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Bipolar Multicriteria Aggregation-Disaggregation Robustness Approach: Theory and Application on European e-Government Benchmarking

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Part of the Studies in Systems, Decision and Control book series (SSDC,volume 407)

Abstract

The aggregation-disaggregation approach is considered as an important tool at the disposal of decision analysts and decision-makers when addressing multiple criteria decision-making problems. This paper proposes a bipolar robustness control approach, implemented in conjunction with the UTASTAR method, where the multicriteria evaluation model is an additive value function. The disaggregation pole of this new algorithm measures and controls the robustness of the evaluation model, as inferred by the decision-maker's preference statements, while the aggregation pole assesses the stability of the results. The bipolar robustness control is complemented with several visualization measures and robustness indicators, the fulfilment of which guarantees the soundness of the model and validates its results. In the end, the methodology is applied to the problem of e-government readiness evaluation in Europe, resulting in the ranking of 22 European countries.

Keywords

  • Multiple criteria
  • Aggregation-disaggregation approach
  • e-government
  • Robustness analysis

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Abbreviations

ARP:

Average Range of Preferential Parameters

ARRI:

Average Range of the Ranking Index

ASI:

Average Stability Index

AVG:

Average

DM:

Decision-Maker

ERA:

Extreme Ranking Analysis

GDP:

Gross Domestic Product

ICT:

Information and Communications Technology

LP:

Linear Programming

R&D:

Research and Development

RARR:

Ratio of the Average Range of the Ranking

ROR:

Robust Ordinal Regression

SPRI:

Statistical Preference Relations Index

UTA:

UTilités Additives

References

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Appendices

Appendix A: Multicriteria Evaluation of European Countries on the Eight e-Government Evaluation Criteria

Country g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8
Belgium 90.00 67.75 2.28 63.00 0.25 36.00 74.33 58.00
Czech Republic 88.00 63.76 1.91 56.00 0.25 21.33 87.67 44.50
Denmark 96.00 72.78 3.05 85.00 0.55 65.33 89.33 65.00
Germany 93.50 71.26 2.94 67.00 0.70 33.33 58.67 45.50
Estonia 89.50 69.50 1.74 87.00 0.76 35.00 80.00 77.50
Ireland 90.00 66.17 1.58 87.00 0.65 43.00 88.00 62.00
Greece 77.50 59.62 0.78 46.00 0.80 27.67 78.33 41.00
Spain 86.00 66.42 1.24 91.00 0.78 36.33 69.00 72.50
France 91.00 68.51 2.23 75.00 0.96 44.00 89.00 68.50
Croatia 82.00 61.67 0.81 53.00 0.33 19.33 81.00 48.00
Italy 85.50 63.35 1.25 77.00 0.78 15.67 69.67 60.50
Hungary 81.50 61.96 1.41 45.00 0.45 34.33 82.33 35.50
Netherlands 98.00 77.40 1.98 82.00 1.00 57.67 80.67 65.50
Austria 89.50 67.41 2.81 86.00 0.63 40.33 80.67 70.50
Poland 84.00 62.50 0.87 76.00 0.49 17.33 81.67 51.00
Portugal 81.00 63.63 1.36 96.00 0.65 30.67 81.00 74.00
Slovenia 87.50 67.39 2.59 68.00 0.39 37.00 85.00 63.00
Slovakia 88.00 63.53 0.83 72.00 0.63 33.00 80.67 30.00
Finland 95.00 75.46 3.32 86.00 0.71 64.00 91.33 71.00
Sweden 94.00 74.94 3.21 83.00 0.61 60.33 90.67 68.50
Norway 95.00 71.42 1.69 78.00 0.69 64.33 84.33 63.50
Un. Kingdom 92.50 72.33 1.63 74.00 0.96 35.00 75.00 51.00

Appendix B: The UTASTAR Disaggregation Algorithm

The UTASTAR method is an improved version of the original UTA method (see [9]). UTASTAR uses a double positive error function, so that the value of each reference action \(a\in {A}_{R}\) can be written as:

$$u^{\prime}[{\mathbf{g}}(a)] = \sum\limits_{i = 1}^{n} {u_{i} [g_{i} (a)] - \sigma^{ + } (a) + \sigma^{ - } (a)} \, \forall a \in A_{R}$$

where \({\sigma }^{+}\) and \({\sigma }^{-}\) are the underestimation and the overestimation error, respectively.

