Mathematical Modeling Midterm Project
Xun (Crystal) Cui/ Yun Tzu (Gloria) Chen
Clear environment and load libraries
# Clear environment of variables and functions
rm(list = ls(all = TRUE))
# Clear environmet of packages
if(is.null(sessionInfo()$otherPkgs) == FALSE)lapply(paste("package:", names(sessionInfo()$otherPkgs), sep=""), detach, character.only = TRUE, unload = TRUE)# Load lpSolve package to demonstrate simple LP
library(lpSolve)
library(lpSolveAPI)
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library(knitr)
library(kableExtra)##
## Attaching package: 'kableExtra'## The following object is masked from 'package:dplyr':
##
## group_rows1 Network Map
Code a “network map” showing the different routes workers and supplies may take to reach the African cities from the United States.
2 Plan 1
2.1 Set up problem
2.1.1 Find out the quieckst route
In order to get the quickest route for each city, we need to find out the time cost of each route by different transportation first. According to the fomula: time = distance/speed, we find out the different time cost by each route as followed:
The quickest routes of nine African cities.
Select the minimum value of the time cost of three stratigic cities as following:
The quickest routes of three stratigic cities.
2.1.2 Which routes appear to have significant time bottlenecks that the IFRC should work to reduce?
Use the same fomula but find out the maximum value of time as followed: It takes 195.09 hours from Jacksonville, FL to Luanda, Angola by ship. In addition, Among the nine cities, the route from New York to Luanda,Angola is the significant time bottlenecks that the IFRC should work to reduce. (It takes 186.46 hours)
2.2 Set up the model
Use code to find the quickest route to arrive at the strategic cities.
# Set up model
sp <- make.lp(0, 35)# Set objective fn and constraints, with 2 dummy nodes as starting point and destination. It ends up having 5 more variables, and 2 more constraints
obj_fn <- c(0,0,20.25,172.11,20.13,17.60,186.46,119.20,19.86,180.83,19.90,17.71,195.09,112.11,10.18,20.64,6.31,5.16,58.32,33.12,1.87,36.84,2.73,3.44,27.78,92.46,5.28,28.30,4.42,3,53.04, 54.50,0,0,0)
set.objfn(sp, obj_fn)#Add constraint
#Dummy start point, NY, FL
add.constraint(sp, c( 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), "=", 1)
add.constraint(sp, c(-1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), "=", 0)
add.constraint(sp, c( 0,-1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), "=", 0)
#Transfer cities
add.constraint(sp, c( 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0), "=", 0)
add.constraint(sp, c( 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0), "=", 0)
add.constraint(sp, c( 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0), "=", 0)
add.constraint(sp, c( 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0), "=", 0)
add.constraint(sp, c( 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0), "=", 0)
add.constraint(sp, c( 0, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0), "=", 0)
#IFRC cities, dummy destination
add.constraint(sp, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1,-1,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0), "=", 0)
add.constraint(sp, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1,-1,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 0, 1, 0), "=", 0)
