A DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP

Abstract

Pairwise comparison matrix (PCM) as a crucial component of Analytic Hierarchy Process (AHP) presents the preference relations among alternatives. However, in many cases, the PCM is difficult to be completed, which obstructs the subsequent operations of the classical AHP. In this paper, based on decision-making and trial evaluation laboratory (DEMATEL) method which has ability to derive the total relation matrix from direct relation matrix, a new completion method for incomplete pairwise comparison matrix (iPCM) is proposed. The proposed method provides a new perspective to estimate the missing values in iPCMs with explicit physical meaning, which is straightforward and flexible. Several experiments are implemented as well to present the completion ability of the proposed method and some insights into the proposed method and matrix consistency.

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Acknowledgements

The authors greatly appreciate the reviewer’s constructive suggestions and the editor’s encouragement. The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61573290, 61503237).

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Correspondence to Yong Deng.

Appendix

Appendix

A Initial iPCMs

$$\begin{aligned} \begin{array}{llll} \mathbf{Order~4 } &{} \\ \hbox {Example 1:} &{} \hbox {Example 2:} &{} \hbox {Example 3:}&{} *\hbox {Example 4:}\\ \left[ {\begin{array}{cccc} 1 &{}\quad 1 &{}\quad 5 &{}\quad 2 \\ 1 &{}\quad 1 &{}\quad 3 &{}\quad 4 \\ 0.20 &{}\quad 0.33 &{}\quad 1 &{}\quad * \\ 0.50 &{}\quad 0.25 &{}\quad * &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{cccc} 1 &{}\quad 0.80 &{}\quad 1.55 &{}\quad 1 \\ 1.25 &{}\quad 1 &{}\quad * &{}\quad 3.65 \\ 0.65 &{}\quad * &{}\quad 1 &{}\quad 1.93 \\ 1 &{}\quad 0.27 &{}\quad 0.52 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{cccc} 1 &{}\quad 0.33 &{}\quad 0.25 &{}\quad 0.11 \\ 3 &{}\quad 1 &{}\quad * &{}\quad 0.14 \\ 4 &{}\quad * &{}\quad 1 &{}\quad 0.25 \\ 9 &{}\quad 7 &{}\quad 4 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{cccc} 1 &{}\quad 2 &{}\quad 4 &{}\quad * \\ 0.50 &{}\quad 1 &{}\quad 2 &{}\quad 4 \\ 0.25 &{}\quad 0.50 &{}\quad 1 &{}\quad 2 \\ * &{}\quad 0.25 &{}\quad 0.50 &{}\quad 1 \end{array}}\right] \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} \mathbf{{Order~5} } &{} \\ \hbox {Example 1:} &{} \hbox {Example 2:} &{}\hbox {Example 3:}&{}*\hbox {Example 