ORIGINAL_ARTICLE
Quasi-Gap and Gap Functions for Non-Smooth Multi-Objective Semi-Infinite Optimization Problems
In this paper, we introduce and study some new single-valued gap functions for non-differentiable semi-infinite multiobjective optimization problems with locally Lipschitz data. Since one of the fundamental properties of gap function for optimization problems is its abilities in characterizing the solutions of the problem in question, then the essential properties of the newly introduced gap functions are established. All results are given in terms of the Clarke subdifferential.
https://mathco.journals.pnu.ac.ir/article_6718_5931fe972a6feb3c383122e430f01982.pdf
2018-01-01
1
12
10.30473/coam.2020.50781.1133
Multiobjective optimization
Semi-Infinite Programming
Gap function
Clarke subdifferential
Atefeh
Hassani Bafrani
a.hassani@pnu.ac.ir
1
Department of Mathematics, Payame Noor University, P.O. Box, 19395-3697, Tehran, Iran.
LEAD_AUTHOR
Ali
Sadeghieh
alijon.sadeghieh@gmail.com
2
Department of Mathematics, Yazd branch, Islamic Azad University, Yazd, Iran
AUTHOR
[1] L. Altangerel, R.I. Bot, G. Wanka, Conjugate duality in vector optimization and some applications to the vector variational inequality, Journal of Mathematical Analysis and Applications 329 (2007) 1010-1035.
1
[2] G. Caristi, N. Kanzi, Karush-Kuhn-Tuker Type Conditions for Optimality of NonSmooth Multiobjective Semi-Infinite Programming, International Journal of Mathematical Analysis 9 (2015) 1929-1938.
2
[3] G. Caristi, N. Kanzi, M., Soleymani-Damaneh, On gap functions for nonsmooth multiobjective optimization problems, Optim. Lett. 12 (2018) 273-286.
3
[4] C.Y. Chen, C.J. Goh, X.Q. Yang, The gap function of a convex multicriteria optimization problem, European Journal of Operational Research 111 (1998) 142-151.
4
[5] T.D. Chuong, D.S. Kim, Nonsmooth semi-infinite multiobjective optimization problems, Journal of Optimization; Theory and Applications 160 (2014) 748-762.
5
[6] F.H. Clarke, Optimization and nonsmooth analysis. Wiley, Interscience (1983).
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[7] D.W. Hearn, The gap function of a convex program, Operations Research Letters 1 (1982) 67-71.
7
[8] N. Kanzi, Necessary and Suffcient Conditions for (Weakly) Effcient of NonDifferentiable Multi-Objective Semi-Infinite Programming Problems, Iran J. Sci. Technol. Trans. Sci. 42 (2018) 1537-1544.
8
[9] N. Kanzi, Karush-Kuhn-Tucker Types Optimality Conditions for Non-Smooth Semi-Infinite Vector Optimization Problems, J. Mathematical Extension 9 (2015) 45-56.
9
[10] N. Kanzi, M. Soleyman-damaneh, Slater CQ, optimality and duality for quasiconvex semi-infinite optimization problems, J. Math.Anal.Appl. 434 (2016) 638-651.
10
[11] N. Kanzi, J. Shaker Ardekani, G. Caristi, Optimality, scalarization and duality in linear vector semi-infinite programming, Optimization. 67 (2018) 523-536.
11
[12] G. Mastroeni, Gap functions for equilibrium problems, J. Glob. Optim. 27 (2003) 411–426.
12
[13] H. Mirzaee, M. Soleimani-damaneh, Optimality, duality and gap function for quasi variational inequality problems, ESAIM Control Optim. Calc. Var. 23 (2017) 297–308.
13
[14] M. Soleimani-damaneh, The gap function for optimization problems in Banach spaces, Nonlinear Analysis 69 (2008) 716-723.
