Proceedings of the 8th World Congress on Intelligent Control and Automation July 6-9 2010, Jinan, China
Multi-Objective Optimization Design Methods Based on Game Theory∗ Rui Meng ,Ye Ye ,Neng-gang Xie School of Mechanical Engineering , Anhui University of Technology Maanshan , Anhui Province, China
[email protected] considering the similarity between multi-objective design and the game, game theory has been used to solve multi-objective design problems, especially concerning some solutions for practical problems in engineering fields. According to the different behaviors of each game player seeking for benefit, the game can be divided into non-cooperative games and cooperative game.
Abstract- The paper presents the game description of multiobjective optimization design problem and takes the design objectives as different players. By calculating the affecting factors of the design variables to objective functions and fuzzy clustering, the design variables are divided into different strategic spaces owned by each player. Then it uses Nash equilibrium game model, coalition cooperative game model and evolutionary game model to solve multi-objective optimization design problem and gives corresponding solving steps. According to the specific game model, a mapping relationship between the game players’ payoff and the objective functions is proposed. Each player takes payoff function of its own as its objective and undertakes single-objective optimization in its own strategy space. Then this player obtains the best strategy versus other players. The best strategies of all players consist of the strategy permutation of a round game and it obtains the final game solutions through multi-round games according to the convergence criterion. Taking two objectives design of four bar joist rack structures for example, the results show that the computational precision of the coalition cooperative game model is the best ,which illustrates that cooperative game has the advantages over non-cooperative game in fulfilling polytropic win-win and collective benefit; while the coalition cooperative game model is the worst from the computational efficiency view, which shows that cooperation spends more time negotiating so as to obtain win-win; considering these factors comprehensively , evolutionary game model is better at computational precision and efficiency.
In non-cooperative game, each player benefits from competitive behavior patterns and the typical models are Nash equilibrium game model and the Stackelberg oligopoly game model. Cooperative game is defined as the 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), “every man for me, and I for all” (unselfish cooperative game model). Using non-cooperative game to solve multi-objective design, Suheyla Ozymldmrmm [1] systematically put forward the noncooperative game based on Nash equilibrium optimization method and got the optimization realization of non-renewable resource model; R.Spallino [2] proposed non-cooperative game optimization method based on evolutionary strategy in the multi-objective design of the composite laminate, which treated each game player as an equal body and eventually found a Nash equilibrium point through negotiation functions. J.Periaux [3] and Wang Jiang-feng [4] applied multi-objective non-cooperative game in some engineering examples such as multi-element airfoil aerodynamic optimization, multi-wing optimization design, multi-standard reverse dynamic optimization. Xie Neng-gang [5-7] 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 and structure of arch –arch ring and luff mechanism of compensative sheave block. In the use of cooperative games to solve multi-objective design, A.K.Dhingra [8] first proposed multi-objective optimization method based on cooperative games in high-speed mechanical devices. Li Chen [9] proposed three-tier twoobjective optimization methods and applied to the manufacture of concurrent product and process optimization; Xie Nenggang [10] adopted a competitive-cooperative game model to conduct a multi-objective optimization design and obtained a good design.
Keywords-Multi-objective optimization; Game; Nash equilibrium game model; Coalition cooperative game model; Evolutionary game model
I. INTRODUCTION Currently the solutions of the multi-objective optimization design models usually include the method of reducing dimension (also called main objective method) and the method of evaluation function which contains linear weighted sum method, min-max method, the ideal point method, the weighted sum of squares method and the virtual target method. The principle of these methods is to convert multi-objective optimization problem to a single-objective optimization problem. In addition to the above methods, other methods such as sequencing method, feasible direction method, the center method and interactive programming method are used to transform multi-objective optimization problem into multiple single-objective optimization problems. In recent years, ∗
This project is supported by Natural Science research project of Anhui provincial Education Department (NO.070414154) and New Century Excellent Talents in University(070003).
978-1-4244-6712-9/10/$26.00 ©2010 IEEE
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(1) Optimize m single objective functions respectively and then obtain optimal solution f1 ( X1* ), f 2 ( X*2 )," , f m ( X*m )
II. GAME DESCRIPTION FOR MULTI-OBJECTIVE OPTIMIZATION PROBLEMS
where X*i = { x1*i , x2*i , ", xni* } (i = 1, 2,", m) .
The mathematics model of multi-objective problems can be written as follows: seeking design variables let objective functions
s.t.
