Study on the evaluation method of attribute coordinate based on morphic computing

Abstract

One core technology in the model of Attribute Comprehensive Evaluation is the application of the barycentric curve which simulates the psychological change of the evaluators, but original method can only select three total score hyperplanes when calculates the barycentric curve, when data set is big, it’s impossible to accurately reflect the psychological change of the evaluators. In order to solve this problem, in this paper morphic computing is applied to calculate the barycentric curve, it can apply more total score hyperplanes when compute the barycentric curve, so makes the results more accurate. The simulation results show the effectiveness of the improved method.

Keywords

Comprehensive evaluation morphic computing morphogenetic system

1. The overview

In the comprehensive evaluation method, there are Delphi method and AHP, etc., whose weight is set by the subjective method, and the weights of each index are set based on the experience of expert. Attribute coordinate evaluation method in essence also belongs to this kind of method, and based on this, the concept of dynamic weight is proposed, the weight is not fixed, but dynamically changes along with the change of total scores of evaluated objects. When calculating the dynamic weight, the original method adopted Lagrange interpolation formula, which fits the polynomial curve according to the remarks of the experts on the three hyperplanes, and then reflect the dynamic weight with this curve. This method is simple, however, when the evaluation data set is big, even if the expert evaluates multiple total score hyperplanes, the method can only select three hyperplanes, and the contributions of other hyperplanes have to be ignored. The core of morphogenetic system is the morphic computing, which uses the form record and reproduce to solve the problem of computing and make the tensor inversion component to be related to a projection operator in the transformation. Therefore, it can take into account all of total score hyperplanes evaluated by experts in the final barycentric curve through morphic computing, so as to make the evaluation result more accurate. In this paper, the attribute coordinate evaluation method of morphic computing is introduced firstly, and then the results are compared with the improved results by simulation experiment.

1.1 Overview of comprehensive evaluation issues

The key of the comprehensive evaluation is to evaluate the optimal and inferior quality of the evaluated objects. First, the indexes of the evaluated object are determined, and then all the evaluated objects are given some kind of uniform dimensional index value. Finally, the evaluated objects are ranked according to the score. There is no optimal principle in the comprehensive evaluation, and the principle of satisfaction is generally adopted. Therefore, in the real decision making model, all we can get is the satisfactory solution instead of the optimal solution. Therefore, the comprehensive evaluation requires only the most satisfactory solution to meet certain weight conditions. The most satisfying solution is expressed as the extreme value of the utility function u of Eq. (1):

$\displaystyle\mathop{\textit{Max}}\limits_{i}\left\{{u=\sum\limits_{j=1}^{m}w_% {j}x_{i,j}}\right\}$ (1)

Among the $0\leqslant x_{ij}\leqslant 1$ is the value of the $i_{th}$ scheme on the agent decision indexes $a_{j}$ , and $W=\left\{{w_{1},\ldots\ldots,w_{m}}\right\}$ is the weight of the decision-maker on the index vector $a=\left\{{a_{1},\ldots\ldots,a_{m}}\right\}$ , which satisfies: $\sum\limits_{j}w_{j}=1$ . In theory, the solution to problem Eq. (1), even if it exists, it is very difficult to find the entire utility value space

$\displaystyle U_{i}=\left\{{u=\sum\limits_{j=1}^{m}w_{j}x_{i,j}}\right\}.$

For the comprehensive evaluation based on subjective empowerment, the question is how to make the preference of the evaluator be reflected in the evaluation model.

1.2 Overview of morphic computing

The morphogenetic system is a system for recording, preserving and recreating all kinds of forms, which can be adopted to explore the morphogenesis process of the target form. It attempts to provide a mathematical model for morphogenesis phenomena and provides a unified mathematical method for problem solving in various fields, including neural network, genetic algorithm, DNA computing, risk analysis, uncertainty analysis, soft science, quantum mechanics, laser, holographic and so on. It supposes sources mainly include morphogenetic and morphic field theory in biology, second source theory, holographic theory and quantum mechanics in physics, and morphic field, internal source and projection operator are its core concepts.