add.constraint(sp, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1,-1,-1,-1,-1,-1, 0, 0, 1), "=", 0)
add.constraint(sp, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1,-1,-1), "=",-1)
# Add names, rows are nodes and capacity constraints, columns are edges
dimnames(sp) <- list(c("Start","NY", "FL","Lusaka", "Liberville", "Nairobi", "Khartoum", "Luanda","Dakar","Niamey","Kosongo","Ndjamena","End"),
c("SNY","SFL","NYLu", "NYLi", "NYNa", "NYKh", "NYLua", "NYDa", "FLLu", "FLLi", "FLNa", "FLKh", "FLLua", "FLda",
"LuNia","LiNia","NaNia","KhNia","LuaNia","DaNia","LuKo","LiKo","NaKo","KhKo","LuaKo","DaKo","LuNd","LiNd","NaNd","KhNd","LuaNd",
"DaNd","NiaE","KoE","NdE"))# Write to view the algebraic formulation
write.lp(sp, "MP_prob2.lp",type = 'lp')
# Solve the model
solve(sp)## [1] 0# Make results and sensitivity table
ps <- get.primal.solution(sp)
obj_sa <- get.sensitivity.obj(sp)
rhs_sa <- get.sensitivity.rhs(sp)
nv <- length(get.variables(sp))
mc <- length(get.constr.type(sp))
ov <- paste0("Objective Value = ", ps[1])
sa_tab <- rbind(ps[2:(nv + mc + 1)],
round(c(rhs_sa$duals[1:mc], obj_fn), 2),
round(c(rhs_sa$dualsfrom[1:mc],obj_sa$objfrom), 2),
round(c(rhs_sa$dualstill[1:mc],obj_sa$objtill), 2))
colnames(sa_tab) <- c(rownames(sp), colnames(sp))
rownames(sa_tab) <- c("solution", "duals/coef", "Sens From", "Sens Till")
sa_tab <- ifelse(sa_tab == -1.000e+30, "-inf", sa_tab)
sa_tab <- ifelse(sa_tab == 1.000e+30, "inf", sa_tab)# Print the table
knitr::kable(sa_tab, format.args = list(big.mark = ",")) %>%
kableExtra::kable_styling(bootstrap_options = c("striped", "bordered")) %>%
kableExtra::add_footnote(label = ov, notation = "none")| Start | NY | FL | Lusaka | Liberville | Nairobi | Khartoum | Luanda | Dakar | Niamey | Kosongo | Ndjamena | End | SNY | SFL | NYLu | NYLi | NYNa | NYKh | NYLua | NYDa | FLLu | FLLi | FLNa | FLKh | FLLua | FLda | LuNia | LiNia | NaNia | KhNia | LuaNia | DaNia | LuKo | LiKo | NaKo | KhKo | LuaKo | DaKo | LuNd | LiNd | NaNd | KhNd | LuaNd | DaNd | NiaE | KoE | NdE | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| solution | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |
| duals/coef | 19.9 | 19.9 | 19.9 | 0.04 | 0 | 0 | 2.3 | 0 | 0 | -0.7 | -0.7 | -0.7 | -0.7 | 0 | 0 | 20.25 | 172.11 | 20.13 | 17.6 | 186.46 | 119.2 | 19.86 | 180.83 | 19.9 | 17.71 | 195.09 | 112.11 | 10.18 | 20.64 | 6.31 | 5.16 | 58.32 | 33.12 | 1.87 | 36.84 | 2.73 | 3.44 | 27.78 | 92.46 | 5.28 | 28.3 | 4.42 | 3 | 53.04 | 54.5 | 0 | 0 | 0 |
| Sens From | 1 | 0 | 0 | 0 | -inf | -inf | 0 | -inf | -inf | 0 | 0 | 0 | -1 | -0.23 | -0.11 | 19.86 | 19.9 | 19.9 | -inf | 19.9 | 19.9 | 18.73 | 19.9 | 17.87 | 17.6 | 19.9 | 19.9 | 0.74 | 0.7 | 0.7 | 3 | 0.7 | 0.7 | 0.74 | 0.7 | 0.7 | 3 | 0.7 | 0.7 | 0.74 | 0.7 | 0.7 | -inf | 0.7 | 0.7 | -2.16 | -0.44 | -inf |
| Sens Till | 1 | 0 | 0 | 0 | inf | inf | 0 | inf | inf | 0 | 0 | 0 | -1 | 0.11 | inf | inf | inf | inf | 17.71 | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | inf | 3.44 | inf | inf | inf | inf | 0.44 |
| Objective Value = 20.6 |
2.3 Result
Fastest route to from departure city to the stategic city: NY - Khartoum - Ndjamena
Fastest route takes 20.6 hrs
3 Plan 2
3.1 Set up problem
How should the IFRC satisfy each African city’s need requirements at minimum cost? Where are the significant bottlenecks in the system that the IFRC should work to reduce? Network Map highlighting the least cost route between the U.S. and African cities.
