Nowadays, one of the most challenging issues for firms, organizations and factories is transportation problems. Major of annual costs is related to these problems. Besides, according to the importance of environmental issues and strict laws for protecting the environment, organizations and firms ought to find the most economical and cleaner ways of transportation. In this paper, we optimized a two-objective fuzzy transportation problem simulated by multi-objective linear integer fuzzy programming technique. The existing fuzzy parameters (transportation costs, demands and supply quantities) are considered triangular fuzzy numbers. The Jimenez method used for Defuzzification the fuzzy model. In addition to principal constraints of transportation problems, the constraints of real-world such as vehicles’ capacity and most shipping time are also included. In order to solve the fuzzy transportation problem, simulated annealing (SA) approach and non-dominated sorting genetic algorithm (NSGA-II) (which has rarely been used) implemented. The results of solving some test problems created in different sizes show that both methods are capable of finding Pareto solutions in a quick time. The results of the comparison between the named algorithms prove that the SA algorithm comes up with more Pareto solutions with better quality in a shorter time than NSGA-II.
HitchcockF.L., The distribution of a product from several sources to numerous localities, J Math Phys20(2) (1941), 224–230.
2.
CharnesA. and CooperW.W., The stepping stone method of explaining linear programming calculations in transportation problems, Management Science1(1) (1954), 49–69.
3.
DantzigG.B., Linear programming and extensions, Princeton University Press, 1965.
4.
ZadehL.A., Fuzzy sets, Information and Control8(3) (1965), 338–353.
5.
ZimmermannH.-J., Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems1(1) (1978), 45–55.
6.
ÓhÉigeartaighM., A fuzzy transportation algorithm, Fuzzy Sets and Systems8(3) (1982), 235–243.
7.
ChanasS., KołodziejczykW. and MachajA., A fuzzy approach to the transportation problem, Fuzzy Sets and Systems13(3) (1984), 211–221.
8.
ChanasS. and KuchtaD., A concept of the optimal solution of the transportation problem with fuzzy cost coefficients, Fuzzy Sets and Systems82(3) (1996), 299–305.
9.
SaadO.M., A parametric study on transportation problem under fuzzy environment, 2002.
10.
LiuS.-T. and KaoC., Solving fuzzy transportation problems based on extension principle, European Journal of Operational Research153(3) (2004), 661–674.
11.
GaniA. and RazakK.A., Two stage fuzzy transportation problem, Journal of Physical Sciences10(1) (2006), 63–69.
12.
GuptaP. and MehlawatM.K., An algorithm for a fuzzy transportation problem to select a new type of coal for a steel manufacturing unit, Top15(1) (2007), 114–137.
13.
DinagarD.S. and PalanivelK., The transportation problem in fuzzy environment, International Journal of Algorithms, Computing and Mathematics2(3) (2009), 65–71.
14.
PandianP. and NatarajanG., A new algorithm for finding a fuzzy optimal solution for fuzzy transportation problems, Applied Mathematical Sciences4(2) (2010), 79–90.
15.
KaurA. and KumarA., A new method for solving fuzzy transportation problems using ranking function, Applied Mathematical Modelling35(12) (2011), 5652–5661.
16.
KeshavarzE. and KhorramE., A fuzzy bi-criteria transportation problem, Computers & Industrial Engineering61(4) (2011), 947–957.
17.
KunduP., KarS. and MaitiM., Fixed charge transportation problem with type-2 fuzzy variables, Information Sciences255 (2014), 170–186.
18.
BellmanR.E. and ZadehL.A., Decision-making in a fuzzy environment, Management Science17(4) (1970), B141–B-164.
19.
WernersB., An interactive fuzzy programming system, Fuzzy Sets and Systems23(1) (1987), 131–147.
20.
ZimmermannH., Fuzzy Set Theory and Its Applications Second, Revised Edition, Springer, 1992.
21.
LaiY. and HwangC., Fuzzy Mathematical Programming: Methods and Applications, Springer, Berlin, 1992.
22.
JiménezM., ArenasM. and BilbaoA., Linear programming with fuzzy parameters: An interactive method resolution, European Journal of Operational Research177(3) (2007), 1599–1609.
23.
JiménezM., Ranking fuzzy numbers through the comparison of its expected intervals, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems4(04) (1996), 379–388.
24.
YagerR.R., A procedure for ordering fuzzy subsets of the unit interval, Information Sciences24(2) (1981), 143–161.
25.
SrinivasN. and DebK., Multi objective optimization using none dominated sorting in genetic algorithms, Evolutionary Computation2(3) (1994), 221–248.
26.
GoldbergD.E., Genetic algorithms in search, optimization, and machine learning, Addison-wesleyReading Menlo Park, Vol. 412, 1989.
27.
FonsecaC.M. and FlemingP.J., Genetic algorithms for multi objective optimization: Formulation discussion and generalization, ICGA93 (1993), 416–423.
28.
HornJ., NafpliotisN. and GoldbergD.E., A niched Pareto genetic algorithm for multi objective optimization, In Evolutionary Computation, 1994 IEEE World Congress on Computational Intelligence, Proceedings of the First IEEE Conference on, 1994, IEEE.
29.
DebK., et al., A fast and elitist multi objective genetic algorithm: NSGA-II, Evolutionary Computation, IEEE Transactions on6(2) (2002), 182–197.
30.
ZitzlerE. and ThieleL., Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, Evolutionary Computation, IEEE Transactions on3(4) (1999), 257–271.
31.
MetropolisN., et al., Combinatorial minimization, J Chem Phys21 (1953), 1087–1092.
32.
KirkpatrickS. and VecchiM., Optimization by simulated annealing, Science220(4598) (1983), 671–680.
33.
HarrisaI., MumfordbC.L. and NaimM.M., A hybrid multi-objective approach to capacitated facility location with flexible store allocation for green logistics modelling, Transportation Research Part E: Logistics and Transportation Review, Volume 66, 2014, pp. 1–22.
34.
BandyopadhyayS. and BhattacharyaR., Applying modified NSGA-II for bi-objective supply chain problem, J Intell Manuf24 (2013), 707–716.
35.
BaykasoğluA. and SubulanK., Constrained fuzzy arithmetic approach to fuzzy transportation problems with fuzzy decision variables, Expert Systems with Applications81 (2017), 193–222.
36.
DuJ., LiX., YuL., DanR. and ZhouJ., Multi-depot vehicle routing problem for hazardous materials transportation: A fuzzy bi level programming, Information Sciences399 (2017), 201–218.
37.
EbrahimnejadA., A new method for solving fuzzy transportation problem with LR flat fuzzy numbers, Information Sciences357 (2016), 108–124.
38.
BaidyaA., BeraU.K. and MaitiM., A solid transportation problem with safety factor under different uncertainty environments, Journal of Uncertainty Analysis and Applications (2013), 1–18.
