J. Software Engineering & Applications, 2010, 3: 409-418
doi:10.4236/jsea.2010.34046 Published Online April 2010 (http://www.SciRP.org/journal/jsea)
Copyright © 2010 SciRes JSEA
Designing a Fuzzy Expert System to Evaluate
Alternatives in Fuzzy Analytic Hierarchy Process
Hamed Fazlollahtabar1, Hamid Eslami2, Hamidreza Salmani2
1Mazandaran Univer si t y of Science and Technology, Babol, Iran; 2Science and Research Campus, Islamic Azad University, Member
of Young Researchers Club, Tehran, Iran.
Email: hamed@ustmb.ac.ir
Received June 23rd, 2009; revised July 18th, 2009; accepted July 25th, 2009.
This paper concerns with proposing a fuzzy logic based expert system to breakthrough the problem of alternatives
evaluation in Analytic Hierarchy Process (AHP). AHP as a multi criteria decision aid helped decision makers for ana-
lyzing and prioritizing the alternatives in a hierarchical structure. During times AHP encountered some problems.
Hence, fuzzy analytic hierarchy process (FAHP) and some other extensions of AHP have been configured to solve those
Keywords: Multi Criteria Decision Making (MCDM), Analytic Hierarchy Pro ces s (AHP), Expert System
1. Introduction
Analytic hierarchy process (AHP) [1] has been widely
used as a useful multiple criteria decision making
(MCDM) tool or a weight estimation technique in many
areas such as selection, evaluation, planning and devel-
opment, decision making, forecasting, and so on [2]. The
AHP is expressed by a unidirectional hierarchical rela-
tionship amongst decision levels. The top element of the
hierarchy is the overall goal for the decision model. The
hierarchy decomposes to a more specific criterion on a
level and each criterion may be related to some subcrite-
ria. The AHP separates complex decision problems into
elements within a simplified hierarchical system.
The AHP usually consists of three stages of problem
solving: decomposition, comparative judgments, and
synthesis of priority. The decomposition stage aims at the
construction of a hierarchical network to represent a de-
cision problem, with the top level representing the over-
all objectives and the lower levels representing the crite-
ria, subcriteria, and alternatives. With comparative
judgments, users are requested to set up a comparison
matrix at each hierarchy by comparing pairs of criteria or
subcriteria. A scale of values ranging from 1 (Equally
Preferred) to 9 (Extremely Preferred), is used to express
the users preferences. Finally, in the synthesis of priority
stage, each comparison matrix is then solved by an ei-
genvector method for determining the importance of the
criteria and alternative performance.
One major advantage of AHP is its applicability to the
problems of group decision-making. In a group decision
setting, each participant is required to set up the prefer-
ence of each alternative by the AHP and the collective
views of the participants are used to obtain an average
weighting of each alternativ e.
The traditional AHP requires crisp judgments. Ho wever,
due to the complexity and uncertainty involved in real
world decision problems, a decision maker (DM) may
sometimes feel more confident to provide fuzzy judgments
than crisp comparisons. A number of methods have been
developed to handle fuzzy comparison matrices. For ex-
ample, Van Laarhoven and Pedrycz [3] suggested a fuzzy
logarithmic least squares method (LLSM) to obtain trian-
gular fuzzy weights from a triangular fuzzy comparison
matrix. Wang et al. [4] presented a modified fuzzy
Buckley [5] utilized the geometric mean method to
calculate fuzzy weights. Chang [6] proposed an extent
analysis method, which derives crisp weights for fuzzy
comparison matrices. Xu [7] brought forward a fuzzy
least squares priority method (LSM). Mikhailov [8] de-
veloped a fuzzy preference programming method (PPM),
which also derives crisp weights from fuzzy comparison
matrices. Csutora and Buckley [9] came up with a
Lambda-Max method, which is the direct fuzzification of
the well-known kmax method.
