Fuzzy AHP method, uncertainty and decision-making.
Buchmeister, Borut ; Polajnar, Andrej ; Pandza, Krsto 等
Abstract: In decision making under uncertainty, risk analysis aims
at minimising the failure to achieve a desired result. In the paper our
original method of uncertainty estimation is presented. Each problem
involving uncertainty and consecutive appearing risk is divided into
identified categories and factors of uncertainty. The basic method used
in the numerical part of the evaluation is the two-pass Fuzzy Analytic
Hierarchy Process (applied: first for the importance and second for the
uncertainty of risk factors). The process is based on the determination
of relations between particular risk categories and factors. The
estimates are derived from pairwise comparison of factors belonging to
each category. By using fuzzy numbers the consideration of possible
errors of the estimator is taken into account. In the following stages
the interval results obtained by this method are used for calculating
the integral uncertainty value, which, in comparison with the boundary
value, defines the risk of the process in question. Based on the
"Uncertainty--Importance" relations special ABC focus diagrams
are created. These diagrams serve for the classification of uncertainty
factors, which provides a decision making part of the systemic approach.
Key words: uncertainty, risk, estimation, fuzzy AHP method,
systematic approach
1. Introduction
Manufacturing is and will remain one of the principal means by
which wealth is created. However manufacturing has changed radically
over the course of the last 20 years and rapid changes are certain to
continue. The emergence of new manufacturing technologies, spurred by
intense competition, will lead to dramatically new products and
processes. New management and labour practices, organizational
structures, and decision making methods will also emerge as complements
to new products and processes. It is essential that the manufacturing
industry be prepared to implement advanced manufacturing methods in time
(Trigeorgis, 2002).
These changes have led organizations to search for new approaches
in organization models and in production management. Uncertainty and
fast changing environment are making long-term planning next to
impossible. This uncertain environment is leaving only time and risk as
means for survival. Decision-making has become one of the most
challenging tasks in these unpredictable global conditions, demanding
competency in understanding these complicated processes (Augier &
Kreiner, 2000; Kremljak, 2004).
Managers employed in industrial companies, the public sector and
service industry cope with high levels of uncertainty in their
decision-making processes, due to rapid, large-scale changes that define
the environment their companies operate in. This means that managers do
not possess complete information on future events, do not know all
possible alternatives or consequences of all possible decisions.
Decision-making in high-risk conditions is becoming a common area for
research within strategic management organizational theory, research and
development management and industrial engineering. These issues have not
been adequately addressed in published research.
Tackling uncertainty involves developing heuristic tools that can
offer satisfactory solutions. The problem of decision-making in
uncertain conditions is only partially presented in relevant literature
(Carpenter & Fredrickson, 2001; Laviolette & Seaman, 1994; Frei
& Harker, 1999). Intensive research in the area of multi-level
decision-making, supported by expert systems is currently under way.
2. Fuzzy AHP method
Used procedure with the application of fuzzy triangular numbers is
described in the following steps (Van Laarhoven & Pedrycz, 1983;
Zadeh, 1965; Kwong & Bai, 2002).
1st step: pairwise AHP comparison (using triangular fuzzy numbers
from [??] to [??]--see Fig. 1) of the elements at the same hierarchy
level. Triangular fuzzy number is
described as M = (a, b, c) and by defining the interval of
confidence level [alpha], we get:
[for all][alpha][member of] [0,1][M.sub.[alpha] = [[a.sup.[alpha]],
[c.sup.[alpha]]] = [(b - a) x [alpha], - (c - b) x [alpha] + c] (1)
[FIGURE 1 OMITTED]
2nd step: constructing the fuzzy comparison matrix A ([a.sub.ij])
with triangular fuzzy numbers:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It is presumed that the evaluator's mistake in quantitative
evaluation might be [+ or -] one class to the left or to the right.
