of infinite or large number of choices. Several authors classify them as i multiple attribute decision-making
MADM, and ii multiple objective decision-making MODM. In this presentation, the discretecontinuous classification is chosen
since it is in accordance with the conventional representation of data in the GIS vector vs. raster and it is more general than the MADMMODM classification.
The other classification depends on the certainty of the decision. If the decision maker has perfect knowledge of the decision environment and the amount of
knowledge available is enough, then the decision is considered as decision under certainty. However, most of the real world decisions involve some aspects that are
nknown and difficult to predict. This type of decisions is referred as decisions can be further subdivided into
u under uncertainty. The decisions under uncertainty
fuzzy and probabilistic decision making Leung, 1988 and Eastman, 1993. The probabilistic decisions are handled by probability theory and statistics. And the
outcome of a stochastic event is either true or false. However, if the situation is ambiguous, the problem is structured as the degree of how much an event belongs
to a class. This type of problems is handled by fuzzy set theory Zadeh, 1965 in Zischg, 2005.
2.6 Spatial Multi-Criteria Decision Making
Spatial multi-criteria decision making MCDM is a process where geographical data is combined and transformed into a decision. In the case of
spatial MCDM, geographical data were used as input to the decision making. Spatial MCDM is more complex and difficult in contrast to conventional MCDM,
19
as large numbers of factors need to be identified and considered, with high correlated relationships among the factors Malczewski 1999.
The goal of spatial MCDM is to achieve solutions for spatial decision problem
ement and analysis component contain a robust set of tools that are available in full-fledged GIS systems. his may include analytical tools for
ex techniques. These techniques can be used to generate data inputs criterion maps
to mult s that take the input from multiple criteria. These criteria, also called
attribute have to be identified very carefully to ensure that the final goal could be achieved. The performance of the objective is measured with the help of the
attributes. In hierarchy, a set of criteria should be decomposable, non-redundant, complete, minimal, and computational Prakash, 2003. The hierarchy serves two
purposes: to provide the overall view of the complex relationships in the situation and to allow decision makers to assess whether they are comparing the issues of
same order or magnitude. The basic reason behind the hierarchical structuring of a decision problem is that the elements being compared should be homogenous.
Spatial data analysis is in many ways the most important part of Geography Information System GIS, because it includes all of the
transformations, manipulations, and methods that can be applied to geographic data to add value to them, to support decisions, and to reveal patterns and
anomalies that are not immediately obvious. It is desirable that the geographical data manag
T ploratory data analysis such as statistical analysis and mathematical modeling
i-criteria decision analysis, to design and explore decision alternatives, as well as tools for sensitivity and uncertainty propagation analysis Malczewski,
1999. The integration of GIS and MCDM techniques could becomes useful in
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evaluating the complex phenomenon, such as the environmental decision problems that are characterized of having multiple and often conflicting
objectives.
Figure 2.5 Loose a and tight b multi-criteria spatial DSS coupling strategies Source: Malczewski, 1999
upled strategy involves accessing MC
the tw therefo
criteria level o
programming skills.
2.7
Thoma analyz
competing objectives Either loose or tight coupling strategies [Jankowski, 1995] can be
implemented for facilitating the integration of GIS and MCDM techniques Figure 2.5. The loose coupling approach combines the capabilities of separate models
for GIS functions and MCDM by transferring files through a file exchange mechanism [Jankowski, 1995]. A tightly co
DM analysis routines from within GIS software [Jankowski, 1995]. It allows o components to run simultaneously and to share a common database;
re, program control remains within the GIS when performing the multi- decision analysis. In general, the tight coupling approach requires a high
f knowledge of the GIS in question and considerable
Analytical Hierarchy Process
Analytical Hierarchy Process AHP was first introduced in 1970’s by s L. Saaty from University of Pittsburg, USA. AHP is a technique for
ing and supporting decisions in which multiple and
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are princip
♦ ecision hierarchy from the general to more
riteria into different gro
teria in its decision outputs. The
tematic procedure for representing hie
the basic rationality by bre
en gui
ents which are
re-examined to express the relative strength or involved and multiple alternatives are available. The method is based on three
les are explained below Saaty, 1980: Decomposition, to decompose a complex decision problem into simpler
decision problems to form a d specific until a level of attributes are reached. Each level must be linked to
the next higher level. Typically a hierarchical structure includes four levels: goal, objectives, attributes and alternatives.
