Intuitionistic Fuzzy Analytical Network Process (IF-ANP

16 Pages·2008·0.81 MB·English

by  Jason Svendstorp

Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi
Criteria Decision Making under Uncertainty

Sara SAEEDI, Mohammadreza MALEK, Mahmoodreza DELAVAR, Amin TAYYBI,
Iran

Key words: Spacial Multi Criteria Decision Making, ANP, Intuitionistic Fuzzy Set, Site
Selection.

SUMMARY

The vagueness of multi criteria decision making (MCDM) is commonly handled through
fuzzy sets theory, by assigning degree of membership. However, the spatial MCDM
(SMCDM) problem encounters ambiguity in assigning the membership function to fuzzy
pairwise comparisons, which is referred to as non-specificity.This paper attempts to reach a
new method in making the comparison matrices in analytical network process (ANP)
approach to consider some aspects of uncertainties and vagueness in the process of SMCDM
based on the rule and logic of intuitionistic fuzzy (IF).
The ANP, a generalized form of the widely used analytic hierarchy process (AHP), has been
proposed as a suitable tool to evaluate the alternatives during the conceptual design phase of
SMCDM. ANP is highly recommended since it allows interdependent influences specified in
the model and generalizes on the supermatrix approach. The local priorities in ANP are
established in the same manner as they are in AHP using pairwise comparisons.
The case study undertaken in this research includes investigation of criteria used in selecting
the public car park lots in a part of Tehran metropolitan area. This paper successfully
demonstrates that the proposed method has the ability to include a number of geospatial data
and constraints to improve reliability of decision making using the Intuitionistic Fuzzy
Analytical Network Process (IF-ANP) in a GIS environment.

TS 6I – GIS Algorithms and Techniques 1/16
Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi
Criteria Decision Making under Uncertainty

Sara SAEEDI, Mohammadreza MALEK, Mahmoodreza DELAVAR, Amin TAYYBI,
Iran

1. INTRODUCTION

Multi criteria decision making (MCDM) techniques can be used to identify a single most
preferred option, to rank options, to list a limited number of options for subsequent detailed
evaluation, or to distinguish acceptable from unacceptable possibilities [8]. Since 80% of data
used by decision makers is related geo-spatially [21], spatial decision making may provide
superior framework in variety of decision making situations. Nowadays as geo-spatial
information systems (GIS) are becoming more widely used for making critical decisions in
many disciplines, more sophisticated approaches heightens in spatial decision making.
Ideally, a GIS with MCDM capabilities should offer decision makers the most information to
aid them in selecting criteria [25]. The ability of a GIS with MCDA capabilities to
simultaneously represent decision spaces and criteria values, as well as allowing the user to
manipulate the displays will provide for the best choices not only on the basis of attribute
data, but also on geography [14]. An active area of research in spatial decision making is
spatial multiple criteria decision analysis (SMCDA), also known as spatial multi criteria
evaluation [14]. A scientific spatial decision making process can be recognized by different
steps as shown in Figure 1 [14].
Problem Definition
Evaluation Criteria Constraints elligence Phase (GIS)
nt
I
Decision Matrix Alternatives
n e P)
gs N
Decision maker's DesiPha F-A
(I
Decision rules

Sensitivity analysis Choice Phase GIS- IF- ANP)
Recommendation (
Figure 1: Framework for SMCDA [14], customized based on the presented IF-ANP

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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
The analytic network process (ANP), a generalized form of the widely used analytic hierarchy
process (AHP), has been proposed as a suitable tool to evaluate the alternatives during the
conceptual design phase of SMCDA [12], [13] and [16]. However, the ANP-based decision
model seems to be ineffective in dealing with the inherent fuzziness or uncertainty in
judgment during the prioritization process. Although the use of the discrete scale of 1–9
represent the verbal judgment in pairwise comparisons, it does not take into account the
uncertainty associated with the mapping of one’s perception to a number. In real-life decision-
making situation, the decision makers or stakeholders could be uncertain about their own
level of preference, due to incomplete information or knowledge, the uncertainty within
evaluated criteria and complexity of the decision environment.

