Bi-Objective Optimization Method and Application of Mechanism Design Based on Pigs' Payoff Game Behavior

Abstract

It takes two design goals as different game players and design variables are divided into strategy spaces owned by corresponding game player by calculating the impact factor and fuzzy clustering. By the analysis of behavior characteristics of two kinds of intelligent pigs, the big pig's behavior is cooperative and collective, but the small pig's behavior is noncooperative, which are endowed with corresponding game player. Two game players establish the mapping relationship between game players payoff functions and objective functions. In their own strategy space, each game player takes their payoff function as monoobjective for optimization. It gives the best strategy upon other players. All the best strategies are combined to be a game strategy set. With convergence and multiround game, the final game solution is obtained. Taking bi-objective optimization of luffing mechanism of compensative shave block, for example, the results show that the method can effectively solve bi-objective optimization problems with preferred target and the efficiency and accuracy are also well.

1. Introduction

Multiobjective optimization problem in actual engineering design is very common. The essential characteristics of multiobjective optimization are as follows: (1) there exist several objective interests; (2) the status of the various objectives are different and have conflicts. The solution methods are diverse; the latest research is as follows: Akbari and Ziarati [1] applied a novel bee swarm optimization method to obtain a uniformly distributed Pareto front. Ismail et al. [2] proposed a new self-organizing genetic algorithm for multiobjective optimization problems to obtain a better value as compared to the existing weighted-sum methods. Lee et al. [3] used the multiobjective fuzzy optimization method to obtain the optimal parameters of rotor experimental apparatus. Ding et al. [4] proposed a new multiobjective optimization algorithm named KSVC-SPEA to effectively achieve the overall performance of injection molding machine.

In recent years, considering the similarity between multiobjective design and the game, game theory has been used to solve multiobjective design problems, especially for practical problems in engineering fields. According to the different behaviors of each game player seeking for benefit, the game can be divided into noncooperative game and cooperative game. In a noncooperative game, each player benefits from competitive behavior patterns and the typical models are Nash equilibrium game model and the Stackelberg oligopoly game model. A cooperative game is defined as game players abiding by a binding agreement, benefiting from cooperative behavior patterns. The typical binding agreements contain three types, which are known as the “self-interest do not harm the others” (competitive and cooperative game model), “You have me, I have you” (coalition cooperative game model), and “all for one and one for all” (unselfish cooperative game model). About noncooperative game to solve multiobjective design, Spallino and Rizzo [5] proposed a noncooperative game optimization method based on evolutionary strategy in the multiobjective design of the composite laminate, which treated each game player as an equal body and eventually found a Nash equilibrium point through negotiation functions. Neng-gang et al. [6] established a multiobjective game design technology roadmap and key indicators based on the Nash equilibrium model and the Stackelberg oligopoly game model and successfully applied to multiobjective optimization design such as gravity dam, structure of arch-arch ring, and luff mechanism of compensative sheave block. In the use of cooperative games to solve multiobjective design, Chen and Li [7] proposed three-tier two-objective optimization method and applied this method to the manufacture of concurrent product and process optimization; Neng-gang et al. [8] adopted a competitive-cooperative game model to conduct a multi-objective optimization design and obtained a good design. However, whether the non-cooperative game methods or the cooperative game methods are used to solve multi-objective design problems, if the game method is selected, behavior modes of all players remain unchanged during the whole process. But this is an ideal situation. Each player’s behavior is diverse in many survival games in nature. Neng-gang et al. [9] proposed a mixed game model according to the diversity of behavior patterns caused by differences in resources and endowment of each player. Through the bionics of the survival mechanisms of reproduction of lizard species, a typical mixed game model is presented, which consists of both competitive behavior patterns and cooperative behavior patterns of “all for one and one for all” and “benefits oneself but do not harm other people”. This method is very good to solve the oneness problem of constructing payoff functions, but there exist two shortcomings as follows. (1) It can only be applied to three objectives or more than three objectives and cannot solve two-objective optimization problems. (2) It cannot solve “principal and subordinate” optimization problems. That is, it cannot solve the optimization problem with target preference. To compensate this deficiency and improve the game method for solving optimization problems, bi-objective optimization method is proposed based on pigs’ payoff behavior, which can be applied in two-objective optimization problem with target preference.