4:}\\ \left[ {\begin{array}{ccccc} 1 &{}\quad 3 &{}\quad 5 &{}\quad 5 &{}\quad 9 \\ 0.33 &{}\quad 1 &{}\quad 3 &{}\quad 4 &{}\quad 6 \\ 0.20 &{}\quad 0.33 &{}\quad 1 &{}\quad * &{}\quad 5 \\ 0.20 &{}\quad 0.25 &{}\quad * &{}1 &{}\quad 5 \\ 0.11 &{}\quad 0.17 &{}\quad 0.20 &{}\quad 0.20 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{ccccc} 1 &{}\quad * &{}\quad 3 &{}\quad 5 &{}\quad 8 \\ * &{}\quad 1 &{}\quad 3 &{}\quad 5 &{}\quad 7 \\ 0.33 &{}\quad 0.33 &{}\quad 1 &{}\quad 0.50 &{}\quad 5 \\ 0.20 &{}\quad 0.20 &{}\quad 2 &{}\quad 1 &{}\quad 3 \\ 0.13 &{}\quad 0.14 &{}\quad 0.20 &{}\quad 0.33 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{ccccc} 1 &{}\quad 0.20 &{}\quad 3 &{}\quad 0.50 &{}\quad 5 \\ 5 &{}\quad 1 &{}\quad * &{}\quad 1 &{}\quad 7 \\ 0.33 &{}\quad * &{}\quad 1 &{}\quad 0.25 &{}\quad 3 \\ 2 &{}\quad 1 &{}\quad 4 &{}\quad 1 &{}\quad 7 \\ 0.20 &{}\quad 0.14 &{}\quad 0.33 &{}\quad 0.14 &{}\quad 1 \\ \end{array}}\right] &{} \left[ {\begin{array}{ccccc} 1 &{}\quad 2 &{}\quad 2 &{}\quad 4 &{}\quad 8 \\ 0.50 &{}\quad 1 &{}\quad 1 &{}\quad 2 &{}\quad * \\ 0.50 &{}\quad 1 &{}\quad 1 &{}\quad 2 &{}\quad 4 \\ 0.25 &{}\quad 0.50 &{}\quad 0.50 &{}\quad 1 &{}\quad 2 \\ 0.13 &{}\quad * &{}\quad 0.25 &{}\quad 0.50 &{}\quad 1 \end{array}}\right] \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} \mathbf {Order~6} &{} \\ \hbox {Example 1:} &{} \hbox {Example 2:}&{}\hbox {Example 3:} &{} *\hbox {Example 4:} \\ \left[ {\begin{array}{ccccccc} 1 &{}\quad 4 &{}\quad 0.33 &{}\quad 4 &{}\quad 7 &{}\quad 0.25 \\ 0.25 &{}\quad 1 &{}\quad * &{}\quad 2 &{}\quad 5 &{}\quad 0.33 \\ 3 &{}\quad * &{}\quad 1 &{}\quad 6 &{}\quad 7 &{}\quad 1 \\ 0.25 &{}\quad 0.50 &{}\quad 0.17 &{}\quad 1 &{}\quad 3 &{}\quad 0.25 \\ 0.14 &{}\quad 0.20 &{}\quad 0.14 &{}\quad 0.33 &{}\quad 1 &{}\quad 0.14 \\ 4 &{}\quad 3 &{}\quad 1 &{}\quad 4 &{}\quad 7 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{ccccccc} 1 &{}\quad 5 &{}\quad * &{}\quad 3 &{}\quad 6 &{}\quad 2 \\ 0.20 &{}\quad 1 &{}\quad 0.33 &{}\quad 0.33 &{}\quad 3 &{}\quad 0.25 \\ * &{}\quad 3 &{}\quad 1 &{}\quad 0.50 &{}\quad 5 &{}\quad 0.33 \\ 0.33 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 5 &{}\quad 2 \\ 0.17 &{}\quad 0.33 &{}\quad 0.20 &{}\quad 0.20 &{}\quad 1 &{}\quad 0.20 \\ 0.50 &{}\quad 4 &{}\quad 3 &{}\quad 0.50 &{}\quad 5 &{}\quad 1 \end{array}}\right] &{}\left[ {\begin{array}{cccccc} 1 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 3 &{}\quad 3 \\ 0.33 &{}\quad 