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July  2021, 17(4): 2203-2215. doi: 10.3934/jimo.2020065

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

 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  July 2021 Early access  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.

Citation: Ru Li, Guolin Yu. Strict efficiency of a multi-product supply-demand network equilibrium model. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2203-2215. 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. [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. [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. [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. [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. [6] F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, Variational Inequalities and Complementarity Problems, Wiley, Chichester, (1980), 151–186. [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. [8] A. Nagurney, Network Economics: A Variational Inequality Approach,Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. doi: 10.1007/978-94-011-2178-1. [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. [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. [11] J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers-Part II, 1, 325–378. [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.

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. [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. [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. [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. [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. [6] F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, Variational Inequalities and Complementarity Problems, Wiley, Chichester, (1980), 151–186. [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. [8] A. Nagurney, Network Economics: A Variational Inequality Approach,Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. doi: 10.1007/978-94-011-2178-1. [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. [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. [11] J. G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the Institution of Civil Engineers-Part II, 1, 325–378. [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.
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