## 5 Conclusions  
In this paper, by incorporating the influence matrix into the M-MOPSO algorithm and establishing a mathematical model, the initial state solution obtained by the traditional method is further optimized to obtain a series of Pareto front solution sets. The fuzzy set theory is then then adopted to select the optimum, and the final reasonable constant  
Figure 17: Main beam bending moment before and after optimization. Figure 18: Stress values of cable before and after optimization. load bridge state of the cable-stayed bridge is obtained. The introduced method is validated in the process of solving the reasonable state of a single-tower cable-stayed-T-frame combination bridge. The following conclusions can be drawn from the full study. 1. The cable force optimization of a cable-stayed bridge can be obtained objectively and accurately by the SOM practical method of fuzzy set theory. Midas/Civil and MATLAB are selected as the structure calculator and cable force optimizer, respectively, and M-MOPSO is used as the calculation kernel to effectively realize the cable force optimization of cable-stayed bridges. This can realize a variety of optimization scheme comparisons, especially suitable for the preliminary design stage. 2. The optimized cable-stayed bridge has more uniform force and more reasonable alignment. The peak vertical displacement of the optimized main girder is reduced from \(-11\) mm to \(-6\) mm, with a reduction of 45%. 3. Compared with the classical algorithm NSGA-II, the modified multiobjective particle swarm optimization algorithm with mutation operation (M-MOPSO algorithm) has advantages in both convergence and distribution in dealing with multiobjective problems. It is an effective algorithm for dealing with multiobjective optimization problems. 4. The M-MOPSO algorithm is influenced by the range of initial tension values, so the SOM method with its kernel has a certain sensitivity to initial populations. Once the initial values are within a reasonable range, even if all the initial values of the diagonal cables are taken to be the same, a more reliable and reasonable bridge state can be obtained. Conceptualization, L.W. and Z.X. ; methodology, L.W. and Z.X. ; software, L.W., Z.X. and M.L. ; validation, L.W. and M.L. ; formal analysis, Z.X. and L.W. ; data curation, M.L. ; writing--review and editing, Z.X. and N.F. ; project administration, L.W. and N.F. ; funding acquisition, L.W. All authors have read and agreed to the published version of the manuscript. This study is supported by four key funds: The Key Research and Development Program of Heilongjiang Province, China (JD22A012, GZ20210206); China Railway 22 Bureau Group Co., Ltd. Science and technology research project (20-01A); The Fundamental Research Funds for the Central Universities, China (2572020 AW53, 2572019 CP02). This study did not require ethical approval. Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. The authors declare no conflict of interest. [PERSON]; [PERSON]; [PERSON]; [PERSON]. Determination of initial cable forces in prestressed concrete cable-stayed bridges for given design deck profiles using the force equilibrium method. _Comput. Struct._**2000**, _74_, 1-9. [CrossRef]
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