Abstract
The current article is devoted to mathematically handling the inherent uncertainties of various practical problems by introducing the concept of LR-type interval-valued intuitionistic fuzzy numbers (LR-type IVIFN). The theoretic development of this concept has also been enhanced by providing various diagrammatic representations of LR-type IVIFNs and establishing the arithmetic operations among these fuzzy numbers. The notion of \(\alpha ,\beta \)-cuts and interval arithmetic have been employed to derive the expressions for the arithmetic operations on LR-type IVIFNs. In addition to this, the total order properties of lexicographic-ranking criteria along with considering seven distinct parameters have been used to define the ordering on LR-type IVIFNs. Further, a linear programming problem (LPP) with both equality and inequality type constraints, all parameters in the form of LR-type IVIFNs, and unrestricted decision variables has been modeled. In order to obtain a unique optimal solution for the proposed LPP, the lexicographic ranking-based solution methodology has been developed in which by introducing some binary variables, the original LPP is converted to an equivalent mixed 0-1 lexicographic non-linear programming problem having seven components in the objective function. Various theorems have also been proved to show the equivalence of the original problem and its different proposed constructions. The model formulation, algorithm, and discussed results have not only developed a new idea but also generalized various well-known related works existing in the literature. Moreover, a numerical illustration has been provided to clarify the step-wise procedure of the proposed solution approach. Additionally, to establish the practical significance of the study, a real-world production planning problem is framed, solved, and analyzed under the LR-type interval-valued intuitionistic fuzzy scenario. In last, a comparative study has also been carried out to prove the relevancy of the developed algorithm.
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References
Akbari MG, Hesamian G (2018) Signed-distance measures oriented to rank interval-valued fuzzy numbers. IEEE Trans Fuzzy Syst 26:3506–3513
Akram M, Allahviranloo T, Pedrycz W, Ali M (2021a) Methods for solving \(LR\)-bipolar fuzzy linear systems. Soft Comput 25:85–108
Akram M, Ullah I, Alharbi MG (2021b) Methods for solving \(LR\)-type Pythagorean fuzzy linear programming problems with mixed constraints. Math Probl Eng 2021:1–29
Akram M, Ullah I, Allahviranloo T, Edalatpanah S (2021c) Fully Pythagorean fuzzy linear programming problems with equality constraints. Comput Appl Math 40:1–30
Akram M, Ullah I, Allahviranloo T, Edalatpanah SA (2021d) \(LR\)-type fully Pythagorean fuzzy linear programming problems with equality constraints. J Intell Fuzzy Syst 41:1975–1992
Akram M, Ullah I, Allahviranloo T (2022a) A new method for the solution of fully fuzzy linear programming models. Comput Appl Math 41(1):55
Akram M, Ullah I, Allahviranloo T (2022b) A new method to solve linear programming problems in the environment of picture fuzzy sets. Iran J Fuzzy Syst 19:29–49
Allahviranloo T, Lotfi FH, Kiasary MK, Kiani N, Alizadeh L (2008) Solving fully fuzzy linear programming problem by the ranking function. Appl Math Sci 2:19–32
Angelov PP (1997) Optimization in an intuitionistic fuzzy environment. Fuzzy Sets Syst 86:299–306
Arana-Jiménez M (2018) Nondominated solutions in a fully fuzzy linear programming problem. Math Methods Appl Sci 41:7421–7430
Arefi M, Taheri SM (2014) Least-squares regression based on Atanassov’s intuitionistic fuzzy inputs-outputs and Atanassov’s intuitionistic fuzzy parameters. IEEE Trans Fuzzy Syst 23:1142–1154
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Atanassov KT (1999) Intuitionistic fuzzy sets: theory and applications. Studies in fuzziness and soft computing. Physica-Verlag, Heidelberg
Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349
Aydın T, Enginoğlu S (2022) Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft matrices and their application to performance-based value assignment to noise-removal filters. Comput Appl Math 41:192
Bharati SK, Singh S (2018) Transportation problem under interval-valued intuitionistic fuzzy environment. Int J Fuzzy Syst 20:1511–1522
Bharati SK, Singh S (2019) Solution of multiobjective linear programming problems in interval-valued intuitionistic fuzzy environment. Soft Comput 23:77–84
Bharati SK, Singh S (2020) Interval-valued intuitionistic fuzzy linear programming problem. New Math Nat Comput 16:53–71
Chen SM, Lee LW (2010) Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. Expert Syst Appl 37:824–833
Chen SM, Wang CY (2013) Fuzzy decision making systems based on interval type-2 fuzzy sets. Inf Sci 242:1–21
Chen SM, Yang MW, Lee LW, Yang SW (2012) Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets. Expert Syst Appl 39:5295–5308
Das S (2021) Optimization of fuzzy linear fractional programming problem with fuzzy numbers. Big Data Comput Vis 1(1):30–35
Das SK, Mandal T, Edalatpanah S (2017) A mathematical model for solving fully fuzzy linear programming problem with trapezoidal fuzzy numbers. Appl Intell 46:509–519
Ebrahimnejad A, Verdegay JL (2018) Fuzzy sets-based methods and techniques for modern analytics. Studies in fuzziness and soft computing, vol 364, 1st edn. Springer, Cham. https://doi.org/10.1007/978-3-319-73903-8
Enginoǧlu S, Arslan B (2020) Intuitionistic fuzzy parameterized intuitionistic fuzzy soft matrices and their application in decision-making. Comput Appl Math 39:325
Ezzati R, Khorram E, Enayati R (2015) A new algorithm to solve fully fuzzy linear programming problems using the MOLP problem. Appl Math Model 39:3183–3193
Farhadinia B (2009) Ranking fuzzy numbers based on lexicographical ordering. Int J Appl Math Comput Sci 5:248–251
Garg H, Rani M, Sharma S, Vishwakarma Y (2014) Intuitionistic fuzzy optimization technique for solving multi-objective reliability optimization problems in interval environment. Expert Syst Appl 41:3157–3167
Giri PK, Maiti MK, Maiti M (2015) Fully fuzzy fixed charge multi-item solid transportation problem. Appl Soft Comput 27:77–91
Gong Z, Zhao W (2018) A novel approach for solving fully fuzzy linear programming problem with LR flat fuzzy numbers. J Comput Anal Appl 24:11–22
Greco S, Figueira J, Ehrgott M (2016) Multiple criteria decision analysis. International series in operations research and management science, vol 233. Springer, New York. https://doi.org/10.1007/978-1-4939-3094-4
Hashemi SM, Modarres M, Nasrabadi E, Nasrabadi MM (2006) Fully fuzzified linear programming, solution and duality. J Intell Fuzzy Syst 17:253–261
Hesamian G (2016) Measuring similarity and ordering based on interval type-2 fuzzy numbers. IEEE Trans Fuzzy Syst 25:788–798
Hesamian G, Akbari MG (2022) A fuzzy empirical quantile-based regression model based on triangular fuzzy numbers. Comput Appl Math 41:267
Hosseinzadeh A, Edalatpanah S (2016) A new approach for solving fully fuzzy linear programming by using the lexicography method. Adv Fuzzy Syst
Hosseinzadeh E, Tayyebi J (2023) A compromise solution for the neutrosophic multi-objective linear programming problem and its application in transportation problem. J Appl Res Ind Eng 10(1):1–10
Ishibuchi H, Tanaka H (1990) Multiobjective programming in optimization of the interval objective function. Eur J Oper Res 48:219–225
Jafar MN, Saeed M, Khan KM, Alamri FS, Khalifa HAEW (2022) Distance and similarity measures using max-min operators of neutrosophic hypersoft sets with application in site selection for solid waste management systems. IEEE Access 10:11220–11235
Kane L, Bado H, Diakite M, Konate M, Kane S, Traore K (2021) Solving semi-fully fuzzy linear programming problems. Int J Res Ind Eng 10(3):251–275
