Multi-Criteria Choosing of Material for Manufacturing a Pressure Relief Valve ^†

Abstract

The paper is dedicated to the multi-criteria choice of an optimal variant of a pressure relief valve with a nominal pressure of 3 bar, manufactured from four different materials. The paper includes the usage of a number of existing methods that are combined in an appropriate way to solve a specific practical problem, and a sequence of steps for their effective application is formulated. The optimization is defined and analyzed, and a seven-stage solution approach is developed. A list of requirements for the product is composed. The requirements are organized into objective groups, and an objective tree is developed. Metrics for measuring the requirements are defined. The “House of Quality” tool is used for correlating the metrics and requirements. Based on these correlations, criteria are selected for the evaluation of alternative variants. A mathematical model of the problem is built, and the evaluation criteria are defined in terms of concrete values for the variants, transforming the criteria into objective functions. A normalization method for the objective functions is selected and a principle of optimality is chosen. Using a known method for defining objective functions’ priorities, the weighting factors for different priority scenarios are obtained. The results of the optimization are shown for the different scenarios in relation to the different priorities (importances) of the selected criteria. Seven optimization problems are solved, and three different solutions are found. The solutions are graphically represented on a radar chart. All solutions found are optimal according to the selected criterion for optimality and calculated weight vectors. The final solution, chosen among the optimal ones found, is selected on the basis of additional decision makers’ considerations.

1. Introduction

Modern design and production technologies, supply chains, storage, etc., enable manufacturers to satisfy to the maximum extent the diverse requirements (functional, economic, operational, ecological, ergonomic, aesthetic, etc.) of consumers.

One of the main prerequisites for satisfying the requirements of the users is the choice of the optimal variant of the products. A number of methods for solving this problem are known [1,2,3,4,5,6,7], but it is necessary to analyze and determine the sequence of their application for the successful solution of a specific problem.

This study is focused on alternative products that can be produced from different materials with the required performance and quality but with fewer production operations and less costs in comparison with the existing ones. The subject of consideration is a pressure relief valve with a nominal pressure of 3 bars (Figure 1).

Valves made of four different materials were analyzed: CuZn40Pb₂ (valve 1), Styrene Acrylonitrile—SAN (valve 2), Polyamide PA66 GF30 (valve 3), and Polymethyl Methacrylate PMMA (valve 4). Design and process documentation have been developed for the four valves, and simulation and experimental performance studies have been conducted.

The purpose of the paper is to present the results of a multi-criteria choice of an optimal variant for the material from which the product “pressure relief valve” is produced under different scenarios related to determining different priorities for selected evaluation criteria.

2. Definition, Analysis and Main Stages for Solving the Problem

The choice of the optimal variant of the product “pressure relief valve” is related to solving the problem:

For a developed set of alternative variants of the product “pressure relieve valve”, a variant should be chosen that best satisfies a predetermined set of requirements regarding the values of its technical and economic characteristics and their relative importance (priority).

Analyzing the defined problem points out some of its typical characteristics and issues that must be taken into consideration when formalizing it and selecting a suitable solution method:

• The defined problem is a multi-criteria optimization problem, since selecting a variant is made after a complex evaluation of the competitive variants according to a set of technical and economic characteristics of the product, which, depending on the specific requirements and goals of the solved problem, must have optimal (minimum or maximum) values. These characteristics form the system of criteria, which determines to a significant extent the properties of the obtained solution. Therefore, determining the set of criteria by which the alternative variants will be evaluated is an important and responsible stage in choosing the optimal one.
• The solution to multi-criteria optimization problems belongs to a set of compromise solutions, according to the optimality principle proposed by Pareto [8]. This set has the property that the solutions in it cannot be improved simultaneously by all criteria. Choosing the “best” alternative from a set of possible trade-offs requires incorporating information about the decision maker’s preferences, as there is no single optimal solution.
• In the considered problem, the possible criteria for evaluating the alternative variants of the product have different physical meanings and are measured in different dimensions and scales—for example: stress—MPa; temperature—°C; price—USD; aging—h; expansion—mm; impact—kJ/m²; hardness—MPa; production time—h; mass—kg, etc.
• The analysis of the data for the considered problem typically shows the existence of a contradiction between some of the possible criteria when choosing an optimal variant. Conflicting criteria are present when different criteria cannot be optimized simultaneously, and trade-offs must be made when selecting a variant.
• Solving multi-criteria optimization problems is associated with a number of specific issues that make their solution difficult. One of the main ones is the choice of optimality principle. It gives an answer to the main question—in what sense does the optimal solution surpass all other admissible solutions? For example, in the conditions of certainty, some of the following principles are applied—of the fair compensation of the absolute values of the normalized criteria, of the fair relative compensation of the criteria, of uniformity (minimax or maximin), of the main criterion, of the “ideal” point, of equality, etc., and in the conditions of risk and uncertainty—of Bayes, Wald, Hurwitz, Laplace, and Savage. Therefore, when solving the problem, it is necessary to choose an appropriate optimality principle.
• Individual criteria can have different relative “importance” (value, weight, priority, significance) when choosing an optimal variant, i.e., one or more criteria can have priority over others. By assigning greater weights to the preferred criteria, the decision maker (DM) can reflect his preferences and priorities, which helps resolve the conflicting criteria issue.

Defining the importance and worth of the evaluations for the individual criteria must be undertaken in consideration of the meaning and content of the defined goals for the designed product, while both objective and subjective information from the DM can be used. Dozens of priority determination methods are described in the specialized literature [9,10,11,12,13,14,15,16,17]. Therefore, to solve the problem, it is necessary to choose an appropriate method that mathematically describes the different importance of individual private criteria.

The problem is solved in the following sequence:

Stage 1. Formulation of the problem.

Stage 2. Selection of a system of criteria for evaluation of alternative variants.

Stage 3. Building a mathematical model of the problem.

Stage 4. Normalization of the criteria.

Stage 5. Determining the priority of the criteria.

Stage 6. Choice of optimality principle.

Stage 7. Solving the problem. Sensitivity analysis.

Following is a short discussion of the main stages.

Stage 1. Formulation of the problem.

This is one of the most responsible stages, and special attention should be paid to it since its results are the basis for the subsequent solving of the problem. In this stage, the requirements for the designed product are specified, and an objective tree is built.

Stage 2. Selection of a system of criteria for evaluation of alternative variants.

In this stage, the set of criteria for evaluating the alternative variants is defined. The criteria are selected from the requirements list for the designed object, and the objectives determined are in Stage 1.

The system of criteria can be represented as a set of objective functions, for some of which a maximum is sought and for the others a minimum:

$$optF(x)=\left\{\begin{array}{c}\mathrm{m}\mathrm{a}\mathrm{x}{F}_1\left(x\right)=\left\{f_k\left(x\right);k\in K_1\right\},\\ \mathrm{m}\mathrm{i}\mathrm{n}{F}_2\left(x\right)=\left\{f_k\left(x\right);k\in K_2\right\}\end{array}\right\}, K_1\cup K_2=K,$$

(1)

where $F\left(x\right)$ is the vector optimality criterion; $F_1\left(x\right)$—the subset of criteria to be maximized; $F_2\left(x\right)$—the subset of criteria to be minimized; $f_k\left(x\right)$—the $k^{-th}$ technical and economic characteristic of the product, for which the optimal value is sought, $k\in K$; $K_1$, $K_2$—respectively, sets of indices of objective functions for which a maximum or minimum value is sought.

Stage 3. Building a mathematical model of the problem.

With selected criteria, a mathematical model of the problem is built at this stage. In the general case, it can be represented as a transformation of the form:

$$\varphi :\left\{X,\hspace{0.33em}F\left(x\right),\hspace{0.33em}\alpha \right\}\to x^{*},$$

(2)

which allows for a known set of admissible alternative solutions $X$, a vector of objective functions $F\left(x\right)$ and the preferences $\alpha$ of the DM to determine the optimal solution, $x^{*}$, $x^{*}\in X$.

