Computers and Structures 136 (2014) 78–89

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An enhanced harmony search algorithm for optimum design of side sway steel frames Mahmoud R. Maheri ⇑, M.M. Narimani Department of Civil Engineering, Shiraz University, Shiraz, Iran

a r t i c l e

i n f o

Article history: Received 18 December 2012 Accepted 8 February 2014 Available online 4 March 2014 Keywords: Metaheuristic optimization algorithms Global search Enhanced harmony search Optimum design of steel frames

a b s t r a c t An enhanced harmony search (EHS) algorithm is developed enabling the HS algorithm to quickly escape from local optima. For this purpose, the harmony memory updating phase is enhanced by considering also designs that are worse than the worst design stored in the harmony memory but are far enough from local optima. The proposed EHS algorithm is utilized to solve four classical weight minimization problems of steel frames. Results indicate that, as far as the quality of optimum design and convergence behavior are concerned, EHS is signiﬁcantly superior or deﬁnitely competitive with other meta-heuristic optimization algorithms including the classical HS. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Development of efﬁcient and robust optimization methods for structural design is one of the most active research ﬁelds in structural engineering. Meta-heuristic optimization algorithms have been used in structural optimization problems with many design variables and large design spaces. The rationale behind meta-heuristic search is the mimicking of natural or physical phenomena such as, for example: biological evolution (e.g., evolution strategies (ESs) [1] and genetic algorithms (GAs) [2]), animal behavior (e.g., taboo search (TS) [3], ant colony (ACO) [4] and particle swarm (PSO) [5]), physical annealing (e.g., simulated annealing (SA) [6]), musical improvisation (harmony search (HS) [7]), evolution of the universe (big bang-big crunch (BB–BC [8]). The main drawback of traditional algorithms such as simple GA or basic SA is the slow convergence rate and high computational cost. More recent algorithms such as TS, ACO, PSO, HS and BB–BC are in general faster but still may be trapped in some local optima. In order to solve very complicated optimization problems, meta-heuristic algorithms were enhanced adopting different strategies: (i) the basic formulation of each algorithm included some improvements (for example, modiﬁed, enhanced, improved, intelligent and multiple-deme GAs [9]); (ii) two or more meta-heuristic algorithms are hybridised by combining their strength points and complementary features (for example, HGAPSO combined genetic algorithms and particle swarm [10], AC-PSO combined hybrid ant colony and particle swarm [11], hybrid particle swarm, ant colony ⇑ Corresponding author. Tel.: +98 9177167274; fax: +98 711 8321353. E-mail address: maheri[email protected] (M.R. Maheri). http://dx.doi.org/10.1016/j.compstruc.2014.02.001 0045-7949/Ó 2014 Elsevier Ltd. All rights reserved.

and harmony search [12], etc.); (iii) developing new heuristic algorithms based on other natural or physical phenomena. Comprehensive reviews on new developments in metaheuristic optimization for engineering problems are given by Saka [13], Lamberti and Pappalettere [14] and Saka and Dogan [15]. The harmony search algorithm, ﬁrst introduced and later developed by Geem and colleagues [7,16–24], is a relatively new metaheuristic optimization algorithm based on the musical process of searching for a perfect state of harmony. An overview of diverse applications of the method is given in [25]. HS performed well in a wide variety of optimization problems [25–31]. However, compared to more established meta-heuristic algorithms, its performance is somewhat less favorable. HS can well identify the high performance regions of the design space in a reasonably small time, but can get into trouble when local searches must be performed to reﬁne design. Hasançebi et al. carried out comparative studies on the performance of different metaheuristic algorithms, including SA, ESs, GAs, TS, ACO, PSO and HS, in optimum design of trusses [32] and frames [33]. In weight minimization problems of truss structures, they found that SA and ESs were the most powerful algorithms: in particular, ESs showed rapid and linear convergence in the early stages of optimization while SA showed slower and gradual convergence; PSO was competitive with SA and ESs; TS and ACO had mediocre performance; HS and simple GA found the worst designs showing slow convergence and unreliable search performance [32]. Despite its inadequate performance in large scale problems, HS was however very efﬁcient in small-scale problems. In optimization of frame structures, Hasançebi et al. [33] found that HS was overall the worst algorithm, even less efﬁcient than

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simple GA. It was concluded that proper reformulation (or reﬁnement) of HS could have improved the performance of this algorithm in large-scale structural optimization problems [33]. More recently, Erdal et al. [34] compared PSO and HS in optimum design problems of cellular beams and again found HS to be less efﬁcient. A number of efforts were made to improve HS performance in large scale structural optimization problems. In [35], several variants of the HS method were considered and the relative merits of those variants were compared in the optimization of three real size steel frames. In [36], another variant of the HS approach, based on the idea of Pareto-dominance-based ranking, was proposed for constrained engineering optimization problems: this algorithm was tested in three engineering design problems. Other modiﬁcations or improvements to HS algorithm were carried out with varying degrees of success [37–42]. A number of hybrid heuristic algorithms involving HS also were developed. One of the earliest studies in this regard is due to Li et al. [43]. They used a PSO with passive congregation (PSOPC) and a harmony search, termed HPSO. The HPSO algorithm handled the problem-speciﬁc constraints using a ﬂy-back mechanism method and the harmony search scheme dealt with constraints on design variables. A mixed-discrete harmony search approach for solving nonlinear optimization problems with integer, discrete and continuous variables, was proposed in [39]. Other HS-based hybrid optimization algorithms may be found in [44–47]. Despite of recent improvements, HS still suffers from getting trapped in local optima. In the present study, a novel approach is adopted to improve the global search capability of HS. For that purpose, in the new algorithm, termed enhanced harmony search (EHS), the updating phase of the harmony search is modiﬁed. The new formulation results in enhanced global search capability and faster convergence. In the following, after an overview of the general HS method, the proposed EHS algorithm is ﬁrst described and then applied to four optimum design problems of side sway steel frames. The results are then compared with those of other meta-heuristic algorithms demonstrating the effectiveness of the proposed approach. 2. Harmony search algorithm

Step 1. In this step, value ranges are set for the different optimization variables and collected in a pool from which the algorithm can select values to form new trial designs. The number of solution vectors in harmony memory (i.e. the size of harmony matrix) (HMS), harmony memory considering rate (HMCR), pitch adjusting rate (PAR) and termination criterion (maximum number of searches) are also selected in this step. Step 2. In this step, the harmony memory matrix (HM) is initialized. Each row of the harmony memory corresponds to a trial design included in a population of HMS designs. The HM matrix has n columns (i.e. equal to the total number of design variables) and HMS rows selected in step 1. The harmony memory matrix, H, has the following form:

2

x1;1 6 x1;2 6 6 6 ... ½H ¼ 6 6 ... 6 6 4 x1;hms1 x1;hms

Step Step Step Step Step

1. 2. 3. 4. 5.

The harmony search parameters are initialized. The harmony memory matrix is initialized. A new harmony vector is improvised. The harmony memory matrix is updated. Steps 3 and 4 are repeated until the termination criterion is satisﬁed.

... ... ... ...

xn1;1 xn1;2

...

... ...

...

...

... ...

...

x2;hms1 x2;hms

. . . . . . xn1;hms1 ... ...

xn1;hms

xn;1 xn;2

3

7 7 7 ... 7 7 ... 7 7 7 xn;hms1 5

ð1Þ

xn;hms

where, xi,j is the value of the ith design variable in the jth randomly selected feasible solution. The feasible solutions in the harmony memory matrix are sorted in descending order according to their objective function value. Step 3. In this step, a new harmony vector is generated from the HM, based on memory consideration, pitch adjustment and randomization. In generating a new harmony matrix, the new value of the ith design variable can be chosen from any discrete value within the range of ith column of the HM with the probability of HMCR which varies between 0 and 1. In other words, the new value of xi can be one of the discrete values of the vector {xi,1, xi,2, . . . , xi,hms}T with the probability of HMCR. The same is applied to all other design variables. In the random selection, the new value of the ith design variable can also be chosen randomly from the entire pool with the probability of 1 HMCR as follows; (

xnew ¼ i The general HS algorithm consists of ﬁve basic steps. Detailed explanation of these steps can be found in Lee and Geem [17]; the main steps can be summarized as follows:

x2;1 x2;2

fxi;1 ; xi;2 ; . . . ; xi;hms gT with probability of HMCR

fx1 ; x2 ; . . . ; xhms gT with probability of ð1 HMCRÞ

ð2Þ

In the above derivation, ns is the total number of values for the design variables stored in the harmony memory. If the new value of the design variable is selected among those of harmony memory matrix, this value is then checked to see whether it should be pitch adjusted. This operation uses the pitch adjustment parameter PAR that sets the rate of adjustment for the pitch chosen from the harmony memory matrix as follows: Pitch adjusting decision for xnew : i

Fig. 1. Schematic representation of the search space in problems containing several regions of multiple local optima.

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vector in the harmony matrix, it is included in the matrix while the worst one is excluded. Step 5. Steps 3 and 4 are repeated until the pre-selected maximum number of cycles is reached. This number is selected large enough such that within this number of design cycles no further improvement is observed in the objective function.

Fig. 2. Illustration of the EHS search strategy for the six-hump camel back problem.

Yes with probability of PAR No

with probability of ð1 PARÞ

ð3Þ

If the new pitch-adjustment decision for xnew came out as ‘yes’ from i the test and if the value selected for xnew from the harmony memory i is the kth element in the general discrete set, then a neighboring value is taken for the new xnew . This operation prevents stagnation and i provides a better chance for the solution to reach the global optimum. Step 4. The cost function is evaluated for the new harmony vector. If this value is better than the value for the worst harmony

In the general HS algorithm, the parameter PAR helps the algorithm to improve local search, whereas, HMCR is designed to improve global search. The HMCR sets the rate of choosing one value from the historic values stored in the HM and (1 HMCR) sets the rate of randomly choosing one value from the possible range of values. In order to make the best use of the harmony vectors in harmony memory, HMCR should ideally be set as 1.0. For improved global solution, however, HMCR is set as less than 1.0 so that other, less favorable, solutions can be added to the current population. For example, a HMCR value of 0.95 indicates that the HS algorithm will choose the decision variable value from historically stored values in the HM (i.e. the current population) with a 95% probability and from the entire possible range with a 5% probability. This is similar to the mutation process in GA. Based on the optimization problem to be solved, different investigators have used values for HMCR ranging from 0.7 to 0.95. The PAR parameter, should be close to 0; high values for this parameter may avoid HS to reach the global optimum. Values ranging from 0.1 to 0.5 have been used in the past for this parameter. 3. The proposed enhanced harmony search (EHS) algorithm In the proposed enhanced harmony search (EHS) algorithm, the harmony memory updating phase of Step 4 is modiﬁed in purpose of improving the global search capability. Other steps of the

Fig. 3. Flowchart of the proposed enhanced harmony search (EHS) method.