♦ Comparative judgment, to perform cardinal rankings for objectives and alternatives are required. It involves 3 steps: 1 development of a
comparison matrix at each level of hierarchy 2 computation of weights for each element of the hierarchy and 3 estimation of consistency ratio
which is mentioned in pairwise comparison section. ♦ Synthesis of priorities, to combine the relative weights of the levels
obtained in the above step to produce composite weights. This is done by means of a sequence of multiplications of the matrices of relative weights
at each level of the hierarchy. The decision making in AHP is a process that continuous from analyzing
the decision environment to understand and arrange the c ups and levels that reach the evaluation of the cri
Analytical Hierarchy Process is a sys rarchically the elements of any problem. It organized
aking down a problem into its smaller and smaller constituent parts and th des decision makers through a series of pairwise comparison judgm
documented and can be
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inte ents in the hierarchy Saaty, 1980. The judgments
tha numbers. The AHP includes procedures
and pr
imates and correspond to what known as hard numbers. AHP has several
uired range, the quality of the judgments should be imp
to its flexibility, easy to use. It is also incorpo
nd the use of 1 to 9 scales can be thought as the disadvantages of this method. The ratio scale makes sense when dealing with something like
nsity of impact of the elem t have been made then translated to
inciples that used to synthesize the many judgments to derive priorities among criteria and down to alternative solutions. The numbers that obtained are
ratio scale est basic steps to be employing as follows Saaty, 1980:
1. identify the problem and determine the goal,
2. structure the hierarchy from the top,
3. construct a set of pairwise comparison matrices,
4. there are nn-12 judgments required to develop each matrix in step 3,
5. determined the consistency using the eigenvalue,
6. horizontal processing,
7. vertical processing,
8. calculate the consistency ratio.
The consistency ratio CR should be about 10 or less to be acceptable. If the CR does not fall in the req
roved. The AHP has widespread use due
rated into GIS environment Eastman, 1993; Jankowski, 1995 and can be used in two distinctive ways within GIS to derive weights and combine them with
attribute map layers and to aggregate the priority for all levels of the hierarchy structures. However, ambiguity in relative importance, inconsistent judgments by
decision maker a
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distanc
2.8
poses by Dr. Lofti A. Zadeh in 1965
ts, which have vagueness and imprecision in their characteristics. At e, or area which is natural ratio scales, but not when dealing with like
comfort, image, or quality of life, for which no clear reference levels exists. Furthermore, for large problems too many pairwise comparisons must be
performed Malczewski, 1999.
Fuzzy Logic in Decision Making
Fuzzy decision making techniques emerged as a result of the normal decision making methods inability to address the imprecision and vagueness. In
the real world decision making process there any many goals, constraints and consequences that are precisely unknown. In such situation fuzzy set theory
becomes useful. Burrough 1989 states that simple Boolean algebraic operations used in
the evaluation process will result in significant loss of information and in this case fuzzy set theory will be useful alternative. Some important aspects of the soil, like
internal heterogeneity, measurement error, complexity, imprecision, etc. are not put into account by normal land evaluation classification.
One method to obtain information on the uncertainties in the risk analysis procedure is the representation of the vagueness related to the risk parameters by
fuzzy number. Theory about “Fuzzy Sets” pro Malczewski, 1999. Fuzzy set is defined as a class of elements or objects without
any definite boundaries between them. The fuzzy logic is useful to define the real world objec
one level, fuzzy logic could be viewed as a language that allows one to translate sophisticated statement from natural language into a mathematical formalism
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Zadeh, 1978. Fuzzy logic is a multi-valued theory where in intermediate values such as “moderate”, “high”, “low” were used instead of yes or no, true or false as
it is used in the conventional crisp theory. The fuzzy sets are defined by the embership functions. The fuzzy sets represent the grade of any element x of X
hav embership to A. The degree to which an element belongs to
a set is
l fuzzy sets representing linguistic concepts such as low, medium, high, a
overcome the inability of AHP to handle imprecision and subjective- ness in
r’s uncerta
m that
e the partial m defined by the value between 0 and 1.
If an element x really belongs to A if Ax = 1, and x not belongs to A if Ax = 0. The higher is the membership value, the greater is the element x belong
to a set A. Severa
nd so on are often employed to define states of a variable. Such a variable is usually called a fuzzy variable. The significance of fuzzy variables is that they
facilitate gradual transitions between states and, consequently, possess a natural capability to express and deal with observation and measurement vagueness.
2.9 Fuzzy AHP