Stage 4. Normalization of the criteria.

Since in the considered problem the individual criteria are measured in different dimensions and scales, this leads to the impossibility of objectively comparing the quality of the alternative variants against each criterion. To overcome this issue, the criteria are translated to a single measure (measurement scale), and their scales are standardized.

The analysis of known methods for normalizing the objective functions, shows that for the present work, it is appropriate to use the so-called full normalization in one of the following ways, depending on the type of extremum:

• For objective functions to be maximized

$${\overline{w}}_k\left(x\right)={\displaystyle \frac{f_k^{*}-f_k\left(x\right)}{f_k^{*}-f_{kmin}}}, k\in K_1.$$

(3)

Since $f_{kmin}\le f_k\left(x\right)\le f_k^{*}$, then $1\ge {\overline{w}}_k\left(x\right)\ge 0$.

• For objective functions to be minimized

$${\overline{w}}_k\left(x\right)={\displaystyle \frac{f_k\left(x\right)-f_k^{*}}{f_{kmax}-f_k^{*}}}, k\in K_2.$$

(4)

Since $f_k^{*}\le f_k\left(x\right)\le f_{kmax}$, then $0\le {\overline{w}}_k\left(x\right)\le 1$.

As seen in (3) and (4), ${\overline{w}}_k\left(x\right)$ represents the relative deviation from the optimal value $f_k^{*}$ of the objective function $f_k\left(x\right)$ for the solution $x$.

Stage 5. Determining the priority of the criteria.

In this stage, the DM determines the relative importance of the criteria. Various ways and dozens of methods for determining the priority of criteria are described in the specialized literature. For the purposes of the present development, the use of a weight vector (vector of weight coefficients) is proposed.

In this way, to consider the priority, each criterion $f_k$, $k\in K$, is assigned a weight coefficient (significance coefficient) ${\alpha }_k$, $k\in K$. The coefficient ${\alpha }_k$ is a real positive number. This number ${\alpha }_k$ determines the relative “weight”, “importance”, “value” of the $k^{-th}$ criterion in relation to the others. The weight vector,

$$\alpha =\left\{{\alpha }_k\right\}, k\in K,$$

(5)

is a $k$ dimensional vector defined in the unit hypercube:

$$\alpha =\{{\alpha }_k:{\alpha }_k\in [0,\hspace{0.33em}1],{\sum }_{k\in K}{\alpha }_k=1\}.$$

(6)

After an analysis of known methods for determining the elements of the weight vector, it is suggested to use the method of binary comparisons of Vojchinskiy and Janson [18]. An algorithm for the method was developed in [19], which was implemented programmatically in [20].

Stage 6. Choice of optimality principle.

The principle of optimality defines the properties of the optimal solution and answers the main question—in what sense it is superior to all other possible solutions—and determines the rule for its search. Various optimality principles have been proposed in the specialized literature, which reflect different approaches to solving multi-criteria optimization problems.

Considering the importance of the problem for choosing an optimal variant and observing the research and recommendations of a number of researchers, it is proposed to use the principle of the guaranteed result (minimax or maximin).

According to this principle, the multi-criteria optimization problem (2) with a given priority is considered solved if a solution $x^{*}$ is found for which:

$${\overline{w}}^{*}=\underset{x\in X}{\min }\hspace{0.33em}\underset{k\in K}{\max }{\overline{w}}_k(x).{\alpha }_k.$$

(7)

Stage 7. Solving the problem. Sensitivity analysis.

At this stage, on the basis of the input data and with the help of the constructed mathematical model and the chosen principle of optimality, the problem is solved under equal criteria, i.e., when they are of equal importance. An analysis of the obtained results follows. In cases where they do not satisfy the DM, the problem is solved by assigning a different priority to the criteria or by using another optimality principle. A sensitivity analysis is performed to assess the impact of changes in criteria weights on the decision results. Decision makers can evaluate different scenarios and examine how the ranking or preferences for alternatives change when the importance of particular criteria changes. This analysis helps to assess the reliability of the decisions and their sensitivity to changes in the importance of the criteria. Decision makers can explore the trade-offs and make an informed choice based on their preferences and understanding of the importance of the criteria.

