Mathematical Apparatus for Selection of Optimal Parameters of Technical, Technological Systems and Materials Based on Vector Optimization

We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling; transformation of mathematical models with uncertainty to the model under certainty; acceptance of an optimal solution with equivalent criteria (the solution of numerical model); acceptance of an optimal solution with the given priority of criterion.


Introduction
The problem of high quality production is associated with the creation of technical, answering to the modern achievements of science and technology. The functioning of technical, technological systems, as well as the structure of materials depend on a set of functional characteristics that must be considered at the design stage. Improvement of one of the characteristics leads to deterioration of other characteristics. And it is necessary for improvement of quality of the produced product improvements of all characteristics in total. For the solution of such problems we use the theory and methods of vector (multi-criteria) optimization [1]- [16]. Development of the theory and methods of vector optimization was carried out both in Russia [1]- [17], and in other countries [18]- [24].
The purpose of this work is to create a methodology for selecting the optimal parameters of technical, technological systems and materials based on vector optimization. The methodology includes the construction of a mathematical model for an object or decision-making system, an algorithm and software for solving a vector problem of mathematical programming.
For realization of a goal in work the following problems are considered and solved. The article presents the construction of a mathematical model as a vector problem of mathematical programming for three types of optimal decision-making problems. The first task is related to the choice of optimal parameters of technical systems, which depend on a set of functional characteristics of the system. The study of this class of problems is presented in [2]- [10] [12]. The second task is bound the choice of optimum parameters of the technological process depending on running characteristics on this process [11] [17]. The third problem is the problem of choosing the optimal structure (components) of the material depending on the functional characteristics of this material [2] [13]. It is assumed that all three tasks are used at the design stage and creation of a new technical object (system) within the framework of information and mathematical concepts [14] [15]. The implementation of the methodology is presented on the solution of numerical problems of decision-making in the engineering systems by two, four parameters. It is assumed that all three tasks are used at the design stage and creation of a new technical object (system) within the framework of information and mathematical concepts [14] [15]. The implementation of the methodology is presented on the solution of numerical problems of decision-making in a technical system by four parameters. x j N = = , in total they represent a vector function:

Setting the Problem of Selection of Optimal Parameters of
The set of characteristics (criteria) to is subdivided into two subsets K 1 and K 2 : min max 1, , where X is the vector of controlled variables (constructive parameters) from (1); To a substantial class of technical systems which can be presented by a vector problem (1a)-(4a), it is possible to refer their rather large number of tasks from various branches of economy of the state: as electro engineering [6], airspace [17] [18], metallurgical (choice of optimal structure of material), chemical [20], etc.
In this article, the technical system is considered in statics. But technical systems can be considered in dynamics [6], using differential-difference methods of transformation, conducting research for a small discrete period t T ∆ ∈ .

Building a Mathematical Process Model in the Form of a Vector Optimization Problem
Please As an object of a research we use "technological process". The problem of decision-making in technology in the production of products is formulated in accordance with the works [11] [25].
We consider a technological process (e.g., Hybrid laser arc welding [25], in which ze41-T5 alloy was chosen as the material to be welded with AZ61 alloy as the filler material). Activity of technological process depends on a particular set of conditions-design data: , (for example: laser powers; speeds of movement; feed rates of a wire; current; frequencies) we Will designate [25]. Let's denote N-set of constructive parameters. Each parameter of the technological process lies in the given limits: where X is the vector of operated variable (design parameters) of the technological process; The set of the functional characteristics of the material K is subdivided into two subsets of K 1 and K 2 : is the vector of the operated variables (a material component) from criterion which each function submits the characteristic (property) of the material which is functionally depending on a vector of variables Y; in (3c) Ratios (1c)-(4c) form mathematical model of the material. It is required to find such vector of the o Y ∈ S parameters at which each component (charac- functions accepts the greatest possible value, and a vector-the In total the mathematical model of the material (1c)-(4c) can be interpreted as systems approach to the study of the material.