3rd step: solving fuzzy eigenvalues [lambda] of matrix, where: A x
x = [lambda] x x (2)
and [??] is a non-zero n x 1 fuzzy vector.
To be able to perform fuzzy multiplication and addition with
interval arithmetic and level of confidence [alpha] the equation (2) is
transferred into:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
for 0<[alpha] [less than or equal to]1 and all i, j, where i = 1
... n, j = 1 ... n.
Degree of satisfaction for the matrix A is estimated by the index
of optimism [mu]. The larger index value [mu] indicates the higher
degree of optimism calculated as a linear convex combination (with upper
and lower limits), defined as:
[[??].sup.[alpha].sub.ij] = [mu] x [a.sup.[alpha].sub.ij] + (1 -
[mu]) x [a.sup.[alpha].sub.ijl], [for all][mu][member of] [0,1] (5)
At optimistic estimates that are above average value ( [mu] >
0,5) [[??].sub.i,j], is higher than the middle triangular value (b) and
vice versa.
While [alpha] is fixed, the following matrix can be obtained after
setting the index of optimism [mu]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
The eigenvector is calculated by fixing the value [mu] and by
identifying the maximal eigenvalue.
4th step: determining total weights. By synthesizing the priorities
over all hierarchy levels the overall importance weights of uncertainty
factors are obtained by varying [alpha] value.
Upper and lower limits of fuzzy numbers considering * are
calculated by application of the appropriate equation, for example:
[[??].sub.[alpha]] = [[1.sup.[alpha]], [5.sup.[alpha]] = [1 + 2 x
[alpha], 5 - 2 x [alpha]] (7)
[[??].sup.-1.sub.[alpha]] = [1 / 5 - 2 x [alpha]], [1 / 1 - 2 x
[alpha]] (8)
2.1 Calculation of the importance of factors
Ratios between categories or factors are expressed with a question:
"How many times is the category/factor i more important than
category/factor j?" By pair wise comparison (Saaty, 1980) of the
factors and categories (according to the AHP estimation scale) and the
use of triangularly distributed fuzzy numbers, we get fuzzy matrices on
all levels of hierarchy.
2.2 Calculation of the uncertainty of factors
Ratios between categories or factors are expressed with a question:
"How many times (max. 9) is the category/factor i more uncertain
than category/factor j?" By pair wise comparison of the factors and
categories (according to the AHP estimation scale, adapted for the level
of uncertainty) and the use of triangularly distributed fuzzy numbers,
we get fuzzy matrices on all levels of hierarchy.
Normally we calculate the importance and uncertainty of categories
and factors at different levels of confidence ([alpha] = 0, 0.5, 1) and
optimism ([mu] = 0.05, 0.5, 0.95). Variations in the results indicate
some possible mistakes of the estimation process (human impact).
Introduction of fuzzy numbers allows the compensation of the possible
errors of the estimator.
3. Uncertainty--importance relations and ABC focus diagrams
Categories and factors are mutually compared according to
importance and uncertainty in the diagram. In the diagram we define
three areas (low, medium, high) with boundaries around [+ or -] 50 %
from average share. The position of every category / factor in one of
the nine fields of the diagram is defined with an ellipsis, whose
boundaries are minimal and maximal shares of category / factor of
importance or uncertainty (see Fig. 2). The diagrams enable selection of
factors that need special attention, or vice versa, the factors which
can be partially neglected, which is shown in ABC focus diagram (see
Fig. 3).
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
4. Integral uncertainty value
We would like to evaluate the problem of uncertainty, represented
with the categories and factors, numerically (Buchmeister et al., 2005).
With the fuzzy AHP we have determined the intervals of the level of
importance and uncertainty for every factor, which have given us the
opportunity for selection of more or less critical factors. The
mentioned intervals of values can be used for:
* Design of the factor importance vector [bar.P], whose elements
are weights of factor importance, obtained as arithmetical mean value
between the lowest and highest value of the importance of the factor
(multiplied by 100),
* Design of the factor uncertainty vector [bar.N], whose elements
are weights of factor uncertainty, obtained also as arithmetical mean
value between the lowest and highest value of the uncertainty of the
factor (multiplied by 100).
