The aim of the article is to introduce a mathematical concept of indecisiveness into the analytic hierarchy process (AHP) framework. Indecisiveness can be useful in two ways: first, decision makers with high indecisiveness (higher than a given threshold value) can be excluded from a decision making process in its early stages as low-competent and replaced by other, more competent DMs; second, indecisiveness along with consistency index C.I. can be used for the calculation of (aposteriori) DMs’ weights without any additional information about DMs’ age, formal knowledge, social status, etc. The proposed approach is demonstrated on examples

Jiri, Mazurek

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Some Notes on Inconsistence and
Indecisiveness in the Analytic Hierarchy
Process
Mazurek Jiri
Silesian University - School of Business Administration
9 November 2012
Online at https://mpra.ub.uni-muenchen.de/42543/
MPRA Paper No. 42543, posted 12 November 2012 10:55 UTC
Some Notes on Inconsistence and Indecisiveness in the Analytic Hierarchy 
Process 
 
Jiří Mazurek 
 
Abstract: The aim of the article is to introduce a mathematical concept of indecisiveness into the analytic 
hierarchy process (AHP) framework. Indecisiveness can be useful in two ways: first, decision makers with high 
indecisiveness (higher than a given threshold value) can be excluded from a decision making process in its early 
stages as low-competent and replaced by other, more competent DMs; second, indecisiveness along with 
consistency index C.I. can be used for the calculation of (aposteriori) DMs’ weights without any additional 
information about DMs’ age, formal knowledge, social status, etc. The proposed approach is demonstrated on 
examples. 
 
Keywords: AHP, inconsistence, decision making, indecisiveness. 
 
JEL classification: D79. 
 
 
1. Introduction 
 
Decision making processes are omnipresent in human life as well as in many areas of 
economics such as marketing, management, human resources, logistics, etc. Decision making 
methods range from the simple majority rule to more sophisticated approaches such as the 
analytic hierarchy/network process (AHP/ANP) by T. L. Saaty, see e.g. [3], [4], [5] or [6], 
computing with words ([7] or [8]) or decision making in the fuzzy environment ([1] or [2]).  
 
In general, a multi-criteria decision making process consists of the following steps: 
1. Identification of the problem. 
2. Analysis of the problem. 
3. Establishing sets of alternatives, criteria and experts who will evaluate alternatives. 
4. Evaluation of alternatives by experts with respect to given criteria. 
5. Selection of the best alternative (which includes aggregation of experts’ preferences and 
ordering of alternatives from the best to the worst). 
In the literature the step 5 is of the most interest, as there is vast number of methods proposed 
for various kinds of decision making processes.  This article focuses on the step 4, and more 
specifically on the problem of ‘quality’ of decision makers’ (DMs’) preferences in the group 
decision making. So far, DMs’ preferences were thoroughly examined in terms of their 
(in)consistence (that is existence of a contradiction) because human judgment is imperfect due 
to many reasons such as time pressure, imprecise information, lack of knowledge, prejudices, 
etc.  
 
DMs’ preferences are crucial part of each decision making, as they represent problem’s input, 
hence they determine problem’s solution. Apart from (in)consistence, which is of great 
importance without no doubt, DMs’ preferences are not studied in a more detail, though there 
are at least two other often neglected features important for problem’s solution associated 
with DMs’ preferences:  
- differences among DMs’ preferences (degree of a conflict among DMs), 
- decisiveness of each DM. 
As for the former, conflicting attitudes of DMs might result in no consensus under some 
circumstances (consider e.g. situation, where two equally strong groups of DMs are in 
opposition). Group decision making is always associated with less or more conflict among 
DMs, which must be resolved finally if a consensus is going to be achieved.   
 
