Multiobjective Optimization of Evacuation Routes in Stadium Using Superposed Potential Field Network Based ACO

Algorithm 4

Superposed potential field based Roulette Wheel Method.

S1. Calculate the distance to neighbor nodes.

  The distance 𝐷_𝑗𝑘 between the current node 𝑠_𝑗 and allowed visit neighbor node 𝑠_𝑘 is given by:

𝐷_𝑗𝑘=𝑥_𝑗−𝑥_𝑘²+𝑦_𝑗−𝑦_𝑘²,𝑠_𝑘∈𝑆_𝐶,

where, 𝑥_𝑘,𝑦_𝑘 is the coordinate of the neighbor node 𝑠_𝑘; 𝑥_𝑗,𝑦_𝑗 is the coordinate of the current node 𝑠_𝑗; 𝐷_𝑗𝑘 is the length of link

𝑗𝑘.

S2. Calculate the congestion degrees of neighbor nodes.

  The allowed visit neighbor node 𝑠_𝑘’s congestion degree CD_𝑘(𝑡) at the 𝑡th time step is given by:

CD_𝑘(𝑡)=𝑁_𝐸𝑘(𝑡)𝑐_𝑘,𝑠_𝑘∈𝑆_𝐶,

where, 𝑁_𝐸𝑘(𝑡) is the number of evacuees in neighbor node 𝑠_𝑘 at the 𝑡th time step. 𝑐_𝑘 is the capacity of node 𝑠_𝑘.

S3. Calculate transition probability from node 𝑠_𝑗 to 𝑠_𝑘

  The transition probability 𝑃^𝑖_𝑗𝑘(𝑚,𝑛,𝑡) from node 𝑠_𝑗 to 𝑠_𝑘 at the 𝑡th time step is given by:

𝑃^𝑖_𝑗𝑘()=𝜏𝑚,𝑛,𝑡^𝛼𝑗𝑘(𝑚,𝑛)𝜂^𝛽_𝑗𝑘(𝑚,𝑛,𝑡_{𝑠𝑤∈𝑆𝐶})𝜏^𝛼_𝑗𝑤(𝑚,𝑛)𝜂^𝛽_𝑗𝑤(𝑚,𝑛,𝑡),𝑠_𝑘∈𝑆_𝐶0,otherwise,

where, 𝑠_𝑗 is the subzone which the ant 𝑖 is in currently. 𝜏^𝛼_𝑖𝑗(𝑚,𝑛) is the pheromone amount on connection 𝑖𝑗 at 𝑚th generation

under 𝑛th pheromone vector; 𝜂^𝛽_𝑖𝑗(𝑚,𝑛,𝑡) is the heuristic information related with link 𝑖𝑗 at 𝑡th time step, under 𝑛th pheromone

vector, at 𝑚th generation; 𝑆_𝐶 is the set of candidate nodes; 𝛼 and 𝛽 are the parameters to control the relative importance between

the pheromone and the heuristic information.

  The heuristic information 𝜂_𝑖𝑗(𝑚,𝑛,𝑡) on link 𝑖𝑗 at 𝑡th time step is given by:

𝜂_𝑖𝑗()=1𝑚,𝑛,𝑡𝐷_𝑖𝑗∗𝑁_𝐸𝑗(𝑚,𝑛,𝑡)+1/𝐶_𝑗+1,

where, 𝐷_𝑖𝑗 is the length of link 𝑖𝑗; 𝑁_𝐸𝑗(𝑚,𝑛,𝑡) is the number of evacuees in node 𝑗 at the 𝑡th time step; 𝐶_𝑗 is the capacity of node 𝑗.

S4. Select one of candidate nodes according to cumulative transition probability

According to roulette wheel selection, the node 𝑠_𝑘 would be selected only and if only when

𝑃𝑃_{𝑗(𝑘−1)}(𝑚,𝑛,𝑡)<rand≤𝑃𝑃_𝑗𝑘(𝑚,𝑛,𝑡), rand is a random real number between 0 and 1. 𝑃𝑃_𝑗𝑘(𝑚,𝑛,𝑡) is the cumulative transition

probability, which is given by:

𝑃𝑃_𝑗𝑘(𝑚,𝑛,𝑡)=∑^𝑘_𝑤=0𝑃_𝑗𝑤(𝑚,𝑛,𝑡),𝑠_𝑘∈𝑆_𝐶

Besides, we rule that 𝑃𝑃_𝑗0(𝑚,𝑛,𝑡)=0.