Find out the cheapest way to deliver the workers and cargo.
3.2 Set up the model
# Set up model
prob3 <- make.lp(0, 30)
# Set objective fn
obj_fn <- c(50,30,55,45,30,32,57,48,61,49,44,56,24,1000,28,22,1000,1000,22,4,25,19,5,5,23,7,2,4,8,9)
set.objfn(prob3, obj_fn)
#Set up requirement node constraints
add.constraint(prob3, c( 150, 240, 150, 150, 240, 240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), "<=", 500000)
add.constraint(prob3, c( 0, 0, 0, 0, 0, 0, 150, 240, 150, 150, 240, 240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), "<=", 500000)
add.constraint(prob3, c(-150, 0, 0, 0, 0, 0,-150, 0, 0, 0, 0, 0, 150, 0, 0, 0, 0, 0, 150, 0, 0, 0, 0, 0, 150, 0, 0, 0, 0, 0), "<=", -150000)
add.constraint(prob3, c( 0,-240, 0, 0, 0, 0, 0,-240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17.7, 0, 0, 0, 0, 0, 17.7, 0, 0, 0, 0), "<=", -100000)
add.constraint(prob3, c( 0, 0,-150, 0, 0, 0, 0, 0,-150, 0, 0, 0, 0, 0, 150, 0, 0, 0, 0, 0, 150, 0, 0, 0, 0, 0, 150, 0, 0, 0), "<=", -120000)
add.constraint(prob3, c( 0, 0, 0,-150, 0, 0, 0, 0, 0,-150, 0, 0, 0, 0, 0, 150, 0, 0, 0, 0, 0, 150, 0, 0, 0, 0, 0, 150, 0, 0), "<=", -90000)
add.constraint(prob3, c( 0, 0, 0, 0,-240, 0, 0, 0, 0, 0,-240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17.7, 0, 0, 0, 0, 0, 17.7, 0), "<=", -130000)
add.constraint(prob3, c( 0, 0, 0, 0, 0,-240, 0, 0, 0, 0, 0,-240, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17.7, 0, 0, 0, 0, 0, 17.7), "<=", -50000)
add.constraint(prob3, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -150, 0, -150, -150, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), "<=", -100000)
add.constraint(prob3, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-150, -17.7, -150, -150, -17.7, -17.7, 0, 0, 0, 0, 0, 0), "<=", -180000)
add.constraint(prob3, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-150, -17.7, -150, -150, -17.7, -17.7), "<=", -80000)
# Set up capacity constraints
add.constraint(prob3, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1), "<=", 840)
add.constraint(prob3, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0), "<=", 200)
add.constraint(prob3, c( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0), "<=", 200)
# Add names, rows are nodes and capacity constraints, columns are edges
dimnames(prob3) <- list(c("NY", "FL","Lusaka", "Liberville", "Nairobi", "Khartoum", "Luanda","Dakar","Niamey","Kosongo","Ndjamena",
"Utruck", "UA1", "UA2"),
c("NYLu", "NYLi", "NYNa", "NYKh", "NYLua", "NYDa", "FLLu", "FLLi", "FLNa", "FLKh", "FLLua", "FLda",
"LuNia","LiNia","NaNia","KhNia","LuaNia","DaNia","LuKo","LiKo","NaKo","KhKo","LuaKo","DaKo","LuNd","LiNd","NaNd","KhNd","LuaNd ","DaNd"))
# Write to view the algebraic formulation
write.lp(prob3, "MP_prob3.lp",type = 'lp')3.3 Solve model
# Solve the model
solve(prob3)## [1] 0# Make solution/sensitivity analysis table