X = [ x1
⎫ ⎪ F( X) = [ f1 ( X) " f m ( X) ] → min ⎪⎪ ai ≤ xi ≤ bi (i = 1, 2, " , n) ⎬ (1) ⎪ hl ( X ) = 0 (l = 1, 2," , p) ⎪ g k ( X ) ≤ 0 (k = 1, 2," , q) ⎪⎭ x2 " xn ]
(2)When the relationship between objective function and design variables is implicit, to arbitrary design variable x j , it is divided into T fragments according to the step length feasible space and
T
Δ ji = ∑ f i ( x1*i ," , x(* j −1) i , x j (t ), x(* j +1) i ," xni* ) − t =1
number of equality constraint ,s non-upper limit and non-low limit; q is the number of inequality constraint ,s non-upper limit and non-low limit.
* 1i
jth strategy. Player
Δ ji =
i , s payoff is ui .
game
," x ） (T ⋅ Δx j )
∂f i ( x1*i ," , x(* j −1) i , x j , x(*j +1) i ," , xni* ) ∂x j
x j = x*ji
(3)
Sometimes, in order to avoid magnitude impact of different objective function, do non-dimensional treatment to Δ ji .
Δ ji = (3)
Classification
Δ = {Δ1 , Δ 2 ," , Δ n } . The mean of
of
Δ ji
(4)
f i ( X i* ) entireness
of
all
samples
are
Δ j is the set of factor index (arbitrary
jth design variable affecting m objective functions). The purpose of their classification is to cluster strong similarity samples into one sample. In this paper, tkl (similarity nearness degree) reflects the nearness degree between samples. To any two samples: Δk and function by normal distribution.
S1 ∪ " ∪ S m = X ； Sa ∩ Sb = 0( a, b = 1," , m； a ≠ b) .
Δl , define fuzzy relation
⎛ ⎞ ⎜ ⎟ Δ ki − Δli ⎟ μi (Δ k , Δl ) = exp ⎜ − m ⎜ 1 ⎟ ⎜ m ∑ Δ ki − Δli ⎟ i =1 ⎝ ⎠ (k , l = 1, 2," , n; k ≠ l; i = 1, 2," , m)
III. COMPUTING GAME PLAYER ,S STRATEGY SPACE The key technology of transforming multi-objective problem into game problem is to divide the variable set X into each player , s strategy space ( S1 ," , S m ). By computing the factor index and fuzzy clustering, it gets each player , s strategy space: follows.
, x j (t − 1), x
(2)
* ni
(i = 1, 2," , m； j = 1, 2," , n)
m players can be written as G = {S1 , " , Sm ; f1 , " , f m } . f1 ," , f m stand for m -design goals( m -game players). S1 = { xi " x j } ," , S m = { xk " xl } stand for strategy sets of m -game players and satisfy the
* ( j +1) i
When the relationship between objective function and design variables is dominant, Δ ji can be calculated as follows:
From the viewpoint of game, the nature of multi-objective design problem is similar to decision-making problem in the game. m -design goals can be regarded as m -game players. Design variables X can be regarded as a collection of all the game strategy S1 , S 2 ," , S m . The feasible region of design variables can be regarded as a viable strategy. Using specific technological means, the variable set X can be divided into each player , s strategy space: S1 , S 2 ," , S m . m -value of the objective function can be regarded as the corresponding benefit in the game. The constraint in multi-objective problem can be regarded as constraint in the game. So
* ( j −1) i
f i ( x ," , x
According to game theory, using G represents one game theory. If G has m players, the set of strategies available to each game player is called strategy set, which is respectively , expressed with S1 , S 2 ," , S m and Si is player i s strategy.
sij is player i , s the
Δ ji (the impact factor of design variable
x j affecting player i ) can be calculated as follows :
,
In the formula, bi and ai are the design variables xi s upper limit and low limit; n is the number of variables; p is the
Δx j in
Where
S1 ," , Sm .The calculating principles are as
between
2221
μi (Δk , Δl )
(5)
is referred to the fuzzy relation
Δk and Δl in the ith objective function.
d (Δ k , Δ l ) =
1 m ∑ μi (Δ k , Δ l ) − 1 m i =1
the corresponding game players according to the average impact factor (design variables of strategy space acting on each game side). When the number of design variables and objective functions are small, can directly divide variable sets X into strategy space S1 ,", S m according to the size of impact factor. When the number design variables and objective functions are large, fuzzy clustering are needed. Meanwhile, according to experience, can first classify variables with strong correlation as a sample to reduce the complexity of clustering analysis.