Among the above approaches, the extent analysis me-
thod has been employed in quite a number of applica-
tions [10-28] due to its computational simplicity. How-
ever, such a method is found unable to derive the true
Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process
weights from a fuzzy or crisp comparison matrix. The
weights determined by th e extent analysis method do not
represent the relative importance of decision criteria or
alternatives at all. Therefore, it should not be used as a
method for estimating priorities from a fuzzy pairwise
comparison matrix. The purpose of this paper is to show
by examples that the priority vectors determined by the
extent analysis method do not represent the relative im-
portance of decision criteria or alternatives and that the
misapplication of the extent analysis method to fuzzy
AHP problems may lead to a wrong decision to be made
and some useful decision information such as decision
criteria and fuzzy comparison matrices not to be consid-
ered. We illustrate these problems to avoid any possible
misapplications in the future. Here, we compare the
Fuzzy AHP with a proposed expert system and illustrate
our proposed expert system in an example.
2. Review of the Extent Analysis Method on Fuzzy
A triangular fuzzy number is represented by
, with the membership function ,
),,( uml )(
, defined
by the expression,
where m is the center, l is the left spread and u is the right
spread. For two triangular fuzzy number
and the fuzzy operations
are defined as follows:
),,( 111 uml),,(
~2222 umlM
~~ 21212121 uummllMM 
21212121 uummllMM 
M 
Consider a triangular fuzzy comparison matrix ex-
pressed by
 nnij
MA )
)1,1,1(...),,(),,( .
where )
aumla , for i,j =
1,…,n and .
To calculate a priority vector of the above triangular
fuzzy comparison matrix, Chang [9] suggested an extent
analysis method, which is summarized as follows.
Firstly, sum up each row of the fuzzy comparison ma-
by fuzzy arithmetic operations:
niumlMRS n
iji   
Secondly, normalize the above row sums by
Thirdly, compute the degree of possibility of ji SS
by the following equation:
lu mmif
ssV ij
where ),,(
iiiiumlS and . The defini-
tion of possibility degree is shown in Figure 1 .
Fourthly, calculate the degree of possibility of i
all the other (n - 1) fuzzy numbers by ,
(jSSVji 
(min);..., ,,...,1 niSSVijn ji
ijnj  
Finally, define the priority vector
wwW ),...,( 1
the fuzzy comparison matrix
It must be pointed out that the normalization formula
is wrong. The correct normalization formula for a set of
triangular fuzzy weights should be as follows:
Figure 1. Definition of the degree of possibility of
Copyright © 2010 SciRes JSEA
Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process411
Although Fuzzy AHP solved some of the problems of
AHP, but still some problems arises:
Problem 1. The extent analysis method may assign a
zero weight to a decision criterion or alternative, leading
to the criterion or alternative not to be considered in de-
cision analysis.
Problem 2. The weights determined by the extent
analysis method do not represent the relative importance
of decision criteria or alternatives and cannot be used as
their priorities.
Problem 3. The extent analysis method may make a
wrong decision and select the worst decision alternative
as the best one when it is misused for solving a fuzzy
AHP problem.
Problem 4. The extent analysis method cannot make
full use of all the fuzzy comparison matrices information
and may cause some useful fuzzy comparison matrices
information to be wasted when it assigns an irrational
zero weight to some useful decision criteria or
Therefore, we propose an expert system which func-
tions based on fuzzy logic, to improve decision making
in uncertainties.
3. Fuzzy Logic
Fuzzy Logic (FL) is a problem-solving control system
methodology that lends itself to implementation in sys-
tems ranging from simple, small, embedded micro-
controllers to large, networked, multi-channel PC or
workstation-based data acquisition and control systems. It
can be implemented in hardware, software, or a combina-
tion of both. FL provides a simple way to arrive at a defi-
nite conclusion based upon vague, ambiguous, imprecise,
noisy, or missing input information. FL’s approach to
control problems mimics how a person would make deci-
sions, only much faster.