14
ORIGINAL_ARTICLE
A General Scalar-Valued Gap Function for Nonsmooth Multiobjective Semi-Infinite Programming
For a nonsmooth multiobjective mathematical programming problem governed by infinitely many constraints, we define a new gap function that generalizes the definitions of this concept in other articles. Then, we characterize the efficient, weakly efficient, and properly efficient solutions of the problem utilizing this new gap function. Our results are based on $(\Phi,\rho)-$invexity, defined by Clarke subdifferential.
https://mathco.journals.pnu.ac.ir/article_6719_59f10887b88b535008100a8c10d94722.pdf
2018-01-01
13
26
10.30473/coam.2019.45495.1110
Semi-Infinite Programming
Multiobjective optimization
Constraint qualification
Optimality conditions
Gap function
Ahmad
Rezayi
a.rezaee70@pnu.ac.ir
1
Department of Mathematics, Payame Noor University, P.O. Box. 19395-3697, Tehran, Iran
LEAD_AUTHOR
[1] Antczak T. (2015) ”Saddle point criteria and Wolfe duality in nonsmooth (Φ, ρ)-invex vector optimization problems with inequality and equality constraints”, International J. Computer Math. 92, 882-907.
1
[2] Antczak T., Stasiak A. (2011) ”(Φ, ρ)-Invexity in Nonsmooth Optimization”, Numer. Func. Anal. Optim.32, 1-25.
2
[3] Ben-Israel A., Mond B. (1986) ”What is invexity?”, J. Australian Math. Sci. 28, 1-9.
3
[4] Caristi G., Kanzi M. and Soleimani-damaneh M. (2017) ”On gap functions for nonsmooth multiobjective optimization problems”, Optim Letters. DOI: 10.1007/s11590-017-1110-4.
4
[5] Caristi G., Ferrara M. and Stefanescu A. (2010) ”Semi-infinite multiobjective programming with generalized invexity”, Mathematical Reports, 62, 217-233.
5
[6] Caristi G., Ferrara M. and Stefanescu A. (2006) ”Mathematical programming with (ρ, Φ)- invexity”, In Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, Vol. 583. (I.V. Konnor, D.T. Luc, and A.M. Rubinov, eds.). Springer, Berlin-Heidelberg-New York, 167-176.
6
[7] Caristi G., Kanzi N. (2015) ”Karush-Kuhn-Tuker Type Conditions for Optimality of NonSmooth Multiobjective Semi-Infinite Programming”, International Journal of Mathematical Analysis 9, 1929-1938.
7
[8] Clarke F. H. (1983) ”Optimization and Nonsmooth Analysis”, Wiley, Interscience.
8
[9] Chen C. Y., Goh C. J., Yang, X. Q. (1998) ”The gap function of a convex multicriteria optimization problem”, Eur. J. Oper. Res. 111, 142–151.
9
[10] Ehrgott M. (2005) ”Multicriteria Optimization”, Springer, Berlin.
10
[11] Gao X. Y. (2013) ”Optimality and duality for non-smooth multiobjective semi-infinite programming”, J. Netw. 8, 413-420.
11
[12] Goberna M.A., Kanzi N. (2017) ”Optimality conditions in convex multiobjective” SIP. Math. Programming. DOI 10.1007/s10107-016-1081-8.
12
[13] Goberna M. A., Guerra-Vazquez F. and Todorov M. I. (2016) ”Constraint qualifications in linear vector semi-infinite optimization”, European J. Oper. Res. 227, 32-40.
13
[14] Goberna M. A., Guerra-Vazquez F. and Todorov M. I. (2013) ”Constraint qualifications in convex vector semi-infinite optimization”, European J. Oper. Res. 249, 12-21.
14
[15] Gopfert A., Riahi H., Tammer C. and Zalinescu C. (2003) ”Variational methods in partial ordered spaces”, Springer, New York.
15
[16] Guerraggio A., Molho E. and Zaffaroni A. (1994) ”On the notion of proper efficiency in vector optimization”, J. Optim. Theory Appl. 82, 1-21.
