Joint chance-constrained multi-objective multi-commodity minimum cost network flow problem with copula theory

This study focuses on a specific problem in network optimization, namely the minimum cost multi-commodity network flow (MCNF) problem. The problem is complicated by the presence of uncertain parameters, including various types of costs associated with each arc in the network. The study presents a multi-objective approach to solving this problem, where the coefficients of the capacity constraints are modelled as random variables with a normal distribution, and the dependence between them is modelled using an Archimedean copula. The capacity constraints are presented as joint chance constraints, and a multi-objective problem is formulated to deal with the uncertainty. This uncertain multi-objective problem is then converted into a certain multi-objective problem using fuzzy programming. The resulting certain multi-objective MCNF problem is converted to a certain single objective problem using second-order cone programming (SOCP), which is solved using either piecewise tangent approximation or piecewise linear approximation methods. The proposed ap- proaches are tested using numerical examples and experimental tests to demonstrate their effectiveness in solving large-scale network problems efficiently. The results show that the proposed approaches are a useful tool for solving uncertain multi-objective MCNF problems in real-world applications.