FL incorporates a simple, rule-based IF X AND Y
THEN Z approach to a solving control problem rather
than attempting to model a system mathematically. The
FL model is empirically-based, relying on an operator’s
experience rather than their technical understanding of
the system.
FL requires some numerical parameters in order to
operate such as what is considered significant error and
significant rate-of-change-of-error, but exact values of
these numbers are usually not critical unless very respon-
sive performance is required in which case empirical
tuning would determine them. For example, a simple
temperature control system could use a single tempera-
ture feedback sensor whose data is subtracted from the
command signal to compute “error” and then time-
differentiated to yield the error slope or rate-of-change-
of-error, hereafter called “error-dot”. Error might have
units of degs F and a small error considered to be 2 F
while a large error is 5 F. The “error-dot” might then have
units of degs/min with a small error-dot being 5 F/min
and a large one being 15 F/min. These values don’t h ave
to be symmetrical and can be “tweaked” once the system
is operating in order to optimize performance. Generally,
FL is so forgiving that the system will probably work the
first time without any tweaking.
FL works as follows:
1) Define the control objectives and criteria: What am
I trying to control? What do I have to do to control the
system? What kind of response do I need? What are the
possible (probable) system failure modes?
2) Determine the input and output relationships and
choose a minimum number of variables for input to the
FL engine (typically error and rate-of-change-of-error).
3) Using the rule-ba sed structure of FL, break the con-
trol problem down into a series of IF X AND/OR Y
THEN Z rules that define the desired system output re-
sponse for given system input conditions. The number
and complexity of rules depends on the number of input
parameters that are to be processed and the number fuzzy
variables associated with each parameter. If possible, use
at least one variable and its time derivative. Although it
is possible to use a single, instantaneous error parameter
without knowing its rate of change, this cripples the sys-
tem’s ability to minimize overshoot for a step inputs.
4) Create FL membership functions that define the
meaning (values) of Input/Output terms used in the rules.
5) Create the necessary pre- and post-processing FL
routines if implementing in S/W, otherwise program the
rules into the FL H/W engine.
6) Test the system, evaluate the results, tune the rules
and membership functions, and retest until satisfactory
results are obtained.
In 1973, Professor Lotfi Zadeh proposed the concept
of linguistic or “fuzzy” variables. Think of them as lin-
guistic objects or words, rather than numbers. The sensor
input is a noun, e.g. “temperature”, “displacement”, “ve-
locity”, “flow”, “pressure”, etc. Since error is just the
difference, it can be thought of the same way. The fuzzy
variables themselves are adjectives that modify the vari-
able (e.g. “large positive” error, “small positive” error,
“zero” error, “small negative” error, and “large negative”
error). As a minimum, one could simply have “positive”,
“zero”, and “negative” variables for each of the parame-
Copyright © 2010 SciRes JSEA
Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process
ters. Additional ranges such as “very large” and “very
small” could also be added to extend the responsiveness
to exceptional or very nonlinear conditions, but aren’t
necessary in a basic system. Here, using fuzzy logic we
define some rules that help the student to select the opti-
mal department, course and teacher based on his age,
average grade and skills.
4. Expert Systems
Knowledge-based systems are systems based on the me-
thods and techniques of Artificial Intelligence. Their core
components are the knowledge base and the inference
mechanisms. Some particular types of knowledge-based
systems are expert systems, case-based reasoning sys-
tems and neural networks.
