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A unified approach to parameter selection in meta-heuristic algorithms for layout optimization
Journal of Constructional Steel Research 67 (2011) 1453–15462
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Journal of Constructional Steel Research
A unified approach to parameter selection in meta-heuristic algorithms for layout optimization A. Kaveh a,⁎, N. Farhoudi b a b
Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran Khajeh Nasir Toosi University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 17 May 2010 Accepted 16 March 2011 Available online 30 April 2011 Keywords: Layout optimization Steel braced frames Convergence factor Heuristic algorithms Genetic algorithm Ant colony optimization Particle swarm Big Bang–Big Crunch
a b s t r a c t Meta-heuristic optimization algorithms have attracted many researchers in the last decade. Adjustment of different parameters of these algorithms is usually a time consuming task which is mostly done by a trial and error approach. In this study an index, namely convergence factor (CF), is introduced that can show the performance of these algorithms. CF of an algorithm provides an estimate of the suitability of the parameters being set and can also enforce the algorithm to adjust its parameters automatically according to a pre-defined CF. In this study GA, ACO, PSO and BB–BC algorithms are used for layout (topology plus sizing) optimization of steel braced frames. Numerical examples show these algorithms have some similarities in common that should be taken into account in solving optimization problems. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Scarcity of structural materials and the need for efficiency in today's competitive world have forced engineers to evince greater interest in economical designs for structures. The meta-heuristic algorithms provide efficient tools for performing structural optimum designs and this is why these methods have been extensively employed in the field of structural engineering. The first meta-heuristic algorithm was genetic algorithm (GA) which was introduced in 1957 [1]. This algorithm uses the Darwin's theory of evaluation and has been employed in solving various types of engineering problems as well as structural design [2–4]. On the other hand, ant colony optimization (ACO) is principally inspired by the rules governing the behavior of real ants in finding their roots. Such a natural process was first simulated in numerical methods in the pioneering work of Dorigo et al. [5]. Since then, its performance has been studied in several optimization problems [6–8]. Particle swarm optimization (PSO), developed in 1995 [9], is another meta-heuristic algorithm that mimics the rules that bees and birds obey, finding their ways to the food source. This algorithm has been used in structural design in the works of [10–15].
⁎ Corresponding author. E-mail address: [email protected] (A. Kaveh). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.03.019
Big Bang–Big Crunch (BB–BC) is one of the most recent algorithms developed by [16]. It simulates the Big Bang theory and recently it is employed in structural design [17–19]. Most of the works applied to structural optimization are on sizing of cross sections of trusses and frames. The present study deals with the layout optimization of dual system of moment frame together with X-bracing. Optimizing such systems, adds two complications to the problem that are not present in the other types of structural systems: 1. In this kind of structural system, two types of elements have to be optimized, bars and beam-columns, where each has its own requirements to be satisfied. 2. Degree of statical indeterminacy is higher compared to the moment frames and trusses. Increase in the degree of indeterminacy may cause the internal forces to be much dependent to each other. Thus when the problem is to optimize topology and sizing together, a small change in topology will cause big changes in the internal forces and hence size of elements. Therefore the ability of the optimization algorithm in finding the globally best answer rather than local one will be important. On the other hand adjusting the algorithm parameters for proper optimization is an important step dealing with GA, ACO, PSO or BB– BC. However, no important suggestion is made up to now for adjusting these parameters and it is mainly done by trial and error approaches. In the present study an index is presented that is called
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the convergence factor (CF). It is shown how CF helps one to set the optimization parameters and how each of the algorithms can adjust themselves properly. In the first part of this paper optimization algorithms and their parameters are introduced. In the second part, formulation for the optimization is presented, and the subsequent part is devoted to the method used for implementing some building code requirements. In the fifth part, the CF that is the main contribution of the present work is introduced. In the sixth part, numerical examples are presented. This part is followed by implementation of the results and conclusion. 2. Meta-heuristic algorithms
4. Probability calculation: aij, or ant decision table is calculated as follows:
ai j ðt Þ =
where α and β are optimization parameters, and νij is the visibility that is defined according to the following formula: vij =
1 : dij
ð4Þ
The probability pkij of ant k choosing a path from i to j at time t is:
The main steps of genetic algorithm are as follows: 1. Initiation: Answers that are mimicked as chromosomes are selected randomly all around the search space. 2. Fitness based selection: Chromosomes that have better fitness are selected for the production of the next generation. 3. Crossover: Selected chromosomes of the previous step as parents make two new chromosomes of the next generation in a way that each gene of every new chromosome is from one of the parents. 4. Mutation: Genes of a chromosome change randomly with a possibility that is called here as “Mutation Rate”. 5. Convergence check: If the convergence criterion is achieved, the algorithm will end. 2.2. Ant colony optimization The main steps of the ant colony optimization algorithm are as follows: 1. Initiation: Answers that are mimicked as the path that ants choose from nest to food source are selected randomly all around the search space. Here it should be noticed that ants trace a kind of hormone called pheromone on their way and as the pheromone on a specific path increases the possibility of the path to be chosen becomes higher. In this algorithm for representing the intensity of pheromone on each path, the τ matrix is defined. The matrix has m rows and n columns where m is the number of alternatives that can be chosen for each path and n is the number of paths. τ is the filling from the first loop of the algorithm to the end and it acts as the memory of the algorithm. In this step, in order to have equivalent chance for each alternative, all arrays of τ matrix are filled by an equivalent initial value of τ0 using the following relationship: ð1Þ
where φ is a parameter between 0 and 1. 2. Evaporation: The pheromone trail evaporation is performed on each edge after it is passed by an ant using the relationship (2): 3. Fitness based pheromone trail intensity adjustment: τ is filled in a way that higher fitness gets higher possibility to be chosen in the next step. Thus the value of Δτij is defined for each ant's crossed way, according to the related answer, in a way that better answers achieve greater Δτij. Therefore the trial intensity is adjusted using the following formula: τij ðt + nÞ = ρ ⋅ τij ðt Þ + Δτij
ð3Þ
l∈allowed
2.1. Genetic algorithm
τij ðt Þ = ð1−φÞ ⋅ τij ðt Þ + φτ0
h iα h iβ τij ðt Þ ⋅ vij h iα h iβ n ∑ τij ðt Þ ⋅ vij
k
ð2Þ
where ρ is a parameter between 0 and 1, chosen such that (1 − ρ) represents the evaporation of pheromone between the time t and t + n (the amount of time required to complete a cycle).
aij ðt Þ
k
pij =
∑
l∈allowedk
ail ðt Þ
:
ð5Þ
5. Convergence check: Once the convergence criterion is satisfied, the algorithm ends. 2.3. Particle swarm The main steps of particle swarm optimization algorithm are: 1. Initiation: Answers that are mimicked as the position of each particle in the swarm are selected randomly all around the search space. 2. Fitness based position update: Velocity vector is defined to update the current position of each particle in the swarm. The velocity vector is updated based on the “memory” gained by each particle, conceptually resembling an autobiographical memory, as well as the knowledge gained by the swarm as a whole. Thus the position of each particle in the swarm is updated based on the social behavior of the swarm which adapts to its environment by returning to promising regions of the space previously discovered and searching for better positions over time. Numerically, the position x of a particle i at iteration k + 1 is updated as i
xk +
i
1
i
= xk + vk + 1 Δt
ð6Þ
where vik + 1 is the corresponding updated velocity vector, and Δt is the time step value. Throughout the present work a unit time step is used. The velocity of each particle is calculated as defined by Eq. (7),
i vk + 1
=
i ωvk
+ c1 r 1
Pki − xik Δt
+ c2 r2
Pkg − xik Δt
ð7Þ
where vik is the velocity vector at iteration k, r1 and r2 represent random numbers between 0 and 1; pik represents the best position of particle i in the kth iteration, and pgk corresponds to the global best position in the swarm up to iteration k. ω is a dynamic parameter that is linearly decreased with each algorithm iteration as the following [20]: ωk + 1 = ωmax −
ω max − ω min k: k max
ð8Þ
c1 and c2 are parameters that have the constant sum of 4. 3. Convergence check. Once the convergence criterion is achieved the algorithm ends.
