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

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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

^* Corresponding author: Hongchun Sun
^* Corresponding author: Hongchun Sun

Received: June 30, 2019

Revised: September 30, 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

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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:

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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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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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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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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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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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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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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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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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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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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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