Patent Publication Number: US-2021174261-A1

Title: Optimum sampling search system and method with risk assessment, and graphical user interface

Description:
This application claims the benefit of Taiwan application Serial No. 108144692, filed Dec. 6, 2019, the disclosure of which is incorporated by reference herein in its entirety. 
     TECHNICAL FIELD 
     The disclosure relates in general to an optimum sampling search system, an optimum sampling search method and a graphical user interface. 
     BACKGROUND 
     In the optimization search technology of many industries, it is usually necessary to meet certain constraints to obtain the sampling parameters that can obtain the best objective function outputting value. Taking the thin film process as an example, the researchers must find a set of process parameters, such as RF energy, the amount of SiH4 and the amount of NH3, so that the film thickness is within a certain specification and the stress is minimized. Taking the chemical process as an example, the researchers must find a set of operating parameters, such as the amount of chemicals added, the cooling water control conditions and the circulating reflux ratio, so that the reaction temperature is within certain limited safety conditions and the product yield is maximized. 
     In these application scenarios, the objective function and the constraint function are unknown black-boxes. Before the experiment, it is impossible to know whether the objective function outputting value is optimal and whether the constraint function outputting value meets the constraint. 
     At present, the conventional optimization search method is easy to confuse the effects of the objective function and the constraint function, and extract too many sampling points that do not meet the constraint, causing problems such as poor search parameter efficiency. Therefore, how to improve the search efficiency of the optimum sampling parameter is one of the research efforts of the researchers. 
     SUMMARY 
     The disclosure is directed to an optimum sampling search system, an optimum sampling search method and a graphical user interface. 
     According to one embodiment, an optimum sampling search system with risk estimation is provided. The optimum sampling search system includes a data acquisition unit, an objective satisfaction score calculation unit, a constraint satisfaction probability calculation unit, a sampling risk evaluation unit and an adjusting unit. The data acquisition unit is used for obtaining at least one objective function outputting value and at least one constraint function outputting value according to at least one executed sampling parameter. The objective satisfaction score calculation unit is used for obtaining an objective satisfaction score model according to the at least one executed sampling parameter and the at least one objective function outputting value. The constraint satisfaction probability calculation unit is used for obtaining a constraint satisfaction probability model according to the at least one sampling parameter and the at least one constraint function outputting value. The sampling risk evaluation unit is used for obtaining a recommended sampling parameter according to the objective satisfaction score model and used for estimating a constraint satisfaction probability of the recommended sampling parameter according to the recommended sampling parameter and the constraint satisfaction probability model. If the constraint satisfaction probability of the recommended sampling parameter is larger than or equal to a first predetermined value, the recommended sampling parameter is outputted. If the constraint satisfaction probability of the recommended sampling parameter is between the first predetermined value and a second predetermined value, the adjusting unit adjusts the recommended sampling parameter to optimize the constraint satisfaction probability model. 
     According to another embodiment, an optimum sampling search method with risk estimation is provided. The optimum sampling search method includes the following steps. At least one objective function outputting value and at least one constraint function outputting value are obtained according to at least one executed sampling parameter. An objective satisfaction score model is obtained according to the at least one executed sampling parameter and the at least one objective function outputting value. A constraint satisfaction probability model is obtained according to the at least one sampling parameter and the at least one constraint function outputting value. A recommended sampling parameter is obtained according to the objective satisfaction score model. A constraint satisfaction probability of the recommended sampling parameter is estimated according to the recommended sampling parameter and the constraint satisfaction probability model. The recommended sampling parameter is outputted, if the constraint satisfaction probability of the recommended sampling parameter is larger than or equal to a first predetermined value. The recommended sampling parameter is adjusted to optimize the constraint satisfaction probability model, if the constraint satisfaction probability of the recommended sampling parameter is between the first predetermined value and a second predetermined value. 
     According to an alternative embodiment, a graphical user interface is provided. The graphical user interface includes an objective satisfaction score model window, a constraint satisfaction probability model window and a sampling information window. The objective satisfaction score model window is used for displaying an objective satisfaction score model. The objective satisfaction score model is obtained according to at least one executed sampling parameter and at least one objective function outputting value. The constraint satisfaction probability model window is used for displaying a constraint satisfaction probability model. The constraint satisfaction probability model is obtained according the at least one executed sampling parameter and at least one constraint function outputting value. The sampling information window is used for displaying a recommended sampling parameter and a constraint satisfaction probability. The recommended sampling parameter is obtained according to the objective satisfaction score model, and the constraint satisfaction probability is obtained according to the recommended sampling parameter and the constraint satisfaction probability model. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows an optimum sampling search system with risk estimation according to one embodiment. 
         FIG. 2  shows a flowchart of an optimum sampling search method with risk estimation according to one embodiment. 
         FIGS. 3 to 12  illustrate the steps in the  FIG. 2 . 
         FIG. 13  shows a graphical user interface according to one embodiment. 
     