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2.4. Big Bang–Big Crunch The main steps of the Big Bang–Big Crunch optimization algorithm are: 1. Initiation: Answers that are mimicked as the position of each mass in the search space are selected randomly all around the search space. The creation of the initial random population is called the Big Bang phase. 2. Center of mass calculation: Fitness of all search space particles is calculated. Center of mass is found according to Eq. (9). In this step the best fit individual can be chosen as the center of mass instead of using the following equation: NC
∑
xcm =
1
∑
i=1
:
ð9Þ
fi
3. Position update: New candidates are calculated around the center of mass by adding or subtracting a normal random number whose value decreases as the iterations elapse. This can be formalized as new
x
= xcm +
αrl k
ð10Þ
2.5. Similarities among GA, ACO, PSO and BB–BC In all these four algorithms, the following features are common: 1. Randomly selected initialization. 2. Refinement of the answers according to the best fitted answers of the previous loop or loops. 3. Addition of some random answers in order to stop the algorithm from an early convergence and preventing it from converging to a local optimum. Here, the optimization parameter adjusting the proportion of these answers is called the Randomness Parameter (RP). 4. Repetition of Steps 2 and 3 until convergence. The randomness parameters can be chosen from different points of view, and in the present study, the randomness parameters are: A. Genetic algorithm: Mutation rate. B. Ant colony optimization: φ (τ0 coefficient) C. Particle swarm optimization: c1 (local best velocity coefficient). It may seem that c1 is not a coefficient for a randomly selected part, but it should be noted, as the sum of c1 and c2 are constant here, an increase in c1 may cause a decrease in c2 that is the coefficient of the global answer. Therefore its role is to prevent the algorithm from converging to the global best answer and can be assumed as RP. It should be mentioned here that He et al. [14] proposed a hybrid PSO with passive congregation (PSOPC). In this method, the velocity was defined as i
i
+ 1
It is noticeable that none of the mutation rates of the GA and φ of the ACO were present in the original form of the algorithms, and passive congregation was not present in the original PSO algorithm.
= ωvk + c1 r1
In a frame structure, once a connection type is selected, the fabrication cost has little effect on the optimum design, since it varies proportionally with the structural weight [21]. Thus, it is a usual practice to seek the minimum weight design for a given loading state and boundary condition. Here, the problem is formulated as follows: Minimize: M
w = ρ ∑ Ai L i i=1
where xcm stands for the center of mass, l is the upper limit of the parameter, r is a normal random number, α is the optimization parameter and k is the iteration step. Then new point xnew is checked to be in the search space. 4. Convergence check: Returns to Step 2 until stopping criteria has been met.
vk
sequence in the range (0, 1). Adding the passive congregation model to the PSO may improve its performance; however this kind of PSO has not been taken into account in the present work because of the complication it adds to the parameter selection. However, if this kind of PSO is taken into account, randomness parameter will be c3, instead of c1. D. Big Bang–Big Crunch: α or random part coefficient. It should be α noticed that here is the real coefficient of the random part. k
3. Formulation of the optimization problem
xi
i = 1 fi N 1
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Pki − xik Δt
+ c2 r 2
Pkg − xik Δt
+ c3 r3
Rik − xik Δt ð11Þ
where Rik is a particle selected randomly from the swarm, c3 is the passive congregation coefficient and r3 is a uniform random
ð12Þ
Subjected to: KU − P = 0
ð13Þ
g1 ≥ 0; g2 ≥ 0; …; gn ≥ 0 where g1, g2, …,gn are constraint functions depending on the element being used in each problem and K, U and P are the stiffness matrix, nodal displacement and force vectors, respectively. In this study, the members should satisfy the following constraint on drift, deflection, compaction, strength and stability coefficients according to the Specification for Structural Steel Buildings [22], Minimum Design Loads for Buildings and Other Structures [23], International Building Code 2006 [24] and Seismic Provisions for Structural Steel Buildings [25]: – Drift Drift ≤ 0:02hsx :
ð14Þ
– Deflection
ΔL b l = 360 : ΔD+L b l = 240
ð15Þ
Consider that according to Table 1604.3 of IBC2006 (Deflection limit), for steel structural members the dead load should be taken as zero. – Compactness For SLRS members these limits are calculated according to Table I8-1 (Limiting Width–Thickness Ratios for Compression Elements) of Seismic Provisions for Structural Steel Buildings [25]. – Strength These constrains are based on both AISC 360–05 specification [22] and Seismic Provisions for Structural Steel Buildings [25]. – Stability θmax b
0:5 : βCd
ð16Þ
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– Irregularity There is no horizontal irregularity, but vertical irregularity limits are taken into consideration according to Table 12.3-2 (Vertical Structural Irregularities) of the ASCE/SEI 7-05[23]. It should be noted that the Vertical Geometric Irregularity has not been considered. This is because this type of irregularity is defined to exist where the horizontal dimension of the seismic force-resisting system in any story is more than 130% that of the adjacent story. As a result, having equal bay sizes in the given structures, this type of irregularity may not let adjacent stories have different numbers of bracings, so that the feasible bracing placement may reduce considerably, thus it is ignored in this study. – Slenderness The AISC specification no longer provides a specific maximum slenderness ratio, as it formerly did. The AISC Commentary (E2) does indicate however that if KL/r is N200 then the critical stress Fcr will be less than 43.43 MPa. That value is based on engineering judgment, practical economies, and the fact that special care has to be taken to protect from damage such a slender member during fabrication, shipping and erection. As a result of these important practical considerations, an engineer using the 2005 AISC Specification will probably select compression members with slenderness values below 200 [26]. Thus this point is also considered in this study. Furthermore, such a constrained formulation is treated in an unconstrained form, using a penalized fitness function as F = F0 −w ⁎ 1 + Kp :V
ð17Þ
s d V = ∑ max g ; 0 +max g ; 0
ð18Þ
NLC
where F0 is a constant taken as zero in the class of considered examples. Kp is the penalty coefficient and V denotes the total constraints' violation considering all nLC loading combinations including g s as the strength and gd for other constraint functions. Calculation of displacements, forces and stresses is based on the second-order elastic behavior of the structure using a finite element structural analysis routine and amplified first-order elastic analysis. 4. Simultaneous design Most of the building codes such as ASCE have some requirements to share loads between members in a reasonable way. For example, ASCE 7–05 defines a dual building frame system as the following: Dual System: A structural system with an essentially complete space frame providing support for vertical loads. Seismic force resistance is provided by the moment resisting frames and the shear walls or braced frames. On the other hand, for a dual system, the moment frames shall be capable of resisting at least 25% of the design seismic forces. Considering these requirements, one is not permitted to design all the members simultaneously. The method presented in this study that is called “simultaneous design of structure for all loads and frame for its loads” helps to better satisfy the building code requirements. Requirements of the essentially complete frame and the entire structure are provided at the same time by the use of this method. In other words, analysis outputs are achieved in two different steps, one after formation of the essential frame and one for the whole structure. After each step, the requirements of the building code are checked.
5. New criteria for optimization performance A convergence factor, CF, is defined as the average possibility of the elitist answer. As an example if the aim is to devote some steel profiles to a structure that has four elements. In the first step, frequency of modal profile of each element should be defined. CF is the mean of these frequencies. Table 1 illustrates an example for calculation of the CF for a structure containing 4 elements. In this study it will be shown how such a factor can help to adapt a meta-heuristic algorithm for our problem. 6. Numerical examples Three frames of 3, 5 and 10 stories are treated in the present work. The following features are common in all the examples. 6.1. Geometry Height of each floor = 3 m Width of the frame = 5.0 m Three degrees of freedom for each joint (x, y-translations and z-rotation) All connections and also supports are fixed. 6.2. Loading condition 1) Uniform distributed dead load of 6.3 kN/m2 in negative y-direction on all beam elements 2) Uniform distributed live load of 1.96 kN/m2 in negative y-direction on all beam elements 3) Earthquake concentrated loads are calculated according to the ASCE 7-05[23], according to the following parameters:
R=7 I=1 Ss = 1.32; S1 = 0.535; Seismic design category = E.
Earthquake loads acting on the given examples are shown in Table 2. 5.3. Material properties The 50 ksi steels are the predominant ones in use today. In fact some of the steel mills charge extra for W-sections if they consist of A36. On the other hand, A992 and A500 are preferred materials for
Table 1 An example for calculation of CF.