Therefore, this study outlined the intuitionistic fuzzy extension of ANP approach (IF-ANP)
which will manage uncertain weights and criteria used in the site selection of multi-level
public car parking construction in a part of Tehran metropolitan area. By combining the IF-
ANP evaluation techniques with GIS, it will be intended that the effective factors would be
evaluated more flexibly and efficient. This paper extends our pervious methodology for
combining ANP and fuzzy set theory to make efficient spatial decision under uncertainty [6].

2. BACKGROUND OF METHODOLOGY

2.1. The ANP and AHP MCDM Approach
In the existing literature of spatial multi criteria analysis (SMCA), studies have utilized AHP
to set up a hierarchical skeleton within which multi-attribute decision problems can be
structured [18] and [22]. However, AHP can only be employed in hierarchical decision
models. For complicated decision problems, the analytical network process (ANP) is highly
recommended since ANP allows interdependent influences specified in the model [15] and
generalizes on the supermatrix approach introduced in [26]. ANP is the first mathematical
theory that makes it possible to deal with all kinds of dependences and feedbacks by replacing
hierarchies with networks [19].

The local priorities in ANP are established in the same manner as they are in AHP using
pairwise comparisons and judgments. However, the supermatrix approach which became
popularly known as the ANP approach is becoming an attractive tool to understand more of
the complex decision problem as it overcomes the limitation of the AHP’s linear hierarchy
structure [19] and [20].

ANP comprises four main steps [3]: 1) Conducting pair-wise comparisons on the elements at
the cluster and sub-cluster levels; 2) Placing the resulting relative importance weights
(eigenvectors) in submatrices within the supermatrix; 3) Adjusting the values in the
supermatrix so that the supermatrix can achieve column stochastic; 4) Raising the supermatrix
to limiting powers until the weights have converged and remain stable. In raising the
supermatrix into a large power, the transmission of influence along all possible paths defined
in the decision structure is captured in the process [20]. The fundamentals of the Analytic
Network Process (ANP) could be found in [20].
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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
2.2. Uncertainty Modeling In ANP Using Fuzzy Set Theory

In conventional AHP/ANP, the pairwise comparison process requires that a crisp number
denoted as aij be assigned to each pair of elements (ei,ej) representing in the perception of
decision maker judgment [20]. This judgment is a subjective belief that may be express by
various inconsistent opinions.
However, the use of these crisp numbers has been criticized because of their inability to
adequately handle the uncertainty and imprecision associated with the mapping of the
decision maker’s perception or subjective judgment [7]. Thus, several studies have proposed
the use of fuzzy sets or fuzzy numbers to represent such uncertain judgments in the fuzzy
pairwise comparison matrix [17].The traditional fuzzy logic has two important problems.
First, for every property and every value we need to assign a crisp membership function.
Second, fuzzy logic does not distinguish between the situations in which there is no
knowledge about a certain statement and a situation that the belief to the statement in favor
and against is the same. Due to this fact, it is not recommended for problems with missing
data and where grades of membership are hard to define [23].
It is expected that human decision making processes and any activities requiring human
expertise and knowledge, which are inevitably imprecise or not totally reliable could be
simulated by intuitionistic fuzzy sets [24]. Therefore, it is attempted to developing the ANP,
as a MCDM approach to consideration of hesitancy in decision maker’s opinions.
The most practical approach to extend fuzzy AHP to the ANP framework is to derive first the
crisp priorities or weights from the triangular fuzzy numbers (TFNs) and fuzzy comparison
matrices [15]. Cheng and Mon [5] proposed fuzzy extension of fuzzy-scale-based algorithm
used alpha-cuts, interval arithmetic, and linear convex combinations defined by optimism
index values to transform the fuzzy comparison matrix into a set of crisp matrices; and then
computed the set of crisp local weights from eigenvector of the said matrix. Several authors
have recently applied this FAHP method [11], [3] and [2]. Some researchers [3] and [9]
applied a fuzzy ANP approach to quality function deployment problems. This method is an
extension of fuzzy AHP approach proposed by [4], which derives crisp local priorities from
fuzzy comparison matrix using the extent analysis method and possibility theory.
While there are some proposed Fuzzy methods to ‘fuzzify’ ANP with different theoretical
foundation, only limited studies had been done to investigate the intuitionistic fuzzy extension
of AHP/ANP to consider the vagueness and hesitancy in decision maker's opinions (see [23]
and [24] for related discussions).