At this stage, it is recommended to develop an appropriate visual (graphical) representation of the results of the problem’s study, e.g., through radar charts depicting the trade-offs between the particular criteria. These diagrams help decision makers navigate the complexity of the problem and balance competing objectives in line with the preferences and priorities of the interested parties.

3. Solving the Problem of Choosing the Optimal Variant of the Product “Pressure Relief Valve”

The choosing is made according to the sequence discussed in Section 2.

3.1. Stage 1. Formulation of the Problem

Based on market research and conversations with customers and manufacturers, a list of requirements for the product was formulated and presented in

Table 1. Requirements List for the product “pressure relief valve”.

No.	Requirement	Priority
1	Lightweight	6
2	Easy to produce	6
3	Low cost	5
4	Durable	6
5	Little change in geometry with temperature change	4
6	Wear resistance	4
7	Sufficient strength	5
8	Resistant to dynamic loading	3
9	High temperature operation	5

.

In Table 1, the priority for individual requirements is also determined. A six-point scale was used, where: 6—highest priority; 5—high priority; 4—relatively high priority; 3—medium high priority; 2—low priority; 1—lowest priority.

In

Table 2. Requirements systematized in objective groups.

Objectives	Costs	Design	Production	Reliability
Requirements	Low cost	Lightweight	Easy to produce	Durable
		Sufficient strength		Little change in geometry with temperature change
		Resistant to dynamic loading		Wear resistance
				High temperature operation

, requirements are organized into objectives. The identified goals were used to create an objective tree (Figure 2). The root (first level) of the tree is the product being developed. The trunk (the second level) is shaped by the goals, and the branches (the third level) are the requirements. The next levels are the leaves, and they answer the question, “What does it mean?”. They clarify the requirements.

3.2. Stage 2. Selection of a System of Criteria for Evaluation of Alternative Variants

Based on the requirements list and the objectives tree, a list of metrics is developed (

Table 3. List of metrics.

No.	Metric	Unit	Short Record
1	Mass	g	M
2	Production time	s	PT
3	Cost	USD	P
4	Aging	h	A
5	Temperature expansion	mm	TE
6	Permissible stress at critical point	MPa	CS
7	Hardness	MPa	H
8	Permissible tensile stress	MPa	TS
9	Impact	kJ/m2	I
10	Operating temperature	deg	WT

). Metrics answer the question “How is a requirement measured?” and clarify how requirements are quantified. The list from

Table 4. House of quality for correlating requirements with metrics for the product “pressure relief valve”.

		Priority		Specifications
			Mass, g	Production Time, s	Cost, USD	Aging, h	Temperature Expansion, mm	Permissible Tensile Stress, MPa	Hardness, MPa	Permissible Tensile Stress, MPa	Impact, kJ/m2	Operating Temperature, deg
Requirements	Lightweight	6	9	1	3
	Easy to produce	6	3	9	1
	Low cost	5		3	9
	Durable	6			3	9		3	3	3	3	3
	Little change in geometry with temperature change	4	1				9					3
	Wear resistance	4				1			9
	Sufficient strength	5						9	1	9	3
	Resistant to dynamic loading	3						3	3	1	9
	High temperature operation	5				3	3					9
Absolute scores			76	75	87	73	51	72	68	66	60	75
Relative scores			8.7	8.6	10.0	8.4	5.9	8.3	7.8	7.6	6.9	8.6

also contains units of measurement for the specified metrics and a short record for more convenient writing of the relevant metric.

