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) |