A global optimization method for multiple response optimization problems

He Huang; Zhen He

doi:10.3934/jimo.2022016

Article Contents

2023, Volume 19, Issue 3: 1755-1769. Doi: 10.3934/jimo.2022016

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A global optimization method for multiple response optimization problems

He Huang^, and
Zhen He

College of Management and Economics, Tianjin University, Tianjin 300072, China

^* Corresponding author: He Huang
^* Corresponding author: He Huang

Received: June 2021

Revised: December 2021

Early access: February 2022

Published: March 2023

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Abstract

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.

Keywords:

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

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Figure 1. Graph of the response yield

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Figure 2. Graph of the response viscosity

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Figure 3. Graph of the response molecular weight

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Figure 4. Graph of the feasible region

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Figure 5. The feasible region for different parts

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Table 1. Designed experiment and response values

Order	Natural variables		Coded variables		Responses
Order	$ \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

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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

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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

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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

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