Integral uncertainty value (IUV) is a scalar product of vectors
[bar.P] and [bar.N]:
IUV = [bar.P] x [bar.N] (9)
Boundary integral uncertainty value can be obtained by using the
same mean weights at all vector components, therefore, with n factors:
[IUV.sub.b] = [n.summation over (i=1)][(100/n) x (100/n)] = n x
100/n x 100/n = 10000/n (10)
In practice, real bounds of IUV depend upon the number of factors
(from 5 to about 100), which gives: [IUV.sub.b] = 100 ... 2000. Integral
uncertainty value, which exceeds boundary value, means that we are
dealing with an activity of higher risk, or vice versa.
5. Systematic approach to managing capability development
The key advantage of the model is its contextual robustness. The
model was developed on the case studies of military logistics and
supplier risk but can be without greater problems transferred in the
environment of business and (narrowly) production systems. Systematic
approach for directing capability development is developed for easier
transfer of the model and its logic between different system
environments, and because of the discussion on its applicative value. In
the following subchapters the systematic approach will be described with
the aim to point out applicability of the developed heuristics.
Systematic approach is composed of three contextual parts: analytical,
process, and decision making part (see Fig. 4).
5.1 Analytical part
The analytical part resembles the project start-up. Before starting
the implementation of the processing and decision part, the basic issues
need to be clarified and all info defining the development of discussed
capabilities acquired. An expert team should be formed to assess the
relevance and uncertainty of the factors. The team members should differ
with respect to experience, education, formal position and age. The
divergence of the team depends on the complexity of the problem. More
complex problem requires more divergent group.
[FIGURE 4 OMITTED]
The analytical part includes a detailed analysis of resources and
capabilities. On the one hand we determine which of them are already
available and where they could be applied, and on the other, which of
them are still lacking. Literature offers a number of systemic
approaches to help us with this analysis. Before the expert team
analyses the development of capabilities and classifies them into
different categories and factors, the types of risks should be
determined. Miliken's classification (1987) into the uncertain
states, effects and responses helps the team members to come to a
uniform understanding of concepts.
The analysis of the discussed process into categories and factors
of the uncertainty represents a domain which differs with respect to
different systemic environments. The expert team dealing with the risk
of the transfer of production to a geographical remote location will
identify different factors and categories than the expert group for a
supply chain which is involved with the vendor selection. The analysis
represents a decomposition of a complex problem and enables the start of
the implementation of the processing part.
5.2 Process part
The processing part is intended for evaluation of analysed factors.
The basic method used is the Fuzzy Analytic Hierarchy Process (basically
used as support to multiple criteria decision-making), based on mutual
determining of relationships between individual categories and factors
of uncertainty (first with regard to importance and in the second stage
with regard to the uncertainty level of factors) and on mathematical
calculation of weights reflecting the aforementioned importance and
level of uncertainty. The advantage of this method lies in mutual
determining of relationships as the evaluation is much easier with
comparison by pairs, and in use of fuzzy numbers when the evaluation
itself includes the possibility of the evaluator's error by a class
(to the left or to the right, a bigger error is virtually excluded) and
the calculation takes it into account as results are presented in
particular intervals. The disadvantage of the method lies in relatively
large number of evaluations required, which can be avoided by grouping
factors in categories already in the analytical part, and in the danger
of excessively diverging evaluations, which would require repeating the
procedure of comparison and adjustment of selected evaluations.
We use the interval results of the fuzzy AHP method for building
the vector of importance of factors and the vector of uncertainty of
factors, the scalar product of which provides us with the integral
uncertainty value, which in comparison with the boundary value
determines the risk of the underlying process.
