3.1. Attribute Importance Measure

Due to the rough set theory is a tool for export concise decision or classification rules, the processing of data are discrete attribute values, and content in the union collected attribute values are continuous, using the theory of rough sets data reduction properties important measure should be first to collect data for discretization of continuous processing [9,10].

The information system is expressed as S=U,A,V,f. U=X1,X2,⋯Xn represents nonempty finite collection of real objects. A represents a set of attributes, including the conditional attribute C noise and the decision attribute D, that is A=C∪D, C∩D=∅, U=∪Va, Va∉A. Va is the range of attribute. f:U×A→V is used to represent an information function. When x∈U, a∈A, get fx,a∈Va. The information system S=U,A containing the conditional attribute C noise and the decision attribute D non is called the decision table.

In this paper, equidistance method is used to discretize attributes. The specific steps of the method are as follows:

Step 1: The interval length of each data attribute is calculated Si max represents the maximum value in the ith data attribute. Si min represents the minimum value in the ith data attribute. ni represents the number of intervals divided. The length of the interval to which the ith data attribute belongs.

Step 2: The scope of the interval is determined. The interval range of the ith data attribute is represented as follows:

Step 3: The quantized value of the attribute according to formulas (1) and (2) is determined. Each attribute has ni interval ranges. If the value zij in the attribute is in the ith interval range, its quantized value can be represented by i.

In order to measure the importance of condition attribute C in attribute set A to decision attribute D, the importance of condition attribute to decision attribute is defined. sigC−cD represents the importance of any condition c to the decision attribute search D.

3.2. Attribute Reduction

In this paper, an improved particle swarm optimization algorithm is used for attribute reduction [11]. The improvement mainly includes the following two aspects. Firstly, the attributes contained in the relative kernel obtained from rough set were put into the initial population and kept unchanged during the whole iteration to improve the convergence speed of the algorithm. Secondly, by introducing penalty function into adaptive value function, the characteristics of attribute reduction problem can be integrated into the optimization process of particle swarm optimization.

3.2.3. Attribute reduction algorithm

The steps of relative attribute reduction algorithm based on particle swarm optimization are as Figure 2.

Figure 2

Attribute reduction process.

Step 1: Create a decision table by reading in raw data. The dependence PaD of each decision attribute D with respect to the conditional attribute C is calculated.

Step 2: Figure out the importance sigC−cDC of each property c∈C relative to D. If sigC−cDC is not equal to 0, then coreDc=coreDc∪c. If KcoreDc=KCD then stops the calculation Otherwise, when KcoreDc<KCD, execute step 3.

Step 3: The control parameters of the particle swarm are initialized. the population size s, the relevant parameters of the termination condition and the adaptive function are set. Using s binary numbers of length L to represent the individuals of the initial population. Set the termination condition of the algorithm. For each attribute c∈C, if c∈ coreDC, the corresponding value is 1; if c∉ coreDC, the corresponding value is 0.

Step 4: For individual x of each population, the dependence value kx of each individual x including the condition attribute relative to the decision attribute D was calculated, and the fitness value Fix of each individual was obtained.

Step 5: Update the position information of each particle, and then calculate the updated particle fitness value. If the new particle fitness value is better than the individual extreme value, the individual extreme value is updated with the current fitness value. Then the global extremum is found according to the individual extremum of each particle.

Step 6: Judge whether the fitness value of the optimal individual satisfies the termination condition of the algorithm after iterative optimization of the continuous t generation. If so, the optimal individual terminated and output is the minimum relative reduction; otherwise, go back to step 2.

4.1. Reliability Function Acquisition Based on Relevance

In practical application, D-S evidence theory requires that the synthesized evidence is independent of each other, which is difficult to be satisfied in most applications. However, most applications directly assume its approximate independence and use D-S evidence theory, which often makes the fusion effect unsatisfactory. Therefore, based on the traditional evidence synthesis, this paper introduces the correlation coefficient between evidences to eliminate the influence caused by the nonindependence of evidences.

If U can represent all the possibilities of X, and all the elements contained in U are incompatible, then U is called the identification framework of X. There is a recognition framework U, and if the function m:2u→0,1 (2u is all subsets of U) satisfies the conditions mφ=0 and ∑A⊂UmA=1, mA is said to be the basic probability assignment used to measure the degree of trust in the proposition A.

In D-S evidence theory, the focus element contained in evidence determines the amount of information it can provide, so the information capacity Es is defined as follows:

|Ai| represents the cardinality of focus Ai. A is the number of focal points. For any I, if mAi=0, then Es=0, which means that evidence S does not contain any useful information. if |Ai|=1 and mΘ=1, then Es=1, which means that evidence S contains the most information.

Es1,s2 is used to represent the correlation between two pieces of evidence. s1 and s2 are two pieces of evidence. m1 and m2 are basic credibility. Ai and Bi are focus elements. If s1 and s2 are related, the degree of correlation is denoted by Dij which basic credibility is denoted by mD. If Es1,s2=0, it means that s1 and s2 are independent of each other. If Es1,s2=1 and Es1<Es2 or Es1>Es2, it means s1 is contained by s2 or s1 contains s2.

