Multi-objective transportation problem under type-2 trapezoidal fuzzy numbers with parameters estimation and goodness of fit
Abstract
The problem of transportation in real-life is an uncertain multi-objective decision-making problem. In particular, by taking into account the conflicting objectives, Decision-Makers (DMs) are looking for the best transport set up to determine the optimum shipping quantity subject to certain capacity constraints on each route. This paper presented a Multi-Objective Transportation Problem (MOTP) where the objective functions are considered as Type-2 trapezoidal fuzzy numbers (T2TpFN), respectively. Demand and supply in constraints are in multi-choice and probabilistic random variables, respectively. Also considered the “rate of increment in Transportation Cost (TC) and rate of decrement in profit on transporting the products from ith sources to jth destinations due to” (or additional cost) of each product due to the damage, late deliveries, weather conditions, and any other issues. Due to the presence of all these uncertainties, it is not possible to obtain the optimum solution directly, so first, we need to convert all these uncertainties from the model into a crisp equivalent form. The two-phase defuzzification technique is used to transform T2TpFN into a crisp equivalent form. Multi-choice and probabilistic random variables are transformed into an equivalent value using Stochastic Programming (SP) approach and the binary variable, respectively. It is assumed that the supply and demand parameter follows various types of probabilistic distributions like Weibull, Extreme value, Cauchy and Pareto, Normal distribution, respectively. The unknown parameters of probabilistic distributions estimated using the maximum likelihood estimation method at the defined probability level. The best fit of the probability distributions is determined using the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), respectively. Using the Fuzzy Goal Programming (FGP) method, the final problem is solved for the optimal decision. A case study is intended to provide the effectiveness of the proposed work.
Keyword : multi-objective optimization, transportation problem, fuzzy goal programming, multi-choice, maximum likelihood estimation, Akaike information criterion, Bayesian information criterion, stochastic programming
How to Cite
Kamal, M., Alarjani, A., Haq, A., Yusufi, F. N. K., & Ali, I. (2021). Multi-objective transportation problem under type-2 trapezoidal fuzzy numbers with parameters estimation and goodness of fit. Transport, 36(4), 317-338. https://doi.org/10.3846/transport.2021.15649
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Appa, G. M. 1973. The transportation problem and its variants, Journal of the Operational Research Society 24(1): 79–99. https://doi.org/10.1057/jors.1973.10
Arsham, H.; Kahn, A. B. 1989. A simplex-type algorithm for general transportation problems: an alternative to stepping-stone, Journal of the Operational Research Society 40(6): 581–590. https://doi.org/10.1057/jors.1989.95
Barik, S. K. 2015. Probabilistic fuzzy goal programming problems involving pareto distribution: some additive approaches, Fuzzy Information and Engineering 7(2): 227–244. https://doi.org/10.1016/j.fiae.2015.05.007
Barik, S. K.; Biswal, M. P.; Chakravarty, D. 2011. Stochastic programming problems involving Pareto distribution, Journal of Interdisciplinary Mathematics 14(1): 40–56. https://doi.org/10.1080/09720502.2011.10700734
Biswal, M. P.; Acharya, S. 2009. Multi-choice multi-objective linear programming problem, Journal of Interdisciplinary Mathematics 12(5): 606–637. https://doi.org/10.1080/09720502.2009.10700650
Biswal, M. P.; Acharya, S. 2011. Solving multi-choice linear programming problems by interpolating polynomials, Mathematical and Computer Modelling 54(5–6): 1405–1412. https://doi.org/10.1016/j.mcm.2011.04.009
Biswal, M. P.; Samal, H. K. 2013. Stochastic transportation problem with Cauchy random variables and multi choice parameters, Journal of Physical Sciences 17: 117–130.