    
    
     In the following detailed description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the disclosed embodiments. It will be apparent, however, that one or more embodiments may be practiced without these specific details. In other instances, well-known structures and devices are schematically shown in order to simplify the drawing. 
     DETAILED DESCRIPTION 
     Please refer to  FIG. 1 , which shows an optimum sampling search system  1000  with risk estimation according to one embodiment. The optimum sampling search system  1000  includes a data acquisition unit  110 , an objective satisfaction score calculation unit  120 , a constraint satisfaction probability calculation unit  130 , a sampling risk evaluation unit  140 , an adjusting unit  170  and a graphical user interface  700 . The data acquisition unit  110  may be a data acquisition card interface, a transmission line, a card reader, a wireless transmission module or a scanner. Each of the objective satisfaction score calculation unit  120 , the constraint satisfaction probability calculation unit  130 , the sampling risk evaluation unit  140  and the adjusting unit  170  may be a circuit, a chip, a circuit board, a plurality of program codes and a storage storing the program codes. The graphical user interface  700  may be a tablet, a mobile phone, a computer, or a head-mounted display. 
     During searching the optimum sampling parameter, the optimum sampling search system  1000  combines the risk assessment mechanism and the probability model optimization technology to improve the search performance of sampling parameters, so as to achieve the industrial application requirements of reducing the number of trials and shortening the adjusting time. The operation of those elements is illustrated via a flowchart. 
     Please refer to  FIGS. 2 to 12 .  FIG. 2  shows a flowchart of an optimum sampling search method with risk estimation according to one embodiment, and  FIGS. 3 to 12  illustrate the steps in the  FIG. 2 . In the step S 110 , the data acquisition unit  110  obtains at least one objective function outputting value (e.g., objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ) in the  FIG. 3 ) and at least one constraint function outputting value (e.g., constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 ) in the  FIG. 3 ) according to at least one executed sampling parameter (e.g., sampling parameters X 1 , X 2 , X 3  in the  FIG. 3 ). The objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ) may be the error values. The lower the error value, the better. The constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 ) may be the changes of the film thickness. The change of the film thickness must higher than a condition value g 0 . 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE I 
               
               
                   
                   
               
               
                   
                   
                   
                 constraint function 
               
               
                   
                   
                   
                 outputting value 
               
               
                   
                 sampling 
                 objective function 
                 (condition 
               
               
                   
                 parameter 
                 outputting value 
                 value g0 = −200) 
               
               
                   
                   
               
             
            
               
                   
                 X1 = −3 
                 f(X1) = 143.9 
                 g(X1) = 230 
               
               
                   
                 X2 = 2.5 
                 f(X2) = 195.1 
                 g(X2) = −190 
               
               
                   
                 X3 = 5.5 
                 f(X3) = 68.6 
                 g(X3) = −420 
               
               
                   
                   
               
            
           
         
       