As for the latter, decision makers can express their preferences on a set of alternatives with a 
different degree of intensity (or confidence). In AHP, the intensity of preference is expressed 
on Saaty’s scale from 1 to 9 (see Table 1). In real decision situations, some decision makers 
can express stronger preferences than others. It seems natural to assign higher weights to a 
DM who has strong opinion on a topic (a DM is strongly decisive possibly due to better 
knowledge or experience), and lower weight to a DM with weak opinion. As shown in 
Section 2, in extreme cases a DM can express preferences that are absolutely consistent (see 
Table 2), but completely indecisive, thus useless. Clearly, inconsistence is not sufficient 
measure of quality DMs’ preferences. 
 
Therefore, the aim of the article is to propose the measure of indecisiveness in the group AHP 
framework. This new measure can be useful in two ways:  
- DMs with high indecisiveness can be excluded from a decision making process as low-
competent immediately in its early stages and replaced by other, more competent DMs. 
- Indecisiveness along with Saaty’s consistency index C.I. can be used for the calculation of 
(aposteriori) DMs’ weights. Decision makers with larger consistency and decisiveness are 
assigned larger weights and vice versa. The advantage of this approach rests in the fact that 
only information given by DMs is used, and no additional knowledge ‘from outside‘ about 
DMs’ age, formal knowledge, social status, etc., is required. 
 
The article is organized as follows: in Section 2 AHP and consistency index C.I. are briefly 
described, in Section 3 the measure of indecisiveness is introduced along with derivation of 
weights of DMs, and in Section 4 numerical examples are provided. Conclusions close the 
article. 
 
2. Analytic hierarchy process  
 
Analytic hierarchy process (AHP) was proposed by T. L. Saaty in 1980 [4]. Its fundamental 
part consists of pair-wise comparisons of objects on the kth level of hierarchy with regard to 
objects on the immediately higher (k – 1)th level. Typically, the highest level is a goal, the 
second level form criteria and the lowest level consists of alternatives. The aim of AHP lies 
in the selection of the best alternative. As the article focuses on indecisiveness of decision 
makers in evaluating alternatives, only comparisons of alternatives with regard to a given set 
of criteria by one or more DMs will be considered.  
 
Alternatives are compared on the scale from 1 to 9, where 1 denotes equal importance and 9 
extreme importance of one alternative over another (see Table 1); sij denotes preference of an 
alternative i over an alternative j. Preferences sij are reciprocal: if an alternative A is 
moderately preferred over an alternative B, then 3=ABs , and by definition 3/1=BAs . 
All pair-wise preferences sij form a reciprocal matrix P with elements 
1
ij
ji
s
s
= ; ji,∀ .  
Pair-wise preferences are consistent, if ij jk iks s s⋅ = , kji ,,∀ ; that is the transitive property of 
DM’s preferences is preserved. 
Consistency of DMs’ preferences is expressed by consistency index C.I. given as [4]: 
max
. .
1
nC I
n
λ −
=
−
,     (1) 
where maxλ  is the largest (positive) eigenvalue of the matrix P, n is the order of P, and 
0.. ≥IC . The value . . 0C I =  indicates absolute consistency of preferences; the larger is C.I., 
the more inconsistent preferences are. According to Saaty [6], human judgment is inconsistent 
by nature, so . . 0.1C I <  is tolerated.   
 
Consistency index is the only measure of ‘quality’ of decision makers’ preferences in AHP. 
However, consider a preference matrix A shown in Table 2. It is easy to verify 
that max ( ) 5Aλ = , so . . 0C I = . Preferences expressed by A are absolutely consistent, but they 
are also  absolutely useless, as no alternative is preferred over any other alternative (because a 
decision maker is absolutely indecisive). This example shows that consistency index alone is 
not a sufficient tool for preferences assessment.  
 
 
Table 1. Saaty’s scale in AHP. Source: [6]. 
 
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
A
 
 
 
 =
 
 
 
 
 
Table 2. An example of preferences of an absolutely indecisive DM. 
 
3. The measure of indecisiveness 
 
As shown in Table 2, a DM is absolutely indecisive, if all sij =1. Let 1 denote the matrix with 
such elements. Then decisiveness of a DM with preferences given by the matrix )( ijaA  can be 
evaluated from an absolute difference matrix )( ijdD : 1−= ijij ad  .  
 
