American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jimo.2020066

A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem

Haodong Chen 1, , Hongchun Sun 1,, and  Yiju Wang 2,

1. 

School of Mathematics and Statistics, Linyi University, Linyi Shandong, 276005, China
2. 

School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China

* Corresponding author: Hongchun Sun

Received  July 2019 Revised  October 2019 Published  March 2020

Fund Project: This work is supported by the Natural Science Foundation of China (Nos. 11671228, 11801309), and the Applied Mathematics Enhancement Program of Linyi University

In this paper, a three-level supply chain network equilibrium problem with direct selling and multi-commodity flow is considered. To this end, we first present equilibrium conditions which satisfy decision-making behaviors for manufacturers, retailers and consumer markets, respectively. Based on this, a nonlinear complementarity model of supply chain network equilibrium problem is established. In addition, we propose a new projection-type algorithm to solve this model without the backtracking line search, and global convergence result and $ R- $linearly convergence rate for the new algorithm are established under weaker conditions, respectively. We also illustrate the efficiency of given algorithm through some numerical examples.

Keywords: Supply chain network equilibrium, complementarity problems, projection type algorithm, global convergence, R-linearly convergence rate.
Mathematics Subject Classification: Primary: 90B20; Secondary: 90C30.
Citation: Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020066
References:
[1]

D. P. Bertsekas, Nonlinear programming, Journal of the Operational Research Society, 48 (1997), 334-334.  doi: 10.1057/palgrave.jors.2600425.  Google Scholar

[2] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, 1992.   Google Scholar
[3]

J. Dong, D. Zhang and A. Nagurney, A supply chain network equilibrium model with random demands, European Journal of Operational Research, 156 (2004), 194-212.  doi: 10.1016/S0377-2217(03)00023-7.  Google Scholar

[4]

F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequality and Complementarity Problems, Springer, 2003.  Google Scholar

[5]

X. Y. Fu and T. G. Chen, Supply chain network optimization based on fuzzy multiobjective centralized decision-making model, Mathematical Problems in Engineering, 2017 (2017), Article ID 5825912, 11pp. doi: 10.1155/2017/5825912.  Google Scholar

[6]

S. Javad, M. M. Seyed and T. A. N. Seyed, Optimizing an inventory model with fuzzy demand, backordering, and discount using a hybrid imperialist competitive algorithm, Applied Mathematical Modelling, 40 (2016), 7318-7335.  doi: 10.1016/j.apm.2016.03.013.  Google Scholar

[7]

W. Liu and C. He, Equilibrium conditions of a logistics service supply chain with a new smoothing algorithm, Asia-Pacific Journal of Operational Research, 35 (2018), 1840003 (22 pages). doi: 10.1142/S0217595918400031.  Google Scholar

[8]

A. Nagurney, J. Dong and D. Zhang, A supply chain network equilibrium model, Transportation Research Part E, 38 (2002), 281-303.  doi: 10.1016/S1366-5545(01)00020-5.  Google Scholar

[9]

A. Nagurney and F. Toyasaki, Supply chain supernetworks and environmental criteria, Transportation Research, Part D (Transport and Environment), 8 (2003), 185-213.  doi: 10.1016/S1361-9209(02)00049-4.  Google Scholar

[10]

A. Nagurney, P. Daniele and S. Shukla, A supply chain network game theory model of cybersecurity investment transportation research part E: Logistics and transportation reviews with nonlinear budget constraints, Annals of Operations Research, 248 (2017), 405-427.  doi: 10.1007/s10479-016-2209-1.  Google Scholar

[11]

M. A. Noor, General variational inequalities, Applied Mathematics Letters, 1 (1988), 119-121.  doi: 10.1016/0893-9659(88)90054-7.  Google Scholar

[12]

H. C. Sun, Y. J. Wang and L. Q. Qi, Global error bound for the generalized linear complementarity problem over a polyhedral cone, Journal of Optimization Theory and Applications, 142 (2009), 417-429.  doi: 10.1007/s10957-009-9509-4.  Google Scholar

[13]

H. C. Sun and Y. J. Wang, Further discussion on the error bound for generalized linear complementarity problem over a polyhedral cone, Journal of Optimization Theory and Applications, 159 (2003), 93-107.  doi: 10.1007/s10957-013-0290-z.  Google Scholar

[14]