     
     As shown in the  FIG. 4  and the table I, for these three sampling parameters X 1 , X 2 , X 3 , the objective function outputting value f(X 3 ) is the lowest, but the constraint function outputting value g(X 3 ) is lower than the condition value g 0 , so the sampling parameter X 3  is not the optimum. The objective function outputting value f(X 1 ) is lower than the objective function outputting value f(X 2 ), and the constraint function outputting value g(X 1 ) is higher than the condition value g 0 , so the sampling parameter X 1  is the optimum among the sampling parameters X 1 , X 2 , X 3 . 
     Only the sampling parameters X 1 , X 2 , X 3  have been obtained so far. 
     Researchers must further search for better sampling parameters. The following steps can be used to modify the model to search for better sampling parameters. 
     Next, in the step S 120 , the objective satisfaction score calculation unit  120  obtains an objective satisfaction score model (e.g., an objective satisfaction score model E 1 ) according to the executed sampling parameter (e.g., the sampling parameters X 1 , X 2 , X 3 ) and the objective function outputting value (e.g., the objective function outputting values f(X 1 ), f(X 2 ), f(X 3 )). As shown in the  FIG. 4 , the sampling parameters X 1 , X 2 , X 3  are inputted into the unknown objective function to obtain the objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ). The unknown objective function may be the curve f 1  in the  FIG. 4 . For the unknown objective function, the sampling parameters X 1 , X 2 , X 3  correspond to the explicit objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ), but the remaining sampling parameters correspond to inexplicit objective function outputting values. The objective satisfaction score calculation unit  120  obtains the objective satisfaction score model E 1  via a Gaussian process (GP) and an Expected Improvement acquisition function (EI acquisition function). In the objective satisfaction score model E 1 , it references to the locations of the sampling parameters X 1 , X 2 , X 3  and the objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ), to show an objective satisfaction score of each of the sampling parameters. The higher the objective satisfaction score, the better able to satisfy the objective function, such as low error value; the lower the objective satisfaction score, the less likely to satisfy the objective function, such as high error value. Therefore, when selecting the sampling parameter, it can be selected from the higher objective satisfaction score. 
     Then, in step S 130 , the constraint satisfaction probability calculation unit  130  obtains a constraint satisfaction probability model (e.g., a constraint satisfaction probability model P 1 ) according to the executed sampling parameter (e.g., the sampling parameters X 1 , X 2 , X 3 ) and the constraint function outputting value (e.g., the constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 )). 
     As shown in the  FIG. 4 , the sampling parameters X 1 , X 2 , X 3  are inputted into the unknown constraint function to obtain the constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 ). The unknown constraint function may be a curve g 1  in the  FIG. 4 . For the unknown constraint function, the sampling parameters X 1 , X 2 , X 3  correspond to the explicit constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 ), but the remaining sampling parameters correspond to inexplicit constraint function outputting values. The constraint satisfaction probability calculation unit  130  obtains the constraint satisfaction probability model P 1  in the  FIG. 6  via a Support Vector Machine (SVM) algorithm. For example, the SVM algorithm may be a one class SVM algorithm or a Multi-class SVM algorithm. 
     As shown in the  FIG. 6 , the constraint satisfaction probability of the constraint satisfaction probability model P 1  is from  0  to  1 . The lower the value, the lower the constraint satisfaction probability; the higher the value, the higher the constraint satisfaction probability. The constraint satisfaction probability can be divided into a possible satisfy region R 1 , an uncertain region R 0  and an impossible satisfy region R 2 . If the constraint satisfaction probability is within the possible satisfy region R 1 , the probability that this sampling parameter satisfies the unknown constraint function is quite high; if the constraint satisfaction probability is within the impossible satisfy region R 2 , the probability that this sampling parameter satisfies the unknown constraint function is quite low; if the constraint satisfaction probability is within the uncertain region R 0 , the probability that this sampling parameter meets the unknown constraint function is uncertain. 
     A first predetermined value V 1  and a second predetermined value V 2  are used to divide the constraint satisfaction probability into the possible satisfy region R 1 , the uncertain region R 0  and the impossible satisfy region R 2 . The first predetermined value V 1  and the second predetermined value V 2  can be set according to a tolerance TR (shown in the  FIG. 7 ). Please refer to the  FIG. 7 , which shows that the first predetermined value V 1  and the second predetermined value V 2  are set according to the tolerance TR. The sampling risk evaluation unit  140  set the first predetermined value V 1  and the second predetermined value V 2  via a discretization algorithm according to the tolerance TR. A difference between the first predetermined value V 1  and the second predetermined value V 2  is the tolerance TR. For example, the first predetermined value V 1  and the second predetermined value V 2  may be calculated according to the following equations (1), (2). 
         V 1=1−0.5*(1− TR )  (1)
 
         V 2=0.5*(1− TR )  (2)
 
