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

  • * Corresponding author: Hongchun Sun

    * Corresponding author: Hongchun Sun 

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
  • 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.

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

    Citation:

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  • Figure 1.  The network structure of the supply chain

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

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

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

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

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

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

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

    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
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