Answer 1 Answer 2 Answer 3 Answer 4 Answer 5 Modal answer Frequency of the modal answer Proportion of the modal answer among all answers CF
Element 1
Element 2
Element 3
Element 4
5 3 4 3 3 3 3 60%
41 36 39 42 41 41 2 40%
22 22 25 22 22 22 4 80%
15 17 16 17 19 17 2 40%
55%
Table 2 Earthquake load acting on different frames in the numerical examples. Floor
Earthquake loads (kN) 3 story
1 2 3 4 5 6 7 8 9 10 Base shear
5 story
120.12 240.24 360.36
10 story
80.08 160.16 240.24 320.32 400.4
720.72
32.73 68.372 105.2 142.83 181.05 219.76 258.88 298.35 338.14 378.2 2023.5
1201.2
Answer/Best Answer
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2.6 2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 1.E+00
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3 story 5 story 10 story
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
mutation rate Fig. 1. GA randomness parameter selection for the 3, 5 and 10 story frames.
Member type
Shape
ASTM designation
Fy(MPa)
Fu(MPa)
Column Beam Bracing
W W HSS Rect.
A992 A992 A500
344.70 344.70 317.20
448.20 448.20 399.90
W-shapes and HSS Rectangular, respectively [26]. Data are selected for the members, according to Table 3 and the following data:
Answer/Best Answer
Table 3 Section types selected for numerical examples.
2.6 2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 1.E+00
3 story 5 story 10 story
1.E-01
2 3 E = 2e8 kN=m Þ; ρ = 76:82 kN=m ; and ν = 0:3:
1.E-02
1.E-03
1.E-04
1.E-05
taw0 coeficient Fig. 2. ACO randomness parameter selection for the 3, 5 and 10 story frames.
5.4. Section list reduction 1.5
1. Sections are checked for slenderness, and compaction limits. Members which do not fall in the feasible region are omitted. 2. Considering Ai as the cross-section area of the ith member in the list, and n as the number of sections in the section list, the following steps are taken to reduce the size of the search space in order to increase the efficiency of the optimization algorithms: Aj −Ai 2.1. All members that satisfy b 0:1, (j = 1:n, i = 1) are put Ai in one group. 2.2. Section lists are classified by repeating step 2.1, considering the members of the last group to be omitted. 3. A new section list is created by substituting all members of each group by the best of them in carrying compression loads. 5.5. Members under optimization
3 story
Answer/Best Answer
The following algorithm is used to lessen the size of section list of bracings:
1.4
5 story 10 story
1.3 1.2 1.1 1 0.0 0.9
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
c1
Fig. 3. PSO randomness parameter selection for the 3, 5 and 10 story frames.
best answer changes more rapidly by an increase in the number of stories). 3. GA and ACO are more parameter dependent in comparison to the PSO and BB–BC.
In this study, all members are determined using optimization methods. 6. Results Figs. 1 to 4 show changes in the best weight achieved from each algorithm by the change in randomness parameter (RP). It should be noticed that in these figures the ratio of the weight achieved by each RP to the best weight achieved after parameter selection (using the best RP) is shown. From these figures it can be shown that: 1. Results are dependent on the RP selection. 2. Dependence of the answers on the RP selection increases by an increase in the size of search space (the ratio of the answer to the
Fig. 4. BB–BC randomness parameter selection for the 3, 5 and 10 story frames.
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According to the results of these figures, if the Best RP is called as RBbest, two groups of answers that are not as well as the best answer can be recognized as: 1. Under speed answers: In this kind of answers RP N RPbest so the randomly selected answers are more than needed, then the algorithm cannot converge in a specified number of loops. 2. Over speed answers: In this kind of answers RP b RPbest so the randomly selected answers are less than needed, then the algorithm converges more rapidly, and the answer will be a local optimum in place of the global one. Figs. 5 to 12 show the variation of the CF and the normalized fitness of some of the frames solved by the algorithms for the following three cases (these figures are selected from 12 optimizations performed on 3, 5 and 10 story frames using four optimization methods): ✓ One of the under speed answers (these answers are named precisely under) ✓ One of the over speed answers (these answers are named precisely over) ✓ The best answer. (These answers are named precisely best.) In these figures normalized fitness describes a kind of normalization for fitness with value changes from 0 to 100%, as the value of both CF and normalized fitness is displayed on the same axes, the following
Fig. 5. Changes of normalized fitness and CF in GA for the 3 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 6. Changes of normalized fitness and CF in GA for the 10 story frame, in “under speed”, “best” and “over speed” cases.
definition is considered for the progress of normalized fitness in loop number i:
Progressi =
8 > > 0 > > <
fitnessi N2 fitnessbest
> > fitnessi > > : 2− ⁎100 fitnessbest
fitnessi ≤2 fitnessbest
ð16Þ
where: fitnessi defines fitness of the best answer of the loop number i, and fitnessbest defines fitness of the best answer achieved at the end of optimization. Progress defines the value of CF for the CF chart. This kind of normalization may omit some initial iterations' results but it will make the most important parts of it more clear. From these figures the following observations can be made: 1. It can be seen from the comparison of the normalized fitness and CF charts of under speed answers in Figs. 5, 14, 7, and 8 that although the fitness does not change in a considerable number of loops, the real convergence that is the frequency of the elitist answer or CF is not more than 60%. This means that achieving a constant answer in a number of loops is not a proper sign of convergence. 2. The rate of the changes of CF is according to the following formula as estimated before: CF Rate Under speed answer b CF Rate Best answer b CF Rate Over speed answer :
Fig. 7. Changes of normalized fitness and CF in ACO for the 3 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 8. Changes of normalized fitness and CF in ACO for the 10 story frame, in “under speed”, “best” and “over speed” cases.
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Fig. 9. Changes of normalized fitness and CF in PSO for the 5 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 11. Changes of normalized fitness and CF in BB for the 5 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 10. Changes of normalized fitness and CF in PSO for the 10 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 12. Changes of normalized fitness and CF in BB for the 10 story frame, in “under speed”, “best” and “over speed” cases.
This means that the convergence rate is not an inherent property of these algorithms, and as the CF chart can be under control by changes in RP, one can change convergence rate as he wishes. 3. Comparing the charts of the CF and the normalized fitness, one can see that most of the time changes in fitness occur when there is a change in the CF chart. This means that: 3.1. When the CF is approximately fixed, one may not expect the algorithm to improve the answers by much. 3.2. When the CF is near 100, there will not be more chance for the CF to promote. Thus the biggest] improvement in the answers has happened and the convergence is achieved. Then there will be no need to wait for a long time to satisfy other convergence criteria such as a constant answer for a number of loops. 3.3. When the answer is constant for a number of loops but CF in not near 100, one can enforce the algorithm to improve the results by decreasing the RP. It may cause the CF and hence the fitness to promote. 4. The value of the CF moves toward 100, in most of the best answers. 5. Refinement in the optimum answers mostly happen when CF changes. Thus a well adjusted algorithm is the one in which: 5.1. CF changes in a reasonable rate. Here a reasonable rate for an algorithm means a rate that is slow enough to refine the answer in each level of the CF, and is fast enough in order not to permit undesirable answers to stay up to the end. 6. BB–BC and PSO are not in the risk of being trapped in local optima resulted from a constant CF not promoted to a value near 100, because their original RP is defined in a way that it decreases during the optimization progress. This is because in BB–BC, the real
α which reduces every loop and in the PSO used k here, there is a dynamic variation of wthat leads the algorithm from a global search to local one.
RP parameter is
7. Modified method According to the results achieved from the previous figures, at this stage a method is presented to enforce the algorithm to adjust its parameters in an adaptive manner. The main idea of this method is to use a method in which: “CF changes with a reasonable rate in a number of iterations that is considered for the optimization and finally reaches a value close to 100.” For achieving this aim the following algorithm is used: 1. Set RP as high as possible. 2. Consider a formula for the CF progress which is named here as PCF (pre-defined convergence factor). 3. Reduce RP in a way that the CF of the algorithm follows the predefined line. In the present work, as the shape of the CF progress is not fixed in all examples, a function PCF is used which is as
loopxi −1 PCFi = 30 + 70 ⁎ : ð200x −1Þ
ð17Þ
Changes in PCF in different iterations of algorithm are shown in Fig. 13 for different values of x. Here PCF starts from the value of 30 because CF of the randomly selected answers achieves this value at
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Fig. 14. Results of the modified method compared with the normal one in the 3 story frame.