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M.R. Maheri, M.M. Narimani / Computers and Structures 136 (2014) 78–89 Table 1 EHS internal parameters selected in the optimization runs. Parameter

HMS HMCR PAR RAC

e* j* *

Table 2 Design groups for the two-bay, three-storey frame problem.

Frame Three-bay, two storey

One-bay, ten storey

Three-bay, 24-storey

744-member, 3D frame

50 0.7 0.4 0.7 2.0 1.0

60 0.8 0.35 0.7 2.0 1.0

100 0.8 0.4 0.7 2.0 1.0

100 0.8 0.38 0.75 2.0 1.0

e and j are penalty function parameters [26].

standard HS are kept unchanged. As it was mentioned earlier, in step 4 of the general algorithm, while HS is updating the harmony memory, if the objective function value of the new harmony vector is better than that of the worst harmony vector in the harmony memory matrix, this new harmony vector would replace the worst harmony in the harmony memory. In other words, the updating process is carried out based only on the value of the objective function. Fig. 1 shows schematically a search space for problems containing several regions of closely-spaced multiple local optima. When the algorithm is searching in one of these local optima regions, the ability of the HMCR parameter in moving the search out of this region and into the more distant local optima clusters will be limited. In other words, searching in a multiple local optima region causes the algorithm to become blind to some other regions of the search space far from the current region, which may contain the global optimum. In the proposed EHS method, the updating process is carried out based not only on the objective function value of the new harmony vector, but also on its distance from the worst harmony vector in the harmony memory matrix. While updating the harmony memory, the new improvised harmony vector is either feasible or infeasible. If it is feasible (i.e. its objective function value is better than the worst harmony vector in memory), it will be included in the harmony memory replacing the worst harmony in the memory and the harmony memory matrix size will remain unchanged. If its cost function value is signiﬁcantly worse than the worst vector in memory (i.e. it is not feasible), the new vector will be discarded. It should be noted that, similar to standard HS, the proposed EHS

Group

Members

1 2

1, 2, 3, 6, 7, 8, 11, 12, 13 4, 5, 9, 10, 14, 15

formulation penalizes the cost function by introducing a penalty function if the trial design turns infeasible. However, if the objective function value of the new harmony is only slightly worse than the worst harmony vector in memory, but it is located at a large distance from the worst harmony vector, we may conclude that this new, slightly not feasible, harmony vector gives some information on a region somewhere far away in the search space with relatively good cost function values and likely enough to be near global optimum. In this case, the harmony memory size is enlarged to include the new slightly infeasible harmony vector. Although this process increases the HMS, which may be thought to result in a decreased convergence rate, the algorithm makes best use of the information gained from this new vector, leading to a fast escape from the multiple local optima region. To further explain the concept of EHS, consider the six-hump camelback function presented below. This function is one of the most frequently used standard test functions in optimization problems [48].

1 f ðxÞ ¼ 4x21 2:1x41 þ x61 þ x1 x2 4x22 þ 4x42 3

ð4Þ

The function depends on two continuous variables x1 and x2 varying between 10 and +10. Therefore, the search space in this problem is a square with sides equal to 20 units. The magniﬁed search space and schematic plot of function values for this function are shown in Fig. 2. As it can be seen in Fig. 2, this function has 4 local optima and 2 global optima which are at either (x1, x2) = (0.08984, 0.71266) or (x1, x2) = (0.08984, 0.71266), each with a corresponding function value equal to f(x1, x2) = 1.0316285. Let us consider that the proposed EHS algorithm is to search this space for the global optimum. As it can be seen in Fig. 2, the region containing points A, B and C is near a local optimum and the algorithm is vulnerable to falling trapped in the nearby local optimum. Suppose point B is the worst

Fig. 4. Geometry and loading details of the two-bay, three-storey frame.

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Table 3 Optimization results obtained for the two-bay, three-storey frame problem. Element group

Beam Column Weight (lb) Number of analyses

AISC W-shapes GA (Pezeshk et al. [50])

ACO (Camp et al. [51])

HS (Degertekin [26])

TLBO (Tog˘an [52])

MMDGA (Safari et al. [53])

EHS (this study)

W24x62 W10x60 18,792 1800

W24x62 W10x60 18,792 3000

W21x62 W10x54 18,292 1853

W24x62 W10x49 17,789 –

W24x62 W10x60 18,792 175

W21x55 W10x68 18,000 220

vector in the harmony memory. At this moment, points C, D and E can be new harmonies with objective function values worse than B. Adding D or E to the harmony memory would evidently be more effective than adding C because they help EHS to get closer to global optima. According to the proposed enhanced harmony search algorithm, since the distances of points D and E to point B are much more than the distance of point C to point B, they are more likely to be added to the harmony memory. Also, as the distances of points D and E to point B are close, of these two points, it is the point with better objective function (point D) which is more likely to be selected for inclusion in the harmony memory. The distance of points in the search space is formulated according to the Euclidean distance; e.g., the distance of point D to B is: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D B 2 D B 2 ðx1 x1 Þ þ ðx2 x2 Þ . The maximum possible distance between qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the points in this search space is: ð10 þ 10Þ2 þ ð10 þ 10Þ2 . Fig. 3 illustrates the ﬂowchart of the proposed enhanced harmony search (EHS) method. The proposed modiﬁcation to the updating step of the HS algorithm is implemented as follows:

xbond ¼ FðHMSÞ FðHMS 1Þ

ð5Þ

x ¼ Fnh FðHMSÞ

ð6Þ

fitp ¼ x=xbond

ð7Þ

distp ¼ y=ybond

ð8Þ

fitpð1 distpÞ 6 RAC

ð9Þ

In the above equations, F(HMS) and F(HMS 1) are the objective function values of, respectively, the worst and the worst but one vectors in harmony memory; Fnh is the objective function value of the new harmony vector; ﬁtp expresses the difference between the new harmony vector objective function and F(HMS); y is the distance of the new harmony vector to the worst vector in the harmony memory and ybond is the maximum possible distance between nodes in the search space. Also, in Eq. (9), RAC

Fig. 5. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the two-bay, three-storey frame problem.

(relative acceptance criterion) is a new parameter in the proposed EHS, denoting the probability of including the new harmony in HM. This parameter should be speciﬁed by the user. An initial investigation carried out by the authors showed that RAC values in the range of 0.65–0.85 result in the best algorithm performance. Eq. (9) sets the criterion for including the new vector in the harmony memory on the basis of being slightly worse than the worst vector in HM, and being far from the last member of HM in the search space. It appears that the acceptance of the new harmony into HM, depends on two parameters: ﬁtp and distp. This equation is developed in such a way that the less the ﬁtp value, the less the new vector is worse than the worst vector in memory, and the more the distp value, the farther the new vector lies in relation to the worst vector in the search space. Therefore, small values of ﬁtp and large values of distp, lead to higher probability for the new vector to be accepted into the memory. Finally, it should be added that, a slight theme of ‘adaptivity’ is introduced in the updating phase of the EHS. When a parameter is said to be changing adaptively during the iterations, it means that not only its value is not constant during the analysis, but also it changes in a way that enhances the operation of the whole algorithm. EHS lets the size of the harmony memory to increase. High HMS values, however, may lower the algorithm speed. Therefore, every single increase in HMS must be carried out carefully. The process for accepting a new harmony vector which increases HMS is designed to depend on the difference between the objective function values of the harmony vectors in the harmony memory. Since this difference decreases in the course of solution, the rate by which HMS increases in size decreases as the solution progresses. In this way, EHS is inherently adaptive. 4. Design examples of side sway steel frames The proposed EHS algorithm is tested in four classical optimization problems of steel frames. Optimization results are compared with those reported in literature for other meta-heuristic optimization algorithms. Similar to other design solutions of the same

Fig. 6. Comparison of convergence curves of different HS algorithms for the twobay, three-storey frame problem.

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Table 1. Optimizations were run in the Matlab 7.8 (2009a) software environment on an Intel CORE i7 1.83 computer.