H. C. Sun, Y. J. Wang, S. J. Li and M. Sun, A sharper global error bound for the generalized linear complementarity problem over a polyhedral cone under weaker conditions, Journal of Fixed Point Theory and Applications, 20 (2018), Art. 75, 19 pp. doi: 10.1007/s11784-018-0556-z.  Google Scholar

[15]

Y. J. Wang, H. X. Gao and W. Xing, Optimal replenishment and stocking strategies for inventory mechanism with a dynamically stochastic short-term price discount, Journal of Global Optimization, 70 (2018), 27-53.  doi: 10.1007/s10898-017-0522-0.  Google Scholar

[16]

N. H. Xiu and J. Z. Zhang, Global projection-type error bound for general variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 213-228.  doi: 10.1023/A:1013056931761.  Google Scholar

[17]

L. P. Zhang, A nonlinear complementarity model for supply chain network equilibrium, Journal of Industrial and Management Optimization, 3 (2007), 727-737.  doi: 10.3934/jimo.2007.3.727.  Google Scholar

[18]

G. T. Zhang, H. Sun, J. S. Hu and G. X. Dai, The closed-loop supply chain network equilibrium with products lifetime and carbon emission constraints in multiperiod planning horizon, Discrete Dynamics in Nature and Society, 2014 (2014), Article ID 784637, 16pp. doi: 10.1155/2014/784637.  Google Scholar

[19] E. H. Zarantonello, Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis, Academic Press, New Youk, 1971.   Google Scholar

show all references

References:
[1]

D. P. Bertsekas, Nonlinear programming, Journal of the Operational Research Society, 48 (1997), 334-334.  doi: 10.1057/palgrave.jors.2600425.  Google Scholar

[2] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, 1992.   Google Scholar
[3]

J. Dong, D. Zhang and A. Nagurney, A supply chain network equilibrium model with random demands, European Journal of Operational Research, 156 (2004), 194-212.  doi: 10.1016/S0377-2217(03)00023-7.  Google Scholar

[4]

F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequality and Complementarity Problems, Springer, 2003.  Google Scholar

[5]

X. Y. Fu and T. G. Chen, Supply chain network optimization based on fuzzy multiobjective centralized decision-making model, Mathematical Problems in Engineering, 2017 (2017), Article ID 5825912, 11pp. doi: 10.1155/2017/5825912.  Google Scholar

[6]

S. Javad, M. M. Seyed and T. A. N. Seyed, Optimizing an inventory model with fuzzy demand, backordering, and discount using a hybrid imperialist competitive algorithm, Applied Mathematical Modelling, 40 (2016), 7318-7335.  doi: 10.1016/j.apm.2016.03.013.  Google Scholar

[7]

W. Liu and C. He, Equilibrium conditions of a logistics service supply chain with a new smoothing algorithm, Asia-Pacific Journal of Operational Research, 35 (2018), 1840003 (22 pages). doi: 10.1142/S0217595918400031.  Google Scholar

[8]

A. Nagurney, J. Dong and D. Zhang, A supply chain network equilibrium model, Transportation Research Part E, 38 (2002), 281-303.  doi: 10.1016/S1366-5545(01)00020-5.  Google Scholar

[9]

A. Nagurney and F. Toyasaki, Supply chain supernetworks and environmental criteria, Transportation Research, Part D (Transport and Environment), 8 (2003), 185-213.  doi: 10.1016/S1361-9209(02)00049-4.  Google Scholar

[10]

A. Nagurney, P. Daniele and S. Shukla, A supply chain network game theory model of cybersecurity investment transportation research part E: Logistics and transportation reviews with nonlinear budget constraints, Annals of Operations Research, 248 (2017), 405-427.  doi: 10.1007/s10479-016-2209-1.  Google Scholar

[11]

M. A. Noor, General variational inequalities, Applied Mathematics Letters, 1 (1988), 119-121.  doi: 10.1016/0893-9659(88)90054-7.  Google Scholar

[12]

H. C. Sun, Y. J. Wang and L. Q. Qi, Global error bound for the generalized linear complementarity problem over a polyhedral cone, Journal of Optimization Theory and Applications, 142 (2009), 417-429.  doi: 10.1007/s10957-009-9509-4.  Google Scholar

[13]