(k , l = 1, 2," , n; k ≠ l ) (6)
d (Δ k , Δ l ) is referred to Hamming distance between Δk and Δl . Where
m
μi (Δ k , Δ l )
i =1
[m + ∑ μi (Δ k , Δ l )]
σ ( Δ k , Δ l ) = 2∑
m
(7)
i =1
(k , l = 1, 2," , n; k ≠ l ) Where
σ (Δ k , Δl )
IV. CLUSTERING ANALYSIS STEPS
is referred to fuzzy nearness degree
(1) Stipulate M (control value of systematic classification) and P (the largest number of samples). Classification entireness is Δ1 , Δ2 ,", Δn and constitute the
between Δk and Δl .
r (Δ k , Δ l ) =
1 m ∑ ξi (Δ k , Δl ) m i =1
(k , l = 1, 2," , n; k ≠ l )
(8)
matrix
(2) tab =
Where r ( Δ k , Δ l ) denotes correlation degree between Δk and Δl ;
ξi ( Δ k , Δ l )
T (0) by computing tkl ( tkl = tlk , tkl > 0 ). max tkl , merge Δa and Δb as a new sample
k , l ∈{1, 2 ,", n}
is referred to correlation coefficient
named Δs . If the sample number of
between Δk and Δl in the ith objective function and can be expressed as:
(0)
min 1 − μi (Δ k , Δl ) + 0.5 max 1 − μi (Δ k , Δl )
i∈{1,2,", m} ξi (Δ k , Δl ) = i∈{1,2,",m} 1 − μi (Δ k , Δ l )) + 0.5 max 1 − μi (Δ k , Δl ))
merge the second largest value in T
nearness degree of
A. Solving method based on Nash equilibrium model 1) Definition The definition of Nash equilibrium is defined as follows: In the game G = {S1， "， S m； u1， "， u m } , each
{
*
*
*
}
player’s strategy can be assembled into a set s1 , s 2 , " , s m , *
if arbitrary player i ’s strategy si is the best strategy to all the other’s strategy set , it exists :
based on tkl (similarity
ui ( s1* ," , si* ," , sm* ) ≤ ui ( s1* ," , sij ," , sm* ) for any
t12 " t1n t22 " t2 n # # #
sij ∈ S i , then we call equilibrium for game G.
tn1 tn 2 " tnn
{s ," s ," , s } * 1
* i
* m
a Nash
2) Solution steps of Nash equilibrium model (1) Obtain S1 ," , Sm attached to each game player by computing impact factor and doing fuzzy clustering.
Classification results of Δ represent the classification results of X because of one to one relationship between Δ = { Δ1 , Δ2 ," , Δn } and X = { x1 , x2 ," , xn } .
(0)
{
(0)
(0)
(0)
}
(2) Generate s = s1 , s 2 , " , s m in each player’s strategy space randomly as initial feasible strategy.
X into strategy space S1 ,", S m
according to fuzzy clustering results, then put
Δc and Δs .
V. GAME ANALYSIS METHODS OF MULTI-OBJECTIVE
(10)
nearness degree) and do fuzzy clustering to matrix T .
(5) Divide variable sets
a
(4) Repeat steps (1), (2), (3) until the system category number is equal to the control value M .
In the formula, ωd 、 ωσ 、 ωr are weight values of Hamming distance, fuzzy nearness degree and correlation degree respectively and their sum is equal to one ( ωd + ωσ + ωr = 1 ).
t11 t21 T= #
Δs as (1)
Establish tkl (similarity nearness degree) based on the above three indicators.
(4) Establish the matrix T
as a new sample.
new classification system, then constitute a new matrix T and tcs = min{tca , tcb } , where tcs is referred to similarity
(9)
i∈{1,2,", m}
( k , l = 1, 2," , n; k ≠ l )
greater than P ,
(3) Combine Δc (c = 1,2,", n; c ≠ a, c ≠ b) with
( k , l = 1, 2," , n; k ≠ l ; i = 1, 2," , m)
tkl = ωd [1 − d ( Δ k , Δ l )] + ωσ σ ( Δ k , Δ l ) + ωr r ( Δ k , Δ l )
Δs is
S1 ,", S m to
2222
(3)
Let
s1( 0) , s 2( 0 ) , " , s m( 0)
supplementary set in strategy set s
(0)
(1) Obtain S1 ," , Sm attached to each game player by computing impact factor and doing fuzzy clustering.
s1( 0) , s 2( 0) , " , s m( 0)
be
. For the ith player
(0)
(i = 1,2,", m) , fix s according to the player’s payoff u i , then begin the single objective optimization design in strategy space S i which belongs to the game player i.
(3)
(0)
}
Let
s1( 0) , s 2( 0 ) , " , s m( 0)
be (0)
s1( 0) , s 2( 0) , " , s m( 0) . For the
ith player
(i = 1,2,", m) , fix s according to the player’s payoff u i , then begin the single objective optimization design in strategy space S i which belongs to the game player i.
ui ( si* , si( 0 ) ) = f i ( si* , si( 0 ) ) → min And satisfies restricted conditions:
*
Seek the best solution: si ∈ S i
hl ( X ) = 0 (l = 1, 2," , p) gk ( X ) ≤ 0
(4) Let strategy set be
s
Let payoff be:
(k = 1, 2," , q)
(1)
* 1
* 2
ui ( si* , si( 0) ) = wii
* m
= s ∪ s ∪ " ∪ s , testes
s (1) feasibility, if it dissatisfies, goes to step 2, otherwise
j =1( j ≠ i )
m
otherwise
f j ( si( 0) , si(0) )
→ min
s (1) = s1* ∪ s 2* ∪ " ∪ s m* , testes
s (1) replaces s ( 0) and goes to step 3 for recycling.
