Analytical Hierarchy Process under Group Decision
Making with Some Induced Aggregation Operators
Binyamin Yusoff1 and José M. Merigó-Lindahl2,3
1
Department of Mathematical Economics and Finance, University of Barcelona,
Av. Diagonal 690, 08034, Barcelona, Spain
2
Manchester Business School, University of Manchester,
Booth Street West, M15 6PB Manchester, UK
3
Risk Center, University of Barcelona,
Av. Diagonal 690, 08034, Barcelona, Spain
[email protected], [email protected]
Abstract. This paper focuses on the extension of Analytical Hierarchy Process
under Group Decision Making (AHP-GDM) with some induced aggregation
operators. This extension generalizes the aggregation process used in AHPGDM by allowing more flexibility in the specific problem under consideration.
The Induced Ordered Weighted Average (IOWA) operator is a promising tool
for decision making with the ability to reflect the complex attitudinal character
of decision makers. The Maximum Entropy OWA (MEOWA) which is based
on the maximum entropy principle and the level of ‘orness’ is a systematic way
to derive weights for decision analysis. In this paper, the focus is given on the
integration of some induced aggregation operators with the AHP-GDM basedMEOWA as an extension model. An illustrative example is presented to show
the results obtained with different types of aggregation operators.
Keywords: AHP-GDM, Maximum Entropy OWA, Induced Generalized OWA.
1
Introduction
Multiple Criteria Decision Making (MCDM) is one of the active topics in Operations
Research. In general, MCDM can be considered as a process of selecting one alternative from a set of discrete alternatives with respect to several criteria. One of the most
widely used MCDM techniques is the Analytic Hierarchy Process (AHP), which was
proposed by Saaty (Saaty, 1977). The AHP is based on the judgment of problems
with multiple criteria by means of the construction of a ratio scale corresponding to
the priorities of alternatives. Since its introduction, AHP has been used in many applications. More details on the analysis and review of AHP and its applications can be
referred for instance in (Vaidya and Kumar, 2006; Saaty, 2013). In order to deal with
complex decision making problems involving multiple experts’ assessments, the AHP
has been extended to AHP-Group Decision Making (AHP-GDM) model. Escobar and
Moreno-Jimerez (2007) and Gargallo et al. (2007) for example, have modified the
conventional AHP method to AHP-GDM and showed the effectiveness and reliability
A. Laurent et al. (Eds.): IPMU 2014, Part I, CCIS 442, pp. 476–485, 2014.
© Springer International Publishing Switzerland 2014
Analytical Hierarchy Process under Group Decision Making
477
of the model. The AHP are popularly used in applications due to some advantages,
include a hierarchy structure by reducing multiple criteria into a pairwise comparison
method for individual or group decision making and it allows the use of quantitative
and qualitative information in evaluation process.
The Ordered Weighted Average (OWA) on the other hand is a family of aggregation procedures and was developed by Yager (Yager, 1988). The OWA operators are
introduced to provide a parameterized class of mean-type aggregation operators.
These include a family of aggregation operators that lie between the ‘and’ (min) and
the ‘or’ (max), and a unified framework for decision making under uncertainty. Subsequently, Yager (2004) generalizes the OWA operator by combining it with other
mean operators and called it the Generalized OWA (GOWA). This includes a wide
range of mean operators, such as the OWA, Ordered Weighted Geometric Average
(OWGA), Ordered Weighted Harmonic Average (OWHA) and Ordered Weighted
Quadratic Average (OWQA) operators. The Induced OWA (IOWA) is another extension of the OWA operator. Yager and Filev (1999) inspired by the work of (Mitchell
and Estrakh, 1997) have introduced the IOWA operator as a general type of aggregation. The main difference between OWA and IOWA is in the reordering step. Instead
of directly order the argument values as in OWA operator, the IOWA operator utilizes
another mechanism called induced variables as a pair of argument values, in which
it uses a first component to induce the argument values of a second component. The
advantage of IOWA is the ability to consider complex attitudinal character of
the decision makers that provide a complex picture of the decision making process.