     If the constraint satisfaction probability is larger than or equal to the first predetermined value V 1 , it is within the possible satisfy region R 1 ; if the constraint satisfaction probability is between the first predetermined value V 1  and the second predetermined value V 2 , it is within the uncertain region R 0 ; if the constraint satisfaction probability is less than or equal to the second predetermined value V 2 , it is within the impossible satisfy region R 2 . 
     Please refer to the  FIG. 8 , which illustrates the SVM algorithm according to one embodiment. In the SVM algorithm, a decision boundary B 0  whose value is 0.5 is used to classify the possible satisfy region R 1  and the impossible satisfy region R 2 . Some points which are not be explicitly classified into the two groups are within the uncertain region R 0 , i.e. the region between the dotted line B 1  and the dotted line B 2 . The constraint satisfaction probability model P 1  can be modified as the amount of data increases, such that the dotted lines B 1 , B 2  gradually approach and the uncertain region R 0  is narrowed. Generally speaking, when the next sampling parameter is close to the decision boundary B 0 , the uncertain region R 0  can be quickly narrowed. 
     With the objective satisfaction score model E 1  and the constraint satisfaction probability model P 1 , a recommended sampling parameter will have the following six situations in Table II. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE II 
               
             
            
               
                   
                   
               
               
                   
                 objective satisfaction 
                   
               
               
                   
                 score model E1 
               
            
           
           
               
               
               
            
               
                   
                 High score 
                 Low score 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                 constraint 
                 possible 
                 Output the 
                 Do not 
               
               
                 satisfaction 
                 satisfy 
                 recommended 
                 output the 
               
               
                 probability 
                 region R1 
                 sampling 
                 recommended 
               
               
                 model P1 
                   
                 parameter 
                 sampling 
               
               
                   
                   
                   
                 parameter 
               
               
                   
                 uncertain 
                 Adjust the 
                 Do not 
               
               
                   
                 region R0 
                 recommended 
                 output the 
               
               
                   
                   
                 sampling 
                 recommended 
               
               
                   
                   
                 parameter to 
                 sampling 
               
               
                   
                   
                 optimize the 
                 parameter 
               
               
                   
                   
                 constraint 
               
               
                   
                   
                 satisfaction 
               
               
                   
                   
                 probability model 
               
               
                   
                 impossible 
                 Do not 
                 Do not 
               
               
                   
                 satisfy 
                 output the 
                 output the 
               
               
                   
                 region R2 
                 recommended 
                 recommended 
               
               
                   
                   
                 sampling 
                 sampling 
               
               
                   
                   
                 parameter 
                 parameter 
               
               
                   
               
            
           
         
       
     
     In the step S 140  of the  FIG. 2 , the sampling risk evaluation unit  140  obtains the recommended sampling parameter according to the objective satisfaction score model E 1 . As shown in Table II, the sampling risk evaluation unit  140  will not output the sampling parameter whose objective satisfaction score is low. As shown in the  FIG. 5 , the sampling risk evaluation unit  140  selects a sampling parameter X 4  whose objective satisfaction score E(X 4 ) is highest. 
     In the step S 150  of the  FIG. 2 , the sampling risk evaluation unit  140  estimates a constraint satisfaction probability (e.g., a constraint satisfaction probability P(X 4 ) in the  FIG. 6 ) according to the recommended sampling parameter (e.g., the sampling parameter X 4 ) and the constraint satisfaction probability model (e.g., the constraint satisfaction probability model P 1 ). As shown in the  FIG. 6  and the table III, the constraint satisfaction probability P(X 4 ) of the sampling parameter X 4  is 0.21. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE III 
               
               
                   
               
               
                   
                   
                 constraint function 
                   
                   
               
               
                   
                   
                 outputting value 
                 objective 
                 constraint 
               
               
                 sampling 
                 objective function 
                 (condition 
                 satisfaction 
                 satisfaction 
               
               
                 parameter 
                 outputting value 
                 value g0 = −200) 
                 score 
                 probability 
               
               
                   
               
             
            
               
                 X1 = −3 
                 f(X1) = 143.9 
                 g(X1) = 230 
                   
                   
               
               
                 X2 = 2.5 
                 f(X2) = 195.1 
                 g(X2) = −190 
               
               
                 X3 = 5.5 
                 f(X3) = 68.6 
                 g(X3) = −420 
               
               
                 X4 = 5.45 
                   
                   
                 E(X4) = 80 
                 P(X4) = 0.21 
               
               
                   
               
            
           
         
       
     
