Uncertainty estimation using two-pass fuzzy AHP method.

Buchmeister, B. ; Pandza, K. ; Kremljak, Z. 等

Abstract: In the paper our original method of uncertainty estimation is presented. Each uncertainty problem is divided into risk categories and factors. The basic method used in the numerical part is the two-pass Fuzzy Analytic Hierarchy Process (first for the importance and second for the uncertainty of risk factors). The estimates are derived from pairwise comparison, and by using fuzzy numbers we account for the errors of the estimator. In the following stages we use the interval results of this method for calculating the integral uncertainty value, which, in comparison with the boundary value, defines the risk of the process in question.

Key words: estimation, fuzzy logic, Analytic Hierarchy Process, uncertainty, risk

1. INTRODUCTION

The world's most developed countries achieve prosperity by employing top levels of technological and scientific development. Globalisation has diminished barriers to technology transfer and deployment. The only obstacles remain market and capital interests. 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. 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.

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 [??] of the elements at the same hierarchy level. Triangular fuzzy number is described as [??] = (a, b, c) and by defining the interval of confidence level [alpha], we get:

[for all][alpha] [member of][0,1][[??].sub.[alpha]] = [[a.sup.[alpha]], [c.sup.[alpha]]] = [(b - a) x [alpha] + a, - (c - b) x [alpha] + c] (1)

[FIGURE 1 OMITTED]

2nd step: constructing the fuzzy comparison matrix [??] ([a.sub.ij]) with triangular fuzzy numbers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

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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 [??] of matrix, where:

[??] 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:

[[a.sup.[alpha].sub.i1l] x [x.sup.[alpha].sub.1l], [a.sup.[alpha].sub.i1u] x [x.sup.[alpha].sub.1u]] [[direct sum] ... [direct sum] [[a.sup.[alpha].sub.inl] x [x.sup.[alpha].sub.nl], [a.sup.[alpha].sub.inu], [a.sup.[alpha].sub.nu]] = [[lambda] x [x.sup.[alpha].sub.il], [lambda] x [x.sup.[alpha].sub.iu] (3)

where:

[??] = [[[??].sub.ij]], [??] = {[[??].sub.1] ... [[??].sub.n]}.sup.T]

[[??].sup.[alpha].sub.ij] = [[a.sup.[alpha].sub.ijl], [a.sup.[alpha].sub.iju]], [[??].sup.[alpha].sub.i] = [[x.sup.[alpha].sub.il], [x.sup.[alpha].sub.iu]] = [[??].sup.[alpha]] = [[[lambda].sup.[alpha].sub.l], [[lambda].sup.[alpha].sub.u] (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 [??] 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.ijn] + (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 [alpha] 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 matrixes 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 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 matrixes 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. INTEGRAL UNCERTAINTY VALUE

We would like to evaluate the problem of uncertainty, represented with the categories and factors, numerically. 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 [??], 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 [??], 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 [??] and [??]

IUV = [??] x [??] (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)] p(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.

4. 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. 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. Human assessment on qualitative attributes is always subjective and thus imprecise. We should take into account the uncertainty associated with the mapping of one's perception or judgment to a number.

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),

* Integral uncertainty value (IUV) and its boundary value represent an original contribution for estimating uncertainty and risk of discussed activities.

5. REFERENCES

Augier, M. & K. Kreiner (2000). Rationality, imagination and intelligence: some boundaries in human decision-making. Industrial and Corporate Change, Vol. 9, No. 4 (659-679)

Carpenter, M. A. & Fredrickson, J. W. (2001). Top management teams, global strategic posture, and the moderating role of uncertainty. Academy of Management Journal, Vol. 44, No. 1 (533-545)

Frei, F. X. & Harker, P. T. (1999). Measuring aggregate process performance using AHP. European Journal of Operational Research, Vol. 116 (436-442)

Kremljak, Z. (2004). Decision Making under Risk. DAAAM International, Vienna.

Kwong, C. K. & Bai, H. (2002). A fuzzy AHP approach to the determination of importance weights of customer requirements in quality function deployment. Journal of Intelligent Manufacturing, Vol. 13 (367-377)

Laviolette, M. & Seaman, J. W. (1994). The efficacy of fuzzy representations of uncertainty. IEEE Transactions on Fuzzy Systems, Vol. 2, No. 1 (4-15)

Saaty, T. L. (1980). The Analytic Hierarchy Process. McGraw-Hill, New York

Van Laarhoven, P. J. M. & Pedrycz, W. (1983). A fuzzy extension of Saaty's priority theory. Fuzzy Sets and Systems, Vol. 11 (229-241)

Zadeh, L. (1965). Fuzzy Sets. Information Control, Vol. 8 (450-456)