16
[17] Hearn D. W., (1982) ”The gap function of a convex program”, Oper. Res. Lett. 1, 67–71.
17
[18] Kanzi N., Shaker Ardekani J., Caristi G. (2018) ”Optimality, scalarization and duality in linear vector semi-infinite programming”, Optimization. Doi: 10.1080/02331934.2018.1454921.
18
[19] Kanzi N. (2017) ”Necessary and sufficient conditions for (weakly) efficient of nondifferentiable multi-objective semi-infinite programming”, Iranian Journal of Science and Technology Transactions A: Science, DOI 10.1007/s40995-017-0156-6.
19
[20] Kanzi N. (2015) ”Karush-Kuhn-Tucker Types Optimality Conditions For Non-Smooth Semi-Infinite Vector Optimization Problems”, Journal of Mathematical Extension 9, 45- 56.
20
[21] Kanzi N. (2015) ”Regularity Conditions for Non-Differentiable Infinite Programming Problems Using Michel-Penot Subdifferential”, Control and Optimization in Applied Mathematics, 1, 21-30.
21
[22] Noor M. A. (2004) ”On generalized preinvex functions and monotonicities”, Journal of Inequalities in Pure and Applied Mathematics. 5, 1–9.
22
[23] Rezayi A. (2018) ”Characterization of Isolated Efficient Solutions in Nonsmooth Multiobjective Semi-infinite Programming”, Iran J Sci Technol Trans Sci, Doi.org/10.1007/s40995- 018-0637-2.
23
ORIGINAL_ARTICLE
MQ-Radial Basis Functions Center Nodes Selection with PROMETHEE Technique
In this paper, we decide to select the best center nodes of radial basis functions by applying the Multiple Criteria Decision Making (MCDM) techniques. Two methods based on radial basis functions to approximate the solution of partial differential equation by using collocation method are applied. The first is based on the Kansa's approach, and the second is based on the Hermite interpolation. In addition, by choosing five sets of center nodes: Uniform grid, Cartesian, Chebyshev, Legendre and Legendre-Gauss-Lobato (LGL) as alternatives and achieving the error, the condition number of interpolation matrix and memory time as criteria, rating of cases with the help of PROMETHEE technique is obtained. In the end, the best center nodes and method is selected according to the rankings. This ranking shows that Hermite interpolation by using non-uniform nodes as center nodes is more suitable than Kansa's approach with each center node.
https://mathco.journals.pnu.ac.ir/article_6720_2370abbdf97f6860667771e922775200.pdf
2018-01-01
27
47
10.30473/coam.2019.46609.1117
Multiple Criteria Decision Making
Radial basis functions
PROMETHEE
Hermite interpolation
Optimal selecting
Farhad
Hadinejad
hadinejadfarhad@gmail.com
1
Phd of Operation Research Management, Allameh Tabataba'i University and Assistant professor, Imam Ali University, Tehran, Iran
LEAD_AUTHOR
Saeed
Kazem
saeedkazem@gmail.com
2
Department of Applied Mathematics, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran
AUTHOR
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88
ORIGINAL_ARTICLE
Characterization of Properly Efficient Solutions for Convex Multiobjective Programming with Nondifferentiable vanishing constraints
This paper studies the convex multiobjective optimization problem with vanishing constraints. We introduce a new constraint qualification for these problems, and then a necessary optimality condition for properly efficient solutions is presented. Finally by imposing some assumptions, we show that our necessary condition is also sufficient for proper efficiency. Our results are formulated in terms of convex subdifferential.
https://mathco.journals.pnu.ac.ir/article_6721_67d7341f87c07e5b97e65519ca245268.pdf
2018-01-01
49
58
10.30473/coam.2019.42442.1094
Multiobjective optimization
Vanishing constraints
Convex optimization
Constraint qualification
Javad
Shaker Ardakani
shaker.ar.yazdstc@gmail.com
1
Department of Mathematics, Payame Noor University (PNU), P.OBox, 19395-4679, Tehran, Iran.
