Intelligent Garbage Can Decision-Making Model Evolution Algorithm for optimization of structural topology of plane trusses

Applied Soft Computing 12 (2012) 2719–2727

Contents lists available at SciVerse ScienceDirect

Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Intelligent Garbage Can Decision-Making Model Evolution Algorithm for optimization of structural topology of plane trusses Hsin-Chuan Kuo a , Jinn-Tong Chiu b,∗ , Ching-Hai Lin a a b

Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Taiwan Department of Mechanical Engineering, De Lin Institute of Technology, Taiwan

a r t i c l e

i n f o

Article history: Received 16 November 2009 Received in revised form 12 January 2012 Accepted 4 March 2012 Available online 11 May 2012 Keywords: Optimum design Differential evolution algorithms Garbage Can Decision-Making Model Topology optimization

a b s t r a c t The optimum design of structural topology of trusses is widely acknowledged as the most difficult and challenging problem in the area of structural optimization. Based on differential evolution algorithms and using the framework of a Garbage Can Decision-Making Model, we proposed an Intelligent Garbage Can Decision-Making Model Evolution Algorithm (IGCMEA) to simulate the decision-making process in human social organizations. In a decision-making process, when faced with issues such as unclear goals and methods, employee turnover and so forth, representatives of all participating parties will communicate, argue, compromise and adapt with each other in order to find a solution to the problems. Group meetings are conducted to choose the best solution in a more objective, reasonable and efficient manner. By applying the differential evolution (DE) algorithm and IGCMEA to perform an optimization test on the 100-dimensional Schwefel Function, we showed that IGCMEA can achieve an efficient and satisfactory result. We also optimized the truss topology using IGCMEA and obtained a better result than when using the Genetic Algorithm as in the literature, thus illustrating the superior power of IGCMEA. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Compared with sectional area optimization, truss topology optimization is the more difficult problem. Sectional area optimization only involves finding the members’ sectional areas, while topology optimization also requires the determining of the connection type between members. Moreover, during the optimization process, structural analysis, optimization model, design space and feasible region may change constantly. These factors all add to the difficulty of the optimization. The structural topology optimization problem was first proposed in 1904 in Michell’s truss theory [1]; later, scholars proposed a series of optimization methods, most of which are aimed at the structure design problem with real continuous variables. The ground structure method proposed by Dorn in 1964 [2] is a good representative of these types of problems. It constructs a set of ground structure nodes, consisting of structure nodes, loading points and supporting points. In a single load condition, considering the stress constraints, it considers the internal forces of members as design variables and solves them with linear programming. In 1969, Dobbs and Felton [3] used the sectional area as design variables, evaluated the constraints of stress and displacement in

∗ Corresponding author. E-mail address: [email protected] (J.-T. Chiu). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2012.03.011

various load conditions, and solved them through non-linear programming. In 1992, Kirsch and Topping [4] divided the topology optimization into two stages. In the first stage, the redundant internal forces and sectional areas of members are taken as design variables, ignoring the deformation coordination conditions and displacement constraints. This simplifies the problem to a linear programming problem that can be easily solved and give a lower bound of the solution. In the second stage, all constraints are respected and non-linear programming is applied around the lower bound found in the first stage, in order to find the members’ sectional areas. However, due to the neglected deformation coordination conditions and displacement constraints, the lower bound derived in the first stage for static structures or displacement constrained structure problems is often poor. Generally speaking, the topology optimization methods for continuous fields can be divided into two categories:

(1) Linear Programming: The members’ internal forces are used as design variables. The model is simple, but it is difficult to incorporate the deformation coordination conditions, displacement constraints and multi-load conditions. (2) Non-Linear programming: The sectional areas are design variables. It is flexible in its ability to incorporate the deformation coordination conditions, displacement constraints and multiload conditions.