The “House of Quality” tool [2] was used to link requirements and metrics and determine a tuple—an arrangement of metrics in order of importance according to their relevance to requirements. The estimates are presented in Table 4. Requirements are written in the rows and specifications in the columns. The strength of the relationship between requirement and specification is recorded in the corresponding row and column intersection cell. The set priority for the requirements is also reported. The absolute rating is obtained according to the following dependence:

$${AR}_j={\displaystyle {\sum }_{i=1}^N}{p}_i{c}_{ij},$$

(8)

where ${AR}_j$ is the absolute importance score of the $j^{-th}$ metric; $j$—the index of the column (metric) for which the score is calculated, $j\in \left[1;M\right]$; $M$—the number of defined metrics; $i$—the row index (requirement), $i\in \left[1;N\right]$; $N$—the number of formulated requirements; $p_i$—the priority of the $i^{-th}$ requirement; $c_{ij}$—the correlation score (strength of relationship) between requirement $i$ and metric $j$.

The assessment of the correlation (strength of the relationship) $c_{ij}$ between a requirement and a metric is determined according to the following scale: 9—strong correlation; 3—medium correlation; 1—weak correlation; empty cell (0)—no correlation.

Relative scores are calculated according to the following relationship:

$${RR}_j={\displaystyle \frac{{AR}_j\max S}{\underset{j}{\max }{AR}_j}},$$

(9)

where ${RR}_j$ is the relative rating of the $j^{-th}$ metric; $\max S$—the maximum score on a selected scale, to which the absolute scores are reduced (for Table 4, the selected scale is from 1 to 10, and the maximum score on it is 10); $\underset{j}{\max }{AR}_j$—the maximum absolute score among those calculated for all metrics.

Calculating the relative score is convenient because it brings the absolute scores to a scale with fewer items, so it is easier to rank metrics. It can be seen that the resulting scores order the metrics into the following tuple:

$$P\succ M\succ PT\approx WT\succ A\succ CS\succ H\succ TS\succ I\succ TE.$$

(10)

Due to the limited number of possible alternative variants and in order to apply the requirement for minimality (economy) of the number of evaluation criteria, it is decided not to use all metrics as evaluation criteria. Thus, the first six metrics from the tuple (10) were selected to evaluate the alternative variants.

3.3. Stage 3. Building a Mathematical Model of the Problem

The following problem is assigned:

For a given set of alternative material options for producing the product “pressure relief valve”, determine the optimal variant so that:

$$\min P\left(x\right),\min M\left(x\right), \min PT\left(x\right), \max WT\left(x\right),\max A\left(x\right),\max CS\left(x\right).$$

(11)

where $P\left(x\right)$—cost, USD; $M\left(x\right)$—mass, g; $PT\left(x\right)$—production time, s; $WT\left(x\right)$—operating temperature, deg; $A\left(x\right)$—aging, h; $CS\left(x\right)$—permissible stress at critical point, MPa.

The values of the techno-economic characteristics, which are used as criteria for evaluating the variants, are given in

Table 5. Values of techno-economic characteristics of the variants.

Criteria↓ Variants→	CuZn40Pb2 x1	Styrene Acrylonitrile (SAN) x2	Polyamide (PA66 GF30) x3	Polymethyl Methacrylate (PMMA) x4
Mass, g	963	205	221	215
Production time, s	280	49	55	52
Cost, USD	185.49	145.71	147.61	146.57
Aging, h	∞	1008	9023	2000
Permissible stress at critical point, MPa	424	43	33	48
Operating temperature, deg	300	100	261	103

.

In Table 5, the evaluation criteria are plotted in the first column, and the alternative options are entered in the first row. Additionally, notations are introduced for variants as follows: CuZn40Pb₂ = x₁, Styrene Acrylonitrile (SAN) = x₂, Polyamide 6.6 (PA66 GF30) = x₃, Polymethyl methacrylate (PMMA) = x₄.

3.4. Stage 4. Normalization of the Criteria

To normalize the functions, applying dependencies (3) and (4), it is necessary to determine the extrema of the objective functions. This is done in

Table 6. Extrema of objective functions.

