A consistency improving method in the analytic hierarchy process based on directed circuit analysis

doi:10.21629/JSEE.2019.06.11

Abstract

Abstract:

Test of consistency is critical for the analytic hierarchy process (AHP) methodology. When a pairwise comparison matrix (PCM) fails the consistency test, the decision maker (DM) needs to make revisions. The state of the art focuses on changing a single entry or creating a new matrix based on the original inconsistent matrix so that the modified matrix can satisfy the consistency requirement. However, we have noticed that the reason that causes inconsistency is not only numerical inconsistency, but also logical inconsistency, which may play a more important role in the whole inconsistency. Therefore, to realize satisfactory consistency, first of all, we should change some entries that form a directed circuit to make the matrix logically consistent, and then adjust other entries within acceptable deviations to make the matrix numerically consistent while preserving most of the original comparison information. In this paper, we firstly present some definitions and theories, based on which two effective methods are provided to identify directed circuits. Four optimization models are proposed to adjust the original inconsistent matrix. Finally, illustrative examples and comparison studies show the effectiveness and feasibility of our method.

Key words: analytic hierarchy process (AHP), pairwise comparison matrix (PCM), logical inconsistency, numerical inconsistency, directed circuit

Shihui WU, Xiaodong LIU, Zhengxin LI, Yu ZHOU. A consistency improving method in the analytic hierarchy process based on directed circuit analysis[J]. Journal of Systems Engineering and Electronics, 2019, 30(6): 1160-1181.

Figures/Tables 19

Fig 1

Fig 2

Fig 3

Fig 4

Fig 5

Fig 6

Table 1

Fig 7

Fig 8

Fig 9

Table 2

Output results for different situations in Example 1"

Situation	Element with logical error	Modified matrix
Situation	Element with logical error	Modified element	Logical consistency	CR	Consistent level	Priority vector $\omega$ and priority order
Situation 1 (Assume one element has logical error)	$a_{13}$ redirect, and other elements keep unchanged	$a_{13}=0.720~2$	Acceptable	0.143 3	Not acceptable	-
	$a_{17}$ redirect, and other elements keep unchanged	$a_{17}=1.762$	Acceptable	0.134 2	Not acceptable	-
	$a_{37}$ redirect, and other elements keep unchanged	$a_{37}=0.464~5$	Acceptable	0.082	Acceptable	{0.174 4, 0.062, 0.102 2, 0.019 3, 0.033 8, 0.041 2, 0.222 3, 0.344 7} 8$ \succ $7$ \succ $1$ \succ $3$ \succ $2$ \succ $6$ \succ $5$ \succ $4
Situation 2 (Assume two elements have logical error)	$a_{13}$ and $a_{17}$ redirect, and other elements keep unchanged	$a_{13}=0.923~9$ $a_{17}= 1.362~5$	Acceptable	0.121 4	Not acceptable	-
	$a_{17}$ and $ a_{37}$ redirect, and other elements keep unchanged	$a_{17}=1.075~9$ $a_{37}=0.575~2$	Acceptable	0.070 3	Acceptable	{0.197 2, 0.063 6, 0.106 7, 0.019 6, 0.034 4, 0.042, 0.18, 0.356 5} 8$\succ$1$\succ$7$\succ$3$\succ$2$\succ$6$\succ$5$\succ$4
	$a_{13}$ and $a_{37}$ redirect, and other elements keep unchanged	$a_{13}=0.999~9$ $a_{37}= 0.561~1$	Acceptable	0.075 9	Acceptable	{0.153 9, 0.063 4, 0.119 4, 0.019 5, 0.034 6, 0.042, 0.217 6, 0.349 4} 8$\succ$7$\succ$1$\succ$3$\succ$2$\succ$6$\succ$5$\succ$4
Situation 3 (Assume elements in the 3-node directed circuit as unknown)	Consider $a_{13}$, $a_{17}$ and $a_{37}$ as unknown $(\delta =0)$	$a_{13}= 1.476~3$ $a_{17}=0.927~1$ $a_{37}= 0.626~4$	Acceptable	0.065 6	Acceptable	{0.178, 0.064 7, 0.117 2, 0.019 8, 0.035, 0.042 7, 0.184 2, 0.358 6} 8$\succ$7$\succ$1$\succ$3$\succ$2$\succ$6$\succ$5$\succ$4
	Consider $a_{13}$, $a_{17}$ and $a_{37}$ as unknown $(\delta =0.5)$	see Table 4	Acceptable	0.037 8	Acceptable	{0.177 6, 0.063 4, 0.117 7, 0.020 3, 0.035 2, 0.041 9, 0.196 6, 0.347 4} 8$\succ$7$\succ$1$\succ$3$\succ$2$\succ$6$\succ$5$\succ$4

