Note on diagonal procedure in analytic hierarchy process

Available online at www.sciencedirect.com

8O,,~NOE ELSEVIER

@ D,.-.cv.

MATHEMATICAL AND COMPUTER MODELLING

Mathematical and Computer Modelling 40 (2004) 1089-]092 www.elsevier.com/locate/mcm

N o t e on Diagonal P r o c e d u r e in A n a l y t i c Hierarchy P r o c e s s HENRY CHUNG-JEN CHAO Department of Civil Engineering National Taiwan University Taiwan, R.O.C. chaohenry©giga, neZ. zw

Kuo-LuNc

YANG

Department of Management Science Chinese Military Academy Taiwan, R.O.C.

PETER CHU Department of Traffic Science Central Police University Taiwan, R.O.C.

(Received and accepted August 2003) A b s t r a c t - - W e study the paper of Finan and Hurley published in the International q'Yansactions of Operational Research. They created a diagonal procedure in which their comparison matrix is rank-order consistent. The purpose of this paper is fourfold. First, we point out their procedure contains questionable results and improve on them. Second, we show that their procedure cannot guarantee that the consistency index is always less than 0.1. Third, we prove that their method preserves the predetermined rank order. Fourth, we offer a simpler procedure to derive the numerical weights. Hence, we suggest that decision-makers still use the comparison matrix of Salty. Numerical examples are included to illustrate our findings. ~) 2004 Elsevier Ltd. All rights reserved.

K e y w o r d s - - A n a l y t i c Hierarchy Process, Comparison Matrix.

1. I N T R O D U C T I O N For a decision-maker who knows the ordinal rank order of objectives and the relative weight between two consecutive objectives, Finan and Hurley [1] have developed the diagonal procedure to c o n s t r u c t a r a n k - o r d e r c o n s i s t e n t m a t r i x . T h e y believe t h a t t h e i r r a n k - o r d e r c o n s i s t e n t m a t r i x w o u l d p r o v i d e t h e m i n i m a l t r a n s i t i v e c o n s i s t e n c y r e q u i r e m e n t s for a p a i r w i s e c o m p a r i s o n m a t r i x a n d p r o v i d e s n u m e r i c a l weights for t h e objectives. However, t h e i r w o r k poses two questions r e g a r d i n g t h e i r use of t h e a n a l y t i c h i e r a r c h y process ( A H P ) . (a) If one a l r e a d y knows t h e o r d i n a l r a n k o r d e r of o b j e c t i v e s , w h y s h o u l d one w a n t t o a p p l y t h e A H P to derive t h e s a m e or a n o t h e r o r d i n a l r a n k o r d e r for t h e s e o b j e c t i v e s ?

0895-7177/04/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2003.08.008

Typeset by .AA/eS-TEX

1090

H. CHAO et al.

(b) If one knows the ordinal rank order of objectives and the relative weights between two consecutive objectives, why would one need to use the AHP to obtain the normalized weights among the objectives? Finan and Hurley [1] did not consider question (a). In this paper, we show that by their diagonal procedure, a rank-order consistent matrix would imply the same ordinal rank order for the objectives. However, for those decision-makers who want the best choice (the highest ordinal rank), for example, the school section in Saaty [3, p. 65], it seems unnecessary to construct a new matrix with some sophisticated procedure and properties to derive the already known ordinal rank order. As for question (b), Finan and Hurley say that their work would not replace the normal Saaty check for the consistency index. We illustrate with an example that their work cannot replace Saaty's check for consistency index because of the patchwork procedure they use when their diagonal procedure fails to pass the consistent test. For a distributive problem, the numerical weights of all the objectives are required to synthesize the final value, for example the expected number of children in [3, p. 92]. On the other hand, using different ratio scales [2] would imply different numerical weights and outcomes. Finan and Hurley [1] developed their procedure for decision-makers knowing the ordinal order and part of the cardinal order ratio (the relative weight between two consecutive objectives) in order to find the normalized numerical weights. One of the great strengths of AHP is that it deals with uncertain objectives and inconsistency. In this paper, we show that Finan and Hurley [1] brought a set of certain as opposed to uncertain ordinal order objectives with sufficient degree of cardinal ratio relationship into the category of AHP. It will be shown that their definitions are incomplete and their lemmas contain questionable results. As a result, we suggest that decision-makers not use Finan and Hurley's diagonal procedure to construct rank-order consistent comparison matrix but continue to use the comparison matrix of Saaty [3]. 2. R E V I E W