AUTHOR
shahriar
Farahmand Rad
sh_fmand@pnu.ac.ir
2
Department of Mathematics, Payame Noor University (PNU), P.OBox, 19395-4679, Tehran, Iran.
AUTHOR
Nader
Kanzi
nad.kanzi@gmail.com
3
Department of Mathematics, Payame Noor University (PNU), P.OBox, 19395-4679, Tehran, Iran.
LEAD_AUTHOR
[1] Achtziger W., Kanzow C. (2007). “ Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications”, Math. Program, 114, 69–99.
1
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[4] Gopfert A., Riahi H., Tammer C., Zalinescu C. (2003). ‘ Variational methods in partial ordered spaces”, Springer, New York.
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[5] Henrion R., Jourani A., Outrata J. (2002). “ On the calmness of a class of multi-functions”, SIAM J. Optim, 13, 603–618.
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[6] Hiriart-Urruty J.B., Lemarechal C. (1991). “ Convex analysis and minimization algorithms”, I. Berlin: Springer.
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[7] Hoheisel T., Kanzow C. (2008). “ Stationarity conditions for mathematical programs with vanishing constraints using weak constraint qualifications”, J. Math. Anal. Appl, 337, 292–310.
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[8] Hoheisel T., Kanzow C., Outrata J. (2010). “ Exact penalty results for mathematical programs with vanishing constraints”, Nonlinear Anal, 72, 2514–2526.
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[9] Hoheisel T., Kanzow C. (2007). “ First– and second–order optimality conditions for mathematical programs with vanishing constraints”, Appl Math, 52, 495–514.
9
[10] Kazemi S., Kanzi N. (2018). “ Constraint Qualifications and Stationary Conditions for Mathematical Programming with Non-differentiable Vanishing Constraints”, Journal of Optimization Theory and Applications, DOI 10.1007/s10957–018–1373–7.
10
[11] Kazemi S., Kanzi N., Ebadian A. (2019). “ Estimating the Frèchet Normal Cone in Optimization Problems with Nonsmooth Vanishing Constraints”, Iranian Journal of Science and Technology, Transactions A: Science, DOI 10.1007/s40995–019–00683–8.
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16
ORIGINAL_ARTICLE
Two Settings of the Dai-Liao Parameter Based on Modified Secant Equations
Following the setting of the Dai-Liao (DL) parameter in conjugate gradient (CG) methods, we introduce two new parameters based on the modified secant equation proposed by Li et al. (Comput. Optim. Appl. 202:523-539, 2007) with two approaches, which use an extended new conjugacy condition. The first is based on a modified descent three-term search direction, as the descent Hestenes-Stiefel CG method. The second is based on the quasi-Newton (QN) approach. Global convergence of the proposed methods for uniformly convex functions and general functions is proved. Numerical experiments are done on a set of test functions of the CUTEr collection and the results are compared with some well-known methods.
https://mathco.journals.pnu.ac.ir/article_6722_a305748baa33534bbbd3520c72a3b261.pdf
2018-01-01
59
76
10.30473/coam.2020.46435.1116
Unconstrained optimization
Modified secant equations
Dai-Liao conjugate gradient method
Saeed
Nezhadhosein
s_nejhadhosein@pnu.ac.ir
1
Department of Applied Mathematics, Payame Noor University, Tehran 193953697, Iran
AUTHOR
Sahar
Mohammadkhan Sartip
saharsartip@gmail.com
2
Department of Applied Mathematics, Payame Noor University, Tehran, 193953697, Iran
LEAD_AUTHOR
[1] Andrei N. (2008). ”An unconstrained optimization test functions collection”, Adv. Model. Optim, 10, 147-161.
