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The pursuit of studying the quadratic minimum spanning tree (QMST) problem has captivated numerous academics because of its distinctive characteristic of taking into account the cost of interaction between pairs of edges. A QMST refers to the minimum spanning tree, which is a graph that is both acyclic and minimally connected. It also includes the interaction cost between a pair of edges in the minimum spanning tree. These interaction costs can occur between any pair of edges, whether they are adjacent or non-adjacent. In the QMST problem, our objective is to minimize both the cost of the edges and the cost of interactions. This eventually classifies the task as NP-hard. The interaction costs, sometimes referred to as quadratic costs, inherently exhibit a contradictory relationship with linear edge costs when solving a multi-objective problem that aims to minimize both linear and quadratic costs simultaneously. This study addresses the bi-objective adjacent only quadratic minimum spanning tree problem (AQMSTP) by incorporating the uncertain nature of the linear and quadratic costs associated with the problem. The focus is on the interaction costs between adjacent edges. Consequently, we have introduced a multi-objective problem called the uncertain adjacent only quadratic minimum spanning tree problem (mUAQMSTP) and formulated it using the uncertain chance-constrained programming technique. Afterwards, two MOEAs—non-dominated sorting genetic algorithm II (NSGAII) and duplicate elimination non-dominated sorting evolutionary algorithm (DENSEA)—and the traditional multi-objective solution approach, the global criterion method, are employed to solve the deterministic transformation of the model. Finally, we provide a suitable numerical illustration to substantiate our suggested framework. Full article

Minimum spanning tree problem (MSTP) has allured many researchers and practitioners due to its varied range of applications in real world scenarios. Modelling these applications involves the incorporation of indeterminate phenomena based on their subjective estimations. Such phenomena can be represented rationally using uncertainty theory. Being a more realistic variant of MSTP, in this article, based on the principles of the uncertainty theory, we have studied a multi-objective minimum spanning tree problem (MMSTP) with indeterminate problem parameters. Subsequently, two uncertain programming models of the proposed uncertain multi-objective minimum spanning tree problem (UMMSTP) are developed and their corresponding crisp equivalence models are investigated, and eventually solved using a classical multi-objective solution technique, the epsilon-constraint method. Additionally, two multi-objective evolutionary algorithms (MOEAs), non-dominated sorting genetic algorithm II (NSGAII) and duplicate elimination non-dominated sorting evolutionary algorithm (DENSEA) are also employed as solution methodologies. With the help of the proposed UMMSTP models, the practical problem of optimizing the distribution of petroleum products was solved, consisting in the search for symmetry (balance) between the transportation cost and the transportation time. Thereafter, the performance of the MOEAs is analyzed on five randomly developed instances of the proposed problem. Full article