C. Solving method based on evolutionary game model 1) Assumptions of definition When adopt either Nash equilibrium model or coalition cooperative mode, behavior of all game players remains the same in the process, which is flawed. It proposes evolutionary game model to solve the deficiency. Evolutionary game model combines research results of classical game theory and biology and takes limited rational game players as research object. It uses of dynamic evolution method to research competition and cooperative behavior as well as examines game Equilibrium by system theory [11-12]. Two most important assumptions in evolutionary game theory are as follows. (1) Dynamics property of game behavior. Each game can change the behavior with the natural environment, survival networks or self-consciousness. Variability of game dynamic behavior has biological base, for example, J. Buhl [13] proposes although locusts are usually timid and like living alone （ Non-cooperative behavior ） with limited hazard, locust will change the original habits and like the gregarious life (cooperative behavior）with great harm when the hind legs of locusts were repeatedly touched to reach a certain threshold value. In a word, locusts can change their behaviors from the single non-cooperative behavior to cooperative herd behavior when the number of locusts increase and the touch with each other increases.
(11)
j =1
u i denotes player i , s absolute non-dimensional
uˆij is relative non-dimensional income. w
f j ( si* , si( 0) )
the before and after strategy set ; if true, the game is over,
B. Solving method based on coalition cooperative model 1) Definition Based on collective rationality in cooperative game, each player complies with a protocol which has constraint force. Each game player determines the strategy through this protocol. The results of each game player may not be the optimal results, but they are relatively good results, which means cooperative game results are Pareto optimal solution. In conspiring cooperative model, it advocates each player considering other players in payoff function to form cooperative mechanism and complies with the agreement of “You have me, I have you”. Each game player adopts tradeoff method and uses weight coefficient to obtain final ui by combining its absolute income with relative income.
(∑ wij = 1, wij = w ji )
j =1( j ≠ i )
wij
computes the distance (a kind norm) to check whether it satisfies convergence criterion s (1) − s ( 0 ) ≤ ε according to
s (1) replaces s ( 0) and goes to step 3 for recycling
m
m
∑
s (1) feasibility, if it dissatisfies, goes to step 2, otherwise
the before and after strategy set; if true, the game is over,
∑ wij uˆ ij
fi ( si* , si( 0) ) + fi ( si( 0) , si( 0) )
(4) Let strategy set be
computes the distance (a kind norm) to check whether it satisfies convergence criterion s (1) − s ( 0 ) ≤ ε according to
Where income.
(0)
(0) i
Let payoff be:
u i = wii u i +
(0)
supplementary set in strategy set s
*
Solve optimal measure: si ∈ S i
otherwise
{
(2) Generate s = s1 , s 2 , " , s m in each player’s strategy space randomly as initial feasible strategy.
(0) i
is weight
wii is large, indicates a low degree of cooperation ; When wii is equal to one, indicates there is only coefficient. When
competition between the games and coalition cooperative model degenerates Nash equilibrium model.
(2) Limited rationality of game players. It means the game players can not make the best response to the environment, but determine the behavior based on an evolutionary game rules.
2) Solution steps of coalition cooperative model
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m
The most well-known rule of evolutionary game is “Tit For Tat” (TFT). In 1984, Axelrod [14-15] collected the game rules from game experts and submitted all rules to computer simulation and the results showed that TFT got the highest score. “TFT” means that game player first takes cooperative strategy and takes identical behavior with the other players in the next round. The current rules of evolution game can be divided into three categories: 1) Passive category. The decision-making power depends on the changes of natural environment and the survival network such as “TFT”. 2) Active category. The decision-making power depends on selfknowledge (The knowledge reflects intelligence and memory of game players). 3) Mixed category. Lin Hai [16] proposes that the decision-making power depends on the objective network structure and subjective memory effect.