2720

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

However, the non-linear programming requires heavy computations and generally cannot find the global optimal solution, or even a satisfactory local optimal solution. In 1975, John H. Holland introduced the concept of natural selection to the algorithms and proposed the so-called Genetic Algorithms (GA) [5]. This heuristic algorithm has become a popular research topic during the recent three decades. Later, other bio-inspired population algorithms gradually emerged [6]. Most scholars studying garbage can decision theory [7] focus on finding an accurate description of the complicated organization and decision-making. They rarely explore the hidden motivations behind the phenomenon. While an accurate description of the decision-making process is certainly necessary, this action is only the starting point for constructing a systematic decision-making theory. Decision-making is an important part of administration. Theoretical models of administrative decision-making include the following three: (1) Classical Model: Proposed by Frederick Winslow Taylor in 1947 [8], its basic concept approaches an optimistic strategy, assuming that decisionmakers are fully rational and use the most promising method to achieve their goals. Their results are perfect. Its main features are: (a) (b) (c) (d) (e)

Identify the problem. Establish goals and objectives. Find possible solutions. Analyze possible consequences. Evaluate all possible solutions based on the established goals and objectives. (f) Select the best option. (g) This decision is practical and assessable.

(2) Administrative Model: Proposed by Simon in 1974 [9], its basic concept approaches a satisfcing strategy and assuming that, limited by the existing knowledge, resources and information, it is impossible for decisionmakers to make perfect decisions. They can only try to achieve satisfactory results. Its main features are: (a) Usually, goals are set before decisions are made. (b) Decision-making is a process of goal-means analysis; decisionmaking is a recursive action; goals may be adjusted during the analysis of consequences. (c) The key measure of a decision is whether it achieves the goals satisfactorily. (d) The problem is explored until a reasonable decision is determined. (e) Theory and experience are both relied upon. (3) Incremental Model: Proposed by Lindblom in 1959 [10], its basic concept is like a constant comparison strategy under a certain threshold. It assumes that, limited by the existing knowledge, resources and evidence, decision-makers can make neither perfect nor satisfactory decisions. Only through constant comparison can they identify the feasible decision that is most suitable for the current situation. Its main features are: (a) Goals and decisions are usually determined simultaneously, which makes the goal-means analysis inappropriate. (b) A good decision means that the decision-makers agree to drop other means for this decision. (c) Many decisions and results are dropped after consideration, leaving only one in line with the current situation.

(d) Analysis is also subject to the differences between the existing situation and the plan. (e) The solution is to avoid theory, constantly choosing the more concrete and practical options. Charles Lindblom said that when the problem to be solved is of high complexity, uncertainty and full of controversy, the incremental model is probably the only viable decision model. This decision model can be likened to crossing a river by feeling the stones. The decision-makers are aware of the goals, but uncertain about the actions required to achieve them. They assess the results after reaching an intermediate stage and then adjust the direction to move on. It is very similar to the real decision-making process in a human social organization. The Garbage Can Model is closest to the incremental model. In this spirit, we can infer that the decisionmaking process in an organization is like an evolving process. Considering the above, this study attempts to interpret the evolution of the population system in population based algorithms by the Garbage Can Model. Based on the differential evolution algorithm framework, we propose a new evolution algorithm. This study applies the new evolution algorithm with shape selection strategies to the optimization design of the structural topology in order to objectively deal with the structural stress and displacement constraints, devise a highly flexible member deletion mechanism and, finally, find the optimal design. The shape selection strategies adopted are: (a) According to Dorn’s concept of ground structure [2], the non ground structures in the population are set to the ground structures closest to the design, forming the ground structure collection of structural nodes, loading points and supporting points. (b) The best average function value from each shape of ground structure as the search reference in the next phase is selected. Analogous to the roulette wheel selection model, the shapes are assigned double the probability of being selected than the original population. 2. Differential evolution (DE) algorithm In 1997, Storn and Price [11] introduced a new system that would use simple calculations in conjunction with the mutation and crossover operators from GA. They termed it the differential evolution algorithm as it related to the population evolving through individuals from different generations competing with each other. From one generation to the next, individuals are replaced or evolved. The three basic operations for individuals are mutation, crossover and selection. DE is comprised of six steps, as follows. DE is comprised of six steps, as follows: (1) Initialization Create NP n-dimensional individuals: XiG = Xmin + i (Xmax − Xmin ),