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[2] Andrei N. (2011). ”Open problem in conjugate gradient algorithms for unconstrained optimization”, Bull Malays Math Sci Soc, 34, 319-330.
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[3] Perry A. (1978). ”A modified conjugate gradient algorithm”, Operations Research, 26, 1073-1078.
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[4] Polyak B. T. (1969). ”The conjugate gradient method in extremal problems”, USSR Computational Mathematics and Mathematical Physics, 9, 94-112.
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[5] Shanno D. F. (1978). ”Conjugate gradient methods with inexact searches”, Mathematics of operations research, 3, 244-256.
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[6] Li D. H., Fukushima M. (2001). ”A modified BFGS method and its global convergence in nonconvex minimization”, Journal of Computational and Applied Mathematics, 129, 15-35.
6
[7] Li D. H., Fukushima M. (2001). ”On the global convergence of the BFGS method for nonconvex unconstrained optimization problems”, SIAM Journal on Optimization, 11, 1054-1064.
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[8] Dolan E. D., Moré J. J. (2002). ”Benchmarking optimization software with performance profiles”, Mathematical programming, 91, 201-213.
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[9] Polak E., Ribiere G. (1969). ”Note sur la convergence de méthodes de directions conjuguées”, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathé-matique et Analyse Numérique, 3, 35-43.
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[10] Li G., Tang C., Wei Z. (2007). ”New conjugacy condition and related new conjugate gradient methods for unconstrained optimization”, Journal of Computational and Applied Mathematics, 202, 523-539.
10
[11] Yabe H., Takano M. (2004). ”Global convergence properties of nonlinear conjugate gradient methods with modified secant condition”, Computational Optimization and Applications, 28, 203-225.
11
[12] Livieris I. E., Pintelas P. (2013). ”A new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization”, Journal of Computational and Applied Mathematics, 239, 396-405.
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[13] Zhang J. Z., Deng N. Y., Chen L. H. (1999). ”New quasi-Newton equation and related methods for unconstrained optimization”, Journal of Optimization Theory and Applications, 102, 147-167.
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[14] Zhang J., Xu, C. (2001). ”Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations”, Journal of Computational and Applied Mathematics, 137, 269-278.
14
[15] Sugiki K., Narushima Y., Yabe H. (2012). ”Globally convergent three-term conjugate gradient methods that use secant conditions and generate descent search directions for unconstrained optimization”, Journal of Optimization Theory and Applications, 153, 733-757.
15
[16] Zhang K., Liu H., Liu Z. (2019). ”A New Dai-Liao Conjugate Gradient Method with Optimal Parameter Choice”, Numerical Functional Analysis and Optimization, 40, 194- 215.
16
[17] Zhou W., Zhang L. (2006). ”A nonlinear conjugate gradient method based on the MBFGS secant condition”, Optimization Methods and Software, 21, 707-714.
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[18] Zhang L., Zhou W., Li D. (2007). ”Some descent three-term conjugate gradient methods and their global convergence”, Optimisation Methods and Software, 22, 697-711.
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[19] Powell M. J. (1984). ”Nonconvex minimization calculations and the conjugate gradient method”. In Numerical analysis, Springer, Berlin, Heidelberg, 122-141.
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[20] Peyghami M. R., Ahmadzadeh H., Fazli A. (2015). ”A new class of effcient and globally convergent conjugate gradient methods in the Dai–Liao family”, Optimization Methods and Software, 30, 843-863.
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[21] Hestenes M. R., Stiefel E. (1952). ”Methods of conjugate gradients for solving linear systems”, Journal of research of the National Bureau of Standards, 49, 409-436.
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22
[23] Fletcher R., Reeves C. M. (1964). ”Function minimization by conjugate gradients”. Thecomputer journal, 7, 149-154.
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[24] Babaie-Kafaki S. (2016). ”On optimality of two adaptive choices for the parameter of Dai–Liao method”, Optimization Letters, 10, 1789-1797.
