Graphical and Numerical analysis

1 Citation

Kevin Kam Fung YUEN (2026). Anchored Regularized Direct Least Squares (ARDLS): Integrating Established Prioritization Operators for Priority Elicitation in the Analytic Hierarchy Process. arXiv preprint arXiv:2608.21187. https://arxiv.org/pdf/2608.21187

2 Import Libraries and Define Global Parameters

# Created by Kevin Kam Fung YUEN
library(ARDLS)  

dls_solve <- function(PRM)
{
    solve_DLS(PRM)$weights
}

POs = list(
  NRS      = NRS,
  NRCS     = NRCS, 
  AMNC     = AMNC, 
  NGMR     = geoMean, 
  EV       = SaatyEigen, 
  SVD      = SVD, 
  CosMax   = CosMax, 
  PIGM     = PIGM,
  DLS   =  dls_solve
)

POList <-function(PRM)
{
    PO_results = list(
          NRS        = NRS(PRM),
          NRCS       = NRCS(PRM), 
          AMNC       = AMNC(PRM), 
          NGMR        = geoMean(PRM), 
          EV        = SaatyEigen(PRM), 
          SVD        = SVD(PRM), 
          CosMax     = CosMax(PRM), 
          PIGM       = PIGM(PRM),
          DLS        = solve_DLS(PRM)$weights
    )
}

3 Case Study 1

3.1 Pairwise Reciprocal Matrix (PRM) Data

PRM1 <- matrix(c(
  1,   2,  6,
  1/2,   1,   3,
  1/6,   1/3, 1
), nrow = 3, byrow = TRUE)

3.2 Consistency Ratio (CR)

ConsistencyRatio(PRM1)

## [1] 0

3.3 Safe Regularizer Verification

ARDLS::safeRegularizer(PRM1,num_starts = 1000)

## Running DLS optimization from 1000 random starting points...
## 
## === DLS Optimization Results ===
## Global Minimum Objective Value: 0.00000000
## Number of Unique Optimal Solutions Found: 1
## Optimal Weight Vectors:
##       w1  w2  w3
## [1,] 0.6 0.3 0.1

## $lambda_star
## [1] 1
## 
## $case
## [1] "Case 1: Unique Convex Minimum (|W*| = 1, Delta_min > 0)"
## 
## $delta_min
## [1] 223.7654
## 
## $delta_max
## [1] 223.7654
## 
## $lambda_tilde
## [1] -111.8827
## 
## $num_solutions
## [1] 1
## 
## $W_star
##       w1  w2  w3
## [1,] 0.6 0.3 0.1

3.4 Comparison of Baseline Prioritization Operators (POs) and ARDLS Across Anchors

options(width = 200)
round(ARDLS::compare_ARDLS_anchors(PRM1, POs, reg_lambda = 1),4)

##               NRS NRCS AMNC NGMR  EV SVD CosMax PIGM DLS
## Base_w1       0.6  0.6  0.6  0.6 0.6 0.6    0.6  0.6 0.6
## Base_w2       0.3  0.3  0.3  0.3 0.3 0.3    0.3  0.3 0.3
## Base_w3       0.1  0.1  0.1  0.1 0.1 0.1    0.1  0.1 0.1
## ARDLS_w1      0.6  0.6  0.6  0.6 0.6 0.6    0.6  0.6 0.6
## ARDLS_w2      0.3  0.3  0.3  0.3 0.3 0.3    0.3  0.3 0.3
## ARDLS_w3      0.1  0.1  0.1  0.1 0.1 0.1    0.1  0.1 0.1
## Objective_Val 0.0  0.0  0.0  0.0 0.0 0.0    0.0  0.0 0.0
## ARDSE         0.0  0.0  0.0  0.0 0.0 0.0    0.0  0.0 0.0
## DSE_ARDLS     0.0  0.0  0.0  0.0 0.0 0.0    0.0  0.0 0.0
## DSE_Base      0.0  0.0  0.0  0.0 0.0 0.0    0.0  0.0 0.0
## ARP           0.0  0.0  0.0  0.0 0.0 0.0    0.0  0.0 0.0
## RMSV_Base     0.0  0.0  0.0  0.0 0.0 0.0    0.0  0.0 0.0
## RMSV_ARDLS    0.0  0.0  0.0  0.0 0.0 0.0    0.0  0.0 0.0

3.5 2D Solution Projection Plot

plot2DSolution_Case1 = plot2DSolution(PRM1, POList(PRM1),filename = "./figures/plot2d_c1.png", 
                                      titleText= "",plot_width = 12, plot_height = 10,,num_digits = 2)

## Plot saved successfully to ./figures/plot2d_c1.png (Width: 12, Height: 10)

options(repr.plot.width = 12, repr.plot.height = 10)
img1 <- png::readPNG("./figures/plot2d_c1.png")
grid::grid.raster(img1)

3.6 Interactive 3D Solution Space

fig_case1 <- plot3DSolution(PRM1, POList(PRM1),z_cap = 5,z_start = -0.1)
fig_case1

4 Case Study 2

4.1 Pairwise Reciprocal Matrix (PRM) Data

PRM2 <- matrix(c(
  1,   1/4, 1/2,
  4,   1,   3,
  2,   1/3, 1
), nrow = 3, byrow = TRUE)

4.2 Consistency Ratio (CR)

ConsistencyRatio(PRM2)

## [1] 0.0157713

4.3 Safe Regularizer Verification

ARDLS::safeRegularizer(PRM2,num_starts = 1000)

## Running DLS optimization from 1000 random starting points...
## 
## === DLS Optimization Results ===
## Global Minimum Objective Value: 0.34914603
## Number of Unique Optimal Solutions Found: 1
## Optimal Weight Vectors:
##         w1    w2    w3
## [1,] 0.148 0.622 0.229

## $lambda_star
## [1] 1
## 
## $case
## [1] "Case 1: Unique Convex Minimum (|W*| = 1, Delta_min > 0)"
## 
## $delta_min
## [1] 130.5419
## 
## $delta_max
## [1] 130.5419
## 
## $lambda_tilde
## [1] -65.27097
## 
## $num_solutions
## [1] 1
## 
## $W_star
##         w1    w2    w3
## [1,] 0.148 0.622 0.229

4.4 Comparison of Baseline Prioritization Operators (POs) and ARDLS Across Anchors

options(width = 200)
round(ARDLS::compare_ARDLS_anchors(PRM2, POs, reg_lambda = 1),4)

##                  NRS   NRCS   AMNC   NGMR     EV    SVD CosMax   PIGM    DLS
## Base_w1       0.1338 0.1433 0.1373 0.1365 0.1365 0.1473 0.1377 0.1476 0.1485
## Base_w2       0.6115 0.6337 0.6232 0.6250 0.6250 0.6302 0.6219 0.6316 0.6220
## Base_w3       0.2548 0.2230 0.2395 0.2385 0.2385 0.2225 0.2404 0.2208 0.2295
## ARDLS_w1      0.1484 0.1485 0.1485 0.1485 0.1485 0.1485 0.1485 0.1485 0.1485
## ARDLS_w2      0.6219 0.6221 0.6220 0.6220 0.6220 0.6221 0.6220 0.6221 0.6220
## ARDLS_w3      0.2296 0.2294 0.2295 0.2295 0.2295 0.2294 0.2295 0.2294 0.2295
## Objective_Val 0.3501 0.3494 0.3494 0.3494 0.3494 0.3493 0.3494 0.3493 0.3491
## ARDSE         0.3501 0.3494 0.3494 0.3494 0.3494 0.3493 0.3494 0.3493 0.3491
## DSE_ARDLS     0.3492 0.3491 0.3491 0.3491 0.3491 0.3491 0.3491 0.3491 0.3491
## DSE_Base      0.7041 0.4211 0.5234 0.5514 0.5514 0.3719 0.5106 0.3806 0.3491
## ARP           0.0010 0.0002 0.0002 0.0002 0.0002 0.0001 0.0002 0.0002 0.0000
## RMSV_Base     0.2797 0.2163 0.2412 0.2475 0.2475 0.2033 0.2382 0.2056 0.1970
## RMSV_ARDLS    0.1970 0.1970 0.1970 0.1970 0.1970 0.1970 0.1970 0.1970 0.1970

4.5 2D Solution Projection Plot

plot2DSolution_Case2 = plot2DSolution(PRM2, POList(PRM2),filename = "./figures/plot2d_c2.png", 
                                      titleText= "",plot_width = 12, plot_height = 10,num_digits = 2)

## Plot saved successfully to ./figures/plot2d_c2.png (Width: 12, Height: 10)

options(repr.plot.width = 12, repr.plot.height = 10)
img2 <- png::readPNG("./figures/plot2d_c2.png")
grid::grid.raster(img2)

4.6 Interactive 3D Solution Space

fig_case2 <- plot3DSolution(PRM2, POList(PRM2),z_cap =2,z_start = 0)
fig_case2

5 Case Study 3

5.1 Pairwise Reciprocal Matrix (PRM) Data

PRM3 <- matrix(c(
  1,   1/4,  4,
  4,   1,   1/4,
  1/4,  4,   1
), nrow = 3, byrow = TRUE)

5.2 Consistency Ratio (CR)

ConsistencyRatio(PRM3)

## [1] 1.939655

5.3 Structural Limitations of Unregularized DLS Solutions

DLS1 = function(PRM) solve_DLS(PRM, c(0.33, 0.33, 0.33))
DLS2 = function(PRM) solve_DLS(PRM,  c(0.34, 0.33, 0.33))
DLS3 = function(PRM) solve_DLS(PRM, c(0.33, 0.34, 0.33))
DLS4 = function(PRM) solve_DLS(PRM, c(0.33, 0.33, 0.34))


POList2 <-function(PRM)
{
    PO_results = list(
          NRS        = NRS(PRM),
          NRCS       = NRCS(PRM), 
          AMNC       = AMNC(PRM), 
          NGMR        = geoMean(PRM), 
          EV        = SaatyEigen(PRM), 
          SVD        = SVD(PRM), 
          CosMax     = CosMax(PRM), 
          PIGM       = PIGM(PRM),
          DLS1     = DLS1(PRM)$weights,
          DLS2      = DLS2(PRM)$weights,
          DLS3      = DLS3(PRM)$weights,
          DLS4     = DLS4(PRM)$weights
    )
}

5.3.1 Local Optimum of DLS

dls1 = DLS1(PRM3)
dls1

## $weights
## [1] 0.3333333 0.3333333 0.3333333
## 
## $obj_val
## [1] 28.6875

5.3.2 Global Optima of DLS

dls2 = DLS2(PRM3)
dls2

## $weights
## [1] 0.4683266 0.3170438 0.2146296
## 
## $obj_val
## [1] 28.44534

dls3 = DLS3(PRM3)
dls3

## $weights
## [1] 0.2146296 0.4683266 0.3170438
## 
## $obj_val
## [1] 28.44534

dls4 = DLS4(PRM3)
dls4

## $weights
## [1] 0.3170438 0.2146296 0.4683266
## 
## $obj_val
## [1] 28.44534

5.4 Evaluation of DLS Local Convexity Bound (\(\Delta_{DLS}\)) for Candidate Solutions

To illustrate the step-by-step calculations presented in Table 2 of the paper, the local convexity bounds (\(\Delta_{DLS}\)) are rounded to three decimal places.

Note: As detailed in the paper, the \(DLS_2\) solution is selected as a representative example to demonstrate the explicit calculation steps for \(\Delta_{DLS}\).

digits =  3

DLSConvexityBound(PRM = PRM3,w = round(dls1$weights,digits))

## [1] -9.018027

DLSConvexityBound(PRM = PRM3,w = round(dls2$weights,digits))

## [1] 29.94059

DLSConvexityBound(PRM = PRM3,w = round(dls3$weights,digits))

## [1] 29.94059

DLSConvexityBound(PRM = PRM3,w = round(dls3$weights,digits))

## [1] 29.94059

As illustrated in the paper, take DLS2 solution as an example for the calculation steps of \(Δ_{DLS}\)

5.4.1 Explicit Evaluation (Direct Formula Substitution)

round(dls2$weights,3)

## [1] 0.468 0.317 0.215

# Set the weights based on DLS2
w1 <- 0.468
w2 <- 0.317
w3 <- 0.215

# ==========================================
# For k = 1 (substituting a21 = 4, a31 = 0.25)
# ==========================================
k1_tm1_pt1 <- (2 * w2) / (w1^4) * (3 * w2 - 2 * 4 * w1)
k1_tm1_pt2 <- (2 * w3) / (w1^4) * (3 * w3 - 2 * 0.25 * w1)
k1_tm2_pt1 <- 2 / (w2^2)
k1_tm2_pt2 <- 2 / (w3^2)

k1_sum_t1 <- k1_tm1_pt1 + k1_tm1_pt2
k1_sum_t2 <- k1_tm2_pt1 + k1_tm2_pt2
delta_k1 <- k1_sum_t1 + k1_sum_t2

cat("For k = 1:\n")