Where
(2) Establish corresponding payoff functions of game according to (12) and (13). (0)
(0)
(0)
}
(4) All game players adopt “Successful” behavior in the m f first round ( u = w f i + ∑ w j ). For the kth (k>1) round: i ii ij f i j =1( j ≠ i ) fj
(5)
when
fi ( k −1) ≤ fi
⇒ ui = wii
when
fi ( k −1) > fi
⇒
Let
m fj ⎫ fi + ∑ wij ⎪ fi j =1( j ≠i ) fj ⎪ ⎬ (14) f ⎪ ui = i ⎪⎭ fi
s1( 0) , s 2( 0 ) , " , s m( 0)
s1( 0) , s 2( 0) , " , s m( 0)
be
supplementary set in strategy set s
(0)
. For the
ith player
(0)
(i = 1,2,", m) , fix si according to the player’s payoff u i , then begin the single objective optimization design in strategy space S i which belongs to the player i. *
Seek the best solution: si ∈ S i *
(0)
Let payoff be: u i ( s i , s i
(6) Let strategy set be s
) → min
(1)
= s1* ∪ s 2* ∪ " ∪ s m* , testes
s (1) feasibility, if it dissatisfies, goes to step 3, otherwise
computes the distance (a kind norm) to check whether it satisfies convergence criterion s
(1)
− s (0) ≤ ε according to
the before and after strategy set ; if true, the game is over, (1) (0) otherwise s replaces s and goes to step 4 for recycling.
(12)
VI．EXAMPLE VERIFICATION—TWO OBJECTIVES DESIGN OF FOUR BAR JOIST RACK STRUCTURES To validate correctness of the three kinds of game models (Nash equilibrium game model, Coalition cooperative game model, Evolutionary game model) to solve multi-objective optimization problems, compile calculation program based on Fortran language independently and does the optimization design aiming to four bar joist rack structures .
Where f is corresponding objective function value of the initial design .
b) “Successful” behavior. Its characteristics is cooperative collectivism and its corresponding function is as follows: m f fi + ∑ wij j fi j =1( j ≠i ) fj
{
(0)
(3) Generate s = s1 , s 2 , " , s m in each player’s strategy space randomly as initial feasible strategy.
3) Behaviors of game players a) “Poor” behavior. Its characteristics is egoism of non-cooperation and its corresponding function is as follows:
ui = wii
= 1 , wii reflects the degree of “Successful”.
If its value is large, the degree of “Successful” behavior is small. 4) Solution steps of evolutionary game model (1) Obtain S1 ," , Sm attached to each game player by computing impact factor and doing fuzzy clustering.
2) Evolution rules of game behavior This article absorbs the essence of Chinese traditional culture (“Successful, they made all the world good. Poor， they made themselves alone good”. ) and formulates the corresponding rules of evolution: (1) When the objective function value of this round is worse than the initial design, it adopts non-cooperative behavior in the next round. (2) When the objective function value of this round is better than the initial design, it adopts cooperative behavior in the next round. (3) All game players adopt cooperative behavior in the first round game. The advantages of this evolution rule lies in four fields: (1) “Autonomy”. Determine the evolution of behavior according to satisfaction degree of their own goals. (2) “Ability to adjust quickly”. It can quickly do adjustment in the next round according to satisfaction degree in the current round. (3) “Friendliness”. All game players first adopt cooperative behavior. (4) “Clarity”. Evolution of the rules are clear and easy to understand.
fi fi
ij
j =1
At present the use of evolutionary game to solve multiobjective optimization problems is very little. Sim [17] proposed co-evolutionary algorithm of evolutionary game theory to solve multi- objective optimization problems but his algorithm was not versatile. Xu Min [18] proposed optimization algorithm of evolutionary game and solved versatile problems by learning evolutionary game and selection mechanism. The basic idea, evolution rules and the algorithm steps which this article describes are very different from the above two methods.
ui =
∑w
(13)
A. Multi-objective optimization model
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Geometric forms and load of our bar joist rack structures are given in figure 1 [19]. Take the structure volume and displacement of node D as the objective functions f1 and f 2 . Take cross-sectional area of each bar as the design variable X = [ x1 , x 2 , x3 , x 4 ] .
Δ3 = (Δ31, Δ32 ) = (
x1 = x
∂f 2 ∂x1
) = (2.0 L,− * 12
x1 = x
2 FL ) 9 E
∂f1 ∂x3
, * x3 = x31
∂f2 ∂x3
) = ( 2 L, 2 * x3 = x32
FL ) E
F F
(3) Fuzzy Clustering In order to avoid magnitude impact of different objective functions, do non-dimensional treatment to impact factors.
Figure 1: four bar joist rack structures
Optimal design model is:
The treatment results are:
Δ1= {2857.143, -0.000080474}, Δ2={2020.305, -0.000113807}
f1( x) = L(2 x1 + 2 x2 + 2 x3 + x4 ) → min
f 2 ( x) = (FL/E)(2 / x1 + 2 2 / x2 − 2 2 / x3 + 2 / x4 ) → min s.t. (F/ σ ) ≤ x1 ≤ 3(F/ σ ) 2 (F/ σ ) ≤ x2 ≤ 3(F/ σ ) 2 (F/ σ ) ≤ x3 ≤ 3(F/ σ ) (F/ σ ) ≤ x4 ≤ 3(F/ σ )
(15)
Δ3= {2020.305, 0.000512132}, Δ4={1428.572, -0.000080474} Because the number of optimization objective function and design variable are two and four respectively, let M (control value of systematic classification) be equal to two, P (the largest number of samples) be equal to three. The process of cluster analysis is as follows under the premise of ωd = ωσ = ωr = 1 3 .
