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

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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).

Author information

Authors and Affiliations

  1. Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Xinyi Zhou & Yong Deng

  2. Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, 13244, USA

    Xinyi Zhou

  3. Big Data Decision Institute, Jinan University, Tianhe, 510632, Guangzhou, China

    Yong Hu & Yong Deng

  4. School of Computer and Information Science, Southwest University, Chongqing, 400715, China

    Yong Deng

  5. Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

    Felix T. S. Chan

  6. Portsmouth Business School, Richmond Building, University of Portsmouth, Portland Street, Portsmouth, PO1 3DE, UK

    Alessio Ishizaka

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  1. Xinyi Zhou
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  2. Yong Hu
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Correspondence to Yong Deng.

Appendix

Appendix

1.1 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}$$

1.2 B Relation between k and CR

See Fig. 4.

Fig. 4
figure 4

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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