Supply Chain Analytics with Vectors, Matrices, and Systems of Equations

Topics

Vectors
Matrices
Systems of Linear Equations

Scenario

A retail company operates three warehouses that distribute products to three sales regions. Management wants to evaluate current inventory levels, shipping patterns, and determine whether existing inventory can satisfy customer demand.

You have been provided the following datasets:

inventory.csv
demand.csv
shipments.csv

inventory <- read.csv("C:/Users/rbron/OneDrive/A New R Class/605/inventory.csv")

demand <- read.csv("C:/Users/rbron/OneDrive/A New R Class/605/demand.csv")

shipments <- read.csv("C:/Users/rbron/OneDrive/A New R Class/605/shipments.csv")

library(tidyverse)

## Warning: package 'tidyverse' was built under R version 4.5.3

## Warning: package 'ggplot2' was built under R version 4.5.3

## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.1.4     ✔ readr     2.1.5
## ✔ forcats   1.0.0     ✔ stringr   1.5.1
## ✔ ggplot2   4.0.3     ✔ tibble    3.3.0
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ✔ purrr     1.1.0     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

Part A: Inventory Analysis

1. Total Inventory Available in Each Region

# Create inventory matrix
inventory <- matrix(
  c(1200, 900, 800,
    1000, 1100, 700,
    900, 700, 1000),
  nrow = 3,
  byrow = TRUE
)

rownames(inventory) <- c("Warehouse A", "Warehouse B", "Warehouse C")
colnames(inventory) <- c("Region1", "Region2", "Region3")

inventory

##             Region1 Region2 Region3
## Warehouse A    1200     900     800
## Warehouse B    1000    1100     700
## Warehouse C     900     700    1000

# Calculate total inventory by region
total_inventory <- colSums(inventory)
total_inventory

## Region1 Region2 Region3 
##    3100    2700    2500

Output:

Region1 Region2 Region3
   3100    2700    2500

Region 1 has the most inventory with 3,100 units, followed by Region 2 with 2,700 units and Region 3 with 2,500 units.

2. Inventory Surplus or Shortage Relative to Demand

# Create demand vector
demand <- c(2800, 2400, 2500)
names(demand) <- c("Region1", "Region2", "Region3")

# Calculate surplus or shortage
surplus_shortage <- total_inventory - demand
surplus_shortage

## Region1 Region2 Region3 
##     300     300       0

Output:

Region1 Region2 Region3
    300     300       0

Regions 1 and 2 each have a surplus of 300 units. Region 3 has neither a surplus nor a shortage because its inventory exactly matches current demand.

3. Region Most at Risk of a Stock Shortage

most_at_risk <- names(which.min(surplus_shortage))
most_at_risk

## [1] "Region3"

Output:

"Region3"

4. Business Interpretation

Region 3 is the most at risk of a future stock shortage because its available inventory exactly matches current demand, leaving no extra inventory as a buffer. Regions 1 and 2 each have a surplus of 300 units, giving them some protection if demand increases unexpectedly. From a supply chain perspective, management should pay particular attention to Region 3 because even a small increase in demand could result in a shortage. Management could also consider whether some of the excess inventory in Regions 1 and 2 could be strategically reallocated. However, moving inventory would also involve transportation and handling costs, so those costs should be considered before making a decision.

Part B: Shipment Matrix Analysis

1. Create a Shipment Matrix

# Create shipment matrix
shipments <- matrix(
  c(300, 400, 500,
    400, 350, 250,
    500, 300, 200),
  nrow = 3,
  byrow = TRUE
)

rownames(shipments) <- c("Warehouse A", "Warehouse B", "Warehouse C")
colnames(shipments) <- c("Region1", "Region2", "Region3")

shipments

##             Region1 Region2 Region3
## Warehouse A     300     400     500
## Warehouse B     400     350     250
## Warehouse C     500     300     200

2. Row and Column Totals

# Total shipments by warehouse
warehouse_totals <- rowSums(shipments)
warehouse_totals

## Warehouse A Warehouse B Warehouse C 
##        1200        1000        1000

# Total shipments by region
region_totals <- colSums(shipments)
region_totals

## Region1 Region2 Region3 
##    1200    1050     950

Warehouse totals:

Warehouse A Warehouse B Warehouse C
       1200        1000        1000

Regional totals:

Region1 Region2 Region3
   1200    1050     950

3. Warehouse That Ships the Most

largest_shipper <- names(which.max(warehouse_totals))
largest_shipper

## [1] "Warehouse A"

Output:

"Warehouse A"

Warehouse A ships the most products, with 1,200 units.

4. Region Receiving the Most Products

region_most <- names(which.max(region_totals))
region_most

## [1] "Region1"

Output:

"Region1"

Region 1 receives the most products, with 1,200 units.

