Dynamic Programming in R - Algorithm for Distance Between Warehouse Locations - r

Question

I am trying to calculate the distance between warehouse locations in rStudio utilizing an algorithm from an academic paper. The formula accounts for dimensions of location width, depth, and side of the aisle. The added complexity comes in when calculating the shortest distance with multiple cross aisles. This is all based on this paper.
This is from a bird's eye view:
I have static values for α = 48, ß = 96, ϒ = 108, Ω = 75, S = 22.
I then have a data frame with i as the key for location number, X for aisle number, y for section number, z for side number, and Cross Aisle is a boolean (0 = not a cross-aisle, 1 = is a cross-aisle). Here is a sample:
i X Y Z Cross Aisle
1 1 1 1 0
2 1 2 1 0
....
357 12 20 2 0
These are the formulas between locations i and j if the warehouse had no cross-aisles and was one rectangular grid:
Items in the same aisle (xi = xj):
dij = |yi - yj| * ß + |zi - zj| * ϒ
If items are in different aisles, there are three different scenarios:
dij = {|xi - xj| * (2α + ϒ) + v} if zi = zj
dij = {(xj - xi) * (2α + ϒ) + ϒ + v} if zi = 1, zj = 2
dij = {(xj - xi) * (2α + ϒ) - ϒ + v} if zi = 2, zj = 1
where v is the "vertical" distance (bird's eye, up-down aisle):
v = min(ß * (2 * S - yi - yj), ß * (yi + yj)) + 2Ω
(*Note: the academic paper has a typo in the v formula. It states 2 - yi - yj in the first bit, but I found another, original source that had it correctly as 2 * S-yi - yj.)
This piece of the formula is not the tricky part. It is a fairly simple if/then function in R to compute. However, this next bit with the cross-aisle gets me.
This is from the academic paper:
The authors state essentially: There are two locations p1 and p2. There are two neighboring cross-aisles, a1 and a2. One is above p1 and the other is below p1. Also, cross-aisles b1 and b2 are found, which are neighboring p2 and lead left. The distance between p1 and p2 are as follows:
d(p1,p2) = min{d(p1,ai) + d(ai,bj) + d(bj,p2),i,j ∈ {1,2}}
I am unsure how to apply this algorithm to my data set and construct the necessary loops, and matrix to find the distances between my warehouse locations. I could really use some help here making sense of it.
Here is my actual data set.
Here is an image of my warehouse to give you a sense of the layout. The "X" locations are cross-aisles.
I was able to get a workable loop without the cross-aisles:
whse.data <- read.xlsx("data set.xlsx", sheet = 1) %>%
as.data.frame()
### CREATE COMBINATION OF LOCATIONS
require(tools)
cmbn.loc <- combinations(n = max(whse.data$i), r = 2, v = whse.data$i,
repeats.allowed = FALSE) %>%
as.data.frame()
### CALCULATE DISTANCE BETWEEN LOCATIONS
LocDst <- function(dc, wc, wa, tr, s, df, comb){
# Creates a distance between various locations
#
# Args:
# dc: Depth of cell (alpha)
# wc: Width of cell (beta)
# wa: Width of aisle (y)
# tr: turning radius (omega)
# s: number of sections (S)
# df: Data Frame with location i, x, y, z, Cross.Aisle data
# comb: Combination of locations to compare
#
# Returns:
# Data frame with distances between each location combination
#
dist.df_total <- data.frame()
for (n in 1:nrow(comb)){
i <- comb[n,1]
j <- comb[n,2]
xi <- df[df$i == i,2]
yi <- df[df$i == i,3]
zi <- df[df$i == i,4]
xj <- df[df$i == j,2]
yj <- df[df$i == j,3]
zj <- df[df$i == j,4]
v <- min(wc * (2 * s - yi - yj), wc * (yi + yj)) + 2 * tr
if(xi == xj){
dij <- abs(yi - yj) * wc + abs(zi - zj) * wa
} else if (zi == zj){
dij <- (abs(xi - xj) * (2 * dc + wa) + v)
} else if (zi == 1 & zj == 2){
dij <- ((xj - xi) * (2 * dc + wa) + wa + v)
} else {
dij <- ((xj - xi) * (2 * dc * wa) - wa + v)
}
dist.df <- data.frame(`i` = i, `j` = j, dist = dij)
dist.df_total <- rbind.data.frame(dist.df_total, dist.df)
}
return(dist.df_total)
}
dist <- LocDst(48, 96, 108, 75, 18, whse.data, cmbn.loc)
I need a workable for loop or something to be run Algorithm 1 above, please.

