Grey Wolf Optimizer (GWO) Algorithm for Minimum Weight Planer Frame Design Subjected to AISC-LRFD Vishwesh Bhensdadia and Ghanshyam Tejani
Abstract In this paper, an optimum planer frame design is achieved using the Grey Wolf Optimizer (GWO) algorithm. The GWO algorithm is a nature involved meta-heuristic which is correlated with grey wolves’ activities in social hierarchy. The objective of the GWO algorithm is to produce minimum weight planer frame considering the material strength requirements specified by American Institute for Steel Construction—Load and Resistance Factor Design (AISC-LRFD). The frame design is produced by choosing the W-shaped cross sections from AISC-LRFD steel sections for a beam and column members. A benchmark problem is investigated in the present work to monitor the success rate in a way of best solution and effectiveness of the GWO algorithm. The result of the GWO algorithm is compared with other meta-heuristics, namely GA, ACO, TLBO and EHS. The results show that the GWO algorithm gives better design solutions compared to other meta-heuristics.
Keywords Structural optimization Meta-heuristics (GWO) algorithm Planer frame design AISC-LRFD
Grey wolf optimizer
1 Introduction In structural engineering problems, the cost of structural material includes more expenses to design a structure compared to other project expenses. Hence to reduce the structural project expenses, the structure raw materials cost parameter is kept in mind during the design stage to take some cost advantages [13]. To fulfill these criteria, it is preferable to reduce the cost of structure by reducing the size (weight) of structural beam and column members up to minimum level with considering certain design constraints [12]. To achieve this goal, choosing an effective V. Bhensdadia G. Tejani (&) School of Engineering, RK University, Rajkot, Gujarat, India e-mail:
[email protected] © Springer Science+Business Media Singapore 2016 S.C. Satapathy et al. (eds.), Proceedings of International Conference on ICT for Sustainable Development, Advances in Intelligent Systems and Computing 409, DOI 10.1007/978-981-10-0135-2_13
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optimization technique is an important task in structural engineering problems. Optimum structure design is achieved by applying efforts towards size reduction during the design stage considering their strength and stiffness parameters during optimization process. Minimum weight of the frame structure was investigated by many researchers in view of considering the requirement of strength and displacement constraints according to AISC-LRFD [1] specifications. Many meta-heuristics such as; upper bound strategy (UBS) [2], ant colony optimization (ACO) [3], Big Bang-Big Crunch (BB-BC) [5, 10, 11], evolution strategy integrated parallel algorithm (ESIPA) [6], bat inspired algorithm (BIA) [8], adaptive harmony search (AHS) [9, 22], charged system search (CSS) [14], enhanced harmony search (EHS) [15], Design Driven Harmony Search (DDHS) [17], Genetic Algorithm (GA) [18, 20, 21] and Teaching-Learning-Based Optimization (TLBO) [23] are used in this field. Hall et al. (1989) [4] presented the combined first-order and second-order design procedure for minimum weight frame design. Hasancebi et al. (2010) [7] compared the seven different meta-heuristics for optimum design of frame structure. This paper presents an optimum design of plane frame structure employing the GWO algorithm, which represents the skills and ability regarding leadership and hunting process of gray wolves in social atmosphere, proposed by Mirjalili et al. (2014) [16]. The rest of paper is formulated as follows: Sect. 2 formulates the optimum frame design problem, Sect. 3 describes brief of the GWO algorithm, Sect. 4 presents planer frame design problem and Sect. 5 concludes the work.
