419 245
Full Length Article
Volume 3 , Issue 2, PP: 54-66 , 2020


An Approach to Solve the Linear Programming Problem Using Single Valued Trapezoidal Neutrosophic Number

Authors Names :   Tuhin Bera   1 *     Nirmal Kumar Mahapatra   2  

1  Affiliation :  Department of Mathematics, Panskura Banamali College, Panskura RS-721152, WB, India

    Email :  tuhin78bera@gmail.com

2  Affiliation :  Department of Mathematics, Panskura Banamali College, Panskura RS-721152, WB, India

    Email :  nirmal_hridoy@yahoo.co.in

Doi   :  10.5281/zenodo.3740647

Abstract :

While making a decision, the neutrosophic set theory includes the uncertainty part beside certainty part (i.e., Yes or No). In the present uncertain socio-economic fashion, this pattern is highly acceptable and hence, the limitations of fuzzy set and intuitionistic fuzzy set are overcome with neutrosophic set theory. The present study provides a modified structure of linear programming problem (LP-problem) and its solution approach in neutrosophic sense. A special type of neutrosophic set defined over the set of real number, viz., single valued trapezoidal neutrosophic number (SVTN-number) is adopted here as the coefficients of the objective function, right-hand side coefficients and decision variables itself of an LP-problem. In order to solve such problem, a parameter based ranking function of SVTN-number is newly constructed from the geometrical configuration of the trapezium. It plays a key role in the development of the solution algorithm. An LP-problem is normally solved under the asset of some given constraints. Besides that, there may be some hidden parameters (e.g., awareness level of nearer society for the smooth run of a clinical pharmacy, ruined structure of road to be met a profit from a bus, etc) of an LP-problem and these affect the solution badly when experts ignore them. This study makes an attempt to solve an LP-problem by giving importance to all these to attain a fair outcome. The efficiency of the proposed concept is illustrated in a real field. A real example is stated and is solved numerically under the present view.

Keywords :

Neutrosophic set; Single valued trapezoidal neutrosophic (SVTN) number; Linear programming problem in neutrosophic sense; Simplex method

References :

[1]       Abbasbandy, S.; Asady, B. ”Ranking of fuzzy numbers by sign distance,” Information Sciences, 2006; vol. 176, pp.2405-2416.

[2]       Atanassov, K. ”Intuitionistic fuzzy sets,” Fuzzy sets and systems, 1986; 20(1), pp. 87-96.

[3]       Bera, T.; Mahapatra, N.K. ”Assignment problem with neutrosophic costs and its solution methodology,” Asia Mathematika, 2019; 3(1), pp. 21-32.

[4]       Bera, T.; Mahapatra, N.K. ”To solve assignment problem by centroid method in neutrosophic environ- ment,” Book chapter for ‘Quadruple Neutrosophic Theory and Applications-Volume I’, Neutrosophic Sci- ence International Association, Brussels, Belgium, 2020; Chapter 7, pp. 84-93.

[5]       Bera, T.; Mahapatra, N.K. ”Optimisation by dual simplex approach in neutrosophic environment,” Int. J. Fuzzy Comput. and Modelling, 2019; 2(4), pp. 334-352.

[6]       Bera, T.; Mahapatra, N.K. Generalised single valued neutrosophic number and its application to neutro- sophic linear programming, Neutrosophic Sets in Decision Analysis and Operations Research, IGI Global, Pennsylvania, 2019; Chapter 9.

[7]       Bera, T.; Mahapatra, N.K. ”Neutrosophic linear programming problem and its application to real life,” Afrika Matematika, accepted, 2019, https://doi.org/10.1007/s13370-019-00754-4.

[8]       Bera, T.; Mahapatra, N.K. ”On solving linear programming problem by duality approach in neutrosophic environment,” Int. J. of Math. in Operation Research, accepted, 2019.

[9]       Biswas, P.; Pramanik, S.;  Giri,  B.C. ”TOPSIS method for multi-attribute group decision making un-  der single- valued neutrosophic environment,” Neural Computing and Applications, Springer, 2015; doi:10.1007/s00521-015-1891-2.

[10]     Chakraborty, A. ”A New Score Function of Pentagonal Neutrosophic Number and its Application in Networking Problem,” International Journal of Neutrosophic Science, 2020; Volume 1 , Issue 1, PP. 40- 51.