Morphogenetic system implemented by morphic computing, morphic computing is composed of writing operations and reading operations. Its meaning is similar to the principle of holography. Writing operation writes the implied information from input field on the internal source. Reading operation can reconstruct input field from projection generated by internal source and basic field.

The main features of the form include direct identification, nonconserving and non-measurable, and repeatability and retention.

(1)
The direct identification morphic is universal and complex. For example, the world around them is full of forms, but any questions posed by the form are not immediately obvious. We recognize them by various perceptual functions.
(2)
The nonconserved and non-measurable. Form is not a vector or a scalar, it is not conserved.
(3)
Repeatable and preservative. For example, seeds grow year after year with the same appearance of plants, and the spiders weave the same mesh generation by generation, and the atoms repeat the same molecular patterns.

2. Attribute coordinate evaluation method construction

2.1 The partial most satisfactory solution under certain constraint conditions

For the solution of problem Eq. (1), can be resolved through for a series of subproblems to solve local most satisfied solution, set for total score equal to

$\displaystyle S_{T}=\left\{{x_{i}=(x_{i1},\ldots,x_{im})|\sum\limits_{j=1}x_{i% ,j}=T}\right\}$

Figure 1.

Local most satisfactory solution.

Figure 2.

Agent scheme of decision attributes one for any node of the graph.

Where $X=(x_{i1},\ldots,x_{im})$ .

The $T$ hyperplane, $S_{T}\subset X$ is a simplex ( $n-1$ ) dimensions (see Fig. 1).

Since the space is convex, the barycenter $b_{i}(x)$ is given by the expression

$\displaystyle b_{i}(x)=\left({\frac{\sum\limits_{h=1}^{t}w_{1h}^{i}x_{1h}}{% \sum\limits_{h=1}^{t}w_{1h}^{i}},\ldots,\frac{\sum\limits_{h=1}^{t}w_{mh}^{i}x% _{mh}}{\sum\limits_{h=1}^{t}w_{mh}^{i}}}\right)$ (2)

Agents decision makers graphs (see Fig. 2).

Graph for the three order tensor for $A_{\alpha\beta\gamma}$ that gives the scheme for one agent for one value of the tensor component. When we change the 27 values of the tensor $A_{\alpha\beta\gamma}$ we have another agent (see Fig. 3).

Figure 3.

Tensor image of the scheme decision attributes for one agent.

Where the agent decision maker $Z=(z_{1},\ldots,z_{n})$ selects the t set from ${\{}x_{k}{\}}$ , ${\{}x_{h},h=1,\ldots,t{\}}$ , which he thinks are more satisfactory, then give the score of ${\{}x_{h},h=1,\ldots,t{\}}$ as $w_{h}(x_{h})$ . For $n$ evaluator groups, the corresponding center of gravity $S$ , is Eq. (3).

$\displaystyle S(x)=\left({\frac{\sum\limits_{i=1}^{n}v_{i}\left(\frac{\sum% \limits_{h=1}^{t}w_{1h}^{i}x_{1h}}{\sum\limits_{h=1}^{t}w_{1h}^{i}}\right)}{% \sum\limits_{i=1}^{n}v_{i}},\ldots,\frac{\sum\limits_{i=1}^{n}v_{i}\frac{\sum% \limits_{h=1}^{t}w_{mh}^{i}x_{1h}}{\sum\limits_{h=1}^{t}w_{mh}^{i}}}{\sum% \limits_{i=1}^{n}v_{i}}}\right)$ (3)

Among them, $(v_{1},\ldots,v_{n})$ is the corresponding weight for each evaluator. Obviously, when the training sample set ${\{}x_{k}{\}}$ is big enough and trained enough times, and solution set $(x_{p1},\ldots,x_{pt})$ selected by evaluators $z_{p}$ is big enough, the barycenter $b_{p}(x)$ will gradually close to the local most satisfied solution $x^{*}|$ . In other words, at the same time, when the number of evaluators is greater than 1, $x^{*}$ is the most local satisfactory solution in $S_{T}\subset X$ (or $\triangle$ A’B’C’) of the evaluator group, which is also the psychological center point of this evaluator group.