H. C. Sun and Y. J. Wang, Further discussion on the error bound for generalized linear complementarity problem over a polyhedral cone, Journal of Optimization Theory and Applications, 159 (2003), 93-107.  doi: 10.1007/s10957-013-0290-z.  Google Scholar

[14]

H. C. Sun, Y. J. Wang, S. J. Li and M. Sun, A sharper global error bound for the generalized linear complementarity problem over a polyhedral cone under weaker conditions, Journal of Fixed Point Theory and Applications, 20 (2018), Art. 75, 19 pp. doi: 10.1007/s11784-018-0556-z.  Google Scholar

[15]

Y. J. Wang, H. X. Gao and W. Xing, Optimal replenishment and stocking strategies for inventory mechanism with a dynamically stochastic short-term price discount, Journal of Global Optimization, 70 (2018), 27-53.  doi: 10.1007/s10898-017-0522-0.  Google Scholar

[16]

N. H. Xiu and J. Z. Zhang, Global projection-type error bound for general variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 213-228.  doi: 10.1023/A:1013056931761.  Google Scholar

[17]

L. P. Zhang, A nonlinear complementarity model for supply chain network equilibrium, Journal of Industrial and Management Optimization, 3 (2007), 727-737.  doi: 10.3934/jimo.2007.3.727.  Google Scholar

[18]

G. T. Zhang, H. Sun, J. S. Hu and G. X. Dai, The closed-loop supply chain network equilibrium with products lifetime and carbon emission constraints in multiperiod planning horizon, Discrete Dynamics in Nature and Society, 2014 (2014), Article ID 784637, 16pp. doi: 10.1155/2014/784637.  Google Scholar

[19] E. H. Zarantonello, Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis, Academic Press, New Youk, 1971.   Google Scholar
Figure 1.  The network structure of the supply chain
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The network structure of $ i- $th manufacturer
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The network structure of $ j $'s retailer
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The network structure of $ k- $th consumer
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The network structure of the supply chain of Example 4.1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The network structure of the supply chain of Example 4.2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The network structure of the supply chain of Example 4.3
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  The network structure of the supply chain for Example 4.5
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Productions from manufacturers to retailers
$ (q_{ij}^1/q_{ij}^2) $ Retailer 1 Retailer 2 Retailer 3
Manufacturer 1 7.9043/7.8023 7.9295/7.8056 7.9800/7.1543
Manufacturer 2 7.9800/7.8100 7.8790/7.8432 8.0305/8.0025
Manufacturer 3 7.9295/7.6265 7.9800/7.6770 8.0305/7.7275
Table Options

Download as excel

$ (q_{ij}^1/q_{ij}^2) $ Retailer 1 Retailer 2 Retailer 3
Manufacturer 1 7.9043/7.8023 7.9295/7.8056 7.9800/7.1543
Manufacturer 2 7.9800/7.8100 7.8790/7.8432 8.0305/8.0025
Manufacturer 3 7.9295/7.6265 7.9800/7.6770 8.0305/7.7275
Table 2.  Productions from manufacturers to consumer markets
$ (\tilde{q}_{ik}^1/\tilde{q}_{ik}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Manufacturer 1 9.4445/9.1415 9.4950/9.1920 9.5455/9.2425
Manufacturer 2 9.4445/9.1415 9.5455/9.2425 9.4950/9.1920
Manufacturer 3 9.5455/9.5455 9.3435/9.3435 9.4950/9.4950
Table Options

Download as excel

$ (\tilde{q}_{ik}^1/\tilde{q}_{ik}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Manufacturer 1 9.4445/9.1415 9.4950/9.1920 9.5455/9.2425
Manufacturer 2 9.4445/9.1415 9.5455/9.2425 9.4950/9.1920
Manufacturer 3 9.5455/9.5455 9.3435/9.3435 9.4950/9.4950
Table 3.  Productions from retailers to consumer markets
$ (\hat{q}_{jk}^1/\hat{q}_{jk}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Retailer 1 1.2475/1.2345 1.2424/1.2323 1.2374/1.2172
Retailer 2 1.2374/1.2475 1.2323/1.2233 1.2273/1.2172
Retailer 3 1.2273/1.2475 1.2222/1.2323 1.2172/1.2172
Table Options