OF

FINAN

AND

HURLEY'S

RESULTS

Finan and Hurley [1] formulated the following definitions. A pairwise comparison matrix with a triple of elements (aij,aik,akj), i < k < j satisfies (a) if MDj =-- (aij -- 1)(aikakj -- 1) < 0, then, A is defined as diagonally inconsistent; (b) if M v j =-- ( a i k - - 1)(a~j(akj) -1 - 1) < 0, then, A is defined as vertically inconsistent; and (c) if MHj =---(akj - 1)(a~j(a~k) -1 - 1) < 0, then, A is defined as horizontally inconsistent; and a pairwise comparison matrix A is defined as rank-order consistent if A is diagonally consistent, vertically consistent and horizontally Consistent. Further, they assumed that given an initial comparison matrix A0 with relabeled objects 1,2,..., n in order of importance, object 1 is more important than object 2; object 2 is more important than object 3, and so on. Let the resulting matrix have the reordering R (A0) and the elements of this matrix be R (aij), then, the following results are given in [1]. LEMMA 1. R(Ao) is diagonally consistent if R(alj) > 1, for all i < j. LEMMA 2. R(Ao) is vertically consistent if R(a3_l,j) <_ R(aj-2,j) < ... <_ R(al,j) for j = 3, 4 , . . . , n. (In their proof, they claimed that "since R (ask) > I by definition," under the condition i < k.) LEMMA 3. R(Ao) is horizontally consistent if R(ai,~+l) <_ R(ai,i+2) < ... < R(ai,n), for i = 1, 2 .... , n - 2. (In their proof, they claimed that "R (a~j) >_ 1 by definition," under the condition k < j.) LEMMA 4. R(Ao) is rank-order consistent if R(aij) _> max {R(ai,j_l), R (ai+l,j)} and R(aij) >_ 1 for all i < j.

Note on Diagonal Procedure

1091

Here, we point out that by the definition of R (A0), we do not have R (a~k) _> 1 with i < k or R(akj) > 1 with k < j. In their Lemma 1 and Lemma 4, the condition of R(aij) > 1 for all i < j is assumed in the sufficient condition of Lemmas 1 and 4, respectively. Hence, it is clear that according to their definition of matrix R (A0), we do not have the property,

R (aij) _> 1,

for all i < j.

3. O U R IMPROVEMENT THEIR THEORETICAL

(1)

FOR PROBLEM

We believe that the simplest way to save their theoretical structure is to put equation (1) into the definition of R (A0), then, the problems with Lemmas 2 and 3 disappear. Similarly, one must remove equation (1) from the sufficient condition of Lemmas 1 and 4. 4.

DIAGONAL

PROCEDURE

OF

FINAN

AND

HURLEY

Next, we review their procedure to construct rank-order consistent matrices. They suggest the following diagonal procedure. 1. Relabel the objects in rank-order (wl >_w2 > ... > Wn). 2, Input elements ai,i+l, i = 1 , 2 , . . . , n - 1. 3. Input all other elements above the diagonal in such a way that a~j > max { a i j - 1 , ai+l,j},

for all i < j.

T h e y point out that in the case where a decision-maker is certain about the rank-order of objects, using the input procedure specified above, followed by a calculation of the consistency index, provides a check for consistency. 5. T H E RANK-ORDER

INDEX RATIO CONSISTENT

OF MATRIX

First, we must point out that in Step 2 of their diagonal procedure, there is no restriction on ai#+l. However, we suggest that a decision-maker inputs ai,i+l, i = 1, 2 , . . . , n - 1 under the condition ai#+l _> 1. Next, we give an example in which if we follow their proposed diagonal procedure, the resulting matrix turns out to be inconsistent. Assume that we have three objects and we know their rank-order to be (wl _ w2 _> w3), and then, we input the elements a12 7 and a23 = 5. From Finan and HuEey's procedure, there are the following three possible choices for a13: a13 = 7, a13 = 8, or a~3 = 9. Since the random index (R.I.) for 3 x 3 matrices is 0.58, we find the maximum eigen value, )~max, the consistency index C.I.= (Amax - 1)/(3 - I)), and the index ratio (C.I./R.I.). We obtain for the three possibilities ~max = 3.295, 3.247 and 3.208, and index ratio = 0.254, 0.213 and 0.180 for a13 = 7, a13 --- 8 and a13 = 9, respectively. Therefore, from Finan and Hurley's procedure, they cannot adjust the value ofal3 so that the resulting matrix has index ratio less than 0. I. Consequently, their sophisticated procedure to derive a rank-order consistent matrix still cannot avoid the problem of inconsistency. Inconsistency and its measurement are vital to the process of adjusting judgment to improve relations and understanding of those relations as a result of perturbation and fuzziness in information and how it is interpreted. =