It is worth pointing out: for each training, it can get a neighborhood with the center as sphere, as radius, which can make any solution become one of the satisfactory solutions to the above training.

2.2 Construct morphological information table and morphic computing projection

The knowledge of morphological system can be expressed by morphological information table (Table 1). It is a two-dimensional table representing the relationship between object and attribute, extending the concept of cross table in formal concept analysis, and expressing the degree of connection between object and attribute. The mixed attribute $S$ is the total score plane and $X^{*}$ is the most satisfactory solution for the total score plane.

Table 1
Morpho information table

Object/attribute	Basic attribute $H_{1}$	Basic attribute $H_{2}$	…	Basic attribute $H_{N}$	Mixed attribute $S$
$X_{1}^{\ast}$	$H_{1,1}$	$H_{1,2}$	…	$H_{1,N}$	$S_{1}$
$X_{2}^{\ast}$	$H_{2,1}$	$H_{2,2}$	…	$H_{2,N}$	$S_{2}$
…	…	…	…	…	…
$X_{T}^{\ast}$	$H_{T,1}$	$H_{T,2}$	…	$H_{T,N}$	$S_{T}$

The information in the table can be represented as

$\displaystyle H=\left[{{\begin{array}[]{*{20}c}{H_{11}}&{H_{12}}&{\ldots}&{H_{% 1N}}\\ {H_{21}}&{H_{22}}&{\ldots}&{H_{2N}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {H_{T1}}&{H_{T2}}&{\ldots}&{H_{TN}}\\ \end{array}}}\right]\text{ with }T>N$

When the matrix $H$ represent a vector basis for the linear superposition

$\displaystyle S=w_{1}\left[{{\begin{array}[]{*{20}c}{H_{11}}\\ {H_{21}}\\ {\ldots}\\ {H_{T1}}\\ \end{array}}}\right]+w_{2}\left[{{\begin{array}[]{*{20}c}{H_{12}}\\ {H_{22}}\\ {\ldots}\\ {H_{T2}}\\ \end{array}}}\right]+\ldots+w_{N}\left[{{\begin{array}[]{*{20}c}{H_{1N}}\\ {H_{2N}}\\ {\ldots}\\ {H_{TN}}\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{H_{11}}&{H_{12}}&{\ldots}% &{H_{1N}}\\ {H_{21}}&{H_{22}}&{\ldots}&{H_{2N}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {H_{T1}}&{H_{T2}}&{\ldots}&{H_{TN}}\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{c_{1}}\\ {c_{2}}\\ {\ldots}\\ {c_{N}}\\ \end{array}}}\right]=HW$ (4)

Given the colon space matrix

Now because $H$ is a rectangular matrix we cannot use the inverse operator to compute the coefficients $C$ to obtain the wanted vector $B$ a linear superposition of the vectors in $H$ . So we must multiply the previous expression by the transport of the matrix $H$ . So we have

$\displaystyle H^{T}S=H^{T}HW$ (6)

Where the matrix $H^{T}H$ is a quadratic $N\times N$ matrix that we can invert when the colons of the matrix $H$ are independent. So in the independent case we can compute the coefficients $W$ can be computed

$\displaystyle W=(H^{T}H)^{-1}S$ (7)

Now with the new weights we can built the vector $B^{\prime}$ in this way

$\displaystyle S^{\prime}=H(H^{T}H)^{-1}H^{T}S=QS$ (8)

Now for $H$ as rectangular matrix we have that

$\displaystyle S\neq QS$ (9)

Bur we can prove that $\left|{S-QS}\right|$ assume the minimum value among all possible linear combination in the colon space of the matrix $H$ . In fact we have that $Q B$ is the orthogonal projection of $B$ into the space which coordinates are the colon vectors of $H$ . To prove that $Q$ is a projection operator we see that

$\displaystyle H(H^{T}H)^{-1}H^{T}S=\textit{QQS}=H(H^{T}H)^{-1}H^{T}H(H^{T}H)^{% -1}H^{T}S=QS$ (10)

The orthogonal projection is

$\displaystyle P=(S-QS)$ (11)

So we have that

$\displaystyle P^{T}QS=(I-Q)^{T}QS=QS-Q^{T}QS=QS-\textit{QQS}=QS-QS=0$ (12)

Figure 4 is the image of the projection for three dimensions.