Download as excel

$ (\hat{q}_{jk}^1/\hat{q}_{jk}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Retailer 1 1.2475/1.2345 1.2424/1.2323 1.2374/1.2172
Retailer 2 1.2374/1.2475 1.2323/1.2233 1.2273/1.2172
Retailer 3 1.2273/1.2475 1.2222/1.2323 1.2172/1.2172
Table 4.  Price from manufacturers to consumer markets
$ (\tilde{\rho}_{ik}^1/\tilde{\rho}_{ik}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Manufacturer 1 198.2578/228.0022 200.0245/230.5247 201.1178/228.7279
Manufacturer 2 198.2445/228.0022 200.8785/227.2225 200.0110/225.7920
Manufacturer 3 196.3135/220.1133 211.8830/234.5435 200.0110/220.4950
Table Options

Download as excel

$ (\tilde{\rho}_{ik}^1/\tilde{\rho}_{ik}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Manufacturer 1 198.2578/228.0022 200.0245/230.5247 201.1178/228.7279
Manufacturer 2 198.2445/228.0022 200.8785/227.2225 200.0110/225.7920
Manufacturer 3 196.3135/220.1133 211.8830/234.5435 200.0110/220.4950
Table 5.  Price from retailers to consumer markets
$ (\hat{\rho}_{jk}^1/\hat{\rho}_{jk}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Retailer 1 220.2273/250.2245 218.2323/248.2475 221.7715/252.2172
Retailer 2 218.2374/248.5455 219.9423/249.1415 220.1920/250.2172
Retailer 3 222.7415/247.1420 222.9435/249.1835 223.1920/251.3570
Table Options

Download as excel

$ (\hat{\rho}_{jk}^1/\hat{\rho}_{jk}^2) $ Consumer market 1 Consumer market 2 Consumer market 3
Retailer 1 220.2273/250.2245 218.2323/248.2475 221.7715/252.2172
Retailer 2 218.2374/248.5455 219.9423/249.1415 220.1920/250.2172
Retailer 3 222.7415/247.1420 222.9435/249.1835 223.1920/251.3570
Table 6.  Price from manufacturers to retailers
$ (\rho_{ij}^1/\rho_{ij}^2) $ Retailer 1 Retailer 2 Retailer 3
Manufacturer 1 157.5213/186.7230 160.1246/187.4950 158.2275/186.2711
Manufacturer 2 155.2117/184.1917 159.3287/190.0058 158.2365/185.5003
Manufacturer 3 156.2226/185.2459 159.4962/189.2234 159.1435/189.3872
Table Options

Download as excel

$ (\rho_{ij}^1/\rho_{ij}^2) $ Retailer 1 Retailer 2 Retailer 3
Manufacturer 1 157.5213/186.7230 160.1246/187.4950 158.2275/186.2711
Manufacturer 2 155.2117/184.1917 159.3287/190.0058 158.2365/185.5003
Manufacturer 3 156.2226/185.2459 159.4962/189.2234 159.1435/189.3872
Table 7.  Consumer market demand price
$ \rho_k^l $ Consumer market 1 Consumer market 2 Consumer market 3
Product 1 236.2227 225.5642 215.2359
Product 2 209.2117 201.4962 189.1165
Table Options

Download as excel

$ \rho_k^l $ Consumer market 1 Consumer market 2 Consumer market 3
Product 1 236.2227 225.5642 215.2359
Product 2 209.2117 201.4962 189.1165
Table 8.  Compared with the results in Example 2([17])
Literature results Results of this paper
Iteration steps 12 64
Running time 0.47 0.43
Table Options

Download as excel

Literature results Results of this paper
Iteration steps 12 64
Running time 0.47 0.43
Table 9.  Compared with the results in Example 4 ([17])
Literature results Results of this paper
Iteration steps 12 58
Running time 0.09 0.07
Table Options

Download as excel

Literature results Results of this paper
Iteration steps 12 58
Running time 0.09 0.07
Table 10.  Compared with the results in Example 5 ([17])
Literature results Results of this paper
Iteration steps 12 109
Running time 0.11 0.086
Table Options

Download as excel

Literature results Results of this paper
Iteration steps 12 109
Running time 0.11 0.086
[1]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[2]

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 & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023
[3]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210
[4]

Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169
[5]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
[6]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29
[7]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[8]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[9]

Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021024
[10]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[11]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[12]

Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021035
[13]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[14]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007
[15]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453
[16]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[17]

Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781
[18]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[19]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[20]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (78)
  • HTML views (411)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences