# A Hierarchical Aggregation Approach for Indicators Based on Data Envelopment Analysis and Analytic Hierarchy Process

## Abstract

**:**

## 1. Introduction

## 2. Methodology

- Computing the composite value of each DMU using one-level DEA-based CI model (4). The computed composite values are applied in three-level DEA-based CI model (6).
- Computing the priority weights of indicators for all DMUs using AHP, which impose weight bounds into model (6).
- Obtaining an optimal set of weights for each DMU using three-level DEA-based CI model (6) (minimum composite loss η).
- Obtaining an optimal set of weights for each DMU using model (6) bounded by AHP (maximum composite loss κ). Note that if the AHP weights are added to model (6), we obtain model (10).
- Measuring the performance of each DMU in terms of the relative closeness to the priority weights of indicators. For this purpose, we develop parameter-distance model (11). Increasing a parameter in a defined range of composite loss we explore how much a DM can achieve its goals. This may result in various ranking positions for a DMU in comparison to the other DMUs.

**Figure 1.**A hierarchical aggregation approach for indicators using a three-level Data Envelopment Analysis (DEA) and Analytic Hierarchy Process (AHP).

#### 2.1. DEA-Based CI Model

#### 2.2. Three-Level DEA-Based CI Model

#### 2.3. Prioritizing Indicator Weights Using AHP

#### 2.4. A Parametric Distance Model

## 3. A Numerical Example: Road Safety Performance Indicators

Countries | Alcohol | Speed | Protective Systems | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

% of Drivers above Legal Alcohol Limit In Roadside Police Tests | % of Alcohol Related Fatalities | Mean Speed | Speed Limit Violation (%) | Seat Belt | Child Restraint | ||||||

Motorways | Rural Roads | Urban Roads | Motorways | Rural Roads | Urban Roads | Daytime Seatbelt Wearing Rate in Front Seats of Light Vehicles (%) | Daytime Seatbelt Wearing Rate in Rear Seats of Light Vehicles (%) | Daytime Usage Rate of Child Restraints (%) | |||

AUT | 0.116 | 0.463 | 0.938 | 0.781 | 0.802 | 0.766 | 0.051 | 0.254 | 0.904 | 0.699 | 0.863 |

BEL | 0.068 | 0.654 | 0.846 | 0.743 | 0.768 | 0.348 | 0.029 | 0.222 | 0.799 | 0.488 | 0.729 |

FIN | 0.593 | 0.136 | 0.963 | 0.729 | 0.907 | 0.409 | 0.023 | 0.323 | 0.911 | 0.922 | 0.716 |

FRA | 0.263 | 0.123 | 0.933 | 0.787 | 0.838 | 0.505 | 0.037 | 0.318 | 1.000 | 1.000 | 0.937 |

HUN | 0.279 | 0.283 | 0.955 | 0.793 | 0.817 | 0.362 | 0.033 | 0.230 | 0.727 | 0.501 | 0.433 |

IRL | 0.237 | 0.119 | 0.945 | 0.762 | 0.724 | 1.000 | 0.032 | 0.223 | 0.901 | 0.914 | 0.857 |

LTU | 0.555 | 0.321 | 1.000 | 0.713 | 0.714 | 0789 | 0.025 | 0.318 | 0.609 | 0.366 | 0.404 |

NLD | 0.081 | 1.000 | 0.899 | 0.740 | 0.881 | 0.454 | 0.020 | 0.234 | 0.959 | 0.890 | 0.758 |

POL | 0.091 | 0.438 | 0.806 | 0.697 | 0.647 | 0.290 | 0.015 | 0.165 | 0.799 | 0.589 | 0.905 |

PRT | 0.137 | 0.610 | 0.847 | 0.618 | 0.919 | 0.302 | 0.014 | 0.360 | 0.881 | 0.574 | 0.591 |

SVN | 0.122 | 0.078 | 0.964 | 1.000 | 0.713 | 0.480 | 1.000 | 0.163 | 0.874 | 0.551 | 0.672 |

SWE | 1.000 | 0.357 | 0.883 | 0.717 | 0.870 | 0.241 | 0.019 | 0.259 | 0.973 | 0.927 | 1.000 |

CHE | 0.277 | 0.230 | 0.943 | 0.757 | 1.000 | 0.710 | 0.043 | 1.000 | 0.887 | 0.805 | 0.895 |

Objective Level | Criteria Level | Sub-Criteria Level | Sub-Sub-Criteria Level |
---|---|---|---|

