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doi: 10.3934/jimo.2021017

## Inverse single facility location problem on a tree with balancing on the distance of server to clients

 Faculty of Mathematical Sciences, Shahrood University of Technology, University Blvd., Shahrood, Iran

* Corresponding author: Jafar Fathali

Received  June 2020 Revised  November 2020 Published  December 2020

We introduce a case of inverse single facility location problem on a tree where by minimum modifying in the length of edges, the difference of distances between the farthest and nearest clients to a given facility is minimized. Two cases are considered: bounded and unbounded nonnegative edge lengths. In the unbounded case, we show the problem can be reduced to solve the problem on a star graph. Then an $O(nlogn)$ algorithm is developed to find the optimal solution. For the bounded edge lengths case an algorithm with time complexity $O(n^2)$ is presented.

Citation: Shahede Omidi, Jafar Fathali. Inverse single facility location problem on a tree with balancing on the distance of server to clients. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021017
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##### References:
The tree $T$ with 10 verteices
The lengths and costs of changing lengths of edges in tree $T$
 $e_i$ $l_i$ $c_i^+$ $c_i^-$ $e_1=(v_1,v_2)$ 3 1 2 $e_2=(v_2,x)$ 3 3 1 $e_3=(x,v_3)$ 5 2 2 $e_4=(v_3,v_{4})$ 2 1 3 $e_5=(v_{4},v_{5})$ 8 1 4 $e_6=(x,v_{6})$ 1 5 3 $e_7=(v_6,v_7)$ 2 3 1 $e_8=(x,v_{8})$ 3 5 2 $e_9=(v_{8},v_{9})$ 4 3 4 $e_{10}=(v_{9},v_{10})$ 3 4 4
 $e_i$ $l_i$ $c_i^+$ $c_i^-$ $e_1=(v_1,v_2)$ 3 1 2 $e_2=(v_2,x)$ 3 3 1 $e_3=(x,v_3)$ 5 2 2 $e_4=(v_3,v_{4})$ 2 1 3 $e_5=(v_{4},v_{5})$ 8 1 4 $e_6=(x,v_{6})$ 1 5 3 $e_7=(v_6,v_7)$ 2 3 1 $e_8=(x,v_{8})$ 3 5 2 $e_9=(v_{8},v_{9})$ 4 3 4 $e_{10}=(v_{9},v_{10})$ 3 4 4
The edge length bounds of the tree $T$
 $e_i$ $l_i$ $\underline{l_i}$ $\bar{l_i}$ $e_1=(v_1,v_2)$ 3 1 4 $e_2=(v_2,x)$ 3 2 6 $e_3=(x,v_3)$ 5 1 7 $e_4=(v_3,v_{4})$ 2 1 5 $e_5=(v_{4},v_{5})$ 8 4 10 $e_6=(x,v_{6})$ 1 1 4 $e_7=(v_6,v_7)$ 2 1 5 $e_8=(x,v_{8})$ 3 1 7 $e_9=(v_{8},v_{9})$ 4 2 8 $e_{10}=(v_{9},v_{10})$ 3 1 6
 $e_i$ $l_i$ $\underline{l_i}$ $\bar{l_i}$ $e_1=(v_1,v_2)$ 3 1 4 $e_2=(v_2,x)$ 3 2 6 $e_3=(x,v_3)$ 5 1 7 $e_4=(v_3,v_{4})$ 2 1 5 $e_5=(v_{4},v_{5})$ 8 4 10 $e_6=(x,v_{6})$ 1 1 4 $e_7=(v_6,v_7)$ 2 1 5 $e_8=(x,v_{8})$ 3 1 7 $e_9=(v_{8},v_{9})$ 4 2 8 $e_{10}=(v_{9},v_{10})$ 3 1 6
The iterations result for Example 2
 Iteration $P_1$ $P_2$ $C_{P_1}$ $C_{P_2}$ $f_1$ $f_2$ 1 $\{p_1\}$ $\{p_2\}$ $c^-_3=2$ $c^+_6=5$ 8 10 2 $\{p_1\}$ $\{p_2,p_3\}$ $c^-_4=3$ - 11 9 3 $\{p_1,p_4\}$ $\{p_2,p_3\}$ $c^-_5+c^-_8=6$ - 23 7 4 $\{p_1,p_4\}$ $\{p_2,p_3\}$ $c^-_5+c^-_9=16$ - 39 5
 Iteration $P_1$ $P_2$ $C_{P_1}$ $C_{P_2}$ $f_1$ $f_2$ 1 $\{p_1\}$ $\{p_2\}$ $c^-_3=2$ $c^+_6=5$ 8 10 2 $\{p_1\}$ $\{p_2,p_3\}$ $c^-_4=3$ - 11 9 3 $\{p_1,p_4\}$ $\{p_2,p_3\}$ $c^-_5+c^-_8=6$ - 23 7 4 $\{p_1,p_4\}$ $\{p_2,p_3\}$ $c^-_5+c^-_9=16$ - 39 5
The optimal edge lengths for Example 2
 $e_i$ optimal $l_i$ $\underline{l_i}$ $\bar{l_i}$ $e_1=(v_1,v_2)$ 3 1 4 $e_2=(v_2,x)$ 3 2 6 $e_3=(x,v_3)$ 1 1 7 $e_4=(v_3,v_{4})$ 1 1 5 $e_5=(v_{4},v_{5})$ 4 4 10 $e_6=(x,v_{6})$ 1 1 4 $e_7=(v_6,v_7)$ 2 1 5 $e_8=(x,v_{8})$ 1 1 7 $e_9=(v_{8},v_{9})$ 2 2 8 $e_{10}=(v_{9},v_{10})$ 3 1 6
 $e_i$ optimal $l_i$ $\underline{l_i}$ $\bar{l_i}$ $e_1=(v_1,v_2)$ 3 1 4 $e_2=(v_2,x)$ 3 2 6 $e_3=(x,v_3)$ 1 1 7 $e_4=(v_3,v_{4})$ 1 1 5 $e_5=(v_{4},v_{5})$ 4 4 10 $e_6=(x,v_{6})$ 1 1 4 $e_7=(v_6,v_7)$ 2 1 5 $e_8=(x,v_{8})$ 1 1 7 $e_9=(v_{8},v_{9})$ 2 2 8 $e_{10}=(v_{9},v_{10})$ 3 1 6
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