The IOWA operator has been studied by a lot of authors in recent years, see for
example (Chiclana et al. 2004; Xu, 2006; Merigo and Casanovas, 2011).
Recently, much attention has been given on the integration of AHP with OWA operator (i.e., concentrated on aggregation process) as inspired by the work of (Yager
and Kelman, 1999). At first, Yager and Kelman (1999) have proposed the extension
of AHP using OWA operator with fuzzy linguistic quantifier. This approach generalizes the weighted average normally used in AHP to OWA based linguistic quantifier.
In addition to OWA based fuzzy linguistic quantifier technique, the Maximum Entropy OWA (MEOWA) operator has been proposed to be used in decision making analysis. O’ Hagan (1988, 1990) has developed a maximum entropy approach, which
formulates the problem as a constraint nonlinear optimization model with a predefined degree of orness as constraint and entropy as objective function, and used it to
determine OWA weights. Subsequently, Filev and Yager (1995) have examined the
analytical properties of MEOWA operators and proposed the analytic approach of
MEOWA operator. Since that the MEOWA has been used in many applications, include in MCDM area (Yager, 2009; Ahn, 2011).
In this paper, the focus is given on the integration of induced aggregation operators
with MEOWA weights in AHP-GDM as an extension model. The reason for doing
this is because there are situations in which it is necessary to aggregate the variables
with an inducing order instead of aggregating with the conventional OWA operator.
For example, such a method is useful when the attitudinal character of the decision
maker is particularly complex or when there are a number of external factors (i.e.,
personal effects on each alternative) affecting the decision analysis. The general
framework model for AHP-GDM which include different types of aggregation operators is proposed. The main advantage of this approach is the possibility of considering
478
B. Yusoff and J.M. Merigó-Lindahl
a wide range of induced aggregation operators. Therefore, the decision makers get a
more complete view of the problem and able to select the alternative that it is in accordance with their interests. These problems are studied in detail by conducting an
extensive analysis of some families of induced aggregation operators.
The remainder of this paper is organized as follows. In Section 2, the different
types of aggregation operators are reviewed in general. Section 3 briefly discusses the
AHP method. Section 4 examines the MEOWA technique. In section 5, the process of
using the IGOWA operators in AHP-GDM is discussed. Section 6 provides an illustrative example of the developed method and Section 7, the conclusions of the paper.
2
Preliminaries
In the following, the basic aggregation operators that are used in this paper are briefly
discussed.
Definition 1 (Yager, 1988). An OWA operator of dimension
is a mapping
:
that has an associated weighting vector
of dimension such that
∑
1 and
0,1 , then:
,
where
is the th largest
∑
,…,
and
(1)
is the set of positive real numbers.
Definition 2 (Yager and Filev, 1999). An IOWA operator of dimension is a mapthat has an associated weighting vector
of dimension
ping
:
such that ∑
1 and
0,1 , then:
,
,
,
,…,
,
∑
(2)
having the th largest ,
is the
value of the IOWA pair
,
is
where
is the argument variable. Note that, in case of
the order-inducing variable and
‘ties’ between argument values, the policy proposed by Yager and Filev (1999) will
be implemented, in which each argument of tied IOWA pair is replaced by their average.
Definition 3 (Merigo and Gil-Lafuente, 2009). An IGOWA operator of dimension
is a mapping
:
that has an associated weighting vector
of dimension such that ∑
1 and
0,1 , then:
,
,
,
,…,
,
∑
⁄
(3)
having the th largest ,
where
is the
value of the IGOWA pair
is the argument variable and is a parameter
is the order inducing variable,
such that
∞, ∞ . With different values of , various type of weighted average
can be derived. For instance, when
1 , IOWHA operator can be derived,
when
2 , the IOWQA operator is derived. The OWA, the IOWA, and the
IGOWA operators are all commutative, monotonic, bounded and idempotent.