Figure 4.

Projection of the operator $S$ .

For

$\displaystyle D^{\prime}=\left|{PB+\delta}\right|=((I-Q)B+\delta)^{T}((I-Q)B+% \delta)=(B^{T}(I-Q)^{T}+\delta^{T})((I-Q)B+\delta)=(B^{T}(I-Q)^{T}(I-Q)B+% \delta^{T}\delta)=D+\delta^{T}\delta$

In Fig. 5, the distance $D=\left|{(1-Q)S}\right|$ is the minimum distance for the vector $S$ projected on the straight line. All the oblique projections (tracted lines) on the same straight line has distance $D^{\prime}\geqslant D$ . So $D$ is the minimum distance.

Figure 5.

The distance $D=\left|{(1-Q)S}\right|$ is the minimum distance for the vector $S$ projected on the straight line.

Because $\delta^{T}\delta\geqslant 0$ for any value of $\delta$ we have that $D\leqslant D^{\prime}$ .

In conclusion the distance $D$ is the minimum distance from original vector $B$ and the linear combination vector $B^{\prime}$ . We remark that in one step method by projection operator $Q$ without to use algorithms or differential equation to compute the minimum we can obtain the best transformation of any type of vector $S$ into another vector $S^{\prime}$ linear combination of the colon vectors in $H$ which distance to $B$ is the minimum distance.

2.3 Inverse barycenter method from linear data to weights by morphic computing projection

Given the matrix

$\displaystyle\left[{{\begin{array}[]{*{20}c}{H_{11}}&{H_{12}}&{\ldots}&{H_{1N}% }\\ {H_{21}}&{H_{22}}&{\ldots}&{H_{2N}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {H_{T1}}&{H_{T2}}&{\ldots}&{H_{TN}}\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{x_{11}}&{x_{12}}&{\ldots}% &{x_{1t}}\\ {x_{21}}&{x_{22}}&{\ldots}&{x_{2t}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {x_{m1}}&{x_{m2}}&{\ldots}&{x_{mt}}\\ \end{array}}}\right]=X$

So the parameters $W$ can be calculate by the expression

$\displaystyle W=(H^{T}H)^{-1}S$ (13)

For the direct barycenter method given by the expression

Where

$\displaystyle\left[{{\begin{array}[]{*{20}c}{x_{11}}&{x_{12}}&{\ldots}&{x_{1t}% }\\ {x_{21}}&{x_{22}}&{\ldots}&{x_{2t}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {x_{m1}}&{x_{m2}}&{\ldots}&{x_{mt}}\\ \end{array}}}\right]$

Are variables and $W=w_{1h}^{i}$ are parameters of the barycenter algorithm, by morphogenetic methods (projection operator) we generate the inverse algorithm that from the known parameters $X$ we can compute the parameters $w_{j}$ of the linear superposition process given by the expression.

The $S$ or mixed attribute vector is

$\displaystyle S_{1}=w_{1}\left[{{\begin{array}[]{*{20}c}{x_{11}}\\ {x_{21}}\\ {\ldots}\\ {x_{m1}}\\ \end{array}}}\right]+w_{2}\left[{{\begin{array}[]{*{20}c}{x_{12}}\\ {x_{22}}\\ {\ldots}\\ {x_{m2}}\\ \end{array}}}\right]+\ldots+w_{t}\left[{{\begin{array}[]{*{20}c}{x_{1t}}\\ {x_{2t}}\\ {\ldots}\\ {x_{mt}}\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{x_{11}}&{x_{12}}&{\ldots}% &{x_{1t}}\\ {x_{21}}&{x_{22}}&{\ldots}&{x_{2t}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {x_{m1}}&{x_{m2}}&{\ldots}&{x_{mt}}\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{w_{1}}\\ {w_{2}}\\ {\ldots}\\ {w_{t}}\\ \end{array}}}\right]=XW$ (15)