Kaur J, Kumar A (2012) Unique fuzzy optimal value of fully fuzzy linear programming problems. Control Cybern 41:497–508
Kaur J, Kumar A (2013) Mehar’s method for solving fully fuzzy linear programming problems with LR fuzzy parameters. Appl Math Model 37:7142–7153
Kaur J, Kumar A (2016) Unique fuzzy optimal value of fully fuzzy linear programming problems with equality constraints having \(LR\) flat fuzzy numbers. In: An introduction to fuzzy linear programming problems: theory, methods and applications, vol 340. Springer, Cham, pp 109–118
Khan IU, Ahmad T, Maan N (2013) A simplified novel technique for solving fully fuzzy linear programming problems. J Optim Theory Appl 159:536–546
Kumar A, Kaur J (2014) Fuzzy optimal solution of fully fuzzy linear programming problems using ranking function. J Intell Fuzzy Syst 26:337–344
Kumar PS, Hussain RJ (2016) Computationally simple approach for solving fully intuitionistic fuzzy real life transportation problems. Int J Syst Assur Eng Manag 7:90–101
Kumar A, Kaur J, Singh P (2011) A new method for solving fully fuzzy linear programming problems. Appl Math Model 35:817–823
Lotfi FH, Allahviranloo T, Jondabeh MA, Alizadeh L (2009) Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Appl Math Model 33:3151–3156
Mahapatra G, Roy T (2009) Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations. World Acad Sci Eng Technol 50:574–581
Mahmoodirad A, Allahviranloo T, Niroomand S (2019) A new effective solution method for fully intuitionistic fuzzy transportation problem. Soft Comput 23:4521–4530
Mahmoudi F, Nasseri SH (2019) A new approach to solve fully fuzzy linear programming problem. J Appl Res Ind Eng 6(2):139–149
Mekawy I (2022) A novel method for solving multi-objective linear fractional programming problem under uncertainty. J Fuzzy Ext Appl 3(2):169–176
Mottaghi A, Ezzati R, Khorram E (2015) A new method for solving fuzzy linear programming problems based on the fuzzy linear complementary problem (FLCP). Int J Fuzzy Syst 17:236–245
Nagoorgani A, Ponnalagu K (2012) A new approach on solving intuitionistic fuzzy linear programming problem. Appl Math Sci 6:3467–3474
Najafi HS, Edalatpanah S (2013) A note on “a new method for solving fully fuzzy linear programming problems’’. Appl Math Model 37:7865–7867
Najafi HS, Edalatpanah S, Dutta H (2016) A nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters. Alex Eng J 55:2589–2595
Niroomand S (2018) A multi-objective based direct solution approach for linear programming with intuitionistic fuzzy parameters. J Intell Fuzzy Syst 35:1923–1934
Ozkok BA, Albayrak I, Kocken HG, Ahlatcioglu M (2016) An approach for finding fuzzy optimal and approximate fuzzy optimal solution of fully fuzzy linear programming problems with mixed constraints. J Intell Fuzzy Syst 31:623–632
Pérez-Cañedo B, Concepción-Morales ER (2019a) A method to find the unique optimal fuzzy value of fully fuzzy linear programming problems with inequality constraints having unrestricted \(LR\) fuzzy parameters and decision variables. Expert Syst Appl 123:256–269
Pérez-Cañedo B, Concepción-Morales ER (2019b) On \(LR\)-type fully intuitionistic fuzzy linear programming with inequality constraints: solutions with unique optimal values. Expert Syst Appl 128:246–255
Rahmani A, Hosseinzadeh Lotfi F, Rostamy-Malkhalifeh M, Allahviranloo T (2016) A new method for defuzzification and ranking of fuzzy numbers based on the statistical beta distribution. Adv Fuzzy Syst
Ranjbar M, Effati S, Miri SM (2022) Fully hesitant fuzzy linear programming with hesitant fuzzy numbers. Eng Appl Artif Intell 114:105047
Rezaei A, Oner T, Katican T, Smarandache F, Gandotra N (2022) A short history of fuzzy, intuitionistic fuzzy, neutrosophic and plithogenic sets. Int J Neutrosophic Sci 18(1): 99–116
Roy SK, Ebrahimnejad A, Verdegay JL, Das S (2018) New approach for solving intuitionistic fuzzy multi-objective transportation problem. Sādhanā 43:1–12
