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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 approach | ε-constraint method | Two-phase method |

Subproblems | Cent-dian problem of the form | For 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 | ||

\(\min \limits _{P\in \mathcal {P}\backslash \mathcal {P}_{SE }}\lambda D^{1}(P)+(1-\lambda)D^{2}(P)\) | ||

Complexity | O(n logn) | O(n logn) |