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

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 asmonoobjective 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.


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.

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 .
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.

The Technology Principle
The design variables: X x 1 , x 2 , . . ., x n ∈ Ω n , let the objective functions be minimized: 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 S 1 , S 2 .The payoff functions are u 1 , u 2 .Hence, the game with 2 players can be written as G m S 1 , S 2 ; u 1 , u 2 .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 S 1 , S 2 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.

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 S 1 , S 2 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 where 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

3.2
To avoid the different functions' self-affecting, make impact factors dimensionless: 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: 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 4 Establish the matrix R based on r kl and do fuzzy clustering to matrix R: R Classification results of Δ represent the classification results of X because of a oneto-one relationship between Δ {Δ 1 , Δ 2 , . . ., Δ n } and X {x 1 , x 2 , . . ., x n }.
5 According to fuzzy clustering, divide the design variables X into strategy subsets S 1 , . . ., 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 S 1 , S 2 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 Δ 1 , Δ 2 , . . ., 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.

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: 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: where m j 1 w 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.

Algorithm Procedures and Flow Chart
1 Obtain strategy subset S 1 , S 2 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.For any player i i 1, 2 , solve the optimal strategy s * i ∈ S i , and make payoff minimum 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 F 1 , then the small pig stands for F 2 and if the big pig stands for F 2 , then the small pig stands for F 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 x 1 , x 2 , x 3 , x 4 , x 5 .Constraints need to meet upper and lower limits of design variables and amplitude range cannot exceed the prescribed range.The objective functions are F 1 stability index and F 2 economic index .

The Objective Function of the Stability Index
Consider 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 L sin ωt − sin α 1 x 3 cos α 1 x 5 − cos ωt x 5 , 4.2 where ω α 2 − α 1 /T is angular velocity, T is the total time of the fluctuation, α 1 is the elevation in R max the maximum luffing , and α 2 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: 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 where Δh t is the deviation relative to the starting point in any time t.

Mathematical Problems in Engineering 9
So, the objective function of the stability index is as follows:

The Objective Function of the Economic Index
The energy consumption is P t in any time t.
where M q t is the torque.For no frame balance system, it is as follows: where y B L cos ωt − x 4 sin ωt x 5 , z B L sin ωt x 4 cos ωt x 5 , G is the gravity of the goods, β arctg z B − x 2 / y B − x 1 , 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: 4.8

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, −0.2.946 kJ.

5.1
The impact factors are shown in Table 2.  3.
According to Table 3, because the maximum value is 90.695, S a S 1 {x 1 , x 2 } is the strategy subset of F 1 .S b S 2 {x 3 , x 4 , x 5 } is the strategy subset of F 2 .

Calculation Results
There exist two kinds of cases.Case 1 is that the big pig stands for F 2 and the small pig stands for F 1 .Case 2 is that the big pig stands for F 1 and the small pig stands for F 2 .
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 S 1 , S 2 as the initial feasible strategies s  a Seek the optimal strategy s * 1 ∈ S 1 and minimize the payoff function, b Seek the optimal strategy s * 2 ∈ S 2 and minimize the payoff function, 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 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, u 1 is constructed according to cooperative behavior mode and u 2 is constructed according to noncooperative behavior mode.  5.
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 .The comparison of deviation trajectory cases 1 and 2, 6, 16, 17 is shown in Figure 4.
According to Table 6, we can know that F 1 in case 1 is better than 6, 16, 17 and case 2, and that F 1 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. F 2 in case 2 is better than 6, 16, 17 and case 1. F 2 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 F 1 and F 2 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 F 2 and the greater the value of w 22 is the cooperative degree is lower , the better the final value of F 2 is.In case 2, the big pig stands for F 1 and the greater the value of w 11 is the cooperative degree is lower , the better the final value of F 1 is.strategy subsets named S 1 , S 2 , 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 S 1 , S 2 and two payoff functions u are constructed based on pigs' payoff game behavior.

Mathematical Problems in Engineering
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.

3
Generate the initial feasible strategies in the strategy set of each player randomly and then form a strategy permutation s 0 {s strategys s(1) -optimal strategy permutation

2
Perform the following two single-objective optimization.

Figure 4 :
Figure 4: The comparison of deviation trajectory.

Table 1 :
The payoff matrix.

Table 2 :
The impact factors.

Table 3 :
The impact factors of strategy subsets to objection functions.M 2, and P 3.Because r 12 1.66393 is the maximum value of matrix R, x 1 and x 2 belong to one class.Namely, S a {x 1 , x 2 } and S b {x 3 , x 4 , x 5 }.The impact factors bof strategy subsets to objection functions are shown in Table

Table 4 :
The iterative process of Case 1.
F 2 28.255 KJ.Iterative process is shown in Table

Table 5 :
The iterative process of Case 2.

Table 6 :
The compared results.

Table 7 :
The influence of w 22 on the final solutions in Case 1.

Table 8 :
The influence of w 11 on the final solutions in Case 2.