A belief rulebased (BRB) system provides a generic nonlinear modeling and inference mechanism. It is capable of modeling complex causal relationships by utilizing both quantitative information and qualitative knowledge. In this paper, a BRB system is firstly developed to model the highly nonlinear relationship between circuit component parameters and the performance of the circuit by utilizing available knowledge from circuit simulations and circuit designers. By using rule inference in the BRB system and clustering analysis, the acceptability regions of the component parameters can be separated from the value domains of the component parameters. Using the established nonlinear relationship represented by the BRB system, an optimization method is then proposed to seek the optimal feasibility region in the acceptability regions so that the volume of the tolerance region of the component parameters can be maximized. The effectiveness of the proposed methodology is demonstrated through two typical numerical examples of the nonlinear performance functions with nonconvex and disconnected acceptability regions and highdimensional input parameters and a realworld application in the parameter design of a track circuit for Chinese highspeed railway.
Tolerance has become a crucial design consideration in integrated and discrete circuit designs due to the demand of improved product quality, longer product lifetimes, and shorter design cycle. Designers have to unceasingly seek a central point with the maximum tolerances in the space of circuit component parameters so as to maximize parametric yield and minimize costs while maintaining compliance with design specifications [
In essence, there are mainly two kinds of methods for tolerance design and yield estimation, that is, the Monte Carlo based statistical methods and the deterministic methods [
This paper develops a novel method of the acceptability region approximation and tolerance optimization to obtain available feasibility region using belief rulebased (BRB) model. In the belief rule base, each possible consequent of a rule is associated with a belief degree. Such a rule base is capable of capturing highly nonlinear and continuous causal relationships between different factors [
In this paper, a BRB system is designed to model the complex nonlinear relationship between circuit component parameters (i.e., input variables to the BRB system) and a performance index of the circuit (i.e., output) by utilizing the limited knowledge from circuit simulations and its designers. Through rule inference in the BRB and clustering analysis, the acceptability regions can be separated from the value domain of component parameters. Then, an optimization method is presented to seek the optimal feasibility region in the acceptability regions to maximize the volume of tolerance region of the circuit parameters. The remainder of this paper is organized as follows. The research issue is expounded in Section
Given a product performance or response specification [
The acceptability region
When given a nominal value
When given the design constraints
As an extension of traditional IFTHEN rules, belief rules are the key parts of a BRB system. In a belief rule, each antecedent attribute takes a referential value, and each possible consequent is related to a belief degree [
Belief rulebased system for circuit performance modelling.
BRB system  Circuit performance function 