2. The Basic Idea

2.1. Pigs’ Payoff Game Model

American economist named Nash (the Nobel economic prize winner) has proposed “Pigs’ Payoff”. It is shown in Figure 1 and is as follows: there are a big pig and a small pig in the pigsty. One side of the pigsty has food slot and the other side has food control button. Whether the big pig or the small pig will pay 2-unit energy cost if it presses the food control button and 10-unit food will fall into food slot in return. If the big pig first arrives in the food slot, the benefit ratio of the big pig to the small pig is 9?:?1. If the big pig and the small pig arrive in the food slot at the same time, the benefit ratio of the big pig to the small pig is 7?:?3. If the small pig first arrives in the food slot, the benefit ratio of the big pig to the small pig is 6?:?4. The payoff matrix is shown in Table 1. In premise of both the big pig and small pig having intelligence, the final game result is that the big pig presses the button and the small pig dose not press the button but chooses to wait [10].

Table 1 

The payoff matrix.

Figure 1 

The picture of pigs’ payoff.

From the result of the behavior, the strategy of waiting is a selfish behavior of noncooperation and the strategy of pressing the button is a collective behavior of cooperation. Hence, two game players (the big pig and the small pig) adopt two different behavior modes and constitute a hybrid game mode. The equilibrium solution (4, 4) is Pareto solution.

2.2. The Technology Principle

Thedesignvariables:𝐗=𝑥₁,𝑥₂,…,𝑥_𝑛∈Ω^𝑛,lettheobjectivefunctionsbeminimized:𝐹(𝐗)=𝐹₁(𝐗),𝐹₂(𝐗)⟶min,subjecttoconstraintconditions:𝑔_𝑘(𝐗)≤0𝑘=1,2,…,𝑞, (2.1) where 𝑛 is the number of design variables. 𝑞 is the number of constraint conditions. Ω^𝑛 is the feasible space of design variables.

Meanwhile, the definition of game is as follows: 𝐺_𝑚 represents one game. If 𝐺_𝑚 has 2 players (Illustration: the implication of number of players is equal to the number of objective functions), the sets of available strategies are denoted by 𝐒₁,𝐒₂. The payoff functions are 𝑢₁,𝑢₂. Hence, the game with 2 players can be written as 𝐺_𝑚=(𝐒₁,𝐒₂;𝑢₁,𝑢₂).

The basic idea for bi-objective optimization method based on game is as follows: (1) there are 2 design objectives, which are seen as 2 players and the design variables 𝐗 are divided into strategy subsets 𝐒₁,𝐒₂ of the corresponding players by certain technical methods. (2) According to the specific game model, mapping relationships are established between the payoff functions 𝑢 and objective functions 𝐹. (3) Each player takes its own payoff function as its objective and gets a single-objective optimal solution in its own strategy subset. So this player obtains the best strategy versus other players. The best strategies of all players form the group strategy in this round. The final equilibrium solutions can be obtained through multiround game according to the convergence criterion.

The payoff function 𝑢 is closely related to the game model. The different behavior characteristics of the big pig and small pig, respectively, are assigned to the corresponding game players based on pigs’ payoff game behavior model; then, the payoff functions 𝑢 is constructed according to the corresponding behavior characteristics.

3. The Key Technology and Structure of the Algorithm

3.1. Game Player’s Strategy Subset Computation

Fuzzy mathematics has been successfully used in the related design fields with the multidisciplinary cross research. Fuzzy mathematics has been successfully applied in filter design [11], T-S fuzzy systems [12, 13], and T-S fuzzy stochastic systems [14] and abundant research results are obtained. In this paper, the design variables are divided into each game players strategy subsets (𝐒₁,𝐒₂) by calculating the impact factor and fuzzy clustering based on fuzzy mathematics.

