Practical Applications of qDEA

Case Study 1: Hospital Efficiency Analysis

The Problem

A hospital administrator wants to evaluate the efficiency of 12 hospitals using:

Inputs: Number of doctors and nurses
Outputs: Outpatient and inpatient treatments

The administrator suspects that 1-2 hospitals may have data quality issues or operate under exceptional circumstances.

Analysis

# Load hospital data
data(CST22)

# Examine the data
print(CST22)
#>    HOSPITAL DOCTORS NURSES OUT_PATIENTS IN_PATIENTS
#> 1         A      20    151          100          90
#> 2         B      19    131          150          50
#> 3         C      25    160          160          55
#> 4         D      27    168          180          72
#> 5         E      22    158           94          66
#> 6         F      55    255          230          90
#> 7         G      33    235          220          88
#> 8         H      31    206          152          80
#> 9         I      30    244          190         100
#> 10        J      50    268          250         100
#> 11        K      53    306          260         147
#> 12        L      38    284          250         120

# Prepare inputs and outputs
X <- as.matrix(CST22[, c("DOCTORS", "NURSES")])
Y <- as.matrix(CST22[, c("OUT_PATIENTS", "IN_PATIENTS")])

# Summary statistics
cat("Input summary:\n")
#> Input summary:
summary(X)
#>     DOCTORS          NURSES     
#>  Min.   :19.00   Min.   :131.0  
#>  1st Qu.:24.25   1st Qu.:159.5  
#>  Median :30.50   Median :220.5  
#>  Mean   :33.58   Mean   :213.8  
#>  3rd Qu.:41.00   3rd Qu.:258.2  
#>  Max.   :55.00   Max.   :306.0
cat("\nOutput summary:\n")
#> 
#> Output summary:
summary(Y)
#>   OUT_PATIENTS    IN_PATIENTS    
#>  Min.   : 94.0   Min.   : 50.00  
#>  1st Qu.:151.5   1st Qu.: 70.50  
#>  Median :185.0   Median : 89.00  
#>  Mean   :186.3   Mean   : 88.17  
#>  3rd Qu.:235.0   3rd Qu.:100.00  
#>  Max.   :260.0   Max.   :147.00

Step 1: Standard DEA Analysis

# Run standard DEA (no outliers allowed)
dea_result <- qDEA(X = X, Y = Y, 
                   orient = "in", 
                   RTS = "VRS", 
                   qout = 0,
                   getproject = TRUE)
#> [1] " on dmu 1 of 12"
#> [1] " on dmu 12 of 12"

# Create results table
results_dea <- data.frame(
  Hospital = CST22$HOSPITAL,
  Efficiency = round(dea_result$effvals, 3),
  Rank = rank(-dea_result$effvals, ties.method = "min")
)

print(results_dea)
#>    Hospital Efficiency Rank
#> 1         A      1.000    2
#> 2         B      1.000    2
#> 3         C      0.896   10
#> 4         D      1.000    1
#> 5         E      0.882   11
#> 6         F      0.939    9
#> 7         G      1.000    2
#> 8         H      0.799   12
#> 9         I      0.989    8
#> 10        J      1.000    2
#> 11        K      1.000    7
#> 12        L      1.000    6

cat("\nEfficient hospitals:", 
    sum(dea_result$effvals >= 0.99), "out of", nrow(X))
#> 
#> Efficient hospitals: 7 out of 12

Step 2: Robust qDEA Analysis

# Run qDEA allowing 10% outliers (≈1 hospital)
qdea_result <- qDEA(X = X, Y = Y,
                    orient = "in",
                    RTS = "VRS",
                    qout = 0.10,
                    getproject = TRUE)
#> [1] " on dmu 1 of 12"
#> [1] " on dmu 12 of 12"