# Get primal solution
ps <- get.primal.solution(prob3)
# Have to re-enter obj fn to get Sens Ana table since cannot pull from model
obj_fn <- c(50,30,55,45,30,32,57,48,61,49,44,56,24,1000,28,22,1000,1000,22,4,25,19,5,5,23,7,2,4,8,9)
# Get sensitivity analysis
obj_sa <- get.sensitivity.obj(prob3)
rhs_sa <- get.sensitivity.rhs(prob3)
n <- length(get.variables(prob3))
m <- length(get.constr.type(prob3))
ov <- paste0("Objective Value = ", ps[1]*1000)
sa_tab <- rbind(ps[2:(n + m + 1)],
c(round(rhs_sa$duals[1:m], 2), obj_fn),
round(c(rhs_sa$dualsfrom[1:m],obj_sa$objfrom), 2),
round(c(rhs_sa$dualstill[1:m],obj_sa$objtill), 2))
colnames(sa_tab) <- c(rownames(prob3), colnames(prob3))
rownames(sa_tab) <- c("solution", "duals/coef", "Sens From", "Sens Till")
sa_tab <- ifelse(sa_tab == -1.000e+30, "-inf", sa_tab)
sa_tab <- ifelse(sa_tab == 1.000e+30, "inf", sa_tab)# Print the table
kable(sa_tab, format.args = list(big.mark = ",")) %>%
kable_styling(bootstrap_options = c("striped", "bordered")) %>%
add_footnote(label = ov, notation = "none")| NY | FL | Lusaka | Liberville | Nairobi | Khartoum | Luanda | Dakar | Niamey | Kosongo | Ndjamena | Utruck | UA1 | UA2 | NYLu | NYLi | NYNa | NYKh | NYLua | NYDa | FLLu | FLLi | FLNa | FLKh | FLLua | FLda | LuNia | LiNia | NaNia | KhNia | LuaNia | DaNia | LuKo | LiKo | NaKo | KhKo | LuaKo | DaKo | LuNd | LiNd | NaNd | KhNd | LuaNd | DaNd | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| solution | 5e+05 | 5e+05 | -150000 | -99999.9999999999 | -120000 | -90000.0000000001 | -130000 | -50000.0000000001 | -1e+05 | -180000 | -80000 | 0 | 0 | 200 | 266.666666666667 | 1166.66666666667 | 0 | 0 | 541.666666666667 | 208.333333333334 | 733.333333333333 | 0 | 1133.33333333333 | 1466.66666666667 | 0 | 0 | 0 | 0 | 0 | 666.666666666667 | 0 | 0 | 0 | 10169.4915254237 | 0 | 0 | 0 | 0 | 0 | 0 | 333.333333333333 | 200 | 0 | 0 |
| duals/coef | -0.05 | 0 | -0.38 | -0.17 | -0.41 | -0.33 | -0.17 | -0.18 | -0.47 | -0.4 | -0.42 | 0 | 0 | -10 | 50 | 30 | 55 | 45 | 30 | 32 | 57 | 48 | 61 | 49 | 44 | 56 | 24 | 1000 | 28 | 22 | 1000 | 1000 | 22 | 4 | 25 | 19 | 5 | 5 | 23 | 7 | 2 | 4 | 8 | 9 |
| Sens From | 5e+05 | -inf | -150000 | -1e+05 | -120000 | -90000 | -130000 | -50000 | -1e+05 | -180000 | -80000 | -inf | -inf | 0 | 48.25 | -5.32 | 54 | 42 | 16.44 | 16.44 | 56 | 41.2 | 51 | 40.65 | 41.2 | 43.2 | 14 | 1000 | 10 | -49 | 1000 | 1000 | 2.65 | -3.04 | -1.35 | 10.65 | 4 | 3.85 | 6 | 4.4 | -8 | -inf | 4.4 | 4.25 |
| Sens Till | 610000 | inf | -40000 | 10000 | 50000 | 130000 | -20000 | 0 | 0 | -70000 | -30000 | inf | inf | 533.33 | 51 | 36.8 | inf | inf | 32.8 | 44.8 | 58.75 | inf | 62 | 52 | inf | inf | inf | inf | inf | 32 | inf | inf | inf | 4.99 | inf | inf | inf | inf | inf | inf | 19 | 14 | inf | inf |
| Objective Value = 310861299.435028 |
3.4 Result
Minimun cost: $310,861,299.43.
