library(igraph)
g <- graph_from_literal(Origin -- One:Two:Three, One:Two:Three -- A:B:C:D, A:B:C:D -- I:II:III, I:II:III -- W:X:Y:Z, W:X:Y:Z -- Destination)
E(g)$weight <- c(0,0,0,4,3,6,5,5,5,7,5,2,5,3,4,6,4,6,5,3,7,3,5,3,4,5,8,4,3,6,5,5,5,7,5,2,5,3,4,0,0,0,0)
E(g)$capacity <- c(200,250,300,rep(c(120,200,200,150),3),rep(c(250,200,150),4),rep(c(50,190,120,200),3),50,190,120,200)

max_flow(g,"Origin","Destination")

[1] 200 210 150   0   0  50 150   0   0 100 110   0   0 150   0   0   0   0   0
[20]   0   0   0 150 150 100  60 100  50   0   0  50   0   0 120  90   0 190   0
[39]  60  50 190 120 200