Computation steps are as follows.(1) Optimize 2 mono-objectives; then obtain optimal solution 𝐹₁(𝐗^∗₁),𝐹₂(𝐗^∗₂), where 𝐗^∗_𝑖=𝑥^∗_1𝑖,𝑥^∗_2𝑖,…,𝑥^∗_𝑛𝑖(𝑖=1,2). (3.1)(2) Every 𝑥_𝑗 is divided into 𝑇 fragments with step length Δ𝑥_𝑗 in its feasible space; Δ_𝑗𝑖 is an impact factor (𝑥_𝑗 affecting the objective 𝑓_𝑖) and is shown as Δ_𝑗𝑖=∑^𝑇_𝑡=1𝐹_𝑖𝑥^∗_1𝑖,…,𝑥^∗_{(𝑗−1)𝑖},𝑥_𝑗(𝑡),𝑥^∗_(𝑗+1)𝑖,…,𝑥^∗_𝑛𝑖−𝐹_𝑖𝑥^∗_1𝑖,…,𝑥^∗_{(𝑗−1)𝑖},𝑥_𝑗(𝑡−1),𝑥^∗_(𝑗+1)𝑖,…,𝑥^∗_𝑛𝑖𝑇⋅Δ𝑥_𝑗. (3.2)?To avoid the different functions’ self-affecting, make impact factors dimensionless: Δ_𝑗𝑖=Δ_𝑗𝑖𝐹_𝑖𝐗^∗_𝑖. (3.3)(3) All samples classification Δ={Δ₁,Δ₂,…,Δ_𝑛}, the classification of 𝑗 is Δ_𝑗={Δ_𝑗1,…,Δ_𝑗2}(𝑗=1,…,𝑛), and Δ_𝑗 means the impact factor set of 𝑗 on all the players. The purpose is classifying highly similar samples as one classification; this paper uses a similar degree approach to reflect the samples’ similarity relation. Select any two samples Δ_𝑘 and Δ_𝑙 and analyze their similarity relation; define a fuzzy relation function by normal distribution: 𝜇_𝑖Δ_𝑘,Δ_𝑙−Δ=exp^𝑘𝑖−Δ_𝑙𝑖(1/𝑚)∑^𝑚_𝑖=1Δ_𝑘𝑖−Δ_𝑙𝑖(𝑘,𝑙=1,2,…,𝑛;𝑘≠𝑙;𝑖=1,2), (3.4)?where 𝜇_𝑖(Δ_𝑘,Δ_𝑙) is the fuzzy relation between Δ_𝑘 and Δ_𝑙 in the 𝑖th objective function.?The correlation degree of Δ_𝑘 and Δ_𝑙 is 𝑟_𝑘𝑙=12²_𝑖=1min_𝑖∈{1,2}Δ_𝑘𝑖−Δ_𝑙𝑖+0.5max_𝑖∈{1,2}Δ_𝑘𝑖−Δ_𝑙𝑖Δ_𝑘𝑖−Δ_𝑙𝑖+0.5max_𝑖∈{1,2}Δ_𝑘𝑖−Δ_𝑙𝑖. (3.5)(4) Establish the matrix 𝐑 based on 𝑟_𝑘𝑙 and do fuzzy clustering to matrix 𝐑: 𝑟𝐑=¹¹𝑟₁₂⋯𝑟_1𝑛𝑟₂₁𝑟₂₂⋯𝑟_2𝑛𝑟⋮⋮⋮⋮_𝑛1𝑟_𝑛2⋯𝑟_𝑛𝑛. (3.6)?Classification results of Δ represent the classification results of 𝐗 because of a one-to-one relationship between Δ={Δ₁,Δ₂,…,Δ_𝑛} and 𝐗={𝑥₁,𝑥₂,…,𝑥_𝑛}.(5) According to fuzzy clustering, divide the design variables 𝐗 into strategy subsets 𝐒₁,…,𝐒_𝑚 and assign the strategy subset to the corresponding player by the average value of impact factors. According to a statistical viewpoint [15], when the number of design variables and objective functions is small, we can directly divide variable sets 𝐗 into strategy space 𝐒₁, 𝐒₂ according to the value of impact factor. When the number design variables and objective functions are large, fuzzy clusterings are needed. Meanwhile, according to experience, we can first classify variables with strong correlation as a sample to reduce the complexity of clustering analysis.