Where $S$ is the linear combination of the colon vector for the matrix $X$ that can write in this way

$\displaystyle b(x)=\left({\begin{array}[]{l}\frac{\sum\limits_{h=1}^{t}{w_{h}x% _{1h}}}{\sum\limits_{h=1}^{t}{w_{h}}}\\ \frac{\sum\limits_{h=1}^{t}{w_{h}x_{2h}}}{\sum\limits_{h=1}^{t}{w_{h}}}\\ \ldots\\ \frac{\sum\limits_{h=1}^{t}{w_{h}x_{mh}}}{\sum\limits_{h=1}^{t}{w_{h}}}\\ \end{array}}\right)=S_{1}=\left[{{\begin{array}[]{*{20}c}{S_{11}}\\ {S_{21}}\\ {\ldots}\\ {S_{m1}}\\ \end{array}}}\right]$ (16)

To computed the best parameters $w_{j}$ to generate

$\displaystyle XW=S$ $\displaystyle X^{T}XW=X^{T}S$ (17) $\displaystyle W=(X^{T}X)^{-1}X^{T}S$ $\displaystyle S^{\prime}=X(X^{T}X)^{-1}X^{T}S=QS$

For the agents we have the expression

$\displaystyle S(x)=\left({\frac{\sum\limits_{i=1}^{n}v_{i}\left(\frac{\sum% \limits_{h=1}^{t}{w_{1h}^{i}x_{1h}}}{\sum\limits_{h=1}^{t}{w_{1h}^{i}}}\right)% }{\sum\limits_{i=1}^{n}{v_{i}}},\ldots,\frac{\sum\limits_{i=1}^{n}v_{i}\frac{% \sum\limits_{h=1}^{t}{w_{mh}^{i}x_{1h}}}{\sum\limits_{h=1}^{t}{w_{mh}^{i}}}}{% \sum\limits_{i=1}^{n}{v_{i}}}}\right)$ (18)

We have $j_{\text{th}}$ agents evaluation for which with the previous computation are given by the expression

$\displaystyle B(x)=v_{1}S_{1}(X_{1})+v_{2}S_{2}(X_{2})+\ldots+v_{n}S_{n}(X_{n}% )=\left[{{\begin{array}[]{*{20}c}{S_{11}}&{S_{12}}&{\ldots}&{S_{1n}}\\ {S_{21}}&{S_{22}}&{\ldots}&{S_{2n}}\\ {\ldots}&{\ldots}&{\ldots}&{\ldots}\\ {S_{m1}}&{S_{m2}}&{\ldots}&{S_{mn}}\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{v_{1}}\\ {v_{2}}\\ {\ldots}\\ {v_{n}}\\ \end{array}}}\right]=Sv$ (19)

So we can compute the weights for the multi agents evaluations by the morphogenetic expression

$\displaystyle Sv=B$ $\displaystyle S^{T}Sv=S^{T}v$ (20) $\displaystyle v=(S^{T}S)^{-1}S^{T}B$ $\displaystyle S^{\prime}=S(S^{T}S)^{-1}S^{T}B=QB$

2.4 Morpho information table of local non linear cubie curve preference

For the previous expression form the data, Table 2 shows the morpho information table of the local preference curve when $N=$ 3.

Table 2
Morpho information table of local preference curve ( $N=$ 3)

Object/attribute	Basic attribute $H_{1}$	Basic attribute $H_{2}$	Basic attribute $H_{3}$	Mixed attribute $S$
$X_{1}^{\ast}$	1	$X_{12}^{\ast}$	$(X_{12}^{\ast})^{2}$	$S_{1}$
$X_{2}^{\ast}$	1	$X_{22}^{\ast}$	$(X_{22}^{\ast})^{2}$	$S_{2}$
…	…	…	…	…
$X_{T}^{\ast}$	1	$X_{T2}^{\ast}$	$(X_{T3}^{\ast})^{2}$	$S_{T}$

Table 3 shows the morpho information table of the local preference curve when $N=$ 4.