i = 1, 2, . . . , NP

(1)

where XiG is the vector of individual i of generation G and i is a random number uniformly distributed between (0, 1). (2) Evaluation For each individual, evaluate the objective value, or fitness. (3) Mutation A disturbance vector is used to achieve mutation. This vector is equivalent to the sum of a base vector and the product of a difference vector, which is calculated as the difference between two or more random vectors, and the mutation factor G and the difference F. As shown in Fig. 1, the base vector is Xr1

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

2721

After mutation and crossover, individuals from a parents generation and a child generation compete with each other. The best ones are kept. (XiG+1 ) = argmin{F(XiG ), F(y)}

(11)

The best individual of the next generation is given by G+1 ) = argmin{F(XiG )} (Xbest

(12)

where F(X) is the value of the objective function, or fitness value. (6) Stop criterion Repeat steps (3)–(5) until either the objective function value reaches the expectation or the number of generations reaches the maximum. 3. Intelligent Garbage Can Decision-Making Model Evolution Algorithm (IGCMEA)

Fig. 1. Generation of a disturbance vector by mutations.

G and X G , where r1, r2, r3 vector is the difference between Xr2 r3 are randomly selected individuals from generation G. With the definitions, the mutation vector formula is G G G + F(Xr2 − Xr3 ) V G+1 = Xr1

(2)

where F is the mutation factor between (0, 2). It is important to note that X best is the best individual in the generation G. DE mutation strategies [11]: (a) DE/best/1 Xbest + F(Xr1 − Xr2 )

(3)

(b) DE/rand/1 Xr1 + F(Xr2 − Xr3 )

(4)

DE/rand-to-best/1 Xr1 + F(Xr2 − Xr3 )

(5)

DE/best/2 Xbest + F(Xr1 + Xr2 − Xr3 − Xr4 )

(6)

(c) DE/rand/2 Xr5 + F(Xr1 + Xr2 − Xr3 − Xr4 )

(7)

From the above strategies for generating mutation vectors, the user should choose the one that is most applicable to the problem. The individual vector of generation G is written as G G G , X2i , . . . , Xni ) XiG = (X1i

(8)

and the mutation vector is denoted by V G+1 = (V1G+1 , V2G+1 , . . . , VnG+1 )

(9)

(4) Crossover Crossover is also conducted at random. The child vector that is generated by crossover between XiG from the parent generation and VG+1 is:



y = (y1 , y2 , . . . , yn ), yj =

VjG+1 , if rj ≤ CR XjiG , if rj > CR

or

j=l

and

j= / l

(10)

where j = 1, 2, 3,. . ., n, random integer l ∈ {1, 2,. . ., n}, random number rj ∈ U (0,1), and crossover constant CR satisfies 0 ≤ CR ≤ 1. (5) Selection

3.1. Garbage Can Decision-Making Model By considering Simon’s bounded rationality [12], March and Cohen argued that decision-makers can never make the perfect decision, due to the limiting issues of inadequate external information and subjective feelings that may affect rational judgment. In 1972 [7], they proposed the Organized Anarchies Model as an alternative decision model for the organization of an actual decision-making process. This non-rational model includes three main characteristics that can lead to ambiguities: (1) Problematic Preferences As decision-makers have inconsistent preferences for the problems and goals which can only be discovered through actions, these preferences cannot be used as the basis for actions. (2) Unclear Technology During the decision-making period, organizers may only be aware that something needs improvement and not know exactly what method and individual should do to improve. The organizers must therefore use their personal knowledge and utilize trial and error methods. (3) Fluid Participation Various issues can present inconsistencies or ambiguities in the decision-making process. For example, if a long-term investigation is needed before making a decision that has controversial aspects, the decision-makers at the beginning and end of this process might not be the same group of people. Also, decision-makers may have varying perspectives, interests and backgrounds. In such situations, we employ the non-rational, decisionmaking model, known as the Garbage Can Model (GCM). This involves regarding each decision-making process as a receptacle or garbage can and representing decision-makers, issues, and solutions as garbage. Decisions are then usually determined by four forces, known as the governing variables of the framework of Garbage Can Model. These forces are problems/objectives, solutions, participants, and opportunities. • To address the aforementioned potential ambiguities in the decision-making process, we clarify the issues under the guidance of our goals so we can propose some solutions. Problematic Preferences: We must use an integrated object model to overcome controversies. • Unclear Technology: We must pursue creative solutions that are not necessarily standard. • Fluid Participation: We must encourage the exchange of ideas and keenly address any opportunity to solve the problem.

