I am currently developing the ompr package to model mixed integer linear programs. In order to debug the package I try to implement various optimization models and see how the package behaves.
Boris Johnson said in an interview that “it would really take me too long to engage in a fully global itinerary of apology to all concerned” [reuters]. Indy100 compiled a map of all countries he has offended so far.
So it seems like this would be a fun example to test a larger TSP model and compute the distance minimal roundtrip visiting all capital cities. I hope nobody feels offended by this :)
First step is to get a list of countries and their capitals that Boris Johnson offended.
library(dplyr)
library(maps)
countries <- c("Canada", "USA", "Guyana", "Belize",
"Ireland", "UK", "France", "Germany", "Netherlands", "Russia",
"Sierra Leone", "Ghana", "Nigeria", "Cameroon", "Congo Democratic Republic",
"Uganda", "Rwanda", "Kenya", "Tanzania", "Zambia",
"Malawi", "Mozambique", "Zimbabwe", "Botswana", "Namibia", "Swaziland",
"Lesotho", "South Africa",
"Turkey", "Syria", "Iran", "Pakistan", "India", "China", "Japan", "Bangladesh",
"Malaysia", "Papua New Guinea",
"Australia", "New Zealand")
capitals <- world.cities %>%
filter(country.etc %in% countries, capital == 1) %>%
arrange(country.etc) %>%
mutate(id = row_number())
Second step is to construct a distance matrix for all capitals. The geosphere
package has a handy method. It uses the Haversine distance in meters for all pairs of points.
distance <- geosphere::distm(as.matrix(dplyr::select(capitals, long, lat))) %>%
round
Instead of the MTZ model we use a model with flow constraints as suggested on OR-exchange. Suprisingly this yields a better LP lower bound and it enables us to solve a TSP of 40 cities without using the classic sub-tour elimination constraints through callbacks (which ompr
does not support yet).
library(ompr)
n <- nrow(capitals)
# Model based on the answer by Alan Erera at http://www.or-exchange.com/questions/11784/solving-tsp-using-solvers/11943
model <- MIPModel() %>%
# we create a variable that is 1 iff we travel from city i to j
add_variable(y[i, j], i = 1:n, j = 1:n, type = "binary") %>%
# flow from i to j
add_variable(x[i, j], i = 1:n, j = 1:n, lb = 0, ub = n - 1) %>%
# minimize travel distance
set_objective(sum_exp(distance[i, j] * y[i, j], i = 1:n, j = 1:n), "min") %>%
# you cannot go to the same city
add_constraint(y[i, i], "==", 0, i = 1:n) %>%
# leave each city
add_constraint(sum_exp(y[i, j], j = 1:n), "==", 1, i = 1:n) %>%
# visit each city
add_constraint(sum_exp(y[i, j], i = 1:n), "==", 1, j = 1:n) %>%
# ensure the flow
add_constraint(x[i, j], "<=", (n - 1) * y[i, j], i = 1:n, j = 1:n) %>%
# each node consumes -1
add_constraint(sum_exp(x[j, i], j = 1:n) , "==", sum_exp(x[i, j], j = 1:n) + 1, i = 2:n) %>%
# source supply
add_constraint(sum_exp(x[1, j], j = 1:n), "==", n - 1)
model
## Mixed linear integer optimization problem
## Variables:
## Continuous: 1600
## Integer: 0
## Binary: 1600
## Search direction: minimize
## Constraints: 1760
We solve the model using Symphony as it seems to be generally faster than GLPK. ompr.roi
is a solver interface to the ROI
package that offers a common interface to various popular solvers.
library(ompr.roi)
library(ROI.plugin.symphony)
# I saved the result in a rds file as it takes a long time to compute the optimal solution. Comment out the following line to start the computation.
#result <- solve_model(model, with_ROI(solver = "symphony", verbose = TRUE))
# saveRDS(result, "opt_result.rds")
result <- readRDS("opt_result.rds")
The following code extracts the results of the y
variable from the solution.
solution <- get_solution(result, y[i, j]) %>%
filter(value > 0)
head(solution)
## variable i j value
## 1 y 27 1 1
## 2 y 13 2 1
## 3 y 12 3 1
## 4 y 31 4 1
## 5 y 8 5 1
## 6 y 38 6 1
Now we link the result to the capitals and convert it to a format that ggplot2’s geom_path
can handle.
paths <- select(solution, i, j) %>%
rename(from = i, to = j) %>%
mutate(trip_id = row_number()) %>%
tidyr::gather(property, idx_val, from:to) %>%
mutate(idx_val = as.integer(idx_val)) %>%
inner_join(capitals, by = c("idx_val" = "id")) %>%
arrange(trip_id)
head(paths)
## trip_id property idx_val name country.etc pop lat
## 1 1 from 27 Port Moresby Papua New Guinea 289861 -9.48
## 2 1 to 1 Canberra Australia 324736 -35.31
## 3 2 from 13 Ni Dilli India 321883 28.60
## 4 2 to 2 Dhaka Bangladesh 6724976 23.70
## 5 3 from 12 Georgetown Guyana 236878 6.79
## 6 3 to 3 Belmopan Belize 14590 17.25
## long capital
## 1 147.18 1
## 2 149.13 1
## 3 77.22 1
## 4 90.39 1
## 5 -58.16 1
## 6 -88.79 1
Finally the map.
library(ggplot2)
library(ggalt)
library(ggthemes)
library(ggmap)
world <- map_data("world") %>% filter(region != "Antarctica")
ggplot(data = paths, aes(long, lat)) +
geom_map(data = world, map = world, aes(long, lat, map_id = region),
fill = "white", color = "darkgray", alpha = 0.8, size = 0.2) +
geom_path(aes(group = trip_id), color = "#E41A1C") +
geom_point(data = capitals, color = "#E41A1C", size = 0.8) +
theme_map() +
coord_proj("+proj=robin +lon_0=0 +x_0=0 +y_0=0 +ellps=WGS84 +datum=WGS84 +units=m +no_defs") +
ggtitle("Shortest fully global itinerary of apology",
subtitle = paste0(format(round(result@objective_value/ 1000), big.mark = ","), " km"))
# projection code from
# http://gis.stackexchange.com/a/186712 by hrbrmstr (https://gis.stackexchange.com/users/29544/hrbrmstr)
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