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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
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Figure 2.  The network structure of $ i- $th manufacturer
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Figure 3.  The network structure of $ j $'s retailer
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Figure 4.  The network structure of $ k- $th consumer
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Figure 5.  The network structure of the supply chain of Example 4.1
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Figure 6.  The network structure of the supply chain of Example 4.2
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Figure 7.  The network structure of the supply chain of Example 4.3
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Figure 8.  The network structure of the supply chain for Example 4.5
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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
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$ (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
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$ (\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
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$ (\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
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$ (\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
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$ (\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
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$ (\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
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$ \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
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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
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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
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Literature results Results of this paper
Iteration steps 12 109
Running time 0.11 0.086
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