American Institute of Mathematical Sciences

May  2022, 5(2): 145-156. doi: 10.3934/mfc.2022001

An optimization model and method for supply chain equilibrium management problem

 1 School of Information Science and Engineering, Linyi University, Linyi, Shandong, 276005, China 2 School of Mathematics and Statistics, Linyi University, Linyi, Shandong, 276005, China

* Corresponding authors: Guirong Pan and Bing Xue

Received  September 2019 Revised  November 2021 Published  May 2022 Early access  February 2022

In this paper, we establish a nonlinear complementarity model and algorithm for supply chain equilibrium management problem consisting of manufacturers, retailers and consumer markets. This work focus on the price of the goods of retailer sell to consumer market in which is a function of the amount of products that are transacted between the retailer and the consumer. Based on this, we investigate the optimizing behavior of the various decision-makers, derive the equilibrium conditions of the manufacturers, the retailers and the consumer markets respectively, and establish a nonlinear complementarity model of this problem. To obtain optimal decision for the problem, we propose a new type of algorithm based on established model, and its global convergence is presented without the assumption of global Lipschitz continuous in detail. The efficiency of given algorithm is also illustrated through some numerical examples.

Citation: Guirong Pan, Bing Xue, Hongchun Sun. An optimization model and method for supply chain equilibrium management problem. Mathematical Foundations of Computing, 2022, 5 (2) : 145-156. doi: 10.3934/mfc.2022001
References:
 [1] D. P. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 1999. [2] J. Dong, D. Zhang and A. Nagurney, A supply chain network equilibrium model with random demands, European J. Oper. Res., 156 (2004), 194-212.  doi: 10.1016/S0377-2217(03)00023-7. [3] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003. doi: 10.1007/b97543. [4] J. Geunes, P. M. Pardalos and H. E. Romeijn, Supply Chain Management: Models, Applications, and Research Directions, Applied Optimization, 62, Springer, Boston, MA, 2002. doi: 10.1007/b106640. [5] S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454.  doi: 10.1007/BF00932654. [6] Y. H. Lee, P. Golinska-Dawson and J.-Z. Wu, Mathematical models for supply chain management, Math. Probl. Engrg., 2016 (2016), 1-4.  doi: 10.1155/2016/6167290. [7] A. Nagurney, J. Cruz, J. Dong and D. Zhang, Supply chain networks, electronic commerce, and supply side and demand side risk, European J. Oper. Res., 164 (2005), 120-142.  doi: 10.1016/j.ejor.2003.11.007. [8] M. A. Noor, General variational inequalities, Appl. Math. Lett., 1 (1988), 119-121.  doi: 10.1016/0893-9659(88)90054-7. [9] D. Simchi-Levi, S. D. Wu, and Z.-J. Shen, Handbook of Quantitative Supply Chain Analysis. Modeling in the E-Business Era, International Series in Operations Research & Management Science, 74, Springer, Boston, MA, 2004. doi: 10.1007/978-1-4020-7953-5. [10] H. Stadtler, C. Kilger and H. Meyr, Supply Chain Management and Advanced Planning, Springer-Verlag, Berlin Heidelberg, 2015. [11] H.-C. Sun and Y.-L. Dong, A new type of solution method for the generalized linear complementarity problem over a polyhedral cone, Internat. J. Automat. Comput., 6 (2009), 228-233.  doi: 10.1007/S11633-009-0228-Y. [12] Y. Wang, F. Ma and J. Zhang, A nonsmooth L-M method for solving the generalized nonlinear complementarity problem over a polyhedral cone, Appl. Math. Optim., 52 (2005), 73-92.  doi: 10.1007/s00245-005-0823-4. [13] N. Xiu and J. Zhang, Some recent advances in projection-type methods for variational inequalities, J. Comput. Appl. Math., 152 (2003), 559-585.  doi: 10.1016/S0377-0427(02)00730-6. [14] E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 237–341. [15] D. Zhang, J. Dong and A. Nagurney, A supply chain network economy: Modeling and qualitative analysis, in Innovations in Financial and Economic Networks, Edward Elgar Publishers, 2003. Available from: https://www.researchgate.net/profile/Anna-Nagurney/publication/41463218_A_supply_Chain_Network_Economy_Modeling_and_Qualitative_Analysis/links/55427f330cf24107d394710c/A-supply-Chain-Network-Economy-Modeling-and-Qualitative-Analysis.pdf. [16] D. Zhang, F. Zou, S. Li and L. Zhang, Green supply chain network design with economies of scale and environmental concerns, J. Adv. Transport., 2017 (2017), 1-14.  doi: 10.1155/2017/6350562. [17] X. Zhang, F. Ma and Y. Wang, A Newton-type algorithm for generalized linear complementarity problem over a polyhedral cone, Appl. Math. Comput., 169 (2005), 388-401.  doi: 10.1016/j.amc.2004.09.057.