Biswas, A.; De, A. K. 2018. A unified method of defuzzification for type-2 fuzzy numbers with its application to multiobjective decision making, Granular Computing 3(4): 301–318. https://doi.org/10.1007/s41066-017-0068-z
Biswas, A.; Modak, N. 2011. A fuzzy goal programming method for solving chance constrained programming with fuzzy parameters, Communications in Computer and Information Science 140: 187–196. https://doi.org/10.1007/978-3-642-19263-0_23
Biswas, A.; Modak, N. 2017. On solving multiobjective transportation problems with fuzzy random supply and demand using fuzzy goal programming, International Journal of Operations Research and Information Systems 8(3): 54–81. https://doi.org/10.4018/ijoris.2017070104
Biswas, P.; Pal, B. B. 2019. A fuzzy goal programming method to solve congestion management problem using genetic algorithm, Decision Making: Applications in Management and Engineering 2(2): 36–53.
Bit, A. K.; Biswal, M. P.; Alam, S. S. 1992. Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy Sets and Systems 50(2): 135–141. https://doi.org/10.1016/0165-0114(92)90212-m
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Chang, C.-T. 2007. Multi-choice goal programming, Omega 35(4): 389–396. https://doi.org/10.1016/j.omega.2005.07.009
Chang, C.-T. 2008. Revised multi-choice goal programming, Applied Mathematical Modelling 32(12): 2587–2595. https://doi.org/10.1016/j.apm.2007.09.008
Charnes, A.; Cooper, W. W. 1954. The stepping stone method of explaining linear programming calculations in transportation problems, Management Science 1(1): 49–69. https://doi.org/10.1287/mnsc.1.1.49
Clímaco, J. N.; Antunes, C. H.; Alves, M. J. 1993. Interactive decision support for multiobjective transportation problems, European Journal of Operational Research 65(1): 58–67. https://doi.org/10.1016/0377-2217(93)90144-C
Current, J.; Marsh, M. 1993. Multiobjective transportation network design and routing problems: taxonomy and annotation, European Journal of Operational Research 65(1): 4–19. https://doi.org/10.1016/0377-2217(93)90140-I
Current, J. Min, H. 1986. Multiobjective design of transportation networks: taxonomy and annotation, European Journal of Operational Research 26(2): 187–201. https://doi.org/10.1016/0377-2217(86)90180-3
Dantzig, G. B. 1963. Linear Programming and Extensions. RAND Corporation. 641 p. https://doi.org/10.7249/R366
Diaz, J. A. 1979. Finding a complete description of all efficient solutions to a multiobjective transportation problem, Ekonomicko-matematickż obzor 15(1): 62–73.
Diaz, J. A. 1978. Solving multiobjective transportation problems, Ekonomicko-matematickż obzor 14(3): 267–274.
Dinagar, D. S.; Palanivel, K. 2009. The transportation problem in fuzzy environment, International Journal of Algorithms, Computing and Mathematics 2(3): 65–71.
Dutta, D.; Murthy, A. S. 2010. Multi-choice goal programming approach for a fuzzy transportation problem, International Journal of Research and Reviews in Applied Sciences 2(2): 132–139.
Ebrahimnejad, A. 2014. A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers, Applied Soft Computing 19: 171–176. https://doi.org/10.1016/j.asoc.2014.01.041
El-Wahed, W. F. A. 2001. A multi-objective transportation problem under fuzziness, Fuzzy Sets and Systems 117(1): 27–33. https://doi.org/10.1016/S0165-0114(98)00155-9
Fazlollahtabar, H. 2018. Operations and inspection cost minimization for a reverse supply chain, Operational Research in Engineering Sciences: Theory and Applications 1(1): 91–107.