4.1. Two-bay, three-storey planar frame

Fig. 7. Geometry and loading details of the one-bay, ten-storey frame.

frames carried out previously by other investigators, in all design examples, the in-plane effective length factors of the column members are calculated as Kx > 0 while the out-of-plane effective length factor is speciﬁed as Ky = 1.0. Each column is considered as non-braced along its length while the unbraced length for beam members is speciﬁed in each problem separately. Also, in line with other works, the shear deformations are ignored. The EHS parameters selected for each of the three examples are listed in

The ﬁrst benchmark problem is a two-bay, three-storey planar frame subject to a single-load case as shown in Fig. 4. This frame was optimized by Hall et al. [49] in accordance with the AISC–LRFD speciﬁcation using an OC method. It was also designed subject to the same speciﬁcation by Pezeshk et al. [50] using a simple GA; by Camp et al. [51] using ACO; by Degertekin [26] using HS; by Tog˘an [52] using teaching–learning based optimization (TLBO) algorithm and by Safari and Maheri [53] using modiﬁed multi-deme GA (MMDGA). The values of the uniform and the point loads in Fig. 4 are factored loads appropriate for direct application of the strength/stability provisions of the AISC–LRFD speciﬁcation. Displacement constraints were not imposed for this design problem. A modulus of elasticity of E = 200 GPa (29,000 ksi), a yield stress of Fy = 248.2 MPa (36 ksi) and a material unit weight of c = 7861 kg/m3 (0.284 lb/in3) were used. The unbraced length factor for each beam member was speciﬁed to be 0.167. Imposed fabrication conditions require that all six beams to be of the same Wshape and that all nine columns have identical sections (Table 2). The beam group section may be chosen from the entire W-shapes of AISC standard list; however, the column group section is limited to W10 sections. Similar to Ref. [26], the harmony memory size was set equal to 50. Degertekin reported that smaller sizes lead to premature convergence and larger sizes resulted in low rates of convergence [26]. The EHS parameters considered were listed in Table 1. Also, similar to the HS solution in [26], 30 different optimization runs were carried out. Details of the best design thus obtained are given in Table 3: the lowest weight is 18,000 lb with a standard deviation of 251.7 lb. This design weighs 18,000 lb and the standard deviation for the 30 designs is evaluated as 251.7 lb. Design histories for the best design and the average of 30 designs for this frame are shown in Fig. 5. Deviation between best and average designs convergence histories is small enough. The robustness of the present algorithm is conﬁrmed by the fact that most of the optimization runs converged to the best design found amongst the 30 optimization runs. The best design found by EHS is compared with literature in Table 3. The present algorithm always designed the lightest structure except for TLBO that converged to a 1.2% lower structural weight than EHS. In order to compare convergence behavior, structural analyses required by the different optimization algorithms are indicated in Table 3. It can be seen that EHS is one order of magnitude faster than all other optimizers but about 26% slower than MMDGA which however designed a heavier structure. Fig. 6 compares the convergence curve relative to the best design found by EHS with its counterpart relative to the HS algorithm described in [26]: there is a substantial increase in convergence speed using the proposed EHS algorithm. It should be noted that since the HS search procedure is substantially different from GA, PSO and ACO, comparing convergence histories would be misleading. In HS, objective function is evaluated only once in each iteration while other meta-heuristic algorithms may involve multiple analyses in the design updating process. Therefore, whilst in HS the number of generations coincides with the number of structural analyses, the same may not be true for other metaheuristic optimizers. In summary, the proposed EHS algorithm seems to be the most efﬁcient optimizer overall in terms of structural weight and computational cost of the optimization process.

84

M.R. Maheri, M.M. Narimani / Computers and Structures 136 (2014) 78–89 Table 4 Design groups for the one-bay, ten-storey frame problem. Group

Members

1 2 3 4 5 6 7 8 9

1, 2, 4, 5 7, 8, 10, 11 13, 14, 16, 17 19, 20, 22, 23 25, 26, 28, 29 3, 6, 9 12, 15, 18 21, 24, 27 30

4.2. One-bay, ten-storey planar frame A one-bay, ten-storey planar frame consisting of 30 members, originally designed by Pezeshk et al. [50] using a standard GA is selected as the second benchmark problem (see Fig. 7). The same frame was also designed by Camp et al. [51] using ACO; by Degertekin [26] using HS; by Kaveh and Talatahari [54] using improved ACO (IACO); by Dog˘an and Saka [55] using PSO; by Tog˘an [52] using TLBO and by Safari and Maheri [53] using MMDGA. The frame was designed according to the AISC-LRFD speciﬁcations and including the following displacement constraint: interstorey drift < storey height/300. The modulus of elasticity was assumed to be E = 200 GPa and the yield stress was set as Fy = 248.2 MPa. Fabrication conditions requiring the same beam section to be used for every three consecutive storeys, beginning at the foundation, as well as, the same column section to be used in every two consecutive storeys were implemented. The element grouping resulted in four beam sections and ﬁve column sections for a total of nine design variables. Each of the four beam element groups could be chosen from the entire W-shapes of the AISC standard list, and the ﬁve column element groups were limited to the W12 and W14 sections (66 W-shapes). For each beam member, the unbraced length was speciﬁed to be as one-ﬁfth of the span. As indicated in literature, frame members can be divided in the nine groups listed in Table 4. The value of HMS was set equal to 60 [26]. Values of EHS internal parameters chosen for this test problem are listed in Table 1. Fifty optimization runs were carried out for different initial populations. The reason for increasing the number of runs for this example compared to that reported in [26] was to increase the probability of ﬁnding better solutions. However, the best solution was achieved within the 30 runs reported in [26]. The best design, weighing 59,514 lb with a standard deviation of 678.2 lb, is shown in Table 5 and compared with literature

Fig. 8. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the one-bay, ten-storey frame problem.

Fig. 9. Comparison of convergence curves of different HS algorithms for the onebay, ten-storey frame problem.

[26,50–55]. The proposed EHS algorithm found the best design overall: 8.6% lighter than simple GA, 4.9% lighter than ACO, 3.8% lighter than basic HS, 3.7% lighter than IACO, 8.4% lighter than PSO, 3.7% lighter than TLBO and 3.0% lighter than MMGA. Looking at the number of structural analyses required in the optimization process, it appears that EHS again is much faster (i.e. between 2 and 6 times) than all other algorithms except for MMGA that was only 27% slower than the present algorithm. Convergence curves relative to the best design and average design are compared

Table 5 Optimization results obtained for the one-bay, ten-storey frame problem. Group No.

AISC W-shapes GA (Pezeshk et al. [50])

ACO (Camp et al. [51])

HS (Degertekin [26])

IACO (Kaveh et al. [54])

PSO (Dog˘an&Saka [55])

TLBO (Tog˘an [52])

MMDGA (Safari et al. [53])

EHS (This study)

1 2 3 4 5 6 7 8 9

W14x233 W14x176 W14x159 W14x99 W12x79 W33x118 W30x90 W27x84 W24x55

W14x233 W14x176 W14x145 W14x99 W12x65 W30x108 W30x90 W27x84 W21x44

W14x211 W14x176 W14x145 W14x90 W14x61 W33x118 W30x90 W24x76 W18x46

W14x233 W14x176 W14x145 W14x90 W12x65 W33x118 W30x90 W24x76 W14x30

W33x141 W14x159 W14x132 W14x99 W14x99 W30x116 W21x68 W14x61 W40x183

W14x233 W14x176 W14x145 W14x99 W12x65 W30x108 W30x90 W27x84 W21x44

W12x230 W14x159 W14x120 W14x90 W12x58 W33x118 W30x108 W24x76 W16x40

W14x159 W14x730 W14x61 W12x87 W14x283 W24x68 W14x99 W21x111 W33x201

Weight (lb) Number of analyses

65,136 3000

62,610 8300

61,864 3690

61,820 –

64,948 7500

61,813 –

61,345 1800

59,514 1412

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Table 6 Design groups for the three-bay, twenty four-storey frame problem. Group

Members

1

5, 7, 12, 14, 19, 21, 26, 28, 33, 35, 40, 42, 47, 49, 54, 56, 61, 63, 68, 70, 75, 77, 82, 84, 89, 91, 96, 98, 103, 105, 110, 112, 117, 119, 124, 126, 131, 133, 138, 140, 145, 147, 152, 154, 159, 161 166, 168 6, 13, 20, 27, 34, 41, 48, 55, 62, 69, 76, 83, 90, 97, 104, 111, 118, 125, 132, 139, 146, 153, 160 167 1, 4, 8, 11, 15, 18 22, 25, 29, 32, 36, 39 43, 46, 50, 53, 57, 60 64, 67, 71, 74, 78, 81 85, 88, 92, 95, 99, 102 106, 109, 113, 116, 120, 123 127, 130, 134, 137, 141, 144 148, 151, 155, 158, 162, 165 2, 3, 9, 10, 16, 17 23, 24, 30, 31, 37, 38 44, 45, 51, 52, 58, 59 65, 66, 72, 73, 79, 80 86, 87, 93, 94, 100, 101 107, 108, 114, 115, 121, 122 128, 129, 135, 136, 142, 143 149, 150, 156, 157, 163, 164

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

4.3. Three-bay, twenty four-storey planar frame

Fig. 10. Geometry and loading details of the three-bay, twenty four-storey frame.

in Fig. 8. Since convergence histories deviated at most by about 10%, the proposed EHS algorithm can be considered robust enough. Fig. 9 compares the EHS convergence history obtained for the best design with its counterpart shown in [26]: the proposed algorithm is clearly superior. In summary, EHS was deﬁnitely the most efﬁcient optimization algorithm also in test problem 2.

The third benchmark problem is the weight optimization of the three-bay twenty four-storey planar steel frame shown in Fig. 10. The structure includes 168 members and is subject to a single load case. This frame was originally designed by Davison and Adams [56]. It was also designed by Camp et al. [51] using ACO, by Degertekin [26] using HS, by Kaveh and Talatahari [54] using improved ACO (IACO), and again by Kaveh and Talatahari [57] using the imperialist competitive algorithm (ICA). The loads considered are W = 5761.85 lb, w1 = 300 lb/ft, w2 = 436 lb/ft, w3 = 474 lb/ft and w4 = 408 lb/ft (1 lb = 4.448 N). This frame was optimized according to the AISC–LRFD speciﬁcation [58] including the following displacement constraint: interstorey drift < storey height/300. The Young modulus E was set as 205 GPa (29,732 ksi) and the yield stress Fy was set as 230.3 MPa (33.4 ksi). The fabrication conditions required grouping the members as shown in Fig. 10, and resulted in 16 column sections and 4 beam sections for a total of 20 design variables. Each of the 4 beam element groups could be chosen from all the W-shapes listed in AISC standard list, while the 16 column element groups were limited to W14 sections. All beams and columns were considered unbraced along their lengths. Frame elements can be divided in the twenty groups listed in Table 6. Due to the relatively larger size of this frame structure, the harmony memory size was set equal to 100. The other internal parameters are listed in Table 1. One hundred optimization runs were carried out for this example. The best design weighed 194,400 lb with a standard deviation of 1981 lb. The best design is quoted in Table 7 and compared with literature [26,51–54]. It can be seen that EHS was the most efﬁcient optimization algorithm improving design by 11.8% with respect to ACO, 9.5% with respect to HS, 10.6% with respect to IACO, 8.6% with respect to ICA, 4.2% with respect to TLBO and 3.7% with respect to MMGA. In terms of the number of structural analyses required in the optimization process, EHS was between 2.8 and 12.3 times faster than the other meta-heuristic algorithms considered in this study as comparison basis. Convergence curves relative to the best design and the average design are compared in Fig. 11. In spite of a 20% difference between

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Table 7 Optimization results obtained for the three-bay, twenty four-storey frame problem. Group No.