# Compare DEA and qDEA results
results_comparison <- data.frame(
  Hospital = CST22$HOSPITAL,
  DEA_Eff = round(dea_result$effvals, 3),
  qDEA_Eff = round(qdea_result$effvalsq, 3),
  Change = round(qdea_result$effvalsq - dea_result$effvals, 3),
  DEA_Rank = rank(-dea_result$effvals, ties.method = "min"),
  qDEA_Rank = rank(-qdea_result$effvalsq, ties.method = "min")
)

print(results_comparison)
#>    Hospital DEA_Eff qDEA_Eff Change DEA_Rank qDEA_Rank
#> 1         A   1.000    1.423  0.423        2         1
#> 2         B   1.000    1.272  0.272        2         2
#> 3         C   0.896    1.000  0.104       10         7
#> 4         D   1.000    1.049  0.049        1         5
#> 5         E   0.882    0.956  0.074       11        10
#> 6         F   0.939    0.984  0.045        9         9
#> 7         G   1.000    1.001  0.001        2         6
#> 8         H   0.799    0.814  0.015       12        11
#> 9         I   0.989    1.000  0.011        8         8
#> 10        J   1.000    1.060  0.060        2         4
#> 11        K   1.000       NA     NA        7        12
#> 12        L   1.000    1.263  0.263        6         3

Step 3: Target Setting

# Calculate targets for inefficient hospitals
targets <- data.frame(
  Hospital = CST22$HOSPITAL,
  Current_Doctors = X[,1],
  Target_Doctors = round(qdea_result$PROJ_DATA$X0HATq[,1], 1),
  Doctor_Reduction = round(X[,1] - qdea_result$PROJ_DATA$X0HATq[,1], 1),
  Current_Nurses = X[,2],
  Target_Nurses = round(qdea_result$PROJ_DATA$X0HATq[,2], 1),
  Nurse_Reduction = round(X[,2] - qdea_result$PROJ_DATA$X0HATq[,2], 1),
  Efficiency = round(qdea_result$effvalsq, 3)
)

# Show only inefficient hospitals
inefficient <- targets[targets$Efficiency < 0.99, ]
print(inefficient)
#>    Hospital Current_Doctors Target_Doctors Doctor_Reduction Current_Nurses
#> 5         E              22           21.0              1.0            158
#> 6         F              55           54.1              0.9            255
#> 8         H              31           25.2              5.8            206
#> NA     <NA>              NA             NA               NA             NA
#>    Target_Nurses Nurse_Reduction Efficiency
#> 5          151.0             7.0      0.956
#> 6          250.9             4.1      0.984
#> 8          167.6            38.4      0.814
#> NA            NA              NA         NA

# Calculate total potential savings
cat("\nTotal potential reductions:\n")
#> 
#> Total potential reductions:
cat("Doctors:", sum(targets$Doctor_Reduction), "\n")
#> Doctors: NA
cat("Nurses:", sum(targets$Nurse_Reduction), "\n")
#> Nurses: NA

Step 4: Peer Benchmarks

# Identify peer hospitals for benchmarking
peers <- qdea_result$PEER_DATA$PEERSq

# Show peers for an inefficient hospital (e.g., Hospital D)
cat("Benchmark hospitals for Hospital D:\n")
#> Benchmark hospitals for Hospital D:
hospital_d_peers <- peers[peers$dmu0 == "D", ]
print(hospital_d_peers[order(-hospital_d_peers$z), ])
#> [1] dmu0 dmuz z   
#> <0 rows> (or 0-length row.names)

Management Report

# Create executive summary
cat("=" , rep("=", 50), "\n", sep="")
#> ===================================================
cat("HOSPITAL EFFICIENCY ANALYSIS - EXECUTIVE SUMMARY\n")
#> HOSPITAL EFFICIENCY ANALYSIS - EXECUTIVE SUMMARY
cat("=" , rep("=", 50), "\n", sep="")
#> ===================================================

cat("\nDATA: 12 hospitals\n")
#> 
#> DATA: 12 hospitals
cat("INPUTS: Doctors, Nurses\n")
#> INPUTS: Doctors, Nurses
cat("OUTPUTS: Outpatients, Inpatients\n")
#> OUTPUTS: Outpatients, Inpatients
cat("METHOD: qDEA with VRS, 10% outlier allowance\n")
#> METHOD: qDEA with VRS, 10% outlier allowance

cat("\n--- EFFICIENCY RESULTS ---\n")
#> 
#> --- EFFICIENCY RESULTS ---
cat("Mean efficiency:", round(mean(qdea_result$effvalsq), 3), "\n")
#> Mean efficiency: NA
cat("Median efficiency:", round(median(qdea_result$effvalsq), 3), "\n")
#> Median efficiency: NA
cat("Efficient hospitals:", sum(qdea_result$effvalsq >= 0.99), "\n")
#> Efficient hospitals: NA
cat("Inefficient hospitals:", sum(qdea_result$effvalsq < 0.99), "\n")
#> Inefficient hospitals: NA

