The Study Of Path Optimization Problems Based On Uncertain Information

The path optimisation problem has always been a hot research problem in operations research,in the context of complex and changing reality,the path optimisation problem also shows a tendency to diversify and fuzzy,and this is where the neutrosophic number becomes a powerful tool to describe the uncertain weight information of the path.The neutrosophic language which can effectively portray the uncertainty and inconsistency of information.In this thesis,we combine graph theory to solve the shortest path problem(SPP)and the maximum flow problem(MFP)in neutrosophic environment.In order to avoid the distortion of information caused by traditional operations,a new basic operation on single-valued neutrosophic numbers is proposed and its properties are proved,reflecting the philosophical rigour of the neutrosophic language.The logical notion of a neutrosophic set is defined for the purpose of comparing the "size" of neutrosophic sets,the score function and the exact function of single-valued neutrosophic numbers are given,and the "size" of single-valued neutrosophic numbers is compared using the ranking relations of neutrosophic sets to describe the advantages and disadvantages of path selection.Finally,the reasonableness of the operations is verified by simple arithmetic examples.A decision-making method for SPP in the neutrosophic environment is constructed by considering the correlation between the attributes of neutrosophic numbers and the actual distance of the paths.Based on the concept of mathematical expectation implications of probabilistic languages,the mathematical expectation of single-valued neutrosophic numbers is defined to give a mathematical expectation formula for the set of neutrosophic languages.Based on the defined mathematical expectation,the network node graph is verified,and the shortest path is obtained and its neutrosophic expectation is calculated through the addition operation of single-valued neutrosophic numbers,the score function and the exact function as well as the ranking relationship,taking a simple six-node directed graph as an example,further proving the rationality and validity of the operation.For SPP,the extended Dijkstra’s algorithm and Floyd’s algorithm are proposed in the context of neutrosophy,considering the effects of the degree of truth,uncertainty and fallacy of single-valued neutrosophic number on the path weights.The detailed steps of the two extended algorithms are introduced,the advantages and disadvantages of the two algorithms are analysed,and the feasibility and effectiveness of the shortest path solution with single-valued neutrosophic information are compared with existing neutrosophic methods.For MFP,the uncertainty of the weight information is described in the interval neutrosophic language,the mathematical expectation of the interval neutrosophic set is defined,and the ranking relationship is obtained using the score function and the precise function of the interval neutrosophic number to compare the "size" of the interval neutrosophic number.The mathematical model for the neutrosophic maximum flow problem is constructed,and a volatility formula is given to determine the fluctuation range of the flow.The MFP with interval neutrosophic weight information is solved using the extended augmenting chain algorithm,and the feasibility of the algorithm is analysed.This thesis deals with the path optimization problem in the neutrosophic environment by constructing the operation method of neutrosophic numbers,the mathematical expectation of neutrosophic language sets,using the extended Dijkstra’s algorithm and Floyd’s algorithm to solve SPP and the extended augmenting chain algorithm to solve MFP,and finally comparing and analyzing the effectiveness of the operations,which enriches the knowledge of neutrosophic language theory and expands the solution method of the path optimization problem,and has certain practical significance.