show all references

References:
 [1] D. P. Bertsekas, Nonlinear Programming, 2$^{nd}$ edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA, 1999. [2] J. Dong, D. Zhang and A. Nagurney, A supply chain network equilibrium model with random demands, European J. Oper. Res., 156 (2004), 194-212.  doi: 10.1016/S0377-2217(03)00023-7. [3] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003. doi: 10.1007/b97543. [4] J. Geunes, P. M. Pardalos and H. E. Romeijn, Supply Chain Management: Models, Applications, and Research Directions, Applied Optimization, 62, Springer, Boston, MA, 2002. doi: 10.1007/b106640. [5] S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454.  doi: 10.1007/BF00932654. [6] Y. H. Lee, P. Golinska-Dawson and J.-Z. Wu, Mathematical models for supply chain management, Math. Probl. Engrg., 2016 (2016), 1-4.  doi: 10.1155/2016/6167290. [7] A. Nagurney, J. Cruz, J. Dong and D. Zhang, Supply chain networks, electronic commerce, and supply side and demand side risk, European J. Oper. Res., 164 (2005), 120-142.  doi: 10.1016/j.ejor.2003.11.007. [8] M. A. Noor, General variational inequalities, Appl. Math. Lett., 1 (1988), 119-121.  doi: 10.1016/0893-9659(88)90054-7. [9] D. Simchi-Levi, S. D. Wu, and Z.-J. Shen, Handbook of Quantitative Supply Chain Analysis. Modeling in the E-Business Era, International Series in Operations Research & Management Science, 74, Springer, Boston, MA, 2004. doi: 10.1007/978-1-4020-7953-5. [10] H. Stadtler, C. Kilger and H. Meyr, Supply Chain Management and Advanced Planning, Springer-Verlag, Berlin Heidelberg, 2015. [11] H.-C. Sun and Y.-L. Dong, A new type of solution method for the generalized linear complementarity problem over a polyhedral cone, Internat. J. Automat. Comput., 6 (2009), 228-233.  doi: 10.1007/S11633-009-0228-Y. [12] Y. Wang, F. Ma and J. Zhang, A nonsmooth L-M method for solving the generalized nonlinear complementarity problem over a polyhedral cone, Appl. Math. Optim., 52 (2005), 73-92.  doi: 10.1007/s00245-005-0823-4. [13] N. Xiu and J. Zhang, Some recent advances in projection-type methods for variational inequalities, J. Comput. Appl. Math., 152 (2003), 559-585.  doi: 10.1016/S0377-0427(02)00730-6. [14] E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 237–341. [15] D. Zhang, J. Dong and A. Nagurney, A supply chain network economy: Modeling and qualitative analysis, in Innovations in Financial and Economic Networks, Edward Elgar Publishers, 2003. Available from: https://www.researchgate.net/profile/Anna-Nagurney/publication/41463218_A_supply_Chain_Network_Economy_Modeling_and_Qualitative_Analysis/links/55427f330cf24107d394710c/A-supply-Chain-Network-Economy-Modeling-and-Qualitative-Analysis.pdf. [16] D. Zhang, F. Zou, S. Li and L. Zhang, Green supply chain network design with economies of scale and environmental concerns, J. Adv. Transport., 2017 (2017), 1-14.  doi: 10.1155/2017/6350562. [17] X. Zhang, F. Ma and Y. Wang, A Newton-type algorithm for generalized linear complementarity problem over a polyhedral cone, Appl. Math. Comput., 169 (2005), 388-401.  doi: 10.1016/j.amc.2004.09.057.
The network structure of the supply chain equilibrium problem
The network structure of $i-$th manufacturer
The network structure of $j-$th retailer
The supply chain equilibrium management problem consists of 2 manufacturers, 1 retailer and 1 consumer market
