This morning, we had our first practicals on network flows, using  an example mentioned in some papers published by Noraini Abdullah and Ting Kien Hua, max flow min cut theorem to minimize traffic congestion in Kota Kinabalu and application of the Shortest Path and Maximum Flow with Bottleneck in Traffic Flow of Kota Kinabalu. From the roads mentioned in the articles, I did try my best to locate the nodes on a map,

m=matrix(c(0,5.995910, 116.105520,
1,5.992737, 116.093718,
2,5.992066, 116.109883,
3,5.976947, 116.095760,
4,5.985766, 116.091580,
5,5.988940, 116.080112,
6,5.968318, 116.080764,
7,5.977454, 116.075460,
8,5.974226, 116.073604,
9,5.969651, 116.073753,
10,5.972341, 116.069270,
11,5.978818, 116.072880),3,12)

we can be visualized below

library(OpenStreetMap)
map = openmap(c(lat= 6.000, lon= 116.06),
c(lat= 5.960, lon= 116.12))
map=openproj(map)
plot(map)
points(t(m[3:2,]),col="black", pch=19, cex=3 )
text(t(m[3:2,]),c("s",1:10,"t"),col="white")

If the source is realistic (up north), I do not feel very confortable with the location of the sink (on the west). But let’s pretend it’s find (to do the maths, at least).

To extract information about edge capacity, on that network use the following code that will extract the three tables from the paper

library(devtools)
install_github("ropensci/tabulizer")
library(tabulizer)
location <- 'http://www.jistm.com/PDF/JISTM-2017-04-06-02.pdf'
out <- extract_tables(location)

with Windows, it seems to be necessary to download another package first

library(devtools)
install_github("ropensci/tabulizerjars")
install_github("ropensci/tabulizer")
library(tabulizer)
location <- 'http://www.jistm.com/PDF/JISTM-2017-04-06-02.pdf'
out <- extract_tables(location)

Now we can get out data frame with capacities

B1=as.data.frame(out[[2]])
B2=as.data.frame(out[[3]])
E=data.frame(from=B1[3:20,"V3"],
to=B1[3:20,"V4"])
E=E[-c(6,8),]
capacity=as.character(B2$V3[-1])
capacity[6]="843"
capacity[4]="2913"
E$capacity=as.numeric(capacity)

We can add those edges on our map (without the arrows to indicate the direction, it would be to heavy to read)

plot(map)
points(t(m[3:2,]),col="black", pch=19, cex=3 )
B=data.frame(i=as.character(c("s",paste("V",1:10,sep=""),"t")),
x=m[3,],y=m[2,])
for(i in 1:nrow(E)){
i1=which(B$i==as.character(E$from[i]))
i2=which(B$i==as.character(E$to[i]))
segments(B[i1,"x"],B[i1,"y"],B[i2,"x"],B[i2,"y"],lwd=3)
}
text(t(m[3:2,]),c("s",1:10,"t"),col="white")

To get the graph with capacities, an alternative is to use

library(igraph)
g=graph_from_data_frame(E)
E(g)$label=E$capacity
plot(g)

but it does not respect geographical locations of nodes. It can actually be done using

plot(g, layout=as.matrix(B[,c("x","y")]))

To get a better understanding of the capacities of the road, use

plot(g, layout=as.matrix(B[,c("x","y")]),
edge.width=E$capacity/200)

From that network with capacities, the goal is to determine maximum flow on that network, from the source to the sink. This can be done with R using

> (m=max_flow(graph=g, source="s", target="t"))
$value
[1] 2571

$flow
[1] 1191 1380 1422 1380 231 0 231 0 1149 1422 1149 0 0 1149 1422
[16] 1149

Our maximum flow is here 2571, which is different from was is actually claimed both in the two papers  max flow min cut theorem to… and application of the Shortest Path… (“the maximum flow for the capacitated network with 12 nodes and 16 edges of the selected scope in this study was 2598 vehicles per hour“) where there are clearly typos since values in the table and on the graph are different. Here I did use the ones from the tables.

E$flux1=m$flow
E(g)$label=E$flux1
plot(g, layout=as.matrix(B[,c("x","y")]),
edge.width=E$flux1/200)

That is nice, but rather odd. Actually, a much simpler flow can be considered, but the same global value

E$flux2=c(1422,1149,1422,1149,0,0,0,0,
1149,1422,1149,0,0,1149,1422,1149)
E(g)$label=E$flux2
plot(g, layout=as.matrix(B[,c("x","y")]),
edge.width=E$flux2/200)

Nice, isn’t it. It is actually possible to do exactly the same on another paper they have, on the same city, traffic congestion problem of road networks in Kota Kinabalu.

location <- 'http://www.worldresearchlibrary.org/up_proc/pdf/999-150486366625-30.pdf'
out <- extract_tables(location)
dim(out[[3]])
B1=as.data.frame(out[[3]])
E=data.frame(from=B1[2:61,"V2"],
to=B1[2:61,"V3"],
capacity=B1[2:61,"V4"])
E$capacity=as.numeric(
as.character(E$capacity))
library(igraph)
g=graph_from_data_frame(E)
m=max_flow(graph=g,
source="S",
target="T")
E$flux1=m$flow
E(g)$label=E$flux1
plot(g,
edge.width=E$flux1/200,
edge.arrow.size=0.15)

Here the value of the maximal flow is 4017, just as they found in the original paper

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