# American Institute of Mathematical Sciences

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

* Corresponding author: Hongchun Sun

Received  July 2019 Revised  October 2019 Published  July 2021 Early access  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.

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.

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##### 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.
The network structure of the supply chain
The network structure of $i-$th manufacturer
The network structure of $j$'s retailer
The network structure of $k-$th consumer
The network structure of the supply chain of Example 4.1
The network structure of the supply chain of Example 4.2
The network structure of the supply chain of Example 4.3
The network structure of the supply chain for Example 4.5
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
 $(q_{ij}^1/q_{ij}^2)$ 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
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
 $(\tilde{q}_{ik}^1/\tilde{q}_{ik}^2)$ 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
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
 $(\hat{q}_{jk}^1/\hat{q}_{jk}^2)$ 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
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
 $(\tilde{\rho}_{ik}^1/\tilde{\rho}_{ik}^2)$ 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
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
 $(\hat{\rho}_{jk}^1/\hat{\rho}_{jk}^2)$ 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
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
 $(\rho_{ij}^1/\rho_{ij}^2)$ 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
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
 $\rho_k^l$ 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
Compared with the results in Example 2([17])
 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
Compared with the results in Example 4 ([17])
 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
Compared with the results in Example 5 ([17])
 Literature results Results of this paper Iteration steps 12 109 Running time 0.11 0.086
 Literature results Results of this paper Iteration steps 12 109 Running time 0.11 0.086
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