A Vector Problem of Mathematical Programming with Conditions of Certainty and Uncertainty
The Mathematical models of a technical system (1a)-(4a), technological process (1b)-(4b) and structures of material (1c)-(4c) are constructed in the assumption that the functional dependence of each criterion (characteristic) and restrictions from parameters of the studied object is known. In real life such functional dependence of criterion from parameters extremely infrequent, i.e. there are conditions of uncertainty. We present a vector optimization problem with certainty and uncertainty conditions.  [14]. For options a) and b) basic data have to be transformed to option c) and are presented in the table form. In work the option c)-with not full data which are, as a rule, obtained from experimental data is investigated.
In real life, the conditions of certainty and uncertainty are combined. The process model should also reflect these conditions. We will present a model of the technological process under certainty and uncertainty in the aggregate: min max 1, , is the vector of controlled variables (input parameters of the studied object); is a vector criterion, each component of which represents a vector of criteria (output characteristics of the studied object). The magnitude of the characteristic (function) depends on the discrete values of the vector of variables X.
( ) ( ) 1 2 , F X F X is the set of functions max and min, respectively; ( ) ( ) 1 2 , I X I X are the set of discrete values of the characteristics max and min, respectively;

Transformation of a Problem of Decision-Making in the Conditions of Uncertainty into a Problem of Vector Optimization in the Conditions of Certainty
Elimination of uncertainty consists in the use of qualitative and quantitative descriptions of the object under study which can be received, for example, on the principle of "input-output". Transformation of such information-initial data into functional dependence is carried out by using mathematical methods (the regression analysis) [7] [10] [13].
In the applied part of the polynomial of the second degree is used.
As a result of this transformation, the source data in (1) and (2) As a result, a vector problem with conditions of certainty and uncertainty (1)-(4) is transformed into a vector problem under conditions of certainty: at restriction min max 1, , where is a vector criterion, each component of which represents a characteristic of the object under study, functionally dependent on the vector of variables X; a subset of the criteria: The vector mathematical programming problem (5)-(8) is analogous to mathematical models (1a)-(4a), (1b)-(4b), (1c)-(4c).

Mathematical Modeling Apparatus: Theory and Methods of Vector Optimization
The theory of vector optimization includes theoretical foundations (axiomatic) and methods of the solution of vector problems with equivalent criteria and with the given criterion priority. The theory is a basis of mathematical apparatus of modeling of "object for optimal decision-making" which allows you to select any point from a set of points, optimum across Pareto and to show why she is optimum. We presented axiomatic and methods for solving vector optimization problems (5)-(8) with equivalent criteria [1] [16].

The Axioms and the Principle of Optimality for Vector Optimization with the Equivalent Criteria
Definition 1. (Definition of the relative assessment of the criterion).
In a vector problem (5)-(8) we will enter designation: K is the relative estimate of a point X ∈ S kth criterion; ( ) k f X -kth criterion at the point X ∈ S ; * k f -value of the kth criterion at the point of optimum We will consider criteria equivalent in vector problems of mathematical programming if in, X ∈ S point when comparing in the numerical size of relative estimates of , among themselves, on each criterion of ( ) 1 , , k f X k K = , and, respectively, relative estimates of ( ) k X λ , isn't imposed conditions about priorities of criteria. Definition 2. (Definition of a minimum level among all relative estimates of criteria).
The relative level λ in a vector problem represents the lower assessment of a point of X S ∈ among all relative estimates of the lower level for performance of a condition (5) in an admissible point of X S ∈ is defined by a formula Ratios (9) and (10) are interconnected. They serve as transition from operation (6) of definition of min to restrictions (9) and vice versa. The level λ allows to unite all criteria in a vector problem one numerical characteristic of λ and to make over her certain operations, thereby, carrying out these operations over all criteria measured in relative units. The level λ functionally depends on the X ∈ S variable, changing X, we can change the lower level-λ . From here we will formulate the rule of search of the optimum decision.