Gupta, A.; Kumar, A. 2012. A new method for solving linear multi-objective transportation problems with fuzzy parameters, Applied Mathematical Modelling 36(4): 1421–1430. https://doi.org/10.1016/j.apm.2011.08.044
Isermann, H. 1979. The enumeration of all efficient solutions for a linear multiple-objective transportation problem, Naval Research Logistics Quarterly 26(1): 123–139. https://doi.org/10.1002/nav.3800260112
Kaliski, J. A.; Ye, Y. 1993. A short-cut potential reduction algorithm for linear programming, Management Science 39(6): 757–776. https://doi.org/10.1287/mnsc.39.6.757
Kamal, M.; Gupta, S.; Chatterjee, P.; Pamucar, D.; Stevic, Z. 2019. Bi-level multi-objective production planning problem with multi-choice parameters: a fuzzy goal programming algorithm, Algorithms 12(7): 143. https://doi.org/10.3390/a12070143
Kamal, M.; Jalil, S. A.; Muneeb, S. M.; Ali, I. 2018. A distance based method for solving multi-objective optimization problems, Journal of Modern Applied Statistical Methods 17(1): 21. https://doi.org/10.22237/jmasm/1532525455
Lee, S. M.; Moore, L. J. 1973. Optimizing transportation problems with multiple objectives, AIIE Transactions 5(4): 333–338. https://doi.org/10.1080/05695557308974920
Li, L.; Lai, K. K. 2000. A fuzzy approach to the multiobjective transportation problem, Computers & Operations Research 27(1): 43–57. https://doi.org/10.1016/S0305-0548(99)00007-6
Liu, S.-T. 2016. Fractional transportation problem with fuzzy parameters, Soft Computing 20(9): 3629–3636. https://doi.org/10.1007/s00500-015-1722-5
Liu, S.-T. 2003. The total cost bounds of the transportation problem with varying demand and supply, Omega 31(4): 247–251. https://doi.org/10.1016/S0305-0483(03)00054-9
Lukovac, V.; Popović, M. 2018. Fuzzy Delphi approach to defining a cycle for assessing the performance of military drivers, Decision Making: Applications in Management and Engineering 1(1): 67–81.
Maity, S.; Roy, S. K. 2019. A new approach for solving type-2-fuzzy transportation problem, International Journal of Mathematical, Engineering and Management Sciences 4(3): 683–696. https://doi.org/10.33889//IJMEMS.2019.4.3-054
Maity, G.; Roy, S. K. 2016. Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand, International Journal of Management Science and Engineering Management 11(1): 62–70. https://doi.org/10.1080/17509653.2014.988768
Maity, G.; Roy, S. K. 2014. Solving multi-choice multi-objective transportation problem: a utility function approach, Journal of Uncertainty Analysis and Applications 2(1): 11. https://doi.org/10.1186/2195-5468-2-11
Maity, G.; Roy, S. K.; Verdegay, J. L. 2016. Multi-objective transportation problem with cost reliability under uncertain environment, International Journal of Computational Intelligence Systems 9(5): 839–849. https://doi.org/10.1080/18756891.2016.1237184
Mahapatra, D. R.; Roy, S. K.; Biswal, M. P. 2013. Multi-choice stochastic transportation problem involving extreme value distribution, Applied Mathematical Modelling 37(4): 2230–2240. https://doi.org/10.1016/j.apm.2012.04.024
Pamucar, D.; Ćirović, G. 2018. Vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty conditions, Decision Making: Applications in Management and Engineering 1(1): 13–37.
Rani, D.; Gulati, T. R. 2014. A new approach to solve unbalanced transportation problems in imprecise environment, Journal of Transportation Security 7(3): 277–287. https://doi.org/10.1007/s12198-014-0143-5
Rani, D.; Gulati, T. R. 2017. Time optimization in totally uncertain transportation problems, International Journal of Fuzzy Systems 19(3): 739–750. https://doi.org/10.1007/s40815-016-0176-y
Ringuest, J. L.; Rinks, D. B. 1987. Interactive solutions for the linear multiobjective transportation problem, European Journal of Operational Research 32(1): 96–106. https://doi.org/10.1016/0377-2217(87)90274-8
Roy, S. K. 2014. Multi-choice stochastic transportation problem involving Weibull distribution, International Journal of Operational Research 21(1): 38–58. https://doi.org/10.1504/IJOR.2014.064021
Roy, S. K.; Mahapatra, D. R.; Biswal, M. P. 2012. Multi-choice stochastic transportation problem with exponential distribution, Journal of Uncertain Systems 6(3): 200–213.