Table 3

Morpho information table of local preference curve ( $N=$ 4)

Object/attribute	Basic attribute $H_{1}$	Basic attribute $H_{2}$	Basic attribute $H_{3}$	Basic attribute $H_{4}$	Mixed attribute $S$
$X_{1}^{\ast}$	1	$X_{12}^{\ast}$	$(X_{12}^{\ast})^{2}$	$(X_{12}^{\ast})^{3}$	$S_{1}$
$X_{2}^{\ast}$	1	$X_{22}^{\ast}$	$(X_{22}^{\ast})^{2}$	$(X_{22}^{\ast})^{3}$	$S_{2}$
…	…	…	…	…	…
$X_{T}^{\ast}$	1	$X_{T2}^{\ast}$	$(X_{T3}^{\ast})^{2}$	$(X_{T3}^{\ast})^{3}$	$S_{T}$

2.5 Internal source calculation

The external source is the basis for calculating the internal source. Therefore, the external source $S_{e}$ of the system is calculated according to the morphological information table, and then the internal source $S_{I}$ is calculated through Eq. (22). The internal source is the contravariant component of vector $X$ in the attribute space.

$\displaystyle S_{e}=H^{T}X$ (21) $\displaystyle S_{i}=\left({H^{T}H}\right)^{-1}S_{e}$ (22)

When $N$ takes 3, $S_{I}=(s_{1},s_{2},s_{3})$ , according to Eq. (23), the local most satisfactory solution line $L\left({{b}^{\prime}\left({\left\{{x_{{}^{h}}\left(Z\right)}\right\}}\right)}\right)$ can be calculated.

$\displaystyle L\left({{b}^{\prime}\left({\left\{{x_{{}^{h}}\left(Z\right)}% \right\}}\right)}\right)=s_{1}+s_{2}X+s_{3}X^{2}$ (23)

When $N$ takes 4, $S_{I}=(s_{1},s_{2},s_{3},s_{4})$ , according to Eq. (24), the local most satisfactory solution $L\left({{b}^{\prime}\left({\left\{{x_{{}^{h}}\left(Z\right)}\right\}}\right)}\right)$ can be calculated.

$\displaystyle L\left({{b}^{\prime}\left({\left\{{x_{{}^{h}}\left(Z\right)}% \right\}}\right)}\right)=s_{1}+s_{2}X+s_{3}X^{2}+s_{4}X^{3}$ (24)

3. Simulation experiment

In order to verify the advantages of the improved method, we adopt a high school score as the test data, which has a total of 3,394 records. In the first step, we use the original method, only 3 hyperplanes of total score are selected, respectively 591, 689, and 750, then the evaluators were asked to rate each points of different hyperplanes. Table 4 shows the score of the hyperplanes of 689 given by the evaluators. According to Eq. (18), the center of gravity of this hyperplane is 122.22, 148.03, 142.24 and 276.51.

Table 4
The score of the hyperplanes of 689 given by the evaluators

ID	Chinese	Math	English	Science	Total	Remark
1990	118	147	145	279	689	8
1994	115	150	147	277	689	10
2070	135	149	135	270	689	9
2121	120	147	139	283	689	8
2195	116	147	141	285	689	8
2912	125	148	143	273	689	8
3048	129	145	141	274	689	7
3051	121	150	146	272	689	9

Table 5

The score of the hyperplanes of 591 given by the evaluators

ID	Chinese	Math	English	Science	Total	Remark
1308	112	127	133	219	591	9
1388	116	98	136	241	591	10
1570	112	121	126	222	591	8
1662	117	104	126	244	591	7
1690	109	128	121	233	591	7
1822	110	116	128	237	591	8
1829	117	123	117	234	591	6

Table 5 shows the score of the hyperplane of 591 given by the evaluators, according to Eq. (2), the center of gravity of the hyperplane was 113.23, 116.02, 127.64 and 232.65.