AISC W-shapes ACO (Camp et al. [51])

HS (Degertekin [26])

IACO (Kaveh et al. [54])

ICA (Kaveh&Tala. [57])

TLBO (Tog˘an [52])

MMDGA (Safari et al. [53])

EHS (This study)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

W30x90 W8x18 W24x55 W8x21 W14x145 W14x132 W14x132 W14x132 W14x68 W14x53 W14x43 W14x43 W14x145 W14x145 W14x120 W14x90 W14x90 W14x61 W14x30 W14x26

W30x90 W10x22 W18x40 W12x16 W14x176 W14x176 W14x132 W14x109 W14x82 W14x74 W14x34 W14x22 W14x145 W14x132 W14x109 W14x82 W14x61 W14x48 W14x30 W14x22

W30x99 W16x26 W18x35 W14x22 W14x145 W14x132 W14x120 W14x109 W14x48 W14x48 W14x34 W14x30 W14x159 W14x120 W14x109 W14x99 W14x82 W14x53 W14x38 W14x26

– – – – – – – – – – – – – – – – – – – –

W30x90 W8x18 W24x62 W6x9 W14x132 W14x120 W14x99 W14x82 W14x74 W14x53 W14x34 W14x22 W14x109 W14x99 W14x99 W14x90 W14x68 W14x53 W14x34 W14x22

W30x90 W8x15 W24x55 W10x15 W14x159 W14x132 W14x90 W14x90 W14x61 W14x48 W14x48 W14x22 W14x109 W14x99 W14x99 W14x74 W14x68 W14x53 W14x26 W14x22

W10x19 W12x190 W6x8.5 W24x370 W14x132 W14x30 W14x99 W14x53 W14x74 W14x26 W14x68 W14x193 W14x145 W14x26 W14x26 W14x43 W14x26 W14x120 W14x426 W14x68

Weight (lb) Number of analyses

220,465 15,500

214,860 14,651

217,464 3500

212,736 7500

203,008 12,000

201,907 4750

194,400 1259

Fig. 11. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the three-bay, twenty four-storey frame problem.

Fig. 12. Comparison of convergence curves of different HS algorithms for the threebay, twenty four-storey frame problem.

convergence histories, the present algorithm can still be considered robust enough. Similar to the other test cases, the present algorithm converged to the optimum design much more quickly than the HS algorithm implemented in [26] (see Fig. 12). In summary, EHS was the most efﬁcient meta-heuristic algorithm also in test problem 3.

including 297 commercial sections, while beam members can be selected from a discrete set of 171 W-shape economical sections. As indicated in [36], the frame is subject to two independent loading conditions of combined gravity (dead, live, and snow loads) and lateral loads (wind loads) evaluated according to ASCE 7-05 (ASCE 2005) [59] and based on the following design values: a design dead load of 60.13 lb/ft (22.88 kN/m2), a design live load of 50 lb/ft (22.39 kN/m2), a ground snow load of 25 lb/ft (21.20 kN/m2) and a basic wind speed of 121 mph (54.09 m/s). Lateral wind loads acting at each ﬂoor level on windward and leeward faces of the frame are given in Table 8. In the ﬁrst load case, gravity loads are applied together with wind loads acting along x axis; in the second load case, they are applied with wind loads acting along y axis. The combined stress, stability, displacement, and geometric constraints are imposed according to the indications of ASD–AISC [60]. Because of the large size of this space frame, harmony memory size was set equal to 100. The other EHS internal parameters are listed in Table 1. One hundred independent optimization runs were carried out for this example. The best design weighed 395,708 lb

4.4. Spatial 744 member steel frame The last test problem considered in this study is the weight minimization of the unbraced spatial steel frame shown in Fig. 13, including 744 elements connected by 315 joints. Considering structural symmetry and practical fabrication requirements, members can be divided in 16 groups. Fig. 13(a)–(d) show side, plan and 3D views of the structure and clarify member grouping. This space frame was optimized by Hasançebi et al. [36] using Adaptive Harmony Search (AHS) which was then compared with standard HS, simple GA, PSO, ACO and Tabu Search. Cross sections of columns can be selected from the complete W-shape proﬁle list

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87

Fig. 13. Schematic of the spatial 744 member steel frame: (a) side view (x–z plane); (b) side view (y–z plane); (c) 3D view; (d) plan view [36].

Table 8 Wind forces acting on the spatial 744 member steel frame. Floor number

1 2 3 4 5 6 7 8

Windward

Leeward

lb/ft

kN/m

lb/ft

kN/m

140.64 171.44 192.49 208.98 222.74 234.65 245.22 127.38

0.63 0.76 0.86 0.93 0.99 1.04 1.09 0.57

159.22 159.22 159.22 159.22 159.22 159.22 159.22 79.61

0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.36

with a standard deviation of 2103.7 lb. The best design is listed in Table 9 and compared with the other meta-heuristic solutions obtained by Hasançebi et al. [36]. It can be seen that the proposed EHS algorithm found exactly the best solution indicated in [36]

for adaptive HS. However, the present algorithm was considerably faster than AHS as it required only 4743 structural analyses vs. the 50,000 analyses declared in [36] for both standard and adaptive HS algorithms (see convergence curves shown in Fig. 14). Convergence curves relative to the best design and the average design are compared in Fig. 15. Since convergence histories deviated at most by about 15%, the proposed EHS algorithm can again be considered robust enough. The HS algorithm internal parameters inﬂuence the optimization results. A sensitivity study was carried out to investigate the effect of HMCR and PAR parameters on the results of the spatial steel frame problem. Table 10 analyzes the sensitivity of the proposed algorithm to internal parameters HMCR and PAR. It can be seen that the best performance was achieved for HMCR = 0.8 and PAR ranging from 0.38 to 0.45, respectively. Remarkably, the EHS algorithm successfully converged to a feasible design in all of the independent optimization runs carried out for each test problem.

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Table 9 Optimum results obtained for the spatial 744 member steel frame problem. Group No.

Simple GA [36]

PSO [36]

ACO [36]

Tabu search [36]

HS [36]

Adaptive HS [36]

EHS (this study)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Weight (lb) Number of analyses

W24X68 W14X120 W8X21 W21X44 W12X65 W12X79 W14X26 W16X26 W8X35 W12X72 W14X26 W14X22 W8X35 W8X35 W8X21 W18X35 424076.52 –

W14X99 W14X99 W8X18 W21X44 W14X48 W14X90 W16X31 W14X26 W8X31 W14X90 W8X18 W16X26 W8X31 W8X31 W8X18 W14X22 412200.98 –

W12X72 W14X109 W14X22 W18X40 W10X49 W12X87 W12X26 W16X26 W12X35 W12X72 W14X22 W16X26 W10X39 W8X35 W8X21 W10X22 405441.65 –

W8X35 W14X145 W16X26 W18X35 W8X24 W14X90 W14X22 W18X35 W10X54 W12X53 W6X20 W16X26 W8X24 W8X40 W8X18 W14X26 401647.84 –

W12X79 W14X132 W14X22 W16X36 W12X72 W12X87 W14X22 W16X26 W16X40 W12X65 W14X22 W16X26 W16X36 W8X35 W10X22 W14X22 420177.57 50,000

W12X87 W14X109 W10X22 W18X35 W12X65 W12X79 W10X22 W16X26 W8X31 W12X65 W14X22 W14X22 W6X20 W12X58 W8X18 W14X22 395708.03 50,000

W12X87 W14X109 W10X22 W18X35 W12X65 W12X79 W10X22 W16X26 W8X31 W12X65 W14X22 W14X22 W6X20 W12X58 W8X18 W14X22 395708.03 4743

Table 10 Effect of internal parameters, HMCR and PAR on optimization results of the spatial 744 member steel frame. HMCR value (PAR = 0.4) Weight (lb) PAR value (HMCR = 0.8) Weight (lb)

0.70

0.75

0.80

0.85

0.90

433,612 0.30

433,612 0.35

395,708 0.40

501,311 0.45

559,543 0.50

433,612

433,612

395,708

395,708

501,311

Table 11 Number of structural analyses and corresponding standard deviation and number of successful optimization runs for the four test problems. Test problem

Fig. 14. Comparison of convergence curves of different HS algorithms for the spatial 744 member frame problem.

Two-bay, threestorey frame One-bay, tenstorey frame Three-bay, twenty fourstorey frame Spatial frame

No. of structural analyses

No. of successful optimization runs

Standard deviation on the number of analyses

220

30

5.06

1412

50

37.9

1259

100

35.02

4743

100

132.63

are listed in Table 11. It should also be noted that although the proposed EHS algorithm performed very well in all test problems, its superiority is more evident in the larger scale problems. Classical HS has been indicated by a number of investigators to perform well in small scale problems but to become less efﬁcient in larger problems. The results of the present study demonstrate that the proposed modiﬁcation of the HS updating phase is very effective in steering the search process towards the global optimum.

Fig. 15. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the spatial 744-member frame problem.