cat("\n--- IMPROVEMENT POTENTIAL ---\n")
#> 
#> --- IMPROVEMENT POTENTIAL ---
cat("If all hospitals achieve target efficiency:\n")
#> If all hospitals achieve target efficiency:
cat("  Doctor reduction:", sum(targets$Doctor_Reduction), 
    "(", round(100*sum(targets$Doctor_Reduction)/sum(X[,1]), 1), "%)\n")
#>   Doctor reduction: NA ( NA %)
cat("  Nurse reduction:", sum(targets$Nurse_Reduction),
    "(", round(100*sum(targets$Nurse_Reduction)/sum(X[,2]), 1), "%)\n")
#>   Nurse reduction: NA ( NA %)

cat("\n--- TOP PERFORMERS ---\n")
#> 
#> --- TOP PERFORMERS ---
top3 <- head(results_comparison[order(-results_comparison$qDEA_Eff), ], 3)
print(top3[, c("Hospital", "qDEA_Eff")])
#>    Hospital qDEA_Eff
#> 1         A    1.423
#> 2         B    1.272
#> 12        L    1.263

cat("\n--- NEEDS IMPROVEMENT ---\n")
#> 
#> --- NEEDS IMPROVEMENT ---
bottom3 <- head(results_comparison[order(results_comparison$qDEA_Eff), ], 3)
print(bottom3[, c("Hospital", "qDEA_Eff")])
#>   Hospital qDEA_Eff
#> 8        H    0.814
#> 5        E    0.956
#> 6        F    0.984

Case Study 2: Retail Store Performance

The Problem

A retail chain wants to evaluate store performance with potential outliers due to: - Special events or temporary factors - Data entry errors - Unique local market conditions

# Load retail data
data(CST21)
print(CST21)
#>   STORE EMPLOYEES FLOOR_AREA SALES
#> 1     A       4.0        3.0     1
#> 2     B       7.0        3.0     1
#> 3     C       8.0        1.0     1
#> 4     D       4.0        2.0     1
#> 5     E       2.0        4.0     1
#> 6     F       5.0        2.0     1
#> 7     G       6.0        4.0     1
#> 8     H       5.5        2.5     1
#> 9     I       6.0        2.5     1

# Prepare data
X <- as.matrix(CST21[, c("EMPLOYEES", "FLOOR_AREA")])
Y <- as.matrix(CST21$SALES)

Sensitivity Analysis: How Many Outliers?

# Test different outlier proportions
qout_values <- c(0, 0.05, 0.10, 0.15, 0.20)

sensitivity_results <- data.frame(
  Store = CST21$STORE
)

for (q in qout_values) {
  result <- qDEA(X = X, Y = Y, orient = "out", RTS = "CRS", qout = q)
  col_name <- paste0("qout_", sprintf("%.2f", q))
  sensitivity_results[[col_name]] <- round(result$effvalsq, 3)
}
#> [1] " on dmu 1 of 9"
#> [1] " on dmu 9 of 9"
#> [1] " on dmu 1 of 9"
#> [1] " on dmu 9 of 9"
#> [1] " on dmu 1 of 9"
#> [1] " on dmu 9 of 9"
#> [1] " on dmu 1 of 9"
#> [1] " on dmu 9 of 9"
#> [1] " on dmu 1 of 9"
#> [1] " on dmu 9 of 9"

print(sensitivity_results)
#>   Store qout_0.00 qout_0.05 qout_0.10 qout_0.15 qout_0.20
#> 1     A        NA     1.167     1.167     1.062     1.062
#> 2     B        NA     1.583     1.583     1.438     1.438
#> 3     C        NA     1.000     1.000     0.500     0.500
#> 4     D        NA     1.000     1.000     0.875     0.875
#> 5     E        NA     1.000     1.000     0.500     0.500
#> 6     F        NA     1.083     1.083     1.000     1.000
#> 7     G        NA     1.667     1.667     1.500     1.500
#> 8     H        NA     1.292     1.292     1.156     1.156
#> 9     I        NA     1.333     1.333     1.219     1.219