Input system’s classification control value is 𝑀 and maximal sample number is 𝑃; each with sample as one classification, the system is Δ₁,Δ₂,…,Δ_𝑛.

The steps of clustering are as follows.(1) Calculate the correlation degree 𝑟_𝑘𝑙 and build matrix 𝐑⁽⁰⁾; attention: 𝑟_𝑘𝑙=𝑟_𝑙𝑘, 𝑟_𝑘𝑙>0.(2) Set maximum value of matrix 𝐑⁽⁰⁾ to be 𝑟_𝑎𝑏, 𝑟_𝑎𝑏=max_{𝑘,𝑙∈{1,2,…,𝑛}}𝑟_𝑘𝑙 and classify Δ_𝑎 and Δ_𝑏 into a new classification Δ_𝑠; if the sample number is larger than 𝑃, then combine the second maximal value of 𝐑⁽⁰⁾.(3) Combine Δ_𝑐(𝑐=1,2,…,𝑛;𝑐≠𝑎,𝑐≠𝑏) and Δ_𝑠 into a new classification system, calculate its correlation degree, and build a new matrix 𝐑⁽¹⁾; the correlation degree of any classification Δ_𝑐 and Δ_𝑠 is 𝑟_𝑐𝑠=min{𝑟_𝑐𝑎,𝑟_𝑐𝑏}.(4) Repeat procedures (1), (2), and (3) until the system classification number equals control value 𝑀.

3.2. Behavior Modes and Construction of Game Payoff Functions

The characteristic of the small pig is competitive behavior mode and its corresponding game payoff function is as follows: 𝑢_𝑖=𝐹_𝑖𝐹_𝑖(𝑖=1,2), (3.7) where 𝐹 is a reference value, which can eliminate the differences in the magnitude for each objective function. In this paper, the initial objective function value is chosen to be 𝐹.

The characteristic of the big pig is cooperative behavior mode and its corresponding game payoff function is as follows: 𝑢_𝑖=𝑤_𝑖𝑖𝐹_𝑖𝐹_𝑖+^𝑚_{𝑗=1(𝑗≠𝑖)}𝑤_𝑖𝑗𝐹_𝑗𝐹_𝑗(𝑖=1,2), (3.8) where ∑^𝑚_𝑗=1𝑤_𝑖𝑗=1 value of 𝑤_𝑖𝑖 reflects the degree of considering its own interest. The greater the value is, the lower the cooperative degree is.

3.3. Algorithm Procedures and Flow Chart

(1) Obtain strategy subset 𝐒₁,𝐒₂ attached to each player through calculating the impact factor and fuzzy clustering.(2) Payoff functions 𝑢_𝑖 to any 𝑖th player (𝑖=1,2) is constructed according to the characteristic of the small pig and big pig proposed by Section 3.2 above.(3) Generate the initial feasible strategies in the strategy set of each player randomly and then form a strategy permutation 𝑠⁽⁰⁾={𝑠₁⁽⁰⁾,𝑠₂⁽⁰⁾}.(4) Let 𝑠₁⁽⁰⁾,𝑠₂⁽⁰⁾ be the corresponding complementary set of 𝑠₁⁽⁰⁾,𝑠₂⁽⁰⁾ in 𝑠⁽⁰⁾. For any player 𝑖(𝑖=1,2), solve the optimal strategy 𝑠^∗_𝑖∈𝐒_𝑖, and make payoff minimum 𝑢_𝑖(𝑠^∗_𝑖,𝑠_𝑖⁽⁰⁾)→min(𝑖=1,2);(5) Define optimal strategy permutation 𝑠⁽¹⁾=𝑠^∗₁∪𝑠^∗₂. Then judge the feasibility of 𝑠⁽¹⁾. If 𝑔_𝑘(𝑠⁽¹⁾)≤0(𝑘=1,2,…,𝑞) does not satisfy, turn to step (3). Otherwise, compute the distance between 𝑠⁽¹⁾ and 𝑠⁽⁰⁾ which is called the Euclidean norm. Then examine whether the distance satisfies the convergence criterion ‖𝑠⁽¹⁾−𝑠⁽⁰⁾‖≤𝜀 or not (𝜀 is a decimal parameter given in advance). If it satisfies, the game is over; if not, let 𝑠⁽¹⁾ displace 𝑠⁽⁰⁾ and turn to step (4) to repeat.

The algorithm chart is shown in Figure 2 (illustration: if the big pig stands for 𝐹₁, then the small pig stands for 𝐹₂ and if the big pig stands for 𝐹₂, then the small pig stands for 𝐹₁).

Figure 2 

The algorithm chart.

4. Bi-Objective Optimization Model of Luff Mechanism of Compensative Sheave Block

4.1. The Design Model

The luff mechanism of compensative sheave block is a working device, which can realize mechanical loading range and is widely used in hoisting machinery. In its working process, there exists the stability goal; namely, the goods need to move along the horizontal path. On the other hand, there also exists the economic goal; namely, it needs less energy consumption. So, design problems have multiobjective optimization issues.

The luff mechanism of compensative sheave block is shown in Figure 3. The design variables are 𝐗=(𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅). Constraints need to meet upper and lower limits of design variables and amplitude range cannot exceed the prescribed range. The objective functions are 𝐹₁ (stability index) and 𝐹₂ (economic index).

Figure 3 

The luff mechanism of compensative sheave block.

4.2. The Objective Function of the Stability Index

Consider 𝑅=𝐿cos𝛼+𝑥₃sin𝛼+𝑥₅+𝑓+𝑟, (4.1) where 𝑅 is the amplitude of fluctuation and 𝛼 is the elevation.

The mechanism in the biggest amplitude is the starting point and the rise quantity relative to the starting point is Δ𝑧(𝑡) in any time 𝑡, Δ𝑧(𝑡)=𝐿sin𝜔𝑡−sin𝛼₁+𝑥₃cos𝛼₁+𝑥₅−cos𝜔𝑡+𝑥₅, (4.2) where 𝜔=(𝛼₂−𝛼₁)/𝑇 is angular velocity, 𝑇 is the total time of the fluctuation, 𝛼₁ is the elevation in 𝑅_max (the maximum luffing), and 𝛼₂ is the elevation in 𝑅_min (the minimum luffing).

The fall quantity relative to the starting point is Δ𝑙(𝑡) in any time 𝑡 due to the rope releasing: (𝑡)=𝑚Δ𝑙^𝑏𝑚_𝑞𝑙₁−𝑙(𝑡), (4.3) where 𝑚_𝑞 is the number of wire rope of lifting pulley and 𝑚_𝑏 is the number of wire rope of compensation pulley, where 𝑙₁=𝐿cos𝛼₁−𝑥₄sin𝛼₁+𝑥₅−𝑥₁²+𝐿sin𝛼₁+𝑥₄cos𝛼₁+𝑥₅−𝑥₂²,𝑙(𝑡)=𝐿cos𝜔𝑡−𝑥₄sin𝜔𝑡+𝑥₅−𝑥₁²+𝐿sin𝜔𝑡+𝑥₄cos𝜔𝑡+𝑥₅−𝑥₂²,Δℎ(𝑡)=Δ𝑧(𝑡)−Δ𝑙(𝑡), (4.4) where Δℎ(𝑡) is the deviation relative to the starting point in any time 𝑡.