2722

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

Fig. 2. Populations of IGCMEA grouping search strategy.

By combining the idea of the group meeting in the decisionmaking process and the evolution algorithm pertaining to GCM that has been developed in this paper, we propose the new evolution algorithm, named as the Intelligent GCM Evolution Algorithm (IGCMEA). 3.2. IGCMEA Intelligent Garbage Can Model Evolution Algorithm takes GCM as the evolution logic and is based on DE. We show the connections between the differential evolution algorithm and the garbage can model below. individuals in a population → participants in an organism; search space → solution (alternatives) space; mutation strategy → garbage can decision model; objective function → satisfcing function; Xbest → the most satisficing solution (alternatives). The most satisfacing decision of the participants in the organization after evolution of several generations is defined as the best individual of the population. IGCMEA uses a diverse selection model for the mutation operations. A ‘mutation-strategy can’ has several mutation strategies placed in it and then, for each generation, a strategy is randomly selected from the can. As the population matures, a grouping search strategy that splits the parent population into many child populations and integrates the search results of each of these child populations is used. In other words, this strategy identifies the promising space of each variable of each child population and then searches separately in each promising space. For example, if a population consisted of 800 individuals and 400 of those individuals were identified as ‘good’, the promising space would be defined by the intervals of each variable of these 400 ‘good’ individuals. Therefore, in this study, we define the evaluated generation number which is then used to determine an amount of generations after which the parent population is divided into three child populations

and each child population is confined by a search space, as shown in Fig. 2. We generally perform this division every few generations. (1) The first child population: The initial search space is defined as U1 . Ui0 = [x- 0i , x¯ i0 ], i = 1, 2, . . . , n. x- 0i , x¯ i0 are the lower and upper bounds, respectively. (2) The second child population: The search space U2 is the promising space p p p Ui = [x- i , x¯ i ], which is subsequently broadened. (3) The third child population: The search space U3 is the promising pr pr pr space which has Ui = [x- i , x¯ i ], broadened. To give an example, we partition each variable into five segments. For each segment, according to the probability model of a roulette wheel, the individuals that have better average function values are assigned a higher probability of being selected. In this paper, we set four parameters to aid us in simulating the decision-making behavior of human society. These parameters are the evaluated generation number, the number of better individuals, the broaden ratio, and the number of partitioned segments. We then set the amount of partitioned segments to 5. Fig. 3 shows the workflow of IGCMEA. Three required definitions are below. (1) Evaluated generation number, NE: To simulate the regular group meetings, the grouping strategy is applied to the population after every NE generations. (2) Number of better individuals, NB: The individuals with better function values are selected from the evaluated population. Their variables, lower bounds and upper bounds are used to confine the promising space, thus simulating the choices in a meeting. (3) Broaden ratio, Br: After the promising space has been determined, it is broadened to simulate flexible decisions in order to improve the population’s optimization performance.

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

2723

Generation of initial Evaluation of individuals

Mutation Crossove

Evaluation

Selection Yes Termination criteria Fig. 4. Plot of 2D Schwefelock function.

No

• • • • • •

Generations %NE=0

Grouping search

Final solution

Convergence condition ε = 1.E−05 Dimension n: 100 Size of population NP:800 (NP = 5∼10 × n) [13] Maximum number of generations MaxGen:6000 Mutation factor F = 0.5∼0.6 [14] Crossover rate CR = 0.3 [14] The parameters for IGCMEA are:

• Evaluated generation number is 200 • Number of better individuals: 400 • Broaden ratio: 10

Fig. 3. Flow chart of IGCMEA.