Roy, S. K.; Maity, G. 2017. Minimizing cost and time through single objective function in multi-choice interval valued transportation problem, Journal of Intelligent & Fuzzy Systems 32(3): 1697–1709. https://doi.org/10.3233/JIFS-151656
Roy, S. K.; Maity, G.; Weber, G.-W. 2017a. Multi-objective twostage grey transportation problem using utility function with goals, Central European Journal of Operations Research 25(2): 417–439. https://doi.org/10.1007/s10100-016-0464-5
Roy, S. K.; Maity, G.; Weber, G. W.; Gök, S. Z. A. 2017b. Conic scalarization approach to solve multi-choice multi-objective transportation problem with interval goal, Annals of Operations Research 253(1): 599–620. https://doi.org/10.1007/s10479-016-2283-4
Safi, M. R.; Ghasemi, S. M. 2017. Uncertainty in linear fractional transportation problem, International Journal of Nonlinear Analysis and Applications 8(1): 81–93.
Sahoo, N. P.; Biswal, M. P. 2005. Computation of some stochastic linear programming problems with Cauchy and extreme value distributions, International Journal of Computer Mathematics 82(6): 685–698. https://doi.org/10.1080/00207160412331336080
Si, A.; Das, S.; Kar, S. 2019. An approach to rank picture fuzzy numbers for decision making problems, Decision Making: Applications in Management and Engineering 2(2): 54–64.
Sinha, B.; Das, A.; Bera, U. K. 2016. Profit maximization solid transportation problem with trapezoidal interval type-2 fuzzy numbers, International Journal of Applied and Computational Mathematics 2(1): 41–56. https://doi.org/10.1007/s40819-015-0044-8
Stojić, G.; Sremac, S.; Vasiljković, I. 2018. A fuzzy model for determining the justifiability of investing in a road freight vehicle fleet, Operational Research in Engineering Sciences: Theory and Applications 1(1): 62–75.
Stone, M. 1979. Comments on model selection criteria of Akaike and Schwarz, Journal of the Royal Statistical Society: Series B (Methodological) 41(2): 276–278. https://doi.org/10.1111/j.2517-6161.1979.tb01084.x
Vilela, M.; Oluyemi, G.; Petrovski, A. 2019. A fuzzy inference system applied to value of information assessment for oil and gas industry, Decision Making: Applications in Management and Engineering 2(2): 1–18.
Xie, F.; Butt, M. M.; Li, Z.; Zhu, L. 2017. An upper bound on the minimal total cost of the transportation problem with varying demands and supplies, Omega 68: 105–118. https://doi.org/10.1016/j.omega.2016.06.007
Zadeh, L. A. 1965. Fuzzy sets, Information and Control 8(3): 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zangiabadi, M.; Maleki, H. R. 2013. Fuzzy goal programming technique to solve multiobjective transportation problems with some non-linear membership functions, Iranian Journal of Fuzzy Systems 10(1): 61–74. https://doi.org/10.22111/ijfs.2013.155
Akaike, H. 1974. A new look at the statistical model identification, in E. Parzen, K. Tanabe, G. Kitagawa (Eds.). Selected Papers of Hirotugu Akaike, 215–222. https://doi.org/10.1007/978-1-4612-1694-0_16
Aneja, Y. P.; Nair, K. P. K. 1979. Bicriteria transportation problem, Management Science 25(1): 73–78. https://doi.org/10.1287/mnsc.25.1.73
Appa, G. M. 1973. The transportation problem and its variants, Journal of the Operational Research Society 24(1): 79–99. https://doi.org/10.1057/jors.1973.10
Arsham, H.; Kahn, A. B. 1989. A simplex-type algorithm for general transportation problems: an alternative to stepping-stone, Journal of the Operational Research Society 40(6): 581–590. https://doi.org/10.1057/jors.1989.95
Barik, S. K. 2015. Probabilistic fuzzy goal programming problems involving pareto distribution: some additive approaches, Fuzzy Information and Engineering 7(2): 227–244. https://doi.org/10.1016/j.fiae.2015.05.007
Barik, S. K.; Biswal, M. P.; Chakravarty, D. 2011. Stochastic programming problems involving Pareto distribution, Journal of Interdisciplinary Mathematics 14(1): 40–56. https://doi.org/10.1080/09720502.2011.10700734
Biswal, M. P.; Acharya, S. 2009. Multi-choice multi-objective linear programming problem, Journal of Interdisciplinary Mathematics 12(5): 606–637. https://doi.org/10.1080/09720502.2009.10700650
Biswal, M. P.; Acharya, S. 2011. Solving multi-choice linear programming problems by interpolating polynomials, Mathematical and Computer Modelling 54(5–6): 1405–1412. https://doi.org/10.1016/j.mcm.2011.04.009
Biswal, M. P.; Samal, H. K. 2013. Stochastic transportation problem with Cauchy random variables and multi choice parameters, Journal of Physical Sciences 17: 117–130.