Expert Systems (ES) are computer programs that are
derived from a branch of computer science research
called Artificial Intelligence (AI). AI’s scientific goal is
to understand intelligence by building computer pro-
grams that exhibit intelligent behavior. It is concerned
with the concepts and methods of symbolic inference, or
reasoning, by a computer, and how the knowledge used
to make those inferences will be represented inside the
Of course, the term intelligence covers many cognitive
skills, including the ability to solve problems, learn, and
understand language; AI addresses all of those. But most
progress to date in AI has been made in the area of prob-
lem solving; concepts and methods for building programs
that reason about problems rather than calculate a solu-
AI programs that achieve expert-level competence in
solving problems in task areas by bringing to bear a body
of knowledge about specific tasks are called knowl-
edge-based or expert systems. Often, the term expert
systems is reserved for programs whose knowledge base
contains the knowledge used by human experts, in con-
trast to knowledge gathered from textbooks or
non-experts. More often, the two terms, expert systems
(ES) and knowledge-based systems (KBS), are used
synonymously. Taken together, they represent the most
widespread type of AI application. The area of human
intellectual endeavor to be captured in an expert system
is called the task domain. Task refers to some
goal-oriented, problem-solving activity. Domain refers to
the area within which the task is being performed. Typi-
cal tasks are diagnosis, planning, scheduling, configura-
tion and design.
Building an expert system is known as knowledge en-
gineering and its practitioners are called knowledge engi-
neers. The knowledge engineer must make sure that the
computer has all the knowledge needed to solve a prob-
lem. The knowledge engineer must choose one or more
forms in which to represent the required knowledge as
symbol patterns in the memory of the computer, that is, he
(or she) must choose a knowledge representation. He
must also ensure that the computer can use the knowledge
efficiently by selecting from a handful of reasoning me-
Every expert system consists of two principal parts:
the knowledge base; and the reasoning, or inference, en-
gine. The knowledge base of expert systems contains
both factual and heuristic knowledge. Factual knowledge
is that knowledge of the task domain that is widely
shared, typically found in textbooks or journals, and
commonly agreed upon by those knowledgeable in the
particular field.
Today there are two ways to build an expert system.
They can be built from scratch, or built using a piece of
development software known as a “tool” or a “shell”.
Before we discuss these tools, let's briefly discuss what
knowledge engineers do. Though different styles and me-
thods of knowledge engineering exist, the basic approach
is the same: a knowledge engineer interviews and ob-
serves a human expert or a group of experts and learns
what the experts know, and how they reason with their
knowledge. The engineer then translates the knowledge
into a computer-usable langua ge, an d desig ns an inference
engine, a reasoning structure, that uses the knowledge
appropriately. He also determines how to integrate the use
of uncertain knowledge in the reasoning process, and
what kinds of explanation would be usef ul to the end user.
Next, the inference engine and facilities for represent-
ing knowledge and for explaining are programmed, and
the domain knowledge is entered into the program piece
by piece. It may be that the inference engine is not just
right; the form of knowledge representation is awkward
for the kind of knowledge needed for the task; and the
expert might decide the pieces of knowledge are wrong.
All these are discovered and modified as the expert sys-
tem gradually gains competence.
The discovery and accumulation of techniques of ma-
chine reasoning and knowledge representation is gener-
ally the work of artificial intelligence research. The dis-
covery and accumulation of knowledge of a task domain
is the province of domain experts. Domain knowledge
consists of both formal, textbook knowledge, and expe-
riential knowledge—the expertise of the experts.
Compared to the wid e variation in domain knowledge,
only a small number of AI methods are known that are
useful in expert systems. That is, currently there are only
a handful of ways in which to re present knowledge, or to
make inferences, or to generate explanations. Thus, sys-
tems can be built that contain these useful methods
without any domain-specific knowledge. Such systems
are known as skeletal systems, shells, or simply AI tools.
Building expert systems by using shells offers signifi-
cant advantages. A system can be built to perform a
unique task by entering into a shell all the necessary
knowledge about a task domain. The inference engine
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Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process
Copyright © 2010 SciRes JSEA
corporate know-how so that it can be widely distributed to
other factories, offices or plants of the company.
that applies the knowledge to the task at hand is built into
the shell. If the pro gram is not v er y co mplica ted and if an
expert has had some training in the use of a shell, the
expert can enter the knowledge himself. Introduction of new products. A good example of a
new product is a pathology advisor sold to clinical pa-
thologists in hospitals to assist in the diagnosis of dis-
eased tissue.