4. Examples and discussion In this section, we use IGCMEA to optimize the Schwefel Function and the structural topology of plane trusses. 4.1. Global optimization of Schwefel Function

function : F(X) =

n 



xi sin(

|xi |)

(13)

Through the average distance between the best found solution of each generation in the population and the best analytical solution, we attempt to understand the dynamic behaviors of the evolving population. Fig. 6 is the optimization convergence diagram. Fig. 7 shows the average distance dpa (%) in each generation. The average distance dpa (%) is defined as follows: dpa (%) =

d¯ pa × 100% L



i=1

Np where: d¯ pa = ( i=1 ||Xi − X ∗ ||/Np ) is the average distance between the population and the best analytical solution X* and

minimum solution: X ∗ = (−420.9687, −420.9687, . . . , −420.9687)

(14)

F(X ∗ ) = −418.983 ∗ n

(15)

This is a multimodal function with the search space Xi ∈ [−500, 500], i = 1,. . ., n. The plot and contour of the twodimensional F(X) are shown in Figs. 4 and 5, respectively. The population evolution analysis of DE and IGCMEA applied on 100-dimentional Schwefel function is as follows: The mutation strategy settings are: • DE: DE/best/1 • IGCMEA: in each generation, randomly select from the following three strategies (1) DE/best/1 (2) DE/rand-to-best/1 (3) DE/rand/2 The parameters are:

(16)

Fig. 5. Contour of 2D Schwefelock function.

2724

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

Fig. 8. Ground structure of plane trusses.

4.2. Optimization of structural topology of plane trusses

Fig. 6. Converges of 100D F(x).

   n L= (¯xi − xi )2 where x¯ i and xi are the lower bound and upper -

-

i=1

bound of ith dimension. From the 100-d optimization problem, we see that through the group discussion mechanism, IGCMEA efficiently leads the population to the correct region. When the appropriate time comes, regular meetings start. Group members inspire each other and search in the confident space. From the results, we notice that: (1) From Fig. 6, we see the IGCMEA converges robustly, leading the population to the good space. Taking F(x) as an example, near the 4000th generation, the average distance IGCMEA approaches is around 27.16%, while DE can only approach 39.02%. (2) For the convergence of the multimodal problem, Fig. 7 shows that although DE has the confluent search capability after adopting the random selection strategy, its randomness cannot lead the population to the good direction. In contrast, IGCMEA has both confluence and best guidance quality. It detects the important extreme points at the beginning of the search and has the potential to find the global optimal solution.

Fig. 8 shows the ground structure of an 11-member plane truss. Below nodes 4 and 6, a 100,000 lb point load is imposed. Young’s modulus is 104 ksi, the density  = 0.1 lb/in3 , the axial stress limited to 25 ksi and the displacement is limited to 2 in. In addition, the critical area in the problem setting is 0.09 in2 . That is, when the converged sectional area of the member is less than 0.09 in2 , the member is removed to reduce the weight of the structure. The optimization model for this problem is: Minimize the weight of the structure minimize 

11 

Ai Li

lb

(17)

i=1

The constraints, as show below, are that the displacements of all components of the structure are less than 2 in, and the stress of all components is less than 25 ksi. subject to ıi ≤ 2 and

i ≤ 25 i = 1, . . . , 11

(18)

To deal with the constrained optimization problems (COP), we use the penalty function: ϕ(x) = 

11 

Ai Li + Ri

11 

i=1

= 1015 ,

i=1

max[0, ıi ] +

11  i=1



max[0, i ] + R2 ˛ (19)

= 1020

where R1 R2 are penalty factors. If the design is not ground structure, ˛ = 1, otherwise ˛ = 0. Finally, this virtual function is set to be the evolution target of the structural topology. For this structural topology optimization problem, the parameters of IGCMEA are as follows. The mutation strategy settings are: • In each generation, randomly select from the following three strategies (1) DE/best/1 (2) DE/rand-to-best/1 (3) DE/rand/2 Parameters are:

Fig. 7. The average distance between the population and the best analytical solution of the 100D F(x).