Axioms and the Principle of Optimality of Vector Optimization with a Criterion Priority
For development of methods of the solution of problems of vector optimization with a priority of criterion we use definitions as follows: priority of one criterion of vector problems, with a criterion priority over other criteria; numerical expression of a priority; the set priority of a criterion; the lower (minimum) level from all criteria with a priority of one of them; a subset of points with priority by criterion (Axiom 2); the principle of optimality of the solution of problems of vector optimization with the set priority of one of the criteria, and related theorems. For more details see [7] [13]. Definition 4. (About the priority of one criterion over the other).
The criterion of q ∈ K in the vector problem of Equations (12) and (13) in a point of X ∈ S has priority over other criteria of , and the relative estimate of ( ) q X λ by this criterion is greater than or equal to relative estimates of ( ) k X λ of other criteria, i.e.: and a strict priority for at least one criterion of t ∈ K , and for other criteria of Introduction of the definition of a priority of criterion q ∈ K in the vector problem of Equations (5)-(8) executed the redefinition of the early concept of a priority. Earlier the intuitive concept of the importance of this criterion was outlined, now this "importance" is defined as a mathematical concept: the higher the relative estimate of the qth criterion compared to others, the more it is important (i.e., more priority), and the highest priority at a point of an optimum is * , From the definition of a priority of criterion of q ∈ K in the vector problem of Equations (5)- (8), it follows that it is possible to reveal a set of points q ⊂ S S that is characterized by However, the answer to whether a criterion of q ∈ K at a point of the set q S has more priority than others remains open. For clarification of this question, we define a communication coefficient between a couple of relative estimates of q and k that, in total, represent a vector: Definition 5. (About numerical expression of a priority of one criterion over another).
In the vector problem of Equations (12) and (13), with priority of the qth criterion over other criteria of 1, k K = , for q X ∀ ∈ S , and a vector of ( ) q P X which shows how many times a relative estimate of ( ), q X q λ ∈ K , is more than other relative estimates of ( ) 1 , , k X k K λ = , we define a numerical expression of the priority of the qth criterion over other criteria of , Definition 6. (About the set numerical expression of a priority of one criterion over another).
In the vector problem of Equations (5)-(8) with a priority of criterion of The vector problem of Equations The λ level is the lowest among all relative estimates with a priority of criterion of q ∈ K such that: The lower level for the performance of the condition in Equation (19) is defined as: Equations (19) and (20) are interconnected and serve as a further transition from the operation of the definition of the minimum to restrictions, and vice versa. In Section 3.1, we gave the definition of a Pareto optimal point o X ∈ S with equivalent criteria. Considering this definition as an initial one, we will construct a number of the axioms dividing an admissible set of S into, first, a subset of Pareto optimal points S°, and, secondly, a subset of points In the vector problem of Equations (12)- (13), the subset of points q ⊂ S S is called the area of priority of criterion of q ∈ K over other criteria, if This definition extends to a set of Pareto optimal points o S that is given by the following definition.
In the following we provide explanations.
We note that the subset of points , to form and choose: a subset of points by priority criterion S q , which is included in a set of points S, q ∀ ∈ K q X ∈ ⊂ S S , (such a subset of points can be used in problems of clustering, but is beyond this article); a subset of points by priority criterion o q S , which is included in a set of Pareto optimal points S o , , Thus, full identification of all points in the vector problem of Equations (12) and (13)   The vector problem of Equations (12) and (13) with the set priority of the qth criterion of ( ), 1, is considered solved if the point X o and maximum level λ o among all relative estimates is found such that: Using the interrelation of Equations (19) and (20), we can transform the maximine problem of Equation (33) into an extreme problem of the form: at restriction We call Equations (22) and (23)  The solution of the λ-problem is the point . This is also the result of the solution of the vector problem of Equations (5) (15), which can be written as: These restrictions are the basis of an assessment of the correctness of the results of a decision in practical vector problems of optimization.
From Definitions 1 and 2, "Principles of optimality", follows the opportunity to formulate the concept of the operation "opt".
In the vector problem of Equations (1)-(4), in which "max" and "min" are part of the criteria, the mathematical operation "opt" consists of the definition of a point X° and the maximum λ° bottom level to which all criteria measured in relative units are lifted: i.e., all criteria of with the indexes , r t ∈ ∈ K K , for which the following strict equality holds: and other criteria are defined by inequalities: Criteria with the indexes , r t ∈ ∈ K K for which the equality of Equation Proof. Similar to Theorem 2 [7].
We note that in Equations (25) and (26), the indexes of criteria , r t ∈ K can coincide with the q ∈ K index.
Consequence of Theorem 1, about equality of an optimum level and relative estimates in a vector problem with two criteria with a priority of one of them.
In a convex vector problem of mathematical programming with two equiva-lent criteria, solved on the basis of normalization of criteria and the principle of the guaranteed result, at an optimum point X o equality is always carried out at a priority of the first criterion over the second: and at a priority of the second criterion over the first:

Mathematical Algorithm of the Solution of a Vector Problem with Equivalent Criteria
To solve of the vector problems of mathematical programming (5) Step 1. The problem (5)-(8) by each criterion separately is solved, i.e. for 1 k ∀ ∈ K is solved at the maximum, and for As a result of the decision we will receive: * k X -an optimum point by the cor- Step 2. We define the worst value of each criterion on S: 0 1, , For what the problem (5)-(8) for each criterion of 1 1, k = K on a minimum is solved: The problem (5)-(8) for each criterion 2 1, k = K maximum is solved: As a result of the decision we will receive: -an optimum point by the corresponding criterion, Step 3. The system analysis of a set of points, optimum across Pareto, for this purpose in optimum points of , are defined sizes of criterion functions of F(X * ) and relative estimates ( ) As a whole on a problem k ∀ ∈ K the relative assessment of Step 4. Creation of the λ-problem. Creation of λ-problem is carried out in two stages: initially built the maximine problem of optimization with the normalized criteria which at the second stage will be transformed to the standard problem of mathematical programming called λ-problem.
For construction maximine a problem of optimization we use definition 2-relative level: The bottom λ level is maximized on X S ∈ , as a result we will receive a maximine problem of optimization with the normalized criteria. (29) At the second stage we will transform a problem (29) to a standard problem of mathematical programming: where the vector of unknown of X has dimension of N + 1: Step 5. Solution of λ-problem. λ-problem (30)-(32) is a standard problem of convex programming and for its decision standard methods are used.
As a result of the solution of λ-problem it is received: -values of the criteria in this point;

Mathematical Method of the Solution of a Vector Problem with Criterion Priority
(Method of the decision in problems of vector optimization with a criterion priority) [7].
Step 1. We solve a vector problem with equivalent criteria. The algorithm of the decision is presented in Section 4.3.
As a result of the decision we obtain: Optimum points by each criterion separately and sizes of criterion functions in these points of ( ) , which represent the boundary of a set of Pareto optimal points; Anti-optimum points by each criterion of and the worst unchangeable part of each criterion of ( ) , an optimum point, as a result of the solution of VPMP at equivalent criteria, i.e., the result of the solution of a maximine problem and the λ-problem constructed on its basis; λ o , the maximum relative assessment which is the maximum lower level for all relative estimates of λ k (X o ), or the guaranteed result in relative units, λ o guarantees that all relative estimates of λ k (X o ) are equal to or greater than λ o : The person making the decision carries out the analysis of the results of the solution of the vector problem with equivalent criteria. If the received results satisfy the decision maker, then the process concludes, otherwise subsequent calculations are performed.
In addition, we calculate: we determine sizes of all criteria of: , and relative estimates Matrices of criteria of F(X * ) and relative estimates of λ(X * ) show the sizes of i.e., on the border of a great number of Pareto.
atan optimum point at equivalent criteria X o we calculate sizes of criteria and relative estimates: This information is also a basis for further study of the structure of a great number of Pareto.
Step 2. Choice of priority criterion of q ∈ K .
From theory (see Theorem 1) it is known that at an optimum point X o there are always two most inconsistent criteria, q ∈ K and v ∈ K , for which in relative units an exact equality holds: S. Others are subject to inequalities: As a rule, the criterion which the decision-maker would like to improve is part of this couple, and such a criterion is called a priority criterion, which we desig- Step 3. Numerical limits of the change of the size of a priority of criterion For priority criterion q ∈ K from the matrix of Equation (34) we define the numerical limits of the change of the size of criterion: in physical units of where ( ) where ( ) As a rule, Equations (36) and (37) are given for the display of the analysis.
Step 4. Choice of the size of priority criterion (decision-making).
The person making the decision carries out the analysis of the results of calculations of Equation (34) and from the inequality of Equation (36) chooses the numerical size f q of the criterion of q ∈ K : For the chosen size of the criterion of f q it is necessary to define a vector of unknown X o . For this purpose, we carry out the subsequent calculations.
Step 5. Calculation of a relative assessment.
For the chosen size of the priority criterion of f q the relative assessment is calculated as: Step 6. Calculation of the coefficient of linear approximation. American Journal of Operations Research Assuming a linear nature of the change of criterion of f q (X) in Equation (36) and according to the relative assessment of ( ) q X λ in Equation (37), using standard methods of linear approximation we calculate the proportionality coefficient between Step 7. Calculation of coordinates of priority criterion with the size f q .
In accordance with Equation (38), the coordinates of the X q priority criterion point lie within the following limits: of a point of priority criterion with the size f q with the relative assessment of Equation (39): Step 8. Calculation of the main indicators of a point x q . For the obtained point x q , we calculate: Any point from Pareto's set S can be similarly calculated. Analysis of results. The calculated size of criterion ally not equal to the set f q . The error of the choice of ( ) fined by the error of linear approximation.