 [1] Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066 [2] Liping Zhang. A nonlinear complementarity model for supply chain network equilibrium. Journal of Industrial and Management Optimization, 2007, 3 (4) : 727-737. doi: 10.3934/jimo.2007.3.727 [3] Yongtao Peng, Dan Xu, Eleonora Veglianti, Elisabetta Magnaghi. A product service supply chain network equilibrium considering risk management in the context of COVID-19 pandemic. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022094 [4] Yeong-Cheng Liou, Siegfried Schaible, Jen-Chih Yao. Supply chain inventory management via a Stackelberg equilibrium. Journal of Industrial and Management Optimization, 2006, 2 (1) : 81-94. doi: 10.3934/jimo.2006.2.81 [5] Wenbin Wang, Peng Zhang, Junfei Ding, Jian Li, Hao Sun, Lingyun He. Closed-loop supply chain network equilibrium model with retailer-collection under legislation. Journal of Industrial and Management Optimization, 2019, 15 (1) : 199-219. doi: 10.3934/jimo.2018039 [6] Amin Aalaei, Hamid Davoudpour. Two bounds for integrating the virtual dynamic cellular manufacturing problem into supply chain management. Journal of Industrial and Management Optimization, 2016, 12 (3) : 907-930. doi: 10.3934/jimo.2016.12.907 [7] Yu-Lin Chang, Jein-Shan Chen, Jia Wu. Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function. Journal of Industrial and Management Optimization, 2013, 9 (1) : 153-169. doi: 10.3934/jimo.2013.9.153 [8] Yunan Wu, Guangya Chen, T. C. Edwin Cheng. A vector network equilibrium problem with a unilateral constraint. Journal of Industrial and Management Optimization, 2010, 6 (3) : 453-464. doi: 10.3934/jimo.2010.6.453 [9] Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial and Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911 [10] Hongming Yang, C. Y. Chung, Xiaojiao Tong, Pingping Bing. Research on dynamic equilibrium of power market with complex network constraints based on nonlinear complementarity function. Journal of Industrial and Management Optimization, 2008, 4 (3) : 617-630. doi: 10.3934/jimo.2008.4.617 [11] Mengmeng Zheng, Ying Zhang, Zheng-Hai Huang. Global error bounds for the tensor complementarity problem with a P-tensor. Journal of Industrial and Management Optimization, 2019, 15 (2) : 933-946. doi: 10.3934/jimo.2018078 [12] 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 [13] Yunan Wu, T. C. Edwin Cheng. Classical duality and existence results for a multi-criteria supply-demand network equilibrium model. Journal of Industrial and Management Optimization, 2009, 5 (3) : 615-628. doi: 10.3934/jimo.2009.5.615 [14] T.C. Edwin Cheng, Yunan Wu. Henig efficiency of a multi-criterion supply-demand network equilibrium model. Journal of Industrial and Management Optimization, 2006, 2 (3) : 269-286. doi: 10.3934/jimo.2006.2.269 [15] Jia Shu, Jie Sun. Designing the distribution network for an integrated supply chain. Journal of Industrial and Management Optimization, 2006, 2 (3) : 339-349. doi: 10.3934/jimo.2006.2.339 [16] I-Lin Wang, Shiou-Jie Lin. A network simplex algorithm for solving the minimum distribution cost problem. Journal of Industrial and Management Optimization, 2009, 5 (4) : 929-950. doi: 10.3934/jimo.2009.5.929 [17] Chunlin Hao, Xinwei Liu. Global convergence of an SQP algorithm for nonlinear optimization with overdetermined constraints. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 19-29. doi: 10.3934/naco.2012.2.19 [18] Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023 [19] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A flame propagation model on a network with application to a blocking problem. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 825-843. doi: 10.3934/dcdss.2018051 [20] Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control and Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307

Impact Factor: