In view of the existing literature panel data factor analysis model in practical application of the deficiency, this paper established the model of factor analysis based on TOPSIS method, which is applied to the analysis of the panel data factor in practice. Compared with the generalized dynamic factor analysis model, the model does not need to satisfy the 4 assumptions of the generalized dynamic factor analysis model at the same time. The model is calculated with regard to every year’s cross section data factor composite scores the highest and lowest, respectively, for the best and worst vector. By TOPSIS theory, the optimal factor scheme approach degree of each research object is obtained. Take the development of China’s service industry as an example; use the optimal factor scheme proximity of model degree to depict the eastern, central, and western development of service industry. The study found that the development of service industry in eastern provinces and in central and western regions differs greatly. In total, China’s service industry has a great development space.

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The length of each clause in a regular (s, c, k) − CN F formula is k. Each argument occurrences s times, among them, positive occurrences (c * s) times and negative occurrences (s − c * s) times, where 0 < c < 1. A regular exact (s, c, k) − SAT question refers to whether there is a set of Boolean variable assignment such that exactly one literal in each clause of the regular (s, c, k) − CN F formula is true. Obviously, the problem is a NP-hard problem. To understand the hardness and the distribution of satisfiable solutions of regular exact (s, c, k) − SAT problem, we introduce a random instance generation model, use the first moment and second moment methods to analyze the satisfiable phase transition of the solutions. Set s * is the phase transition point, we show that the stochastic regular exact (s, c, k) − SAT problem instance is satisfiable with high probability if s < s * and unsatisfiable with high probability if s > s * , among them, s * is a function about a parameter c. Further, we discuss the phase transition point of the random regular exact (s, r, k) − SAT problem-the difference between the positive and negative occurrences of the variable is r-is s *. Finally, through several groups of experimental data to verify, the experimental results are consistent with the theory, and prove the correctness of the theory. INDEX TERMS first moment method, phase transition, satisfiability problem, second moment method.

In a general Markov decision progress system, only one agent’s learning evolution is considered. However, considering the learning evolution of a single agent in many problems has some limitations, more and more applications involve multi-agent. There are two types of cooperation, game environment among multi-agent. Therefore, this paper introduces a Cooperation Markov Decision Process (CMDP) system with two agents, which is suitable for the learning evolution of cooperative decision between two agents. It is further found that the value function in the CMDP system also converges in the end, and the convergence value is independent of the choice of the value of the initial value function. This paper presents an algorithm for finding the optimal strategy pair (πk0,πk1) in the CMDP system, whose fundamental task is to find an optimal strategy pair and form an evolutionary system CMDP(πk0,πk1). Finally, an example is given to support the theoretical results.