Figure 1.
The network structure of the supply chain
-
Previous Article
Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters
- JIMO Home
- This Issue
-
Next Article
Strict efficiency of a multi-product supply-demand network equilibrium model
July
2021, 17(4): 2217-2242.
doi: 10.3934/jimo.2020066
A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem
| 1. | School of Mathematics and Statistics, Linyi University, Linyi Shandong, 276005, China |
| 2. | School of Management Science, Qufu Normal University, Rizhao Shandong, 276800, China |
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 and Management Optimization,
2021, 17
(4)
: 2217-2242.
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. |
| [2] |
R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, 1992.
|
| [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. |
| [4] |
F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequality and Complementarity Problems, Springer, 2003. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
M. A. Noor,
General variational inequalities, Applied Mathematics Letters, 1 (1988), 119-121.
doi: 10.1016/0893-9659(88)90054-7. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
E. H. Zarantonello, Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis, Academic Press, New Youk, 1971.
|
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. |
| [2] |
R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, 1992.
|
| [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. |
| [4] |
F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequality and Complementarity Problems, Springer, 2003. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
M. A. Noor,
General variational inequalities, Applied Mathematics Letters, 1 (1988), 119-121.
doi: 10.1016/0893-9659(88)90054-7. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
E. H. Zarantonello, Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis, Academic Press, New Youk, 1971.
|
Figure 2.
The network structure of $ i- $ th manufacturer
Figure 3.
The network structure of $ j $ 's retailer
Figure 4.
The network structure of $ k- $ th consumer
Figure 5.
The network structure of the supply chain of Example 4.1
Figure 6.
The network structure of the supply chain of Example 4.2
Figure 7.
The network structure of the supply chain of Example 4.3
Figure 8.
The network structure of the supply chain for Example 4.5
Table 1.
Productions from manufacturers to retailers
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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 |
| Literature results | Results of this paper | |
| Iteration steps | 12 | 64 |
| Running time | 0.47 | 0.43 |
| Literature results | Results of this paper | |
| Iteration steps | 12 | 64 |
| Running time | 0.47 | 0.43 |
| Literature results | Results of this paper | |
| Iteration steps | 12 | 58 |
| Running time | 0.09 | 0.07 |
| Literature results | Results of this paper | |
| Iteration steps | 12 | 58 |
| Running time | 0.09 | 0.07 |
| [1] |
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 |
| [2] |
Yazheng Dang, Fanwen Meng, Jie Sun. Convergence analysis of a parallel projection algorithm for solving convex feasibility problems. Numerical Algebra, Control and Optimization, 2016, 6 (4) : 505-519. doi: 10.3934/naco.2016023 |
| [3] |
Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial and Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333 |
| [4] |
Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems and Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077 |
| [5] |
X. X. Huang, D. Li, Xiaoqi Yang. Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints. Journal of Industrial and Management Optimization, 2006, 2 (3) : 287-296. doi: 10.3934/jimo.2006.2.287 |
| [6] |
Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030 |
| [7] |
Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2553-2566. doi: 10.3934/jimo.2021080 |
| [8] |
Xueling Zhou, Meixia Li, Haitao Che. Relaxed successive projection algorithm with strong convergence for the multiple-sets split equality problem. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2557-2572. doi: 10.3934/jimo.2020082 |
| [9] |
Mingzheng Wang, M. Montaz Ali, Guihua Lin. Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks. Journal of Industrial and Management Optimization, 2011, 7 (2) : 317-345. doi: 10.3934/jimo.2011.7.317 |
| [10] |
Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993 |
| [11] |
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 |
| [12] |
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 |
| [13] |
Do Sang Kim, Nguyen Ngoc Hai, Bui Van Dinh. Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 63-78. doi: 10.3934/naco.2021051 |
| [14] |
Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053 |
| [15] |
Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks and Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61 |
| [16] |
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 |
| [17] |
Yan Tang. Convergence analysis of a new iterative algorithm for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2020, 16 (2) : 945-964. doi: 10.3934/jimo.2018187 |
| [18] |
Eric Cancès, Claude Le Bris. Convergence to equilibrium of a multiscale model for suspensions. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 449-470. doi: 10.3934/dcdsb.2006.6.449 |
| [19] |
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 |
| [20] |
Dang Van Hieu. Projection methods for solving split equilibrium problems. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2331-2349. doi: 10.3934/jimo.2019056 |
2021 Impact Factor: 1.411
Tools
Metrics
Other articles
by authors
[Back to Top]