Prioritizing road user behavior | Alcohol ${w}_{1}=$ 0.2727 | % of drivers above legal alcohol limit ${e}_{11}=$ 0.333 | % of drivers above legal alcohol limit ${f}_{111}=$ 1.000 |

% of alcohol-related fatalities ${e}_{12}=$ 0.667 | % of alcohol-related fatalities ${f}_{121}=$ 1.000 | ||

Speed ${w}_{2}=$ 0.5454 | Mean speed ${e}_{21}=$ 0.60 | Mean speed of vehicles on motorways ${f}_{211}=$ 0.081 | |

Mean speed of vehicles on rural roads, ${f}_{212}=$ 0.342 | |||

Mean speed of vehicles on urban roads, ${f}_{213}=$ 0.577 | |||

Speed limit violations ${e}_{22}=$ 0.40 | % of vehicles exceeding the speed limit on motorways ${f}_{221}=$ 0.081 | ||

% of vehicles exceeding the speed limit on rural roads ${f}_{222}=$ 0.342 | |||

% of vehicles exceeding the speed limit on urban roads ${f}_{223}=$ 0.577 | |||

Protective systems ${w}_{3}=$ 0.1818 | Seat belt ${e}_{31}=$ 0.40 | Daytime seatbelt wearing rate in front seats of light vehicles (%) ${f}_{311}=$ 0.60 | |

Daytime seatbelt wearing rate in rear seats of light vehicles (%) ${f}_{312}=$ 0.40 | |||

Child Restraint ${e}_{32}=$ 0.60 | Daytime usage rate of child restraints (%) ${f}_{321}=$ 1.000 |

Weights of Categories | Weights of Sub-Categories | Weights of Sub-Sub-Categories |
---|---|---|

${p}_{1}\text{\hspace{0.17em}}=$ 0.0361 | ${p}_{11}^{\prime}=$ 0.0000 | ${u}_{111}^{\prime}=$ 0.0000 |

${p}_{12}^{\prime}=$ 0.0361 | ${u}_{121}^{\prime}=$ 0.0361 | |

${p}_{2}=$ 0.9415 | ${p}_{21}^{\prime}=$ 0.9415 | ${u}_{211}^{\prime}=$ 0.9415 |

${u}_{212}^{\prime}=$ 0.0000 | ||

${u}_{213}^{\prime}=$ 0.0000 | ||

${p}_{22}^{\prime}=$ 0.0000 | ${u}_{221}^{\prime}=$ 0.0000 | |

${u}_{222}^{\prime}=$ 0.0000 | ||

${u}_{223}^{\prime}=$ 0.0000 | ||

${p}_{3}=$ 0.1160 | ${p}_{31}^{\prime}=$ 0.0000 | ${u}_{311}^{\prime}=$ 0.0000 |

${u}_{312}^{\prime}=$ 0.0000 | ||

${p}_{32}^{\prime}=$ 0.1160 | ${u}_{321}^{\prime}=$ 0.1160 | |

$\eta =0.0000$ |

Countries | $C{{I}_{k}}^{*}$ | η | κ |
---|---|---|---|

AUT | 1.000 | 0.000 | 0.158 |

BEL | 0.938 | 0.000 | 0.127 |

FIN | 1.000 | 0.000 | 0.173 |

FRA | 1.000 | 0.000 | 0.184 |

HUN | 1.000 | 0.000 | 0.284 |

IRL | 1.000 | 0.000 | 0.254 |

LTU | 1.000 | 0.000 | 0.270 |

NLD | 1.000 | 0.000 | 0.023 |

POL | 0.955 | 0.000 | 0.224 |

PRT | 0.978 | 0.000 | 0.139 |

SVN | 1.000 | 0.000 | 0.208 |

SWE | 1.000 | 0.000 | 0.040 |

CHE | 1.000 | 0.000 | 0.000 |

Weights of Categories | Weights of Sub-Categories | Weights of Sub-Sub-Categories |
---|---|---|