A general point regarding the performance of the proposed EHS algorithms is that although some optimized designs slightly violated some optimization constraints, all the independent optimization runs carried out for each test problem reached feasible designs. The number of structural analyses, number of successful optimization runs and the standard deviation on the number of structural analyses for the four test problems discussed above

5. Conclusions This paper described an enhanced harmony search (EHS) algorithm where the harmony memory updating phase was enhanced by including in [HM] also those trial designs that although worse than the worst design currently included in the population are far enough from local optima. This allowed the global search capability of HS to be signiﬁcantly improved. The new algorithm was tested in four classical optimization problems of side sway frames and compared with other state-of-the-art meta-heuristic

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

An enhanced harmony search algorithm for optimum design of side sway steel frames Mahmoud R. Maheri ⇑, M.M. Narimani Department of Civil Engineering, Shiraz University, Shiraz, Iran

a r t i c l e

i n f o

Article history: Received 18 December 2012 Accepted 8 February 2014 Available online 4 March 2014 Keywords: Metaheuristic optimization algorithms Global search Enhanced harmony search Optimum design of steel frames

a b s t r a c t An enhanced harmony search (EHS) algorithm is developed enabling the HS algorithm to quickly escape from local optima. For this purpose, the harmony memory updating phase is enhanced by considering also designs that are worse than the worst design stored in the harmony memory but are far enough from local optima. The proposed EHS algorithm is utilized to solve four classical weight minimization problems of steel frames. Results indicate that, as far as the quality of optimum design and convergence behavior are concerned, EHS is signiﬁcantly superior or deﬁnitely competitive with other meta-heuristic optimization algorithms including the classical HS. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Development of efﬁcient and robust optimization methods for structural design is one of the most active research ﬁelds in structural engineering. Meta-heuristic optimization algorithms have been used in structural optimization problems with many design variables and large design spaces. The rationale behind meta-heuristic search is the mimicking of natural or physical phenomena such as, for example: biological evolution (e.g., evolution strategies (ESs) [1] and genetic algorithms (GAs) [2]), animal behavior (e.g., taboo search (TS) [3], ant colony (ACO) [4] and particle swarm (PSO) [5]), physical annealing (e.g., simulated annealing (SA) [6]), musical improvisation (harmony search (HS) [7]), evolution of the universe (big bang-big crunch (BB–BC [8]). The main drawback of traditional algorithms such as simple GA or basic SA is the slow convergence rate and high computational cost. More recent algorithms such as TS, ACO, PSO, HS and BB–BC are in general faster but still may be trapped in some local optima. In order to solve very complicated optimization problems, meta-heuristic algorithms were enhanced adopting different strategies: (i) the basic formulation of each algorithm included some improvements (for example, modiﬁed, enhanced, improved, intelligent and multiple-deme GAs [9]); (ii) two or more meta-heuristic algorithms are hybridised by combining their strength points and complementary features (for example, HGAPSO combined genetic algorithms and particle swarm [10], AC-PSO combined hybrid ant colony and particle swarm [11], hybrid particle swarm, ant colony ⇑ Corresponding author. Tel.: +98 9177167274; fax: +98 711 8321353. E-mail address: maheri[email protected] (M.R. Maheri). http://dx.doi.org/10.1016/j.compstruc.2014.02.001 0045-7949/Ó 2014 Elsevier Ltd. All rights reserved.

and harmony search [12], etc.); (iii) developing new heuristic algorithms based on other natural or physical phenomena. Comprehensive reviews on new developments in metaheuristic optimization for engineering problems are given by Saka [13], Lamberti and Pappalettere [14] and Saka and Dogan [15]. The harmony search algorithm, ﬁrst introduced and later developed by Geem and colleagues [7,16–24], is a relatively new metaheuristic optimization algorithm based on the musical process of searching for a perfect state of harmony. An overview of diverse applications of the method is given in [25]. HS performed well in a wide variety of optimization problems [25–31]. However, compared to more established meta-heuristic algorithms, its performance is somewhat less favorable. HS can well identify the high performance regions of the design space in a reasonably small time, but can get into trouble when local searches must be performed to reﬁne design. Hasançebi et al. carried out comparative studies on the performance of different metaheuristic algorithms, including SA, ESs, GAs, TS, ACO, PSO and HS, in optimum design of trusses [32] and frames [33]. In weight minimization problems of truss structures, they found that SA and ESs were the most powerful algorithms: in particular, ESs showed rapid and linear convergence in the early stages of optimization while SA showed slower and gradual convergence; PSO was competitive with SA and ESs; TS and ACO had mediocre performance; HS and simple GA found the worst designs showing slow convergence and unreliable search performance [32]. Despite its inadequate performance in large scale problems, HS was however very efﬁcient in small-scale problems. In optimization of frame structures, Hasançebi et al. [33] found that HS was overall the worst algorithm, even less efﬁcient than

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simple GA. It was concluded that proper reformulation (or reﬁnement) of HS could have improved the performance of this algorithm in large-scale structural optimization problems [33]. More recently, Erdal et al. [34] compared PSO and HS in optimum design problems of cellular beams and again found HS to be less efﬁcient. A number of efforts were made to improve HS performance in large scale structural optimization problems. In [35], several variants of the HS method were considered and the relative merits of those variants were compared in the optimization of three real size steel frames. In [36], another variant of the HS approach, based on the idea of Pareto-dominance-based ranking, was proposed for constrained engineering optimization problems: this algorithm was tested in three engineering design problems. Other modiﬁcations or improvements to HS algorithm were carried out with varying degrees of success [37–42]. A number of hybrid heuristic algorithms involving HS also were developed. One of the earliest studies in this regard is due to Li et al. [43]. They used a PSO with passive congregation (PSOPC) and a harmony search, termed HPSO. The HPSO algorithm handled the problem-speciﬁc constraints using a ﬂy-back mechanism method and the harmony search scheme dealt with constraints on design variables. A mixed-discrete harmony search approach for solving nonlinear optimization problems with integer, discrete and continuous variables, was proposed in [39]. Other HS-based hybrid optimization algorithms may be found in [44–47]. Despite of recent improvements, HS still suffers from getting trapped in local optima. In the present study, a novel approach is adopted to improve the global search capability of HS. For that purpose, in the new algorithm, termed enhanced harmony search (EHS), the updating phase of the harmony search is modiﬁed. The new formulation results in enhanced global search capability and faster convergence. In the following, after an overview of the general HS method, the proposed EHS algorithm is ﬁrst described and then applied to four optimum design problems of side sway steel frames. The results are then compared with those of other meta-heuristic algorithms demonstrating the effectiveness of the proposed approach. 2. Harmony search algorithm

Step 1. In this step, value ranges are set for the different optimization variables and collected in a pool from which the algorithm can select values to form new trial designs. The number of solution vectors in harmony memory (i.e. the size of harmony matrix) (HMS), harmony memory considering rate (HMCR), pitch adjusting rate (PAR) and termination criterion (maximum number of searches) are also selected in this step. Step 2. In this step, the harmony memory matrix (HM) is initialized. Each row of the harmony memory corresponds to a trial design included in a population of HMS designs. The HM matrix has n columns (i.e. equal to the total number of design variables) and HMS rows selected in step 1. The harmony memory matrix, H, has the following form:

2

x1;1 6 x1;2 6 6 6 ... ½H ¼ 6 6 ... 6 6 4 x1;hms1 x1;hms

Step Step Step Step Step

1. 2. 3. 4. 5.

The harmony search parameters are initialized. The harmony memory matrix is initialized. A new harmony vector is improvised. The harmony memory matrix is updated. Steps 3 and 4 are repeated until the termination criterion is satisﬁed.

... ... ... ...

xn1;1 xn1;2

...

... ...

...

...

... ...

...

x2;hms1 x2;hms

. . . . . . xn1;hms1 ... ...

xn1;hms

xn;1 xn;2

3

7 7 7 ... 7 7 ... 7 7 7 xn;hms1 5

ð1Þ

xn;hms

where, xi,j is the value of the ith design variable in the jth randomly selected feasible solution. The feasible solutions in the harmony memory matrix are sorted in descending order according to their objective function value. Step 3. In this step, a new harmony vector is generated from the HM, based on memory consideration, pitch adjustment and randomization. In generating a new harmony matrix, the new value of the ith design variable can be chosen from any discrete value within the range of ith column of the HM with the probability of HMCR which varies between 0 and 1. In other words, the new value of xi can be one of the discrete values of the vector {xi,1, xi,2, . . . , xi,hms}T with the probability of HMCR. The same is applied to all other design variables. In the random selection, the new value of the ith design variable can also be chosen randomly from the entire pool with the probability of 1 HMCR as follows; (

xnew ¼ i The general HS algorithm consists of ﬁve basic steps. Detailed explanation of these steps can be found in Lee and Geem [17]; the main steps can be summarized as follows:

x2;1 x2;2

fxi;1 ; xi;2 ; . . . ; xi;hms gT with probability of HMCR

fx1 ; x2 ; . . . ; xhms gT with probability of ð1 HMCRÞ

ð2Þ

In the above derivation, ns is the total number of values for the design variables stored in the harmony memory. If the new value of the design variable is selected among those of harmony memory matrix, this value is then checked to see whether it should be pitch adjusted. This operation uses the pitch adjustment parameter PAR that sets the rate of adjustment for the pitch chosen from the harmony memory matrix as follows: Pitch adjusting decision for xnew : i

Fig. 1. Schematic representation of the search space in problems containing several regions of multiple local optima.

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vector in the harmony matrix, it is included in the matrix while the worst one is excluded. Step 5. Steps 3 and 4 are repeated until the pre-selected maximum number of cycles is reached. This number is selected large enough such that within this number of design cycles no further improvement is observed in the objective function.

Fig. 2. Illustration of the EHS search strategy for the six-hump camel back problem.

Yes with probability of PAR No

with probability of ð1 PARÞ

ð3Þ

If the new pitch-adjustment decision for xnew came out as ‘yes’ from i the test and if the value selected for xnew from the harmony memory i is the kth element in the general discrete set, then a neighboring value is taken for the new xnew . This operation prevents stagnation and i provides a better chance for the solution to reach the global optimum. Step 4. The cost function is evaluated for the new harmony vector. If this value is better than the value for the worst harmony

In the general HS algorithm, the parameter PAR helps the algorithm to improve local search, whereas, HMCR is designed to improve global search. The HMCR sets the rate of choosing one value from the historic values stored in the HM and (1 HMCR) sets the rate of randomly choosing one value from the possible range of values. In order to make the best use of the harmony vectors in harmony memory, HMCR should ideally be set as 1.0. For improved global solution, however, HMCR is set as less than 1.0 so that other, less favorable, solutions can be added to the current population. For example, a HMCR value of 0.95 indicates that the HS algorithm will choose the decision variable value from historically stored values in the HM (i.e. the current population) with a 95% probability and from the entire possible range with a 5% probability. This is similar to the mutation process in GA. Based on the optimization problem to be solved, different investigators have used values for HMCR ranging from 0.7 to 0.95. The PAR parameter, should be close to 0; high values for this parameter may avoid HS to reach the global optimum. Values ranging from 0.1 to 0.5 have been used in the past for this parameter. 3. The proposed enhanced harmony search (EHS) algorithm In the proposed enhanced harmony search (EHS) algorithm, the harmony memory updating phase of Step 4 is modiﬁed in purpose of improving the global search capability. Other steps of the

Fig. 3. Flowchart of the proposed enhanced harmony search (EHS) method.

