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Sains Malaysiana 53(11)(2024):
3803-3815
http://doi.org/10.17576/jsm-2024-5311-22
A Linearization
based on Taylor Expansion to Multi-Objective Linear Fractional Program with
Fuzzy Coefficients and Fuzzy Decision Variables
(Linearisasi berdasarkan Pengembangan Taylor kepada Program Pecahan Linear Pelbagai Objektif dengan Pekali Kabur dan Pemboleh Ubah Keputusan Kabur)
MOJTABA
BORZA & AZMIN SHAM RAMBELY*
Department of
Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi,
Selangor, Malaysia
Received: 23
January 2024/Accepted: 27 September 2024
Abstract
Reaching an efficient
solution for multi-objective programming problem (MOPP) is not easy and may
encompass some hardships due to existing more than one objective. The aim of
this research was to introduce a new efficient method to tackle fully fuzzy multi-objective
linear fractional programming problem (FFMOLFPP) i.e., a multi-objective linear
fractional programming problem (MOLFPP) with fuzzy coefficients and fuzzy
decision variables. To construct the approach, the of the fuzzy numbers, variable
transformations, the first-order Taylor series, the membership functions, and
the weighted sum method are used. In two
phases, this method alters the fully fuzzy problem into linear programming
problem (LPP) which its solution is at least a weakly efficient for the main
problem. Numerical examples are compared
to an existing method and the outcomes demonstrate that our proposed method is
much more accurate.
Keywords: Fuzzy
numbers; membership functions; Taylor series; the weighted sum method
Abstrak
Mencapai penyelesaian yang cekap untuk masalah pengaturcaraan berbilang objektif (MOPP) bukanlah mudah dan mungkin merangkumi beberapa kesukaran kerana terdapat lebih daripada satu objektif sedia ada. Matlamat penyelidikan ini adalah untuk memperkenalkan kaedah baharu yang cekap untuk menangani masalah pengaturcaraan pecahan linear berbilang objektif kabur sepenuhnya (FFMOLFPP) iaitu masalah pengaturcaraan pecahan linear berbilang objektif (MOLFPP) dengan pekali kabur dan pemboleh ubah keputusan kabur. Untuk membina pendekatan ini, potongan α nombor kabur, transformasi pemboleh ubah, siri Taylor tertib pertama, fungsi keahlian dan kaedah jumlah wajaran digunakan. Dalam dua fasa, kaedah ini mengubah masalah kabur sepenuhnya kepada masalah pengaturcaraan linear (LPP) yang penyelesaiannya sekurang-kurangnya ϵ-cekap untuk masalah utama. Contoh berangka dibandingkan dengan kaedah sedia ada dan hasilnya menunjukkan bahawa kaedah cadangan kami adalah lebih tepat.
Kata kunci: Fungsi keahlian; kaedah hasil tambah wajaran; nombor kabur; siri Taylor
References
Ahmad, S., Ullah, A., Akgül,
A. & Baleanu, D. 2020. Analysis of the fractional
tumour-immune-vitamins model with Mittag-Leffler
kernel. Results in Physics 19: 103559.
Antunes, C.H., Alves, M.J. & Clímaco, J. 2016. Multiobjective Linear and Integer Programming. New York: Springer.
Arya, R.,
Singh, P., Kumari, S. & Obaidat, M.S. 2020. An approach for solving fully
fuzzy multi-objective linear fractional optimization problems. Soft
Computing 24: 9105-9119.
Borza, M. & Rambely, A.S. 2023. A new efficient approach to tackle
multi objective linear fractional problem with flexible constraints. Journal
of Industrial and Management Optimization 19(6): 4180-4198.
Borza, M. & Rambely, A.S. 2022a. Tackling the fuzzy multi-objective linear
fractional problem using a parametric approach. Journal of Intelligent
& Fuzzy Systems 43(1): 721-734.
Borza, M. & Rambely, A.S. 2022b. A parametric approach to fuzzy
multi-objective linear fractional program: An alpha cut based method. Journal
of Intelligent & Fuzzy Systems 42(6): 5639-5652.
Borza, M. & Rambely, A.S. 2022c. An approach based on cuts and max-min
technique to linear fractional programming with fuzzy coefficients. Iranian
Journal of Fuzzy Systems 19(1): 153-169.
Borza, M. & Rambely, A.S. 2021a. A new method to solve multi-objective
linear fractional problems. Fuzzy Information and Engineering 13(3):
323-334.
Borza, M. & Rambely,
A.S. 2021b. A linearization to the sum of linear ratios programming problem. Mathematics 9(9): 1004.
Borza, M., Rambely, A.S. & Edalatpanah,
S.A. 2023. A linearization to the multi-objective linear plus linear fractional
program. Operations Research Forum 4: 82.
Borza, M., Rambely,
A. & Saraj, M. 2012. A Stackelberg solution to a two-level linear fractional programming problem with interval
coefficients in the objective functions. Sains Malaysiana 41(12): 1651-1656.
Chakraborty, M. & Gupta,
S. 2002. Fuzzy mathematical programming for multi objective linear fractional
programming problem. Fuzzy Sets and Systems 125(3): 335-342.
Chanas, S. & Kuchta, D. 1994. Linear programming problem with fuzzy
coefficients in the objective function. In Fuzzy Optimization, Recent
Advances, edited by Delgado, M., Kacprzyk, J., Verdegay, J.L. & Vila, M.A. Wurzburg: Physica-Verlag. pp.148-157.
Charnes, A. & Cooper, W.W. 1962.
Programming with linear fractional functionals. Naval Research
Logistics Quarterly 9(3‐4): 181-186.
