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doi: 10.3934/jimo.2020065

Strict efficiency of a multi-product supply-demand network equilibrium model

Ru Li 1, and  Guolin Yu 1,2,,

1. 

School of Management, Hefei University of Technology, Hefei, 230009, China
2. 

Institute of Applied Mathematics, Beifang University of Nationalities, Yinchuan, 750021, China

* Corresponding author: Guolin Yu

Received  July 2019 Revised  November 2019 Published  March 2020

In this paper, we consider a kind of proper efficiency, namely strict efficiency, of a multi-product supply-demand network equilibrium model. We prove that strict equilibrium pattern flows with both a single criterion and multiple criteria are equivalent to vector variational inequalities. In the case of multiple criteria, we provide necessary and sufficient conditions for strict efficiency in terms of vector variational inequalities by using Gerstewitz's function without any convexity assumptions.

Keywords: Strict efficiency, Network equilibrium, Strict equilibrium pattern flow, Vector variational inequality, Gerstewitz's function.
Mathematics Subject Classification: Primary: 90C29, 90C35; Secondary: 90B06, 90B10.
Citation: Ru Li, Guolin Yu. Strict efficiency of a multi-product supply-demand network equilibrium model. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020065
References:
[1]

Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space, Mathematical Methods of Operations Research, 50 (1999), 373-384.  doi: 10.1007/s001860050076.  Google Scholar

[2]

G. Y. Chen, C. J. Goh and X. Q. Yang, Vector network equilibrium problems and nonlinear scalarization methods, Mathematical Methods of Operations Research, 49 (1999), 239-253.  doi: 10.1007/s001860050023.  Google Scholar

[3]

T. C. E. Cheng and Y. N. Wu, A multi-product, multi-criterion supply-demand network equilibrium model, Operations Research, 54 (2006), 544-554.  doi: 10.1287/opre.1060.0284.  Google Scholar

[4]

G. Y. Chen and N. D. Yen, On the variational inequality model for network equilibrium, Internal Report, Department of Mathematics, University of Pisa, 196 (1993), 724–735. Google Scholar

[5]

G. Y. Chen and X. Q. Yang, Characterizations of variable domination structures via nonlinear scalarization, Journal of Optimization Theory and Applications, 112 (2002), 97-110.  doi: 10.1023/A:1013044529035.  Google Scholar

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F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, Variational Inequalities and Complementarity Problems, Wiley, Chichester, (1980), 151–186.  Google Scholar

[7]

X. H. Gong, Efficiency and Hening efficiency for vector equilibrium problems, Journal of Optimization Theory and Applications, 108 (2001), 139-154.  doi: 10.1023/A:1026418122905.  Google Scholar

[8]

A. Nagurney, Network Economics: A Variational Inequality Approach,Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[9]

Y. N. Wu and T. C. E. Cheng, Benson efficiency of a multi-criterion network equilibrium model, Pacific Journal of Optimization, 5 (2009), 443-458.  doi: 10.1016/j.obhdp.2009.08.002.  Google Scholar

[10]

Y. N. Wu, Y. C. Peng, L. Peng and L. Xu, Super efficiency of multicriterion network equilibrium model and vector variational inequality, Journal of Optimization Theory and Applications, 153 (2012), 485-496.  doi: 10.1007/s10957-011-9950-z.  Google Scholar

[11]

J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers-Part II, 1, 325–378. Google Scholar

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G. L. Yu, Y. Zhang and S. Y. Liu, Strong duality with strict efficiency in vector optimization involving nonconvex set-valued maps, Journal of Mathematics, 37 (2017), 223-230.   Google Scholar

show all references

References:
[1]

Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space, Mathematical Methods of Operations Research, 50 (1999), 373-384.  doi: 10.1007/s001860050076.  Google Scholar

[2]

G. Y. Chen, C. J. Goh and X. Q. Yang, Vector network equilibrium problems and nonlinear scalarization methods, Mathematical Methods of Operations Research, 49 (1999), 239-253.  doi: 10.1007/s001860050023.  Google Scholar

[3]

T. C. E. Cheng and Y. N. Wu, A multi-product, multi-criterion supply-demand network equilibrium model, Operations Research, 54 (2006), 544-554.  doi: 10.1287/opre.1060.0284.  Google Scholar

[4]

G. Y. Chen and N. D. Yen, On the variational inequality model for network equilibrium, Internal Report, Department of Mathematics, University of Pisa, 196 (1993), 724–735. Google Scholar

[5]

G. Y. Chen and X. Q. Yang, Characterizations of variable domination structures via nonlinear scalarization, Journal of Optimization Theory and Applications, 112 (2002), 97-110.  doi: 10.1023/A:1013044529035.  Google Scholar

[6]

F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, Variational Inequalities and Complementarity Problems, Wiley, Chichester, (1980), 151–186.  Google Scholar

[7]

X. H. Gong, Efficiency and Hening efficiency for vector equilibrium problems, Journal of Optimization Theory and Applications, 108 (2001), 139-154.  doi: 10.1023/A:1026418122905.  Google Scholar

[8]

A. Nagurney, Network Economics: A Variational Inequality Approach,Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[9]

Y. N. Wu and T. C. E. Cheng, Benson efficiency of a multi-criterion network equilibrium model, Pacific Journal of Optimization, 5 (2009), 443-458.  doi: 10.1016/j.obhdp.2009.08.002.  Google Scholar

[10]

Y. N. Wu, Y. C. Peng, L. Peng and L. Xu, Super efficiency of multicriterion network equilibrium model and vector variational inequality, Journal of Optimization Theory and Applications, 153 (2012), 485-496.  doi: 10.1007/s10957-011-9950-z.  Google Scholar

[11]

J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers-Part II, 1, 325–378. Google Scholar

[12]

G. L. Yu, Y. Zhang and S. Y. Liu, Strong duality with strict efficiency in vector optimization involving nonconvex set-valued maps, Journal of Mathematics, 37 (2017), 223-230.   Google Scholar

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