Biswas, A.; De, A. K. 2018. A unified method of defuzzification for type-2 fuzzy numbers with its application to multiobjective decision making, Granular Computing 3(4): 301–318. https://doi.org/10.1007/s41066-017-0068-z
Biswas, A.; Modak, N. 2011. A fuzzy goal programming method for solving chance constrained programming with fuzzy parameters, Communications in Computer and Information Science 140: 187–196. https://doi.org/10.1007/978-3-642-19263-0_23
Biswas, A.; Modak, N. 2017. On solving multiobjective transportation problems with fuzzy random supply and demand using fuzzy goal programming, International Journal of Operations Research and Information Systems 8(3): 54–81. https://doi.org/10.4018/ijoris.2017070104
Biswas, P.; Pal, B. B. 2019. A fuzzy goal programming method to solve congestion management problem using genetic algorithm, Decision Making: Applications in Management and Engineering 2(2): 36–53.
Bit, A. K.; Biswal, M. P.; Alam, S. S. 1992. Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy Sets and Systems 50(2): 135–141. https://doi.org/10.1016/0165-0114(92)90212-m
Chakraborty, A.; Chakraborty, M. 2010. Cost-time minimization in a transportation problem with fuzzy parameters: a case study, Journal of Transportation Systems Engineering and Information Technology 10(6): 53–63. https://doi.org/10.1016/S1570-6672(09)60071-4
Chang, C.-T. 2007. Multi-choice goal programming, Omega 35(4): 389–396. https://doi.org/10.1016/j.omega.2005.07.009
Chang, C.-T. 2008. Revised multi-choice goal programming, Applied Mathematical Modelling 32(12): 2587–2595. https://doi.org/10.1016/j.apm.2007.09.008
Charnes, A.; Cooper, W. W. 1954. The stepping stone method of explaining linear programming calculations in transportation problems, Management Science 1(1): 49–69. https://doi.org/10.1287/mnsc.1.1.49
Clímaco, J. N.; Antunes, C. H.; Alves, M. J. 1993. Interactive decision support for multiobjective transportation problems, European Journal of Operational Research 65(1): 58–67. https://doi.org/10.1016/0377-2217(93)90144-C
Current, J.; Marsh, M. 1993. Multiobjective transportation network design and routing problems: taxonomy and annotation, European Journal of Operational Research 65(1): 4–19. https://doi.org/10.1016/0377-2217(93)90140-I
Current, J. Min, H. 1986. Multiobjective design of transportation networks: taxonomy and annotation, European Journal of Operational Research 26(2): 187–201. https://doi.org/10.1016/0377-2217(86)90180-3
Dantzig, G. B. 1963. Linear Programming and Extensions. RAND Corporation. 641 p. https://doi.org/10.7249/R366
Diaz, J. A. 1979. Finding a complete description of all efficient solutions to a multiobjective transportation problem, Ekonomicko-matematickż obzor 15(1): 62–73.
Diaz, J. A. 1978. Solving multiobjective transportation problems, Ekonomicko-matematickż obzor 14(3): 267–274.
Dinagar, D. S.; Palanivel, K. 2009. The transportation problem in fuzzy environment, International Journal of Algorithms, Computing and Mathematics 2(3): 65–71.
Dutta, D.; Murthy, A. S. 2010. Multi-choice goal programming approach for a fuzzy transportation problem, International Journal of Research and Reviews in Applied Sciences 2(2): 132–139.