The larger are elements dij, that is differences 1−ija , the higher is DM’s decisiveness.  For a 
decisiveness evaluation, the entrywise p-norm applied on D can be used:   
1/
1 1
p
n n
p
ijp
i j
D d
= =
 
=  
 
∑∑       (2) 
where p N∈ . The special case p = 2 is called Frobenius norm (it is an equivalent of the 
Euclidean norm for vectors), and will be used henceforward. 
 
The larger is 
p
D  the greater is the decisiveness of a decision maker. It is possible to state 
some minimal value of 
p
D  (the threshold of decisiveness – τ) dependent on n, so that DMs 
with decisiveness  
p
D  smaller than τ would be excluded from a decision making process. 
 
Weights wi of decision makers can be derived with the use of formulas (1) and (2) combined  
into one overall index. Weights of DMs are functions of indecisiveness and consistency 
index: ( , . .)
p
w w D C I= , and should satisfy the following properties:  
- The function w is non-negative. 
- The function w is strictly increasing in 
p
D . 
- The function w is strictly decreasing in C.I. 
In addition, because C.I. acts as a penalization factor in w, it is reasonable to require no 
penalization for absolute consistency (C.I. = 0). This condition is satisfied e.g. for the 
exponential function ( 0 1e = ). One simple function w satisfying all three conditions, and 
additional condition mentioned above, can be given as:  
10 . .
2
C Iw D e−= ⋅      (3) 
The factor of 10 in the exponential function was chosen to magnify the value of the 
inconsistency index given by (1), which is by definition much smaller than matrix elements in 
most cases. Also, it follows from (3) that for the limit value of acceptable consistency C.I. = 
0.1, the weight w takes especially simple form: 2
D
w
e
= .  
Usually, DMs compare alternatives with regard to more than one criterion. In such a case, 
indecisiveness and consistency index C.I. is estimated for each criterion separately, and then 
the average indecisiveness along with average C.I. is used to calculate (average) DMs’ 
weights by relation (3).   
 
Numerical examples illustrating weight evaluation with the use of (3) are provided in the 
following section. Nowdays, there are many software tools which facilitate computation of 
C.I. or maxλ :  
- ExpertChoice (www.expertchoice.com),  
- Mathematica (www.wolfram.com/mathematica/), 
- Meta-numerics (www.meta-numerics.net),  
- WolframAlpha (www.wolframalpha.com), etc. 
ExpertChoice is a commercial product for AHP solutions; it computes C.I. directly. Other 
mathematical tools are capable of computing an eigenvalue system of the matrix P. The last 
two products are free of charge. 
 
 
4. Numerical examples 
 
In this Section two numerical examples are provided. In Example 1 alternatives are compared 
by one criterion and in Example 2 by two criteria. In both cases consistency index C.I., 
indecisiveness 
p
D and weights w of all DMs are evaluated. 
Example 1. Let A and B be preference matrices of decision makers DM1 and DM2 
respectively on three alternatives (see Table 3) by a given criterion. We will evaluate weights 
of both decision makers using the formula (3) and the Frobenius norm (p = 2). 
 
1 9 7
1/ 9 1 3
1/ 7 1/ 3 1
A
 
 
=  
 
 
 
1 2 3
1 / 2 1 2
1 / 3 1 / 2 1
B
 
 
=  
 
 
 
Table 3. Preferences of  DM1 (matrix A) and DM2 (matrix B). 
 