Table 2

Table 3

Table 4

Table 5

Output results for different situations in Example 2"

Situation	Element with logical error	Modified matrix
Situation	Element with logical error	Modified element	Logical consistency	CR	Consistent level	Priority vector $\omega$ and priority order
Situation 1 (Assume one element has logical error)	$a_{37}$ redirect, and other elements keep unchanged	$a_{37}=0.358 8$	Acceptable	0.230 3	Not acceptable	-
Situation 2 (Assume two elements have logical error)	$a_{47}$ and $a_{37}$ redirect, and other elements keep unchanged	$a_{47}=1.268~7$ $a_{37}= 0.468~8$	Acceptable	0.206	Not acceptable	-
	$a_{35}$ and $a_{37}$ redirect, and other elements keep unchanged	$a_{35}=1.794~5$ $a_{37}= 0.504~7$	Acceptable	0.190 9	Not acceptable	-
	$a_{34}$ and $a_{35}$ redirect, and other elements keep unchanged	$a_{34}= 1.393~7$ $a_{35}= 3.837~6$	Acceptable	0.180 3	Not acceptable	-
	$a_{47}$ and $a_{35}$ redirect, and other elements keep unchanged	$a_{47}= 2.003~2$ $a_{35}= 2.416~4$	Acceptable	0.187 5	Not acceptable	-
	$a_{47}$ and $a_{57}$ redirect, and other elements keep unchanged	$a_{47}=2.826~5$ $a_{57}= 1.002~4$	Acceptable	0.214 8	Not acceptable	-
Situation 3 (Assume three elements have logical error)	$a_{34}$, $a_{37}$ and $a_{35}$ redirect, and other elements keep unchanged	$a_{34}=1.000~1$ $a_{37}=0.787~9$ $a_{35}= 2.755~8$	Acceptable	0.160 8	Not acceptable	-
	$a_{47}$, $a_{37}$ and $a_{35}$ redirect, and other elements keep unchanged	$a_{47}= 1.460~8$ $a_{37}=0.711$ $a_{35}= 1.867~1$	Acceptable	0.163 6	Not acceptable	-
	$a_{47}$, $a_{34}$ and $a_{35}$ redirect, and other elements keep unchanged	$a_{47}= 1.396~8$ $a_{34}=1.002~1$ $a_{35}=3.419~1$	Acceptable	0.154 3	Not acceptable	-
	$a_{47}$, $a_{57}$ and $a_{35}$ redirect, and other elements keep unchanged	$a_{47}= 2.524~3$ $a_{57}=1.000~1$ $a_{35}=1.947~6$	Acceptable	0.171 7	Not acceptable	-
	$a_{47}$, $a_{37}$ and $a_{57}$ redirect, and other elements keep unchanged	$a_{47}= 2.069~2$ $a_{37}=0.736~6$ $a_{57}=1.000~1$	Acceptable	0.191 3	Not acceptable	-
Situation 4 (Assume four elements have logical error)	$a_{34}$, $a_{47}$, $a_{35}$ and $a_{57}$ redirect, and other elements keep unchanged	$a_{34}= 1.000~1$ $a_{47}= 2.043~2$ $a_{35}= 2.546~4$ $a_{57}= 1.000~1$	Acceptable	0.143 7	Not acceptable	-
Situation 5 (Assume elements in the 3-node directed circuits as unknown)	Consider $a_{34}$, $a_{47}$, $a_{37}$, $a_{35}$ and $a_{57}$ as unknown $(\delta =0)$	$a_{34}= 0.935~2$$a_{47}= 1.772~4$$a_{37}= 2.8$$a_{35}= 2.819~1$$a_{57}=0.587~7$	Acceptable	0.140 2	Not acceptable	-
	Consider $a_{34}$, $a_{47}$, $a_{37}$, $a_{35}$ and $a_{57}$ as unknown $(\delta =0.5)$	See Table 6	Acceptable	0.097 8	Acceptable	{0.437 8, 0.172 3, 0.103 7, 0.134, 0.042 4, 0.043 9, 0.065 9} 1$ \succ $2$ \succ $4$ \succ $3$\succ$7$\succ$6$\succ$5