Chinnadurai, V. & Muthukumar, S. 2016. Solving the linear fractional
programming problem in a fuzzy environment: Numerical approach. Applied
Mathematical Modelling 40(11-12): 6148-6164.
Das, S.K., Edalatpanah, S.A. & Mandal, T. 2020. Application of
linear fractional programming problem with fuzzy nature in industry
sector. Filomat 34(15): 5073-5084.
De, P.K. & Deb, M. 2015.
Solution of multi objective linear fractional programming problem by Taylor
series approach. 2015 International Conference on Man and Machine
Interfacing (MAMI), Bhubaneswar,
India. pp. 1-5. doi: 10.1109/MAMI.2015.7456582
Deb, M. 2018. A study of
fully fuzzy linear fractional programming problems by signed distance ranking
technique. In Optimization Techniques for Problem Solving in
Uncertainty, edited by Tilahun, S.L. & Ngnotchouye,
J.M.T. Hershey: IGI Global. pp. 73-115.
Dinkelbach, W. 1967. On nonlinear
fractional programming. Management Science 13(7): 492-498.
Garai, T. & Garg, H. 2019.
Multi‐objective linear fractional inventory model with possibility and
necessity constraints under generalised intuitionistic fuzzy set
environment. CAAI Transactions on Intelligence Technology 4(3):
175-181.
Garg, H., Mahmoodirad,
A. & Niroomand, S. 2021. Fractional two-stage transshipment problem under uncertainty: Application of the
extension principle approach. Complex & Intelligent Systems 7(2):
807-822.
Kaufmann, A. & Gupta,
M.M. 1988. Fuzzy Mathematical Models in Engineering and Management
Science. Elsevier Science Inc.
Kumar, A., Kaur, J. &
Singh, P. 2011. A new method for solving fully fuzzy linear programming
problems. Applied Mathematical Modelling 35(2): 817-823.
Liu, X., Gao, Y.L., Zhang, B.
& Tian, F.P. 2019. A new global optimization algorithm for a class of
linear fractional programming. Mathematics 7(9): 867.
Mahmoodirad, A., Garg, H. & Niroomand, S. 2022. Solving fuzzy linear fractional set covering
problem by a goal programming based solution
approach. Journal of Industrial & Management Optimization 18(1):
439-456.
Mehra, A., Chandra, S. & Bector, C.R. 2007. Acceptable optimality in linear
fractional programming with fuzzy coefficients. Fuzzy Optimization and
Decision Making 6: 5-16.
Moore, R.E., Kearfott, R.B. & Cloud, M.J. 2009. Introduction
to Interval Analysis. Philadelphia: Society for Industrial and Applied
Mathematics.
Nayak, S. & Ojha, A.K.
2019. Multi-objective linear fractional programming problem with fuzzy
parameters. In Soft Computing for Problem Solving. Advances in
Intelligent Systems and Computing, edited by Bansal, J., Das, K., Nagar,
A., Deep, K. & Ojha, A. Singapore: Springer. pp. 79-90.
Pal, B.B., Moitra, B.N. & Maulik, U. 2003.
A goal programming procedure for fuzzy multiobjective linear fractional programming problem. Fuzzy Sets and Systems 139(2):
395-405.
Radhakrishnan, B. & Anukokila, P. 2014. Fractional goal programming for fuzzy
solid transportation problem with interval cost. Fuzzy Information and
Engineering 6(3): 359-377.
Rao, P.P.B. 2017. Ranking fuzzy numbers
using alpha cuts and centroids. Journal of Intelligent & Fuzzy
Systems 33(4): 2249-2258.
Rashmanlou, H. & Borzooei,
R.A. 2016. Vague graphs with application. Journal of Intelligent &
Fuzzy Systems 30(6): 3291-3299.
Stancu-Minasian, I.M. 1997. Fractional
Programming: Theory, Methods and Applications. Springer Science &
Business Media.
StanojeviÄ, B. & StanojeviÄ, M. 2013. Solving method for linear fractional optimization
problem with fuzzy coefficients in the objective function. International
Journal of Computers Communications & Control 8(1): 146-152.
Toksarı, M.D. 2008. Taylor series
approach to fuzzy multiobjective linear fractional
programming. Information Sciences 178(4): 1189-1204.
Vafamand, A., Vafamand,
N., Zarei, J., Razavi-Far,
R. & Saif, M. 2021. Multi-objective NSBGA-II
control of HIV therapy with monthly output measurement. Biomedical
Signal Processing and Control 68: 102561.
Vafamand, A., Vafamand,
N., Zarei, J., Razavi-Far,
R. & Dragičević, T. 2020. Intelligent multiobjective NSBGA-II control of power converters in DC
microgrids. IEEE Transactions on Industrial Electronics 68(11):
10806-10814.
Veeramani, C. & Sumathi, M. 2014.
Fuzzy mathematical programming approach for solving fuzzy linear fractional
programming problem. RAIRO-Operations Research 48(1): 109-122.
Wang, L.X. 1996. A
Course in Fuzzy Systems and Control. New Jersey: Prentice-Hall, Inc.
Wang, Y., Liu, L., Guo, S., Yue, Q. &
Guo, P. 2019. A bi-level multi-objective linear fractional programming for
water consumption structure optimization based on water shortage risk. Journal
of Cleaner Production 237: 117829.
Zapata, H., Perozo,
N., Angulo, W. & Contreras, J. 2020. A hybrid swarm algorithm for
collective construction of 3D structure. International Journal of
Artificial Intelligence 18(1): 1-18.
Zimmermann, H.J. 2001. Fuzzy Set
Theory - and Its Applications. 4th ed. Springer Science & Business
Media.
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