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Table 1 Summary of results

From: An algorithm for solving the bi-objective median path-shaped facility on a tree network

 Bi-objective location problem 
 Algorithm proposed in [23]Algorithm proposed in this paper
Model\(\min \limits _{P\in \mathcal {P}}(M^{1}(P),D^{2}(P)),\) where\(\min \limits _{P\in \mathcal {P}}(D^{1}(P),D^{2}(P)),\) where
 \(M^{1}(P)=\max \limits _{v_{j} \in V} w_{j}^{1}d(v_{j},P),\)\(D^{1}(P)=\sum \limits _{v_{j} \in V} w_{j}^{1}d(v_{j},P),\)
 \(D^{2}(P)=\sum \limits _{v_{j} \in V} w_{j}^{2}d(v_{j},P)\)\(D^{2}(P)=\sum \limits _{v_{j} \in V} w_{j}^{2}d(v_{j},P)\)
Used approachε-constraint methodTwo-phase method
SubproblemsCent-dian problem of the formFor supported Pareto paths \(\mathcal {P}_{SE }\)
 \(\begin {array}{c}\min \limits _{P \in \mathcal {P}} D^{2}(P) ~~~~~~ \\\text {s.t.} M^{1}(P)=\epsilon \end {array} \)\(\min \limits _{P \in \mathcal {P}} \lambda D^{1}(P)+(1-\lambda)D^{2}(P)\)
  For unsupported Pareto paths
  \(\min \limits _{P\in \mathcal {P}\backslash \mathcal {P}_{SE }}\lambda D^{1}(P)+(1-\lambda)D^{2}(P)\)
ComplexityO(n logn)O(n logn)