NewYork needs 267 flights to Lusaka, Zambia, 1167 ships to Libreville, Gabon, 542 ships to Luanda, Angola, and 209 ships to Dakar, Senegal.
Florida needs 734 flights to Lusaka, Zambia, 1134 flights to Nairobi, Kenya, and 1147 flights to Khartoum, Sudan.
Khartoum, Sudan needs 667 flights Niamey, Niger, and 200 flights to Ndjamena, Chad.
Libreville, Gabon needs 10170 trucks to Kosongo, D.R. Congo.
Nairobi, Kenya needs 200 flights to Ndjamena, Chad.
With an additional flight increase in the route from Khartoum, Sudan to Ndjamena, Chad can decrease $10000 in cost. This would be the bottelneck in the system that IFRC can try to reduce the cost.
4 Plan 3
4.1 Set up problem
How can the IFRC maximize the total amount of cargo that reaches Africa? Where are the significant bottlenecks in the system that the IFRC should work to reduce? Provide a table and/or network map highlighting the maximum cargo and routes between the U.S. and African cities.
Maximize the total amount of cargo that reaches Africa
4.2 Set up the model
# Set up model
prob4 <- make.lp(0, 30)#set objective function to min
lp.control(prob4,sense="max")## $anti.degen
## [1] "fixedvars" "stalling"
##
## $basis.crash
## [1] "none"
##
## $bb.depthlimit
## [1] -50
##
## $bb.floorfirst
## [1] "automatic"
##
## $bb.rule
## [1] "pseudononint" "greedy" "dynamic" "rcostfixing"
##
## $break.at.first
## [1] FALSE
##
## $break.at.value
## [1] 1e+30
##
## $epsilon
## epsb epsd epsel epsint epsperturb epspivot
## 1e-10 1e-09 1e-12 1e-07 1e-05 2e-07
##
## $improve
## [1] "dualfeas" "thetagap"
##
## $infinite
## [1] 1e+30
##
## $maxpivot
## [1] 250
##
## $mip.gap
## absolute relative
## 1e-11 1e-11
##
## $negrange
## [1] -1e+06
##
## $obj.in.basis
## [1] TRUE
##
## $pivoting
## [1] "devex" "adaptive"
##
## $presolve
## [1] "none"
##
## $scalelimit
## [1] 5
##
## $scaling
## [1] "geometric" "equilibrate" "integers"
##
## $sense
## [1] "maximize"
##
## $simplextype
## [1] "dual" "primal"
##
## $timeout
## [1] 0
##
## $verbose
## [1] "neutral"# Set objective finction
obj_fn <- c(1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)
set.objfn(prob4, obj_fn)#Add aid requirement constraints for each of cities
add.constraint(prob4, c(1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),"<=", 500000)
add.constraint(prob4, c(0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),"<=", 500000)
add.constraint(prob4, c(1,0,0,0,0,0,1,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 50000)
add.constraint(prob4, c(0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 100000)
add.constraint(prob4, c(0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,0,0), "<=", 130000)
add.constraint(prob4, c(0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0),"<=", 150000)
add.constraint(prob4, c(0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0), "<=", 90000)
add.constraint(prob4, c(0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1), "<=", 120000)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0), "<=", 100000)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0), "<=", 180000)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1), "<=", 80000) #Add Airline Constraints
add.constraint(prob4, c(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 300*150)
