## 7 Simulation Result
### Mscoco  
Analyzing the baseline performance (B) in Tables 4 and 5, it is evident that Qwen-2 achieves the highest performance across most evaluation metrics. This is followed by Pixtral, Llama 3.2, and Phi-3.5 in descending order of performance. Notably, Llama 3.2 exhibits the best results in semantic similarity metrics (S-BERT, MoverScore), suggesting that the generated captions are likely to be more semantically appropriate. Meanwhile, Phi-3.5 achieves a higher CIDEr score than Pixtral and Llama 3.2, although it records the lowest performance in other metrics.  
However, when applying the Greedy PES, the optimal model \(m^{*}\) and optimal prompt engineering strategy \(p^{*}\) are found to be Phi-3.5 with the base RAG technique. This indicates that although Phi-3.5 initially exhibited the lowest baseline performance, it outperforms all other models when Greedy PES is applied. This underscores the significant impact of PE on MLLM performance. Additionally, it is noteworthy that Phi-3.5, despite being the smallest model at 4.2B parameters, achieves superior performance compared to larger models when optimized using Greedy PES. Furthermore, following Phi-3.5, the models rank in performance as Qwen-2, Pixtral, and Llama 3.2. Interestingly, Qwen-2, which had the lowest baseline performance, demonstrates the second-best performance under Greedy PES. The BLEU score improvement for Qwen-2 through Greedy PES is nearly tenfold, highlighting the effectiveness of prompt engineering optimization. On the other hand, for the CIDEr metric, the ToT technique proves to be the most effective, with Qwen-2 emerging as the best-performing model.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Method** & **Phi** & **Llama** & **Pixtral** & **Qwen** & **Phi** & **Llama** & **Pixtral** & **Qwen** & **Phi** & **Llama** & **Pixtral** & **Qwen** \\ \hline
**Metric** & \multicolumn{4}{c}{**BLEU**} & \multicolumn{4}{c}{**ROUGE**} & \multicolumn{4}{c}{**METEOR**} \\ \hline B & 1.5927 & 2.4344 & 2.9209 & 3.7367 & 0.072 & 0.1085 & 0.1793 & 0.2487 & 0.1076 & 0.2057 & 0.2587 & 0.2712 \\ \hline I & 17.7844 & **14.6007** & 3.3469 & 6.787 & 0.436 & **0.4217** & 0.1689 & 0.245 & 0.401 & **0.4086** & 0.3237 & 0.2702 \\ \hline I, S & 8.8645 & 5.4292 & 1.5041 & 5.7771 & 0.3107 & 0.2054 & 0.0812 & 0.2283 & 0.315 & 0.2924 & 0.164 & 0.2321 \\ \hline I, T & 9.3105 & 2.8614 & 3.1159 & 7.735 & 0.3255 & 0.1548 & 0.1621 & 0.2188 & 0.3209 & 0.2393 & 0.3087 & 0.2509 \\ \hline C & 14.9965 & 8.8183 & 7.8643 & 12.9066 & 0.4574 & 0.296 & 0.3082 & 0.4551 & 0.4987 & 0.3827 & 0.3785 & 0.4933 \\ \hline C, I & 15.4373 & 5.7162 & 4.0372 & 15.3999 & 0.4748 & 0.2609 & 0.2184 & 0.4695 & 0.5114 & 0.3529 & 0.3134 & 0.4955 \\ \hline C, S & 14.4889 & 5.5974 & 12.3729 & 11.0606 & 0.4549 & 0.2188 & 0.433 & 0.4087 & 0.5045 & 0.3144 & 0.4836 & 0.4507 \\ \hline C, T & 15.018 & 4.385 & 1.5935 & 15.391 & 0.4558 & 0.1986 & 0.085 & 0.467 & 0.4919 & 0.2823 & 0.1383 & 0.5121 \\ \hline S & 15.1778 & 4.6746 & 1.8906 & 12.3664 & 0.4532 & 0.2049 & 0.1125 & 0.451 & 0.5018 & 0.3039 & 0.1965 & 0.4989 \\ \hline T & 16.6347 & 4.4104 & **13.4563** & **17.5892** & 0.4788 & 0.1796 & **0.4629** & **0.4869** & 0.5112 & 0.2819 & **0.4974** & **0.5411** \\ \hline R(B) & **27.8553** & 10.1012 & 5.0772 & 3.0467 & **0.5559** & 0.3387 & 0.2108 & 0.1355 & **0.5325** & 0.3803 & 0.3794 & 0.1237 \\ \hline R(C) & 14.0141 & 8.4676 & 4.3829 & 2.982 & 0.443 & 0.3169 & 0.1953 & 0.1317 & 0.4767 & 0.3749 & 0.3413 & 0.1319 \\ \hline R(S) & 2.2911 & 6.0903 & 2.8862 & 4.1465 & 0.1148 & 0.33 & 0.1479 & 0.1408 & 0.1253 & 0.3382 & 0.2539 & 0.1623 \\ \hline R(T) & 13.3791 & 3.5638 & 4.4854 & 3.1019 & 0.4161 & 0.1586 & 0.1981 & 0.1195 & 0.4306 & 0.2428 & 0.362 & 0.1186 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performances of BLEU, ROUGE, and METEOR according to MLLM models and prompt engineering techniques on the MSCOCO dataset (The boldfaced numbers indicate the highest performance for each language model across different PES variants).