Consistency index C.I. (1) for both DMs: 
max ( ) 3.206Aλ = , n = 3 ⇒  1
3.206 3
. .( ) 0.103
2
C I DM −= = . 
max ( ) 3.008Bλ = , n = 3 ⇒  1
3.008 3
. .( ) 0.004
2
C I DM −= = . 
Hence, DM2 is more consistent in his evaluation of alternatives. On the other hand, DM1 is 
much more decisive than DM2: 
( )
1/22 2 2
2 2 2
12
8 6 28 6 2 10.29
9 7 3
D DM
      
= + + + + + =      
       
 
( )22D DM = 2.64 
Finally, we compute weights w1 and w2, and normalized weights w1´ and w2´ of DM1 and 
DM2 respectively using (3): 
10 0.103
1 10.29 3.67w e
− ⋅
= ⋅ = . 
10 0.004
2 2.64 2.63w e
− ⋅
= ⋅ = . 
1´ 0.583w = , 2´ 0.417w = . 
The first decision maker is assigned slightly larger weight due to his greater decisiveness.  
 
Example 2. Let A and B be the same preference matrices of decision makers DM1 and DM2 
respectively on three alternatives (see Table 3) by a criterion C1 as in Example 1. Let C and D 
be preference matrices of decision makers DM1 and DM2 respectively on three alternatives by 
a criterion C2 (see Table 4). 
 
1 2 3
1/ 3 1 4
1/ 2 1/ 4 1
C
 
 
=  
 
 
 
1 4 1/ 2
1/ 4 1 1/ 5
2 5 1
D
 
 
=  
 
 
 
Table 4. Preferences of  DM1 (matrix C) and DM2 (matrix D) with regard to the criterion C2. 
 
Consistency index C.I. (1) of preferences with regard to the criterion C2: 
max ( ) 3.107Aλ = , n = 3 ⇒  1
3.107 3
. .( ) 0.054
2
C I DM −= = . 
max ( ) 3.025Bλ = , n = 3 ⇒  1
3.025 3
. .( ) 0.013
2
C I DM −= = . 
Again, DM2 is more consistent in his evaluation. As for decisiveness, DM2 is more decisive:  
( )
1/22 2 2
2 2 2
12
2 1 31 2 3 3.906
3 2 4
D DM
      
= + + + + + =      
       
, 
( )22D DM = 5.239. 
Weights  (2)1w  and 
(2)
2w , and normalized weights 
(2)
1 ´w  and 
(2)
2 ´w  of DM1 and DM2 
respectively, are: 
(2) 10 0.054
1 3.906 2.276w e
− ⋅
= ⋅ = . 
10 0.013
2 5.239 4.600w e
− ⋅
= ⋅ = . 
(2)
1 ´ 0.331w =
(2)
2 ´ 0.669.w =  
From Example 1 we know, that weights of DMs with regard to the criterion C1 are: 
(1)
1 ´ 0.583w = , 
(1)
2 ´ 0.417w =  
By averaging we can assign final weights to both DMs: 1´ 0.457w = , 2´ 0.543w = . 
 
With computed weights the decision making can proceed into the aggregation phase and then 
to the selection of the best alternative. 
 
5. Conclusions 
 
The aim of the article was to introduce a concept of indecisiveness into the group analytic 
hierarchy process (AHP) framework and to show how weights of decision makers can be 
derived only from their preferences with the use of indecisiveness and Saaty’s consistency 
index C.I. The advantage of this approach is that no additional information about decision 
makers’ experience or knowledge ‘from outside’ is required. Further work may focus on 
differences (conflicts) among decision makers’ preferences in the group AHP and the 
influence of the differences on a possibility of an existence of a group consensus.   
 
