Multiobjective Optimization Method and Application of Tolerance Allocation for the Steam Turbine Based on Cooperative Game Theory

Abstract

Aiming at the optimization problem of multiple objectives with contradictions and conflicts in the process of allocating the tolerance for complex products, taking advantage of the features of coordinating, and balancing contradictions and conflicts of cooperative game theory, this paper uses cooperative game theory to solve the multiobjective optimization problem of tolerance allocation. The quality requirements and cost requirements of assembly products are used as the game decision parties, and the fuzzy clustering method is used to group the design variables of tolerance allocation problem of the steam turbine to form the strategic space of game parties. Take the quality level and cost level of the assembly product as the optimization goals, complete the calculation of the utilities of the two game parties, and establish the multiobjective optimization model of tolerance allocation based on cooperative game theory. Finally, the Shapley value method based on cooperative game theory, the Nash equilibrium method based on noncooperative game theory, and the traditional single-objective optimization method with the quality as the constraint and the cost as the optimization objective are used to solve the tolerance allocation problem of steam turbine. The solution results show that the method of cooperative game realized the balance, coordination, and comprehensive optimization of the quality and cost from the perspective of collective interests, overcame the shortcomings of the traditional single-objective optimization method, and obtained better result than the Nash equilibrium method.

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1. Introduction

Tolerance allocation is a core issue of tolerance design theory; it mainly studies how to scientifically and reasonably allocate the design tolerance of the closed ring to each component ring [1] and realizes the balanced coordination and comprehensive optimization of some indicators such as processing cost of the product, assembly quality, and assembly robustness under the premise of ensuring certain assembly success rate [1]. The key issue of tolerance allocation is how to establish a balance among its contradictory and conflicting goals, especially between the quality and cost, consider the mutual influence and conflict between the two goals, and finally get a scientific, reasonable, and balanced and coordinated tolerance allocation optimization program. At present, the widely used tolerance allocation method is single-objective optimization method with quality as the constraint and cost as the optimization goal [2, 3], or both quality and cost are considered, and a multiobjective comprehensive weighted evaluation function is constructed to complete the establishment of the tolerance allocation optimization model. In addition, various analytical methods or some intelligent optimization algorithms are adopted to solve [4].

Over these years, many optimization algorithms have been developed, divided into deterministic algorithms and stochastic algorithms [5]. Traditional optimization algorithms are usually deterministic, because they run multiple times to output the same results. Therefore, a majority of mathematical programming methods are based on the gradient of the objective function and constraint conditions [5]. Many researchers have proved the applicability of deterministic optimization methods, such as linear programming [6] and nonlinear programming [7], which are used to solve the most basic tolerance allocation problem [8]. Later, due to the advantages that the stochastic algorithms are not restricted by the gradient information of the objective function, compared with the deterministic algorithms, they can handle more complex tolerance allocation models, so they have been more widely used. Among the stochastic algorithms, there are some more common algorithms, such as simulated annealing (SA) [9], genetic algorithm (GA) [10], particle swarm optimization (PSO) [11], and ant colony algorithm; besides, some less common algorithms are also used for tolerance-cost optimization, such as the imperial competition algorithm [12], self-organizing migration algorithm [13], bat algorithm [14], artificial bee colony algorithm [15], and cuckoo search [16]. In addition, the application of hybrid algorithm in the field of tolerance allocation is also studied. Hybrid algorithm mainly combines stochastic and deterministic or another stochastic optimization algorithm to improve the optimization effect [17]. Although the above intelligent algorithms have certain advantages, sometimes the calculation efficiency is not high and the construction of the evaluation function is difficult. Moreover, the traditional tolerance allocation method which takes quality as the constraint and takes the cost as the optimal goal is difficult to find a scientific and reasonable equilibrium solution among multiple goals. The tolerance allocation method, which relies on experience to determine the weight of quality and cost and form a comprehensive evaluation function of quality and cost, cannot scientifically measure the mutual influence and conflict among different goals.

The abovementioned algorithms can effectively solve single-objective or multiobjective optimization problems, such as particle swarm optimization algorithm of solving multiobjective optimization problems, but it is difficult to deal with the optimization problem of multiple objectives with contradictions and conflicts. The essence of game theory solving the problem of multiobjective optimization is to find a balanced solution among multiple goals with contradictions and conflicts, and to achieve the balanced coordination and comprehensive optimization of multiple goals. Therefore, game theory has been well applied in all walks of life. Zhu and Başar studied game theory and realized the tradeoff between security and usability of computing systems [18]. In the book of game theory for wireless communications and networking, game theory appeared as a new tool for the wireless engineer to tackle spectrum sharing, power control, resource allocation, transmission strategy, and security and network etiquette issues [19]. It can be seen that game theory has achieved good optimization effects in the allocation of resources and the balance of objectives.

The study found that the tolerance allocation optimization problem is a multiobjective optimization problem with contradictions and conflicts among multiple objectives. Therefore, in order to meet the continuous increase need of users for the quality and the continuous decrease need of enterprises for the manufacturing cost of complex products, this paper introduces the cooperative game theory and makes the most of its characteristics of coordination, balancing conflicts and contradictions to establish a set of tolerance allocation multiobjective optimization models based on cooperative game theory. By solving and analyzing the game utility matrix to obtain the tolerance allocation optimization program, we can realize the comprehensive optimization and balanced coordination of product quality and cost and achieve the multiobjective optimization goals of high quality and low cost for complex products.

2. Cooperative Game Decision-Making Method of Solving the Multiobjective Optimization Problem of Steam Turbine Tolerance Allocation

The steam turbine is a typical representative of complex products and major equipment. It not only has a large number of parts, but also has a large size and weight. More importantly, it has very strict requirements on the accuracy of the flow gap between the moving and static parts. If the gap is too large, it is not conducive to the efficiency of power generation of steam turbine; however, if the gap is too small, it is not conducive to the safe operation of the steam turbine. Therefore, the steam turbine is not only a complex and important equipment, but also a country’s important equipment, and its assembly quality and manufacturing cost cannot be ignored. In addition to ensuring the flow gap of the steam turbine, respectively, by certain processing and assembly processes, more importantly, it is also necessary to optimize the tolerance allocation during the initial tolerance design process, which can ensure that the flow gap fluctuates within the design tolerance range and achieve the improvement of the flow gap assembly qualification rate, the enhancement of the assembly robustness, and the optimization of the manufacturing cost. As a core technology of tolerance design theory, tolerance distribution determines the assembly quality and manufacturing cost of steam turbines fundamentally. When assigning tolerances for steam turbines, quality and cost are two indicators that cannot be ignored; they are also two contradictory and conflicting goals. Therefore, the essence of the tolerance distribution optimization problem of steam turbines is the multiobjective optimization problem with contradictions and conflicts in the tolerance distribution of complex products. Game theory has the advantages of reconciling conflicts and contradictions, as well as flexible and convenient modeling characteristics; it is suitable for solving multiobjective optimization problems in the engineering field where contradictions and conflicts exist. Therefore, this paper introduces game theory and establishes a set of multiobjective optimization models of steam turbine tolerance allocation based on cooperative game theory.

2.1. Construction of the Cooperative Game Model of Steam Turbine Tolerance Distribution

2.1.1. Modeling Process

The flow gap is an important assembly quality indicator of the steam turbine, and it is also the final quality indicator to be guaranteed during the assembly process of the steam turbine. Besides, the flow gap is also a terminal closed ring formed by manufacturing deviation transmission, coupling, and accumulation. The remaining dimensions that form the closed ring (the flow gap) of the steam turbine are the component ring dimensions. The tolerances of these component ring dimensions are the design variables involved in the tolerance distribution of the steam turbine. The optimization goal of the tolerance distribution is the assembly quality and manufacturing cost of the steam turbine. The design tolerance of the flow gap is allocated to each component ring constituting the flow gap of the steam turbine, and a multiobjective optimization model of steam turbine tolerance allocation that takes into account quality and cost is established, as shown in the following equation:$$\begin{array}{}\text{(1)}& \begin{array}{cc}\min & \begin{array}{c}{T}_0=T_0{T}_1,T_2,\dots ,T_n,\\ C=C{T}_1,T_2,\dots ,T_n,\end{array}\\ \text{s.t.}& \begin{array}{c}{l}_i\le T_i\le h_{i}1\le i\le n,\\ l_0\le T_0\le h_0,\end{array}\end{array}\end{array}$$where $L=l_1,l_2,\dots ,l_n\text{\hspace{0.17em}and\hspace{0.17em}}H=h_1,h_2,\dots ,h_n$ are the upper and lower deviation of each component ring tolerance and $l_0\text{\hspace{0.17em}and\hspace{0.17em}}{h}_0$ are the design tolerance of assembly quality (the closed ring).

Figure 1 shows the construction process of the optimization model of steam turbine tolerance allocation based on cooperative game theory. We took assembly quality and processing cost as the two game parties, adopted the fuzzy cluster analysis method to determine the strategy space of each game party, assembly quality level, and processing cost level as the optimization goals, calculated the utility of the two game parties, and established a set of tolerance distribution multiobjective optimization models for the steam turbines based on the cooperative game theory. In the game model, the strategies of each player are independent of each other and do not interfere with each other. Therefore, when the game model is established, each player must be divided into its own strategy vector. When optimizing tolerance allocation that takes into account both quality and cost goals, the design tolerances of each component ring are the design variables shared by the quality and cost optimization objective functions. Therefore, according to the relationship between the design variables and the game parties, the design variables, which are more closely related to the quality game player, are assigned to form the strategy space of the quality game player, and the remaining design variables become the strategy space of the cost game player. Fuzzy cluster analysis can analyze the relationship between the samples and categories, describe the relationship between the samples and each category, divide the samples into the corresponding number of groups according to the relationship between the samples and each category, and finally divide the samples into the corresponding categories. Therefore, the fuzzy cluster analysis method is used to solve the degree of influence of design variables on each game party, and the design variables that are closely related to the quality game party are allocated to the quality game party, and the remaining design variables are allocated to the cost game party, form the strategic space of each game party, and finally complete the establishment of the game model of steam turbine tolerance allocation.

[figure omitted; refer to PDF]

The game utility matrix of alliance {P1} is shown in Figure 7. According to von Neumann’s “Minimum Maximum Criterion,” the characteristic function of the alliance {P1} is $\nu 1=-0.0588$.

[figure omitted; refer to PDF]

The game utility matrix of alliance {P2} is shown in Figure 8. According to von Neumann’s “Minimum Maximum Criterion,” the characteristic function of the alliance {P2} is $\nu 2=1.0152$.

[figure omitted; refer to PDF]

The game utility matrix of alliance {P1, P2} is shown in Figure 9. According to von Neumann’s “Minimum Maximum Criterion,” the characteristic function of the alliance {P1, P2} is $\nu \mathrm{1,2}=-0.5882$.

[figure omitted; refer to PDF]

$x=x_1,x_2,\dots ,x_n={\phi }_1\nu ,{\phi }_2\nu ,\dots ,{\phi }_n\nu$ is a distribution plan of the alliance game. For the Shapley vector $x_i$, each player will accept this resource allocation, because the utility of each player in cooperative game is not inferior to his own efforts, namely, ${\phi }_i\nu <\nu i,i=\mathrm{1,2}$, and the specific calculation process is shown in (14) and (15). In this manuscript, whether it is a tolerance value or a cost value, the smaller the value, the better.$$\begin{array}{c}\text{(14)}& \nu 1=-0.0588,\nu 2=1.0152,\\ \nu \mathrm{1,2}=\min u_1+u_2=1+-1.5882=-\mathrm{0.5882.}\\ \text{(15)}& x=x_1,x_2,={\phi }_1\nu ,{\phi }_2\nu \\ =-\text{0.}8311\text{,0.}2429\\ -\text{0.}8311+\text{0.}2429=-\text{0.}5882\text{.}\end{array}$$

The Shapley value vector is $\mathrm{\Phi }={\phi }_1,{\phi }_2=-0.8311,0.2429$; we calculated all the two-norm of all the effective combination and the Shapley value and obtained the optimal utility combination is $u_1^{\ast },u_2^{\ast }=-0.8235,1.0045$, the corresponding optimal strategy combination is $s_{1-3}^{\ast },s_{1-9}^{\ast },s_{1-10}^{\ast },s_{1-12}^{\ast },s_{2-1}^{\ast },s_{2-2}^{\ast },s_{2-4}^{\ast },s_{2-7}^{\ast },s_{2-8}^{\ast },s_{2-11}^{\ast }=0.1,0.1,0.025,0,-0.05,-0.05,0.025,0.05,0.1,0.05$, and $T_0=0.35$, $0.25<T_0<0.55$, where $T_0$ is within the design range of the flow gap. It meets the tolerance constraint requirements.

6.3. Solution and Analysis of the Game Model of Steam Turbine Tolerance Distribution

In this paper, the Shapley value method, the Nash equilibrium method, and the quality-constrained cost optimization method were used to solve the optimization model of the steam turbine tolerance allocation. The solution results are shown in Table 1.

Table 1

The solution and comparative analysis of the optimization model of steam turbine tolerance distribution.

Optimization methods	Tolerance design variables (tolerances assigned to each component ring)										Utility		Optimization objectives
	$S_{1–3}^{\ast }$	$S_{1–9}^{\ast }$	$S_{1–10}^{\ast }$	$S_{1–12}^{\ast }$	$S_{2–1}^{\ast }$	$S_{2–2}^{\ast }$	$S_{2–4}^{\ast }$	$S_{2–7}^{\ast }$	$S_{2–8}^{\ast }$	$S_{2–11}^{\ast }$	$u_1$	$u_2$	T0 (mm)	$C$
Shapley value method	0.1	0.1	0.025	0	−0.05	−0.05	0.025	0.05	0.1	0.05	−0.8235	1.0045	0.3500	9.8668
Nash equilibrium method	0.2	0.1	0.025	0	0.05	0.05	0.05	0.05	0.1	0.05	−1.5882	1.0000	0.6750	9.8226
Quality-constrained method	0.2	0.1	0.025	0	−0.05	0	0.05	0.05	0.1	0.05	−1.2353	1.0015	0.5250	9.8373

By comparing the solution results of the three methods in Table 1, the following conclusions could be obtained: the closed ring tolerance obtained by the Nash equilibrium method was 0.6750 mm, which was beyond the design range of the closed ring tolerance of $0.25\le T_{bl}\le 0.55$, and could not meet the design requirements. However, the solution results solved by the Shapley method and the traditional method of taking quality as constraint were, respectively, 0.3500 mm and 0.5250 mm, which were all within the design range of the closed-loop tolerance. The cost obtained by the above three methods was almost the same, but the quality solutions obtained by them were significantly different. Although the closed-loop tolerances obtained by the Shapley value method and the quality-constrained method were within the design range, the closed-loop tolerance obtained by the Shapley value method was 0.35 mm, and the corresponding product quality is significantly better than that of the quality-constrained method. For the flow gap of the steam turbine, when the cost is almost the same, and the tolerance value is within the design range, the smaller the tolerance value of the flow gap, the higher the working efficiency of the product, and the quality of the product is higher. Therefore, compared with the noncooperative game method and the traditional solution method of quality-constrained cost optimization, the Shapley value method in the cooperative game method that takes into account both quality and cost had the best solution effect.

7. Conclusions

(1) Aiming at the multiobjective optimization problem with contradictions and conflicts of the tolerance allocation of complex products, we took advantage of the features of coordinating and balancing contradictions and conflicts of cooperative game theory. The quality requirements and cost requirements of assembly products were used as the game decision party, and the fuzzy clustering method was used to group the design variables of tolerance allocation problem of the steam turbine to form the strategic space belonging to each game party. Taking the quality level and cost level of the assembly product as the optimization goals, we completed the calculation of the utility belonging to the two game parties and established a set of multiobjective optimization model of steam turbine tolerance allocation based on cooperative game theory.

(2) Taking the solution of the multiobjective optimization model of steam turbine tolerance allocation as an example, the Shapley value method in cooperative game theory, the Nash equilibrium method in noncooperative game theory, and the traditional single-objective optimization method with quality as the constraint and the cost as the optimization objective were used to solve the problem. Comparing the cost solution results of the three optimization methods, the product cost obtained by the three methods was almost the same. Comparing the quality solution results, the Nash equilibrium solution still could be further optimized, and the Shapley value method cooperated from the perspective of collective interests and obtained a better result than the Nash equilibrium method. Moreover, the quality solution of the Shapley value method, which could coordinate and balance the optimization goals, was also significantly better than the traditional single-objective optimization method that took quality as the constraint and the lowest cost as the optimization goal. Therefore, the example of the multiobjective optimization of steam turbine tolerance allocation showed that the cooperative game method balanced and coordinated quality and cost and realized the comprehensive optimization for the two goals from the perspective of collective interests, overcame the disadvantages of the traditional single-objective optimization method, and obtained a better result than Nash equilibrium method.

Additional Points

Aiming at the optimization problem of multiobjective with contradictions and conflicts of the tolerance allocation of complex products, we established a set of multiobjective optimization models of tolerance allocation based on cooperative game theory. And, the cooperative game solving method was used to solve the above established game model, and we achieved a good solution effect.

Acknowledgments

The authors gratefully acknowledge the financial supports by the National Key Research and Development Program of China under Grant no. 2019YFB1703800.

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