# Calculate how efficiency changes with qout
sensitivity_results$Range <- apply(
  sensitivity_results[, -1], 1, 
  function(x) max(x) - min(x)
)

cat("\nStores most sensitive to outlier allowance:\n")
#> 
#> Stores most sensitive to outlier allowance:
print(sensitivity_results[order(-sensitivity_results$Range), 
                          c("Store", "Range")])
#>   Store Range
#> 1     A    NA
#> 2     B    NA
#> 3     C    NA
#> 4     D    NA
#> 5     E    NA
#> 6     F    NA
#> 7     G    NA
#> 8     H    NA
#> 9     I    NA

Recommended Approach

# Use moderate outlier allowance
result_retail <- qDEA(X = X, Y = Y,
                      orient = "out",
                      RTS = "VRS",
                      qout = 0.10,
                      getproject = TRUE)
#> [1] " on dmu 1 of 9"
#> [1] " on dmu 9 of 9"

# Performance report
performance <- data.frame(
  Store = CST21$STORE,
  Employees = X[,1],
  Floor_Area = X[,2],
  Actual_Sales = Y[,1],
  Target_Sales = round(result_retail$PROJ_DATA$Y0HATq[,1], 0),
  Sales_Gap = round(result_retail$PROJ_DATA$Y0HATq[,1] - Y[,1], 0),
  Efficiency = round(result_retail$effvalsq, 3)
)

print(performance)
#>   Store Employees Floor_Area Actual_Sales Target_Sales Sales_Gap Efficiency
#> 1     A       4.0        3.0            1            1         0          1
#> 2     B       7.0        3.0            1            1         0          1
#> 3     C       8.0        1.0            1            1         0          1
#> 4     D       4.0        2.0            1            1         0          1
#> 5     E       2.0        4.0            1            1         0          1
#> 6     F       5.0        2.0            1            1         0          1
#> 7     G       6.0        4.0            1            1         0          1
#> 8     H       5.5        2.5            1            1         0          1
#> 9     I       6.0        2.5            1            1         0          1

# Classify stores
performance$Category <- ifelse(
  performance$Efficiency >= 0.95, "Excellent",
  ifelse(performance$Efficiency >= 0.85, "Good",
         ifelse(performance$Efficiency >= 0.75, "Needs Improvement",
                "Critical"))
)

cat("\nStore Classification:\n")
#> 
#> Store Classification:
table(performance$Category)
#> 
#> Excellent 
#>         9

Case Study 3: Dealing with Outliers

Identifying Outliers

data(CST11)
X <- as.matrix(CST11$EMPLOYEES)
Y <- as.matrix(CST11$SALES_EJOR)

# Run with very restrictive outlier allowance
strict <- qDEA(X = X, Y = Y, orient = "out", RTS = "CRS", qout = 0.01)
#> [1] " on dmu 1 of 8"
#> [1] " on dmu 8 of 8"

# Run with moderate outlier allowance  
moderate <- qDEA(X = X, Y = Y, orient = "out", RTS = "CRS", qout = 0.15)
#> [1] " on dmu 1 of 8"
#> [1] " on dmu 8 of 8"

# Stores with big efficiency changes are likely outliers
outlier_check <- data.frame(
  Store = CST11$STORE,
  Strict = round(strict$effvalsq, 3),
  Moderate = round(moderate$effvalsq, 3),
  Change = round(moderate$effvalsq - strict$effvalsq, 3)
)

print(outlier_check)
#>   Store Strict Moderate Change
#> 1     A  2.667    1.600 -1.067
#> 2     B  1.000    0.600 -0.400
#> 3     C  2.000    1.200 -0.800
#> 4     D  1.778    1.067 -0.711
#> 5     E  1.667    1.000 -0.667
#> 6     F  3.333    2.000 -1.333
#> 7     G  2.667    1.600 -1.067
#> 8     H  2.133    1.280 -0.853

# Flag potential outliers (large efficiency changes)
outlier_check$Potential_Outlier <- outlier_check$Change > 0.10

cat("\nPotential outliers identified:\n")
#> 
#> Potential outliers identified:
print(outlier_check[outlier_check$Potential_Outlier, ])
#> [1] Store             Strict            Moderate          Change           
#> [5] Potential_Outlier
#> <0 rows> (or 0-length row.names)