So, the objective function of the stability index is as follows: 𝐹₁=sup_{[]𝑡∈0,𝑇}Δℎ(𝑡)−inf_{[]𝑡∈0,𝑇}Δℎ(𝑡). (4.5)

4.3. The Objective Function of the Economic Index

The energy consumption is 𝑃(𝑡) in any time 𝑡.𝑃(𝑡)=𝑀_𝑞(𝑡)𝜔, (4.6) where 𝑀_𝑞(𝑡) is the torque. For no frame balance system, it is as follows: 𝑀_𝑞(𝑡)=𝐺9.8𝐿cos𝜔𝑡+𝑥₃sin𝜔𝑡+𝑥₅−𝑚+𝑟^𝑏𝑚_𝑞𝐺𝑧9.8^𝐵cos𝛽−𝑦_𝐵+𝐺sin𝛽^′9.8𝐿𝜉cos𝜔𝑡, (4.7) where 𝑦_𝐵=𝐿cos𝜔𝑡−𝑥₄sin(𝜔𝑡+𝑥₅), 𝑧_𝐵=𝐿sin𝜔𝑡+𝑥₄cos(𝜔𝑡+𝑥₅), 𝐺 is the gravity of the goods, 𝛽=arct𝑔((𝑧_𝐵−𝑥₂)/(𝑦_𝐵−𝑥₁)), 𝐺^′ is the gravity of the arm frame, and 𝜉 is the ratio of the distance (center of gravity of the arm frame from 𝑂 point in Figure 3) to arm length-𝐿.

So, the objective function of the economic index is as follows: 𝐹₂=^𝑇₀𝑃(𝑡)𝑑𝑡=𝜔^𝑇₀𝑀_𝑞(𝑡)𝑑𝑡. (4.8)

5. The Application in Mechanism Design

5.1. Calculation Statement

The paper takes the luff mechanism of compensative sheave block (shown in Figure 3) as application object. 𝐺=31360?N, 𝐺^′=13720?N, 𝐿=14?m, 𝑓=0.7?m, 𝜉=0.5, 𝑟=0.2?m, 𝑚_𝑏=6, 𝑚_𝑞=2. 𝑅_max=12?m, 𝑅_min=5.8?m, −0.4≤𝑥₁≤0.5, 3≤𝑥₂≤8, 0.4≤𝑥₃≤0.8, 0.3≤𝑥₄≤0.7, 0≤𝑥₅≤0.43633. The total fluctuation time is 40 seconds (𝑇=40 seconds). The smallest unit time is 1 second. The more-detailed mechanism instructions can refer to [16]. Meanwhile, a group of realistic optimization design parameters (𝑥₁=−0.030, 𝑥₂=4.040, 𝑥₃=0.550, 𝑥₄=0.370, 𝑥₅=0.160) is listed [16].

5.2. Single-Objective Optimization Results

Consider 𝐗^∗₁=(0.01897,4.08848,0.51854,0.53969,0.33220),𝐹₁𝐗^∗₁𝐗=0.01558m,_∗2=(0.00954,4.83460,0.77453,0.57673,0.31204),𝐹₂𝐗^∗₂=2.946kJ. (5.1)

The impact factors are shown in Table 2.

Table 2 

The impact factors.

5.3. Fuzzy Clustering

Consider Δ₁=(95.53,17.12), Δ₂=(85.86,13.99), Δ₃=(29.75,5.34), Δ₄=(21.57,5.27), Δ₅=(1.78,0.36), ∗𝐑=∗1.663931.547281.529191.54697∗1.523361.504501.53111∗1.346021.54624Symmetry∗1.60735, (5.2)𝑀=2, and 𝑃=3. Because 𝑟₁₂=1.66393 is the maximum value of matrix 𝐑, 𝑥₁ and 𝑥₂ belong to one class. Namely, 𝐒_𝑎={𝑥₁,𝑥₂} and 𝐒_𝑏={𝑥₃,𝑥₄,𝑥₅}. The impact factors bof strategy subsets to objection functions are shown in Table 3.