Table 5

Table 6

Table 7

Table 8

Comparison between our method and method in Example 2 of [11]"

Situation	Element with logical error	Modified matrix in our method (also see Table 5)		Modified matrix by method in [11]
Situation	Element with logical error	Modified element	CR	Modified element	CR	Logical consistency	Consistent level
Situation 1 (Assume one element has logical error)	$a_{37}$	$a_{37}=0.358~8$	0.230 3	$a_{73}=6.52$ $(a_{37}=0.153~4)$	0.240 1	Acceptable	Not acceptable
Situation 2 (Assume two elements have logical error)	$a_{47}$, $a_{37}$	$a_{47}=1.268~7$ $a_{37}= 0.468~8$	0.206	$a_{47}=3.96$$a_{73}=6.52$$(a_{37}=0.153~4)$	0.244 3	Acceptable	Not acceptable
	$a_{35}$, $a_{37}$	$a_{35}=1.794~5$ $a_{37}= 0.504~7$	0.190 9	$a_{73}=6.52$$(a_{37}=0.153~4)$$a_{35}= 4.493~3$	0.226 3	Acceptable	Not acceptable
	$a_{34}$, $a_{35}$	$a_{34}= 1.393~7$ $a_{35}= 3.837~6$	0.180 3	$a_{34}= 2.540~7$$a_{35}= 4.493~3$	0.184 8	Acceptable	Not acceptable
	$a_{47}$, $a_{35}$	$a_{47}= 2.003~2$ $a_{35}= 2.416~4$	0.187 5	$a_{47}= 3.96$$a_{35}= 4.493~3$	0.198 7	Acceptable	Not acceptable
	$a_{47}$, $a_{57}$	$a_{47}=2.826~5$ $a_{57}= 1.002~4$	0.214 8	$a_{47}= 3.96$$a_{57}= 2.238~5$	0.223 1	Acceptable	Not acceptable
Situation 3 (Assume three elements have logical error)	$a_{34}$, $a_{37}$, $a_{35}$	$a_{34}= 1.000~1$ $a_{37}=0.787~9$$a_{35}= 2.755~8$	0.160 8	$a_{34}= 2.540~7$$a_{73}=6.52$$(a_{37}=0.153~4)$$a_{35}= 4.493~3$	0.220 5	Acceptable	Not acceptable
	$a_{47}$, $a_{37}$, $a_{35}$	$a_{47}= 1.460~8$$a_{37}=0.711$$a_{35}= 1.867~1$	0.163 6	$a_{47}= 3.96$$a_{73}=6.52$$(a_{37}=0.153~4)$$a_{35}= 4.493~3$	0.233 3	Acceptable	Not acceptable
	$a_{47}$, $a_{34}$, $a_{35}$	$a_{47}= 1.396~8$ $a_{34}=1.002~1$ $a_{35}=3.419~1$	0.154 3	$a_{47}= 3.96$ $a_{34}=2.540~7$ $a_{35}= 4.493~3$	0.181 9	Acceptable	Not acceptable
	$a_{47}$, $a_{57}$, $a_{35}$	$a_{47}= 2.524~3$ $a_{57}=1.000~1$$a_{35}=1.947~6$	0.171 7	$a_{47}= 3.96$$a_{57}= 2.238~5$$a_{35}= 4.493~3$	0.201 1	Acceptable	Not acceptable
	$a_{47}$, $a_{37}$, $a_{57}$	$a_{47}= 2.069~2$$a_{37}=0.736~6$$a_{57}=1.000~1$	0.191 3	$a_{47}= 3.96$$a_{73}=6.52$$(a_{37}=0.153~4)$$a_{57}= 2.238~5$	0.256 2	Acceptable	Not acceptable
Situation 4 (Assume four elements have logical error)	$a_{34}$, $a_{47}$, $a_{35}$, $a_{57}$	$a_{34}= 1.000~1$ $a_{47}= 2.043~2$$a_{35}= 2.546~4$$a_{57}= 1.000~1$	0.143 7	$a_{34}= 2.540~7$$a_{47}= 3.96$$a_{35}= 4.493~3$$a_{57}= 2.238~5$	0.185 4	Acceptable	Not acceptable