Ebrahimnejad, A. 2014. A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers, Applied Soft Computing 19: 171–176. https://doi.org/10.1016/j.asoc.2014.01.041
El-Wahed, W. F. A. 2001. A multi-objective transportation problem under fuzziness, Fuzzy Sets and Systems 117(1): 27–33. https://doi.org/10.1016/S0165-0114(98)00155-9
Fazlollahtabar, H. 2018. Operations and inspection cost minimization for a reverse supply chain, Operational Research in Engineering Sciences: Theory and Applications 1(1): 91–107.
Gupta, A.; Kumar, A. 2012. A new method for solving linear multi-objective transportation problems with fuzzy parameters, Applied Mathematical Modelling 36(4): 1421–1430. https://doi.org/10.1016/j.apm.2011.08.044
Isermann, H. 1979. The enumeration of all efficient solutions for a linear multiple-objective transportation problem, Naval Research Logistics Quarterly 26(1): 123–139. https://doi.org/10.1002/nav.3800260112
Kaliski, J. A.; Ye, Y. 1993. A short-cut potential reduction algorithm for linear programming, Management Science 39(6): 757–776. https://doi.org/10.1287/mnsc.39.6.757
Kamal, M.; Gupta, S.; Chatterjee, P.; Pamucar, D.; Stevic, Z. 2019. Bi-level multi-objective production planning problem with multi-choice parameters: a fuzzy goal programming algorithm, Algorithms 12(7): 143. https://doi.org/10.3390/a12070143
Kamal, M.; Jalil, S. A.; Muneeb, S. M.; Ali, I. 2018. A distance based method for solving multi-objective optimization problems, Journal of Modern Applied Statistical Methods 17(1): 21. https://doi.org/10.22237/jmasm/1532525455
Lee, S. M.; Moore, L. J. 1973. Optimizing transportation problems with multiple objectives, AIIE Transactions 5(4): 333–338. https://doi.org/10.1080/05695557308974920
Li, L.; Lai, K. K. 2000. A fuzzy approach to the multiobjective transportation problem, Computers & Operations Research 27(1): 43–57. https://doi.org/10.1016/S0305-0548(99)00007-6
Liu, S.-T. 2016. Fractional transportation problem with fuzzy parameters, Soft Computing 20(9): 3629–3636. https://doi.org/10.1007/s00500-015-1722-5
Liu, S.-T. 2003. The total cost bounds of the transportation problem with varying demand and supply, Omega 31(4): 247–251. https://doi.org/10.1016/S0305-0483(03)00054-9
Lukovac, V.; Popović, M. 2018. Fuzzy Delphi approach to defining a cycle for assessing the performance of military drivers, Decision Making: Applications in Management and Engineering 1(1): 67–81.
Maity, S.; Roy, S. K. 2019. A new approach for solving type-2-fuzzy transportation problem, International Journal of Mathematical, Engineering and Management Sciences 4(3): 683–696. https://doi.org/10.33889//IJMEMS.2019.4.3-054
Maity, G.; Roy, S. K. 2016. Solving a multi-objective transportation problem with nonlinear cost and multi-choice demand, International Journal of Management Science and Engineering Management 11(1): 62–70. https://doi.org/10.1080/17509653.2014.988768
Maity, G.; Roy, S. K. 2014. Solving multi-choice multi-objective transportation problem: a utility function approach, Journal of Uncertainty Analysis and Applications 2(1): 11. https://doi.org/10.1186/2195-5468-2-11
Maity, G.; Roy, S. K.; Verdegay, J. L. 2016. Multi-objective transportation problem with cost reliability under uncertain environment, International Journal of Computational Intelligence Systems 9(5): 839–849. https://doi.org/10.1080/18756891.2016.1237184
Mahapatra, D. R.; Roy, S. K.; Biswal, M. P. 2013. Multi-choice stochastic transportation problem involving extreme value distribution, Applied Mathematical Modelling 37(4): 2230–2240. https://doi.org/10.1016/j.apm.2012.04.024
Pamucar, D.; Ćirović, G. 2018. Vehicle route selection with an adaptive neuro fuzzy inference system in uncertainty conditions, Decision Making: Applications in Management and Engineering 1(1): 13–37.