Analytical Hierarchy Process under Group Decision Making
3
479
The Analytical Hierarchy Process Method
The AHP is introduced based on the weighted average model for complex decision
making problems (Saaty, 1977; Saaty, 2013) or also known as multiplicative preference relation. The AHP can be divided into three major steps; developing the AHP
hierarchy, pairwise comparison of elements of the hierarchical structure and constructing an overall priority rating. Specifically, the pairwise comparison matrix for
each level has the following form: Let
where
is pairwise comparis
ison rating for components i and components j ( ,
1,2, … , . The matrix
reciprocal, such that
for
and all its diagonal elements are unity,
1,
. In order to measure the degree of consistency, calculate the consistency index (CI) as follows:
(4)
is the biggest eigenvalue that can be obtained once we have its assowhere
ciated eigenvector and
is the number of columns of matrix . Further, we can
calculate the consistency ratio (CR), which is defined as follows:
(5)
where RI is the random index, the consistency index of a randomly generated pairwise
comparison matrix. It can be shown that RI depends on the number of elements being
compared. The table for RI can be referred in Saaty (1977). The consistency ratio
(CR) is designed in such a way that if
0.10 then the ratio indicates a reasonable level of consistency in the pairwise comparison.
For the given hierarchical structure, the overall evaluation score,
of the ith al∑
. The performance of alterternative is calculated as follows:
with respect to criteria
is described by a set of criteria values
natives
;
0,1 for
1,2, … , and
1,2, … , . The evaluation process in the
AHP uses a simple weighted linear combination to calculate the local scores of each
alternative.
4
Maximum Entropy OWA
Various approaches have been suggested for obtaining the weights in decision making
process. Motivated by the maximum entropy principle, O’Hagan (1988, 1990) has
developed a way to generate OWA weights by maximizing the entropy which subject
to the weight constraint and the value of the attitudinal character (or degree of orness).
The methodology is based on the mathematical programming problem and have come
to be known as the Maximum Entropy OWA weights. It can be noticed that MEOWA
weights used to spread the weights as uniformly as possible and at the same time
satisfying the attitudinal character.
480
B. Yusoff and J.M. Merigó-Lindahl
Filev and Yager (1995) obtained an analytic solution for the determination of the
MEOWA weights. In particular, the authors showed that the MEOWA weights for an
aggregation of degree n can be expressed as:
⁄
,
⁄
∑
1,2, … ,
(6)
where
∞, ∞ is a parameter dependent upon the value , which is desired
attitudinal character. Specifically, they showed that
1 ln
, where
⁄
0.
as a positive solution of the equation ∑
1
In what follows, the MEOWA weight will be used to be integrated with AHPGDM in the next section. In this case, the value
is computed based on the weight
of AHP pairwise comparison (criteria weight or relative importance of criteria). The
attitude of expert in differentiate each criterion under consideration determine the
degree of orness. Then, based on the degree of orness, MEOWA weight, or defined as
ordered weight can be calculated.
5
An Extension of the AHP-Group Decision Making Method
In this section, an extension of the AHP method under group decision making is presented. In what follows, the proposed method is represented step by step as in the
consequence algorithm. Assume
, ,
1,2, … ,
comprise a finite set of
, ,
1,2, … ,
and
,
,
1,2, … , ;
alternatives. Let
1,2, … ,
are the criteria and sub-criteria under consideration, respectively. Then, let
, ( ,
1,2, … , , be a group of experts, with each expert
presenting
his/her preferences or opinions for rating the alternatives , and weighting the crite(or sub-criteria
). Based on the above concepts, the algorithm for the
ria
IGOWA AHP-GDM consists of the following steps.
1,2, … , , compares the
Step 1: Each decision maker or expert ( ,
natives
,
1,2, … ,
and provides a pairwise comparison matrix:
, ,
with (
,
for each expert
1,2, … ,
1) and
,
can be calculated as follow:
∑
∑
alter-
∑
(7)
. Then, the alternatives values
, ,
1,2, … ,
(8)
Step 2: Compute the judgment matrix for a group of experts
. Consider Θ as
,
1,2, … ,
has in forming the group decision
the weight that the th expert
Θ
0; ∑
Θ
1 .