Many commercial shells are available today, ranging
in size from shells on PCs, to shells on workstations, to
shells on large mainframe computers. They range in price
from hundreds to tens of thousands of dollars, and range
in complexity from simple, forward-chained, rule-based
systems requiring two days of training to those so com-
plex that only highly trained knowledge engineers can
use them to advantage. They range from general-purpose
shells to shells custom-tailored to a class of tasks, such as
financial planning or real-time process control.
An expert system tool, or shell, is a software develop-
ment environment containing the basic components of
expert systems. Associated with a shell is a prescribed
method for building applications by configuring and in-
stantiating these components. Some of the generic com-
ponents of a shell are shown in Figure 2 and described
below. The core components of expert systems are the
knowledge base and the reasoning engine.
Knowledge base: A store of factual and heuristic
knowledge. An ES tool provid es one or more knowledge
representation schemes for expressing knowledge about
the application domain. Some tools use both frames (ob-
jects) and IF-THEN rules. In PROLOG the knowledge is
represented as logical statements.
Although shells simplify progr amming, in general they
don't help with knowledge acquisition. Knowledge ac-
quisition refers to the task of endowing expert systems
with knowledge, a task currently performed by knowl-
edge engineers. The choice of reasoning method, or a
shell, is important, but it isn’t as important as the accu-
mulation of high-quality knowledge. The power of an
expert system lies in its store of knowledge about the
task domain—the more knowledge a system is given, the
more competent it becomes. Primarily, the benefits of
ESs to end users include:
Reasoning engine: Inference mechanisms for ma-
nipulating the symbolic information and knowledge in
the knowledge base to form a line of reasoning for solv-
ing a problem. The inference mechanism can range from
simple modus pones backward chaining of IF-THEN
rules to case-based reasoning.
A speed-up of human professional or semi-professional Knowledge acquisition subsystem: A subsystem to
help experts build knowledge bases. Collecting knowl-
edge needed to solve problems and build the knowledge
base continues to be the biggest bottleneck in building
expert systems.
work—typically by a factor of ten and sometimes by a
factor of a hundred or more.
Within companies, major internal cost savings. For
small systems, savings are sometimes in the tens or hun-
dreds of th ousan ds of do llars; bu t for large syste ms, often
in the tens of millions of dollars and as high as hundreds
of millions of dollars. These cost savings are a result of
quality improvement, a major motivation for employing
expert system technology.
Explanation subsystem: A subsystem that explains
the system’s actions. The explanation can range from
how the final or intermediate solutions were arrived at to
justifying the need for additional data.
User interface: The means of communication with the
user. The user interface is generally not a part of the ES
technology, and was not given much attention in the past.
However, it is now widely accepted that the user inter-
face can make a critical difference in the perceived utility
of a system regardless of the system’s performance.
Improved quality of decision making. In some cases,
the quality or correctness of decisions evaluated after the
fact show a ten-fold improv ement.
Preservation of scarce expertise. ESs are used to pre-
serve scarce know-how in organizations, to capture the
expertise of individuals who are retiring, and to preserve
Figure 2. Basic components of expert system tools
Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process
5. Comparison between Expert System and
Fuzzy AHP
Expert system has been applied for ranking [29]. Expert
system in comparison with fuzzy AHP has the following
The alternatives are analyzed using quantitative and
qualitative criteria without normalization process
More than seven alternatives can be processed
against AHP which encountered problems in pairwise
comparisons [30].
By entrance of new alternatives, the ranking of the
alternatives do not change
The fuzzy expert system is able to consider a stan-
dard in evaluating the alternatives
It is possible to apply group decision making of the
experts in evaluating the alternatives
The capability of sensitivity analysis for all the al-
No limitation for evaluating many criteria
The mistakes in computations such as the zero result
will not occur in expert system
The possibility of evaluating the alternatives using
both quantitative and qualitative criteria
The possibility of evaluating the alternatives while
some information about some criteria are missing
The possibility of keeping the same membership
during the process of decisio n m a king [31] .