Where: F = 10 kN ， E = 2 × 10 3 GPa ， L = 2 m ， σ = 100 MPa .
The matrix
B. Theoretical solution Stadler [20] proved x3 was a constant and its value was
2(F/ σ ) = 2 cm 2 and the efficient theoretical solution set can be written to the equation of parameter t . ⎧ 2 ⎪ (1) F/σ ≤ t ≤ F/σ： x1 = F/σ, x2 = 2F/σ, x3 = 2F/σ, x4 = 2t 2 ⎪ ⎪⎪ 3 2 F/σ： x1 =t, x2 = 2t, x3 = 2F/σ, x4 = 2t ⎨ (2)F/σ ≤ t ≤ 2 ⎪ ⎪ 32 F/σ ≤t ≤ 3F/σ： x1 =t， x2 = 3F/σ， x3 = 2F/σ, x4 = 3F/σ ⎪ (3) 2 ⎪⎩
T (0)
x 2 can be classified to a class with any other design variables including x1 , x3 and x 4 . Besides, x1 and x 4 belong to a class. x3 and x1 does not belong to a class because of t13 = 1.4212040 ; x3 and x 4 does not belong to a class because of t 34 = 1.4212036 . By analysing impact factor, knows that
]
Δ1 , Δ2
Δ4 are positively relevant and they are negatively relevant with Δ3 .
* * * * X 1* = [ x11 , x 21 , x31 , x 41 ] = 1.0, 2 , 2 ,1.0 × 10 −4 m 2 ,
f1 ( X 1* ) = 1.4 ×10−3 m3 ;
[
⎧* 1.4212048 1.4212040 1.4212048 ⎫ ⎪ * 1.4212048 1.4212048 ⎪⎪ ⎪ =⎨ ⎬ * 1.4212036 ⎪ ⎪ ⎪⎩ * ⎭⎪ (0)
,
[
T (0) based on similarity nearness degree is:
There are four identical maximum values in T as follows: t12 = t14 = t23 = t24 = 1.4212048 . Design variables
(16)
C. Computing game player s strategy space (1) Single-objective optimization results
and
Considering these factors comprehensively, knows that strategy space of f 2 is x3 and strategy space of f1 is
]
* * * X 2* = [ x12 , x22 , x32 , x*42 ] = 3.0,3.0, 2,3.0 ×10−4 m2 ,
f 2 ( X 2* ) =
, * 11
∂f ∂f 2 FL Δ 4 = (Δ 41, Δ 42 ) = ( 1 , 2 ) = (1.0L,− ) ∂x4 x = x* ∂x4 x = x* 9 E 4 41 4 42
L
L
∂f1 ∂x1
∂f ∂f 2 2 FL Δ2 = (Δ21, Δ22 ) = ( 1 , 2 ) = ( 2 L,− ) ∂x2 x = x* ∂x2 x = x* 9 E 2 21 2 22
D
F
L
Δ1 = ( Δ11 , Δ12 ) = (
x1 , x2 and x 4 .
2( 2 − 1) × 10 − 4 m . 3
D. The results of game analysis 1) The results based on Nash equilibrium game model Table I shows the calculation results.
(2) Calculation results of the impact factors
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TABLE I.
THE RESULTS BASED ON NASH EQUILIBRIUM GAME MODEL -3
Game round
f1/10 m
Initial strategy Bout 1 Bout 2 Bout 3
2.3314 1.4298 1.4193 1.4193
-4
3
-4
f2/10 m
x1/10 m
2.0000 3.8471 3.8667 3.8667
2.0000 1.0363 1.0220 1.0220
2
-4
x2/10 m
2
-4
x3 /10 m
2.0000 1.4622 1.4220 1.4220
2
2.0000 1.4142 1.4142 1.4142
-4
x4 /10 m
w11 = w22 = 0.75 ， w12 = w21 = 0.25 , Algorithm starts
from the initial strategy X 0 = [2,2,2,2]× 10 −4 m 2 and obtains convergence value after three round game. Solution is
2
2.0000 1.0087 1.0413 1.0413
strategy
equivalent to parameter t = 7 F/ 2σ and exact values of design variables are Xˆ = ⎡ 7 2, 14 2, 2, 14 2 ⎤ ×10−4 m 2 . ⎣ ⎦
and exact values of design variable are
Putting Xˆ into (15), the theoretical value of f1 and f2 are 1.8325×10-3m3 and 2.0928×10-4m respectively. The corresponding errors are 0.56% and 1.41% respectively.