24
[25] Babaie-Kafaki S., Ghanbari R. (2014). ”A descent family of Dai–Liao conjugate gradient methods”, Optimization Methods and Software, 29, 583-591.
25
[26] Babaie-Kafaki S., Ghanbari R., Mahdavi-Amiri N. (2010). ”Two new conjugate gradient methods based on modified secant equations”, Journal of computational and applied Mathematics, 234, 1374-1386. 1374-1386.
26
[27] Babaie-Kafaki S., Ghanbari R. (2014). ”The Dai–Liao nonlinear conjugate gradient method with optimal parameter choices”, European Journal of Operational Research, 234, 625-630.
27
[28] Babaie-Kafaki S., Ghanbari R. (2014). ”Two modified three-term conjugate gradient methods with sufficient descent property”, Optimization Letters, 8, 2285-2297.
28
[29] Babaie-Kafaki S., Ghanbari R. (2015). ”Two optimal Dai–Liao conjugate gradient methods”, Optimization, 64, 2277-2287.
29
[30] Sun W., Yuan Y. X. (2006). ”Optimization theory and methods: nonlinear programming”, Springer Science & Business Media.
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[31] Hager W. W., Zhang H. (2005). ”A new conjugate gradient method with guaranteed descent and an efficient line search”, SIAM Journal on optimization, 16, 170-192.
31
[32] Hager W. W., Zhang H. (2006). ”A survey of nonlinear conjugate gradient methods”, Pacific journal of Optimization, 2, 35-58.
32
[33] Dai Y. H., Kou C. X. (2013). ”A nonlinear conjugate gradient algorithm with an optimal property and an improved Wolfe line search”, SIAM Journal on Optimization, 23, 296-320.
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[34] Dai Y., Han J., Liu G., Sun D., Yin H., Yuan Y. X. (2000). ”Convergence properties of nonlinear conjugate gradient methods”, SIAM Journal on Optimization, 10, 345-358.
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[35] Dai Y. H., Liao L. Z. (2001). ”New conjugacy conditions and related nonlinear conjugate gradient methods”, Applied Mathematics and Optimization, 43, 87-101.
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[37] Narushima Y., Yabe H., Ford J. A. (2011). ”A three-term conjugate gradient method with sufficient descent property for unconstrained optimization”, SIAM Journal on Optimization, 21, 212-230.
37
[38] Aminifard Z., Babaie-Kafaki S. (2019). ”An optimal parameter choice for the Dai–Liao family of conjugate gradient methods by avoiding a direction of the maximum magnification by the search direction matrix”. 4OR, 17, 317-330.
38
[39] Wei Z., Li G., Qi L. (2006). ”New quasi-Newton methods for unconstrained optimization problems”, Applied Mathematics and Computation, 175, 1156-1188.
39
[40] Wei Z., Yu G., Yuan G., Lian Z. (2004). ”The super-linear convergence of a modified BFGS-type method for unconstrained optimization”, Computational optimization and applications, 29, 315-332.
40
ORIGINAL_ARTICLE
A Fully Fuzzy Method of Network Data Envelopment Analysis for Assessing Revenue Efficiency Based on Ranking Functions
The purpose of this paper is to evaluate the revenue efficiency in the fuzzy network data envelopment analysis. Precision measurements in real-world data are not practically possible, so assuming that data is crisp in solving problems is not a valid assumption. One way to deal with imprecise data is fuzzy data. In this paper, linear ranking functions are used to transform the full fuzzy efficiency model into a precise linear programming problem and, assuming triangular fuzzy numbers, the fuzzy revenue efficiency of decision makers is measured. In the end, a numerical example shows the proposed method.
https://mathco.journals.pnu.ac.ir/article_6723_9de7a236e8cddde916d6cae4a73d36c6.pdf
2018-01-01
77
96
10.30473/coam.2019.46226.1115
Network data envelopment analysis
Revenue efficiency
Full fuzzy linear programming
Ranking function
Mohsen
Rostamy-Malkhalifeh
mohsen_rostamy@yahoo.com
1
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran
LEAD_AUTHOR
Elham
Poudineh
e_poodineh@yahoo.com
2
Department of mathematics, kerman Branch, Islamic azad university, kerman.