Criteria↓ Extrema→	min	max
Mass, g	205	963
Production time, s	49	280
Cost, USD	145.71	185.49
Aging, h	1008	90230
Permissible stress at critical point, MPa	33	424
Operating temperature, deg	100	300

. When compiling the table, the aging criterion value for variant x₁ is replaced by the tenfold increased value of the criterion for variant x₃. This is necessary because otherwise normalization will result in undefined values due to the substitution by infinity in dependencies (3) and (4).

Applying dependencies (3) and (4) to the information from Table 5 and Table 6, the normalized values for the criteria are calculated (

Table 7. Normalized values of evaluation criteria.

Criteria↓ Variants→	CuZn40Pb2 x1	Styrene Acrylonitrile (SAN) x2	Polyamide (PA66 GF30) x3	Polymethyl Methacrylate (PMMA) x4
Mass, g	1.000	0.000	0.021	0.013
Production time, s	1.000	0.000	0.026	0.013
Cost, USD	1.000	0.000	0.048	0.022
Aging, h	0.000	1.000	0.910	0.989
Permissible stress at critical point, MPa	0.000	0.974	1.000	0.962
Operating temperature, deg	0.000	1.000	0.195	0.985

).

3.5. Stage 5. Determining the Priority of the Criteria

The problem will be solved by finding a compromise solution with equal priority for the objective functions, and then solutions with increased priority for each of the objective functions will be explored. The weighting factors are determined by the chosen method. The calculated values are presented in

Table 8. Weighting coefficients when setting priority for individual objective functions.

Problem	Goal	Criteria
		M	PT	P	A	CS	WT
1	Compromise solution	Equal priority
2	Priority M	0.248	0.123	0.123	0.123	0.173	0.208
3	Priority PT	0.145	0.251	0.175	0.099	0.210	0.120
4	Priority P	0.145	0.175	0.251	0.099	0.210	0.120
5	Priority A	0.130	0.134	0.134	0.243	0.174	0.184
6	Priority CS	0.130	0.163	0.136	0.147	0.242	0.183
7	Priority WT	0.099	0.193	0.193	0.119	0.144	0.250

.

3.6. Stage 6. Choice of Optimality Principle

The optimality principle chosen is the minimax criterion (7), which is applied to the normalized values of the objective functions, i.e., a solution with a minimum deviation from the optimum is sought when evaluating against the maximum deviations of the variants from the optimum.

3.7. Stage 7. Solving the Problem. Sensitivity Analysis

Seven problems are solved. A trade-off problem with equal priority objective functions and six problems with a priority set according to Table 8, rows 2 to 7. The results are presented in

Table 9. Solutions to the problems.

(Problem 1) Compromise Solution
Variants	Evaluation (max)	Optimum (min)
CuZn40Pb2 x1	1.000
Styrene Acrylonitrile (SAN) x2	1.000
Polyamide (PA66 GF30) x3	1.000
Polymethyl Methacrylate (PMMA) x4	0.989	0.989
(Problem 2) Priority for the criterion mass M
Variants	Evaluation (max)	Optimum (min)
CuZn40Pb2 x1	0.248
Styrene Acrylonitrile (SAN) x2	0.208
Polyamide (PA66 GF30) x3	0.173	0.173
Polymethyl Methacrylate (PMMA) x4	0.205
(Problem 3) Priority for the criterion production time PT
Variants	Evaluation (max)	Optimum (min)
CuZn40Pb2 x1	0.251
Styrene Acrylonitrile (SAN) x2	0.205
Polyamide (PA66 GF30) x3	0.210
Polymethyl Methacrylate (PMMA) x4	0.202	0.202
(Problem 4) Priority for the criterion cost P
Variants	Evaluation (max)	Optimum (min)
CuZn40Pb2 x1	0.251
Styrene Acrylonitrile (SAN) x2	0.205
Polyamide (PA66 GF30) x3	0.210
Polymethyl Methacrylate (PMMA) x4	0.202	0.202
(Problem 5) Priority for the criterion aging A
Variants	Evaluation (max)	Optimum (min)
CuZn40Pb2 x1	0.134	0.134
Styrene Acrylonitrile (SAN) x2	0.243
Polyamide (PA66 GF30) x3	0.221
Polymethyl Methacrylate (PMMA) x4	0.240
(Problem 6) Priority for the criterion permissible stress at critical point CS
Variants	Evaluation (max)	Optimum (min)
CuZn40Pb2 x1	0.163	0.163
Styrene Acrylonitrile (SAN) x2	0.235
Polyamide (PA66 GF30) x3	0.242
Polymethyl Methacrylate (PMMA) x4	0.232
(Problem 7) Priority for the criterion operating temperature WT
Variants	Evaluation (max)	Optimum (min)
CuZn40Pb2 x1	0.193
Styrene Acrylonitrile (SAN) x2	0.250
Polyamide (PA66 GF30) x3	0.144	0.144
Polymethyl Methacrylate (PMMA) x4	0.246