Saghi S, Nazemi A, Effati S, Ranjbar M (2023) Simplex algorithm for hesitant fuzzy linear programming problem with hesitant cost coefficient. Iran J Fuzzy Syst 20:137–152
Şahin R (2016) Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets. Soft Comput 20:2557–2563
Sidhu SK, Kumar A (2019) Mehar methods to solve intuitionistic fuzzy linear programming problems with trapezoidal intuitionistic fuzzy numbers. In: Deep K, Jain M, Salhi S (eds) Performance prediction and analytics of fuzzy, reliability and queuing models. Asset Analytics. Springer Berlin, pp 265–282
Singh SK, Yadav SP (2015) Modeling and optimization of multi objective non-linear programming problem in intuitionistic fuzzy environment. Appl Math Model 39:4617–4629
Singh V, Yadav SP (2017) Development and optimization of unrestricted LR-type intuitionistic fuzzy mathematical programming problems. Expert Syst Appl 80:147–161
Suresh M, Vengataasalam S, Prakash KA (2014) Solving intuitionistic fuzzy linear programming problems by ranking function. J Intell Fuzzy Syst 27:3081–3087
Tadesse A, Acharya M, Sahoo M, Acharya S (2021) Fuzzy linear programming problem with fuzzy decision variables: a geometrical approach. J Stat Manag Syst 24:853–863
Tamilarasi G, Paulraj S (2022) An improved solution for the neutrosophic linear programming problems based on Mellin’s transform. Soft Comput 26(17):8497–8507
Tanaka H, Asai K (1984) Fuzzy solution in fuzzy linear programming problems. IEEE Trans Syst Man Cybern 2:325–328
Voskoglou M (2020) Assessment and linear programming under fuzzy conditions. https://doi.org/10.22105/jfea.2020.253436.1024
Wan SP, Wang F, Lin LL, Dong JY (2015) An intuitionistic fuzzy linear programming method for logistics outsourcing provider selection. Knowl Based Syst 82:80–94
Wang P, Lin Y, Fu M, Wang Z (2023) VIKOR method for plithogenic probabilistic linguistic MAGDM and application to sustainable supply chain financial risk evaluation. Int J Fuzzy Syst 25:780–793
Yang X, Lin TY, Yang J, Li Y, Yu D (2009) Combination of interval-valued fuzzy set and soft set. Comput Math Appl 58:521–527
Zadeh LA (1965) Information and control. Fuzzy Sets 8:338–353
Zhang SF, Liu SY, Zhai RH (2011) An extended GRA method for MCDM with interval-valued triangular fuzzy assessments and unknown weights. Comput Ind Eng 61:1336–1341
Zhao J, Li B, Rahman AU, Saeed M (2023) An intelligent multiple-criteria decision-making approach based on sv-neutrosophic hypersoft set with possibility degree setting for investment selection. Manag Decis 61:472–485
Zimmermann HJ (1975) Description and optimization of fuzzy systems. Int J Gen Syst 2:209–215
Zulqarnain RM, Siddique I, Ali R, Jarad F, Iampan A (2023) Aggregation operators for interval-valued Pythagorean fuzzy hypersoft set with their application to solve MCDM problem. CMES Comput Model Eng Sci 135(1):619–651
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The authors would like to sincerely thank the Editor-in-Chief and anonymous referees for their insightful comments and recommendations which have significantly enhanced both the quality and clarity of the paper. The first author is also grateful to the Ministry of Human Resource Development, India, for financial support, to carry out this research work.
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Malik, M., Gupta, S.K. & Arana-Jiménez, M. Developing solution algorithm for LR-type fully interval-valued intuitionistic fuzzy linear programming problems using lexicographic-ranking method. Comp. Appl. Math. 42, 274 (2023). https://doi.org/10.1007/s40314-023-02408-5
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DOI: https://doi.org/10.1007/s40314-023-02408-5
Keywords
- Fuzzy mathematical programming
- Interval-valued intuitionistic fuzzy number
- LR-type fuzzy number
- Lexicographic ranking criterion
- Score function
- Accuracy function