July 2006 Volume 9 Number 3 - CiteSeerX

To analyze the convergence behavior of the particles, we testify whether the swarm evolves to the same
optimization goal. We propose the information entropy for measuring the similarity convergence among the
particles as follows. Let pij be the binary value of the jth bit for the ith particle, i = 1, 2, …, R, and j = 1, 2, …,
NK, where R is the swarm size. We can calculate probj as the conditional probability that value one happens at
the jth bit given the total number of bits that take value one in the entire swarm as follows.
prob
R
∑i=
∑ ∑
j = R
p
1 ij .
NK
i=
1 h=
1
The particle entropy can be then defined as
Entropy = −
NK
prob log prob .
∑ j=
1
p
ih
j
2
( )
j
The particle entropy is smaller if the probability distributions are denser. As such, the variations of particle
entropy during the swarm evolution measure the convergence about the similarity among all particles. If the
particles are highly similar to one another, the values of the non-zero probj would be high, resulting in denser
probability distributions and less entropy value. This also means the swarm particles reach the consensus about
which test items should be selected for composing the test sheets.
Figure 6 shows the variations of particle entropy as the number of generations increases. It is observed that the
entropy value drops drastically during the first 18 generations since the particles exchange information by
referring to the swarm’s best solution. After this period, the entropy value is relatively fixed due to the good
quality solutions found and the high similarity among the particles, meaning the particles are resorting to the
same high quality solution as the swarm converges.
6. Conclusions and Future work
Figure 6. The particle entropy as the number of generations increases
In this paper, a particle swarm optimization-based approach is proposed to cope with the serial test sheet
composition problems. The algorithm has been embedded in an intelligent tutoring, evaluation and diagnosis
system with large-scale test banks that are accessible to students and instructors through the World-Wide Web.
To evaluate the performance of the proposed algorithm, a series of experiments have been conducted to compare
the execution time and the solution quality of three solution-seeking strategies on twelve item banks.
Experimental results show that serial test sheets with near-optimal average difficulty to a specified target value
can be obtained with reasonable time by employing the novel approach.
For further application, collaborative plans with some local e-learning companies are proceeding, in which the
present approach is used in the testing and assessment of students in elementary school and junior high schools.
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