## For k = 1:

cat(sprintf("= %.3f + %.3f + %.3f + %.3f = %.3f + %.3f = %.3f\n\n", 
            k1_tm1_pt1, k1_tm1_pt2, k1_tm2_pt1, k1_tm2_pt2, 
            k1_sum_t1, k1_sum_t2, delta_k1))

## = -36.913 + 3.684 + 19.903 + 43.267 = -33.229 + 63.169 = 29.941

# ==========================================
# For k = 2 (substituting a12 = 0.25, a32 = 4)
# ==========================================
k2_tm1_pt1 <- (2 * w1) / (w2^4) * (3 * w1 - 2 * 0.25 * w2)
k2_tm1_pt2 <- (2 * w3) / (w2^4) * (3 * w3 - 2 * 4 * w2)
k2_tm2_pt1 <- 2 / (w1^2)
k2_tm2_pt2 <- 2 / (w3^2)

k2_sum_t1 <- k2_tm1_pt1 + k2_tm1_pt2
k2_sum_t2 <- k2_tm2_pt1 + k2_tm2_pt2
delta_k2 <- k2_sum_t1 + k2_sum_t2

cat("For k = 2:\n")

## For k = 2:

cat(sprintf("= %.3f - %.3f + %.3f + %.3f = %.3f + %.3f = %.3f\n\n", 
            k2_tm1_pt1, abs(k2_tm1_pt2), k2_tm2_pt1, k2_tm2_pt2, 
            k2_sum_t1, k2_sum_t2, delta_k2))

## = 115.447 - 80.524 + 9.131 + 43.267 = 34.923 + 52.398 = 87.321

# ==========================================
# For k = 3 (substituting a13 = 4, a23 = 0.25)
# ==========================================
k3_tm1_pt1 <- (2 * w1) / (w3^4) * (3 * w1 - 2 * 4 * w3)
k3_tm1_pt2 <- (2 * w2) / (w3^4) * (3 * w2 - 2 * 0.25 * w3)
k3_tm2_pt1 <- 2 / (w1^2)
k3_tm2_pt2 <- 2 / (w2^2)

k3_sum_t1 <- k3_tm1_pt1 + k3_tm1_pt2
k3_sum_t2 <- k3_tm2_pt1 + k3_tm2_pt2
delta_k3 <- k3_sum_t1 + k3_sum_t2

cat("For k = 3:\n")

## For k = 3:

cat(sprintf("= %.3f + %.3f + %.3f + %.3f = %.3f + %.3f = %.3f\n\n", 
            k3_tm1_pt1, k3_tm1_pt2, k3_tm2_pt1, k3_tm2_pt2, 
            k3_sum_t1, k3_sum_t2, delta_k3))

## = -138.423 + 250.277 + 9.131 + 19.903 = 111.853 + 29.034 = 140.888

# ==========================================
# Global Local Curvature (Minimum)
# ==========================================
delta_DLS <- min(delta_k1, delta_k2, delta_k3)

cat("Taking the minimum across all coordinates yields Δ_DLS:\n")

## Taking the minimum across all coordinates yields Δ_DLS:

cat(sprintf("Δ_DLS  = min(%.3f, %.3f, 140.888) = %.3f\n", delta_k1, delta_k2, delta_DLS))

## Δ_DLS  = min(29.941, 87.321, 140.888) = 29.941

5.4.2 Functional Implementation for Curvature Evaluation

# Using the precise unrounded vector from the solver for exact matching

#w <- round(dls1$weights,3)  # -9.018
w <- round(dls2$weights,3)   # 29.941
#w <- round(dls3$weights,3)
#w <- round(dls4$weights,3)

A = PRM3
# Function to calculate Delta_DLS^(k)
calc_delta_k <- function(k, w, A) {
  n <- length(w)
  
  # Term 1: sum_{i != k} [ (2 * w_i / w_k^4) * (3 * w_i - 2 * A[i,k] * w_k) ]
  term1 <- 0
  for (i in 1:n) {
    if (i != k) {
      term1 <- term1 + (2 * w[i] / w[k]^4) * (3 * w[i] - 2 * A[i,k] * w[k])
    }
  }
  
  # Term 2: sum_{j != k} [ 2 / w_j^2 ]
  term2 <- 0
  for (j in 1:n) {
    if (j != k) {
      term2 <- term2 + (2 / w[j]^2)
    }
  }
  
  return(term1 + term2)
}

# Evaluate curvature across all coordinates k = 1, 2, 3
delta_k1 <- calc_delta_k(1, w, A)
delta_k2 <- calc_delta_k(2, w, A)
delta_k3 <- calc_delta_k(3, w, A)

cat(sprintf("Delta_DLS^(1) = %.3f\n", delta_k1))