The correlation between evidence s1 and s2 is defined as follows:

The correlation coefficient of evidence s1 and s2 is defined as follows:

The correlation coefficient of evidence s2 and s1 is defined as follows:

Assuming that the basic credibility of evidence s1 to s2 in the same identification framework is m1,m2,⋯mn and the focus element set is Ai,i=1,2,3,⋯,k, Bj,j=1,2,3,⋯l⋯, Zk,r=1,2,3,⋯,n, the basic credibility of the new joint is as follows:

The size of K1 reflects the degree of evidence conflict. 11−K1 is called a normalized factor, which is used to avoid the probability of nonzero existence in the synthetic space-time set. If K1≠1, the basic probability assignment can be determined by m. If K1=1, the basic probability assignment is considered contradictory and there is no joint basic probability assignment.

4.2. Decisions Based on Trust Allocation

By taking the data set after attribute reduction as evidence, each data source can provide a set of propositions and establish appropriate reliability functions accordingly, so that multi-source data fusion is conducted under the same framework to synthesize different evidence bodies into a new evidence body [13]. Different application scenarios determine different ways to make decisions after evidence combination. This paper introduces the concept of minimum risk to make final decisions on the basis of evidence combination.

Set ∃A1,A2⊂U meet:

If

5.1. Data Discretization

The process data collected in the field are converted into discrete data representation by discretization of continuous attributes with equal distance method, as shown in Table 1. The quality judgment decision-making was established as shown in Table 2. The condition attributes represent the temperature of air blast hood, air blast bellow, air draft hood, air draft bellow, preheat hood I, preheat bellow I, preheat hood II, preheat bellow II, feed layer thickness. d stands for decision attribute, and 1, 2, 3 represent good, qualified and unqualified of the three decision levels.

5.2. Attribute Reduction

The rough set attribute reduction method based on particle swarm optimization algorithm is applied to the collected data. The value of each parameter in the calculation is m=20,α=16,β=2,k=0.9. The maximum number of iterations is set as n=400. The reduction of the minimum attribute of the conditional attribute to the decision attribute can be obtained as follows:

• core: a5,a6;

• gbest: a1,a2,a5,a6, a3,a5,a6,a8∘

5.3. Basic Probability Assignment

md1,md2,md3 is used to represent the basic reliability distribution of the indicators of production quality belonging to 1, 2 and 3. Taking the temperature of the air blast hood as an example, the basic reliability is assigned. The basic reliability of other data attributes is assigned in a similar way, and no specific constructor is given.

When a1<50, md3=0.9, md2=0, md1=0.

When 50≤a1<70, md3=1−a1−50100∗0.9, md2=a1−50100∗0.9, md1=0, mϕ=0.1.

When 70≤a1<80, md3=0md2=1−a1−70100∗0.9, md1=a1−70100∗0.9, mϕ=0.1.

When 80≤a1<100, md3=0, md2=a1−80100∗0.9, md1=1−a1−80100∗0.9, mϕ=0.1.

When a1≥100, md3=0, md2=0, md1=1, mϕ=0.1.

Assume that the collected information is a1=75,a4=80,a5=90,a6=70,a8=50, and use it as input to make decision judgment. Take a1,a2,a5,a6 as evidence r1,r2,r3,r4, and synthesize evidence r. Take a3,a5,a6,a8 as evidence r1′,r2′,r3′,r4′, and synthesize evidence r′. The three decision levels of good, qualified and unqualified are taken as the identification framework element B.

For a1,a2,a5,a6, the basic probability assignment is shown in Table 3.

According to the evidence synthesis formula of Equation (20), the combined results of a1,a2,a5,a6 are shown in Table 4.

For a3,a5,a6,a8, the basic probability assignment is shown in Table 5.

According to the evidence synthesis formula of Equation (20), the combined results of a3,a5,a6,a8 are shown in Table 6.

Combining evidence r and r′, the results are shown in Table 7.

It can be seen from Table 7, when a single evidence r or t is used for combination calculation, the credibility value of the evidence uncertainty changes from greater than or equal to 0.1 of a single evidence to 0.003 of the combined evidence. The uncertainty credibility value has been greatly reduced. On the basis of obtaining the credibility, the formula (22) is used to make the final decision. The threshold value is set as ε1=0.1,ε2=0.01, obviously mϕ=0.003<0.01 and md1−md2>0.1, so the final decision type can be judged as good.

5.4. Method Comparison

The different method were selected to compare the decision accuracy and algorithm timeliness for three kinds of query cycle data, as shown in Tables 8 and 9. The first method relies on artificial experience to make decision. The second method uses general D-S evidence theory in reference [14] without data reduction. The third methods uses the general attribute reduction method and D-S evidence theory method in reference [14,15]. The fourth methods uses the improved attribute reduction method and D-S evidence theory method in reference [16,17]. The fifth method is to use the method proposed in this paper. The validity of data fusion method is verified by fusion accuracy, which is equal to the ratio of the number of correct decisions to the total number of all decisions. The algorithm delay time is the average time to calculate the correct result.

It can be seen from Tables 8 and 9 that the data fusion method without attribute reduction is not only less accurate but also poor timeliness. Compared with the traditional method and the improved method, the method proposed in this paper is both accurate and timely.