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M.R. Maheri, M.M. Narimani / Computers and Structures 136 (2014) 78–89 Table 1 EHS internal parameters selected in the optimization runs. Parameter

HMS HMCR PAR RAC

e* j* *

Table 2 Design groups for the two-bay, three-storey frame problem.

Frame Three-bay, two storey

One-bay, ten storey

Three-bay, 24-storey

744-member, 3D frame

50 0.7 0.4 0.7 2.0 1.0

60 0.8 0.35 0.7 2.0 1.0

100 0.8 0.4 0.7 2.0 1.0

100 0.8 0.38 0.75 2.0 1.0

e and j are penalty function parameters [26].

standard HS are kept unchanged. As it was mentioned earlier, in step 4 of the general algorithm, while HS is updating the harmony memory, if the objective function value of the new harmony vector is better than that of the worst harmony vector in the harmony memory matrix, this new harmony vector would replace the worst harmony in the harmony memory. In other words, the updating process is carried out based only on the value of the objective function. Fig. 1 shows schematically a search space for problems containing several regions of closely-spaced multiple local optima. When the algorithm is searching in one of these local optima regions, the ability of the HMCR parameter in moving the search out of this region and into the more distant local optima clusters will be limited. In other words, searching in a multiple local optima region causes the algorithm to become blind to some other regions of the search space far from the current region, which may contain the global optimum. In the proposed EHS method, the updating process is carried out based not only on the objective function value of the new harmony vector, but also on its distance from the worst harmony vector in the harmony memory matrix. While updating the harmony memory, the new improvised harmony vector is either feasible or infeasible. If it is feasible (i.e. its objective function value is better than the worst harmony vector in memory), it will be included in the harmony memory replacing the worst harmony in the memory and the harmony memory matrix size will remain unchanged. If its cost function value is signiﬁcantly worse than the worst vector in memory (i.e. it is not feasible), the new vector will be discarded. It should be noted that, similar to standard HS, the proposed EHS

Group

Members

1 2

1, 2, 3, 6, 7, 8, 11, 12, 13 4, 5, 9, 10, 14, 15

formulation penalizes the cost function by introducing a penalty function if the trial design turns infeasible. However, if the objective function value of the new harmony is only slightly worse than the worst harmony vector in memory, but it is located at a large distance from the worst harmony vector, we may conclude that this new, slightly not feasible, harmony vector gives some information on a region somewhere far away in the search space with relatively good cost function values and likely enough to be near global optimum. In this case, the harmony memory size is enlarged to include the new slightly infeasible harmony vector. Although this process increases the HMS, which may be thought to result in a decreased convergence rate, the algorithm makes best use of the information gained from this new vector, leading to a fast escape from the multiple local optima region. To further explain the concept of EHS, consider the six-hump camelback function presented below. This function is one of the most frequently used standard test functions in optimization problems [48].

1 f ðxÞ ¼ 4x21 2:1x41 þ x61 þ x1 x2 4x22 þ 4x42 3

ð4Þ

The function depends on two continuous variables x1 and x2 varying between 10 and +10. Therefore, the search space in this problem is a square with sides equal to 20 units. The magniﬁed search space and schematic plot of function values for this function are shown in Fig. 2. As it can be seen in Fig. 2, this function has 4 local optima and 2 global optima which are at either (x1, x2) = (0.08984, 0.71266) or (x1, x2) = (0.08984, 0.71266), each with a corresponding function value equal to f(x1, x2) = 1.0316285. Let us consider that the proposed EHS algorithm is to search this space for the global optimum. As it can be seen in Fig. 2, the region containing points A, B and C is near a local optimum and the algorithm is vulnerable to falling trapped in the nearby local optimum. Suppose point B is the worst

Fig. 4. Geometry and loading details of the two-bay, three-storey frame.

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Table 3 Optimization results obtained for the two-bay, three-storey frame problem. Element group

Beam Column Weight (lb) Number of analyses

AISC W-shapes GA (Pezeshk et al. [50])

ACO (Camp et al. [51])

HS (Degertekin [26])

TLBO (Tog˘an [52])

MMDGA (Safari et al. [53])

EHS (this study)

W24x62 W10x60 18,792 1800

W24x62 W10x60 18,792 3000

W21x62 W10x54 18,292 1853

W24x62 W10x49 17,789 –

W24x62 W10x60 18,792 175

W21x55 W10x68 18,000 220

vector in the harmony memory. At this moment, points C, D and E can be new harmonies with objective function values worse than B. Adding D or E to the harmony memory would evidently be more effective than adding C because they help EHS to get closer to global optima. According to the proposed enhanced harmony search algorithm, since the distances of points D and E to point B are much more than the distance of point C to point B, they are more likely to be added to the harmony memory. Also, as the distances of points D and E to point B are close, of these two points, it is the point with better objective function (point D) which is more likely to be selected for inclusion in the harmony memory. The distance of points in the search space is formulated according to the Euclidean distance; e.g., the distance of point D to B is: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D B 2 D B 2 ðx1 x1 Þ þ ðx2 x2 Þ . The maximum possible distance between qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the points in this search space is: ð10 þ 10Þ2 þ ð10 þ 10Þ2 . Fig. 3 illustrates the ﬂowchart of the proposed enhanced harmony search (EHS) method. The proposed modiﬁcation to the updating step of the HS algorithm is implemented as follows:

xbond ¼ FðHMSÞ FðHMS 1Þ

ð5Þ

x ¼ Fnh FðHMSÞ

ð6Þ

fitp ¼ x=xbond

ð7Þ

distp ¼ y=ybond

ð8Þ

fitpð1 distpÞ 6 RAC

ð9Þ

In the above equations, F(HMS) and F(HMS 1) are the objective function values of, respectively, the worst and the worst but one vectors in harmony memory; Fnh is the objective function value of the new harmony vector; ﬁtp expresses the difference between the new harmony vector objective function and F(HMS); y is the distance of the new harmony vector to the worst vector in the harmony memory and ybond is the maximum possible distance between nodes in the search space. Also, in Eq. (9), RAC

Fig. 5. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the two-bay, three-storey frame problem.

(relative acceptance criterion) is a new parameter in the proposed EHS, denoting the probability of including the new harmony in HM. This parameter should be speciﬁed by the user. An initial investigation carried out by the authors showed that RAC values in the range of 0.65–0.85 result in the best algorithm performance. Eq. (9) sets the criterion for including the new vector in the harmony memory on the basis of being slightly worse than the worst vector in HM, and being far from the last member of HM in the search space. It appears that the acceptance of the new harmony into HM, depends on two parameters: ﬁtp and distp. This equation is developed in such a way that the less the ﬁtp value, the less the new vector is worse than the worst vector in memory, and the more the distp value, the farther the new vector lies in relation to the worst vector in the search space. Therefore, small values of ﬁtp and large values of distp, lead to higher probability for the new vector to be accepted into the memory. Finally, it should be added that, a slight theme of ‘adaptivity’ is introduced in the updating phase of the EHS. When a parameter is said to be changing adaptively during the iterations, it means that not only its value is not constant during the analysis, but also it changes in a way that enhances the operation of the whole algorithm. EHS lets the size of the harmony memory to increase. High HMS values, however, may lower the algorithm speed. Therefore, every single increase in HMS must be carried out carefully. The process for accepting a new harmony vector which increases HMS is designed to depend on the difference between the objective function values of the harmony vectors in the harmony memory. Since this difference decreases in the course of solution, the rate by which HMS increases in size decreases as the solution progresses. In this way, EHS is inherently adaptive. 4. Design examples of side sway steel frames The proposed EHS algorithm is tested in four classical optimization problems of steel frames. Optimization results are compared with those reported in literature for other meta-heuristic optimization algorithms. Similar to other design solutions of the same

Fig. 6. Comparison of convergence curves of different HS algorithms for the twobay, three-storey frame problem.

M.R. Maheri, M.M. Narimani / Computers and Structures 136 (2014) 78–89

83

Table 1. Optimizations were run in the Matlab 7.8 (2009a) software environment on an Intel CORE i7 1.83 computer.