Antecedent attributes  Parameter inputs 


The set of referential values 
Referential values of the design parameter 


Antecedent in the 
The 


Consequent in the 



Rule weights 
Relative importance of the 


Attribute weights 
Relative importance of 
Corresponding to Table
To use the BRB system to approximate acceptability region
The initial belief rules can be established in the following four ways [
In the tolerance design, we first need to specify the upper and lower bounds of each design parameter
As an illustrative example suppose there are two inputs
Selection of referential values of parameter inputs and performance output.
As a result, we totally select 64 points (including 29 EB, 23 IB, and 12 inside points) as the referential values of parameter input. The sets of referential values can be listed as
Next, we need to calculate the bounds of performance output as
In Table
Note that, in practice, there may be more than one acceptability region in the parameter space
Section
Having determined the activation weight of each rule in the rule base, the ER approach can be directly applied to combine the rules and generate final conclusions [
Although the initial belief rules can be constructed by the limited simulations and designers’ knowledge and the performance outputs can be estimated by ER inference of the initial belief rules, the estimation accuracy can be improved if the parameters in the belief rules are finetuned through learning from some selected training samples. The adjustable parameters in a rule base include belief degrees (
When we search for the available feasibility region
For example, in the case of twodimensional inputs as shown in Figure
In this section, the proposed BRB can be used to identify the optimal feasibility region from the acceptability regions based on the design criterion of maximizing volume of tolerance region.
Firstly, we have to select an available point and its tolerance from the acceptability region as the initial solution of optimization, since an arbitrary initial point and its tolerance may cause the optimization process which is timeconsuming and even becomes trapped in the local optima [
The first way is to seek the initial solution directly from the inside points and inside boundary (IB) points that we have obtained by circuit simulations.
For
Let us take the 2dimensional space as an example. Figure
Searching for the initial solution of optimization in the inside points.
Instead of circuit simulations, we use the proposed BRB to estimate more inside points so as to increase the density of the cast points in the acceptability region. Hence, the second way is to search for the initial solution from these estimated points using the above procedure given in the first way. Obviously, the initial solution got by the second way is more accurate than that got by the first way, but it needs more computational loads.
Note that, if there are multiple disconnected acceptability regions in the parameter space
According to the criterion of maximizing volume of tolerance region, the objective function can be defined as
In this section, two numerical studies and an industrial case are given to illustrate the procedure of using the BRB system to solve tolerance design problem.
The Rosenbrock function is a wellknown benchmark for assessing nonlinear numerical optimization algorithms [
Set the acceptability regions as
The acceptability regions
According to the procedure of building BRB system given in Section
The partial parameters of the initial rule base.











1  −0.4  −0.15  0  0  0  0  0.215  0.785  0 
2  −0.4  −0.1  0  0  0  0.64  0.36  0  0 
3  −0.4  −0.05  0  0  0.815  0.185  0  0  0 
4  −0.4  0  0  0.74  0.26  0  0  0  0 
5  −0.4  0.3  0.04  0.96  0  0  0  0  0 
6  −0.4  0.35  0  0.215  0.785  0  0  0  0 
7  −0.4  0.4  0  0  0.14  0.86  0  0  0 
8  −0.35  −0.2  0  0  0  0  0  0.8884  0.1116 










121  0.35  −0.1  0  0.3134  0.6866  0  0  0  0 
122  0.35  −0.05  0.3009  0.6991  0  0  0  0  0 
123  0.35  0.3  0.2134  0.7866  0  0  0  0  0 
124  0.35  0.35  0  0.2009  0.7991  0  0  0  0 
125  0.35  0.4  0  0  0  0.9384  0.0616  0  0 
126  0.4  −0.2  0  0  0  0  0  0.34  0.66 
127  0.4  −0.15  0  0  0  0.015  0.985  0  0 
128  0.4  −0.1  0  0  0.44  0.56  0  0  0 
In Table
Next, set the 79 central points of the BG cells as training points and then obtain their performance outputs by circuit simulations as the training samples as shown in Figure
The partial parameters in the trained rule base.












1  −0.4  −0.15  1  0.0177  0.0005  0  0  0  0.9818  0 
2  −0.4  −0.1  1  0  0  0  1  0  0  0 
3  −0.4  −0.05  0.9789  0.0411  0.03  0.9289  0  0  0  0 
4  −0.4  0  1  0.0982  0.9018  0  0  0  0  0 
5  −0.4  0.3  1  0.0118  0.9882  0  0  0  0  0 
6  −0.4  0.35  1  0.0185  0.11  0.8715  0  0  0  0 
7  −0.4  0.4  0.9990  0.0185  0  0.9815  0  0  0  0 
8  −0.35  −0.2  1  0.0283  0  0  0  0  0.9152  0.0565 











121  0.35  −0.1  1  0.0308  0.0004  0  0.2358  0.7332  0  0 
122  0.35  −0.05  0.9746  0.0625  0  0.9375  0  0  0  0 
123  0.35  0.3  0.9990  0.5379  0.4620  0  0  0  0  0 
124  0.35  0.35  1  0.0156  0.0587  0.9254  0  0  0  0 
125  0.35  0.4  1  0.0158  0  0.1587  0.8255  0  0  0 
126  0.4  −0.2  0.9677  0.0391  0.9609  0  0  0  0  0 
127  0.4  −0.15  0.9878  0.0429  0.0039  0  0  0  0.2058  0.7474 
128  0.4  −0.1  1  0.0308  0.0004  0  0.2355  0.7333  0  0 
We uniformly select 404 sample points from
The relative errors of the initial BRB and the trained BRB.
After obtaining the trained BRB, the next step is to optimize tolerances by the proposed method in Section
Comparisons of the optimization results for