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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
3. THE PROPOSED INTUITIONISTIC FUZZY ANP

To realize the implementation of IF-ANP, basic definitions of IFS refer to Atanassov [1], can
be denoted as:
(1)

The membership ( ) and non-membership ( ) functions for an element x are defined as:
(2)

where X defines the possible range of values for a variable x, and ‘A’ is an IFS defined over
X using membership and non- membership functions. For any value of x in an IFS ‘A’, the
expression holds true. However, if the expression reduces to = 1,
the IFS becomes an ordinary fuzzy set. For IFS, the degree of non-determinacy (or non-
specificity) of the element x in IFS ‘A’ is defined as follows:
(3)
Therefore, for ordinary fuzzy sets the degree of non-determinacy = 0.
In practice the decision-maker usually gives some or all pair-to-pair comparison values with
an uncertainty degree rather than precise ratings, but pair wise comparison values are the
judgments obtained from an appropriate semantic scale [21]. In order to deal with vagueness
intuitionistic fuzzy sets introduced by Atanassov [1] among several higher-order fuzzy sets.
The algorithm of the proposed IF-ANP (inspired from [5]) can be described in the following
steps:

Step 1: Decompose the multiple criteria evaluation problem into a strongly connected
hierarchical network with a finite set of clusters and elements.

Step 2: Develop intuitionistic fuzzy judgment comparisons matrix. Intuitionistic fuzzy
judgment matrix is generated using pairwise comparisons during the evaluation; the
decision maker can specify the vagueness (the TFNs) and his/her degree of belief (bel) for the
pairwise comparisons. This belief is represented by a membership function bel, and
if there is a degree of non-belief about his/her evaluation, it is represented by a
nonmembership function while . So there is a fuzzification factor
is introduced, which may or may not have the same value as of . Therefore, the triangular
IFS (figure 2) can be written as A=[(L , M , U ; ) , (L ',M ',U '; )]
A A A A A A
that:
and (4)
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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
L'A LA m UA U'A

Figure 2. Triangular intuitionistic fuzzy sets (IFS) "a"

Then, construct the fuzzy pairwise comparison matrix such that:
(5)
where .

Step 3: Solve the fuzzy eigenvalue problem:
or (6-a)
(6-b)
where is a IF matrix containing n(cid:1)n IF numbers is a positive n(cid:1)1 IF vector
containing IF numbers and is the IF eigenvalue. operators denote the IF
multiplications and addition respectively according to the equation (7) and (8).

(7)

(8)

To solve the equation (6), instead of performing IF multiplications and additions, we apply
two different fuzzy scenario; one of them utilize and the other utilize
which incorporates the decision maker’s confidence over his/her judgment. Using alpha-cut
allows the algorithm to alter fuzzy sets to interval arithmetic.
I) , , (9-a)
II) , , (9-b)

Step 4: Assign an optimism index value to combine upper and lower limits of the pairwise
comparisons by the linear convex defined as
I) (10-a)
II) (10-a)

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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
The use of optimism index allows the algorithm to reflect the decision maker’s attitude
towards the fuzziness of judgment. For example, an ‘optimistic’ decision ( ) is apt to
assign higher values of the crisp interval.

Step 5: construct matrices and after setting the parameter:
and (11)

Step 6: From Eq. (11), calculate the local priorities or the normalized principal right
eigenvector for all a-cut values [0,1] at a given fixed o value. Computationalwise, the
principal right eigenvector can be approximated from the following equation [18]:
(12)

Where eT=[1, 1, ..., 1] is a unit row vector, c is a constant and is the eigenvector
corresponding to the principal eigenvalue of the primitive matrix . Thus, the computation of
principal eigenvector or local priorities which are needed to populate the initial supermatrix of
network can easily be implemented in a spreadsheet (e.g., Microsoft Excel 2007).