 
References: 
[1] Carlsson, C. Fullér, R. Fuzzy multiple criteria decision making: Recent developments. 
Fuzzy Sets and Systems 78(2), 1996, p. 139-153. 
[2] Fodor, J., Roubens, M. Fuzzy preference modelling and multicriteria decision support, 
Dordrecht: Kluwer 1994.  
[3] Ramík, J. Perzina, R. Moderní metody hodnocení a rozhodování, SU OPF: 2008.  ISBN 
978-80-7248-497-3. 
[4] Saaty T. L. The Analytic Hierarchy Process: Planning, Priority Setting, Resource 
Allocation, ISBN 0-07-054371-2, McGraw-Hill, 1980. 
[5] Saaty T. L. The Analytic hierarchy and Analytic Network Measurement Processes: 
Aplications to the Decisions under Risk. European Journal of Pure and Applied 
Mathematics, (1)1, 2008, p. 122-196. 
[6] Saaty T. L. Relative Measurement and its Generalization in Decision Making – Why 
Pairwise Comparisons are Central in Mathematics for the Measurement of Intanglible 
Factors, The Analytic Hierarchy/Network Process. Rev. R. Acad. Cien. Serie A, Mat., Vol. 
102(2), 2008, p. 251-318.  
[7] Wang, P. P. Computing with words. John Wiley, NewYork, 2001. 
[8] Zadeh, L. A. Fuzzy logic-computing with words. IEEE Transactions on Fuzzy Systems, 4, 
1996, p. 103-111. 
 
 
Mgr. Jiří Mazurek, Ph.D. 
Assistant Professor 
Department of Mathematical Methods in Economics 
School of Business Administration in Karviná, 
Silesian University in Opava 
e-mail: mazurek@opf.slu.cz 
 
 