Based on the above, the UTASTAR algorithm may be summarized in the following steps:

Step 1.

Express the global value of reference actions \(u\left[{\varvec{g}}\left({a}_{k}\right)\right]\), \(k=\mathrm{1,2},\dots m\), first in terms of the marginal values \({u}_{i}\left({g}_{i}\right)\), and then in terms of the variables \({w}_{ij}\):

$$\left\{ \begin{gathered} u_{i} (g_{i}^{1} ) = 0 \,\,\, \forall i = 1,2, \ldots ,n \hfill \\ u_{i} (g_{i}^{j} ) = \sum\limits_{t = 1}^{j - 1} {w_{it} } \,\,\, \forall i = 1,2, \ldots ,n{\text{ and }}j = 2,3, \ldots ,\alpha_{i} \hfill \\ \end{gathered} \right.$$

Step 2.

Introduce two error functions \({\sigma }^{+}\) and \({\sigma }^{-}\) on \({A}_{R}\) by writing for each pair of consecutive actions in the ranking of the analytic expressions:

$$\Delta (a_{k} ,a_{k + 1} ) = u[{\mathbf{g}}(a_{k} )] - \sigma^{ + } (a_{k} ) + \sigma^{ - } (a_{k} ) - u[{\mathbf{g}}(a_{k + 1} )] + \sigma^{ + } (a_{k + 1} ) - \sigma^{ - } (a_{k + 1} )$$

Step 3.

Solve the following LP:

$$[\min ]z = \sum\limits_{k = 1}^{m} {[\sigma^{ + } (a_{k} ) + \sigma^{ - } (a_{k} )]}$$

Subject to:

$$\begin{gathered} \left. \begin{gathered} \Delta (a_{k} ,a_{k + 1} ) \ge \delta {\text{ if }}a_{k} \succ a_{k + 1} \hfill \\ \Delta (a_{k} ,a_{k + 1} ) = 0{\text{ if }}a_{k} \sim a_{k + 1} \hfill \\ \end{gathered} \right\} \,\,\, \forall k \hfill \\ \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{{\alpha_{i} - 1}} {w_{ij} } = 1} \hfill \\ w_{ij} \ge 0, \, \sigma^{ + } (a_{k} ) \ge 0, \,\,\, \sigma^{ - } (a_{k} ) \ge 0 \,\,\, \forall i,j{\text{ and }}k \hfill \\ \end{gathered}$$

where \({a}_{k}\) and \({a}_{k+1}\) are two successive actions in the DM’s ranking and \(\delta\) is a small positive number, indicating the preference threshold between the two actions.

Step 4.

Test the existence of multiple or near optimal solutions of the LP (stability/robustness analysis); in case of non-uniqueness, find the mean additive value function as the most representative (barycenter) of those (near) optimal solutions which maximize/minimize the objective functions:

$$u_{i} (g_{i}^{j} ) = \sum\limits_{{t = 1}}^{{j - 1}} {w_{{it}} } \;{\text{for}}\,i = 1,2, \ldots ,n\,{\text{and}}\,j = 2,3, \ldots ,\alpha _{i}.$$

on the polyhedron of the constraints of the previous LP bounded by the new constraint:

$$\sum\limits_{k = 1}^{m} {\left[ {\sigma^{ + } (a_{k} ) + \sigma^{ - } (a_{k} )} \right]} \le z^{*} + \varepsilon$$

where \(z^{*}\) is the optimal value of the LP in step 3 and \(\varepsilon\) is a very small positive number.

The number of LPs that have to be solved in this step (and the corresponding value functions obtained) is \(2 \cdot \sum\nolimits_{i = 1}^{n} {(\alpha_{i} - 1)}\). In most of the UTASTAR applications, one usually seeks value functions that are free of errors (all error variables \(\sigma\) are zero), since no relaxation from the minimal error is allowed \((\varepsilon = 0)\).

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Siskos, E., Kourousias, G., Siskos, Y. (2022). Bipolar Multicriteria Aggregation-Disaggregation Robustness Approach: Theory and Application on European e-Government Benchmarking. In: Kulkarni, A.J. (eds) Multiple Criteria Decision Making. Studies in Systems, Decision and Control, vol 407. Springer, Singapore. https://doi.org/10.1007/978-981-16-7414-3_7

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