${p}_{1}=$ 0.4089 | ${p}_{11}^{\prime}=$ 0.1362 | ${u}_{111}^{\prime}=$ 0.1362 |

${p}_{12}^{\prime}=$ 0.2727 | ${u}_{121}^{\prime}=$ 0.2727 | |

${p}_{2}=$ 0.8178 | ${p}_{21}^{\prime}=$ 0.4907 | ${u}_{211}^{\prime}=$ 0.0397 |

${u}_{212}^{\prime}=$ 0.1678 | ||

${u}_{213}^{\prime}=$ 0.2831 | ||

${p}_{22}^{\prime}=$ 0.3271 | ${u}_{221}^{\prime}=$ 0.0265 | |

${u}_{222}^{\prime}=$ 0.1119 | ||

${u}_{223}^{\prime}=$ 0.1888 | ||

${p}_{3}=$ 0.2726 | ${p}_{31}^{\prime}=$ 0.1090 | ${u}_{311}^{\prime}=$ 0.0654 |

${u}_{312}^{\prime}=$ 0.0436 | ||

${p}_{32}^{\prime}=$ 0.1636 | ${u}_{321}^{\prime}=$ 0.1636 | |

$\alpha =$ 1.4993 |

**Table 6.**The ranking position of each country based on the minimum distance to priority weights of SPIs.

Countries | ${Z}^{*}(\eta )$ | Rank |
---|---|---|

AUT | 0.332 | 6 |

BEL | 0.705 | 11 |

FIN | 0.438 | 9 |

FRA | 0.338 | 7 |

HUN | 0.591 | 10 |

IRL | 0.331 | 5 |

LTU | 0.381 | 8 |

NLD | 0.021 | 2 |

POL | 0.718 | 12 |

PRT | 0.782 | 13 |

SVN | 0.166 | 4 |

SWE | 0.035 | 3 |

CHE | 0.000 | 1 |

**Figure 4.**The relative closeness to the priority weights of indicators [∆(θ)], versus composite loss (θ) for each country.

## 4. Conclusions

## Conflicts of Interest

## Appendix A

**Table A1.**The measure of relative closeness to the priority weights of hierarchical SPIs [${\Delta}_{k}(\theta )$ ] vs. composite loss [θ] for each country.

θ | AUT | BEL | FIN | FRA | HUN | IRL | LTU | NLD | POL | PRT | SVN | SWE | CHE |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.0000 |

Rank | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | N/A | 1 |

0.01 | 0.0788 | 0.3002 | 0.0808 | 0.0595 | 0.0784 | 0.0450 | 0.0419 | 0.4367 | 0.1951 | 0.2454 | 0.0492 | 0.2521 | 1.0000 |

Rank | 8 | 3 | 7 | 10 | 9 | 12 | 13 | 2 | 6 | 5 | 11 | 4 | 1 |

0.02 | 0.1479 | 0.4018 | 0.1497 | 0.1190 | 0.1295 | 0.0863 | 0.0823 | 0.8733 | 0.2640 | 0.3572 | 0.0983 | 0.5041 | 1.0000 |

Rank | 8 | 4 | 7 | 10 | 9 | 12 | 13 | 2 | 6 | 5 | 11 | 3 | 1 |

0.03 | 0.2160 | 0.4852 | 0.2148 | 0.1784 | 0.1793 | 0.1269 | 0.1223 | 1.0000 | 0.3268 | 0.4286 | 0.1473 | 0.7562 | 1.0000 |

Rank | 7 | 4 | 8 | 10 | 9 | 12 | 13 | 1 | 6 | 5 | 11 | 3 | 1 |

0.04 | 0.2838 | 0.5605 | 0.2797 | 0.2378 | 0.2272 | 0.1675 | 0.1624 | 1.0000 | 0.3845 | 0.5133 | 0.1963 | 1.0000 | 1.0000 |