Methodology for Selecting Optimal Parameters of Engineering Systems under Conditions of Certainty and Uncertainty Based on Vector Optimization
As the object of the study we consider "Engineering Systems," for which the con- The methodology includes a number of stages. 1) Formation of the technical specification (source data) for numerical modeling and choice of optimum parameters of a system. The initial data is formed by the designer who projects the system.
2) Construction of mathematical and numerical models of the technical system in terms of certainty and uncertainty.
3) The solution of the vector problem of mathematical programming (VPMP) -a model of the Engineering system at equivalent criteria. 4) Creation of geometrical interpretation of results of the decision in a three-dimensional coordinate system in relative units.
5) The solution of a vector problem of mathematical programming-a model of theEngineering system at the given priority of the criterion.
6) Geometrical interpretation of results of the decision in a three-dimensional coordinate system in physical units.

Methodology for Selecting of the Optimal Parameters of Technical System under Conditions of Certainty and Uncertainty Based on Vector Optimization
The problem of numerical modeling and simulation of a technical system in which data on a certain set of functional characteristics (conditions of certainty), discrete values of characteristics (conditions of uncertainty) and restrictions imposed on the functioning of the technical system are known is considered [2] [5] [8] [16]. The numerical problem of modeling a technical system is considered with equivalent criteria and with a given criterion priority.
Stage 1. Formation of technical specifications (initial data) It is given. We're investigating the technical (engineering) system. The functioning of the technical system is determined by four parameters which represent the vector of controlled variables. The parameters of the technical system are set within the following limits: The operation of the technical system is determined by four characteristics (criteria): The uncertainty condition. For the first, second and third characteristic the results of experimental data are known: the values of the parameters and corresponding characteristics. Numerical values of parameters X and characteristics of y 1 (X), y 2 (X) and y 3 (X) are presented in Table 1.
Decision, assessment size of the first and the third characteristic (criterion) is possible to receive above: ; for the second and fourth characteristic is possible below: Note. The author developed in the Matlab system the software for the decision of vector problem of mathematical programming. The vector problem includes four variables (parameters of technical system): But for each new data (new system) the program is configured individually. In the soft- Table 1 they are provided as a part of { } at restriction Table 1. Numerical values of parameters and characteristics of the system.
x 1 x 2 x 3 x 4 y 1 (X) → max y 2 (X) → min y 3 (X) → max         (8) is non-empty and is a compact: These data are used further at creation of mathematical model of technical system.
Construction in the conditions of not certainty.
Construction in the conditions of uncertainty consists in use of the qualitative and quantitative descriptions of technical system received by the principle "input-output" in Table 1. Transformation of information (basic data of ( ) ( ) ( ) , , y X y X y X ) to a functional type of ( ) ( ) ( ) 1 2 3 , , f X f X f X is carried out by use of mathematical methods (the regression analysis). Basic data of Table 1 are created in MATLAB system in the form of a matrix: 1  2  3  4  2  3 , , , , , 1, 2, 3