Fig. 13. Variation of the PCF function for different values of x.
the first loop. This is because when the answers are selected randomly, there are some roles for bracing placements that do not permit a story to have more than 3 bracings, thus many alternatives for bracing fills will be zero (no bracing). The following algorithm is used for the changes in RP at the ith iteration: tmp =
CFi − PCFi CFi
RP = RP ⁎ RPCoef RP = RP = RPCoef
ð18Þ tmp b −0:02 tmp N 0:1
ð19Þ
in which RPCoef is a coefficient that is multiplied to RP to adjust it and here it is taken as 0.95 in order to make smooth changes in RP in each loop. However, one can change it according to the algorithm being used. Results of the modified method compared to the standard methods are shown in Figs. 14 to 16. In these figures it can be seen that for all the methods and all the considered frames, the modified method achieves better or equal answers compared to the standard ones, and also:
Fig. 15. Results of the modified method compared with the normal one in the 5 story frame.
a) In complicated problems, modified methods achieve better results in comparison with standard methods. b) In small problems, although there may not be real difference between the results of the two methods, the number of tries for adjustment of the optimization parameters may considerably decrease in this way. The variation of CF and PCF for each frame, using the modified method, is shown in Figs. 17 to 19. From these figures one can conclude that:
Fig. 16. Results of the modified method compared with the normal one in the 10 story frame.
1. In most of the cases the best PCF is a linear one. 2. When the search space is too small to be under control, no value of RP can stop it from convergence. Thus the algorithm may show its inherent behavior. 3. The possibility of getting the convergence rate under control increases by an increase in the size of the search space. Figs. 20 to 22 show the five best placements achieved for each frame. It can be seen that in these figures one cannot expect an optimum form for the placement of the bracings. 8. Concluding remarks In this study layout optimization of the braced frames is studied using GA, ACO, PSO and BB–BC algorithms. The results of more than 700 frames designed by these methods show that: 1. There are some similarities between the 4 meta-heuristic algorithms investigated in the present study that cause them to obey the same rules. The most important similarity is that they all select the next loop generation based on the two main steps: a. Generation of new answers according to the best answers achieved in the previous loops.
Fig. 17. CF and predefined CF for the 3 story frame.
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b. Generation of some random answers in order to prevent the algorithm from being trapped in local optima. The present study shows that for changes in the proportion of the answers of phase b in comparison with those chosen from phase a, one can control the convergence rate. Therefore, the convergence rate will not be considered as an essentially inherent property of these algorithms and can be adjusted. Parameters of the algorithms defining the proportion of phase b to those of phase a which are called the Randomness Parameters (RP) in this paper are: In GA: Mutation rate In ACO: φ(τ0coefficient). In PSO: c1 (c3 if passive congregation is used) In BB–BC: α. 2. Reaching an answer in a pre-defined number of loops one cannot be sure of the convergence of the algorithm. Thus an index namely convergence factor (CF) is defined as the average possibility of the elitist answer. This index can show the real convergence of an algorithm. 3. Convergence rate of all 4 algorithms can be changed by adjusting the CF. 4. CF can be changed in all 4 algorithms by the change in RP. 5. Refinement in the optimum answers mostly happens when CF changes. And one cannot expect much from the algorithm that has a fixed CF. Therefore, a well adjusted algorithm is the one, in which:
Fig. 18. CF and predefined CF for the 5 story frame.
a. CF changes in a reasonable rate. Here a reasonable rate for an algorithm means a rate that is slow enough to refine the answer in each level of CF and is fast enough in order not to permit undesirable answers to stay to the end. b. If CF reaches near the value of 100 at the end of algorithm, it means a real convergence. 6. Optimization parameter adjustment is an important step using all four algorithms used here, so real run time of each algorithm should contain the time for parameter selection as well. This becomes more important when the size of search space increases. 7. Convergence rate should be decreased by an increase in the size of search space. In other words, RP should be increased by an increase in the size of search space. 8. PSO and BB used here are modified essentially in a way that the RP decreases by time so the algorithm may not be under the risk of being trapped in a constant value of CF.
Fig. 19. CF and predefined CF for the 10 story frame.
1 W=86.2314 kN
2 W=87.7729 kN
3 W=87.7729 kN
4 W=89.7264 kN
5 W=90.1184 kN
Fig. 20. Optimum placement of bracing in the 3 story frame.
1 W=158.5415 kN
2 W=158.5415 kN
3 W=158.5415 kN
4 W=158.5415 kN
Fig. 21. Optimum placement of bracing in the 5 story frame.
5 W=158.5415 kN
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1 W=392.519 kN
2 W=393.9542 kN
3 W=397.4314 kN
4 W=397.4979 kN
5 W=399.1922 kN
Fig. 22. Optimum placement of bracing in the 10 story frame.
9. Having the knowledge of desirable behavior of an optimization algorithm, one can enforce the algorithm to behave as desired by dynamic optimization parameters. This is the most important achievement of the study. And using this method, optimization parameters can be set automatically. Modified methods introduced in the present work utilize this ability using a pre-defined CF function namely PCF and enforcing the algorithm to obey this function. Comparing the modified method with the non-adaptive approach, it can be seen that: a. In complicated problems, modified methods achieve better results compared to the non-adaptive ones. b. In small problems, although there may not be a real difference between the answers of the two methods, the number of tries for adjustment of optimization parameters may considerably decrease in this way. 10. Using modified method one should be aware that: a. When the search space is too small to be under control, no value of RP can stop it from early convergence. Thus the algorithm may behave as its inherent behavior. b. The possibility of getting the convergence rate under control increases by an increase in the size of the search space. c. Almost all 4 algorithms behave well when the PCF is linear, and the only exception is the ACO. The results of ACO depend on the convergence rate more than other algorithms since the ACO employs a memory that selects answers according to the frequency of them in all loops from beginning to end. If an undesirable answer remains for a long time in the algorithm, it may stay to the end. However, in all 3 other algorithms a slow convergence rate may not harm the results considerably. 11. For a frame of general configuration one cannot find a special optimum layout for the bracings. Suggestion: In some studies 2-phase optimization algorithms are used [7,16]. In these methods as the size of search space differs in different phases; the need for a parameter selection may be time consuming. The present method can help the multi-phase optimization methods to be used more efficiently. Acknowledgment The first author is grateful to Iran National Science Foundation for the support.