6. THE

WEIGHTS

FROM

THE

DIAGONAL

PROCEDURE

Finan and Hurley [I] considered that a decision-maker can specify the rank-order of objects (Wl _> w2 _> --. > wn) in an AHP pairwise comparison matrix and be uncertain about the precise weights corresponding to each object. If the decision-maker follows the diagonal procedure to create a rank-order consistent matrix, say (aij)n× n and assumes the eigenvector corresponding to Amax as (vl, v2, ... ,vn) then does the eigenvector have the same predetermined order as vl > v2 >_ ." _> vn? In what follows, we prove that vl _> v2 _> ... > vn holds.

1092

H. CHAOet al.

THEOREM

i. The eigenveetor corresponding to Amax of a rank-order consistent matrix preserves the predetermined order as vl >_ v2 >_ ... >_ Vn.

PROOF. First, we consider the upper half triangle of (aij)nxn with i < j. From our improved procedure we have ai#+l _> 1 for i = 1, 2 , . . . , n - 1. By Step 3 of the diagonal procedure with vertical consistency, we have aij > a~+1,9, for i = 1,... , j - 1. Since (a~j)~x~ is reciprocal, we know for i = 1 , . . . , n - 1 and j = 1 , . . . , n that

aij ~ ai+l,j.

(2)

Now, we consider Combining equations (2) and (3) with Xm~x >__n > 0, we have, 1

~)i

-

-

Vi+l •

n

)kma x E

j

(aij

-

ai+l,j) vj > O.

-

(4)

Equation (4) shows that the diagonal procedure preserves rank-order.

7. A S I M P L E P R O C E D U R E T O DERIVE NUMERICAL WEIGHTS Next, we consider the meaning of the diagonal procedure. Before they perform the diagonal procedure, Finan and Hurley [1] already know the order of the objectives as wl _> w2 >_ .-- >_ w~ and the value of the relative ratio between w~ and wi+l as (wi)/(w~+l) = ai#+l. After they run the diagonal procedure, they still obtain the same rank-order. There is no reason to perform a sophisticated procedure to derive the already known results. On the other hand, if Finan and Hurley [1] really want to know the numerical relation between every objective, we suggest the following elementary procedure. From wi/w~+l = ai,i+l, we have w i / w ~ = ~Ij=i aj,j+l, and the normalized weights are given by n--1

n--1

I-[ aj,j+~

j=l n--1 n--1

1 JI- E

/=1

II azj+l

'

j=2 n--1 n--1

an--l, n '''''

H aj,j+l 1-~ E 1-I aj,j+l j=i i=1 j=i

n--1 n--1

1-~ E

i=1

1 '

n--1 n--1

,

(5)

1-~ aj,j+l l--~ E I-~ aj,j+ 1 j=i i=1 j=i

which is easy to compute and exactly expresses the relationship among the objectives and avoids the problem of inconsistency in the diagonal procedure of Finan and Hurley [1].

8. C O N C L U S I O N We have examined the diagonal procedure (to construct a rank-order consistent comparison matrix) of Finan and Hurley [1] and proposed improvements in their definitions, lemmas, and procedure. We gave an example to show that their rank-order consistent m a t r i x still has a problem with inconsistency. Moreover, we offered a simple procedure to derive the numerical weights for those decision-makers who know the rank-order of the objectives and the relative weight between two consecutive objectives. As a result, we conclude that the decision-makers should use the comparison matrix of Saaty [3] and not refer to what Finan and Hurley [1] have proposed.

REFERENCES 1. J.S, Finan and W.J. Hurley, A note on a method to ensure rank-order consistency in the analytic hierarchy process, Int. Trans. Opl Res. 3 (1), 99-103, (1996). 2. P.T. Harker and L.G. Vargas, The theory of ratio scale estimation: Saaty's anaiytic hierarchy process, Management Science 33 (11), 1383-1403, (1987). 3. T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, (1990).