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

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

Buchmeister, B.; Pandza, K.; Kremljak, Z. & Polajnar, A. (2005). Uncertainty estimation using two-pass fuzzy AHP method. Proceedings of the 16th DAAAM Symposium, Katalinic, B. (Ed.), pp. 45-46, Opatija, Croatia

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, pp. 533-545

Frei, F. X. & Harker, P. T. (1999). Measuring aggregate process performance using AHP, European Journal of Operational Research, Vol. 116, pp. 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, pp. 367-377

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

Milliken, F. J. (1987). Three types of perceived uncertainty about the environment: state effect, and response uncertainty. Academy of Management Review, Vol. 12, No. 1, pp. 133-144

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

Trigeorgis, L. (2002). Real options--Managerial flexibility and strategy in resource allocation. MIT Press, Cambridge

Van Laarhoven, P. J. M. & Pedrycz, W. (1983). A fuzzy extension of Saaty's priority theory, Fuzzy Sets and Systems, Vol. 11, No. 1, pp. 229-241

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

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], k.pandza@leeds.ac.uk, zvonko.kremljak@s5.net

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