References [1] Fraser AS. Simulation of genetic systems by automatic digital computers. I. Introduction. Aust J Biol Sci 1957;10:484–91. [2] Rajeev S, Krishnamoorthy CS. Discrete optimization of structures using genetic algorithms. J Struct Eng, ASCE 1992;118(5):1233–50. [3] Kaveh A, Kalatjari V. Genetic algorithm for discrete sizing optimal design of trusses using the force method. Int J Numer Methods Eng 2002;55:55–72. [4] Affenzeller M. A generic evolutionary computation approach based upon genetic algorithms and evolution strategies. J Syst Sci 2002;28(4):59–71. [5] Dorigo M, Gambardella LM. Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput 1997;1:53–66. [6] Bullnheimer B, Hartl RF, Strauss C. A new rank-based version of the ant system: a computational study. Tech. Rep. POM-03/97. Austria: Institute of Management Science, University of Vienna; 1997. [7] Camp CV, Bichon BJ. Design of space trusses using ant colony optimization. J Struct Eng 2004;130(5):741–51. [8] Kaveh A, Shojaee S. Optimal design of skeletal structures using ant colony optimisation. Int J Numer Methods Eng 2007;70(5):563–81. [9] Kennedy J, Eberhart R. Particle swarm optimization. Proceedings of IEEE International Conference on Neural Networks. IV; 1995. p. 1942–8. [10] Kaveh A, Talatahari S. Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Comput Struct 2009;87:267–83. [11] Kaveh A, Talatahari S. A discrete particle swarm ant colony optimization for design of steel frames. Asian J Civil Eng Building Housing 2007;9:563–75. [12] Gholizadeh S, Salajegheh E. Optimal design of structures subjected to time history loading by swarm intelligence and an advanced meta-model. Comput Methods Appl Mech Eng 2009;198:2936–49. [13] Li LJ, Ren FM, Liu F, Wu QH. An improved particle swarm optimization method and its application in civil engineering. The Eighth International Conference on Computation and Structures Technology. Las Palmas de Gran Canaria, Spain; Paper 42; 2006. [14] He S, Wu QH, Wen JY, Saunders JR, Paton RC. A particle swarm optimizer with passive congregation. Biosystem 2004;78:135–47. [15] Perez RE, Behdinan K. Particle swarm approach for structural design optimization. Comput Struct 2007;85:1579–88. [16] Erol OK, Eskin I. A new optimization method: Big Bang–Big Crunch. Adv Eng Softw 2007;37:106–11. [17] Camp CV. Design of trusses using Big Bang–Big Crunch optimization. J Struct Eng 2007;999:133–7. [18] Kaveh A, Talatahari S. Size optimization of space trusses using Big Bang–Big Crunch algorithm. Comput Struct 2009;87:1129–40. [19] Kaveh A, Talatahari S. Optimal design of Schwedler and ribbed domes via hybrid Big Bang–Big Crunch algorithm. Constr Steel Res 2010;66:412–9. [20] Shi Y, Eberhart R. A modified particle swarm optimizer. IEEE international conference on evolutionary computation. Piscataway, NJ: IEEE Press; 1998. p. 69–73. [21] Jarmai K, Farkas J, Kurobane Y. Optimum seismic design of a multi-storey steel frame. J Eng Struct 2006;28(7):1038–48. [22] ANSI/AISC 360-05. Specification for Structural Steel Buildings, American Institute of Steel Construction, Chicago, Illinois 60601-1802; March 9 2005. [23] ASCE/SEI 7-05. Minimum Design Loads for Buildings and Other Structures. American Society of Civil Engineers; 2009. [24] International Building Code 2006. International Code Council, INC; 2006. [25] ANSI/AISC 341-05. Seismic Provisions for Structural Steel Buildings, American Institute of Steel Construction, Chicago, Illinois 60601-1802; March 9, 2005. [26] McCormac J. Structural steel design, Pearson International Edition; 2008.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
A unified approach to parameter selection in meta-heuristic algorithms for layout optimization A. Kaveh a,⁎, N. Farhoudi b a b
Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran Khajeh Nasir Toosi University of Technology, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 17 May 2010 Accepted 16 March 2011 Available online 30 April 2011 Keywords: Layout optimization Steel braced frames Convergence factor Heuristic algorithms Genetic algorithm Ant colony optimization Particle swarm Big Bang–Big Crunch
a b s t r a c t Meta-heuristic optimization algorithms have attracted many researchers in the last decade. Adjustment of different parameters of these algorithms is usually a time consuming task which is mostly done by a trial and error approach. In this study an index, namely convergence factor (CF), is introduced that can show the performance of these algorithms. CF of an algorithm provides an estimate of the suitability of the parameters being set and can also enforce the algorithm to adjust its parameters automatically according to a pre-defined CF. In this study GA, ACO, PSO and BB–BC algorithms are used for layout (topology plus sizing) optimization of steel braced frames. Numerical examples show these algorithms have some similarities in common that should be taken into account in solving optimization problems. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Scarcity of structural materials and the need for efficiency in today's competitive world have forced engineers to evince greater interest in economical designs for structures. The meta-heuristic algorithms provide efficient tools for performing structural optimum designs and this is why these methods have been extensively employed in the field of structural engineering. The first meta-heuristic algorithm was genetic algorithm (GA) which was introduced in 1957 [1]. This algorithm uses the Darwin's theory of evaluation and has been employed in solving various types of engineering problems as well as structural design [2–4]. On the other hand, ant colony optimization (ACO) is principally inspired by the rules governing the behavior of real ants in finding their roots. Such a natural process was first simulated in numerical methods in the pioneering work of Dorigo et al. [5]. Since then, its performance has been studied in several optimization problems [6–8]. Particle swarm optimization (PSO), developed in 1995 [9], is another meta-heuristic algorithm that mimics the rules that bees and birds obey, finding their ways to the food source. This algorithm has been used in structural design in the works of [10–15].
⁎ Corresponding author. E-mail address: [email protected] (A. Kaveh). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.03.019
Big Bang–Big Crunch (BB–BC) is one of the most recent algorithms developed by [16]. It simulates the Big Bang theory and recently it is employed in structural design [17–19]. Most of the works applied to structural optimization are on sizing of cross sections of trusses and frames. The present study deals with the layout optimization of dual system of moment frame together with X-bracing. Optimizing such systems, adds two complications to the problem that are not present in the other types of structural systems: 1. In this kind of structural system, two types of elements have to be optimized, bars and beam-columns, where each has its own requirements to be satisfied. 2. Degree of statical indeterminacy is higher compared to the moment frames and trusses. Increase in the degree of indeterminacy may cause the internal forces to be much dependent to each other. Thus when the problem is to optimize topology and sizing together, a small change in topology will cause big changes in the internal forces and hence size of elements. Therefore the ability of the optimization algorithm in finding the globally best answer rather than local one will be important. On the other hand adjusting the algorithm parameters for proper optimization is an important step dealing with GA, ACO, PSO or BB– BC. However, no important suggestion is made up to now for adjusting these parameters and it is mainly done by trial and error approaches. In the present study an index is presented that is called
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the convergence factor (CF). It is shown how CF helps one to set the optimization parameters and how each of the algorithms can adjust themselves properly. In the first part of this paper optimization algorithms and their parameters are introduced. In the second part, formulation for the optimization is presented, and the subsequent part is devoted to the method used for implementing some building code requirements. In the fifth part, the CF that is the main contribution of the present work is introduced. In the sixth part, numerical examples are presented. This part is followed by implementation of the results and conclusion. 2. Meta-heuristic algorithms
4. Probability calculation: aij, or ant decision table is calculated as follows:
ai j ðt Þ =
where α and β are optimization parameters, and νij is the visibility that is defined according to the following formula: vij =
1 : dij
ð4Þ
The probability pkij of ant k choosing a path from i to j at time t is:
The main steps of genetic algorithm are as follows: 1. Initiation: Answers that are mimicked as chromosomes are selected randomly all around the search space. 2. Fitness based selection: Chromosomes that have better fitness are selected for the production of the next generation. 3. Crossover: Selected chromosomes of the previous step as parents make two new chromosomes of the next generation in a way that each gene of every new chromosome is from one of the parents. 4. Mutation: Genes of a chromosome change randomly with a possibility that is called here as “Mutation Rate”. 5. Convergence check: If the convergence criterion is achieved, the algorithm will end. 2.2. Ant colony optimization The main steps of the ant colony optimization algorithm are as follows: 1. Initiation: Answers that are mimicked as the path that ants choose from nest to food source are selected randomly all around the search space. Here it should be noticed that ants trace a kind of hormone called pheromone on their way and as the pheromone on a specific path increases the possibility of the path to be chosen becomes higher. In this algorithm for representing the intensity of pheromone on each path, the τ matrix is defined. The matrix has m rows and n columns where m is the number of alternatives that can be chosen for each path and n is the number of paths. τ is the filling from the first loop of the algorithm to the end and it acts as the memory of the algorithm. In this step, in order to have equivalent chance for each alternative, all arrays of τ matrix are filled by an equivalent initial value of τ0 using the following relationship: ð1Þ
where φ is a parameter between 0 and 1. 2. Evaporation: The pheromone trail evaporation is performed on each edge after it is passed by an ant using the relationship (2): 3. Fitness based pheromone trail intensity adjustment: τ is filled in a way that higher fitness gets higher possibility to be chosen in the next step. Thus the value of Δτij is defined for each ant's crossed way, according to the related answer, in a way that better answers achieve greater Δτij. Therefore the trial intensity is adjusted using the following formula: τij ðt + nÞ = ρ ⋅ τij ðt Þ + Δτij
ð3Þ
l∈allowed
2.1. Genetic algorithm
τij ðt Þ = ð1−φÞ ⋅ τij ðt Þ + φτ0
h iα h iβ τij ðt Þ ⋅ vij h iα h iβ n ∑ τij ðt Þ ⋅ vij
k
ð2Þ
where ρ is a parameter between 0 and 1, chosen such that (1 − ρ) represents the evaporation of pheromone between the time t and t + n (the amount of time required to complete a cycle).
