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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.
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.
![]() |








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 |
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 |
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 |
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 |
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 |
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 |
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 |
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