Algorithm
starts
from
the
initial
X = [2,2,2,2]× 10 m and obtains convergence value after three round game. Solution is equivalent to parameter −4
0
t = 2 F/ 2σ
2
Xˆ = ⎡1.0, 2, 2,1.0⎤ ×10−4 m 2 . ⎣ ⎦
4) The results analysis Comparison results of three kinds methods above are shown in Table IV.
Putting Xˆ into (15), the theoretical value of f1 and f2 are and 4.0000×10-4m respectively. The 1.4000×10-3m3 corresponding errors are 1.38% and 3.33% respectively.
TABLE IV.
2) The results based on coalition cooperative game model Table II shows the calculation results. TABLE II.
Nash equilibrium game model coalition cooperative game model
MODEL Game round
-
-
f1/10 3 3 m
-
f2/10 m
-
x1/10 2 m
4
-
-
x3 /10 2 m
x2/10 4 2 m
4
x4 /10 2 m
4
evolutionary game model
2.3314
2.0000
2.0000
2.0000
2.0000
2.0000
1.9531
1.7749
1.4358
2.0360
1.4142
2.0149
Bout 2
1.8428
2.0638
1.3482
1.8668
1.4142
1.8777
Bout 3
1.7176
2.4580
1.2642
1.7357
1.4142
1.6047
Bout 4
1.6077
2.8567
1.1430
1.5697
1.4142
1.5326
Bout 5
1.5467
3.1285
1.0483
1.5926
1.4142
1.3844
Bout 6
1.4754
3.4607
1.0279
1.4188
1.4142
1.3145
Bout 7 Bout 8
1.4337 1.4337
3.7134 3.7134
1.0006 1.0006
1.4161 1.4161
1.4142 1.4142
1.1646 1.1646
2
Putting Xˆ into (15), the theoretical value of f1 and f2 are 1.4345×10-3m3 and 3.7056×10-4m respectively. The corresponding errors are 0.06% and 0.21% respectively. 3) The results based on evolutionary game model Table III shows the calculation results. THE RESULTS BASED ON EVOLUTIONARY GAME MODEL
Objective behaviors -3 3 f1/10 m
Initial strategy
2.3314
functions
and
-
-
-
4
x2/10 2 m
4
x3/10 2 m
4
2.0000
2.0000
2.2000
2.2000
2.0000
Bout 1
1.9091 (“successful”)
1.8850 (“successful”)
1.4016
1.9728
1.4142
1.9524
Bout 2
1.8428 (“successful”)
2.0638 (“successful”)
1.3482
1.8668
1.4142
1.8777
Bout 3
1.8428 (“successful”)
2.0638 (“poor”)
1.3482
1.8668
1.4142
1.8777
f2 /10 m
3.33%
3 round
0.06%
0.21%
8 round
0.56%
1.41%
3 round
(2) Propose the method of computing game player , s strategy space. The key technology of transforming multi-objective problems into game problems is to divide the variable set X into each player , s strategy space S1 ," , S m .
-
x1/10 4 2 m
-4
1.38%
(1) Propose multi-objective optimization problems described in the game. The nature of multi-objective design problems is similar to decision-making problems in the game. m design objectives can be considered as m game players. Design variables X can be regarded as a collection of all the game strategies S1 , S 2 ," , S m . The feasible region of design variables can be regarded as a viable strategy . Using specific technological means, the variable set X can be divided into each player , s strategy space: S1 , S 2 ," , S m . m -value of the objective function can be regarded as the corresponding benefit in the game. The constraint in multi-objective problem can be regarded as constraint in the game.
equivalent to parameter t = 11 F/ 4σ and exact values of design variable are Xˆ = ⎡1, 2, 2, 11 8 ⎤ × 10−4 m 2 . ⎣ ⎦
Game round
f2
VII．CONCLUSION
from the initial strategy X = [2,2,2,2]× 10 m and obtains convergence value after eight round game. Solution is
TABLE III.
f1
The results show that the computational precision of the coalition cooperative game model is the best ,which illustrates that cooperative game has the advantages over noncooperative game in fulfilling polytropic win-win and collective benefit; but the coalition cooperative game model is the worst in view of computational efficiency , which shows that cooperation spends more time negotiating so as to obtain win-win; considering these factors comprehensively, evolutionary game model is better at computational precision and efficiency.
w11 = w22 = 0.75 ， w12 = w21 = 0.25 , Algorithm starts −4
Calculation efficiency
4
Initial strategy Bout 1
0
RESULTS
The error between calculated and theoretical value
Game model
THE RESULTS BASED ON COALITION COOPERATIVE GAME
COMPARISON
x4/10 2 m
Get each player , s strategy space by computing the factor index and fuzzy clustering which includes correlation analysis and clustering analysis.