Step 7: Compute the overall priorities from principal column eigenvector of the initial
supermatrix of the hierarchical network. The supermatrix is initially populated by the
pertinent eigenvectors computed from step 6. This column eigenvector can be approximated
from an algorithm which can also be easily implemented in a spreadsheet using the following
equation:
(13)

Where eT is a unit row vector, q is a constant and is the eigenvector corresponding to the
principal eigenvalue of the primitive irreducible supermatrix . Then, the meaningful overall
priorities can be derived from the column eigenvector normalized according to pertinent
clusters or levels.

4. CASE STUDY: PARKING SITE SELECTION

Public parking spaces as one of the important parts of a modern urban transportation system,
plays an essential role in decreasing the load of heavy traffic [10]. Suitable site selection for
public parking spaces not only increases the parking efficiency, but also decreases marginal
car parking and so results in increase of streets' width and traffic fluency. Considering all
effective parameters in site selection is almost impossible and site selection is done by just
considering some limited factors like land price. Therefore, it is necessary to employ GIS-
based spatial multi criteria decision analysis tools which have the ability to analyze a number
of criteria and constraints simultaneously, in the parking site selection process [6].
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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
4.1. Assessment of the Criteria

The first step in assessing the criteria is to determine the factors affecting the suitable site
selection on the basis of an analysis of existing studies and knowledge. Layers representing
the criteria are referred to as criterion maps. Effective parameters are selected according to the
result of our previous research on parking site selection [3]. These criteria are listed as
follows:
• Physical site features (A), which includes area (A ) and shape (A2) for site
1
condition;
• Absorbing excursion spaces (B), which contains distance from commercial (B ),
1
administrative (B ), educational (B ), tourist centres (B ) as well as transportation station (B );
2 3 4 5
• Traffic flow (C), which consists of average daily traffic (C ) and street type (C );
1 2
• Economic parameters (D), which includes land ownership (D ), local price (D )
1 2
and construction quality (D );
3
• Legal restrictions: This group includes limitations considered for parking site
selection, which include land use plan, green spaces, historical, educational and cultural
centres, specific buffer for hospital and mosques, and other pious legacies mentioned in the
comprehensive plan of Tehran.

The evaluation criteria are related to geo-spatial entities that are inherently uncertain and
ambiguous. Vagueness (imprecision) in geospatial data refers to lack of definite or sharp
distinction, whereas ambiguity is due to non specified distinction of the various indicator
limits for the criteria. Therefore, in this phase, we can divide our objectives on two categories.
First group of objectives can be expressed by IF triangular linguistic memberships (such as
traffic flow and economic parameter) to define the membership function for modelling of the
criteria based on the critic opinions. Another part of the objectives has no sense of fuzzy
variables (such as legal restriction) which are considered as crisp constraints. The list of the
clusters and elements are mentioned in Table 1.

Table 1. Introducing effective fuzzy criteria, elements and indicators for parking site selection
Linguistic Observed
Clusters Elements
Indicators Indicators
a11= Very small 0 to 750 (m2)
a12= Small 750 to 1100 (m2)
a1= Area a13= Big 1100 to 1400
(m2)
A=Physical site feature
a14=Very big >1400 (m2)
(Linguistic variables:
a21= Appropriate > 1
Excellent,
a22= Probably 0.45 to 0.70
Proper, Medium, Poor) a2= Shape
appropriate
(area-perimeter
a23= Probably 0.10 to 0.45
ratio)
inappropriate
a24= Inappropriate 0 to 0.10
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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008
b11= near 0 to 90 (m)
b1= Administrative b12= Probably near 100 to 180 (m)
centers b13= Probably far 200 to 290 (m)
b14= far >300 (m)
b21= near 0 to 80 (m)
b2= Commercial b22= Probably near 95 to 160 (m)
centers b23= Probably far 180 to 250 (m)
B= Distance from b24= far >270 (m)
absorbing excursion b31= near 0 to 160 (m)
spaces b3= Educational b32= Probably near 160 to 320 (m)
(Linguistic variables: centers b33= Probably far 320 to 400 (m)
Very near, near, b34= far >400 (m)
approximately far, Far) b41= near 0 to 110 (m)
b42= Probably near 110 to 230 (m)
b4= Tourist centers
b43= Probably far 230 to 350 (m)
b44= far >350 (m)
b51= near 0 to 50 (m)
b5= Transportation b52= Probably near 65 to 95 (m)
station b53= Probably far 110 to 150 (m)
b54= far >170 (m)
c11= Infrequent 0 to 5500
c12= Probably 5000 to 7500
c1= Average daily
Infrequent
traffic
c13= Probably 7500 to 10000
(Vehicles/Day)
Dense
c14= Dense >10000
c21= principal -
C= Traffic flow
arterial
(Linguistic variables:
c2= Street type c22= minor arterial -
Infrequent, Moderately
c23= collector street -
Dense, Dense)
c24= local street -
c31= Infrequent 0 to 100
c32= Probably 100 to 225
c3= Dynamic
Infrequent
population
c33= Probably 225 to 350
(person per hectare)
Dense
c34= Dense >350
D= Economic d11= Ruined >50 years
d1= Construction
parameters d12= Maintainable 15 to 50 years
quality
(Linguistic variables: d13= New 0-15 years
economical, d21= Public -
reasonable, Slightly d2= Land ownership d22= personal -
Expensive, d23= vaghf -
costly) d3= Local price d31= Expensive >25000 ($/ m2)
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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
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Stockholm, Sweden 14-19 June 2008
d32= Probably 18000 to 25000
Expensive ($/ m2)
d33= Probably 10000 to 18000
Cheap ($/ m2)
d34= Cheap 0 to 10000 ($/
m2)