aij ðt Þ
k
pij =
∑
l∈allowedk
ail ðt Þ
:
ð5Þ
5. Convergence check: Once the convergence criterion is satisfied, the algorithm ends. 2.3. Particle swarm The main steps of particle swarm optimization algorithm are: 1. Initiation: Answers that are mimicked as the position of each particle in the swarm are selected randomly all around the search space. 2. Fitness based position update: Velocity vector is defined to update the current position of each particle in the swarm. The velocity vector is updated based on the “memory” gained by each particle, conceptually resembling an autobiographical memory, as well as the knowledge gained by the swarm as a whole. Thus the position of each particle in the swarm is updated based on the social behavior of the swarm which adapts to its environment by returning to promising regions of the space previously discovered and searching for better positions over time. Numerically, the position x of a particle i at iteration k + 1 is updated as i
xk +
i
1
i
= xk + vk + 1 Δt
ð6Þ
where vik + 1 is the corresponding updated velocity vector, and Δt is the time step value. Throughout the present work a unit time step is used. The velocity of each particle is calculated as defined by Eq. (7),
i vk + 1
=
i ωvk
+ c1 r 1
Pki − xik Δt
+ c2 r2
Pkg − xik Δt
ð7Þ
where vik is the velocity vector at iteration k, r1 and r2 represent random numbers between 0 and 1; pik represents the best position of particle i in the kth iteration, and pgk corresponds to the global best position in the swarm up to iteration k. ω is a dynamic parameter that is linearly decreased with each algorithm iteration as the following [20]: ωk + 1 = ωmax −
ω max − ω min k: k max
ð8Þ
c1 and c2 are parameters that have the constant sum of 4. 3. Convergence check. Once the convergence criterion is achieved the algorithm ends.
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2.4. Big Bang–Big Crunch The main steps of the Big Bang–Big Crunch optimization algorithm are: 1. Initiation: Answers that are mimicked as the position of each mass in the search space are selected randomly all around the search space. The creation of the initial random population is called the Big Bang phase. 2. Center of mass calculation: Fitness of all search space particles is calculated. Center of mass is found according to Eq. (9). In this step the best fit individual can be chosen as the center of mass instead of using the following equation: NC
∑
xcm =
1
∑
i=1
:
ð9Þ
fi
3. Position update: New candidates are calculated around the center of mass by adding or subtracting a normal random number whose value decreases as the iterations elapse. This can be formalized as new
x
= xcm +
αrl k
ð10Þ
2.5. Similarities among GA, ACO, PSO and BB–BC In all these four algorithms, the following features are common: 1. Randomly selected initialization. 2. Refinement of the answers according to the best fitted answers of the previous loop or loops. 3. Addition of some random answers in order to stop the algorithm from an early convergence and preventing it from converging to a local optimum. Here, the optimization parameter adjusting the proportion of these answers is called the Randomness Parameter (RP). 4. Repetition of Steps 2 and 3 until convergence. The randomness parameters can be chosen from different points of view, and in the present study, the randomness parameters are: A. Genetic algorithm: Mutation rate. B. Ant colony optimization: φ (τ0 coefficient) C. Particle swarm optimization: c1 (local best velocity coefficient). It may seem that c1 is not a coefficient for a randomly selected part, but it should be noted, as the sum of c1 and c2 are constant here, an increase in c1 may cause a decrease in c2 that is the coefficient of the global answer. Therefore its role is to prevent the algorithm from converging to the global best answer and can be assumed as RP. It should be mentioned here that He et al. [14] proposed a hybrid PSO with passive congregation (PSOPC). In this method, the velocity was defined as i
i
+ 1
It is noticeable that none of the mutation rates of the GA and φ of the ACO were present in the original form of the algorithms, and passive congregation was not present in the original PSO algorithm.
= ωvk + c1 r1
In a frame structure, once a connection type is selected, the fabrication cost has little effect on the optimum design, since it varies proportionally with the structural weight [21]. Thus, it is a usual practice to seek the minimum weight design for a given loading state and boundary condition. Here, the problem is formulated as follows: Minimize: M
w = ρ ∑ Ai L i i=1
where xcm stands for the center of mass, l is the upper limit of the parameter, r is a normal random number, α is the optimization parameter and k is the iteration step. Then new point xnew is checked to be in the search space. 4. Convergence check: Returns to Step 2 until stopping criteria has been met.
vk
sequence in the range (0, 1). Adding the passive congregation model to the PSO may improve its performance; however this kind of PSO has not been taken into account in the present work because of the complication it adds to the parameter selection. However, if this kind of PSO is taken into account, randomness parameter will be c3, instead of c1. D. Big Bang–Big Crunch: α or random part coefficient. It should be α noticed that here is the real coefficient of the random part. k
3. Formulation of the optimization problem
xi
i = 1 fi N 1
1455
Pki − xik Δt
+ c2 r 2
Pkg − xik Δt
+ c3 r3
Rik − xik Δt ð11Þ
where Rik is a particle selected randomly from the swarm, c3 is the passive congregation coefficient and r3 is a uniform random
ð12Þ
Subjected to: KU − P = 0
ð13Þ
g1 ≥ 0; g2 ≥ 0; …; gn ≥ 0 where g1, g2, …,gn are constraint functions depending on the element being used in each problem and K, U and P are the stiffness matrix, nodal displacement and force vectors, respectively. In this study, the members should satisfy the following constraint on drift, deflection, compaction, strength and stability coefficients according to the Specification for Structural Steel Buildings [22], Minimum Design Loads for Buildings and Other Structures [23], International Building Code 2006 [24] and Seismic Provisions for Structural Steel Buildings [25]: – Drift Drift ≤ 0:02hsx :
ð14Þ
– Deflection
ΔL b l = 360 : ΔD+L b l = 240
ð15Þ
Consider that according to Table 1604.3 of IBC2006 (Deflection limit), for steel structural members the dead load should be taken as zero. – Compactness For SLRS members these limits are calculated according to Table I8-1 (Limiting Width–Thickness Ratios for Compression Elements) of Seismic Provisions for Structural Steel Buildings [25]. – Strength These constrains are based on both AISC 360–05 specification [22] and Seismic Provisions for Structural Steel Buildings [25]. – Stability θmax b
0:5 : βCd
ð16Þ
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– Irregularity There is no horizontal irregularity, but vertical irregularity limits are taken into consideration according to Table 12.3-2 (Vertical Structural Irregularities) of the ASCE/SEI 7-05[23]. It should be noted that the Vertical Geometric Irregularity has not been considered. This is because this type of irregularity is defined to exist where the horizontal dimension of the seismic force-resisting system in any story is more than 130% that of the adjacent story. As a result, having equal bay sizes in the given structures, this type of irregularity may not let adjacent stories have different numbers of bracings, so that the feasible bracing placement may reduce considerably, thus it is ignored in this study. – Slenderness The AISC specification no longer provides a specific maximum slenderness ratio, as it formerly did. The AISC Commentary (E2) does indicate however that if KL/r is N200 then the critical stress Fcr will be less than 43.43 MPa. That value is based on engineering judgment, practical economies, and the fact that special care has to be taken to protect from damage such a slender member during fabrication, shipping and erection. As a result of these important practical considerations, an engineer using the 2005 AISC Specification will probably select compression members with slenderness values below 200 [26]. Thus this point is also considered in this study. Furthermore, such a constrained formulation is treated in an unconstrained form, using a penalized fitness function as F = F0 −w ⁎ 1 + Kp :V
ð17Þ
s d V = ∑ max g ; 0 +max g ; 0
ð18Þ
NLC
where F0 is a constant taken as zero in the class of considered examples. Kp is the penalty coefficient and V denotes the total constraints' violation considering all nLC loading combinations including g s as the strength and gd for other constraint functions. Calculation of displacements, forces and stresses is based on the second-order elastic behavior of the structure using a finite element structural analysis routine and amplified first-order elastic analysis. 4. Simultaneous design Most of the building codes such as ASCE have some requirements to share loads between members in a reasonable way. For example, ASCE 7–05 defines a dual building frame system as the following: Dual System: A structural system with an essentially complete space frame providing support for vertical loads. Seismic force resistance is provided by the moment resisting frames and the shear walls or braced frames. On the other hand, for a dual system, the moment frames shall be capable of resisting at least 25% of the design seismic forces. Considering these requirements, one is not permitted to design all the members simultaneously. The method presented in this study that is called “simultaneous design of structure for all loads and frame for its loads” helps to better satisfy the building code requirements. Requirements of the essentially complete frame and the entire structure are provided at the same time by the use of this method. In other words, analysis outputs are achieved in two different steps, one after formation of the essential frame and one for the whole structure. After each step, the requirements of the building code are checked.