Impact of Outlier Removal

# Compare DEA vs qDEA to see impact of outlier allowance
impact <- data.frame(
  Store = CST11$STORE,
  DEA = round(strict$effvals, 3),
  qDEA = round(moderate$effvalsq, 3),
  Difference = round(moderate$effvalsq - strict$effvals, 3)
)

print(impact)
#>   Store   DEA  qDEA Difference
#> 1     A 2.667 1.600     -1.067
#> 2     B 1.000 0.600     -0.400
#> 3     C 2.000 1.200     -0.800
#> 4     D 1.778 1.067     -0.711
#> 5     E 1.667 1.000     -0.667
#> 6     F 3.333 2.000     -1.333
#> 7     G 2.667 1.600     -1.067
#> 8     H 2.133 1.280     -0.853

cat("\nMean efficiency:\n")
#> 
#> Mean efficiency:
cat("DEA (no outliers):", round(mean(strict$effvals), 3), "\n")
#> DEA (no outliers): 2.156
cat("qDEA (15% outliers):", round(mean(moderate$effvalsq), 3), "\n")
#> qDEA (15% outliers): 1.293

Workflow: Complete Analysis Template

Here’s a complete workflow you can adapt:

# ==========================================
# COMPLETE qDEA ANALYSIS WORKFLOW
# ==========================================

# 1. Load and examine data
data(CST22)  # Replace with your data
X <- as.matrix(CST22[, c("DOCTORS", "NURSES")])
Y <- as.matrix(CST22[, c("OUT_PATIENTS", "IN_PATIENTS")])

# Check data quality
summary(X)
summary(Y)
# Look for: missing values, extreme values, data entry errors

# 2. Run standard DEA baseline
baseline <- qDEA(X = X, Y = Y, 
                 orient = "in",    # Choose: in, out, inout
                 RTS = "VRS",      # Choose: CRS, VRS, DRS, IRS
                 qout = 0)

# 3. Run robust qDEA
robust <- qDEA(X = X, Y = Y,
               orient = "in",
               RTS = "VRS",
               qout = 0.10,       # Adjust based on expected outliers
               nqiter = 3,        # Iterative refinement
               getproject = TRUE) # Get targets

# 4. Compare results
comparison <- data.frame(
  Unit = rownames(X),
  DEA = round(baseline$effvals, 3),
  qDEA = round(robust$effvalsq, 3),
  Change = round(robust$effvalsq - baseline$effvals, 3)
)

# 5. Identify outliers
potential_outliers <- comparison$Unit[abs(comparison$Change) > 0.10]

# 6. Calculate targets
targets <- data.frame(
  Unit = rownames(X),
  Efficiency = round(robust$effvalsq, 3),
  Current_Input1 = X[,1],
  Target_Input1 = round(robust$PROJ_DATA$X0HATq[,1], 2),
  Current_Input2 = X[,2],
  Target_Input2 = round(robust$PROJ_DATA$X0HATq[,2], 2)
)

# 7. Generate report
# Export to CSV
write.csv(comparison, "efficiency_comparison.csv", row.names = FALSE)
write.csv(targets, "efficiency_targets.csv", row.names = FALSE)

# 8. Optional: Bootstrap for confidence intervals
boot_result <- qDEA(X = X, Y = Y,
                    orient = "in",
                    RTS = "VRS",
                    qout = 0.10,
                    nboot = 1000,
                    seedval = 12345)

boot_ci <- data.frame(
  Unit = rownames(X),
  Efficiency = round(boot_result$effvalsq, 3),
  BC_Efficiency = round(boot_result$BOOT_DATA$effvalsq.bc, 3),
  Bias = round(boot_result$effvalsq - boot_result$BOOT_DATA$effvalsq.bc, 3)
)

Best Practices Summary

1. Data Preparation

✓ Check for missing values
✓ Verify all values are positive
✓ Look for extreme outliers or data entry errors
✓ Ensure comparable units (scale if necessary)
✓ Document data sources and definitions

2. Model Selection

✓ Choose orientation based on managerial control
✓ Use VRS unless scale efficiency is of interest
✓ Start with qout = 0.10 (10% outliers)
✓ Test sensitivity to qout selection

3. Interpretation

✓ Efficiency scores are relative, not absolute
✓ Compare units within same analysis only
✓ Consider context (outliers may be legitimate)
✓ Verify targets are achievable
✓ Use peers for benchmarking

4. Reporting

✓ Document methodology clearly
✓ Report both DEA and qDEA results
✓ Explain outlier allowance rationale
✓ Provide actionable recommendations
✓ Include sensitivity analysis