4.2. Assigning criteria weights based on the IF-ANP

After describing effective criteria in parking site selection and composing the ANP network,
pairwise comparison weights should be assigned to the criteria. In this paper we allow for IF
pairwise comparison, each pair of alternatives is explicitly characterized by a scaled number
(1-9) and two membership functions value (μ , (cid:2)) between 0 and 1 that are evaluated from the
membership function (μ) and degree of non-determinacy ( ). μ and (cid:2) define a degree to which
the first alternative (i) is preferred to the second one (j) and a degree to which the first
alternative is not preferred to the second one by a specific scaled number (a ). In addition for
ij
this case we choose (cid:1)=0.5 and to obtain the
Conducting pair-wise comparisons of various criteria and elements at the cluster and sub-
cluster levels produces more than 100 different IF comparison matrices. The super-matrix has
105 rows and 105 columns. After the process of raising the super-matrix to limiting powers
until the weights have converged, assigned weights are obtained based on two super matrix A
and A' (Eq. 11). Assigned weights and parameters are listed in the Table 1.

Table 1. Consistent ANP weights (Network model) derived from IF-ANP aproach
Parameters and ANP models and assigned ANP models and assigned
elements weights (Supermatrix A) weights (Supermatrix A’)
Parking Site
P= 0.34A+0.48B+0.18C P= 0.38A+0.52B+0.11C
Selection:
A= 0.58a1+0.42a2 A= 0.56a1+0.44a2
Physical Site Feature:
A= 0.53B+0.47C A= 0.57B+0.43C
Area A1= A1=
0.09a11+0.18a12+0.29a13+0.4 0.11a11+0.16a12+0.31a13+0.4
4a14 3a14
Shape A2= A2=
0.39a21+0.29a22+0.21a23+0.1 0.36a21+0.28a22+0.22a23+0.1
1a24 4a24
Distance from B=0.28b1+0.19b2+0.07b3+0.1 B=0.20b1+0.22b2+0.12b3+0.1
absorbing Excursion 3b4+0.33b5 4b4+0.33b5
Spaces: B= 0.35A+0.65C B= 0.43A+0.57C
Administrative Centers B1= B1=
0.57b11+0.22b12+0.15b13+0.0 0.43b11+0.24b12+0.18b13+0.1
6b14 5b14
Commercial Centers B2= B2=
0.52b21+0.26b22+0.13b23+0.0 0.47b21+0.26b22+0.13b23+0.1
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Sara Saeedi, Mohammad Reza Malek, Mahmood Reza Delavar and Amin Tayyebi
Intuitionistic Fuzzy Analytical Network Process (IF-ANP) for Spatial Multi Criteria Decision Making under
Uncertainty

Integrating Generations
FIG Working Week 2008
Stockholm, Sweden 14-19 June 2008