Rank | 7 | 4 | 8 | 9 | 10 | 12 | 13 | 1 | 6 | 5 | 11 | 1 | 1 |

0.05 | 0.3512 | 0.6312 | 0.3444 | 0.2970 | 0.2679 | 0.2081 | 0.2023 | 1.0000 | 0.4418 | 0.5846 | 0.2452 | 1.0000 | 1.0000 |

Rank | 7 | 4 | 8 | 9 | 10 | 12 | 13 | 1 | 6 | 5 | 11 | 1 | 1 |

0.06 | 0.4185 | 0.7113 | 0.4091 | 0.3562 | 0.3073 | 0.2486 | 0.2422 | 1.0000 | 0.4832 | 0.6491 | 0.2941 | 1.0000 | 1.0000 |

Rank | 7 | 4 | 8 | 9 | 10 | 12 | 13 | 1 | 6 | 5 | 11 | 1 | 1 |

0.07 | 0.4857 | 0.7813 | 0.4736 | 0.4152 | 0.3454 | 0.2891 | 0.2819 | 1.0000 | 0.5195 | 0.7121 | 0.3428 | 1.0000 | 1.0000 |

Rank | 7 | 4 | 8 | 9 | 10 | 12 | 13 | 1 | 6 | 5 | 11 | 1 | 1 |

0.08 | 0.5527 | 0.8464 | 0.5379 | 0.4740 | 0.3818 | 0.3296 | 0.3215 | 1.0000 | 0.5556 | 0.7739 | 0.3913 | 1.0000 | 1.0000 |

Rank | 7 | 4 | 8 | 9 | 11 | 12 | 13 | 1 | 6 | 5 | 10 | 1 | 1 |

0.09 | 0.6196 | 0.8991 | 0.6019 | 0.5325 | 0.4163 | 0.3699 | 0.3609 | 1.0000 | 0.5915 | 0.8331 | 0.4397 | 1.0000 | 1.0000 |

Rank | 6 | 4 | 7 | 9 | 11 | 12 | 13 | 1 | 8 | 5 | 10 | 1 | 1 |

0.1 | 0.6861 | 0.9396 | 0.6655 | 0.5906 | 0.4497 | 0.4102 | 0.4001 | 1.0000 | 0.6273 | 0.8858 | 0.4879 | 1.0000 | 1.0000 |

Rank | 6 | 4 | 7 | 9 | 11 | 12 | 13 | 1 | 8 | 5 | 10 | 1 | 1 |

0.11 | 0.7520 | 0.9692 | 0.7283 | 0.6481 | 0.4829 | 0.4504 | 0.4391 | 1.0000 | 0.6628 | 0.9290 | 0.5358 | 1.0000 | 1.0000 |

Rank | 6 | 4 | 7 | 9 | 11 | 12 | 13 | 1 | 8 | 5 | 10 | 1 | 1 |

0.12 | 0.8166 | 0.9869 | 0.7896 | 0.7046 | 0.5160 | 0.4905 | 0.4778 | 1.0000 | 0.6980 | 0.9645 | 0.5833 | 1.0000 | 1.0000 |

Rank | 6 | 4 | 7 | 8 | 11 | 12 | 13 | 1 | 9 | 5 | 10 | 1 | 1 |

0.13 | 0.8777 | 1.0000 | 0.8476 | 0.7596 | 0.5490 | 0.5304 | 0.5161 | 1.0000 | 0.7328 | 0.9838 | 0.6309 | 1.0000 | 1.0000 |

Rank | 14 | 1 | 15 | 16 | 19 | 20 | 21 | 1 | 17 | 11 | 18 | 1 | 1 |

0.14 | 0.9271 | 1.0000 | 0.8967 | 0.8115 | 0.5818 | 0.5701 | 0.5540 | 1.0000 | 0.7669 | 1.0000 | 0.6784 | 1.0000 | 1.0000 |