2 Formulation of the Optimum Design of Steel Frame The objective of the design problem is to minimize the weight (Eq. 2) of the frame structure subjected to the strength constraints (Eqs. 3 and 4) specified by AISC-LRFD [1] by selecting cross-sectional area (W-Shapes) for beam and column members as a design variables as shown in Eq. (1). The design variables, i.e. cross sections, for beam members and column members are considered according to AISC-LRFD [1]. This consideration turns into the optimum structure design problem, which is given as per follows equation [23]: Find; Z ¼ ½A1 ; A2 ; A3 ; . . .; And
ð1Þ
To minimize the weight ‘W’ of the frame structure which is expressed as: WðZÞ ¼
nd X i¼1
Ai
mt X j¼1
qi Li
ð2Þ
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Subjected to, Ckr 0, where k = 1,…,na 1 Ai mk; where i ¼ 1; . . .; nd where, ‘Z’ represents the design variables; ‘nd’ is the total number of design groups in the frame structure; ‘mt’ is the total number of members in group ‘i’ of frame structure; ‘ρj’ and ‘Lj’ are mass density and length of member ‘j’, respectively. ‘Ai’ is cross-sectional area of member group ‘i’ of frame structure. The inequalities parameter, i.e. ≤ 0 represent the strength constraints specified by the AISC-LRFD [1] specification. Since ‘Ai’ is the W-shaped cross-sectional area which is chosen from standard structure design manual, i.e. AISC-LRFD [1]. ‘na’ represents the number of beam and columns in frame structure; ‘mk’ shows the total number of W-shaped cross-sectional area which is considered for structure design in group ‘i’. The strength constraints, Ckr 0, for frame members subjected to axial force and bending as per AISC-LRFD [1] specification are given as follows: Ckr Ckr
Pu 8 Mux Muy ¼ þ þ 1 ; pn 9 ;b Mnx ;b Mny
Pu Mux Muy ¼ þ þ 1 2 ; pn ;b Mnx ;b Mny
; if
Pu 0:2 ; pn
; if
Pu 0:2 ; pn
ð3Þ ð4Þ
where, ‘Pu ’ and ‘Pn ’ are the required and nominal axial strength (compression or tension), respectively; ‘Mux’ and ‘Muy’ are the required flexural strengths about the major and the minor axes, respectively; ‘Mnx’ and ‘Mny’ are the nominal flexural strength about the major and the minor axis, respectively (for 2-D frames, ‘Mny’ = 0); ‘ ;’ is the resistance factor shown as ‘ ;c ’ for compression members (equal to 0.85) and ‘ ;t ’ for tension members (equal to 0.90), respectively; ‘ ;b ’ is the flexural resistance factor, with a value of 0.90 [1]. The penalty function used in this study is known as the Kaveh–Zolghadr technique, which is expressed as follows [19]: f ðXÞ ¼ ð1 þ e1 tÞe2
ð5Þ
where, ‘f’ represents the value of penalized function; and are taken as 2. t¼
na X i¼1
crki si , where si ¼ 1 r cki;all
ð6Þ
where, ‘na’ and represents the number of beam and columns and strength constraints violation, respectively. At a design solution ‘x’, if the ith constraint is not violated, then the value is zero; otherwise, is as per Eq. (6) [19].
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3 Grey Wolf Optimizer (GWO) Algorithm 3.1
Inspiration
Grey wolves are in the top level of the food chain in social atmosphere. Grey wolves organize the hunt activity in a pack of average 5–12 members. The social hierarchy of grey wolf consists of four levels from top to bottom, such as Alpha (α), Beta (β), Delta (δ) and Omega (ω), respectively [16]. Alpha is responsible for taking decisions about hunting, time to wake, sleeping place, to command a pack and so on. Beta provides help to alpha during the stage of decision and other activities regarding hunting. It can be consider as the best candidate to replace the alpha in case of it die. The duty of delta is to submit the activity and work regarding hunting possessed by him to alpha and beta but they dominate the omega. Omega is always responsible to submit activity and work regarding hunting to alpha, beta and delta. The grey wolf hunting activities have following main three stages [16]: (i) To track, chase and approach the prey, (ii) To pursue, encircle and harass the prey until it becomes stable, (iii) Attack in direction to the prey.
3.2
GWO Mathematical Model
The GWO mathematical models are categorized into the social hierarchy, encircling, hunting, attacking and search for prey activities, which are described as follows: [16]: To represent the social hierarchy of grey wolves in mathematical form during designing the GWO algorithm, the Mirjalili et al. (2014) [16] considered the best solution represented by alpha candidate. The beta and delta are considered as second and third best candidates to search the prey respectively. The remaining candidates are considered as an omega. The grey wolves encircle activity towards prey during the hunting process is proposed by Mirjalili et al. (2014) [16] in the form of equation as follows: D ¼ CXp ðtÞ XðtÞ
ð7Þ
Xðt þ 1Þ ¼ Xp ðtÞ AD
ð8Þ
where, ‘t’ shows the current iteration, ‘Xp’ and ‘X’ indicate the position vector of the prey and the grey wolf, respectively and ‘A’ and ‘C’ represent the coefficient vectors. The vectors ‘A’ and ‘C’ are evaluated by using following equations [16]: A ¼ 2ar1 a
ð9Þ
Grey Wolf Optimizer (GWO) Algorithm for Minimum Weight Planer …
C ¼ 2r2
147
ð10Þ
where, components of ‘a’ are linearly decreased from value 2 to 0 over the course of iterations, and ‘r1’ and ‘r2’ are random vectors in [0, 1] [16]. The hunting activity of grey wolves requires the skills and ability to locate the prey and encircle them, which is guided by the alpha. In search space initially, there is no idea regarding the optimum prey location. Now in mathematical form, we consider that the alpha is the best search agent, and beta and delta have better idea regarding the optimum prey location. So, we consider the first three search agents as best solutions (alpha, beta and delta) and remaining search agents must update their positions with respect to alpha, beta and delta. The following equations are proposed by Mirjalili et al. (2014) [16] in this regard as follows: Da ¼ C1 Xa X; Db ¼ C2 Xb X; Dd ¼ C3 Xd X
ð11Þ
X1 ¼ Xa A1 Da ; X2 ¼ Xb A2 Db ; X3 ¼ Xd A3 Dd
ð12Þ
Xðt þ 1Þ ¼
X1 þ X2 þ X3 3
ð13Þ
To mathematically model the grey wolves attacking activity towards the prey, the vector ‘A’ (fluctuation range) decreased in range of [−a, a], with decreasing the vector ‘a’ from value 2 to 0. If condition |A| < 1 occurs then grey wolves attack towards the prey [16]. Grey wolves mostly search the prey by diverging their positions with respect to the alpha, beta and delta in a pack. To convert this divergence concept in mathematical form, vector ‘A’ utilize with random values in a range of 1 to −1 [16]. This condition permits the GWO algorithm to search the optimum prey location globally.