     Next, in the step S 160 , the sampling risk evaluation unit  140  determines the relationship among the constraint satisfaction probability of the recommended sampling parameter (e.g., the constraint satisfaction probability P(X 4 ) of the sampling parameter X 4 ), the first predetermined value V 1  and the second predetermined value V 2 . In this case, the constraint satisfaction probability P(X 4 ) of the sampling parameter X 4  is less than the second predetermined value V 2 , and within the impossible satisfy region R 2 . As shown in the table II, the sampling risk evaluation unit  140  will not output the sampling parameter X 4 . So, the process backs to the step S 140  to get another recommended sampling parameter. 
     As shown in the  FIG. 5 , the sampling risk evaluation unit  140  obtains the next recommended sampling parameter (e.g., the sampling parameter X 5  in the  FIG. 5 ) according to the objective satisfaction score model E 1 . Further, in the step S 150  of the  FIG. 2 , the sampling risk evaluation unit  140  estimates the constraint satisfaction probability (e.g., the constraint satisfaction probability P(X 5 )) according to the recommended sampling parameter (e.g., the sampling parameter X 5 ) and the constraint satisfaction probability model P 1 . As shown in the  FIG. 6  and the table IV, the constraint satisfaction probability P(X 5 ) of the sampling parameter X 5  is 0.225. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE IV 
               
               
                   
               
               
                   
                   
                 constraint function 
                   
                   
               
               
                   
                   
                 outputting value 
                 objective 
                 constraint 
               
               
                 sampling 
                 objective function 
                 (condition 
                 satisfaction 
                 satisfaction 
               
               
                 parameter 
                 outputting value 
                 value g0 = −200) 
                 score 
                 probability 
               
               
                   
               
             
            
               
                 X1 = −3 
                 f(X1) = 143.9 
                 g(X1) = 230 
                   
                   
               
               
                 X2 = 2.5 
                 f(X2) = 195.1 
                 g(X2) = −190 
               
               
                 X3 = 5.5 
                 f(X3) = 68.6 
                 g(X3) = −420 
               
               
                 X4 = 5.45 
                   
                   
                 E(X4) = 80 
                 P(X4) = 0.21 
               
               
                 X5 = 5.05 
                   
                   
                 E(X5) = 75 
                 P(X5) = 0.225 
               
               
                   
               
            
           
         
       
     
     Next, in the step S 160 , the sampling risk evaluation unit  140  determines the relationship among the constraint satisfaction probability of the recommended sampling parameter (e.g., the constraint satisfaction probability P(X 5 ) of the sampling parameter X 5 ), the first predetermined value V 1  and the second predetermined value V 2 . In this case, the constraint satisfaction probability P(X 5 ) of the sampling parameter X 5  is still less than the second predetermined value V 2 , and within the impossible satisfy region R 2 . According to the table II, the sampling risk evaluation unit  140  will not output the sampling parameter X 5 . So, the process backs to the step S 140  to get another recommended sampling parameter. 
     As shown in the  FIG. 5 , the sampling risk evaluation unit  140  obtains the next recommended sampling parameter (e.g., the sampling parameter X 6  in the  FIG. 5 ) according to the objective satisfaction score model E 1 . Further, in the step S 150  of the  FIG. 2 , the sampling risk evaluation unit  140  estimates the constraint satisfaction probability (e.g., the constraint satisfaction probability P(X 6 )) according to the recommended sampling parameter (e.g., the sampling parameter X 6 ) and the constraint satisfaction probability model P 1 . As shown in the  FIG. 6  and the table V, the constraint satisfaction probability P(X 6 ) of the sampling parameter X 6  is 0.251. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE V 
               
               
                   
               
               
                   
                   
                 constraint function 
                   
                   
               
               
                   
                   
                 outputting value 
                 objective 
                 constraint 
               
               
                 sampling 
                 objective function 
                 (condition 
                 satisfaction 
                 satisfaction 
               
               
                 parameter 
                 outputting value 
                 value g0 = −200) 
                 score 
                 probability 
               
               
                   
               
             
            
               
                 X1 = −3 
                 f(X1) = 143.9 
                 g(X1) = 230 
                   
                   
               
               
                 X2 = 2.5 
                 f(X2) = 195.1 
                 g(X2) = −190 
               
               
                 X3 = 5.5 
                 f(X3) = 68.6 
                 g(X3) = −420 
               
               
                 X4 = 5.45 
                   
                   
                 E(X4) = 80 
                 P(X4) = 0.21 
               
               
                 X5 = 5.05 
                   
                   
                 E(X5) = 75 
                 P(X5) = 0.225 
               
               
                 X6 = 4.9 
                   
                   
                 E(X6) = 72 
                 P(X6) = 0.251 
               
               
                   
               
            
           
         
       
     