Rank | 14 | 1 | 15 | 16 | 19 | 20 | 21 | 1 | 17 | 1 | 18 | 1 | 1 |

0.15 | 0.9681 | 1.0000 | 0.9289 | 0.8577 | 0.6145 | 0.6096 | 0.5913 | 1.0000 | 0.8007 | 1.0000 | 0.7260 | 1.0000 | 1.0000 |

Rank | 14 | 1 | 15 | 16 | 19 | 20 | 21 | 1 | 17 | 1 | 18 | 1 | 1 |

0.16 | 1.0000 | 1.0000 | 0.9592 | 0.8998 | 0.6468 | 0.6489 | 0.6279 | 1.0000 | 0.8340 | 1.0000 | 0.7735 | 1.0000 | 1.0000 |

Rank | 14 | 1 | 15 | 16 | 20 | 19 | 21 | 1 | 17 | 1 | 18 | 1 | 1 |

0.17 | 1.0000 | 1.0000 | 0.9896 | 0.9409 | 0.6789 | 0.6880 | 0.6635 | 1.0000 | 0.8665 | 1.0000 | 0.8211 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 7 | 8 | 12 | 11 | 13 | 1 | 9 | 1 | 10 | 1 | 1 |

0.18 | 1.0000 | 1.0000 | 1.0000 | 0.9820 | 0.7104 | 0.7269 | 0.6987 | 1.0000 | 0.8973 | 1.0000 | 0.8687 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 8 | 12 | 11 | 13 | 1 | 9 | 1 | 10 | 1 | 1 |

0.19 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.7415 | 0.7655 | 0.7335 | 1.0000 | 0.9253 | 1.0000 | 0.9162 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 12 | 11 | 13 | 1 | 9 | 1 | 10 | 1 | 1 |

0.2 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.7724 | 0.8036 | 0.7677 | 1.0000 | 0.9538 | 1.0000 | 0.9638 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 12 | 11 | 13 | 1 | 10 | 1 | 9 | 1 | 1 |

0.21 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.8029 | 0.8408 | 0.8015 | 1.0000 | 0.9736 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 12 | 11 | 13 | 1 | 10 | 1 | 1 | 1 | 1 |

0.22 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.8329 | 0.8772 | 0.8349 | 1.0000 | 0.9919 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 13 | 11 | 12 | 1 | 10 | 1 | 1 | 1 | 1 |

0.23 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.8621 | 0.9136 | 0.8681 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 13 | 11 | 12 | 1 | 1 | 1 | 1 | 1 | 1 |

0.24 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.8905 | 0.9501 | 0.9013 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 13 | 11 | 12 | 1 | 1 | 1 | 1 | 1 | 1 |

0.25 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9180 | 0.9865 | 0.9345 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 13 | 11 | 12 | 1 | 1 | 1 | 1 | 1 | 1 |

0.26 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9432 | 1.0000 | 0.9677 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 13 | 1 | 12 | 1 | 1 | 1 | 1 | 1 | 1 |

0.27 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9669 | 1.0000 | 0.9999 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 13 | 1 | 12 | 1 | 1 | 1 | 1 | 1 | 1 |

0.28 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 0.9906 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

0.29 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

Rank | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

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**MDPI and ACS Style**

Pakkar, M.S.
A Hierarchical Aggregation Approach for Indicators Based on Data Envelopment Analysis and Analytic Hierarchy Process. *Systems* **2016**, *4*, 6.
https://doi.org/10.3390/systems4010006

**AMA Style**

Pakkar MS.
A Hierarchical Aggregation Approach for Indicators Based on Data Envelopment Analysis and Analytic Hierarchy Process. *Systems*. 2016; 4(1):6.
https://doi.org/10.3390/systems4010006

**Chicago/Turabian Style**

Pakkar, Mohammad Sadegh.
2016. "A Hierarchical Aggregation Approach for Indicators Based on Data Envelopment Analysis and Analytic Hierarchy Process" *Systems* 4, no. 1: 6.
https://doi.org/10.3390/systems4010006