[11]     Deli, I.; Subas, Y. A ”ranking method of single valued neutrosophic numbers and its application to multi-attribute decision making problems,” Int. J. Mach. Learn. and Cyber., (February, 2016), DOI 10.1007/s13042-016-0505-3.

[12]     Edalatpanah, S.A. ”A Direct Model for Triangular Neutrosophic Linear Programming,” International Journal of Neutrosophic Science; 2020, Volume 1 , Issue 1, PP. 19-28.

[13]     Gani, A.N.; Ponnalagu, K.”A method based on intuitionistic fuzzy linear programming for investment strategy, Int. J. Fuzzy Math. Arch., 2016; 10(1), pp. 71-81.

[14]     Hussian, A.; Mohamed, M.; Baset, M.; Smarandache, F. ”Neutrosophic linear programming problem,” Mathematical Sciences Letters, 2017; 3(6), pp. 319-324, DOI 10.18576/msl/060315.

[15]     Khalid, H.E. ”An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation Equa- tions (FNRE) with Geometric Programming (GP),” Neutrosophic Sets and Systems, 2015; vol. 7, pp. 3-7.

[16]     Khalid, H.E. ”The Novel Attempt for Finding Minimum Solution in Fuzzy Neutrosophic Relational Geometric Programming (FNRGP) with (max, min) Composition,” Neutrosophic Sets and Systems, 2016; vol. 11, pp. 107-111.


[17]     Khalid, H.E.; Smarandache, F.; Essa, A.K. ”The Basic Notions for (over, off, under) Neutrosophic Geo- metric Programming Problems,” Neutrosophic Sets and Systems, 2018; Vol. 22, pp. 50-62.

[18]     Khalid, H.E. ”Geometric Programming Dealt with a Neutrosophic Relational Equations Under the Op- eration. Neutrosophic Sets in Decision Analysis and Operations Research,” IGI Global Publishing House, 2020; Chapter 4.

[19]     Khalid, H.E. ”Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An Innovative Model,” Neutrosophic Sets and Systems; 2020, vol. 32, pp. 269-281.

[20]     Khalid, H.E. ”Neutrosophic Geometric Programming (NGP) Problems Subject to Operator; the Mini- mum Solution,” Neutrosophic Sets and Systems; 2020, vol. 32, pp. 15-24.

[21]     Li, D.F. ”A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems,” Comput. Math. Appl., 2010; Vol. 60, pp. 1557–1570.

[22]     Mukherjee, S.; Basu, K. ”Application of fuzzy ranking method for solving assignment problems with fuzzy costs,” Int. J. of Computational and App. Math., 2010; 5(3), pp. 359-368.

[23]     Mullai, M.; Sangeetha, K.; Surya, R.; Madhankumar, G.; Jeyabalan, R.; Broumi, S. ”A Single Valued Neutrosophic Inventory Model with Neutrosophic Random Variable,” International Journal of Neutro- sophic Science, 2020; Volume 1, Issue 2, PP. 52-63.

[24]     Pramanik, S. ”Neutrosophic multi-objective linear programming,” Global Journal of Engineering Science and Research Management, 2016; 3(8), DOI: 10.5281/zenodo.59949.

[25]     Rao, P.P.B.; Shankar, N.R. ”Ranking fuzzy numbers with an area method using circumferance of centroid, Fuzzy Information and Engineering,” 2013; Vol. 1, pp. 3-18.

[26]     Roy, R.; Das, P. ”A multi-objective production planning problem based on neutrosophic linear program- ming approach,” Int. J. of Fuzzy Mathematical Archive, 2015; 8(2), pp. 81-91.

[27]     Smarandache, F. ”Neutrosophic set, A generalisation of the intuitionistic fuzzy sets,” Inter. J. Pure Appl. Math., 2005; Vol. 24, pp. 287-297.

[28]     Wang, H.; Zhang, Y.; Sunderraman, R.; Smarandache, F. ”Single valued neutrosophic sets, Fuzzy Sets,” Rough Sets and Multivalued Operations and Applications, 2011; 3(1), pp. 33-39.

[29]     Yao, J.S.; Wu, K. ”Ranking of fuzzy numbers based on decomposition principle and signed distance,” Fuzzy Sets and System, 2000; Vol. 16, pp. 275-288.

[30]     Zadeh, ”L.A. Fuzzy sets,” Information and control, 1965; Vol. 8, pp. 338-353.