5. Common Mistakes to Avoid

✗ Comparing efficiency across different analyses
✗ Using CRS when scale varies significantly
✗ Setting qout too high (> 0.25)
✗ Ignoring data quality issues
✗ Over-interpreting small efficiency differences

Exporting Results

To CSV

# Prepare comprehensive results
results_export <- data.frame(
  Unit = CST22$HOSPITAL,
  Input1 = X[,1],
  Input2 = X[,2],
  Output1 = Y[,1],
  Output2 = Y[,2],
  DEA_Efficiency = round(baseline$effvals, 4),
  qDEA_Efficiency = round(robust$effvalsq, 4),
  Target_Input1 = round(robust$PROJ_DATA$X0HATq[,1], 2),
  Target_Input2 = round(robust$PROJ_DATA$X0HATq[,2], 2)
)

# Export
write.csv(results_export, "qDEA_results.csv", row.names = FALSE)

To Excel (requires openxlsx package)

library(openxlsx)

# Create workbook
wb <- createWorkbook()

# Add worksheets
addWorksheet(wb, "Efficiency Scores")
addWorksheet(wb, "Targets")
addWorksheet(wb, "Peers")

# Write data
writeData(wb, "Efficiency Scores", comparison)
writeData(wb, "Targets", targets)
writeData(wb, "Peers", robust$PEER_DATA$PEERSq)

# Save
saveWorkbook(wb, "qDEA_analysis.xlsx", overwrite = TRUE)

Visualization Examples

Efficiency Distribution

data(CST22)
X <- as.matrix(CST22[, c("DOCTORS", "NURSES")])
Y <- as.matrix(CST22[, c("OUT_PATIENTS", "IN_PATIENTS")])

result <- qDEA(X = X, Y = Y, orient = "in", RTS = "VRS", qout = 0.10)
#> [1] " on dmu 1 of 12"
#> [1] " on dmu 12 of 12"

# Histogram
hist(result$effvalsq, 
     breaks = 10,
     col = "lightblue",
     border = "white",
     main = "Distribution of Efficiency Scores",
     xlab = "Efficiency",
     ylab = "Frequency")
abline(v = mean(result$effvalsq), col = "red", lwd = 2, lty = 2)
legend("topleft", 
       legend = paste("Mean =", round(mean(result$effvalsq), 3)),
       col = "red", lty = 2, lwd = 2)

Efficiency Comparison

# Compare DEA and qDEA
dea <- qDEA(X = X, Y = Y, orient = "in", RTS = "VRS", qout = 0)
#> [1] " on dmu 1 of 12"
#> [1] " on dmu 12 of 12"
qdea <- qDEA(X = X, Y = Y, orient = "in", RTS = "VRS", qout = 0.10)
#> [1] " on dmu 1 of 12"
#> [1] " on dmu 12 of 12"

# Scatter plot
plot(dea$effvals, qdea$effvalsq,
     xlim = c(0.4, 1.2), ylim = c(0.4, 1.2),
     xlab = "DEA Efficiency",
     ylab = "qDEA Efficiency",
     main = "DEA vs qDEA Efficiency Scores",
     pch = 19, col = "blue")
abline(0, 1, col = "red", lty = 2)  # 45-degree line
text(dea$effvals, qdea$effvalsq, 
     labels = CST22$HOSPITAL, 
     pos = 3, cex = 0.8)
grid()

Practical Applications of qDEA

Joe Atwood

2026-04-07

Introduction

Case Study 1: Hospital Efficiency Analysis

The Problem

Analysis

Step 1: Standard DEA Analysis

Step 2: Robust qDEA Analysis

Step 3: Target Setting

Step 4: Peer Benchmarks

Management Report

Case Study 2: Retail Store Performance

The Problem

Sensitivity Analysis: How Many Outliers?

Recommended Approach

Case Study 3: Dealing with Outliers

Identifying Outliers

Impact of Outlier Removal

Workflow: Complete Analysis Template

Best Practices Summary

1. Data Preparation

2. Model Selection

3. Interpretation

4. Reporting

5. Common Mistakes to Avoid

Exporting Results

To CSV

To Excel (requires openxlsx package)

Visualization Examples

Efficiency Distribution

Efficiency Comparison

Conclusion

Further Reading