  Hani Hagras,et al.  Towards the Wide Spread Use of Type-2 Fuzzy Logic Systems in Real World Applications , 2012, IEEE Computational Intelligence Magazine.

[13]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[14]  Darko Bozanic,et al.  Green logistic vehicle routing problem: Routing light delivery vehicles in urban areas using a neuro-fuzzy model , 2014, Expert Syst. Appl..

[15]  M. MendelJ. Type-2 Fuzzy Sets and Systems , 2007 .

[16]  Dusan Teodorovic,et al.  The fuzzy set theory approach to the vehicle routing problem when demand at nodes is uncertain , 1996, Fuzzy Sets Syst..

[17]  R. K. Ursem Multi-objective Optimization using Evolutionary Algorithms , 2009 .

[18]  Jerry M. Mendel,et al.  On KM Algorithms for Solving Type-2 Fuzzy Set Problems , 2013, IEEE Transactions on Fuzzy Systems.

[19]  Woei Wan Tan,et al.  Towards an efficient type-reduction method for interval type-2 fuzzy logic systems , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[20]  Gilbert Laporte,et al.  The Pollution-Routing Problem , 2011 .

[21]  Jaber Jemai,et al.  An NSGA-II Algorithm for the Green Vehicle Routing Problem , 2012, EvoCOP.

[22]  Ismail Karaoglan,et al.  The green vehicle routing problem: A heuristic based exact solution approach , 2016, Appl. Soft Comput..

[23]  Pranab K. Muhuri,et al.  Semi-elliptic membership function: Representation, generation, operations, defuzzification, ranking and its application to the real-time task scheduling problem , 2017, Eng. Appl. Artif. Intell..

[24]  Teodor Gabriel Crainic,et al.  Multi-start Heuristics for the Two-Echelon Vehicle Routing Problem , 2011, EvoCOP.

[25]  F. Jolai,et al.  A green vehicle routing problem with customer satisfaction criteria , 2016 .

[26]  Sai Ho Chung,et al.  Survey of Green Vehicle Routing Problem: Past and future trends , 2014, Expert Syst. Appl..

[27]  Elise Miller-Hooks,et al.  A Green Vehicle Routing Problem , 2012 .

[28]  Christian Wagner,et al.  On transitioning from type-1 to interval type-2 fuzzy logic systems , 2015, 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[29]  Hani Hagras,et al.  Interval Type-2 Fuzzy Logic Congestion Control for Video Streaming Across IP Networks , 2009, IEEE Transactions on Fuzzy Systems.

[30]  Hani Hagras,et al.  A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots , 2004, IEEE Transactions on Fuzzy Systems.

[31]  Michel Gendreau,et al.  A unified solution framework for multi-attribute vehicle routing problems , 2014, Eur. J. Oper. Res..

[32]  Manoj Kumar Tiwari,et al.  Multi-objective modeling of production and pollution routing problem with time window: A self-learning particle swarm optimization approach , 2016, Comput. Ind. Eng..

[33]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[34]  Robert Ivor John,et al.  Type 2 Fuzzy Sets: An Appraisal of Theory and Applications , 1998, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[35]  Jerry M. Mendel,et al.  Simplified Interval Type-2 Fuzzy Logic Systems , 2013, IEEE Transactions on Fuzzy Systems.