Fuzzy comprehensive evaluation adopts fuzzy mathematical theory to obtain quantitative evaluation result in view of the complexity of teaching evaluation object and fuzziness of the evaluation index. The evaluation result has certain objectivity. The process of fuzzy comprehensive evaluation is shown in Figure 2.

Figure 2 Fuzzy comprehensive evaluation for network based interuniversity collaborative learning

Step 1. Set up evaluation factor set

The first level of evaluation factor set *U* = (*U*_{1}, *U*_{2}, *U*_{3}, *U*_{4}) is set up, which represents initial state, learning attitude, exchange and cooperation, and resource utilization.

Set up the second level of evaluation factor set *U*_{1} = (*u*_{11}, *u*_{12}, *u*_{13}, *u*_{14}), *U*_{2} = (*u*_{21}, *u*_{22},. . ., *u*_{29}), *U*_{3} = (*u*_{31},*u*_{32},. . ., *u*_{37}), *U*_{4} = (*u*_{41}, *u*_{42},. . ., *u*_{46}).

Step 2. Determine the evaluation set. *V* = (*v*_{1}, *v*_{2},..., *v*_{n}), *v*_{i}(*i* = 1, 2,..., *n*) represents all possible evaluation result.

We select five grades as evaluation level of student network learning, which is excellent, good, medium, pass, and fail. The corresponding evaluation sets is *V*=excellent, good, medium, pass, fail. *V* = {*v*_{1}, *v*_{2}, *v*_{3}, *v*_{4}, *v*_{5}}. The score of *v*_{1} is from 90 to 100, the score of *v*_{2} is from 80 to 90, the score of *v*_{3} is from 70 to 80, the score of *v*_{4} is from 60 to 70 and the score of *v*_{5} is from 0 to 60.

Step 3. Determine weight of evaluation index

Weight is used to measure the role and status of an index in the whole learning evaluation index system. After the indexes are established, we consider its relative importance in the index system. If weight of an index is larger, its value change has large impact on the evaluation results. For different evaluation results, we should scientifically determine the index weight to reasonably carry out the analysis and evaluation. Because, there may be many evaluation indexes, in the selection of index, a kind of attribute reduction method based on difference matrix [19] is adopted to choose the most suitable indexes. The difference matrix is *m*_{ij}. *S* = (*U*, *C* ∪ *D*, *V*, *f*) represents the information system decision table, *U*represents comment domain, *V* represents set of attribute value, *f* : *U* × *A* → *V* is an information function, *C* and *D* represents condition attribute and decision attribute respectively.
$${m}_{ij}=\left\{\begin{array}{rlrl}& \{a\in C:f({x}_{i},a)\ne f({x}_{j},a)\},& & \text{\hspace{0.17em}when\hspace{0.17em}}f({x}_{i},D)\\ & & & \phantom{\rule{2em}{0ex}}\ne f({x}_{j},D)\\ & \varnothing ,& & \text{otherwise}\end{array}\right.$$

Firstly, difference matrix of the decision table is worked out. Secondly, the weight value *w*(*a*_{k}) of each attribute *a*_{k} is calculated. We also work out the number of element containing *a*_{k} in the difference matrix labeled as *card*(*a*_{k}). *R* ← ∅, *W* ← 0. Thirdly, for *k* from 1 to *n* that represents the number of attribute, if *w*(*a*_{k}) > *W*, *W* ← *w*(*a*_{k}). If *w*(*a*_{k}) = *W* and *card*(*a*_{k}) is larger *W* ← *w*(*a*_{k}). Fourthly, *R* ← *R* ∪ {*a*_{k}} and the element containing *a*_{k} in the difference matrix is deleted. Fifthly, if the difference matrix is not empty, calculate the each left *w*(*a*_{k}) and turn to step 3.

Analytic hierarchy process combines the expert’s experience knowledge with mathematical method. When judging matrix is established, the two comparing methods are used, which greatly reduce the uncertainty factors. According to 1 to 9 scale method, index relative important degree value is assigned and comparative value can be obtained. Based on comparative value, judgment matrix can be obtained. If there are *n* elements, we obtain a matrix of *n* x *n* after the two comparing as shown in .

Table 1 Comparative table

Suppose the weight factor vector of evaluation set is *A* = (*a*_{1}, *a*_{2}, . . .,*a*_{n}), *a*_{i} represents the weight of evaluation factor *u*_{i} in the total evaluation factor. *a*_{i} ≥ 0, Σ *a*_{i} = 1. According to the idea of AHP method, we develop the two comparative matrix of network learning behavior evaluation factors by means of expert meeting. The weight of the first level of index is described as follows.

Calculate the sum of each column.