Some Notes on Inconsistence and Indecisiveness in the Analytic Hierarchy Process

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveSome Notes on Inconsistence andIndecisiveness in the Analytic HierarchyProcessMazurek JiriSilesian University - School of Business Administration9. November 2012Online at http://mpra.ub.uni-muenchen.de/42543/MPRA Paper No. 42543, posted 12. November 2012 10:55 UTCSome Notes on Inconsistence and Indecisiveness in the Analytic Hierarchy Process  Jiří Mazurek  Abstract: The aim of the article is to introduce a mathematical concept of indecisiveness into the analytic hierarchy process (AHP) framework. Indecisiveness can be useful in two ways: first, decision makers with high indecisiveness (higher than a given threshold value) can be excluded from a decision making process in its early stages as low-competent and replaced by other, more competent DMs; second, indecisiveness along with consistency index C.I. can be used for the calculation of (aposteriori) DMs’ weights without any additional information about DMs’ age, formal knowledge, social status, etc. The proposed approach is demonstrated on examples.  Keywords: AHP, inconsistence, decision making, indecisiveness.  JEL classification: D79.   1. Introduction  Decision making processes are omnipresent in human life as well as in many areas of economics such as marketing, management, human resources, logistics, etc. Decision making methods range from the simple majority rule to more sophisticated approaches such as the analytic hierarchy/network process (AHP/ANP) by T. L. Saaty, see e.g. [3], [4], [5] or [6], computing with words ([7] or [8]) or decision making in the fuzzy environment ([1] or [2]).   In general, a multi-criteria decision making process consists of the following steps: 1. Identification of the problem. 2. Analysis of the problem. 3. Establishing sets of alternatives, criteria and experts who will evaluate alternatives. 4. Evaluation of alternatives by experts with respect to given criteria. 5. Selection of the best alternative (which includes aggregation of experts’ preferences and ordering of alternatives from the best to the worst). In the literature the step 5 is of the most interest, as there is vast number of methods proposed for various kinds of decision making processes.  This article focuses on the step 4, and more specifically on the problem of ‘quality’ of decision makers’ (DMs’) preferences in the group decision making. So far, DMs’ preferences were thoroughly examined in terms of their (in)consistence (that is existence of a contradiction) because human judgment is imperfect due to many reasons such as time pressure, imprecise information, lack of knowledge, prejudices, etc.   DMs’ preferences are crucial part of each decision making, as they represent problem’s input, hence they determine problem’s solution. Apart from (in)consistence, which is of great importance without no doubt, DMs’ preferences are not studied in a more detail, though there are at least two other often neglected features important for problem’s solution associated with DMs’ preferences:  - differences among DMs’ preferences (degree of a conflict among DMs), - decisiveness of each DM. As for the former, conflicting attitudes of DMs might result in no consensus under some circumstances (consider e.g. situation, where two equally strong groups of DMs are in opposition). Group decision making is always associated with less or more conflict among DMs, which must be resolved finally if a consensus is going to be achieved.    As for the latter, decision makers can express their preferences on a set of alternatives with a different degree of intensity (or confidence). In AHP, the intensity of preference is expressed on Saaty’s scale from 1 to 9 (see Table 1). In real decision situations, some decision makers can express stronger preferences than others. It seems natural to assign higher weights to a DM who has strong opinion on a topic (a DM is strongly decisive possibly due to better knowledge or experience), and lower weight to a DM with weak opinion. As shown in Section 2, in extreme cases a DM can express preferences that are absolutely consistent (see Table 2), but completely indecisive, thus useless. Clearly, inconsistence is not sufficient measure of quality DMs’ preferences.  Therefore, the aim of the article is to propose the measure of indecisiveness in the group AHP framework. This new measure can be useful in two ways:  - DMs with high indecisiveness can be excluded from a decision making process as low-competent immediately in its early stages and replaced by other, more competent DMs. - Indecisiveness along with Saaty’s consistency index C.I. can be used for the calculation of (aposteriori) DMs’ weights. Decision makers with larger consistency and decisiveness are assigned larger weights and vice versa. The advantage of this approach rests in the fact that only information given by DMs is used, and no additional knowledge ‘from outside‘ about DMs’ age, formal knowledge, social status, etc., is required.  