 









(−0.026, 0.266)  (0.186, 0.033)  (−0.0255, 0.265)  (0.191, 0.035)  4.4%  (−0.02, 0.26)  (0.15, 0.04)  20.3% 
Comparisons of the optimization results for


 









(0.0164, −0.234)  (0.182, 0.033)  (0.0195, 0.234)  (0.185, 0.034)  2.4%  (0.02, −0.24)  (0.14, 0.04)  22.1% 
The approximation of
In the yield estimation of integrated circuits, the highdimensional quadratics or higher order polynomials are usually used to model the performances of interest. Here, we give an eightdimensional quadratic function to test the proposed method. Without loss of generality, assume a circuit performance can be expressed as an 8dimension quadratic function whose symmetric matrixes are diagonal or block diagonal [
Suppose the numbers of the referential values uniformly selected for
By testing the selected 1373 sample points in
Comparisons of the optimization results for


 










1  4.9929  0.0738  4.9891  0.0743  5.8%  4.99  0.065  43.2% 
2  4.9913  0.0887  4.9896  0.0889  4.99  0.065  
3  4.9994  0.0157  4.9969  0.0164  4.99  0.045  
4  4.9925  0.0774  4.9891  0.0778  4.99  0.065  
5  4.9922  0.0802  4.9893  0.0808  4.99  0.065  
6  4.9928  0.0747  4.9891  0.0753  4.99  0.055  
7  4.9915  0.0868  4.9895  0.0872  4.99  0.055  
8  4.9940  0.0642  4.9888  0.0652  4.99  0.05 
Railway track circuit is an essential component of information transmission system between track and vehicle and the automatic train control system [
The railway track is divided into different sections. Each one of them has a specific ZPW 2000A consisting of main track circuit and short track circuit [
System structure of ZPW 2000A.
Figure
The short track circuit of G2.
We uniformly select
Similar with example 2, the given initial BRB is accurate enough for tolerance design, so the training process is cancelled. Figure
Comparisons of the optimization results for


 










1  97.0506  0.5335  97.0153  0.5402  6.5%  96.9  0.6  11.1% 
2  91.0297  0.8900  91.1083  0.8917  91  0.9  
3  262.2203  1.1877  262.2363  1.2363  262.2  1.2 
The approximated
In this paper, a BRB system is designed to model the acceptability region and optimize the feasibility region of circuit parameters so that the volume of the tolerance region of the circuit can be maximized. Its virtues can be demonstrated by several examples in this paper.
The main advantages of this new method are as follows.
The physical meanings of the parameters and structures of the BRB system are transparent and intuitively easy to understand by experts and engineers, so they can participate in the main steps of system modeling (e.g., determining the number of rules by considering the inside points and IB points, choosing the training samples from the BG cells, and determining the attribute weights and rule weights).
The proposed BRB system is applicable to complex cases, such as highly nonlinear performance function and nonconvex and disconnected feasibility regions.
The proposed optimization algorithm provides alternative ways to obtain the initial solutions of the optimization problem so as to avoid local optima with a higher chance and improve the efficiency of the algorithm.
Like all other deterministic methods, the proposed method also suffers from exponential explosion of computational costs when the dimension of a design parameter space increases. However, when there are strong correlations between the design parameters, the correlation analysis methods can be used to find out less independent variables or principal components so as to reduce the dimension of design parameter space as analyzed in [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the NSFC (no. 61374123, 61433001), the Zhejiang Province Research Program Project of Commonweal Technology Application (no. 2012C21025), and the Program for Excellent Talents of Chongqing Higher School (no. 201418).