{ }
As a result of calculations of coefficients of , 3 k A k = , we received the f 1 (X), f 2 (X) and f 3 (X) function:   , , f X f X f X slightly differ from experimental data.
The index of correlation and coefficients of determination are presented in the lower lines of Table 1. Results of the regression analysis (54)-(55) are used further at creation of mathematical model of technical system.
Construction of a numerical model of the system under certainty and uncertainty.
For creation of numerical model of the system we used: the functions received conditions of definiteness (9) and uncertainty (53), (54), (55); parametrical restrictions (50).We considered functions (49) and (53), (54), (55) as the criteria defining focus of functioning of the system. A set of criteria K = 4 included two criteria of ( ) ( ) 1 3 , max f X f X → and two ( ) ( ) 2 4 , min f X f X → . As a result model of functioning of the system was presented a vector problem of mathematical programming: The vector problem of mathematical programming (56)-(60) represents the model decision making under certainty and uncertainty in the aggregate. The solution of a vector problem (56)-(60) with was submitted as sequence of steps.
As a result of calculation for each criterion we received optimum points:  , , , X X X X in coordinates {x 1 , x 2 } are presented on Figure 1.
Step 2. We defined the worst unchangeable part of each criterion (anti-optimum):  *  *  *  1  2  3  4 , , , X X X X X = the relative assessment is equal to unit.
Other criteria there is much less than unit. It is required to find such point (parameters) at which relative estimates are closest to unit. The steps 4, 5 are directed on the solution of this problem.
Step 4. Creation of λ-problem is carried out in two stages: originally the maximine problem ofoptimization with the normalized criteria is under construction: which at the second stage was transformed to a standard problem of mathematical programming (λ-problem): at restrictions where the vector of unknown had dimension of N + 1: x x x x an optimum point-design data of the system, point X o is presented in Figure 1; and for other criteria is defined as an inequality: Stage 4. Creation of geometrical interpretation of results of the decision in a three-dimensional coordinate system in relative units.
In an admissible set of points of S formed by restrictions (74), optimum points , X X X X united in a contour, presented a set of points, optimum across Pareto, to o ⊂ S S, Figure 1.
Coordinates of these points, and also characteristics of technical system in relative units of ( ) ( ) ( ) ( ) Figure 2 in three measured space, where the third axis of λ-a relative assessment.
Discussion. Looking at a Figure 2, we can provide changes of all functions of ( ) ( ) ( ) ( ) in four measured space. We will consider, for example, an optimum point of  But we know that the relative assessment of ( ) f X function on the third step is equal to unit, we will designate it as ( ) Figure 2 by a red point. The difference between ( ) The decision maker is usually the system designer.
Step 1. We solve a vector problem with equivalent criteria. The algorithm of the decision is presented in Stage 3. Numerical results of the solution of the vector problem are given above.
Pareto's great number of o S S ⊂ lies between optimum points We will carry out the analysis of a great number of Pareto o S S ⊂ . For this purpose we will connect auxiliary points: * * * * *  Figure 1. These coordinates are shown in three measured space {x 1 , x 2 , λ} in Figure 2 where the third axis of λ-a relative assessment. Restrictions of a set of points, optimum across Pareto, in Figure 14 it is lowered to −0.5 (that restrictions were visible). This information is also a basis for further research of structure of a great number of Pareto. The person making decisions, as a rule, is the designer of the system. If results of the solution of a vector problem with equivalent criteria don't satisfy the person making the decision, then the choice of the optimal solution is carried out from any subset of points of 1  , , , Step 2. Choice of priority criterion of q ∈ K . From the theory (see Theorem 1) it is known that in an optimum point of X o always there are two most inconsistent criteria, q ∈ K and p ∈ K for which in relative units exact equality is carried out: S, and for the others it is carried out inequalities: In model of the system (54)-(58) and the corresponding λ-problem (67)- (71) such criteria are the first and third: We will show the λ 1 (X) and λ 3 (X) functions separately in Figure 3 from an Here all points and data about which it was told in Figure 2 are shown. As a rule, the criterion which the decision-maker would like to improve gets out of couple of contradictory criteria. Such criterion is called "priority criterion", we will designate it 3 q = ∈ K . This criterion is investigated in interaction with the first criterion of 1 q = ∈ K . We will allocate these two criteria from all set of the criteria 4 = K shown in Figure 3. On the display the message is given: q = input ('Enter priority criterion (number) of q = ')-Have entered: q = 3.
Step 3. Numerical limits of change of size of a priority of criterion of  In relative units the criterion of q = 2 changes in the following limits: These data it is analyzed.
Step 4. Choice of size of priority criterion. q ∈ K . (Decision-making).
The message is displayed: "Enter the size of priority criterion f q = "-we enter, for example, 80 q f = .
Step 5. Calculation of a relative assessment.
Step 6. Calculation of coefficient of linear approximation.
Assuming linear nature of change of criterion of f q (X) in (79) and according to a relative assessment of λ q (X) in (80), using standard methods of linear approximation, we will calculate proportionality coefficient between λ q (X o ), λ q , which we will call ρ : Step 7. Calculation of coordinates of priority criterion with the size f q .
{ } 1 2 , 3 q X x x q = = we will determine coordinates of a point of priority criterion with the size f q with a relative assessment (80): As a result of calculations we have received point coordinates: Step 8. Calculation of the main indicators of a point of X q . For the received X q point, we will calculate: all criteria in physical units all relative estimates of criteria the parameters of the technical system at a given priority criterion q = 2: { } in the two-dimensional coordinate system x 1 , x 2 on Figure 1, three-dimensional coordinate system x 1 , x 2 and λ in Figure 2. We also present these parameters in physical units for each technical system characteristic (criterion):   Figure 6. Similarly same characteristic in relative units of ( ) 2 X λ is shown in Figure 7.
Indicators of the second ( ) ( ) of characteristics of the system (are highlighted in red color) define transition errors from four-dimensional x x = to system of coordinates.