## Delta_DLS^(1) = 29.941

cat(sprintf("Delta_DLS^(2) = %.3f\n", delta_k2))

## Delta_DLS^(2) = 87.321

cat(sprintf("Delta_DLS^(3) = %.3f\n", delta_k3))

## Delta_DLS^(3) = 140.888

# Find the global minimum curvature for this solution
delta_DLS <- min(delta_k1, delta_k2, delta_k3)
cat(sprintf("Local Delta_DLS = %.3f\n\n", delta_DLS))

## Local Delta_DLS = 29.941

# Compute the Safe Regularization Parameter (lambda^*)
delta_DLS_max <- delta_DLS
alpha <- 1.2

lambda_bound <- alpha * delta_DLS_max

cat(sprintf("Lower Bound (alpha * Delta_max) = %.3f\n", lambda_bound))

## Lower Bound (alpha * Delta_max) = 35.929

cat(sprintf("Selected Safe Regularizer (lambda^*) = %d\n", ceiling(lambda_bound)))

## Selected Safe Regularizer (lambda^*) = 36

5.5 Safe Regularizer Package Function Call

ARDLS::safeRegularizer(PRM3,num_starts = 1000)

## Running DLS optimization from 1000 random starting points...
## 
## === DLS Optimization Results ===
## Global Minimum Objective Value: 28.44534182
## Number of Unique Optimal Solutions Found: 3
## Optimal Weight Vectors:
##         w1    w2    w3
## [1,] 0.468 0.317 0.215
## [2,] 0.215 0.468 0.317
## [3,] 0.317 0.215 0.468

## $lambda_star
## [1] 35.9287
## 
## $case
## [1] "Case 2: Multiple Convex Minima (|W*| > 1, Delta_min > 0)"
## 
## $delta_min
## [1] 29.94059
## 
## $delta_max
## [1] 29.94059
## 
## $lambda_tilde
## [1] -14.97029
## 
## $num_solutions
## [1] 3
## 
## $W_star
##         w1    w2    w3
## [1,] 0.468 0.317 0.215
## [2,] 0.215 0.468 0.317
## [3,] 0.317 0.215 0.468

5.6 Comparison of Baseline Prioritization Operators (POs) and ARDLS Across Anchors

options(width = 200)
round(ARDLS::compare_ARDLS_anchors(PRM3, POs, reg_lambda = 36),4)

##                   NRS    NRCS    AMNC    NGMR      EV     SVD  CosMax    PIGM     DLS
## Base_w1        0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333
## Base_w2        0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333
## Base_w3        0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333
## ARDLS_w1       0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333
## ARDLS_w2       0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333
## ARDLS_w3       0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333  0.3333
## Objective_Val 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875
## ARDSE         28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875
## DSE_ARDLS     28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875
## DSE_Base      28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875 28.6875
## ARP            0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000
## RMSV_Base      1.7854  1.7854  1.7854  1.7854  1.7854  1.7854  1.7854  1.7854  1.7854
## RMSV_ARDLS     1.7854  1.7854  1.7854  1.7854  1.7854  1.7854  1.7854  1.7854  1.7854

5.7 2D Solution Projection Plot

plot2DSolution_Case3 = plot2DSolution(PRM3, POList2(PRM3),filename = "./figures/plot2d_c3.png", 
                                      titleText= "",plot_width = 12, plot_height = 10,,num_digits = 2)

## Plot saved successfully to ./figures/plot2d_c3.png (Width: 12, Height: 10)

options(repr.plot.width = 12, repr.plot.height = 10)
img3 <- png::readPNG("./figures/plot2d_c3.png")
grid::grid.raster(img3)

5.8 Interactive 3D Solution Surface

5.8.1 Interactive 3D Plot for Unregularized DLS

fig_case3a <- plot3DSolution(PRM3, POList2(PRM3),z_cap = 1.5,z_start = 28.3)
fig_case3a

5.8.2 Interactive 3D Plot for ARDLS

fig_case3b <- plot3DARDLS(
  PRM      = PRM3, 
  POs      = POList2(PRM3), 
  lambda   = 36,                 # Regularization penalty
  w_anchor = POList(PRM3)$PIGM,    # Anchor weights
  z_cap    = 1.5,                  # Height cutoff for the surface
  z_start  = 28.5                    # Base of the Z-axis
)
fig_case3b