• • • • • •

dimension n: 11 size of population NP: 450 maximum number of generations MaxGen: 600 mutation factor F = 0.5∼0.6 crossover rate CR = 0.3 search space: the lower bound and upper bound of all dimensions are 10−7 in2 and 35 in2

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

2725

Table 1 Evolution result of IGCMEA applying to the truss topology. Evolution stage

Optimum shape

Characters

1

Number of group Percentage of group Average mass

1 0.22% 3.00E+06

2

Number of group Percentage of group Average mass

8 1.78% 6.56E+03

3

Number of group Percentage of group Average mass

15 3.33% 5.34E+03

4

Number of group Percentage of group Average mass

1 0.22% 4.99E+03

5

Number of group Percentage of group Average mass

130 28.89% 5.02E+03

6

Number of group Percentage of group Average mass

270 60.00% 4.96E+03

7

Number of group Percentage of group Average mass

382 84.89% 4.94E+03

8

Number of group Percentage of group Average mass

415 92.22% 4.92E+03

9

Number of group Percentage of group Average mass

1 0.22% 4.89E+03

10

Number of group Percentage of group Average mass

430 95.56% 4.90E+03

2726

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

Fig. 9. Convergence diagram of topology optimization of plane trusses.

Table 2 Comparison of IGCMEA and [15] for optimization of truss topology. Member No.

Deb [15]

IGCMEA

IGCMEA + shape selection strategy

No. 2 (in2 ) No. 3 (in2 ) No. 4 (in2 ) No. 6 (in2 ) No. 7 (in2 ) No. 10 (in2 ) F(x) (lb)

22.07 21.44 29.68 15.30 21.29 6.09 4899.15

22.21 22.65 27.98 15.58 21.14 5.93 4899.12

23.95 20.69 28.65 15.90 20.88 5.80 4878.52

• evaluated generation number: 20 • number of better individuals: 200 • broaden ratio: 10 The optimization results of shape selection strategy are shown in Figs. 9 and 10. In the following, the generations are divided into ten stages, where every 60 generations constitutes a stage, and the population evolution data of the ten stages is listed in Table 1. In the first stage, the members in the generation are mainly distributed around the mean value. Only one individual has all members’ sectional areas approximate to the upper bound. The reason that the structure’s average weight reaches 3.0E+06 is probably because the sectional area of the primary supporting member is too small. In the second stage, the percentage of the optimal shape in the first stage grows to 1.78%, while the average weight decreases to 6.56E+03. In the third stage, there are already 15 individuals that have evolved to the global optimal result. Despite the number of optimal shape in the second stage decreasing to 8 individuals, we can see that the average weight of the structure decreases from 3.0E+06 to 6.56E+03. In the fifth stage, the percentage of the global optimal shape in the population has grown to 28.89%. In the sixth stage, the percentage comes to 60.00%. From the seventh stage to the eighth stage, the percentage becomes 84.89% and 92.22%,

Fig. 10. The deformation of the optimal structure of plane trusses.

respectively. When evolving to the ninth stage, although another optimal shape is generated, the percentage of the global optimal shape in the population still slightly grows to 94.44%. Finally, in the tenth stage, the average value of the global optimal shape has approached to 4.90E+03, and the optimal shape in the last stage deteriorates. From the ten stages of truss structure evolution shown in Fig. 9 and Table 2, we can see the following. (1) The optimization method is similar to the one of Dobbs and Felton [3]. (2) By the third stage, the percentage of the global optimal shape design in the population evolves from 3.33% to 95.56%. (3) When observing from the eighth stage to the tenth stage, we note that although in the eighth stage the selected optimal shape is already the global optimal design, evolving to the ninth stage shows that the intelligent population evolution with shape selection strategy can flexibly jump out of the regional optimal solution. (4) From the experiment, if only IGCMEA is used for structural optimization, we can also achieve a result of 4899.12 lb which is an improvement over Deb and Gulati [15]. This means that even for computationally intensive and difficult structural topology optimization problem, the IGCMEA can still achieve more satisfactory design than the GA does in the literature [15]. The structure of the optimal plane truss topology design is shown in Fig. 10. After calculating, we know the maximal stress of the optimal design is 24.37 ksi and the maximal displacement is 1.99 in. The comparisons of theories for stability of truss structure before and after optimization are listed in Table 3. The dimensions of truss structure before and after optimization are 11 and 7, respectively. After optimization, the truss structure is also stable.