∏
,
1,2, … ,
(9)
Analytical Hierarchy Process under Group Decision Making
481
Step 3: Calculate the pairwise comparison matrix for the criteria
, ,
,
,
1,2, … , ;
1,2, … , . Then derive the
1,2, … ,
and sub-criteria
and sub-criteria weights
using the same formulation as for
criteria weights
alternatives, equations (8) and (9).
and sub-criteria .
Step 4: Calculate the composite weights of the criteria
(10)
Step 5: Compute the orness value
calculate the ordered weight, .
using the Maximum Entropy OWA to
∑
where
⁄
is the reordered th criteria weight,
.
with
Step 6: Find a positive solution
∑
1
(11)
(or composite weight) associated
of the algebraic equation.
⁄
0
1
(12)
Step 7: Compute the ordered weight
using the equation (6), where 0
1
and ∑
1.
of the ith alternative is defined as the
Step 8: Finally, the overall score
summation of the product of weight of each criterion by the performance of the alternative with respect to that criterion,
∑
where
is the
value of the OWA pair
,
(13)
having the th largest
of
is the argument variable. Besides,
is the
the order-inducing variable, and
ordered weight based on Maximum Entropy OWA and is a parameter such that
∞, ∞ , with different values of
reflect various types of weighted average.
Note that, when
, the IGOWA-AHP-GDM turn to GOWA-AHP-GDM.
Similarly, when
values are not arranged using OWA function, then the method
turn to conventional AHP-GDM.
6
Illustrative Example
An illustrative example is given to implement the methodologies discussed in the
previous sections. For this purpose, let consider an investment selection problem
where a company is looking for an optimal investment. There are five possible alternatives to be considered as follows:
is a computer company;
is a chemical
is a food company;
is a car company;
is a TV company.
company;
In order to evaluate these alternatives, a group of experts must make a decision
according to the following four attributes:
= risk analysis;
= growth analysis;
= social-political impact analysis; and
= environmental impact analysis.
482
B. Yusoff and J.M. Merigó-Lindahl
In this case, assume that three experts involved and the weight vector for the experts is
Θ
0.5,0.3,0.2 ,
1,2,3. Due to the fact that the attitudinal character is very
complex because it involves the opinion of different members of the board of directors,
the experts use order inducing variables to represent it as shown in Table 1.
Table 1. Inducing variables
25
12
22
31
30
18
34
13
24
25
24
18
28
14
23
16
22
21
20
16
First, let all the experts agreed to provide pairwise comparison of criteria as shown
in Table 2. In this case, no sub-criteria considered. Hence, based on criteria weight
technique, the weight for each criterion can be derived and consistency ratio is then
computed to check the consistency of pairwise comparison.
Table 2. Pairwise comparison matrix and the weight ratio of criteria
1
2
0.5
0.25
0.5
1
0.5
0.3333
2
2
1
0.5
0.3111
0.4064
0.1824
0.1001
4
3
2
1
CR=0.036
Subsequently, the weights
are proceed to be calculated with the MEOWA. The
is
results of this weight are presented in Table 3, where α is the measure of orness,
the positive solution of algebraic equation, is the value that relates to the weights
and the measure of orness, and
is the weight of MEOWA.
Table 3. The α,
,
values of criteria weights and MEOWA weights
α
0.5682
0.5339
0.4232
0.6154
0.6429
1.1793
1.0849
0.8301
1.3275
1.4262
0.4948
0.2445
-0.5586
0.8499
1.0650
0.3148
0.2813
0.1850
0.3639
0.3941
0.2669
0.2593
0.2229
0.2741
0.2763
0.2263
0.2390
0.2685
0.2065
0.1937
0.1919
0.2203
0.3235
0.1555
0.1358
Next, each expert provides rating (or pairwise comparison) for all alternatives with
respect to each criterion in order to get relative performance of alternatives in specific
criterion. The results of standardized performance of each expert are shown in Tables
4, 5 and 6, respectively.
