In a previous post I showed how a single warehouse location problem can be approached by locating the warehouse near its center of mass. In this post we want to locate multiple warehouses at their center of mass.

We will use the functions already defined in previous posts.

In https://www.supplychaindataanalytics.com/single-warehouse-problem-locating-warehouse-at-center-of-mass-using-r/ I provided a function for locating a single warehouse at its center of mass.

In https://www.supplychaindataanalytics.com/proximity-based-spatial-customer-grouping-in-r I showed how to group customers based on spatial proximity clustering.

```
# example see here: https://www.supplychaindataanalytics.com/single-warehouse-problem-locating-warehouse-at-center-of-mass-using-r/
center_of_mass <- function(x,y,w){
c(crossprod(x,w)/sum(w),crossprod(y,w)/sum(w))
}
# example see here: https://www.supplychaindataanalytics.com/proximity-based-spatial-customer-grouping-in-r
initial_centers <- function(customers,centers){
quantiles <- c()
for(i in 1:centers){
quantiles <- c(quantiles,i*as.integer(nrow(customers)/centers))
}
quantiles
}
```

We will combine these two approaches to solve a multiple warehouse location problem using center of mass theory. This approach is applicable when we already know how many warehouses we want to operate. The approach in the post at hand will not suffice to solve a problem in which we do not know yet how many warehouses we actually want to operate.

I start by creating a dataframe with 1000 randomly distributed customers with randomly distributed demand

```
customer_df <- as.data.frame(matrix(nrow=1000,ncol=3))
colnames(customer_df) <- c("lat","long","demand")
customer_df$lat <- runif(n=1000,min=-90,max=90)
customer_df$long <- runif(n=1000,min=-180,max=180)
customer_df$demand <- runif(n=1000,min=0,max=10)
head(customer_df)
```

```
## lat long demand
## 1 -42.50378 137.62188 9.608067
## 2 47.79308 101.30536 9.510299
## 3 -14.17326 24.38595 1.610305
## 4 -85.34352 -151.29061 6.394425
## 5 -26.31244 112.75030 6.972434
## 6 55.01428 58.17198 2.797564
```

Next, I group the customers using the approach shown in a previous post and applying the defined function initial_centers. I want to locate 20 warehouses so I group customers into 20 groups.

```
centeroids <- initial_centers(customer_df[,-3],20)
cluster_obj <- kmeans(customer_df[,-3],centers = customer_df[centeroids,-3])
customer_df$group <- cluster_obj$cluster
head(customer_df)
```

```
## lat long demand group
## 1 -42.50378 137.62188 9.608067 2
## 2 47.79308 101.30536 9.510299 4
## 3 -14.17326 24.38595 1.610305 6
## 4 -85.34352 -151.29061 6.394425 8
## 5 -26.31244 112.75030 6.972434 18
## 6 55.01428 58.17198 2.797564 16
```

As displayed above I added the clustering-based group index to the customer dataframe.

Now, I will define a function that will loop through every customer group and identify the center of mass. The requirement is that a dataframe must be inputted, containing a latitude, longitude, demand and group column – in exactly this format:

```
multiple_centers_of_mass <- function(df){
result_df <- as.data.frame(matrix(nrow=nrow(df),ncol=6))
colnames(result_df) <- c("lat","long","demand","group","com_lat","com_long")
result_df[,c(1,2,3,4)] <- df
for(i in 1:length(unique(df[,4]))){
sub_df <- result_df[result_df$group==i,]
com <- center_of_mass(sub_df$lat,sub_df$long,sub_df$demand)
result_df$com_lat[result_df$group==i] <- com[1]
result_df$com_long[result_df$group==i] <- com[2]
}
result_df
}
```

Let us test the multiple_centers_of_mass functio that I just defined:

```
com_df <- multiple_centers_of_mass(customer_df)
head(com_df)
```

```
## lat long demand group com_lat com_long
## 1 -42.50378 137.62188 9.608067 2 -25.97973 158.17382
## 2 47.79308 101.30536 9.510299 4 63.58158 84.91329
## 3 -14.17326 24.38595 1.610305 6 -21.20417 26.80993
## 4 -85.34352 -151.29061 6.394425 8 -64.12072 -145.48419
## 5 -26.31244 112.75030 6.972434 18 -33.15564 99.15738
## 6 55.01428 58.17198 2.797564 16 35.04988 44.42388
```

Lets visualize the test results using a scatterplot from the ggplot2 R package. Below you see the warehouse locations (centers of mass):

```
library(ggplot2)
lat_wh_vc <- unique(com_df$com_lat)
long_wh_vc <- unique(com_df$com_long)
warehouse_df <- as.data.frame(matrix(nrow=length(lat_wh_vc),ncol=2))
warehouse_df[,1] <- lat_wh_vc
warehouse_df[,2] <- long_wh_vc
colnames(warehouse_df) <- c("lat","long")
ggplot(warehouse_df) + geom_point(mapping = aes(x=lat,y=long)) + xlim(-90,90) + ylim(-180,180)
```

The customers are grouped as shown below:

`library(viridis)`

`## Warning: package 'viridis' was built under R version 3.5.3`

`## Loading required package: viridisLite`

```
ggplot(com_df) + geom_point(mapping = aes(x=lat,y=long,color=group,size=demand)) +
xlim(-90,90) + ylim(-180,180) + scale_color_viridis(discrete = FALSE, option = "D") + scale_fill_viridis(discrete = FALSE)
```

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