     Then, in the step S 160 , the sampling risk evaluation unit  140  determines the relationship among the constraint satisfaction probability of the recommended sampling parameter (e.g., the constraint satisfaction probability P(X 6 ) of the sampling parameter X 6 ), the first predetermined value V 1  and the second predetermined value V 2 . In this case, the constraint satisfaction probability P(X 6 ) of the sampling parameter X 6  is between the first predetermined value V 1  and the second predetermined value V 2 , and within the uncertain region R 0 . According to the table II, the constraint satisfaction probability model P 1  is needed to be adjusted, so the process proceeds to the step S 170 . 
     In the step S 170 , the adjusting unit  170  adjusts the sampling parameter, to optimize the constraint satisfaction probability model P 1 . In one embodiment, as shown in the  FIG. 9 , the adjusting unit  170  obtains the next recommended sampling parameter via an Active Learning algorithm with uncertainty sampling. For example, a sampling parameter X 7  which is close to the decision boundary B 0  is selected to optimize the constraint satisfaction probability model P 1  and narrow the uncertain region R 0 . As shown in the  FIG. 9  and the table VI, the constraint satisfaction probability P(X 7 ) of the sampling parameter X 7  is 0.51, which is close to the decision boundary B 0 . 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE VI 
               
               
                   
               
               
                   
                   
                 constraint function 
                   
                   
               
               
                   
                   
                 outputting value 
                 objective 
                 constraint 
               
               
                 sampling 
                 objective function 
                 (condition 
                 satisfaction 
                 satisfaction 
               
               
                 parameter 
                 outputting value 
                 value g0 = −200) 
                 score 
                 probability 
               
               
                   
               
             
            
               
                 X1 = −3 
                 f(X1) = 143.9 
                 g(X1) = 230 
                   
                   
               
               
                 X2 = 2.5 
                 f(X2) = 195.1 
                 g(X2) = −190 
               
               
                 X3 = 5.5 
                 f(X3) = 68.6 
                 g(X3) = −420 
               
               
                 X4 = 5.45 
                   
                   
                 E(X4) = 80 
                 P(X4) = 0.21 
               
               
                 X5 = 5.05 
                   
                   
                 E(X5) = 75 
                 P(X5) = 0.225 
               
               
                 X6 = 4.9 
                   
                   
                 E(X6) = 72 
                 P(X6) = 0.251 
               
               
                 X7 = 3.9 
                 f(X7) = 73.9 
                 g(X7) = −550 
                   
                 P(X7) = 0.51 
               
               
                   
               
            
           
         
       
     
     After obtaining the sampling parameter X 7 , the user can execute the sampling parameter X 7 . When performing this process again, in the step S 110 , as shown in the  FIG. 10 , the data acquisition unit  110  obtains the objective function outputting value (e.g., the objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ), f(X 7 )) and the constraint function outputting value (e.g., the constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 ), g(X 7 )) according to the executed sampling parameter (e.g., the sampling parameters X 1 , X 2 , X 3 , X 7 ). As shown in the table VI, the four sampling parameters X 1 , X 2 , X 3 , X 7 , the objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ), f(X 7 ) and the constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 ), g(X 7 ) are obtained. 
     Afterwards, in the step S 120 , the objective satisfaction score calculation unit  120  obtains an objective satisfaction score model E 2  of the  FIG. 11  via the GP and the EI acquisition function. In the objective satisfaction score model E 2 , it references the location of the sampling parameters X 1 , X 2 , X 3 , X 7  and the objective function outputting values f(X 1 ), f(X 2 ), f(X 3 ), f(X 7 ) to show an objective satisfaction score of each of the sampling parameters. Comparing with the objective satisfaction score model E 1 , the objective satisfaction score model E 2  is more accurate. 
     Then, in the step S 130 , the constraint satisfaction probability calculation unit  130  obtains a constraint satisfaction probability model P 2  according to the executed sampling parameters (e.g., the sampling parameters X 1 , X 2 , X 3 , X 7 ) and the constraint function outputting values (e.g., the constraint function outputting values g(X 1 ), g(X 2 ), g(X 3 ), g(X 7 )). As shown in the  FIG. 12 , the sampling parameter X 7  is close to the decision boundary B 0 , so the uncertain region R 0  in the constraint satisfaction probability model P 2  may be quickly narrowed. As shown in  FIGS. 9 and 12 , the dotted pattern shows the uncertainty region R 0 . It is clear that the uncertain region R 0  in the constraint satisfaction probability model P 2  is greatly narrowed. As such, the next selected sampling parameter having higher objective satisfaction score is more likely to fall into the possible satisfy region R 1 . 
     Then, in the step S 140 , the sampling risk evaluation unit  140  obtains another recommended sampling parameter (e.g., the sampling parameter X 8 ) according to the objective satisfaction score model E 2 . Further, in the step S 150 , the sampling risk evaluation unit  140  estimates the constraint satisfaction probability (e.g., the constraint satisfaction probability P(X 8 )) according to the recommended sampling parameter (e.g., the sampling parameter X 8 ) and the constraint satisfaction probability model P 2 . As shown in the  FIG. 12 , the constraint satisfaction probability P(X 8 ) of the sampling parameter X 8  is 0.76. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE VII 
               