After obtaining the centers of gravity coordinates of three hyperplanes, the barycentric curves of four attributes were respectively calculated by Lagrange interpolation. Figure 6 shows the barycentric curves of each attribute, among which the dotted line stands for Chinese, the real line stands for mathematics, and the hexagonal star stands for English.

Figure 6.

Barycentric curves.

Then we do the same experiment by using the improved method, namely the morphic computing method, the morphic computing method allows to add more total points hyperplanes, also can more accurately reflect the changes of evaluator’s preference curve. Based on the previous experiment, the score of the hyperplanes of 628 is selected and remarked by the evaluators, as shown in Table 6, the corresponding center of gravity of this hyperplane is 116.07, 130.58, 136.06 and 245.29.

Table 6

The score of the hyperplanes of 628 given by the evaluators

ID	Chinese	Math	English	Science	Total	Remark
1957	115	128	129	256	628	7
2114	127	129	136	236	628	9
2165	104	130	135	259	628	6
2347	118	133	130	247	628	10
2498	119	132	136	241	628	8
2502	113	136	140	239	628	7
2623	117	133	138	240	628	7
2829	113	133	143	239	628	8
3167	113	120	139	256	628	7

Figure 7.

Comparison of the barycentric curves of Chinese attribute.

Then, the score of the hyperplanes of 641 was added, after that the evaluators remark the sample points on this hyperplane. The corresponding center of gravity was 116.14, 135.42, 135.59 and 253.84.

The next step is to construct the morpho information table for each index according to Table 2. Table 7 is a Chinese morpho information table.

Table 7

Chinese morpho information table

Chinese	Basic attribute $H_{1}$	Basic attribute $H_{2}$	Basic attribute $H_{3}$	Mixed attribute $S$
C1	1	113.23	12821.03	591
C2	1	116.07	13472.24	628
C3	1	116.14	13488.5	641
C4	1	122.22	14937.73	689
C5	1	150	22500	750

By using Eq. (8), we get the barycentric curve of Chinese, which is the real line in Fig. 7, and the dashed line is the barycentric curve obtained by using the original method. It can be clearly seen from the figure that the barycentric curve obtained by using the morphic computing can more accurately reflect the change of the preference of the evaluators.

Table 8

Mathematical morpho information table

Math	Basic attribute $H_{1}$	Basic attribute $H_{2}$	Basic attribute $H_{3}$	Mixed attribute $S$
M1	1	116	13456	591
M2	1	130	16900	628
M3	1	135	18225	641
M4	1	148	21904	689
M5	1	150	22500	750

Table 9

The foreign language morpho information table

English	Basic attribute $H_{1}$	Basic attribute $H_{2}$	Basic attribute $H_{3}$	Mixed attribute $S$
E1	1	127	16129	591
E2	1	136	18496	628
E3	1	135	18225	641
E4	1	142	20164	689
E5	1	150	22500	750

Figure 8.

Comparison of the barycentric curves of mathematical.

Table 8 is the morpho information table for mathematics. Figure 8 shows the barycentric curve of the evaluator based on the mathematical morpho information table, and the dashed line is the barycentric curve obtained by using the original method which obviously has major defects. One is barycentric curve was supposed to be monotone increasing, but there is an extremum points between 600 and 650, it means the barycentric curve isn’t is monotone increasing, this case is not reasonable, on the contrary, the barycentric curve obtained by morpho computing is monotone increasing.

The other advantage of improved method is it take into account hyperplanes of 628 and 641, the original method obviously ignores both of them.

Table 9 is the foreign language morpho information table. The real line is the center of gravity curve drawn from the attribute table of foreign language, and the dotted line is the center of gravity curve obtained by using the original method. It can be seen from the figure that the improved algorithm reflects the influence of the scoring results of 628 and 641 total point plane experts on the center of gravity curve. In the total score of 625, the original center of gravity curve has a large error, and the improved algorithm error is relatively reduced.