In an expert system a membership function is pro-
posed for criteria regarding to the experts idea. To pro-
pose an expert system the following steps should be tak-
1) Determining the objective, alternatives and criteria
2) Identifying the input and output variables
3) Proposing membership functions for input and out-
put variables
4) Proposing rules to determine the relations between
inputs and outputs
5) Selecting an appropriate inference mechanism
6) Placement of alternatives corresponding to each
7) Extracting the evaluation result by the proposed
expert system
8) Sensitivity analysis of evaluated alternatives
Net section presents a numerical illustration to indicate
the application of the proposed expert system.
6. Numerical Illustrations
Here, we illustrate the proposed expert system in priori-
tizing four br and s of mobile p hon e. We analyze HAD, IC,
TA, HAM as alternatives using the criteria services,
power of antenna, prestige, and price. The hierarchy of
the model is shown in Figure 3.
The linguistic variables for criteria and their corresponded
membership functions are as follows (Figures 4-9).
Figure 3. The hierarchy of the model
Figure 4. The inputs and outp uts
Considering the experts the price has a Gaussian
membership function with minimum price of 5000 and
maximum of 700000.
For the power of antenna linguistic triangular fuzzy
number (high, medium, low) is considered.
Membership Value
Figure 5. Price membership function
Membership Value
Figure 6. The power of antenna membership function
Membership Value
Figure 7. Services membership function
Copyright © 2010 SciRes JSEA
Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process
Copyright © 2010 SciRes JSEA
5-If services
is low Then rating low.
6-If services
is high Then rating high.
7-If the_power_of_antenna is high Then rating
Membership Value
8-If the_power_of_antenna is low Then rating
the_power_of_antenna is medium
Then rat-
ing low.
10-If price is medium Then rating high.
11-If price is very high Then rating low.
Figure 8. Prestige membership function 12-If price is low Then rating high.
13-If prestige is very low Then rating low.
4-If prestige is medium Then rating me-
Membership Value
15-If prestige is very high Then rating high
Regarding to the proposed rules of the expert system,
we evaluate the alternatives as follows. To facilitate the
computations MATLAB package has been applied (Ap-
pendix A).
Price: 200
Services: medium
Figure 9. Rating membership function Power of antenna: high
Prestige: medium
The output of the system which is the evaluation result,
is a combined linguistic fuzzy number with a Gaussian
membership function for medium and triangular fuzzy
membership function for high and low.
Output: 0.518
The graphical presentation is shown in Figure 10.
Price: 15
Regarding to the experts and taking the criteria into
considerations, the following rules are derive d: Services: low
Power of antenna: high
1-If price
is high Then rating medium. Prestige: low
2-If price is very low Then rating low. Output: 0.51
3-If services
is high Then rating high. The graphical presentation is shown in Figure 11.
4-If services
is medium Then rating low.
Figure 10. The rule viewer for TA
Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process
Figure 11. The rule viewer for IC
Figure 12. The rule viewer for HAD
Figure 13. The rule viewer for HAM
Copyright © 2010 SciRes JSEA
Designing a Fuzzy E xpert System to Ev al uate Alternati ves in Fuzzy Analyti c Hi e ra r c hy Process417
Price: 500
Services: high
Power of antenna: high
Prestige: very high
Output: 0.654
The graphical presentation is shown in Figure 12.
Price: 200
Services: high
Power of antenna: high
Prestige: high
Output: 0.641
The graphical presentation is shown in Figure 13.
The ranking indicates the importance degree of each mo-
bile brand.
7. Conclusions
In this paper, we developed a fuzzy expert system on the
basis of rule base fuzzy logic to overcome the problems
in AHP and Fuzzy AHP. The advantages of using expert
system to prioritize the alternatives in comparison with
fuzzy AHP are discussed. To present the validity and
effectiveness of the proposed expert system a numerical
example is illustrated.
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Appendix A
Here, some useful MATLAB commands to work with the
proposed fuzzy in ference system (FIS) which is based on
Mamdani are presented:
Name='AHP mobile'