Y. Mashunin American Journal of Operations Research
The third characteristic of technical system ( ) 3 f X in x 1 , x 2 coordinates is shown in Figure 8. Similarly same characteristic in relative units of ( ) 3 X λ is shown in Figure 9.
Indicators of the third ( ) ( )      Collectively, the submitted version: • relative estimates of ( ) There is an optimum decision at equivalent criteria (characteristics), and procedure of receiving is adoption of the optimum decision at equivalent criteria (characteristics).
• point-X q ; characteristics of ( ) ( ) ( ) ( ) ( ) • relative estimates of ( ) ( ) ( ) ( ) ( ) • maximum λ o relative level such that There is an optimal solution at the set priority of the q-th criterion (characteristic) in relation to other criteria. Procedure of receiving a point is X q adoption of the optimal solution at the set priority of the second criterion.
Theory of vector optimization, methods of solution of the vector problems with equivalent criteria and given priority of criterion can choose any point from the set of points, optimum across Pareto, and show the optimality of this point.

Methodology for Selecting of the Optimal Parameters of Technological Process under Conditions of Certainty and Uncertainty Based on Vector Optimization
We study a technological process for which data are known about a certain set , , ,  Table 2.
In the decision taken, the evaluation value for the first, second and third characteristics (criteria) are desirable to get as high as possible: vary within the following limits: Note. The author has developed software for four parameters:  , , y X y X y X ) into a functional form ( ) 4 f X is carried out by using mathematical methods (the regression analysis). The initial data of Table   2 are formed as a matrix I in the MATLAB system: 1  2  1  2  3 , , , , , , 1,