On the basis of original diagrams
"Uncertainty/Importance", we create the ABC diagram of
attention, which is used for selecting and classifying uncertainty
factors on which the decision-making part of the systematic approach
builds on.
5.3 Decision-making part
The decision-making part represents the most important part of the
approach with regard to the model's applicability criterion. In
this part the model's logic was transferred to real-life decisions.
The decision-making model begins by classifying factors. "C"
factors are those, which do not require larger attention with regard to
the levels of importance and uncertainty. When such factors require
certain decisions or measures, they can be implemented quickly and
effectively without worrying how uncertainty may affect the consequences
of the decision. "A" factors are the opposite pole of factors
with regard to importance and uncertainty. These factors require extra
attention. The instructions regarding proposing of final decisions
should be elaborated in greater detail. These instructions should not be
interpreted as a proposal for not taking any decisions. The instructions
say that in such case no decisions can be taken, which would be final
and prevent the flexibility of action. These decisions should be
directed towards creating a wide range of options, enabling reaction in
case of different development scenarios. Before adopting decisions,
which create future options, the wide range of available options shall
be identified. "B" factors lie between "A" and
"C" factors with regard to their importance and the level of
uncertainty. These are by all means factors, which require adequate
level of attention. The proportion of attention naturally also depends
on the number of factors being defined as "A" level factors.
If we get a large number of "A" level factors, the attention
given to "B" level factors will be slightly lower than in
cases where there are only few "A" level factors. In case
factors are classified in the "B" category due to the fact
that they are important but not subject to uncertainty, the same
measures as for "C" factors can be adopted. In case of
increased uncertainty, decisions enabling flexibility of future actions
shall also be adopted for "B" factors.
6. Conclusion
The companies are exposed to various risks every day. Risk
management in the quickly changing environment is essential, for it
contributes to achieving the strategic advantage of the company. In
decision making under uncertainty, risk analysis aims at minimising the
failure to achieve a desired result.
The article encompasses the original synthesis of risk management,
modelling uncertainty, method of analytic hierarchy process and fuzzy
logic, and it represents a contribution to the construction of tools for
decision-making support in organisational systems.
Based on the "Uncertainty--Importance" relations special
ABC focus diagrams are created. These diagrams serve for the
classification of uncertainty factors, which provides a decision making
part of the systemic approach. The original contribution in this article
is comprised by:
* Completion of heuristic approach for effective interpretation of
numerical results and their support to decision-making process,
* Use of fuzzy AHP method for determining uncertainty level is an
entirely original idea, for the abovementioned method is used only for
defining the importance (weights),
* Original combined diagrams 'Importance--Uncertainty'
and ABC diagram of attention enable the selection and classification of
factors,
* Integral uncertainty value (IUV) and its boundary value represent
an original contribution for estimating uncertainty and risk of
discussed activities.
7. References
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Authors' data: Associate Prof. Dr. Buchmeister B.[orut] *,
Prof. Dr. Polajnar A.[ndrej] *, Assistant Prof. Dr. Pandza K[rsto] **,
Dr. Kremljak Z[vonko] *** * University of Maribor, Faculty of Mechanical
Engineering, Maribor, Slovenia, ** Leeds University Business School,
Leeds, United Kingdom, *** Ministry of the Economy, Government of the
Republic of Slovenia, borut.buchmeister@uni-mb.[s.sub.i],
andrej.polajnar@uni-mb.[s.sub.i],
[email protected],
[email protected]
This Publication has to be referred as: Buchmeister, B.; Polajnar,
A.; Pandza, K. & Kremljak, Z. (2006). Fuzzy AHP Method, Uncertainty
and Decision-Making, Chapter 08 in DAAAM International Scientific Book
2006, B. Katalinic (Ed.), Published by DAAAM International, ISBN 3-901509-47-X, ISSN 1726-9687, Vienna, Austria
DOI: 10.2507/daaam.scibook.2006.08