Abstract
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.
Highlights
The quest for studying network optimization problems has been emphasized for long time
It can be inferred that for all the instances of uncertain multi-objective minimum spanning tree problem (UMMSTP), the medians of the performance metrics are better for DENSEA compared to non-dominated sorting genetic algorithm II (NSGAII)
We propose a UMMSTP, which optimized the uncertain parameters in terms of ξ cij and ξ tij in a minimum spanning tree
Summary
The quest for studying network optimization problems has been emphasized for long time. The existence of sufficient historical data for any problem parameter motivates us to possibly estimate its distribution function and the analysis of the problem can be performed by following the fundamental principle of probability theory In this context, for the first time, Ishii et al [16] modeled an MSTP with edges having random weights whose probability distributions are not known. Uncertainty theory has been progressively developed as an important area of mathematics, which can express and model human uncertainty In this aspect, to tackle optimization problems with uncertain parameters, the concept of uncertain programming is presented by Liu [23]. A multi-objective minimum spanning tree problem with indeterminate parameters based on uncertainty theory is studied. We summarize all the abbreviations related to this study in Appendix A
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