On solving bi-objective constrained minimum spanning tree problems

This paper investigates two approaches for solving bi-objective constrained minimum spanning tree problems. The first seeks to minimize the tree weight, keeping the problem’s additional objective as a constraint, and the second aims at minimizing the other objective while constraining the tree weight. As case studies, we propose and solve bi-objective generalizations of the Hop-Constrained Minimum Spanning Tree Problem (HCMST) and the Delay-Constrained Minimum Spanning Tree Problem (DCMST). First, we present an Integer Linear Programming (ILP) formulation for the HCMST. Then, we propose a new compact mathematical model for the DCMST based on the well-known Miller–Tucker–Zemlin subtour elimination constraints. Next, we extend these formulations as bi-objective models and solve them using an Augmented $$\epsilon $$ ϵ -constraints method. Computational experiments performed on classical instances from the literature evaluated two different implementations of the Augmented $$\epsilon $$ ϵ -constraints method for each problem. Results indicate that the algorithm performs better when minimizing the tree weight while constraining the other objective since this implementation finds shorter running times than the one that minimizes the additional objective and constrains the tree weight.