1 &{}\quad 0.50 &{}\quad 0.33 &{}\quad 2 &{}\quad 1 \\ 0.50 &{}\quad 2 &{}\quad 1 &{}\quad 0.50 &{}\quad * &{}\quad 2 \\ 1 &{}\quad 3 &{}\quad 2 &{}\quad 1 &{}\quad 3 &{}\quad 2 \\ 0.25 &{}\quad 0.50 &{}\quad * &{}\quad 0.33 &{}\quad 1 &{}\quad 0.50 \\ 0.33 &{}\quad 1 &{}\quad 0.50 &{}\quad 0.50 &{}\quad 2 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{ccccccc} 1 &{}\quad 1.50 &{}\quad 2.25 &{}\quad * &{}\quad 5.06 &{}\quad 7.60 \\ 0.67 &{}\quad 1 &{}\quad 1.50 &{}\quad 2.25 &{}\quad 3.38 &{}\quad 5.06 \\ 0.44 &{}\quad 0.67 &{}\quad 1 &{}\quad 1.50 &{}\quad 2.25 &{}\quad 3.38 \\ * &{}\quad 0.44 &{}\quad 0.67 &{}\quad 1 &{}\quad 1.50 &{}\quad 2.25 \\ 0.20 &{}\quad 0.30 &{}\quad 0.44 &{}\quad 0.67 &{}\quad 1 &{}\quad 1.50 \\ 0.13 &{}\quad 0.20 &{}\quad 0.30 &{}\quad 0.44 &{}\quad 0.67 &{}\quad 1 \end{array}}\right] \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} \mathbf{Order~7 } &{} \\ \hbox {Example 1:} &{} \hbox {Example 2:}&{}\hbox {Example 3:}&{} *\hbox {Example 4:} \\ \left[ {\begin{array}{ccccccc} 1 &{}\quad 9 &{}\quad 5 &{}\quad 2 &{}\quad 1 &{}\quad 1 &{}\quad 0.50 \\ 0.11 &{}\quad 1 &{}\quad 0.33 &{}\quad 0.11 &{}\quad 0.11 &{}\quad 0.11 &{}\quad 0.11 \\ 0.20 &{}\quad 3 &{}\quad 1 &{}\quad 0.33 &{}\quad * &{}\quad 0.33 &{}\quad 0.11 \\ 0.50 &{}\quad 9 &{}\quad 3 &{}\quad 1 &{}\quad 0.50 &{}\quad 1 &{}\quad 0.33 \\ 1 &{}\quad 9 &{}\quad * &{}\quad 2 &{}\quad 1 &{}\quad 2 &{}\quad 0.50 \\ 1 &{}\quad 9 &{}\quad 3 &{}\quad 1 &{}\quad 0.50 &{}\quad 1 &{}\quad 0.33 \\ 2 &{}\quad 9 &{}\quad 9 &{}\quad 3 &{}\quad 2 &{}\quad 3 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{ccccccc} 1 &{}\quad 0.25 &{}\quad 5 &{}\quad 0.14 &{}\quad 0.33 &{}\quad 0.50 &{}\quad 0.50 \\ 4 &{}\quad 1 &{}\quad * &{}\quad 0.33 &{}\quad 0.33 &{}\quad 0.25 &{}\quad 0.33 \\ 0.20 &{}\quad * &{}\quad 1 &{}\quad 0.14 &{}\quad 0.14 &{}\quad 0.17 &{}\quad 0.33 \\ 7 &{}\quad 3 &{}\quad 7 &{}\quad 1 &{}\quad 0.50 &{}\quad 2 &{}\quad 3 \\ 3 &{}\quad 3 &{}\quad 7 &{}\quad 2 &{}\quad 1 &{}\quad 2 &{}\quad 3 \\ 2 &{}\quad 4 &{}\quad 6 &{}\quad 0.50 &{}\quad 0.50 &{}\quad 1 &{}\quad 2 \\ 2 &{}\quad 3 &{}\quad 3 &{}\quad 0.33 &{}\quad 0.33 &{}\quad 0.50 &{}\quad 1 \end{array}}\right] &{}\left[ {\begin{array}{ccccccc} 1 &{}\quad 3 &{}\quad 0.33 &{}\quad * &{}\quad 0.25 &{}\quad 0.33 &{}\quad 3 \\ 0.33 &{}\quad 1 &{}\quad 0.14 &{}\quad 0.14 &{}\quad 0.17 &{}\quad 0.33 &{}\quad 2 \\ 3 &{}\quad 7 &{}\quad 1 &{}\quad 0.50 &{}\quad 2 &{}\quad 3 &{}\quad 