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

Haodong Chen; Hongchun Sun; Yiju Wang

doi:10.3934/jimo.2020066

Article Contents

2021, Volume 17, Issue 4: 2217-2242. Doi: 10.3934/jimo.2020066

This issue Previous Article Strict efficiency of a multi-product supply-demand network equilibrium model Next Article Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

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
^* Corresponding author: Hongchun Sun

Received: July 2019

Revised: October 2019

Early access: March 2020

Published: July 2021

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

Abstract / Introduction Full Text(HTML) Figure(8) / Table(10) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 90B20; Secondary: 90C30.

Citation:

Full Text(HTML)

Figure 1. The network structure of the supply chain

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image PowerPoint slide

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

Download: Full-size image 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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.