Figure 9.

Comparison of the barycentric curves of foreign language attributes.

Respectively through Eq. (15) using two methods on 3394 sample data was received, the sequencing result difference rate was 2.29%, that is to say, there are 2.29% of results obtained the correction, also show the effectiveness of the improved method.

4. Conclusion

In this paper, the method of attribute coordinate evaluation of morphological calculation is studied and the corresponding mathematical model is given. The experimental results show that the method is not only improved the monotonicity of the original fitting curve problem, also more evaluation planes can be applied to the calculation of dynamic weight, so as to make the evaluation results more accurate and reasonable.

References

Saaty

T.L.

, The analytic hierarchy and analytic network measurement processes: Applications to decisions under Risk, European Journal of Pure and Applied Mathematics 1(1) (2008), 122–196.

D.F.

, Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment, Fuzzy Optimization and Decision Making 7(1) (2008), 17–34.

Opricovic

and Tzeng

G.H.

, Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS, European Journal of Operational Research 156(2) (2004), 445–455.

Jorm

A.F.

, Using the Delphi expert consensus method in mental health research, Australian & New Zealand Journal of Psychiatry 49(10) (2015), 887–897.

Boender

C.G.E.

de Graan

J.G.

and Lootsma

F.A.

, Multicriteria decision analysis with fuzzy pairwise comparisons, Fuzzy Sets and Systems 29(2) (1989), 133–143.

Wang

S.C.

G.L.

and Du

R.J.

, Restricted Bayesian classification networks, Science China Information Sciences 56(7) (2013), 1–15.

Resconi

X.L.

and Xu

G.L.

, Introduction to Morphogenetic Computing, Springer, 2017.

Resconi

and Van der Wal

A.J.

, Morphogenic neural networks encode abstract rules by data, Information Sciences 142(1) (2002), 249–273.

Resconi

and Nikravesh

, Morphic computing, Applied Soft Computing 8(3) (2008), 1164–1177.

10.

Feng

J.L.

, Qualitative mapping orthogonal system induced by subdivision of qualitative criterion and biomimetic pattern recognition, Chinese Journal of Electronics 15(4A) (2006), 850–856.

11.

Tian

, Morphogenetic Estimation and its Application in Updating Probabilistic Risk, Ph.D. Dissertation, Beijing Normal University, 2012.

12.

G.L.

and Xu

X.L.

, Study on comprehensive evaluation model of attribute coordinate based on evaluation sample selection by K-means, Journal of Computational Methods in Sciences and Engineering 17(3) (2017), 463–472.

13.

G.L.

and Min

, Research on multi-agent comprehensive evaluation model based on attribute coordinate, in: Granular Computing (GrC), IEEE International Conference, 2012, pp. 556–562.

14.

X.L.

G.L.

and Feng

J.L.

, Study on Updating Algorithm of Attribute Coordinate Evaluation Model, in: International Conference on Intelligent Computing, Springer, Cham, 2017, pp. 653–662.

15.

Feng

J.L.

G.L.

and Xu

X.L.

, The research on the application of qualitative mapping in MapReduce, Annals of Data Science 2(3) (2015), 271–280.

16.

Bothos

Apostolou

and Mentzas

, Recommender systems for nudging commuters towards eco-friendly decisions, Intelligent Decision Technologies 9(3) (2015), 295–306.

Study on the evaluation method of attribute coordinate based on morphic computing

Abstract

Keywords

1. The overview

1.1 Overview of comprehensive evaluation issues

2.1 The partial most satisfactory solution under certain constraint conditions

Table 1 Morpho information table

Table 2 Morpho information table of local preference curve ( N = 3)

Table 4 The score of the hyperplanes of 689 given by the evaluators

References

Table 1
Morpho information table

Table 2
Morpho information table of local preference curve ( $N=$ 3)

Table 4
The score of the hyperplanes of 689 given by the evaluators