3 \\ * &{}\quad 7 &{}\quad 2 &{}\quad 1 &{}\quad 2 &{}\quad 3 &{}\quad 5 \\ 4 &{}\quad 6 &{}\quad 0.50 &{}\quad 0.50 &{}\quad 1 &{}\quad 2 &{}\quad 5 \\ 3 &{}\quad 3 &{}\quad 0.33 &{}\quad 0.33 &{}\quad 0.50 &{}\quad 1 &{}\quad 3 \\ 0.33 &{}\quad 0.50 &{}\quad 0.33 &{}\quad 0.20 &{}\quad 0.20 &{}\quad 0.33 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{ccccccc} 1 &{}\quad 0 &{}\quad 1 &{}\quad 2 &{}\quad 4 &{}\quad 8 &{}\quad 8 \\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 2 &{}\quad 4 &{}\quad 8 &{}\quad 8 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 2 &{}\quad 4 &{}\quad 8 &{}\quad 8 \\ 0.50 &{}\quad 0.50 &{}\quad 0.50 &{}\quad 1 &{}\quad 2 &{}\quad 4 &{}\quad 4 \\ 0.25 &{}\quad 0.25 &{}\quad 0.25 &{}\quad 0.50 &{}\quad 1 &{}\quad 2 &{}\quad 2 \\ 0.13 &{}\quad 0.13 &{}\quad 0.13 &{}\quad 0.25 &{}\quad 0.50 &{}\quad 1 &{}\quad 1 \\ 0.13 &{}\quad 0.13 &{}\quad 0.13 &{}\quad 0.25 &{}\quad 0.50 &{}\quad 1 &{}\quad 1 \end{array}}\right] \\ \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{llll} \mathbf{Order~8 } \\ \hbox {Example 1:}&{} \hbox {Example 2:} &{}\hbox {Example 3: } &{}*\hbox {Example 4:} \\ \left[ {\begin{array}{cccccccc} 1 &{}\quad 5 &{}\quad 5 &{}\quad 0.14 &{}\quad 0.33 &{}\quad 0.50 &{}\quad 0.50 &{}\quad 2 \\ 0.20&{}\quad 1 &{}\quad 3 &{}\quad 0.33 &{}\quad 0.33 &{}\quad 0.25 &{}\quad 0.33 &{}\quad 3 \\ 0.20&{}\quad 0.33 &{}\quad 1 &{}\quad 0.14 &{}\quad 0.14 &{}\quad 0.17 &{}\quad 0.33 &{}\quad 2 \\ 7 &{}\quad 3 &{}\quad 7 &{}\quad 1 &{}\quad 0.50 &{}\quad 2 &{}\quad 3 &{}\quad 3 \\ 3 &{}\quad 3 &{}\quad 7 &{}\quad 2 &{}\quad 1 &{}\quad 2 &{}\quad 3 &{}\quad 5 \\ 2 &{}\quad 4 &{}\quad 6 &{}\quad 0.50 &{}\quad 0.50 &{}\quad 1 &{}\quad 2 &{}\quad * \\ 2 &{}\quad 3 &{}\quad 3 &{}\quad 0.33 &{}\quad 0.33 &{}\quad 0.50&{}\quad 1 &{}\quad 3 \\ 0.50&{}\quad 0.33 &{}\quad 0.50&{}\quad 0.33 &{}\quad 0.20&{}\quad * &{}\quad 0.33 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{cccccccc} 1 &{}\quad 5 &{}\quad 3 &{}\quad 7 &{}\quad 6 &{}\quad 6 &{}\quad 0.33 &{}\quad 0.25 \\ 0.20 &{}\quad 1 &{}\quad 0.50 &{}\quad 5 &{}\quad * &{}\quad 3 &{}\quad 0.14 &{}\quad 0.14 \\ 0.33 &{}\quad 2 &{}\quad 1 &{}\quad 4 &{}\quad 3 &{}\quad 3 &{}\quad 0.17 &{}\quad 0.17 \\ 0.14 &{}\quad 0.20 &{}\quad 0.25 &{}\quad 1 &{}\quad 1 &{}\quad 0.25 &{}\quad 0.11 &{}\quad 0.13 \\ 0.17 &{}\quad * &{}\quad 0.33 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0.20 &{}\quad 0.11 \\ 0.17 &{}\quad 0.33 &{}\quad 0.33 &{}\quad 4 &{}\quad 1 &{}\quad 1 &{}\quad 0.11 &{}\quad 0.17 \\ 3 &{}\quad 7 &{}\quad 6 &{}\quad 9 &{}\quad 5 &{}\quad 9 &{}\quad 1 &{}\quad 0.50\\ 4 &{}\quad 7 &{}\quad 6 &{}\quad 8 &{}\quad 9 &{}\quad 6 &{}\quad 2 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{cccccccc} 1 &{}\quad 0.78 &{}\quad 2.73 &{}\quad 0.66 &{}\quad 2.48 &{}\quad 3.65 &{}\quad 7.78 &{}\quad 9 \\ 1.28 &{}\quad 1 &{}\quad 2.89 &{}\quad 3.70 &{}\quad 2.89 &{}\quad 5 &{}\quad 8 &{}\quad 8.12 \\ 0.37 &{}\quad 0.35 &{}\quad 1 &{}\quad 1.66 &{}\quad 2 &{}\quad 2.65 &{}\quad 7.37 &{}\quad 8.85 \\ 1.52 &{}\quad 0.27 &{}\quad 0.60 &{}\quad 1 &{}\quad * &{}\quad 3.18 &{}\quad 8.81 &{}\quad 7.22 \\ 0.40 &{}\quad 0.35 &{}\quad 0.50 &{}\quad * &{}\quad 1 &{}\quad 1.42 &{}\quad 4 &{}\quad 7.75 \\ 0.27 &{}\quad 0.20 &{}\quad 0.38 &{}\quad 0.31 &{}\quad 0.70 &{}\quad 1 &{}\quad 3 &{}\quad 5 \\ 0.13 &{}\quad 0.13 &{}\quad 0.14 &{}\quad 0.11 &{}\quad 0.25 &{}\quad 0.33 &{}\quad 1 &{}\quad 4 \\ 0.11 &{}\quad 0.12 &{}\quad 0.11 &{}\quad 0.14 &{}\quad 0.13 &{}\quad 0.20 &{}\quad 0.25 &{}\quad 1 \end{array}}\right] &{} \left[ {\begin{array}{cccccccc} 1 &{}\quad 2 &{}\quad 0.50 &{}\quad 2 &{}\quad 0.50 &{}\quad 2 &{}\quad 0.50 &{}\quad 2 \\ 0.50 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 \\ 2 &{}\quad 4 &{}\quad 1 &{}\quad 4 &{}\quad 1 &{}\quad 4 &{}\quad 1 &{}\quad 4 \\ 0.50 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 \\ 2 &{}\quad 4 &{}\quad 1 &{}\quad 4 &{}\quad 1 &{}\quad 4 &{}\quad 1 &{}\quad 4 \\ 0.50 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 \\ 2 &{}\quad 4 &{}\quad 1 &{}\quad 4 &{}\quad 1 &{}\quad 4 &{}\quad 1 &{}\quad * \\ 0.50 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad 0.25 &{}\quad 1 &{}\quad * &{}\quad 1 \end{array}}\right] \\ \end{array} \end{aligned}$$

B Relation between k and CR

See Fig. 4.

Fig. 4
figure4

k versus CR. a Order 4: Example 2, b Order 4: Example 3, c Order 5: Example 2, d Order 5: Example 3, e Order 6: Example 2, f Order 6: Example 3, g Order 7: Example 2, h Order 7: Example 3, i Order 8: Example 2, j Order 8: Example 3

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Zhou, X., Hu, Y., Deng, Y. et al. A DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP. Ann Oper Res 271, 1045–1066 (2018). https://doi.org/10.1007/s10479-018-2769-3

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Keywords

  • AHP
  • Pairwise comparison matrix
  • Completion method
  • Missing values
  • DEMATEL