Each item in the matrix is divided by its corresponding column sum, then the standard two comparative matrix is obtained.

Calculate the arithmetic average of each row of the standard two comparative matrix and this average value is called the weight. It is impossible to judge exactly right in the two compared matrices, sometimes there will be error or even contradiction. The higher the judgment order, the more difficult the determination is, and the deviation will increase. In fact, almost all of the two compared matrices have a certain degree of inconsistency. In order to solve the problem of the consistency, the AHP provides a way to measure the degree of consistency. If the degree of consistency cannot meet the requirements, we should review and modify the two compared matrices. Judgment matrix and weight of index in the first level is shown in .

Table 2 Judgment matrix and weight of index in the first level

$CI=\frac{{\lambda}_{max}-n}{n-1}$ is defined as consistency index, *n* represents the number of compared index and *λ*_{max} represents maximum characteristic root of judgment matrix. The weighted vector is [0.232, 0.528, 2.279, 1.130]. *λ*_{max} = (0.232/0.057 + 0.528/0.131 + 2.279/0.540 + 1.130/0.272)/4 = 4.119

*CR* = 0.044 < 0.1 means that judgment matrix has the characteristic of consistency and the weight is reasonable. The second level of index and corresponding weight is worked out in the same way. 16 teachers evaluate one student comprehensively, the obtained survey statistics is shown in .

Table 3 The survey statistics table of interuniversity collaborative learning evaluation

*u*_{11} represents whether the learners current level has the condition for acquiring new learning content. *u*_{12} represents whether the learners can adapt to study under interuniversity collaborative learning. *u*_{13} represents learning condition of the learner. *u*_{14} represents learning motivation. *u*_{21} represents the case whether a student can complete learning task according to curriculum plane. *u*_{22} represents the case in which a student takes part in network teaching activities actively. *u*_{23} represents the case in which a students is likely to interact with other partners. *u*_{24} represents the case in which a students can complete some challenging tasks. *u*_{25} represents the autonomous learning ability. *u*_{26} represents the autonomous learning note. *u*_{27} represents a student, who can take part in the necessary tutorial. *u*_{28} represents a student, who can timely submit course assignments. *u*_{29} represents the case in which a student has been cheating. *u*_{31} represents the case in which a student usually asks a teacher. *u*_{32} represents the case in which a student often announces opinion corresponding to curriculum in the discussion area. *u*_{33} represents the case in which a student can extract useful information from others opinion. *u*_{34} represents the case in which a student often talk questions with teachers or students. *u*_{35} represents the case in which a student proposes constructive advice for teacher’s work. *u*_{36} represents the case in which a student can answer the questions timely. *u*_{37} represents the case in which a student can complete subject with other actively learning partners. *u*_{41} represents the case in which a student uploads valuable resource on the system platform. *u*_{42} represents the case in which a student queries information in the resource library. *u*_{43} represents the case in which a student often makes notes. *u*_{44} represents the case in which a student can choose all kinds of resources to study. *u*_{45} represents the case in which a student has strong information ability. *u*_{46} represents the case in which a student can use his knowledge to solve actual problem.

Process of network based interuniversity collaborative learning evaluation is shown in Figure 3. According to , we can obtain fuzzy evaluation matrix of the second level of index,${R}_{1}^{(2)},{R}_{2}^{(2)},{R}_{3}^{(2)}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{R}_{4}^{(2)}$. According to fuzzy evaluation matrix ${R}_{i}^{(2)}(i=1,2,\dots ,n)$ and corresponding weight set ${W}_{i}^{(2)}$, we obtain evaluation result set of each factor in the second level of index.
$${A}_{i}^{(2)}={({W}_{i}^{(2)})}^{T}\cdot {R}_{i}^{(2)}$$

Figure 3 Process of network based interuniversity collaborative learning evaluation

Then we get fuzzy evaluation matrix of the first level of index ${R}^{(1)}={({A}_{1}^{(2)},{A}_{2}^{(2)},{A}_{3}^{(2)},{A}_{4}^{(2)})}^{T}$. According to *A*^{(1)} = (*W* ^{(1)})^{T}· *R*^{(1)}, we work out the comprehensive evaluation result set for the student.

After normalization, *A*^{(1)} = (0.149, 0.342, 0.282, 0.159, 0.068). Then weighted average method is used to determine the comprehensive evaluation results. We need to assign level parameter to each evaluation level, and then level parameter matrix is obtained. The fuzzy comprehensive evaluation result is *S* = *A*^{(1)} · *V*^{T}. *V* = (90, 80, 70, 60, 50). *S* = *A*^{(1)} · *V*^{T} = (0.149, 0.342, 0.282, 0.159, 0.068) · *V*^{T} = 73.

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