4 Design Problem In present study, optimization of two-bay three-story planar steel frame is done by using the GWO algorithm. This frame consists of 15 members (6-Beams and 9-columns). Frame geometry, loading conditions and boundary conditions of two-bay three-story steel frame, are presented in Fig. 1. The frame problem was studied by the many researchers using various algorithms, namely ACO [3], EHS [15], GA [18] and TLBO [23]. The material properties are assumed as the modulus of elasticity (E) is 29000 ksi and yield stress (Fy) is 36 ksi. In frame design, 268W shaped sections for beam members and 18-W shaped sections (W10 sections only) for column members are considered as per AISC-LRFD [1, 23]. The out of plane effective length factor for column and beam members are specified as 1.0 and 0.167, respectively [4].
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Fig. 1 Geometry and loading conditions of two-bay three-story frame
In this study, to get best tuning of algorithm parameters, population size are assumed to be 40, 20 and 10, respectively and corresponding generations are assumed to be 40, 80 and 160, respectively. Hence, in each case maximum function evolutions (FE) consumed to be 1600 . In addition, statistical results are measured by 30 independent runs and best results are displayed. Table 1 compares the results of the GWO algorithm and the best designs (AISC W-Shapes and Weight) obtained in the literature. It can be seen that a design obtained by using the GWO algorithm, W21 × 57 for beams and W10 × 49 for columns, is the best design in terms of optimum weight without violating the stated constraints. The corresponding weight achieved for this design is of 16686.5702 lb. The best frame design weight for the GWO algorithm is approximately 11.20, 11.20, 6.20 and 7.30 % less than the best design weight for the GA [18], ACO [3], TLBO [23] and EHS [15], respectively. Table 2 represents the statistical results of the GWO algorithm for two-bay three-story frame design problem for different population and generation. Optimum weight of frame is of 16686.57 lb achieved in each set of algorithm parameter. It should be also noticed that best result is achieved 28 times out of 30 runs, and hence the GWO algorithm reported 93.333 % success rate to get best solution.
Table 1 Optimum results obtained for the two-bay three-story frame design problem AISC W-Shapes Pezeshk et al. (2000) [18]
Camp et al. (2005) [3]
Vedat (2012) [23]
Maheri and Narimani (2014) [15]
This study GWO
Method
GA
ACO
TLBO
EHS
Beam (10–15)
W24 × 62
W24 × 62
W24 × 62
W21 × 55
W21 × 57
Column (1–9)
W10 × 60
W10 × 60
W10 × 49
W10 × 68
W10 × 49
Weight (lb)
18792
18792
17789
18000
16686.5702
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Table 2 Statistical results of the GWO algorithm Population size
Generation
FE
Minimum (lb)
Maximum (lb)
Standard deviation
Success rate to get best solution (%)
40 20 10
40 80 160
1600 1600 1600
16686.57 16686.57 16686.57
17115.37 17115.37 21280.89
16686.57 16686.57 16686.57
90.00 93.333 93.333
29000
Best
27000
Average
Weight
25000 23000 21000 19000 17000 15000 1
31
61
91
121
151
Generations Fig. 2 Convergence history of two-bay three-story frame design using the GWO algorithm
The convergence of the two-bay three-story frame for the best weight and average weight of 30 independent runs is represented in Fig. 2. However, the GWO algorithm gives the presented result at a population size of 10, resulting in 1600 FE. This graph indicates that the optimum weight is reduced with increasing the generation, and best value of weight is achieved within 60 generations.
5 Conclusions The present work proposed the GWO algorithm to optimize two-bay three-story frame structure according to AISC-LRFD. The GWO algorithm is inspired by leadership hierarchy of grey wolves. The obtained results are compared with results of GA, ACO, TLBO and EHS. The achieved optimum weight of frame is of 16686.57 lb in each set of algorithm parameter with considering the material and performance constraints according to AISC-LRFD. It is stated from the results that the best design weight of frame obtained using the GWO algorithm is 11.20, 11.20, 6.20 and 7.30 % less than the best design weight obtained by GA, ACO, TLBO and EHS, respectively. The best result is achieved 28 times out of 30 runs, hence the GWO algorithm reported 93.333 % success rate to get best solution. The GWO algorithm represents better computational efforts and ability over the other state-of-the-art algorithms.
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