5. New criteria for optimization performance A convergence factor, CF, is defined as the average possibility of the elitist answer. As an example if the aim is to devote some steel profiles to a structure that has four elements. In the first step, frequency of modal profile of each element should be defined. CF is the mean of these frequencies. Table 1 illustrates an example for calculation of the CF for a structure containing 4 elements. In this study it will be shown how such a factor can help to adapt a meta-heuristic algorithm for our problem. 6. Numerical examples Three frames of 3, 5 and 10 stories are treated in the present work. The following features are common in all the examples. 6.1. Geometry Height of each floor = 3 m Width of the frame = 5.0 m Three degrees of freedom for each joint (x, y-translations and z-rotation) All connections and also supports are fixed. 6.2. Loading condition 1) Uniform distributed dead load of 6.3 kN/m2 in negative y-direction on all beam elements 2) Uniform distributed live load of 1.96 kN/m2 in negative y-direction on all beam elements 3) Earthquake concentrated loads are calculated according to the ASCE 7-05[23], according to the following parameters:
R=7 I=1 Ss = 1.32; S1 = 0.535; Seismic design category = E.
Earthquake loads acting on the given examples are shown in Table 2. 5.3. Material properties The 50 ksi steels are the predominant ones in use today. In fact some of the steel mills charge extra for W-sections if they consist of A36. On the other hand, A992 and A500 are preferred materials for
Table 1 An example for calculation of CF.
Answer 1 Answer 2 Answer 3 Answer 4 Answer 5 Modal answer Frequency of the modal answer Proportion of the modal answer among all answers CF
Element 1
Element 2
Element 3
Element 4
5 3 4 3 3 3 3 60%
41 36 39 42 41 41 2 40%
22 22 25 22 22 22 4 80%
15 17 16 17 19 17 2 40%
55%
Table 2 Earthquake load acting on different frames in the numerical examples. Floor
Earthquake loads (kN) 3 story
1 2 3 4 5 6 7 8 9 10 Base shear
5 story
120.12 240.24 360.36
10 story
80.08 160.16 240.24 320.32 400.4
720.72
32.73 68.372 105.2 142.83 181.05 219.76 258.88 298.35 338.14 378.2 2023.5
1201.2
Answer/Best Answer
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2.6 2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 1.E+00
1457
3 story 5 story 10 story
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
mutation rate Fig. 1. GA randomness parameter selection for the 3, 5 and 10 story frames.
Member type
Shape
ASTM designation
Fy(MPa)
Fu(MPa)
Column Beam Bracing
W W HSS Rect.
A992 A992 A500
344.70 344.70 317.20
448.20 448.20 399.90
W-shapes and HSS Rectangular, respectively [26]. Data are selected for the members, according to Table 3 and the following data:
Answer/Best Answer
Table 3 Section types selected for numerical examples.
2.6 2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 1.E+00
3 story 5 story 10 story
1.E-01
2 3 E = 2e8 kN=m Þ; ρ = 76:82 kN=m ; and ν = 0:3:
1.E-02
1.E-03
1.E-04
1.E-05
taw0 coeficient Fig. 2. ACO randomness parameter selection for the 3, 5 and 10 story frames.
5.4. Section list reduction 1.5
1. Sections are checked for slenderness, and compaction limits. Members which do not fall in the feasible region are omitted. 2. Considering Ai as the cross-section area of the ith member in the list, and n as the number of sections in the section list, the following steps are taken to reduce the size of the search space in order to increase the efficiency of the optimization algorithms: Aj −Ai 2.1. All members that satisfy b 0:1, (j = 1:n, i = 1) are put Ai in one group. 2.2. Section lists are classified by repeating step 2.1, considering the members of the last group to be omitted. 3. A new section list is created by substituting all members of each group by the best of them in carrying compression loads. 5.5. Members under optimization
3 story
Answer/Best Answer
The following algorithm is used to lessen the size of section list of bracings:
1.4
5 story 10 story
1.3 1.2 1.1 1 0.0 0.9
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
c1
Fig. 3. PSO randomness parameter selection for the 3, 5 and 10 story frames.
best answer changes more rapidly by an increase in the number of stories). 3. GA and ACO are more parameter dependent in comparison to the PSO and BB–BC.
In this study, all members are determined using optimization methods. 6. Results Figs. 1 to 4 show changes in the best weight achieved from each algorithm by the change in randomness parameter (RP). It should be noticed that in these figures the ratio of the weight achieved by each RP to the best weight achieved after parameter selection (using the best RP) is shown. From these figures it can be shown that: 1. Results are dependent on the RP selection. 2. Dependence of the answers on the RP selection increases by an increase in the size of search space (the ratio of the answer to the
Fig. 4. BB–BC randomness parameter selection for the 3, 5 and 10 story frames.
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According to the results of these figures, if the Best RP is called as RBbest, two groups of answers that are not as well as the best answer can be recognized as: 1. Under speed answers: In this kind of answers RP N RPbest so the randomly selected answers are more than needed, then the algorithm cannot converge in a specified number of loops. 2. Over speed answers: In this kind of answers RP b RPbest so the randomly selected answers are less than needed, then the algorithm converges more rapidly, and the answer will be a local optimum in place of the global one. Figs. 5 to 12 show the variation of the CF and the normalized fitness of some of the frames solved by the algorithms for the following three cases (these figures are selected from 12 optimizations performed on 3, 5 and 10 story frames using four optimization methods): ✓ One of the under speed answers (these answers are named precisely under) ✓ One of the over speed answers (these answers are named precisely over) ✓ The best answer. (These answers are named precisely best.) In these figures normalized fitness describes a kind of normalization for fitness with value changes from 0 to 100%, as the value of both CF and normalized fitness is displayed on the same axes, the following
Fig. 5. Changes of normalized fitness and CF in GA for the 3 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 6. Changes of normalized fitness and CF in GA for the 10 story frame, in “under speed”, “best” and “over speed” cases.
definition is considered for the progress of normalized fitness in loop number i:
Progressi =
8 > > 0 > > <
fitnessi N2 fitnessbest
> > fitnessi > > : 2− ⁎100 fitnessbest
fitnessi ≤2 fitnessbest
ð16Þ
where: fitnessi defines fitness of the best answer of the loop number i, and fitnessbest defines fitness of the best answer achieved at the end of optimization. Progress defines the value of CF for the CF chart. This kind of normalization may omit some initial iterations' results but it will make the most important parts of it more clear. From these figures the following observations can be made: 1. It can be seen from the comparison of the normalized fitness and CF charts of under speed answers in Figs. 5, 14, 7, and 8 that although the fitness does not change in a considerable number of loops, the real convergence that is the frequency of the elitist answer or CF is not more than 60%. This means that achieving a constant answer in a number of loops is not a proper sign of convergence. 2. The rate of the changes of CF is according to the following formula as estimated before: CF Rate Under speed answer b CF Rate Best answer b CF Rate Over speed answer :
Fig. 7. Changes of normalized fitness and CF in ACO for the 3 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 8. Changes of normalized fitness and CF in ACO for the 10 story frame, in “under speed”, “best” and “over speed” cases.
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Fig. 9. Changes of normalized fitness and CF in PSO for the 5 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 11. Changes of normalized fitness and CF in BB for the 5 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 10. Changes of normalized fitness and CF in PSO for the 10 story frame, in “under speed”, “best” and “over speed” cases.
Fig. 12. Changes of normalized fitness and CF in BB for the 10 story frame, in “under speed”, “best” and “over speed” cases.