.

With equal objective functions, production time priority, and cost priority, option x₄ Polymethyl methacrylate (PMMA) is optimal. With mass priority and operating temperature priority, variant x₃ Polyamide 6.6 (PA66 GF30) is optimal. With priority of aging and priority of permissible stress at critical point, variant x₁ CuZn40Pb₂ is optimal.

Summarized and graphically, the obtained solutions to problems 2 to 7 are presented in Figure 3. In this way of representation, the graph can be likened to a target; the closer to its center a given solution is, the closer it is to the ideal (the center of the target). It can be seen that solutions x₁ CuZn40Pb₂, x₃ Polyamide 6.6 (PA66 GF30), and x₄ Polymethyl methacrylate (PMMA) are optima when solving problems 5 and 6, problems 2 and 7, and problems 3 and 4, respectively. In addition, with equal priority target functions, a compromise solution is x₄ Polymethyl methacrylate (PMMA).

In Figure 4, only the optimal solutions to problems 1–7 are presented. In this representation, each polygon represents an optimal solution, and the target rays are the evaluation criteria. Again, the closer the vertices of the polygon are to the center, the closer the solution is to the optimum for the corresponding criterion. Due to the coincidence of optima for some problems, only three polygons are visible in Figure 4. It can be seen that the optimum for problems 1, 3, and 4 (PMMA) is close to the center in terms of mass, production time, and cost; the optimum for problems 2 and 7 (PA66 GF30) is close to the center in terms of operating temperature, mass, production time, and cost; and the optimum for problems 5 and 6 is close to the center in terms of operating temperature, permissible stress at the critical point, and aging criteria.

4. Conclusions

In this paper, the choice of an optimal variant of material for the production of a pressure relief valve is presented. After researching the market and technological processes for production, four possible variants are proposed, for which ten techno-economic characteristics have been determined. They are determined based on the requirements for the product related to costs, production processes, construction, and reliability. A statement of the problem is made, and characteristic features are indicated. Based on this, a solution sequence consisting of seven stages is proposed. The proposed stages are applied to the problem of choosing the optimal variant of material for the production of a pressure relief valve. A list of requirements for the product is formulated, and they are grouped and clarified through an objective tree. Metrics have been defined for the specified requirements by which they can be quantified. The House of Quality tool is applied to determine the ranking of the metrics by importance, with the aim of reducing the evaluation criteria due to the limited number of variants (four variants, ten criteria). A mathematical model is built, and the values of the objective functions are determined. Due to the different measurement units, a normalization of the objective functions is performed. The priority of the criteria is determined, and seven problems are formulated that offer a complete study of the problem. An optimality principle is chosen, which is often applied to problems where guaranteed success is sought even with degraded results. The formulated problems are solved, and an analysis of the obtained solutions is carried out. The solutions found are presented to the DM for subsequent decision making.

In the paper, a number of existing methods are used, and they are combined in an appropriate way to solve a specific practical problem, in which a sequence of steps for their effective application is formulated.

The proposed combination of methods and application sequence is sufficiently general and flexible, which allows its application in problems of choosing an optimal variant of a different nature.

Subject to future development is the automation of the process through software applications.