5. Visualization

# Create grouped bar chart
par(mar = c(5, 5, 4, 10))

barplot(
  shipments,
  beside = TRUE,
  main = "Shipments by Warehouse and Region",
  xlab = "Region",
  ylab = "Number of Products",
  col = c("lightblue", "lightgreen", "lightpink"),
  ylim = c(0, 550)
)

# Add legend outside the plotting area
legend(
  "topright",
  inset = c(-0.30, 0),
  legend = rownames(shipments),
  fill = c("lightblue", "lightgreen", "lightpink"),
  bty = "n",
  xpd = TRUE
)

Figure 1: Shipments by Warehouse and Region

The visualization shows the shipment volumes from each warehouse to each region. Warehouse A has the largest total shipment volume at 1,200 units. Region 1 receives the most products overall, with 1,200 units, while Region 3 receives the fewest at 950 units. Warehouse C sends the largest shipment to Region 1 at 500 units, while Warehouse A sends the largest shipment to Region 3 at 500 units. The shipment patterns show that warehouses contribute different amounts to each region and that Warehouse A plays an important role in the company’s overall distribution network.

Part C: Resource Allocation Model

The system of equations is:

x + y + z = 18
2x + y = 26
y + 2z = 22

1. Express the System in Matrix Form

The system can be written as:

\[Ax=b\]

where:

A =
[ 1  1  1 ]
[ 2  1  0 ]
[ 0  1  2 ]

x =
[ x ]
[ y ]
[ z ]

b =
[ 18 ]
[ 26 ]
[ 22 ]

In R:

# Coefficient matrix
A <- matrix(
  c(1, 1, 1,
    2, 1, 0,
    0, 1, 2),
  nrow = 3,
  byrow = TRUE
)

# Variable vector
x <- c("x", "y", "z")

# Constants vector
b <- c(18, 26, 22)

A

##      [,1] [,2] [,3]
## [1,]    1    1    1
## [2,]    2    1    0
## [3,]    0    1    2

## [1] "x" "y" "z"

## [1] 18 26 22

2. Solve the System Computationally

# Check the determinant of the coefficient matrix
det(A)

## [1] 0

# Attempt to solve the system
solution <- tryCatch(
  solve(A, b),
  error = function(e) {
    paste("No solution:", e$message)
  }
)

solution

## [1] "No solution: Lapack routine dgesv: system is exactly singular: U[3,3] = 0"

The system does not produce a valid solution because the equations are inconsistent. The determinant of the coefficient matrix is zero, indicating that the matrix is singular.

The second and third equations are:

2x + y = 26
y + 2z = 22

Adding these equations gives:

2x + 2y + 2z = 48

Dividing by 2 gives:

x + y + z = 24

However, the first equation states:

x + y + z = 18

Therefore, the equations contradict each other and the system has no solution as written.

3. Verify the Solution

Because there is no solution, the verification demonstrates the contradiction.

# Add equations 2 and 3
equation_sum <- c(2, 2, 2)
constant_sum <- 26 + 22

# Divide both sides by 2
implied_coefficients <- equation_sum / 2
implied_total <- constant_sum / 2

implied_coefficients

## [1] 1 1 1

implied_total

## [1] 24

Output:

[1] 1 1 1
[1] 24

This demonstrates:

x + y + z = 24

But the first equation requires:

x + y + z = 18

Therefore, the system has no solution as written.

4. Business Interpretation

The current resource allocation model cannot determine how many truckloads each warehouse should ship because the three equations are inconsistent. The second and third equations imply that the total number of truckloads must be 24, while the first equation states that only 18 truckloads are available. Management should therefore review the transportation capacity or replenishment requirements before using the model to make a shipping decision.

The inventory analysis provides additional information for planning. Warehouse A has the largest inventory with 2,900 units, followed by Warehouse B with 2,800 units and Warehouse C with 2,600 units. Warehouse A also ships the most products, with 1,200 units. At the regional level, Region 3 is the most vulnerable because its inventory exactly matches demand, leaving no surplus buffer. These results suggest that Warehouse A may have an important role in supporting future shipments, particularly if additional inventory is needed for regions at greater risk of shortages.

Although the current system cannot provide a valid allocation, systems of linear equations are useful in supply chain planning because they allow management to consider multiple constraints simultaneously. Once the equations are corrected, the model could help determine how available transportation capacity should be distributed among warehouses while meeting business requirements.

405 HW 1 A

Rebecca Bronstein

2026-09-12

Supply Chain Analytics with Vectors, Matrices, and Systems of Equations

Topics

Scenario

Part A: Inventory Analysis

1. Total Inventory Available in Each Region

2. Inventory Surplus or Shortage Relative to Demand

3. Region Most at Risk of a Stock Shortage

4. Business Interpretation

Part B: Shipment Matrix Analysis

1. Create a Shipment Matrix

2. Row and Column Totals

3. Warehouse That Ships the Most

4. Region Receiving the Most Products

5. Visualization

Part C: Resource Allocation Model

1. Express the System in Matrix Form

2. Solve the System Computationally

3. Verify the Solution

4. Business Interpretation