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A global optimization method for multiple response optimization problems

  • * Corresponding author: He Huang

    * Corresponding author: He Huang 
Abstract Full Text(HTML) Figure(5) / Table(4) Related Papers Cited by
  • The multiple response optimization problem has been studied extensively. However, most existing methods only find locally optimal solutions to the concerned optimization problem. Several methods were proposed which tried to find a globally optimal solution of the problem, but there is no theoretical guarantee to obtain a globally optimal solution. In this paper, we investigate a global optimization method for the problem of a chemical process studied by Myers et al. which involves two input variables and three responses of interest. Based on the fitted polynomial functions of three responses, this multiple response problem is reformulated as a polynomial optimization problem where the primary response is objective while the other responses are put into constraints. We obtain a globally optimal solution to the concerned polynomial optimization problem when requirements of non-primary responses and experimental region are given. The satisfactory optimal designs can be obtained by adjusting non-primary responses appropriately. The method we proposed can be implemented easily, and it obtains a globally optimal solution to the multiple response optimization problem we considered.

    Mathematics Subject Classification: Primary: 62K20, 90B50, 90C90.

    Citation:

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  • Figure 1.  Graph of the response yield

    Figure 2.  Graph of the response viscosity

    Figure 3.  Graph of the response molecular weight

    Figure 4.  Graph of the feasible region

    Figure 5.  The feasible region for different parts

    Table 1.  Designed experiment and response values

    Order Natural variables Coded variables Responses
    $ \xi_1 $ $ \xi_2 $ $ x_1 $ $ x_2 $ $ y_1 $ $ y_2 $ $ y_3 $
    1 80 170 -1 -1 76.5 62 2940
    2 90 170 1 -1 78.0 66 3680
    3 80 180 -1 1 77.0 60 3470
    4 90 180 1 1 79.5 59 3890
    5 77.93 175 -1.414 0 75.6 71 3020
    6 92.07 175 1.414 0 78.4 68 3360
    7 85 167.93 0 -1.414 77.0 57 3150
    8 85 182.07 0 1.414 78.5 58 3630
    9 85 175 0 0 79.9 72 3480
    10 85 175 0 0 80.3 69 3200
    11 85 175 0 0 80.0 68 3410
    12 85 175 0 0 79.7 70 3290
    13 85 175 0 0 79.8 71 3500
     | Show Table
    DownLoad: CSV

    Table 2.  The numerical results of problem (16)

    $ Num $ $ Va $ $ Vb $ $ Vc $ $ Vd $ $ x_1^* $ $ x_2^* $ $ \hat{y}_1^* $ $ \hat{y}_2^* $ $ \hat{y}_3^* $
    1 -1.6 0.0 -1.6 0.0 0.0 -0.6224 79.229 68 3275.8
    2 -1.6 0.0 0.0 1.6 -0.3710 0.5068 79.341 68 3400
    3 0.0 1.6 -1.6 0.0 0.2789 -0.6390 79.325 68 3330.1
    4 0.0 1.6 1.6 0.0 F F F F F
    5 -1.6 1.6 -1.6 1.6 -0.3710 0.5068 79.341 68 3400
     | Show Table
    DownLoad: CSV

    Table 3.  The numerical results (Ⅰ) of problem (17)

    $ Num $ $ \alpha $ $ \beta_1 $ $ \beta_2 $ $ \gamma $ $ x_1^* $ $ x_2^* $ $ \hat{y}_1^* $ $ \hat{y}_2^* $ $ \hat{y}_3^* $
    1 N N N N 0.3885 0.3085 80.213 68.7539 3520.6
    2 N N N 3500 0.3389 0.2496 80.207 69.1067 3500
    3 N N N 3400 0.0987 -0.0363 80.004 70.0076 3400
    4 N N N 3300 -0.1416 -0.3222 79.511 69.5634 3300
    5 N N N 3200 -0.3819 -0.6081 78.731 67.7740 3200
    6 N N N 3100 -0.6220 -0.8942 77.662 64.6379 3100
    7 N N 68 N 0.4130 0.4228 80.2 68 3545.9
    8 N N 67 N 0.4372 0.5445 80.157 67 3572.5
    9 N N 66 N 0.4566 0.64698 80.098 66 3594.6
    10 N N 65 N 0.4733 0.7370 80.029 65 3614
    11 N N 64 N 0.4881 0.8184 79.953 64 3631.5
    12 N N 63 N 0.5016 0.8932 79.871 63 3647.5
    13 N N 62 N 0.5140 0.9628 79.784 62 3662.4
     | Show Table
    DownLoad: CSV

    Table 4.  The numerical results (Ⅱ) of problem (17)

    $ Num $ $ \alpha $ $ \beta_1 $ $ \beta_2 $ $ \gamma $ $ x_1^* $ $ x_2^* $ $ \hat{y}_1^* $ $ \hat{y}_2^* $ $ \hat{y}_3^* $
    1 N N 68 3400 -0.3710 0.5068 79.341 68 3400
    2 N N 68 3300 0.1260 -0.6315 79.296 68 3300
    3 N N 68 3200 -0.3817 -0.6083 78.731 67.7726 3200
    4 N N 67 3400 0.2537 -0.7616 79.079 67 3303.1
    5 N N 67 3300 0.2377 -0.7607 79.079 67 3300
    6 N N 67 3200 -0.2906 -0.7136 78.706 67 3200
    7 N N 66 3400 0.2338 -0.8644 78.850 66 3280.8
    8 N N 66 3200 -0.1910 -0.8287 78.621 66 3200
    9 N N 65 3400 0.2168 -0.9548 78.631 65 3261.3
    10 N N 65 3200 -0.1046 -0.9287 78.5 65 3200
    11 N N 64.9 3400 0.2152 -0.9633 78.610 64.9 3259.5
    12 N N 64.9 3200 -0.0965 -0.9381 78.486 64.9 3200
    13 N N 64 3400 0.2018 -1.0364 78.419 64 3243.7
    14 N N 64 3200 -0.0271 -1.0183 78.353 64 3200
     | Show Table
    DownLoad: CSV
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