4.1. Two-bay, three-storey planar frame

Fig. 7. Geometry and loading details of the one-bay, ten-storey frame.

frames carried out previously by other investigators, in all design examples, the in-plane effective length factors of the column members are calculated as Kx > 0 while the out-of-plane effective length factor is speciﬁed as Ky = 1.0. Each column is considered as non-braced along its length while the unbraced length for beam members is speciﬁed in each problem separately. Also, in line with other works, the shear deformations are ignored. The EHS parameters selected for each of the three examples are listed in

The ﬁrst benchmark problem is a two-bay, three-storey planar frame subject to a single-load case as shown in Fig. 4. This frame was optimized by Hall et al. [49] in accordance with the AISC–LRFD speciﬁcation using an OC method. It was also designed subject to the same speciﬁcation by Pezeshk et al. [50] using a simple GA; by Camp et al. [51] using ACO; by Degertekin [26] using HS; by Tog˘an [52] using teaching–learning based optimization (TLBO) algorithm and by Safari and Maheri [53] using modiﬁed multi-deme GA (MMDGA). The values of the uniform and the point loads in Fig. 4 are factored loads appropriate for direct application of the strength/stability provisions of the AISC–LRFD speciﬁcation. Displacement constraints were not imposed for this design problem. A modulus of elasticity of E = 200 GPa (29,000 ksi), a yield stress of Fy = 248.2 MPa (36 ksi) and a material unit weight of c = 7861 kg/m3 (0.284 lb/in3) were used. The unbraced length factor for each beam member was speciﬁed to be 0.167. Imposed fabrication conditions require that all six beams to be of the same Wshape and that all nine columns have identical sections (Table 2). The beam group section may be chosen from the entire W-shapes of AISC standard list; however, the column group section is limited to W10 sections. Similar to Ref. [26], the harmony memory size was set equal to 50. Degertekin reported that smaller sizes lead to premature convergence and larger sizes resulted in low rates of convergence [26]. The EHS parameters considered were listed in Table 1. Also, similar to the HS solution in [26], 30 different optimization runs were carried out. Details of the best design thus obtained are given in Table 3: the lowest weight is 18,000 lb with a standard deviation of 251.7 lb. This design weighs 18,000 lb and the standard deviation for the 30 designs is evaluated as 251.7 lb. Design histories for the best design and the average of 30 designs for this frame are shown in Fig. 5. Deviation between best and average designs convergence histories is small enough. The robustness of the present algorithm is conﬁrmed by the fact that most of the optimization runs converged to the best design found amongst the 30 optimization runs. The best design found by EHS is compared with literature in Table 3. The present algorithm always designed the lightest structure except for TLBO that converged to a 1.2% lower structural weight than EHS. In order to compare convergence behavior, structural analyses required by the different optimization algorithms are indicated in Table 3. It can be seen that EHS is one order of magnitude faster than all other optimizers but about 26% slower than MMDGA which however designed a heavier structure. Fig. 6 compares the convergence curve relative to the best design found by EHS with its counterpart relative to the HS algorithm described in [26]: there is a substantial increase in convergence speed using the proposed EHS algorithm. It should be noted that since the HS search procedure is substantially different from GA, PSO and ACO, comparing convergence histories would be misleading. In HS, objective function is evaluated only once in each iteration while other meta-heuristic algorithms may involve multiple analyses in the design updating process. Therefore, whilst in HS the number of generations coincides with the number of structural analyses, the same may not be true for other metaheuristic optimizers. In summary, the proposed EHS algorithm seems to be the most efﬁcient optimizer overall in terms of structural weight and computational cost of the optimization process.

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M.R. Maheri, M.M. Narimani / Computers and Structures 136 (2014) 78–89 Table 4 Design groups for the one-bay, ten-storey frame problem. Group

Members

1 2 3 4 5 6 7 8 9

1, 2, 4, 5 7, 8, 10, 11 13, 14, 16, 17 19, 20, 22, 23 25, 26, 28, 29 3, 6, 9 12, 15, 18 21, 24, 27 30

4.2. One-bay, ten-storey planar frame A one-bay, ten-storey planar frame consisting of 30 members, originally designed by Pezeshk et al. [50] using a standard GA is selected as the second benchmark problem (see Fig. 7). The same frame was also designed by Camp et al. [51] using ACO; by Degertekin [26] using HS; by Kaveh and Talatahari [54] using improved ACO (IACO); by Dog˘an and Saka [55] using PSO; by Tog˘an [52] using TLBO and by Safari and Maheri [53] using MMDGA. The frame was designed according to the AISC-LRFD speciﬁcations and including the following displacement constraint: interstorey drift < storey height/300. The modulus of elasticity was assumed to be E = 200 GPa and the yield stress was set as Fy = 248.2 MPa. Fabrication conditions requiring the same beam section to be used for every three consecutive storeys, beginning at the foundation, as well as, the same column section to be used in every two consecutive storeys were implemented. The element grouping resulted in four beam sections and ﬁve column sections for a total of nine design variables. Each of the four beam element groups could be chosen from the entire W-shapes of the AISC standard list, and the ﬁve column element groups were limited to the W12 and W14 sections (66 W-shapes). For each beam member, the unbraced length was speciﬁed to be as one-ﬁfth of the span. As indicated in literature, frame members can be divided in the nine groups listed in Table 4. The value of HMS was set equal to 60 [26]. Values of EHS internal parameters chosen for this test problem are listed in Table 1. Fifty optimization runs were carried out for different initial populations. The reason for increasing the number of runs for this example compared to that reported in [26] was to increase the probability of ﬁnding better solutions. However, the best solution was achieved within the 30 runs reported in [26]. The best design, weighing 59,514 lb with a standard deviation of 678.2 lb, is shown in Table 5 and compared with literature

Fig. 8. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the one-bay, ten-storey frame problem.

Fig. 9. Comparison of convergence curves of different HS algorithms for the onebay, ten-storey frame problem.

[26,50–55]. The proposed EHS algorithm found the best design overall: 8.6% lighter than simple GA, 4.9% lighter than ACO, 3.8% lighter than basic HS, 3.7% lighter than IACO, 8.4% lighter than PSO, 3.7% lighter than TLBO and 3.0% lighter than MMGA. Looking at the number of structural analyses required in the optimization process, it appears that EHS again is much faster (i.e. between 2 and 6 times) than all other algorithms except for MMGA that was only 27% slower than the present algorithm. Convergence curves relative to the best design and average design are compared

Table 5 Optimization results obtained for the one-bay, ten-storey frame problem. Group No.

AISC W-shapes GA (Pezeshk et al. [50])

ACO (Camp et al. [51])

HS (Degertekin [26])

IACO (Kaveh et al. [54])

PSO (Dog˘an&Saka [55])

TLBO (Tog˘an [52])

MMDGA (Safari et al. [53])

EHS (This study)

1 2 3 4 5 6 7 8 9

W14x233 W14x176 W14x159 W14x99 W12x79 W33x118 W30x90 W27x84 W24x55

W14x233 W14x176 W14x145 W14x99 W12x65 W30x108 W30x90 W27x84 W21x44

W14x211 W14x176 W14x145 W14x90 W14x61 W33x118 W30x90 W24x76 W18x46

W14x233 W14x176 W14x145 W14x90 W12x65 W33x118 W30x90 W24x76 W14x30

W33x141 W14x159 W14x132 W14x99 W14x99 W30x116 W21x68 W14x61 W40x183

W14x233 W14x176 W14x145 W14x99 W12x65 W30x108 W30x90 W27x84 W21x44

W12x230 W14x159 W14x120 W14x90 W12x58 W33x118 W30x108 W24x76 W16x40

W14x159 W14x730 W14x61 W12x87 W14x283 W24x68 W14x99 W21x111 W33x201

Weight (lb) Number of analyses

65,136 3000

62,610 8300

61,864 3690

61,820 –

64,948 7500

61,813 –

61,345 1800

59,514 1412

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85

Table 6 Design groups for the three-bay, twenty four-storey frame problem. Group

Members

1

5, 7, 12, 14, 19, 21, 26, 28, 33, 35, 40, 42, 47, 49, 54, 56, 61, 63, 68, 70, 75, 77, 82, 84, 89, 91, 96, 98, 103, 105, 110, 112, 117, 119, 124, 126, 131, 133, 138, 140, 145, 147, 152, 154, 159, 161 166, 168 6, 13, 20, 27, 34, 41, 48, 55, 62, 69, 76, 83, 90, 97, 104, 111, 118, 125, 132, 139, 146, 153, 160 167 1, 4, 8, 11, 15, 18 22, 25, 29, 32, 36, 39 43, 46, 50, 53, 57, 60 64, 67, 71, 74, 78, 81 85, 88, 92, 95, 99, 102 106, 109, 113, 116, 120, 123 127, 130, 134, 137, 141, 144 148, 151, 155, 158, 162, 165 2, 3, 9, 10, 16, 17 23, 24, 30, 31, 37, 38 44, 45, 51, 52, 58, 59 65, 66, 72, 73, 79, 80 86, 87, 93, 94, 100, 101 107, 108, 114, 115, 121, 122 128, 129, 135, 136, 142, 143 149, 150, 156, 157, 163, 164

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

4.3. Three-bay, twenty four-storey planar frame

Fig. 10. Geometry and loading details of the three-bay, twenty four-storey frame.

in Fig. 8. Since convergence histories deviated at most by about 10%, the proposed EHS algorithm can be considered robust enough. Fig. 9 compares the EHS convergence history obtained for the best design with its counterpart shown in [26]: the proposed algorithm is clearly superior. In summary, EHS was deﬁnitely the most efﬁcient optimization algorithm also in test problem 2.

The third benchmark problem is the weight optimization of the three-bay twenty four-storey planar steel frame shown in Fig. 10. The structure includes 168 members and is subject to a single load case. This frame was originally designed by Davison and Adams [56]. It was also designed by Camp et al. [51] using ACO, by Degertekin [26] using HS, by Kaveh and Talatahari [54] using improved ACO (IACO), and again by Kaveh and Talatahari [57] using the imperialist competitive algorithm (ICA). The loads considered are W = 5761.85 lb, w1 = 300 lb/ft, w2 = 436 lb/ft, w3 = 474 lb/ft and w4 = 408 lb/ft (1 lb = 4.448 N). This frame was optimized according to the AISC–LRFD speciﬁcation [58] including the following displacement constraint: interstorey drift < storey height/300. The Young modulus E was set as 205 GPa (29,732 ksi) and the yield stress Fy was set as 230.3 MPa (33.4 ksi). The fabrication conditions required grouping the members as shown in Fig. 10, and resulted in 16 column sections and 4 beam sections for a total of 20 design variables. Each of the 4 beam element groups could be chosen from all the W-shapes listed in AISC standard list, while the 16 column element groups were limited to W14 sections. All beams and columns were considered unbraced along their lengths. Frame elements can be divided in the twenty groups listed in Table 6. Due to the relatively larger size of this frame structure, the harmony memory size was set equal to 100. The other internal parameters are listed in Table 1. One hundred optimization runs were carried out for this example. The best design weighed 194,400 lb with a standard deviation of 1981 lb. The best design is quoted in Table 7 and compared with literature [26,51–54]. It can be seen that EHS was the most efﬁcient optimization algorithm improving design by 11.8% with respect to ACO, 9.5% with respect to HS, 10.6% with respect to IACO, 8.6% with respect to ICA, 4.2% with respect to TLBO and 3.7% with respect to MMGA. In terms of the number of structural analyses required in the optimization process, EHS was between 2.8 and 12.3 times faster than the other meta-heuristic algorithms considered in this study as comparison basis. Convergence curves relative to the best design and the average design are compared in Fig. 11. In spite of a 20% difference between

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Table 7 Optimization results obtained for the three-bay, twenty four-storey frame problem. Group No.