AUTHOR
Ali
Payan
apayan_srb@yahoo.com
3
Department of mathematics, zahedan branch, Islamic azad university, zahedan.
AUTHOR
[1] Aghayi, N., “Revenue efficiency measurement with undesirable data in fuzzy DEA”, 7th International Conference on Intelligent systems, Modelling and Simulation, 2016.
1
[2] Banihashemi. S., Tohidi G., 2013. “Allocation efficiency in the network DEA”, International Journal of Data Envelopment Analysis, 1 (2), 85-97.
2
[3] Charnes, A., Cooper, W.W., Rhodes, E., 1978. “Measuring the efficiency of decision-making units”, European Journal of Operational Research, 2, 429–444.
3
[4] Chen, S.M., 1994. “Fuzzy system reliability analysis using fuzzy number arithmetic operations”, Fuzzy Sets and Systems, 66, 31-38.
4
[5] Coelli, T., Rao, D.S.P. and Battese, G., “An introduction to efficiency and productivity analysis (2nd Edition)”, Springer US( 2005).
5
[6] Cooper, W.W., Seiford, L.M., Tone, K., “Data envelopment analysis: a comprehensive text with models, applications, references and DEA-solver software”, 2nd Edition. Springer, New York (2007).
6
[7] Fare. R., Grosskopf. S., “Intertemporal production frontiers: with Dynamic DEA”, Kulwer Academic publishers, Boston, 2000.
7
[8] Kordrostami, S., Jahani S. N., 2019, “Fuzzy revenue efficiency in sustainable supply chains”, International Journal of Applied Operation Research, 9(1), 63-70.
8
[9] Lewis. H. F., Sexton. T. R., 2004. “Network DEA: Performance analysis of an organization with a comprehensive internal structure”, Computers & Operations Research, 31, 1365- 1410.
9
[10] Maleki, H.R., 2002. “Ranking functions and their applications to fuzzy linear programming”, Far East Journal of Mathematical Sciences, 4, 283-301.
10
[11] Nasseri, S.H., Behmanesh, E., Taleshian, F., Abdolalipoor, M., Taghi-Nezhad, N.A ., 2013.“ Fully fuzzy linear programming with inequality constraints”, International Journal of Industrial Mathematics, 5 (4), 309-316.
11
[12] Paryab, K., Tavana, M., Shiraz, R.K., 2014. “Convex and non-convex approaches for cost efficiency models with fuzzy data”, International Journal of Data Mining Modeling and Management (in press).
12
[13] Puri, J., Yadav, SP., 2016. “A fully fuzzy, DEA approach for cost and revenue efficiency measurements in the presence of undesirable outputs and its application to the banking sector in India”, International Journal of Fuzzy Systems, 18 (2), 212-226.
13
[14] Sexton. T. R. Lewis. H. F., 2003. “Two stage DEA: An application to major league baseball”, Journal of Productivity Analysis, 19, 227-249.
14
[15] Tone, K., 2002. “A strange case of cost and allocation efficiencies in DEA”, Journal of the Operational Research Society, 53 (11), 1225-1231.
15
[16] Tone, K., Tsutsui. M., 2009. “Network DEA: A slacks-based measure approach”, European Journal of Operational Research, 197, 243-252.
16
[17] Zadeh, L.A., 1965. “Fuzzy sets”, Information and Control, 8, 338-353.
17
[18] Zimmermann, H.J., “Fuzzy set theory and its applications”, 3rd Edition. Kluwer-Nijhoff Publishing, Boston, 1996.
18