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Article Contents

# Dynamic network flow location models and algorithms for quickest evacuation planning

• * Corresponding author: Urmila Pyakurel
• Dynamic network flow problems have wide applications in evacuation planning. From a given subset of arcs in a directed network, choosing the suitable arcs for facility location with a given objective is very important in the optimization of flow in emergency cases. Because of the decrease in capacity of an arc by placing a facility in it, there may be a reduction in the maximum flow or increase in the quickest time. In this work, we consider a problem of identifying the optimal facility locations so that the increase in the quickest time is minimum. Introducing the quickest FlowLoc problem, we give strongly polynomial time algorithms to solve the single facility case. Realizing NP-hardness of the multi-facility case, we develop a mixed integer programming formulation of it and propose two polynomial time heuristics for its solution. Because of the growing concerns of arc reversals in evacuation planning, we introduce the quickest ContraFlowLoc problem and present exact algorithms to solve the single-facility case and heuristics to solve the multi-facility case, with polynomial time complexity. The solutions thus obtained here are practically important, particularly in evacuation planning, to systematize traffic flow with facility allocation minimizing evacuation time.

Mathematics Subject Classification: Primary: 90B10, 90C27, 68Q25; Secondary: 90B06, 90B20.

 Citation:

• Figure 1.  Evacuation network $N$ with arc labels (capacity, travel time)

Figure 2.

Figure 3.  Auxiliary network $\bar{N}$ of network $N$ in Figure 1 with arc labels (capacity, travel time)

Table 1.  Maximum dynamic FlowLoc decisions (cf. Figure 1)

 $T$ maximum dynamic flow value when facility is placed on Location decision (2, 3) (2, 4) 4 0 1 (2, 4) 5 4 5 (2, 4) 6 11 11 (2, 4) or (2, 3) 7 18 17 (2, 3)

Table 2.  Quickest FlowLoc decisions (cf. Figure 1)

 $F$ quickest time when facility is placed on Location decision (2, 3) (2, 4) 1 4.25 4 (2, 4) 5 5.14 5 (2, 4) 11 6 6 (2, 4) or (2, 3) 21 7.43 7.67 (2, 3)

Table 4.  Computational results for some instances with $F = 20000$

 $|L|$ $|P|$ Algorithm 4 Algorithm 5 MILP R. time $T^*$ R. time $T^*$ R. time $T^*$ 5 5 0.14 3209.3 0.53 2717.6 0.87 2713.5 5 7 0.16 2831.9 0.49 2707.6 1.39 2707.1 6 7 0.13 3222.1 0.65 2881.2 1.93 2878.1 8 10 0.15 2580.0 0.58 2575.0 2.92 2575.0 8 8 0.14 3189.3 0.47 3189.3 2.01 3189.3 9 10 0.15 2725.9 0.66 2593.3 3.46 2593.3 10 10 0.17 2587.8 0.73 2585.6 8.01 2585.6 11 10 0.11 2847.5 0.81 2573.9 5.69 2573.9 11 11 0.17 2586.7 0.83 2580.6 15.16 2580.6 11 15 0.20 2721.8 0.72 2710.6 52.52 2707.6 12 11 0.15 2708.2 0.73 2585.0 6.7 2585.0 12 15 0.17 2877.5 0.69 2881.2 114.77 2877.5 13 15 0.12 2588.3 0.51 2576.1 2.82 2576.1 14 15 0.24 2578.3 0.72 2579.4 645.67 2576.1 14 21 0.11 2861.9 0.69 2712.4 134.76 2595.0 15 17 0.16 3005.3 0.98 2843.1 224.36 2843.1 15 20 0.16 3036.7 1.10 2733.5 47.37 2733.5 16 17 0.16 2736.5 1.06 2722.4 221.88 2722.4 17 19 0.19 2851.9 1.29 2717.1 379.54 2716.5 18 19 0.20 2581.7 0.97 2583.3 214.78 2581.1 18 19 0.16 2589.4 1.18 2585.6 822.64 2585.6 19 20 0.24 2595.0 1.93 2607.2 - - 19 24 0.59 3032.7 1.76 2868.8 - - 20 25 0.18 2707.6 1.25 2707.6 - - 21 25 0.15 2577.8 1.21 2577.8 - - 23 25 0.14 2692.9 1.47 2575.0 - - 23 26 0.14 2705.3 1.05 2705.3 - - 24 26 0.20 3026.0 1.87 2863.1 - - 26 29 0.14 2704.7 1.42 2704.7 - - 28 40 0.20 2585.0 1.14 2585.0 - - 28 39 0.17 2586.7 1.20 2586.7 - - 29 34 0.16 2851.2 1.42 2851.2 - - 30 40 0.24 2588.3 1.55 2588.3 - - 31 38 0.17 2851.2 1.75 2851.2 - - 31 37 0.22 2600.0 1.57 2600.0 - - 32 40 0.17 2706.5 1.54 2706.5 - - 37 43 0.22 2701.8 1.77 2701.8 - - 38 41 0.21 2597.2 1.44 2598.9 - - 39 50 0.25 2710.0 1.95 2710.0 - - 40 50 0.09 2573.3 1.76 2571.7 - - 41 48 0.20 2710.6 2.19 2710.6 - -

Table 3.  Percentage Deviation from the MILP objective function values

 Algorithm 4 Algorithm 5 Maximum deviation 21.55% 5.31 % Average deviation 3.48 % 0.18%

Table 5.  Quickest time calculations (cf. Example 4)

 Quickest time, $F = 109$ Facility placed on Before contraflow After contraflow $(2,1)$ 20 15 $(1,3)$ 25.8 14.27 Location Decision $(2,1)$ $(1,3)$
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