Table 3 

The impact factors of strategy subsets to objection functions.

According to Table 3, because the maximum value is 90.695, 𝐒_𝑎=𝐒₁={𝑥₁,𝑥₂} is the strategy subset of 𝐹₁. 𝐒_𝑏=𝐒₂={𝑥₃,𝑥₄,𝑥₅} is the strategy subset of 𝐹₂.

5.4. Calculation Results

There exist two kinds of cases. Case 1 is that the big pig stands for 𝐹₂ and the small pig stands for 𝐹₁. Case 2 is that the big pig stands for 𝐹₁ and the small pig stands for 𝐹₂.

We take case 1; for example, the detailed calculation steps are as follows.(1) Take the corresponding values of the initial design in strategy subsets 𝐒₁, 𝐒₂ as the initial feasible strategies 𝑠₁⁽⁰⁾, 𝑠₂⁽⁰⁾. Then, form a strategy permutation 𝑠⁽⁰⁾={𝑠₁⁽⁰⁾,𝑠₂⁽⁰⁾}.(2) Perform the following two single-objective optimization.(a)Seek the optimal strategy 𝑠^∗₁∈𝐒₁ and minimize the payoff function, 𝑢₁𝑠^∗₁,𝑠₂⁽⁰⁾=𝐹₁𝑠^∗₁,𝑠₂⁽⁰⁾𝐹₁𝑠₁⁽⁰⁾,𝑠₂⁽⁰⁾⟶min. (5.3)(b)Seek the optimal strategy 𝑠^∗₂∈𝐒₂ and minimize the payoff function, 𝑢₂𝑠₁⁽⁰⁾,𝑠^∗₂=𝑤₂₂×𝐹₂𝑠₁⁽⁰⁾,𝑠^∗₂𝐹₂𝑠₁⁽⁰⁾,𝑠₂⁽⁰⁾+𝑤₂₁×𝐹₁𝑠₁⁽⁰⁾,𝑠^∗₂𝐹₁𝑠₁⁽⁰⁾,𝑠₂⁽⁰⁾→min. (5.4)(3) Define strategy permutation 𝑠⁽¹⁾=𝑠^∗₁∪𝑠^∗₂. Then, justify the feasibility of 𝑠⁽¹⁾. If 𝑠⁽¹⁾ does not satisfy constraint conditions, turn to step (1). Otherwise, compute ∑⁵_𝑗=1(((𝑥_𝑗⁽¹⁾−𝑥_𝑗⁽⁰⁾)/𝑥_𝑗⁽⁰⁾)²/5) and examine whether it satisfies the convergence precision 𝜀 (𝜀 is 0.0001 in this paper). If it satisfies, the game is over; if not, let 𝑠⁽⁰⁾=𝑠⁽¹⁾ and turn to step (2) to iteration loop.

Illustration: for case 2, 𝑢₁ is constructed according to cooperative behavior mode and 𝑢₂ is constructed according to noncooperative behavior mode.

(1) For case??1, 𝑤₂₂=0.5, 𝑤₂₁=0.5, calculation starts from the initial strategy 𝐗⁰=(0.02968,4.09656,0.59710,0.56809,0.04602) and obtains convergence value 𝐗^∗=(−0.00605,4.06225,0.58250,0.49698,0.02102) after six rounds game and 𝐹₁=0.01581?m, 𝐹₂=33.438?KJ. Iterative process is shown in Table 4.(2) For case??2, 𝑤₁₁=0.5, 𝑤₁₂=0.5, calculation starts from the initial strategy 𝐗⁰=(0.19247,4.17230,0.42101,0.31284,0.31592) and obtains convergence value 𝐗^∗=(0.01724,4.08923,0.40303,0.30493,0.43130) after four rounds game and 𝐹₁=0.14471?m, 𝐹₂=28.255?KJ. Iterative process is shown in Table 5.