6 Case Study 4

6.1 Pairwise Reciprocal Matrix (PRM) Data

PRM4 <- matrix(c(
    1,   4,   3,   1,   3,   4,
  1/4,   1,   7,   3, 1/5,   1,
  1/3, 1/7,   1, 1/5, 1/5, 1/6,
    1, 1/3,   5,   1,   1, 1/3,
  1/3,   5,   5,   1,   1,   3,
  1/4,   1,   6,   3, 1/3,   1
), nrow = 6, ncol = 6, byrow = TRUE)

6.2 Consistency Ratio (CR)

ConsistencyRatio(PRM4)

## [1] 0.2290113

6.3 Safe Regularizer Verification

ARDLS::safeRegularizer(PRM4,num_starts = 1000)

## Running DLS optimization from 1000 random starting points...
## 
## === DLS Optimization Results ===
## Global Minimum Objective Value: 60.01548404
## Number of Unique Optimal Solutions Found: 1
## Optimal Weight Vectors:
##         w1   w2    w3   w4   w5    w6
## [1,] 0.184 0.22 0.037 0.15 0.21 0.197

## $lambda_star
## [1] 1
## 
## $case
## [1] "Case 1: Unique Convex Minimum (|W*| = 1, Delta_min > 0)"
## 
## $delta_min
## [1] 1300.278
## 
## $delta_max
## [1] 1300.278
## 
## $lambda_tilde
## [1] -650.1389
## 
## $num_solutions
## [1] 1
## 
## $W_star
##         w1   w2    w3   w4   w5    w6
## [1,] 0.184 0.22 0.037 0.15 0.21 0.197

6.4 Comparison of Baseline Prioritization Operators (POs) and ARDLS Across Anchors

options(width = 200)
round(ARDLS::compare_ARDLS_anchors(PRM4, POs, reg_lambda = 1),4)

##                   NRS    NRCS    AMNC    NGMR      EV      SVD  CosMax     PIGM     DLS
## Base_w1        0.2421  0.3812  0.3047  0.3160  0.3208   0.4010  0.2926   0.4150  0.1845
## Base_w2        0.1884  0.1052  0.1486  0.1391  0.1395   0.1032  0.1554   0.0936  0.2204
## Base_w3        0.0309  0.0447  0.0382  0.0360  0.0348   0.0412  0.0395   0.0348  0.0371
## Base_w4        0.1312  0.1312  0.1414  0.1251  0.1285   0.1212  0.1465   0.1123  0.1504
## Base_w5        0.2321  0.2106  0.2208  0.2360  0.2374   0.2112  0.2144   0.2190  0.2103
## Base_w6        0.1753  0.1271  0.1463  0.1477  0.1391   0.1221  0.1517   0.1253  0.1973
## ARDLS_w1       0.1845  0.1847  0.1846  0.1846  0.1846   0.1847  0.1846   0.1847  0.1845
## ARDLS_w2       0.2204  0.2203  0.2203  0.2203  0.2203   0.2203  0.2203   0.2203  0.2204
## ARDLS_w3       0.0371  0.0371  0.0371  0.0371  0.0371   0.0371  0.0371   0.0371  0.0371
## ARDLS_w4       0.1504  0.1504  0.1504  0.1504  0.1504   0.1504  0.1504   0.1504  0.1504
## ARDLS_w5       0.2104  0.2104  0.2104  0.2104  0.2104   0.2104  0.2104   0.2104  0.2103
## ARDLS_w6       0.1973  0.1972  0.1972  0.1972  0.1972   0.1972  0.1973   0.1972  0.1973
## Objective_Val 60.0212 60.0728 60.0379 60.0431 60.0452  60.0826 60.0335  60.0914 60.0155
## ARDSE         60.0212 60.0728 60.0379 60.0431 60.0452  60.0826 60.0335  60.0914 60.0155
## DSE_ARDLS     60.0155 60.0155 60.0155 60.0155 60.0155  60.0156 60.0155  60.0156 60.0155
## DSE_Base      74.9265 95.1335 77.3477 85.2792 89.8472 107.7832 73.1656 138.2288 60.0155
## ARP            0.0057  0.0572  0.0224  0.0276  0.0297   0.0670  0.0180   0.0758  0.0000
## RMSV_Base      1.4427  1.6256  1.4658  1.5391  1.5798   1.7303  1.4256   1.9595  1.2912
## RMSV_ARDLS     1.2912  1.2912  1.2912  1.2912  1.2912   1.2912  1.2912   1.2912  1.2912