               
                   
               
               
                   
                   
                 constraint function 
                   
                   
               
               
                   
                   
                 outputting value 
                 objective 
                 constraint 
               
               
                 sampling 
                 objective function 
                 (condition 
                 satisfaction 
                 satisfaction 
               
               
                 parameter 
                 outputting value 
                 value g0 = −200) 
                 score 
                 probability 
               
               
                   
               
             
            
               
                 X1 = −3 
                 f(X1) = 143.9 
                 g(X1) = 230 
                   
                   
               
               
                 X2 = 2.5 
                 f(X2) = 195.1 
                 g(X2) = −190 
               
               
                 X3 = 5.5 
                 f(X3) = 68.6 
                 g(X3) = −420 
               
               
                 X4 = 5.45 
                   
                   
                 E(X4) = 80 
                 P(X4) = 0.21 
               
               
                 X5 = 5.05 
                   
                   
                 E(X5) = 75 
                 P(X5) = 0.225 
               
               
                 X6 = 4.9 
                   
                   
                 E(X6) = 72 
                 P(X6) = 0.251 
               
               
                 X7 = 3.9 
                 f(X7) = 73.9 
                 g(X7) = −550 
                   
                 P(X7) = 0.51 
               
               
                 X8 = 2.9 
                   
                   
                   
                 P(X8) = 0.76 
               
               
                   
               
            
           
         
       
     
     Next, in the step S 160 , the sampling risk evaluation unit  140  determines the relationship among the constraint satisfaction probability of the recommended sampling parameter (e.g., the constraint satisfaction probability P(X 8 ) of the sampling parameter X 8 ), the first predetermined value V 1  and the second predetermined value V 2 . In this case, the constraint satisfaction probability P(X 8 ) of the sampling parameter X 8  is larger than the first predetermined value V 1 , and within the possible satisfy region R 1 . According to the table II, the sampling risk evaluation unit  140  will output the sampling parameter X 8 . So, the process proceeds to the step S 180  to output the sampling parameter X 8 . 
     According to the embodiment, the optimum sampling search system  1000  combines the risk assessment mechanism and the probability model optimization technology to improve the search performance of sampling parameters, so as to achieve the industrial application requirements of reducing the number of trials and shortening the adjusting time. 
     The above embodiment is described with a single parameter model. 
     However, in another embodiment, the technology can be applied to a multi-parameter model. 
     The above technology can be installed on a mobile device or computer via an application (APP), and the searching process can be shown on the mobile device or computer. Please refer to  FIG. 13 , which shows a graphical user interface  700  according to one embodiment. The graphical user interface  700  includes an objective satisfaction score model window  710 , a constraint satisfaction probability model window  720 , a sampling information window  730 , a tolerance setting window  740 , a score window  750  and a probability window  760 . The objective satisfaction score model window  710  is used for displaying the objective satisfaction score model, such as the objective satisfaction score model E 1  or E 2 . The constraint satisfaction probability model window  720  is used for displaying the constraint satisfaction probability model, such as the constraint satisfaction probability model P 1  or P 2 . The sampling information window  730  is used for displaying the recommended sampling parameter, such as the sampling parameter X 7  or X 8 , and the constraint satisfaction probability, such as the constraint satisfaction probability P(X 7 ) or P(X 8 ). The tolerance setting window  740  is used for displaying the tolerance TR, the score window  750  is used for displaying the objective function outputting value of the recommended sampling parameter, the probability window  760  is used for displaying the constraint satisfaction probability of the recommended sampling parameter. 
     It will be apparent to those skilled in the art that various modifications and variations can be made to the disclosed embodiments. It is intended that the specification and examples be considered as exemplary only, with a true scope of the disclosure being indicated by the following claims and their equivalents.