AISC W-shapes ACO (Camp et al. [51])

HS (Degertekin [26])

IACO (Kaveh et al. [54])

ICA (Kaveh&Tala. [57])

TLBO (Tog˘an [52])

MMDGA (Safari et al. [53])

EHS (This study)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

W30x90 W8x18 W24x55 W8x21 W14x145 W14x132 W14x132 W14x132 W14x68 W14x53 W14x43 W14x43 W14x145 W14x145 W14x120 W14x90 W14x90 W14x61 W14x30 W14x26

W30x90 W10x22 W18x40 W12x16 W14x176 W14x176 W14x132 W14x109 W14x82 W14x74 W14x34 W14x22 W14x145 W14x132 W14x109 W14x82 W14x61 W14x48 W14x30 W14x22

W30x99 W16x26 W18x35 W14x22 W14x145 W14x132 W14x120 W14x109 W14x48 W14x48 W14x34 W14x30 W14x159 W14x120 W14x109 W14x99 W14x82 W14x53 W14x38 W14x26

– – – – – – – – – – – – – – – – – – – –

W30x90 W8x18 W24x62 W6x9 W14x132 W14x120 W14x99 W14x82 W14x74 W14x53 W14x34 W14x22 W14x109 W14x99 W14x99 W14x90 W14x68 W14x53 W14x34 W14x22

W30x90 W8x15 W24x55 W10x15 W14x159 W14x132 W14x90 W14x90 W14x61 W14x48 W14x48 W14x22 W14x109 W14x99 W14x99 W14x74 W14x68 W14x53 W14x26 W14x22

W10x19 W12x190 W6x8.5 W24x370 W14x132 W14x30 W14x99 W14x53 W14x74 W14x26 W14x68 W14x193 W14x145 W14x26 W14x26 W14x43 W14x26 W14x120 W14x426 W14x68

Weight (lb) Number of analyses

220,465 15,500

214,860 14,651

217,464 3500

212,736 7500

203,008 12,000

201,907 4750

194,400 1259

Fig. 11. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the three-bay, twenty four-storey frame problem.

Fig. 12. Comparison of convergence curves of different HS algorithms for the threebay, twenty four-storey frame problem.

convergence histories, the present algorithm can still be considered robust enough. Similar to the other test cases, the present algorithm converged to the optimum design much more quickly than the HS algorithm implemented in [26] (see Fig. 12). In summary, EHS was the most efﬁcient meta-heuristic algorithm also in test problem 3.

including 297 commercial sections, while beam members can be selected from a discrete set of 171 W-shape economical sections. As indicated in [36], the frame is subject to two independent loading conditions of combined gravity (dead, live, and snow loads) and lateral loads (wind loads) evaluated according to ASCE 7-05 (ASCE 2005) [59] and based on the following design values: a design dead load of 60.13 lb/ft (22.88 kN/m2), a design live load of 50 lb/ft (22.39 kN/m2), a ground snow load of 25 lb/ft (21.20 kN/m2) and a basic wind speed of 121 mph (54.09 m/s). Lateral wind loads acting at each ﬂoor level on windward and leeward faces of the frame are given in Table 8. In the ﬁrst load case, gravity loads are applied together with wind loads acting along x axis; in the second load case, they are applied with wind loads acting along y axis. The combined stress, stability, displacement, and geometric constraints are imposed according to the indications of ASD–AISC [60]. Because of the large size of this space frame, harmony memory size was set equal to 100. The other EHS internal parameters are listed in Table 1. One hundred independent optimization runs were carried out for this example. The best design weighed 395,708 lb

4.4. Spatial 744 member steel frame The last test problem considered in this study is the weight minimization of the unbraced spatial steel frame shown in Fig. 13, including 744 elements connected by 315 joints. Considering structural symmetry and practical fabrication requirements, members can be divided in 16 groups. Fig. 13(a)–(d) show side, plan and 3D views of the structure and clarify member grouping. This space frame was optimized by Hasançebi et al. [36] using Adaptive Harmony Search (AHS) which was then compared with standard HS, simple GA, PSO, ACO and Tabu Search. Cross sections of columns can be selected from the complete W-shape proﬁle list

M.R. Maheri, M.M. Narimani / Computers and Structures 136 (2014) 78–89

87

Fig. 13. Schematic of the spatial 744 member steel frame: (a) side view (x–z plane); (b) side view (y–z plane); (c) 3D view; (d) plan view [36].

Table 8 Wind forces acting on the spatial 744 member steel frame. Floor number

1 2 3 4 5 6 7 8

Windward

Leeward

lb/ft

kN/m

lb/ft

kN/m

140.64 171.44 192.49 208.98 222.74 234.65 245.22 127.38

0.63 0.76 0.86 0.93 0.99 1.04 1.09 0.57

159.22 159.22 159.22 159.22 159.22 159.22 159.22 79.61

0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.36

with a standard deviation of 2103.7 lb. The best design is listed in Table 9 and compared with the other meta-heuristic solutions obtained by Hasançebi et al. [36]. It can be seen that the proposed EHS algorithm found exactly the best solution indicated in [36]

for adaptive HS. However, the present algorithm was considerably faster than AHS as it required only 4743 structural analyses vs. the 50,000 analyses declared in [36] for both standard and adaptive HS algorithms (see convergence curves shown in Fig. 14). Convergence curves relative to the best design and the average design are compared in Fig. 15. Since convergence histories deviated at most by about 15%, the proposed EHS algorithm can again be considered robust enough. The HS algorithm internal parameters inﬂuence the optimization results. A sensitivity study was carried out to investigate the effect of HMCR and PAR parameters on the results of the spatial steel frame problem. Table 10 analyzes the sensitivity of the proposed algorithm to internal parameters HMCR and PAR. It can be seen that the best performance was achieved for HMCR = 0.8 and PAR ranging from 0.38 to 0.45, respectively. Remarkably, the EHS algorithm successfully converged to a feasible design in all of the independent optimization runs carried out for each test problem.

88

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Table 9 Optimum results obtained for the spatial 744 member steel frame problem. Group No.

Simple GA [36]

PSO [36]

ACO [36]

Tabu search [36]

HS [36]

Adaptive HS [36]

EHS (this study)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Weight (lb) Number of analyses

W24X68 W14X120 W8X21 W21X44 W12X65 W12X79 W14X26 W16X26 W8X35 W12X72 W14X26 W14X22 W8X35 W8X35 W8X21 W18X35 424076.52 –

W14X99 W14X99 W8X18 W21X44 W14X48 W14X90 W16X31 W14X26 W8X31 W14X90 W8X18 W16X26 W8X31 W8X31 W8X18 W14X22 412200.98 –

W12X72 W14X109 W14X22 W18X40 W10X49 W12X87 W12X26 W16X26 W12X35 W12X72 W14X22 W16X26 W10X39 W8X35 W8X21 W10X22 405441.65 –

W8X35 W14X145 W16X26 W18X35 W8X24 W14X90 W14X22 W18X35 W10X54 W12X53 W6X20 W16X26 W8X24 W8X40 W8X18 W14X26 401647.84 –

W12X79 W14X132 W14X22 W16X36 W12X72 W12X87 W14X22 W16X26 W16X40 W12X65 W14X22 W16X26 W16X36 W8X35 W10X22 W14X22 420177.57 50,000

W12X87 W14X109 W10X22 W18X35 W12X65 W12X79 W10X22 W16X26 W8X31 W12X65 W14X22 W14X22 W6X20 W12X58 W8X18 W14X22 395708.03 50,000

W12X87 W14X109 W10X22 W18X35 W12X65 W12X79 W10X22 W16X26 W8X31 W12X65 W14X22 W14X22 W6X20 W12X58 W8X18 W14X22 395708.03 4743

Table 10 Effect of internal parameters, HMCR and PAR on optimization results of the spatial 744 member steel frame. HMCR value (PAR = 0.4) Weight (lb) PAR value (HMCR = 0.8) Weight (lb)

0.70

0.75

0.80

0.85

0.90

433,612 0.30

433,612 0.35

395,708 0.40

501,311 0.45

559,543 0.50

433,612

433,612

395,708

395,708

501,311

Table 11 Number of structural analyses and corresponding standard deviation and number of successful optimization runs for the four test problems. Test problem

Fig. 14. Comparison of convergence curves of different HS algorithms for the spatial 744 member frame problem.

Two-bay, threestorey frame One-bay, tenstorey frame Three-bay, twenty fourstorey frame Spatial frame

No. of structural analyses

No. of successful optimization runs

Standard deviation on the number of analyses

220

30

5.06

1412

50

37.9

1259

100

35.02

4743

100

132.63

are listed in Table 11. It should also be noted that although the proposed EHS algorithm performed very well in all test problems, its superiority is more evident in the larger scale problems. Classical HS has been indicated by a number of investigators to perform well in small scale problems but to become less efﬁcient in larger problems. The results of the present study demonstrate that the proposed modiﬁcation of the HS updating phase is very effective in steering the search process towards the global optimum.

Fig. 15. Comparison of the EHS convergence curves relative to the best design and average optimization runs for the spatial 744-member frame problem.

A general point regarding the performance of the proposed EHS algorithms is that although some optimized designs slightly violated some optimization constraints, all the independent optimization runs carried out for each test problem reached feasible designs. The number of structural analyses, number of successful optimization runs and the standard deviation on the number of structural analyses for the four test problems discussed above

5. Conclusions This paper described an enhanced harmony search (EHS) algorithm where the harmony memory updating phase was enhanced by including in [HM] also those trial designs that although worse than the worst design currently included in the population are far enough from local optima. This allowed the global search capability of HS to be signiﬁcantly improved. The new algorithm was tested in four classical optimization problems of side sway frames and compared with other state-of-the-art meta-heuristic

M.R. Maheri, M.M. Narimani / Computers and Structures 136 (2014) 78–89

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