{ }
For each experimental data set 1 3 , , k y k = the regression function is constructed using the least squares method ( ) , max f X f X → and K 2 = 2 at minimization: As a result, the model of the functioning of the technological process will be represented by the vector problem of mathematical programming:   The set of points that are Pareto optimal, S o represents the area of the set of points that lie between the points of the optimum * * *  *  1  2  3  4   ,  , , X X X X . We see that in this problem, the set of admissible points S and the set of points optimal in Pareto, S o , are equal to each other: S = S o .
Step 2. The worst unchanging part of each criterion is determined (an- X Step 3 , , , X X X X X = (diagonally) the relative estimate is equal to one. The remaining criteria are significantly less than one. It is required to find a point (parameters) at which the relative estimates are closest to unity. The solution of this problem is aimed at solving the λ-problem-step 4.
Step 4. The construction of the λ-problem is carried out in two stages: a maximin optimization problem with normalized criteria is initially constructed: At the second stage, the maximin problem (98) is transformed into a standard mathematical programming problem (λ-problem): at restrictions where the vector of unknowns has dimension of N + 1: Step 5. Solution of the λ-problem. For this purpose we use the function fmincon(…) [19]: In the vector problem of mathematical programming, as a rule, for two criteria, equality is satisfied: and for other criteria it is defined as inequality: , , , X X X X X = , which are shown in Figure 12, combined into a contour, represent the set of Pareto optimal points, o ⊂ S S . The coordinates of these points, as well as the characteristics of the technological process in relative units Figure 13 in the three-dimensional space x 1 , x 2 and λ, where the third axis is the relative estimate λ. The decision maker is usually the process designer.
Step 1. We solve a vector problem (89) Step 2. Choice of priority criterion of q ∈ K . From the theory (the Theorem 2) it is known that in an optimum point of X o there are always two most contradictory criteria: q ∈ K and v ∈ K for which in the relative unit's precise equality is carried out: and for the others it is carried out inequalities:  Figure 14.
As a rule, from a pair of the contradictory criteria, a criterion chosen by the decision maker would be improved. Such criterion is called "priority criterion", we will designate it 3 q = ∈ K . This criterion is investigated in interaction with the first criterion of 1 k = ∈ K .
On the display the message is given: q = input ('Enter priority criterion (number) of q =')-Entered: q = 3.
In the relative units the criterion of q = 3 changes in the following limits: This data is analyzed.
Step 4. Choice of size of the priority criterion of q ∈ K . (Decision-making). On the message: "Enter the size of priority criterion f q = "-we enter, the size of the characteristic Step 5. The relative assessment is calculated. Step 6. Let's calculate coefficient of the linear approximation Assuming the linear nature of the change of the criterion of f q (X) in (109) and according to the relative assessment of λ q , using standard linear approximation techniques, we will calculate the proportionality coefficient between λ q (X o ), λ q which we will call ρ : We represent these parameters in a two-dimensional x 1 , x 2 and three dimensional coordinate system x 1 , x 2 and λ in Figures 12-14, and also in physical units for each function f 1 (X), f 2 (X), f 3 (X), f 4 (X) on Figures 15-18 respectively.
The first characteristic f 1 (X) in physical units show in Figure 15.
The second characteristic f 2 (X) in physical units show in Figure 16.
The third characteristic f 3 (X) in physical units show in Figure 17.
The four characteristic f 4 (X) in physical units show in Figure 18.    In the aggregate, the first option presented: 100 y y y y + + + = .     It is required. Construction in the conditions of indeterminacy consists in use of qualitative, quantitative descriptions of material by the principle "input-output" to Table 3.
Using methods of the regression analysis, input data of The vector problem of mathematical programming (112)-(117) represents the model of optimal decision making, i.e. the choice of the optimal structure of the material in conditions of certainty and uncertainty in the total. Step 1. Decides problem (112)-(117) by each criterion separately, at the same time the function fmincon (…) of the Matlab system is used, the appeal to the function fmincon (…) is considered in [16]. As a result of calculation for each criterion we receive optimum points: The result the solution of a problem of non-linear programming (112), The result the solution of a problem of non-linear programming (114),    , , , X X X X . are shown in Figure 19, ..., Figure 22 respectively.
Step 3. Systems analysis of a set of points that are Pareto-optimal is per-    Step 2. Choice of priority criterion of q ∈ K .
From the theory (the Theorem 2 [4]) it is known that in an optimum point of X o there are always two most contradictory criteria: q ∈ K and v ∈ K for which in the relative units precise equality is carried out: Let's show the first and third in Figure 7 on the basis of which we will conduct their further research.
As a rule, from a pair of conflicting criteria, a criterion chosen by the decision maker would be improved. Such criterion is called "priority criterion", we will designate it 3 q = ∈ K . This criterion is investigated in interaction with the first criterion of 1 q = ∈ K . On the display the message is given: q = input ('Enter priority criterion (number) of q = ')-Entered: q = 3.
Step 3. Numerical limits of change of size of a priority of criterion of  These data it is analyzed.
Step 4. Choice of size of priority criterion of q ∈ K . (Decision-making). On the message: "Enter the size of priority criterion f q = "-we enter, the size of the characteristic defining structure of material: 138.2 q f = .
Step 5. The relative assessment is calculated.  100 summa X x x x x = + + + = .

Conclusions
The problem of developing mathematical methods of vector optimization and making optimal decisions based on them in a complex engineering system for a set of experimental data and functional characteristics is one of the most important tasks of system analysis and design.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.