The article is organized as follows: in Section 2 AHP and consistency index C.I. are briefly described, in Section 3 the measure of indecisiveness is introduced along with derivation of weights of DMs, and in Section 4 numerical examples are provided. Conclusions close the article.  2. Analytic hierarchy process   Analytic hierarchy process (AHP) was proposed by T. L. Saaty in 1980 [4]. Its fundamental part consists of pair-wise comparisons of objects on the kth level of hierarchy with regard to objects on the immediately higher (k – 1)th level. Typically, the highest level is a goal, the second level form criteria and the lowest level consists of alternatives. The aim of AHP lies in the selection of the best alternative. As the article focuses on indecisiveness of decision makers in evaluating alternatives, only comparisons of alternatives with regard to a given set of criteria by one or more DMs will be considered.   Alternatives are compared on the scale from 1 to 9, where 1 denotes equal importance and 9 extreme importance of one alternative over another (see Table 1); sij denotes preference of an alternative i over an alternative j. Preferences sij are reciprocal: if an alternative A is moderately preferred over an alternative B, then 3=ABs , and by definition 3/1=BAs . All pair-wise preferences sij form a reciprocal matrix P with elements 1ijjiss= ; ji,∀ .  Pair-wise preferences are consistent, if ij jk iks s s⋅ = , kji ,,∀ ; that is the transitive property of DM’s preferences is preserved. Consistency of DMs’ preferences is expressed by consistency index C.I. given as [4]: max. .1nC Inλ −=−,     (1) where maxλ  is the largest (positive) eigenvalue of the matrix P, n is the order of P, and 0.. ≥IC . The value . . 0C I =  indicates absolute consistency of preferences; the larger is C.I., the more inconsistent preferences are. According to Saaty [6], human judgment is inconsistent by nature, so . . 0.1C I <  is tolerated.    Consistency index is the only measure of ‘quality’ of decision makers’ preferences in AHP. However, consider a preference matrix A shown in Table 2. It is easy to verify that max ( ) 5Aλ = , so . . 0C I = . Preferences expressed by A are absolutely consistent, but they are also  absolutely useless, as no alternative is preferred over any other alternative (because a decision maker is absolutely indecisive). This example shows that consistency index alone is not a sufficient tool for preferences assessment.    Table 1. Saaty’s scale in AHP. Source: [6].  1 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 1A    =     Table 2. An example of preferences of an absolutely indecisive DM.  3. The measure of indecisiveness  As shown in Table 2, a DM is absolutely indecisive, if all sij =1. Let 1 denote the matrix with such elements. Then decisiveness of a DM with preferences given by the matrix )( ijaA  can be evaluated from an absolute difference matrix )( ijdD : 1−= ijij ad  .   The larger are elements dij, that is differences 1−ija , the higher is DM’s decisiveness.  For a decisiveness evaluation, the entrywise p-norm applied on D can be used:   1/1 1pn npijpi jD d= = =   ∑∑       (2) where p N∈ . The special case p = 2 is called Frobenius norm (it is an equivalent of the Euclidean norm for vectors), and will be used henceforward.  The larger is pD  the greater is the decisiveness of a decision maker. It is possible to state some minimal value of pD  (the threshold of decisiveness – τ) dependent on n, so that DMs with decisiveness  pD  smaller than τ would be excluded from a decision making process.  Weights wi of decision makers can be derived with the use of formulas (1) and (2) combined  into one overall index. Weights of DMs are functions of indecisiveness and consistency index: ( , . .)pw w D C I= , and should satisfy the following properties:  - The function w is non-negative. - The function w is strictly increasing in pD . - The function w is strictly decreasing in C.I. In addition, because C.I. acts as a penalization factor in w, it is reasonable to require no penalization for absolute consistency (C.I. = 0). This condition is satisfied e.g. for the exponential function ( 0 1e = ). One simple function w satisfying all three conditions, and additional condition mentioned above, can be given as:  10 . .2C Iw D e−= ⋅      (3) The factor of 10 in the exponential function was chosen to magnify the value of the inconsistency index given by (1), which is by definition much smaller than matrix elements in most cases. Also, it follows from (3) that for the limit value of acceptable consistency C.I. = 0.1, the weight w takes especially simple form: 2Dwe= .  Usually, DMs compare alternatives with regard to more than one criterion. In such a case, indecisiveness and consistency index C.I. is estimated for each criterion separately, and then the average indecisiveness along with average C.I. is used to calculate (average) DMs’ weights by relation (3).    