Table 4 

The iterative process of Case 1.

Table 5 

The iterative process of Case 2.

The compared results are shown in Table 6. (Illustration: multiobjective fuzzy optimization method is adopted in [17] and multiobjective Nash equilibrium game method is adopted in [6]).

Table 6 

The compared results.

The comparison of deviation trajectory (cases 1 and 2, [6, 16, 17]) is shown in Figure 4.

Figure 4 

The comparison of deviation trajectory.

According to Table 6, we can know that 𝐹₁ in case 1 is better than [6, 16, 17] and case 2, and that 𝐹₁ in case 2 is the worst. According to Figure 4, the deviation trajectory in case 1 is better than [6, 16, 17] and case 2. 𝐹₂ in case 2 is better than [6, 16, 17] and case 1. 𝐹₂ in case 1 is the worst. The results show that the method can effectively solve bi-objective optimization problems with preferred target and that multiobjective fuzzy optimization method [17] is an effective method without preferred target (both 𝐹₁ and 𝐹₂ are better than realistic optimization results [16]).

By analyzing the results, we can know three conclusions as follows. (1) The game player with noncooperative characteristic of the small pig has greater advantage in the pursuit of its own interests than the game player with cooperative characteristic of the big pig. (2) If the designers have target preference, they need to take the preferred target as the small pig side and take another target as the big pig side. (3) The satisfactory equilibrium solution can be obtained through less iteration rounds because the design variables are decomposed into the strategy subset owned by 2 game players.

To reveal the influence of 𝑤_𝑖𝑖 on the final solutions, 𝑤_𝑖𝑖=0.1,0.3,0.5,0.7, respectively. The results are shown in Tables 7 and 8. In case 1, the big pig stands for 𝐹₂ and the greater the value of 𝑤₂₂ is (the cooperative degree is lower), the better the final value of 𝐹₂ is. In case 2, the big pig stands for 𝐹₁ and the greater the value of 𝑤₁₁ is (the cooperative degree is lower), the better the final value of 𝐹₁ is.

Table 7 

The influence of 𝑤₂₂ on the final solutions in Case 1.

Table 8 

The influence of 𝑤₁₁ on the final solutions in Case 2.

6. Conclusions

(1) One new bi-objective optimization game method is proposed. Two design goals can be regarded as two game players, the design variables set can be regarded as strategy subsets named 𝐒₁, 𝐒₂, and the constraints in multiobjective problems can be regarded as constraints in the game method. Through the specific technological means, the design variables can be divided into each game players strategy subsets (𝐒₁, 𝐒₂) and two payoff functions 𝑢 are constructed based on pigs’ payoff game behavior.(2) The solution step of game player’s strategy subset is presented. The design variables are divided into strategy spaces owned by the corresponding game player by calculating the impact factor and fuzzy clustering.(3) The big pig’s behavior is cooperative but the small pig’s behavior is noncooperative. The different behavior characteristics of the big pig and small pig, respectively, are assigned to the corresponding game players based on pigs’ payoff game behavior. The paper constitutes a hybrid game mode and proposes the detailed solution steps.(4) For optimization problems with preferred target, the designers need to emphasize one design goal. For this problem, there exist traditional methods such as weighting method (by adjusting the weight of each goal), hierarchical sequence method (by adjusting the objective optimization order), and goal programming method. In this paper, one new bi-objective optimization game method is proposed based on pigs’ payoff game behavior for solving optimization problems with preferred target. It takes bi-objective optimization of luffing mechanism of compensative shave block; for example, the results show that the method can effectively solve the bi-objective optimization problems with preferred target (designers need to take the preferred target as the small pig side and take another target as the big pig side), the efficiency and accuracy are well, and the solution is obtained only through fewer game rounds.

Acknowledgments

The project was supported by the Natural Science Foundation of Anhui Province of China (Grant no. 11040606?M119), State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (Grant no. 2010491011), and the Foundation for Young Talents in College of Anhui Province (Grant no. 2011SQRL034).