Table 8

Table 9

Table 10

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Linguistic description	Scale	Tolerance deviations interval
Extreme importance	9	[8, 10]
Demonstrated importance	7	[6, 8)
Strong importance	5	[4, 6)
Moderate importance	3	[2, 4)
Equal importance	1	[1, 2)

Method	Modified matrix
Method	Modified element	Logical consistency	CR	Consistent level	$\varepsilon $
[5]	All elements are modified with maximum modification $\sigma =1.845$	Not acceptable with directed circuit 1$ \to $3$ \to $7$ \to $1	0.097	Acceptable	0.589
[6]	All elements are modified with maximum modification $\sigma =1.713$	Not acceptable with directed circuit 1$ \to $3$\to $7$ \to $1	0.099 7	Acceptable	0.448
[27] and [7]	$a_{37}$ is modified from 6 to 0.5	Acceptable	0.082 24	Acceptable	0
Our method	$a_{37}$ is modified from 6 to 0.464 5	Acceptable	0.082 19	Acceptable	0

Element	Original value	Modification bound	Modified value
a₁₂	5	[4.5, 5.5]	4.5
a₁₃	3	[1/9, 9]	1.425 7
a₁₄	7	[6.5, 7.5]	7.496 1
a₁₅	6	[5.5, 6.5]	5.778
a₁₆	6	[5.5, 6.5]	5.5
a₁₇	0.333 3	[1/9, 9]	0.869 1
a₁₈	0.25	[1/4.5, 1/3.5]	0.285 7
a₂₃	0.333 3	[1/3.5, 1/2.5]	0.4
a₂₄	5	[4.5, 5.5]	4.5
a₂₅	3	[2.5, 3.5]	2.5
a₂₆	3	[2.5, 3.5]	2.5
a₂₇	0.2	[1/5.5, 1/4.5]	0.222 2
a₂₈	0.142 9	[1/7.5, 1/6.5]	0.153 8
a₃₄	6	[5.5, 6.5]	6.278 4
a₃₅	3	[2.5, 3.5]	3.326 6
a₃₆	4	[3.5, 4.5]	3.5
a₃₇	6	[1/9, 9]	0.590 4
a₃₈	0.2	[1/5.5, 1/4.5]	0.222 2
a₄₅	0.333 3	[1/3.5, 1/2.5]	0.4
a₄₆	0.25	[1/4.5, 1/3.5]	0.285 7
a₄₇	0.142 9	[1/7.5, 1/6.5]	0.133 3
a₄₈	0.125	[1/8.5, 1/7.5]	0.117 6
a₅₆	0.5	[1/2.5, 1/1.5]	0.666 7
a₅₇	0.2	[1/5.5, 1/4.5]	0.181 8
a₅₈	0.166 7	[1/6.5, 1/5.5]	0.153 8
a₆₇	0.2	[1/5.5, 1/4.5]	0.205 5
a₆₈	0.166 7	[1/6.5, 1/5.5]	0.153 8
a₇₈	0.5	[1/2.5, 1/1.5]	0.606 5