Answer 1

I was able to get something to work. If anyone has anything more straightforward, I am all ears. Maybe this will be helpful to someone out there!
I had to use Excel to calculate the distance between the cross-aisles. There's probably a code for it, but it wasn't value-add for me at this time. Here's a sample of that data:
V1 V2 Dist
7 18 672
7 19 780
7 33 204
....
341 342 108
where V1 represents the first location number and V2 the second for all cross-aisle combinations.
Everything else should be computed within the code (beyond what put above):
require(dplyr)
require(openxlsx)
require(tools)
whse.data <- read.xlsx("data set.xlsx", sheet = 1) %>%
as.data.frame()
### CREATE COMBINATION OF LOCATIONS
cmbn.loc <- combinations(n = max(whse.data$i), r = 2, v = whse.data$i,
repeats.allowed = FALSE) %>%
as.data.frame()
# CROSS-AISLES IN EACH SHELF
ca.shelf <- cross.aisles %>%
group_by(Shelf) %>%
summarise(No.Cross.Aisles = sum(Cross.Aisle)) %>%
as.data.frame()
# DISTANCE BETWEEN CROSS AISLES
cmbn.cross.aisle <- combinations(n = nrow(cross.aisles),
r = 2,
v = cross.aisles$i,
repeats.allowed = FALSE) %>%
as.data.frame()
dist.cross.aisle <- read.xlsx("Combination of Cross-Aisles v3.xlsx", sheet = 1) %>%
as.data.frame()
# CROSS AISLE FUNCTION
CrsAisDst <- function(dc, wc, wa, tr, s, no.sh, df, comb, ca.m, d.m){
# Creates a distance between various locations
#
# Args:
# dc: Depth of cell (alpha)
# wc: Width of cell (beta)
# wa: Width of aisle (y)
# tr: turning radius (omega)
# s: number of sections (S)
# no.sh: number of shelves
# df: Data Frame with location i, x, y, z, Cross.Aisle data
# comb: Combination of locations to compare
# ca.m: Cross-aisles matrix
# d.m: Distances between cross-aisles
#
# Returns:
# Data frame with distances between each location combination
#
dist.df_total <- data.frame()
for (n in 1:nrow(comb)){
i <- comb[n,1]
j <- comb[n,2]
xi <- df[df$i == i,2]
yi <- df[df$i == i,3]
zi <- df[df$i == i,4]
xj <- df[df$i == j,2]
yj <- df[df$i == j,3]
zj <- df[df$i == j,4]
v <- min(wc * (2 * s - yi - yj), wc * (yi + yj)) + 2 * tr
if(xi == xj){
min.dij <- abs(yi - yj) * wc + abs(zi - zj) * wa
} else {
shi <- df[df$i == i,6]
shj <- df[df$i == j,6]
### CROSS-AISLES
ca.i <- #ca.m[ca.m$Shelf == shi,1]
data.frame(`i` = ca.m[ca.m$Shelf == shi,1])
ca.j <- #ca.m[ca.m$Shelf == shj,1]
data.frame(`j` = ca.m[ca.m$Shelf == shj,1])
## JOIN DISTANCES
dist.df_total.i <- data.frame()
dist.df_total.j <- data.frame()
#
for (m in 1:nrow(ca.i)){
i.i <- i
j.i <- ca.i[m,]
xi.i <- df[df$i == i.i,2]
yi.i <- df[df$i == i.i,3]
zi.i <- df[df$i == i.i,4]
xj.i <- df[df$i == j.i,2]
yj.i <- df[df$i == j.i,3]
zj.i <- df[df$i == j.i,4]
dij.i <- abs(yi.i - yj.i) * wc + abs(zi.i - zj.i) * wa
dist.df.i <- data.frame(`i` = i.i, `j` = j.i, dist = dij.i)
dist.df_total.i <- rbind.data.frame(dist.df_total.i, dist.df.i)
}
for (l in 1:nrow(ca.j)){
i.j <- j
j.j <- ca.j[l,]
xi.j <- df[df$i == i.j,2]
yi.j <- df[df$i == i.j,3]
zi.j <- df[df$i == i.j,4]
xj.j <- df[df$i == j.j,2]
yj.j <- df[df$i == j.j,3]
zj.j <- df[df$i == j.j,4]
dij.j <- abs(yi.j - yj.j) * wc + abs(zi.j - zj.j) * wa
dist.df.j <- data.frame(`i` = i.j, `j` = j.j, dist = dij.j)
dist.df_total.j <- rbind.data.frame(dist.df_total.j, dist.df.j)
}
min.i <- dist.df_total.i %>% slice(which.min(dist))
min.j <- dist.df_total.j %>% slice(which.min(dist))
aisle <- data.frame(V1=min.i$j,V2=min.j$j)
dist.aisle <- semi_join(d.m, aisle, by = c("V1", "V2"))
# CALCULATING DISTANCE WITHOUT CROSS-AISLES
if (zi == zj){
dij <- (abs(xi - xj) * (2 * dc + wa) + v)
} else if (zi == 1 & zj == 2){
dij <- ((xj - xi) * (2 * dc + wa) + wa + v)
} else {
dij <- ((xj - xi) * (2 * dc * wa) - wa + v)
}
min.dij <- min(dij, (min.i$dist + min.j$dist + dist.aisle$Dist))
}
dist.df <- data.frame(`i` = i, `j` = j, dist = min.dij)
dist.df_total <- rbind.data.frame(dist.df_total, dist.df)
}
return(dist.df_total)
}
aisle.dist <- CrsAisDst(48, 96, 108, 75, 18, 23, whse.data, cmbn.loc, cross.aisles,
dist.cross.aisle)
Output looks like this:
i j dist
7 18 672
7 19 780
7 33 204
....
341 342 108
(Note: this last same I ran was just among cross-aisles, which is why the numbers look the same. I have tested it, though, and it will use the regular formula if it is less distance.)