This means that the convergence rate is not an inherent property of these algorithms, and as the CF chart can be under control by changes in RP, one can change convergence rate as he wishes. 3. Comparing the charts of the CF and the normalized fitness, one can see that most of the time changes in fitness occur when there is a change in the CF chart. This means that: 3.1. When the CF is approximately fixed, one may not expect the algorithm to improve the answers by much. 3.2. When the CF is near 100, there will not be more chance for the CF to promote. Thus the biggest] improvement in the answers has happened and the convergence is achieved. Then there will be no need to wait for a long time to satisfy other convergence criteria such as a constant answer for a number of loops. 3.3. When the answer is constant for a number of loops but CF in not near 100, one can enforce the algorithm to improve the results by decreasing the RP. It may cause the CF and hence the fitness to promote. 4. The value of the CF moves toward 100, in most of the best answers. 5. Refinement in the optimum answers mostly happen when CF changes. Thus a well adjusted algorithm is the one in which: 5.1. CF changes in a reasonable rate. Here a reasonable rate for an algorithm means a rate that is slow enough to refine the answer in each level of the CF, and is fast enough in order not to permit undesirable answers to stay up to the end. 6. BB–BC and PSO are not in the risk of being trapped in local optima resulted from a constant CF not promoted to a value near 100, because their original RP is defined in a way that it decreases during the optimization progress. This is because in BB–BC, the real
α which reduces every loop and in the PSO used k here, there is a dynamic variation of wthat leads the algorithm from a global search to local one.
RP parameter is
7. Modified method According to the results achieved from the previous figures, at this stage a method is presented to enforce the algorithm to adjust its parameters in an adaptive manner. The main idea of this method is to use a method in which: “CF changes with a reasonable rate in a number of iterations that is considered for the optimization and finally reaches a value close to 100.” For achieving this aim the following algorithm is used: 1. Set RP as high as possible. 2. Consider a formula for the CF progress which is named here as PCF (pre-defined convergence factor). 3. Reduce RP in a way that the CF of the algorithm follows the predefined line. In the present work, as the shape of the CF progress is not fixed in all examples, a function PCF is used which is as
loopxi −1 PCFi = 30 + 70 ⁎ : ð200x −1Þ
ð17Þ
Changes in PCF in different iterations of algorithm are shown in Fig. 13 for different values of x. Here PCF starts from the value of 30 because CF of the randomly selected answers achieves this value at
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Fig. 14. Results of the modified method compared with the normal one in the 3 story frame.
Fig. 13. Variation of the PCF function for different values of x.
the first loop. This is because when the answers are selected randomly, there are some roles for bracing placements that do not permit a story to have more than 3 bracings, thus many alternatives for bracing fills will be zero (no bracing). The following algorithm is used for the changes in RP at the ith iteration: tmp =
CFi − PCFi CFi
RP = RP ⁎ RPCoef RP = RP = RPCoef
ð18Þ tmp b −0:02 tmp N 0:1
ð19Þ
in which RPCoef is a coefficient that is multiplied to RP to adjust it and here it is taken as 0.95 in order to make smooth changes in RP in each loop. However, one can change it according to the algorithm being used. Results of the modified method compared to the standard methods are shown in Figs. 14 to 16. In these figures it can be seen that for all the methods and all the considered frames, the modified method achieves better or equal answers compared to the standard ones, and also:
Fig. 15. Results of the modified method compared with the normal one in the 5 story frame.
a) In complicated problems, modified methods achieve better results in comparison with standard methods. b) In small problems, although there may not be real difference between the results of the two methods, the number of tries for adjustment of the optimization parameters may considerably decrease in this way. The variation of CF and PCF for each frame, using the modified method, is shown in Figs. 17 to 19. From these figures one can conclude that:
Fig. 16. Results of the modified method compared with the normal one in the 10 story frame.
1. In most of the cases the best PCF is a linear one. 2. When the search space is too small to be under control, no value of RP can stop it from convergence. Thus the algorithm may show its inherent behavior. 3. The possibility of getting the convergence rate under control increases by an increase in the size of the search space. Figs. 20 to 22 show the five best placements achieved for each frame. It can be seen that in these figures one cannot expect an optimum form for the placement of the bracings. 8. Concluding remarks In this study layout optimization of the braced frames is studied using GA, ACO, PSO and BB–BC algorithms. The results of more than 700 frames designed by these methods show that: 1. There are some similarities between the 4 meta-heuristic algorithms investigated in the present study that cause them to obey the same rules. The most important similarity is that they all select the next loop generation based on the two main steps: a. Generation of new answers according to the best answers achieved in the previous loops.
Fig. 17. CF and predefined CF for the 3 story frame.
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b. Generation of some random answers in order to prevent the algorithm from being trapped in local optima. The present study shows that for changes in the proportion of the answers of phase b in comparison with those chosen from phase a, one can control the convergence rate. Therefore, the convergence rate will not be considered as an essentially inherent property of these algorithms and can be adjusted. Parameters of the algorithms defining the proportion of phase b to those of phase a which are called the Randomness Parameters (RP) in this paper are: In GA: Mutation rate In ACO: φ(τ0coefficient). In PSO: c1 (c3 if passive congregation is used) In BB–BC: α. 2. Reaching an answer in a pre-defined number of loops one cannot be sure of the convergence of the algorithm. Thus an index namely convergence factor (CF) is defined as the average possibility of the elitist answer. This index can show the real convergence of an algorithm. 3. Convergence rate of all 4 algorithms can be changed by adjusting the CF. 4. CF can be changed in all 4 algorithms by the change in RP. 5. Refinement in the optimum answers mostly happens when CF changes. And one cannot expect much from the algorithm that has a fixed CF. Therefore, a well adjusted algorithm is the one, in which:
Fig. 18. CF and predefined CF for the 5 story frame.
a. CF changes in a reasonable rate. Here a reasonable rate for an algorithm means a rate that is slow enough to refine the answer in each level of CF and is fast enough in order not to permit undesirable answers to stay to the end. b. If CF reaches near the value of 100 at the end of algorithm, it means a real convergence. 6. Optimization parameter adjustment is an important step using all four algorithms used here, so real run time of each algorithm should contain the time for parameter selection as well. This becomes more important when the size of search space increases. 7. Convergence rate should be decreased by an increase in the size of search space. In other words, RP should be increased by an increase in the size of search space. 8. PSO and BB used here are modified essentially in a way that the RP decreases by time so the algorithm may not be under the risk of being trapped in a constant value of CF.
Fig. 19. CF and predefined CF for the 10 story frame.
1 W=86.2314 kN
2 W=87.7729 kN
3 W=87.7729 kN
4 W=89.7264 kN
5 W=90.1184 kN
Fig. 20. Optimum placement of bracing in the 3 story frame.
1 W=158.5415 kN
2 W=158.5415 kN
3 W=158.5415 kN
4 W=158.5415 kN
Fig. 21. Optimum placement of bracing in the 5 story frame.
5 W=158.5415 kN
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1 W=392.519 kN
2 W=393.9542 kN
3 W=397.4314 kN
4 W=397.4979 kN
5 W=399.1922 kN
Fig. 22. Optimum placement of bracing in the 10 story frame.
9. Having the knowledge of desirable behavior of an optimization algorithm, one can enforce the algorithm to behave as desired by dynamic optimization parameters. This is the most important achievement of the study. And using this method, optimization parameters can be set automatically. Modified methods introduced in the present work utilize this ability using a pre-defined CF function namely PCF and enforcing the algorithm to obey this function. Comparing the modified method with the non-adaptive approach, it can be seen that: a. In complicated problems, modified methods achieve better results compared to the non-adaptive ones. b. In small problems, although there may not be a real difference between the answers of the two methods, the number of tries for adjustment of optimization parameters may considerably decrease in this way. 10. Using modified method one should be aware that: a. When the search space is too small to be under control, no value of RP can stop it from early convergence. Thus the algorithm may behave as its inherent behavior. b. The possibility of getting the convergence rate under control increases by an increase in the size of the search space. c. Almost all 4 algorithms behave well when the PCF is linear, and the only exception is the ACO. The results of ACO depend on the convergence rate more than other algorithms since the ACO employs a memory that selects answers according to the frequency of them in all loops from beginning to end. If an undesirable answer remains for a long time in the algorithm, it may stay to the end. However, in all 3 other algorithms a slow convergence rate may not harm the results considerably. 11. For a frame of general configuration one cannot find a special optimum layout for the bracings. Suggestion: In some studies 2-phase optimization algorithms are used [7,16]. In these methods as the size of search space differs in different phases; the need for a parameter selection may be time consuming. The present method can help the multi-phase optimization methods to be used more efficiently. Acknowledgment The first author is grateful to Iran National Science Foundation for the support.
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