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:
where
is the vector optimality criterion;
—the subset of criteria to be maximized;
—the subset of criteria to be minimized;
—the
technical and economic characteristic of the product, for which the optimal value is sought,
;
,
—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:
which allows for a known set of admissible alternative solutions
, a vector of objective functions
and the preferences
of the DM to determine the optimal solution,
,
.
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:
Since , then .
Since , then .
As seen in (3) and (4), represents the relative deviation from the optimal value of the objective function for the solution .
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
,
, is assigned a weight coefficient (significance coefficient)
,
. The coefficient
is a real positive number. This number
determines the relative “weight”, “importance”, “value” of the
criterion in relation to the others. The weight vector,
is a
dimensional vector defined in the unit hypercube:
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
is found for which:
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.
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.