Rani, D.; Gulati, T. R. 2014. A new approach to solve unbalanced transportation problems in imprecise environment, Journal of Transportation Security 7(3): 277–287. https://doi.org/10.1007/s12198-014-0143-5
Rani, D.; Gulati, T. R. 2017. Time optimization in totally uncertain transportation problems, International Journal of Fuzzy Systems 19(3): 739–750. https://doi.org/10.1007/s40815-016-0176-y
Ringuest, J. L.; Rinks, D. B. 1987. Interactive solutions for the linear multiobjective transportation problem, European Journal of Operational Research 32(1): 96–106. https://doi.org/10.1016/0377-2217(87)90274-8
Roy, S. K. 2014. Multi-choice stochastic transportation problem involving Weibull distribution, International Journal of Operational Research 21(1): 38–58. https://doi.org/10.1504/IJOR.2014.064021
Roy, S. K.; Mahapatra, D. R.; Biswal, M. P. 2012. Multi-choice stochastic transportation problem with exponential distribution, Journal of Uncertain Systems 6(3): 200–213.
Roy, S. K.; Maity, G. 2017. Minimizing cost and time through single objective function in multi-choice interval valued transportation problem, Journal of Intelligent & Fuzzy Systems 32(3): 1697–1709. https://doi.org/10.3233/JIFS-151656
Roy, S. K.; Maity, G.; Weber, G.-W. 2017a. Multi-objective twostage grey transportation problem using utility function with goals, Central European Journal of Operations Research 25(2): 417–439. https://doi.org/10.1007/s10100-016-0464-5
Roy, S. K.; Maity, G.; Weber, G. W.; Gök, S. Z. A. 2017b. Conic scalarization approach to solve multi-choice multi-objective transportation problem with interval goal, Annals of Operations Research 253(1): 599–620. https://doi.org/10.1007/s10479-016-2283-4
Safi, M. R.; Ghasemi, S. M. 2017. Uncertainty in linear fractional transportation problem, International Journal of Nonlinear Analysis and Applications 8(1): 81–93.
Sahoo, N. P.; Biswal, M. P. 2005. Computation of some stochastic linear programming problems with Cauchy and extreme value distributions, International Journal of Computer Mathematics 82(6): 685–698. https://doi.org/10.1080/00207160412331336080
Si, A.; Das, S.; Kar, S. 2019. An approach to rank picture fuzzy numbers for decision making problems, Decision Making: Applications in Management and Engineering 2(2): 54–64.
Sinha, B.; Das, A.; Bera, U. K. 2016. Profit maximization solid transportation problem with trapezoidal interval type-2 fuzzy numbers, International Journal of Applied and Computational Mathematics 2(1): 41–56. https://doi.org/10.1007/s40819-015-0044-8
Stojić, G.; Sremac, S.; Vasiljković, I. 2018. A fuzzy model for determining the justifiability of investing in a road freight vehicle fleet, Operational Research in Engineering Sciences: Theory and Applications 1(1): 62–75.
Stone, M. 1979. Comments on model selection criteria of Akaike and Schwarz, Journal of the Royal Statistical Society: Series B (Methodological) 41(2): 276–278. https://doi.org/10.1111/j.2517-6161.1979.tb01084.x
Vilela, M.; Oluyemi, G.; Petrovski, A. 2019. A fuzzy inference system applied to value of information assessment for oil and gas industry, Decision Making: Applications in Management and Engineering 2(2): 1–18.
Xie, F.; Butt, M. M.; Li, Z.; Zhu, L. 2017. An upper bound on the minimal total cost of the transportation problem with varying demands and supplies, Omega 68: 105–118. https://doi.org/10.1016/j.omega.2016.06.007
Zadeh, L. A. 1965. Fuzzy sets, Information and Control 8(3): 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zangiabadi, M.; Maleki, H. R. 2013. Fuzzy goal programming technique to solve multiobjective transportation problems with some non-linear membership functions, Iranian Journal of Fuzzy Systems 10(1): 61–74. https://doi.org/10.22111/ijfs.2013.155