Analytical Hierarchy Process under Group Decision Making
483
Table 4. Standardized performance ratings of Expert 1
0.1866
0.3069
0.0573
0.3069
0.1422
0.2412
0.1353
0.0743
0.1353
0.4137
0.2618
0.0892
0.1528
0.0526
0.4436
0.2571
0.0881
0.1539
0.4129
0.0881
Table 5. Standardized performance ratings of Expert 2
0.0881
0.4129
0.0881
0.2571
0.1539
0.1618
0.2760
0.1054
0.0596
0.3971
0.0604
0.1382
0.3972
0.0954
0.3088
0.0890
0.1579
0.2976
0.2976
0.1579
Table 6. Standardized performance ratings of Expert 3
0.0890
0.2976
0.1579
0.2976
0.1579
0.1042
0.3902
0.0588
0.1505
0.2962
0.0743
0.1353
0.2412
0.1353
0.4137
0.0986
0.1611
0.4162
0.0624
0.2618
Then, the results for each expert can be aggregated to form a matrix for group consensus. Table 7 presented the aggregated performance of experts.
Table 7. Aggregated performance of experts
0.1285
0.3334
0.0798
0.2893
0.1487
0.1809
0.2071
0.0788
0.1081
0.3823
0.1311
0.1105
0.2230
0.0760
0.3924
0.1544
0.1184
0.2288
0.2565
0.1305
of each alternative ith is derived as the summation
Finally, the overall score
of the product of MEOWA weights by the aggregated performance of experts. With
this information, different results are obtained using different types of IGOWA operators. The final results of the aggregation process with different operators are shown in
Tables 8 and 9. Meanwhile the ordering of investments is shown in Table 10.
484
B. Yusoff and J.M. Merigó-Lindahl
Table 8. Aggregated results 1
AM
0.149
0.192
0.153
0.182
0.263
WA
0.147
0.226
0.128
0.193
0.258
OWA
0.151
0.220
0.122
0.186
0.281
OWHA
0.148
0.180
0.098
0.137
0.219
OWQA
0.152
0.239
0.139
0.207
0.307
IOWA
0.146
0.189
0.146
0.200
0.258
OWG
0.139
0.199
0.108
0.160
0.250
IOWG
0.145
0.171
0.128
0.175
0.229
Table 9. Aggregated results 2
IOWHA
0.143
0.157
0.112
0.150
0.204
IOWQA
0.148
0.208
0.163
0.219
0.285
GM
0.147
0.173
0.134
0.157
0.232
WG
0.145
0.205
0.112
0.167
0.229
Table 10. Ranking of the investments
Ranking
7
Ranking
AM
IOWHA
WA
IOWQA
OWA
GM
OWHA
WG
OWQA
OWG
IOWA
IOWG
Conclusions
This paper has presented an extension of the Analytical Hierarchy Process method
under Group Decision Making (AHP-GDM) with some induced aggregation operators. The Maximum Entropy OWA (MEOWA) has been proposed to derive weights
in the AHP-GDM model. First, some modifications have been made to generalize the
aggregation process used in AHP-GDM with some Induced Generalized Ordered
Weighted Average (IGOWA) operators. The main advantages of this approach are
the ability to deal with the complex attitudinal character of the decision makers and
the aggregation of the information with a particular reordering process. Therefore, the
decision makers get a more complete view of the problem and able to select the alternative that it is in accordance with their interests. The procedure of the AHP-GDM
method with IGOWA operators has been discussed in detail. A numerical example on
investment selection problem has been given to exemplify the feasibility of the proposed method. The comparison of some induced aggregation operators has also been
made.
Analytical Hierarchy Process under Group Decision Making
485
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