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[15] Axelord R, Dion D, “The further evolution of cooperation,” Science, Vol. 12,pp.1385~1390,1988. [16] Lin Hai ， WU Chengxu, “Evolution of strategies based on genetic algorithm in the iterated prisoner's dilemma on complex networks,” Acta Physica sinica , Vol. 56 NO. 8, pp.4313~4318,2007. [17] Kwee-Bo Sim, Dong-Wook L ee, J i-Yoon Kim, “Game theory based coevolutionary algorithm: a new computational coevolutionary approach,” International Journal of Control, Automation, and Systems, Vol.2 NO. 4, pp.463~474, 2004. [18] Xu Min, Zhang Min, Wang Xufa, “Evolutionary Game Based for Multiobjective Optimization Problems Optimizaiton Algorithm,” Journal of chinese computer systems，Vol.28 NO. 4, pp.640~644,2007. [19] Li Xingsi , Zhang Qi , Tan Tao, “Method of centers for structure optimization with multiple objectives,” Chinese journal of theoretical and applied mechanics, Vol.37 NO. 5, pp.606~610，2005. [20] Stadler W， Dauer J, “Multi-criteria Optimization in Engineering, A Tutorial and Survey, Structural Optimization: Status and Future,” American Institute of Aeronautics and Astronautics, 1992.
(3) Propose Nash equilibrium game model according to non-cooperative game theory and list the corresponding algorithm; Propose coalition cooperative game model according to cooperative game theory and list the corresponding algorithm; Propose evolutionary game model based on evolutionary game theory and adopt the rules of evolution (“Successful, they made all the world good. Poor, they made themselves alone good.”) and list the corresponding algorithm. (4) Taking four bar joist rack structures for example, compile calculation program based on FORTRAN language independently. The results reflect the validity of these methods.
ACKNOWLEDGMENT This project is supported by Natural Science research project of Anhui provincial Education Department (NO.070414154) and New Century Excellent Talents in University (070003). REFERENCES [1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12] [13]
[14]
Suheyla Ozymldmrmm, Nedim M. Alemdar, “Learning the optimum as a Nash equilibrium,” Journal of Economic Dynamics & Control, Vol.24, pp.483~499, 2000. R. Spallino, S. Rizzo, “Multi-objective discrete optimization of laminated structures,” Mechanics Research Communications, Vol. 29, pp.17~25, 2002. J. Periaux, H.Q.Chen, B.Mantel, M.Sefrioui, H.T.Sui, “Combining game theory and genetic algorithms with application to DDM-nozzle optimization problems,” Finite Elements in Analysis and Design, Vol.37 NO.5,pp.417~429,2001. Wang Jiangfeng, Wu Yizhao, Périaux J, “Decentralized multi-point optimization algorithms for multi-airfoil design in aerodynamics,” Journal of Astronautics, Vol.24 NO. 1,pp.71~77,2003. Xie Nenggang, Na Shi, Jiahan Bao, Hao Fang, “Analysis and application of multi-object decision design based on game theory,”6th World Congresses of Structural and Multidisciplinary Optimization,Rio de Janeiro, Brazil,30 May - 03 June 2005. Xie Nenggang，Fang Hao，Bao Jiahan, Zhao Lei, “Analysis of multiobjective design on mechanism of compensative sheave block ,” Journal of Mechanical Strength，Vol.27 NO. 2, pp.202~206,2005. Xie Nenggang，Hao Fang，Bao Jiahan，Zhao Lei, “multi- objective game design and application of mechanism with objective predilection ,” Journal of system simulation，Vol.19 NO. 1 ,pp.138~140,2007. A.K.Dhingra, S.S.Rao, “A cooperative fuzzy game theoretic approach to multiple objective design optimization,” European Journal of Operational Research, Vol. 83, pp.547~567,1995. Li Chen, Simon Li, “A computerized team approach for concurrent product and process design optimization ,”Computer-Aided Design, Vol. 34, pp.57~69,2002. Xie Nenggang ， Sun LinSong ， Bao Jiahan ， Fang Hao,“Multiobjective game design of arch dam shape ,” Acta mechanica solida sinica，Vol. 27 NO. 2 , pp.200~206,2007. Wang Li, “A computing simulation and analysis of evolutionary game Theory,” Journal of capital university of economics and business，Vol. 4, pp.5~8,2005 Hauert C, “Spatial effects in social dilemmas,”Journal of Theoretical Biology, Vol.240 NO. 4, pp.627~636,2006. J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller, and S. J. Simpson, “As predicted by statistical models, swarms of locusts undergo rapid transitions from disordered to ordered collective motion as their density increases,”Science, Vol. 6, pp.1402~1406,2006. Axelord R. The Evolution of Cooperation [M]. Arts &Licensing International Inc , 1984.
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