Numerical examples illustrating weight evaluation with the use of (3) are provided in the following section. Nowdays, there are many software tools which facilitate computation of C.I. or maxλ :  - ExpertChoice (www.expertchoice.com),  - Mathematica (www.wolfram.com/mathematica/), - Meta-numerics (www.meta-numerics.net),  - WolframAlpha (www.wolframalpha.com), etc. ExpertChoice is a commercial product for AHP solutions; it computes C.I. directly. Other mathematical tools are capable of computing an eigenvalue system of the matrix P. The last two products are free of charge.   4. Numerical examples  In this Section two numerical examples are provided. In Example 1 alternatives are compared by one criterion and in Example 2 by two criteria. In both cases consistency index C.I., indecisiveness pD and weights w of all DMs are evaluated. Example 1. Let A and B be preference matrices of decision makers DM1 and DM2 respectively on three alternatives (see Table 3) by a given criterion. We will evaluate weights of both decision makers using the formula (3) and the Frobenius norm (p = 2).  1 9 71/ 9 1 31/ 7 1/ 3 1A  =     1 2 31 / 2 1 21 / 3 1 / 2 1B  =     Table 3. Preferences of  DM1 (matrix A) and DM2 (matrix B).  Consistency index C.I. (1) for both DMs: max ( ) 3.206Aλ = , n = 3 ⇒  13.206 3. .( ) 0.1032C I DM −= = . max ( ) 3.008Bλ = , n = 3 ⇒  13.008 3. .( ) 0.0042C I DM −= = . Hence, DM2 is more consistent in his evaluation of alternatives. On the other hand, DM1 is much more decisive than DM2: ( )1/22 2 22 2 2128 6 28 6 2 10.299 7 3D DM      = + + + + + =              ( )22D DM = 2.64 Finally, we compute weights w1 and w2, and normalized weights w1´ and w2´ of DM1 and DM2 respectively using (3): 10 0.1031 10.29 3.67w e− ⋅= ⋅ = . 10 0.0042 2.64 2.63w e− ⋅= ⋅ = . 1´ 0.583w = , 2´ 0.417w = . The first decision maker is assigned slightly larger weight due to his greater decisiveness.   Example 2. Let A and B be the same preference matrices of decision makers DM1 and DM2 respectively on three alternatives (see Table 3) by a criterion C1 as in Example 1. Let C and D be preference matrices of decision makers DM1 and DM2 respectively on three alternatives by a criterion C2 (see Table 4).  1 2 31/ 3 1 41/ 2 1/ 4 1C  =     1 4 1/ 21/ 4 1 1/ 52 5 1D  =     Table 4. Preferences of  DM1 (matrix C) and DM2 (matrix D) with regard to the criterion C2.  Consistency index C.I. (1) of preferences with regard to the criterion C2: max ( ) 3.107Aλ = , n = 3 ⇒  13.107 3. .( ) 0.0542C I DM −= = . max ( ) 3.025Bλ = , n = 3 ⇒  13.025 3. .( ) 0.0132C I DM −= = . Again, DM2 is more consistent in his evaluation. As for decisiveness, DM2 is more decisive:  ( )1/22 2 22 2 2122 1 31 2 3 3.9063 2 4D DM      = + + + + + =             , ( )22D DM = 5.239. Weights  (2)1w  and (2)2w , and normalized weights (2)1 ´w  and (2)2 ´w  of DM1 and DM2 respectively, are: (2) 10 0.0541 3.906 2.276w e− ⋅= ⋅ = . 10 0.0132 5.239 4.600w e− ⋅= ⋅ = . (2)1 ´ 0.331w =(2)2 ´ 0.669.w =  From Example 1 we know, that weights of DMs with regard to the criterion C1 are: (1)1 ´ 0.583w = , (1)2 ´ 0.417w =  By averaging we can assign final weights to both DMs: 1´ 0.457w = , 2´ 0.543w = .  With computed weights the decision making can proceed into the aggregation phase and then to the selection of the best alternative.  5. Conclusions  The aim of the article was to introduce a concept of indecisiveness into the group analytic hierarchy process (AHP) framework and to show how weights of decision makers can be derived only from their preferences with the use of indecisiveness and Saaty’s consistency index C.I. The advantage of this approach is that no additional information about decision makers’ experience or knowledge ‘from outside’ is required. Further work may focus on differences (conflicts) among decision makers’ preferences in the group AHP and the influence of the differences on a possibility of an existence of a group consensus.     References: [1] Carlsson, C. Fullér, R. Fuzzy multiple criteria decision making: Recent developments. Fuzzy Sets and Systems 78(2), 1996, p. 139-153. [2] Fodor, J., Roubens, M. Fuzzy preference modelling and multicriteria decision support, Dordrecht: Kluwer 1994.  [3] Ramík, J. Perzina, R. Moderní metody hodnocení a rozhodování, SU OPF: 2008.  ISBN 978-80-7248-497-3. [4] Saaty T. L. The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, ISBN 0-07-054371-2, McGraw-Hill, 1980. [5] Saaty T. L. The Analytic hierarchy and Analytic Network Measurement Processes: Aplications to the Decisions under Risk. European Journal of Pure and Applied Mathematics, (1)1, 2008, p. 122-196. [6] Saaty T. L. Relative Measurement and its Generalization in Decision Making – Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intanglible Factors, The Analytic Hierarchy/Network Process. Rev. R. Acad. Cien. Serie A, Mat., Vol. 102(2), 2008, p. 251-318.  [7] Wang, P. P. Computing with words. John Wiley, NewYork, 2001. [8] Zadeh, L. A. Fuzzy logic-computing with words. IEEE Transactions on Fuzzy Systems, 4, 1996, p. 103-111.   Mgr. Jiří Mazurek, Ph.D. Assistant Professor Department of Mathematical Methods in Economics School of Business Administration in Karviná, Silesian University in Opava e-mail: mazurek@opf.slu.cz   

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Some Notes on Inconsistence and Indecisiveness in the Analytic Hierarchy Process

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