Element	Original value	Modification bound	Modified value
$a_{12}$	7	[6.5, 7.5]	6.5
$a_{13}$	3	[2.5, 3.5]	3.5
$a_{14}$	5	[4.5, 5.5]	4.5
$a_{15}$	9	[8.5, 9]	9
$a_{16}$	3	[2.5, 3.5]	3.5
$a_{17}$	5	[4.5, 5.5]	5.378 7
$a_{23}$	3	[2.5, 3.5]	2.5
$a_{24}$	3	[2.5, 3.5]	2.5
$a_{25}$	5	[4.5, 5.5]	4.5
$a_{26}$	3	[2.5, 3.5]	3.5
$a_{27}$	3	[2.5, 3.5]	2.5
$a_{34}$	0.2	[1/9, 9]	0.838 4
$a_{35}$	0.333 3	[1/9, 9]	2.735 9
$a_{36}$	3	[2.5, 3.5]	2.679 4
$a_{37}$	3	[1/9, 9]	2.5
$a_{45}$	9	[8.5, 9]	8.5
$a_{46}$	3	[2.5, 3.5]	3.169 2
$a_{47}$	0.333 3	[1/9, 9]	1.837 1
$a_{56}$	3	[2.5, 3.5]	2.5
$a_{57}$	0.2	[1/9, 9]	0.577 8
$a_{67}$	0.333 3	[1/3.5, 1/2.5]	0.4

Model being used	Optimization algorithm	Modified matrix
Model being used	Optimization algorithm	Modified element	Logical consistency	CR	Total perturbation $\varepsilon $	Maximum deviation $\sigma$	Priority vector $\omega$ and priority order
Model 3	Improved pattern search algorithm	$\mathit{\boldsymbol{X}}_1$={6.5, 3.1, 4.664 4, 9, 3.5, 5.003 7, 2.501 7, 2.5, 4.969 7, 3.5, 2.825, 0.838 4, 2.735 9, 2.987 3, 2.5, 8.5, 3.006 2, 1.837 1, 2.5, 0.577 8, 0.4}	Acceptable	0.1	0.222 7	0.5	{0.433 4, 0.177 3, 0.106 9, 0.131 7, 0.041 8, 0.043 6, 0.065 3} 1$ \succ $2$ \succ $4$ \succ $3$ \succ $7$ \succ $6$ \succ $5
Model 3	fmincon	$\mathit{\boldsymbol{X}}_{2}$={6.514 7, 3.485 2, 4.514 7, 9, 3.485 2, 5.367 6, 2.515, 2.515, 4.514 7, 3.485 2, 2.515, 0.838 4, 2.735 9, 2.689, 2.515, 8.514 5, 3.164 2, 1.837 1, 2.515, 0.577 8, 0.368 57}	Acceptable	0.1	0.255 5	0.485 5	{0.437 3, 0.172 5, 0.103 8, 0.133 6, 0.042 4, 0.043 6, 0.066 9} 1$ \succ $2$\succ $4$\succ $3$\succ $7$\succ $6$\succ $5
Model 4	fmincon	$\mathit{\boldsymbol{X}}_{3}$={6.529 2, 3.470 9, 4.529 2, 9, 3.470 9, 5.378 7, 2.529 2, 2.529 2, 4.529 2, 3.470 9, 2.529 2, 0.838 4, 2.735 9, 2.679 4, 2.529 2, 8.529 2, 3.169 2, 1.837 1, 2.529 2, 0.577 8, 0.395 4}	Acceptable	0.1	0.255 1	0.470 9	{0.437 6, 0.172 9, 0.103 7, 0.133 7, 0.042 5, 0.043 9, 0.065 7} 1$\succ $2$\succ $4$\succ $3$\succ $7$\succ $6$\succ $5