Table 3 Comparisons of theories for stability of truss structure.

Before optimization After optimization

Members (b)

Reaction forces (r)

Joints (J)

Statically indeterminate (b + r − 2J)

Stability

11 7

4 4

6 5

2 1

Stable Stable

H.-C. Kuo et al. / Applied Soft Computing 12 (2012) 2719–2727

5. Conclusion The proposed IGCMEA is tested on the Schwefelock Function, showing its stability in searching for the global optimum. When applying this novel idea to the constrained optimization of structural topology of plane truss, it achieves a satisfactory result. (1) In this study, we use a penalty function to deal with the constrained optimization problem of truss structural topology. The roulette wheel selection model is used as the shape selection (structural topology) strategy, by which good structure shapes are selected to evolve, avoiding blind population evolution, and therefore implementing a directive strategy. (2) We successfully applied the IGCMEA to the design optimization of truss structure topology, achieving a slightly better result than the GA method in [15]. If considered in relation to the effect of shape selection strategy, the result is better than [15]. The convergence diagram shows that the shape selection strategy converges slowly, but it improves the result. Acknowledgement We are grateful to National Science of Council (Republic of China) for the financial support to the study with a grant number of NSC 99-2221-E-109-052. References [1] A.G.M. Michell, The limits of economy of material in framed structures, Philosophical Magazine (series 6) 8 (1904) 589–597.

2727

[2] W.S. Dorn, R.E. Gomory, H.J. Greenberg, Automatic design of optimal structures, In Journal de Mecanique 3 (1) (1964) 25–52. [3] M.W. Dobbs, L.P. Felton, Optimization of truss geometry, Journal of the Structural Division ASCE 95 (1969) 2105–2118. [4] U. Kirsch, B.H.V. Topping, Minimum weight design of structural topologies, Journal Structural Engineering 118 (7) (1992) 1770–1785. [5] J.H. Holland, Adaptation in Natural and Artificial System: an Introductory Analysis with Applications to Biology, Control and Artificial Intelligence, University of Michigan Press, 1975. [6] M. Kim, Solving traveling salesman problem using harmony search, ENCE 677 (Fall) (2003). [7] M. Cohen, J. March, J. Olson, A garbage can model of organizational choice, Administrative Science Quarterly (1972) 1–25. [8] F.W. Taylor, Scientific Management: Comprising Shop Management, the Principles of Scientific Management, Testimony before the Special House Committee, Harper & Row, New York, 1947. [9] H. Simon, How big is a chunk? Science (1974) 482–488. [10] Charles E. Lindblom, Public Administration Review 19 (2) (1959) 79–88. [11] R. Storn, K. Price, Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces, Global Optimization (1997) 341–359. [12] H.A. Simon, Some further notes on a class of skew distribution functions, Information and Control 3 (1960) 80–88. [13] H. Tim, M. Laszlo, V. Jozsef, A. Moonis, A combined swarm differential evolution algorithm for optimization problems, in: International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, Budapest, 2001, pp. 11–18. [14] Hui-Yuan Fan, Jouni Lampinen, A trigonometric mutation operation to differential evolution, Journal of Global Optimization (2003) 105–129. [15] K. Deb, S. Gulati, Design of Truss-Structures for Minimum Weight Using Genetic Algorithms, Department of Mechanical Engineering, Indian Institute of Technology, 2001.