# V.I. Levin MODELS OF OPTIMIZATION SYSTEMS IN CONDITIONS OF INTERVAL UNCERTAINTY

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## V.I. Levin MODELS OF OPTIMIZATION SYSTEMS IN CONDITIONS OF INTERVAL UNCERTAINTY

The numerous application tasks of optimization, decision making, handle, control, measurement etc. in conditions is inexact of known parameters of investigated systems result in operations with numbers given as intervals of possible values, i.e. with interval numbers (IN). Thus above IN it is necessary to execute not only algebraic operations: addition, subtraction, multiplying, division , but also to compare them to the help of the ratioes. In operation the system of matchings IN, based on the entered axioms similar to the appropriate axioms of matching of material numbers is constructed. It gives sets of the ratioes IN, similar to the ratioes of material numbers and having the same properties It opens path to solution of the above-stated tasks. Our approach to these tasks is oriented not to any one combination of values of parameters of the system inside appropriate intervals (worst combination with the pessimistic strategy, best - with optimistic, expected with probability), and on all set of possible values of parameters. According to it IN are considered as set of material numbers, the subjects to matching under the ratioes - are more, less, it is more or equally etc. However to compare directly IN under some ratio on the basis of that, whether there are in this respect numbers, included in them, it is impossible, as one pairs numbers can be in this respect, and others - is not present. Therefore matching IN under some ratio P we execute indirectly, substituting this ratio in the equivalent operation allowing to select from compared IN what should stand at the left (on the right) from P. This operation is entered as theoretical-multiple generalization of appropriate operation above material numbers taken in conditions, when these numbers run sets of all possible values inside appropriate IN. For example operation of a capture of a maximum IN A and B, equivalent to the ratio « is more or equally » between them: .

Thus, if , IN A it is more or equally IN B. The matching IN A and B under the given ratio is thus executed. The detailed researches have shown [2, 3]: IN are equal only if their low bounds are equal and their high bounds are equal; IN A one boundary A - lower or upper - more appropriate boundary B is more IN B only if even, thus the second boundary A should be more or equal of the same boundary B; IN A the low bound A is more or equally IN B only more if or is equal low bounds B, and the high bound A more or is equal high bounds B; IN A the low bound A less low bound B or high bound A less high bound B or A = B is no more IN B only if; IN A and B are not comparable under the ratio « more or equally » only if the interval A covers an interval B or on the contrary; In the system IN some IN A - maximum (minimum) only if the low bound A is a maximum (minimum) of low bounds all IN, and high bound A - maximum (minimum) of high bounds all IN. These results make a mathematical basis for solution of the mentioned above application tasks with is inexact known parameters given to within intervals.

References

1. Alefeld G., Hercberger U., Introduction to interval calculations. - M.: Mir, 1987.

2. V.I. Levin. Discrete optimization in conditions of interval uncertainty // Avtomatika i telemehanika. - 1992. -¹ 7.

3. V.I. Levin. The theory of matching and optimization of interval values and it application in the tasks of measurement // Izmeritelnaja tehnika. - 1998. - ¹ 5.

## MEANING, COMPREHENSION, TRANSITIVITY

The Analytic Hierarchy Process (AHP) is well known as a practical systematic approach for decision making. It allows to rank alternatives with ratio scale but decision–makers’ (DMs) preferences may be given with a verbal scale. This kind of scales can be successfully used either for individual and group decision.

The different kind of scales can be used by a DM to make parawise comparisons or direct judgments. For verbal scale it can be point two sides: the numeric interpretation and the substantive interpretation. To offer a DM any kind of verbal scale we must be sure that a DM understand the correspondence between numbers and sound behind the scale values.

This paper considers verbal scales and their numeric and meaningful interpretation for AHP: 1-9 numeric scale by Saaty (1980), the geometric scale by Finan and Hurley (1999) and other similar scales (Lootsma, 1996) . The underlying DM’s notions by using different kind of verbal scales are inspected.

The transitivity of scale values is investigated. As a number of verbal scale values is finite the inconsistency of DM’s judgment may be take place (Murphy, 1993). To keep the transitivity of judgments a DM need to have some resource of verbal scale values. From the other side human thinking cannot capture accurately a very great or very small difference. Hence to provide the transitivity of judgment a DM should use the verbal scale with certain characteristics. Three additional characteristics of verbal scales are offered: the comprehension index, the relative range and the ratio difference. Those characteristics help to clarify the DM’s embarrassment by using the verbal scale.

References

1. Finan J.S., Hurley W.J., 1999. Transitive calibration of the AHP verbal scale. European Journal of Operational Research 112, 367-372.

2. Lootsma F.A., 1996. A model for the relative importance of the criteria in the Multiplicative AHP and SMART. European Journal of Operational Research 94, 467-476.

3. Murphy, C.K., 1993. Limits on the analytic hierarchy process from its inconsistency index. European Journal of Operational Research 67, 138-139.

4. Saaty, T.L., 1980. The analytic hierarchy process, McGraw-Hill, New York.

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