(-1 c

ISAHP 1999, Kobe, Japan, August 12-14, 1999

DECOMPOSITION OF DATA AND VARIATION IN THE ANALYTIC HIERARCHY PROCESS

Shin'ya Nagasawa

College of Business Administration, Ritsumeikan University 1-1-1 Noji-kita, Kusatsu, Shiga 525-8577 Japan snt01497@basitsumei.ac.jp

Abstract: In order to improve Analytic Hierarchy Process (AHP), this paper proposes the decomposition of data in the AHP by the logarithmic linear model. Data are decomposed to a main effect, an order effect, a combination effect, personal interactions and errors by the proposed mode. The proposed model is considered to be useful to analyze details of decision making by the AHP.

In order to improve Saaty's Analytic Hierarchy Process (AHP) this paper proposes the decomposition of data and variation in the AHP by the following logarithmic linear model;

xijk.(coilcopX(wiklatjk)x cijx(d X dk)x eijk

where i,j :alternative k : person xijk : paired comparison data (ratio) cot : main effect (weight) (II CO ik =1) co& : personal interaction of weight (TI ik=1)

Cu : combination effect ( -1 c ij=1)

d : order effect

dk : personal order effect (idk=1)

eijk :error

By logarithmic least squares method, data and variation are decomposed to a main effect, an order effect, a combination effect, personal interactions and errors by the proposed mode. The proposed model is

considered to be useful to analyze details of decision making by the AHP.

Table 1 Inverse order ARP data by 3 persons

1 2 34

P1

2o

2-1

2-1

1 P 2 1 2 21 21

p3

21 1 1 21

P1 23 2 P 2 20

2? 2? 1 2-2 2-2

P 3 22

2? 2-2

P, 23 21

21

3 P 2 23 21 1 21

P 3 22 22

24

P, 23 23 21

4 P 2 23 23 21 1 22 21 20

P3

Table 2 Analysis of geometric mean data (1)

(``,4 1 1 1

2 3 4 weight 2-0.50 2.200 2-2.00 0.095

2

1 2333 2?133 0.143

3

1 2?3" 0.294

4

1 0.467

C.I.=0.013, C.R.=0.014

Table 3 Analysis of geometric mean data (2)

i\J 1 2 3 4 weight

1 1 2?33 2.133 2433 0.095

2 2133 1

2'133 0.143

3 2/67 21.00 1 213? 0.294

4 22.67 22.33 2367 1 0.467

CI.=0.430, C.R.=0.477

-- 209 --

Table 4 Decomposition of Data in AHP

(a) ratio of main effect (weight) wilaki

1

2

3

4

weight

P1

2 -0.58 2 -1.62 2 -2.29

1

P2

1

P3

2 -o.ss 1 1.62 1129 2 -0ss 2 -1.62 2 -229

0.09 5

P1

2 0.58

11.04 24.71

2

2055. P2

1

2 -119) 2 421 0.143

P 3 2 828

1104 2 4.71

P1

2 1.62 2 1.04

2 a67

3

P2

2 1.62 2 1.04 1

2 4167 0.294

P3

2 1.62 2 104

2 4167

P1

2 2.29 2131 2 0.67

4

P2

2 22.9 2131 2067.

1

0.467

P3

2 2.29 2131 2067.

C.I. 0, C.R. = 0

i\./ 1 2 3

4

(d) order effect d

1

P1

2

3

2.44 2 a"

4 2 0'44

weight

P2

1

2a" 2 ?." 2 a"

0.25

P 3

2 844 2 844 2 a"

P 1 2044

2' 2044

P2 2 ?." 1

P3 2 8'14

2 044 2044

2 Q" 20.44

0.25

P1

2 0A4 2 0'"

2 a44

P2 2 ? '" 2 a"

1

2 ?." 0.25

P3 2 1144 2 11"

2,8.44

P1 2 1144 2 844 2 044

P 2 2 ?444 2"1" 2 a"

1

0.25

P 3 224" 2 ? '" 2 8.44

C.I. = 0.360, CR. = 0.400

(b) ratio of personal interaction culla cojk

i\j

1

2

3

4

weight

P1

2047 2.012 2 -0.21

1

'2

1

P3

2 o.33 2 0.33

2 -038 1 0.46 2 0.50 2 0.67

0 . 25

P1

2 0.67

2 0.54 2 0.46

2 P2 1 2 4)23 2' 2' 9 0.25

P3

2 .033

2017.

2 033

P1

2 0.12 2 .054

2 408

3

P2 P3

2 ?38 2 an 1

2 4158 2417

2 408 0 . 25

? ,,on

P 1

2021.

2-0.46 2 0.08

4

2 0A6 20.79 2008. "2

1

p 3 2 4.67 2 -0.33 2-0.17

0.25

CI = 0, C.R. = 0

iNi 1 2 3

4

(e) personal order effect dk

. P 1

P 2 P3 p1

p 2

P3

P1 p 2

P3

pi

1

2

3

4

2 047 20.47 2047.

1

2 .828 r ?28 Tau

2 4.19 2 -019 2 4.19

2 0.47 1 028 1

2 0.47 2 047 2 -828 2 4128

2 4)19

2 4119 2 4119

2047.

2 0.47

20.47

2-0.28 24.28 1

2 419 24.19

2 4128

2047 2 0.47 2 0.47

weight 0.25 0.25 0.25

P2 2 428 2'028 2 428 1

0.25

P3 2 019 2 4119 2 4319

C.I. = 0, C.R. = 0

(c) combination effect c i

1

2

3

4

weight

P1

2 0a8 2' 038 2 029

1

P2

1

P3

2o.os 2 0.08

2 438 2 029 2-038 2 0.29

0.25

P1 2 41"

2 021 2 4.12

2 P 2 10 " 1

2 0.21 2412 0 . 25

P3

2 0.21 2 4/12

P

2 038 2 421

2 417

3

P 2 2p.ss roil

1

2 -017 0 . 25

P3

2 0.38 2 4.21

2 4/17

2 -0.29 2 0.12 2 0.17

4

P2

2 4.29 20.12 20.17 1

P-4

2 429 2 0.12 2 0.17

0.25

C.I. = 0.013, C.R. = 0.014

i\i 1

2

3

4

(I) error e ijk

1

2

3

4

2 oas 20.21 2029. P 1

P 2

1

2 0.00 2 0.21 2029.

2092.

2025 2008.

P3

p i 2 0.92

24.62 2046.

weight 0.287

1 P2 2 4133 241.62 2046. 0.256

p 3 2 0.58

2 0442 2 .075

p 1 2 -0.04 2 4.21

2 -1130

P2

2046.

21 1

2 4)25 0.222

p 3 2025 2 0.08

2 4158

p i 2-012 2071.

2-0.83

P2

2818 2021 24.08 1

p

20.42 2.075 24.92

0.235

CI. = 0.038, C.R. = 0.042

Nagasawa, S.(1993)Pcoceedings of the Autumn Conference of OW, pp.220-221.(in Japanese)

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