add.constraint(prob4, c(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 500*150)
add.constraint(prob4, c(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 500*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 500*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 700*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 600*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0), "<=", 200*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0), "<=", 0*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0), "<=", 300*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0), "<=", 140*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0), "<=", 40*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0), "<=", 80*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0), "<=", 0*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1), "<=", 300*150)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0), "<=", 40*150)#Add Truck Constraints
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 300*17.7)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0), "<=", 250*17.7)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 700*17.7)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 160*17.7)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0), "<=", 240*17.7)
add.constraint(prob4, c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), "<=", 450*17.7)dimnames(prob3) <- list(c(“NY”, “FL”,“Lusaka”, “Liberville”, “Mairobi”, “Khartoum”, “Luanda”,“Dakar”,“Niamey”,“Kosongo”,“Ndjamena”, “Utruck”, “UA1”, “UA2”), c(“NYLu”, “NYLi”, “NYNa”, “NYKh”, “NYLua”, “NYDa”, “FLLu”, “FLLi”, “FLNa”, “FLKh”, “FLLua”, “FLda”, “LuNia”,“LiNia”,“NaNia”,“KhNia”,“LuaNia”,“DaNia”,“LuKo”,“LiKo”,“NaKo”,“KhKo”,“LuaKo”,“DaKo”,“LuNd”,“LiNd”,“NaNd”,“KhNd”,“LuaNd”,“DaNd”))
# Add names, rows are nodes and capacity constraints, columns are edges
dimnames(prob4) <- list(c("NY", "FL","Dakar","Liberville","Lusaka","Luanda", "Khartoum","Nairobi", "Niamey","Kosongo","Ndjamena",
"UA1","UA2","UA3","UA4","UA5","UA6","UA7","UA8","UA9","UA10","UA11","UA12","UA13","UA14","UA15","Utruck1","Utruck2","Utruck3","Utruck4","Utruck5","Utruck6"),
c("NYDa", "NYLi", "NYLua", "NYLu", "NYKh", "NYNa", "FLDa", "FLLi", "FLLua", "FLLu", "FLKh", "FLNa",
"DaNia","DaKo","DaNd","LiNia","LiKo","LiNd","LuaNia","LuaKo","LuaNd","LuNia","LuKo","LuNd","KhNia","KhKo","KhNd","NaNia","NaKo","NaNd"))
# Write to view the algebraic formulation
write.lp(prob4, "MP_prob4.lp",type = 'lp')solve(prob4)## [1] 0get.objective(prob4)## [1] 816170# Make results and sensitivity table
ps <- get.primal.solution(prob4)
obj_sa <- get.sensitivity.obj(prob4)
rhs_sa <- get.sensitivity.rhs(prob4)
nv <- length(get.variables(prob4))
mc <- length(get.constr.type(prob4))
ov <- paste0("Objective Value = ", ps[1])
sa_tab <- rbind(ps[2:(nv + mc + 1)],
round(c(rhs_sa$duals[1:mc], obj_fn), 2),
round(c(rhs_sa$dualsfrom[1:mc],obj_sa$objfrom), 2),
round(c(rhs_sa$dualstill[1:mc],obj_sa$objtill), 2))
colnames(sa_tab) <- c(rownames(prob4), colnames(prob4))
rownames(sa_tab) <- c("solution", "duals/coef", "Sens From", "Sens Till")
sa_tab <- ifelse(sa_tab == -1.000e+30, "-inf", sa_tab)
