To address the issues of uneven initial positions and low search accuracy in the traditional Puma (puma optimizer), TPuma (Tent multi-strategy puma optimizer) was proposed. The Tent chaotic mapping was adopted to replace the traditional pseudo-random number generation for the initial population, thereby enhancing the diversity of the population. On the basis of the original position update formula, a new update mechanism was introduced to enable individuals to effectively escape from local optima during the iterative process. Additionally, the Levy flight mechanism was integrated into the original algorithm to balance the selection probabilities in the exploration and exploitation phases, allowing the algorithm to quickly find the optimal value. The Wilcoxon test was used to determine whether there are significant differences between Puma and TPuma, and the steady-state distribution histogram of the Markov chain was used to visually observe the probability distribution of different states. Benchmark test functions and two classic engineering design problems were used for simulation experiments and compared with other popular algorithms. The simulation results show that after adding the Tent chaotic mapping and the Levy flight mechanism, the algorithm exhibits superior convergence speed and higher optimization accuracy.