Research and development of decision-making methods for creating 3 d objects in mechanical engineering

Let us consider the problem of alternatives ranking [1,3,5] from set Z with respect to criteria k. The result of alternative zz ∈ ZZ can be estimated by the indicated quality criterion. The evaluation of the alternative can be represented in the form of a conjunctive form and fuzzy analysis will be denoted as GGii(aa): WW: < WW1 ∪ ...∪ WWii ∪ ...∪ WWnn >≡ ZZ1(zz) = GG1(aa) ∪ ...∪ ZZii(zz) = GGii(aa) ∪ ...∪ ZZnn(zz) = GGnn(aa) (1)

The following necessary step is ranking.Let us apply an approach that is based on using the theoretically possible ideal alternative  0 .
Expression (3) takes the following form: In order to evaluate the truth of the correspondence between the expressed alternative of the introduced ideal one, let us check the probability of truth (3) to (4): Let us establish the rule for ranking; q(z) stands for the index of the integral adjacency of the membership function of the fuzzy truth evaluation and also the values absolute and true: To exclude the exponential complexity of the algorithm for calculating expressions, we apply the following property: (, ) =  � ((  ,   )). ( We denote by T the extended T-norm by means of the generalization principle.(  ,   ) is evaluation of the truth of the alternative according to the chosen criterion.
Figure 1 shows the interpretation of the criterion of integral adjacency of the fuzzy truth value to the value of absolute truth.It is obvious from Fig. 1 that (5) allows us to apply the method both for a small and for a large number of criteria.
With increasing number of criteria, it is effective to use calculation paralleling.However, in this case, there is a need to harmonize solutions, both on the tiers inside the branch, and on neighboring branches.Besides it is necessary to coordinate all areas of the decision-making system.
To facilitate the work of users and simplify the matching procedure, there are methods [6] of finding the best compromise.
Let a group of several objects is involved in the decision making process.Let us consider the process of harmonizing the evaluation of an object by the members of this group.In this case, let's assume that each object evaluates and makes a decision on a certain criterion, let us also assume that between the members of the process the criteria groups and rating scales are distributed.
So, it is necessary to obtain an agreed evaluation of the object in accordance with the priorities of the decision-making objects.
Depending on the importance of the criterion, each object in the decision-making group can set the criterion for the absence of artefacts as 18 and greater, and the file size cannot be more than the current one by 56%.Each object forms an ideal rating of the criterion in accordance with (9): Here by   , we denote an ideal estimate of the i th criterion, which was expressed by the j th object of decision-making.
For example, an object can claim that the presence of artefacts is not critical and can reach 99%; therefore, the likelihood of their absence is below the threshold; it can also be assumed that the quality of the file is not essential and can be equal to the source file.
where   =  1 or  2 etc.In order to evaluate the truth of the statement of the j th object according to the ideal estimate formed by it, it is necessary to verify the truth of ( 8) and ( 6): As is known, when analyzing fuzzy values for each fuzzy truth evaluation, a criterion of correspondence is required as an absolutely true one.A variant of the solution and achievement of this criterion is the introduction of the index of integral adjacency of the membership function of the fuzzy truth evaluation of the prime statement Wj and the absolute truth value.
and in the case of a discrete representation of membership functions: where   −1 must satisfy the monotonicity condition and max   () = 1.The introduced index qt is numerically equal to the area of the shaded part in Fig. 1.With the help of (4.20) it is possible to establish an order on fuzzy truth values:

MATEC
The evaluation, which was made by the first decision-making object, will be the result of the approval procedure.
To exclude the exponential complexity of the expression calculating algorithm, let us apply the following property: We denote by  � the extended T-norm by means of the generalization principle.�  ,   � is an assessed truth value of the alternative according to the chosen criterion.
To provide greater flexibility to this method, it is desirable to use criterion weights; let us write (16) as:

Conclusion
Summarizing the above, it can be noted that we have studied the procedure for agreeing object evaluations on the basis of the formation of a hypothetical ideal criterion evaluation, using fuzzy truth evaluation to identify an agreed evaluation.This method is optimal for the stated task.

Fig. 1 .
Fig. 1.Interpretation of the criterion of integral adjacency of the fuzzy truth value to the value of absolute truth.