sa_tab <- ifelse(sa_tab == 1.000e+30, "inf", sa_tab)# Print the table
knitr::kable(sa_tab, format.args = list(big.mark = ",")) %>%
kableExtra::kable_styling(bootstrap_options = c("striped", "bordered")) %>%
kableExtra::add_footnote(label = ov, notation = "none")| NY | FL | Dakar | Liberville | Lusaka | Luanda | Khartoum | Nairobi | Niamey | Kosongo | Ndjamena | UA1 | UA2 | UA3 | UA4 | UA5 | UA6 | UA7 | UA8 | UA9 | UA10 | UA11 | UA12 | UA13 | UA14 | UA15 | Utruck1 | Utruck2 | Utruck3 | Utruck4 | Utruck5 | Utruck6 | NYDa | NYLi | NYLua | NYLu | NYKh | NYNa | FLDa | FLLi | FLLua | FLLu | FLKh | FLNa | DaNia | DaKo | DaNd | LiNia | LiKo | LiNd | LuaNia | LuaKo | LuaNd | LuNia | LuKo | LuNd | KhNia | KhKo | KhNd | NaNia | NaKo | NaNd | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| solution | 5e+05 | 316170 | 50000 | 1e+05 | 130000 | 120000 | 90000 | 120000 | 1e+05 | 40125 | 66045 | 45000 | 66000 | 63000 | 75000 | 105000 | 90000 | 0 | 0 | 45000 | 0 | 6000 | 12000 | 0 | 45000 | 6000 | 5310 | 4425 | 12390 | 2832 | 4248 | 7965 | 79185 | 108142 | 138673 | 45000 | 63000 | 66000 | 46170 | 0 | 0 | 75000 | 90000 | 105000 | 55000 | 12390 | 7965 | 0 | 5310 | 2832 | 0 | 4425 | 4248 | 0 | 0 | 0 | 45000 | 12000 | 6000 | 0 | 6000 | 45000 |
| duals/coef | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Sens From | 420815 | -inf | 3830 | 53830 | 83830 | -inf | 43830 | 73830 | 53830 | -inf | -inf | 0 | -inf | -inf | 0 | 96000 | 78000 | -inf | 0 | 0 | -inf | 0 | 0 | -inf | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | -inf | -inf | 0 | 1 | 1 | 0 | -1 | -1 | -inf | -1 | -1 | -inf | -1 | -1 | -inf | -inf | -inf | 0 | -1 | -1 | -inf | -1 | -1 |
| Sens Till | 546170 | inf | 233830 | 179185 | 209185 | inf | 102000 | 129000 | 283830 | inf | inf | 75000 | inf | inf | 105000 | 151170 | 136170 | inf | 9000 | 57000 | inf | 15000 | 24000 | inf | 54000 | 18000 | 84495 | 83610 | 152265 | 16787 | 18203 | 21920 | 1 | 1 | 1 | inf | 1 | 1 | 1 | 1 | 1 | inf | inf | inf | 0 | inf | inf | 0 | inf | inf | 0 | inf | inf | 1 | 0 | 0 | inf | inf | inf | inf | inf | inf |
| Objective Value = 816170 |
4.3 Result
The IFRC can maximize the total amount of cargo of 816,170 tons that reaches Africa.
The amount of cargo will follow below requirement and routes to reach Africa:
NY - Dakar, Senegal [79185 tons], NY - Libreville, Gabon [108142 tons], NY - Luanda, Angola [138673 tons], NY - Lusaka, Zambia [45000 tons],
NY - Khartoum, Sudan [63000 tons], NY - Nairobi, Kenya [66000 tons]
FL - Dakar, Senegal [46170 tons], FL - Lusaka, Zambia [75000 tons], FL - Khartoum, Sudan [90000 tons], FL - Nairobi, Kenya [105000 tons]
Dakar - Niamey [55000 tons], Dakar - Kosongo [12390 tons], Dakar - Ndjamena [7965 tons], Libreville - Kosongo [5310 tons], Libreville - Ndjamena [2832 tons] , Luanda - Kosongo [4425 tons], Luanda - Ndjamena [4248 tons], Khartoum - Niamey [45000 tons], Khartoum - Kosongo [12000 tons], Khartoum - Ndjamena [6000 tons], Nairobi - Kosongo [6000 tons], Nairobi - Ndjamena [45000 tons]
For the bottleneck of plan 3, Dakar and Khartoum are taking supplies from both NY and FL, and distributing all supplies to three different african strategic cities. IFRC might need to consider the time and arrangement of this two cities.
