Efficiency analysis using DEA Meta-Frontiers

Payment terms at Purchasing departments, by different industries or services (DMU’s) in Argentina

Executive Summary: Efficiency Using DEA (Data Envelopment Analysis) Method

• Objective: To evaluate the negotiation capacity of different Decision-Making Units (DMUs). In this case, the different purchasing departments of various industries in Argentina.

• Main Metric: Payment terms to suppliers (days) –> Financial Output of the Purchasing Process. The inputs used were the number of buyers, the annual purchase value, and the lines per purchase order. These three inputs define the workforce, the economic value managed by each DMU, and the demand structure, respectively.

• Methodology: DEA - Efficient Frontier Analysis using Convex Hull. This allows us to identify which units best optimize their payment terms in relation to the resources consumed (annual expenditure, lines per purchase order, and number of buyers).

• • Impact: Facilitates the detection of inefficiencies and the establishment of best corporate practices in the area of purchasing across different industries - Benchmarking.

• Conclusions: The graph shows the relationship between the outputs inputs of each DMU by industry, automatically defining a benchmark and allowing visualization of the most efficient purchasing areas. The black lines represent the boundaries, the limit of what is possible with current technology. To move beyond this boundary, innovation and/or improvement of both processes and technology is necessary; otherwise, the boundaries will establish a limit. The DMUs closest to this limit are the most efficient. The further from the boundaries, the more opportunities for improvement the processes of those industries have, according to those DMUs. The DMUs closest to the boundaries are the example to follow. However, note that there are approximately four convex regions in each graph. This indicates that each group within that region shares similar structural characteristics, whether related to demand, labor force, technology, or purchased values. These graphs do not show absolute values, only values in comparisons relative to the frontier. The frontier serves as the benchmark for comparison between DMUs.Finally you can see a link showing the dynamics of the system in 3D, depicting the animation of DMU’s and meta frontier during the period 2000 - 2025 according to exogenous variables such as PBI/capita and inflation of Argentina. Clearly, this shows that as the GDP contracted, payment terms tightened and negotiations became increasingly difficult.

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# Verificación e instalación de paquetes
if(!require(tidyverse)) install.packages("tidyverse")
if(!require(ggrepel)) install.packages("ggrepel")
if(!require(readxl)) install.packages("readxl")

# Carga en memoria
library(tidyverse)
library(ggrepel)
library(readxl)

# Lectura de la base de datos
datos <- read_excel("datos_dmu_3.xlsx")

# Transformación de los datos
df_long <- datos %>%
  mutate(
    Gasto_kUSD_año = Valor_Comprado / 1000,
    Lineas_xOC = Lineas_xOC,
    Compradores = Compradores,
    Output_Plazo = Plazo_Pago_Proveedores,
    Cohorte = as.factor(Cohorte)
  ) %>%
  pivot_longer(
    cols = c(Gasto_kUSD_año, Lineas_xOC, Compradores),
    names_to = "Tipo_Input",
    values_to = "Valor_Input"
  )

3. Cálculo de Fronteras de Eficiencia

Se calcula la envolvente convexa para delimitar la frontera de eficiencia en cada una de las variables de entrada analizadas.

df_fronteras_facet <- df_long %>%
  group_by(Tipo_Input) %>%
  do({
    d <- .
    # Se calcula el índice de los puntos que conforman la envolvente convexa
    indices <- chull(d$Valor_Input, d$Output_Plazo)
    d[indices, ]
  }) %>%
  arrange(Tipo_Input, Valor_Input)

4. Visualización de Meta-Fronteras

Se genera el gráfico final. Se implementa un etiquetado exhaustivo para asegurar la identificación unívoca de cada DMU en el reporte.

ggplot(df_long, aes(x = Valor_Input, y = Output_Plazo)) +
  # Polígono de frontera
  geom_polygon(data = df_fronteras_facet, fill = "#8e44ad", alpha = 0.1) +
  geom_path(data = df_fronteras_facet, color = "black", linewidth = 0.8) +
  
  # Puntos de dispersión
  geom_point(aes(color = Cohorte), alpha = 0.6, size = 1.5) +
  
  # Etiquetado  para mostrar todas las DMUs
  geom_text_repel(
    aes(label = DMU),
    size = 2.2,                  
    max.overlaps = Inf,          
    force = 2,                   
    point.padding = 0.2,         
    box.padding = 0.3,           
    min.segment.length = 0,      
    segment.color = "grey50",
    segment.linewidth = 0.2
  ) +
  
  # Facetado y estética
  facet_wrap(~ Tipo_Input, scales = "free_x") +
  scale_color_brewer(palette = "Dark2") +
  labs(
    title = "Identificación Completa de DMUs en Meta-Fronteras",
    subtitle = "Output: Plazo de Pago a Proveedores | Análisis de Capacidad de Negociación",
    x = "Valor del Recurso (Input)",
    y = "Plazo de Pago (Días)"
  ) +
  theme_minimal() +
  theme(
    legend.position = "bottom", 
    strip.text = element_text(face = "bold")
  )

#The following link displays the trajectory of DMUs between 2000 and 2025 relative to Argentina’s GDP per capita and inflation rates. This visualization illustrates the evolution of the metafrontier (represented by the mesh) and the positioning of individual DMUs (indicated by the dots). 

#Copy and paste in google brower the next link

# https://gemini.google.com/share/b1e7bed85052

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