discrete optimization truss structures by multi ... · the discrete nonlinear truss optimization...
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European International Journal of Science and Technology ISSN: 2304-9693 www.eijst.org.uk
200
Discrete optimization truss structures by Multi-combination and
Bi-combination Methods
Burak KAYMAK*a
, Mehmet Tevfik BAYER*
*Dumlupinar University,
Engineering Faculty,
Department of Civil Engineering,
43100, Kutahya, TURKEY
aCorresponding Author
Abstract
Truss design problem is a discrete nonlinear optimization problem. This problem can be solved by the
proposed Multi-combination Method (MCM) which is a two phase optimization method. In the first phase of
the method the continuous optimum solution of the nonlinear truss design problem is solved by Sequential
Linear Programming (SLP) method. In the second phase of the method, in the vicinity of the continuous
optimum solution point, the discrete optimum solution is searched for by using the combinations of the
neighbouring profile areas. As the number of design variables is increased, then the number of combinations
increases exponentially. For these cases an approximate solution method which is called Bi-combination
Method (BCM) is proposed. Sample problems, which are taken from literature are solved and their solutions
are compared with MCM and BCM. For the sample problems it is observed that the discrete optimum
solutions can be calculated with sufficient accuracy byBCM.
Keywords: truss optimization, two phase optimization, discrete variables, combinatorial optimization.
1. Introduction
Truss structures are commonly used in practice as bridges, roof structures, pilons, industrial buildings and
parts of aircrafts and spacecrafts etc. Most of these truss structures are made from commercially available
profiles which means that member cross-sectional areas must be discrete variables. Therefore in truss design
problems there are discrete variables which are member cross-sectional areas and also there are continuous
variables which are member stresses and nodal displacements. In the definition of the truss design problems
there are also member forces which are defined by multiplying the member areas by the member stresses, that
makes the problem nonlinear. The geometry of the truss structure is given and it carries given load cases. For
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the truss structure there are constraints on the member cross-sectional areas, on the member stresses and also
on the nodal displacements. Under these conditions the least weight truss design is required. This truss design
problem is a discrete nonlinear optimization problem and it can be formulated as 0 - 1 nonlinear optimization
problem Kaymak (2011).
These optimization problems can be solved by either mathematical programming methods or by heuristic
methods.
There are various mathematical programming methods shuch as; Gomory's cutting plane method, penalty
function method, langrangian relaxation method which are given by Arora et al (1994) and branch and bound
method which is used by (Land and Doig, 1960; Dakin, 1965; Toakley, 1968; Cella and Logher, 1971). In
additon to these methods there are also two phase optimization methods. In the first phase of these methods a
nonlinear continuous optimization problem is solved by various methods and the continuous optimum
solution is calculated. Then in the second phase, in the vicinity of this continuous optimum solution point, the
discrete optimum solution is searched for. This search takes place in a relatively small solution domain which
makes these methods attractive (Olsen and Vanderplaats, 1989; Chai and Sun, 1997; Huang and Arora,
1997;Tong and Liu, 2001; Souza and Fonseca, 2008).
In this study a new mathematical programming method is presented. This new method is a two phase
optimization method which is called Multi-combination Method(MCM), Kaymak (2011). By reducing the
number of combinations of MCM, a new approximate method called Bi-combination Method(BCM) is also
developed Kaymak (2011). The details of MCM and BCM are given in section 3. For these methods
computer codes are developed which are used in solving the sample problems in section 5. Discussions and
conclusions are presented in section 6.
2.Formulation of the discrete nonlinear truss optimızation problem
The nonlinear truss optimization problem with discrete variables described in relations(1- 11) is a
variation of Baugh Jr. et al (1997) formulation which is given by Kaymak (2011):
��������� = ∑ ��� ����� (1)
Subject to:
∑ ������ = ������� (2)
��
��∑ ������ = ���
���� (3)
��� = �σ�� (4)
���� ≤ ��� ≤ ���
(5)
���� ≤ ��� ≤ ���
(6)
0 ≤ �� ≤ � ≤ �
(7)
"� � = "� ∑ #��#$#�� (8)
∑ ��# = 1$#�� (9)
��# ∈ {0,1} (10)
"� ∈ {0,1} (11)
� = 1, … , �, + = 1, … , �, , = 1, … , -
In these relations, l is the number of load cases, m denotes the number of members used in the truss
structure, n denotes the degree of freedom, and q denotes the number of available profile sections. ���is the
direction cosine related to the degree of freedom i and the internal force of bar j with any load case. ���is
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the node load at degree of freedom i under load case k. In equation Eq. 3, ���, is the stress of member j
under load case k. ��� is the axial force of member j in load case k, � is the cross-sectional area of
member j. ��� is the displacement belonging to freedom i under load case k. ���� and ���
are the lower bound
and upper bound for the displacement i under load case k, respectively. ���� and���
are the lower bound and
the upper bound for the stress of member j under load case k, respectively. While �� and �
are the lower
bound and the upper bound for the cross-sectional area of member j respectively. ��is the length of the
member j, � is the material density of truss member j. W is the total weight of the truss structure. �, ���,
���, "� and ��# are the variables of the truss optimization problem. "�is a 0-1 variable and is equal to 1 when
member j is equal to any available profile area value, otherwise it is equal to 0. ��# is also a 0-1 variable and
according to Eq. 9, only one ��# variable can be equal to 1, and the other ��# variables should be 0 for the
related member j. # is the cross-sectional area value of the profile s in the profile table. If the area of
member j is equal to profile area value # then the related ��# and "� variables in relations Eq. 8 and Eq. 9
should be equal to 1.
3.Combination Methods
3.1.Multi-combination Methods (MCM)
3.1.1. First Phase Calculations for MCM
The discrete nonlinear truss optimization problem presented in relations(1 - 11), is a nonlinear problem
due to Eq. 4. In the first phase ofMCM the variables ��# and "� are fixed to 0 which converts the problem to a
nonlinear continuous truss optimization problem. Thus this problem can be solved by using the sequential
linear programming(SLP) method Kaymak(2011).
At the end of the first phase of MCM the continuous optimum solution of the nonlinear truss optimization
problem is calculated. It is obvious that the discrete optimum weight of the truss structure can not be less than
this continuous optimum weight.
3.1.2. Second Phase calculations for MCM
3.1.2.1.Combination problem
At the end of the first phase of the MCM the continuous optimum solution of the nonlinear truss
optimizaiton problem is calculated. Then the second phase of the method starts where the discrete optimum
solution of the problem is calculated. In the second phase of the method the member area values calculated at
the continuous optimum solution, are taken as the reference areas.
If �. is the number of design variables of the truss optimization problem and if q is the total number of
neighbouring profile areas then it can be stated that the number of combination problems(N) which is present
in the related MCM is defined as/ = 0�1. From this definition it is clear that the total number of
neighbouring profile areas (q) must be kept as small as possible. Otherwise the number of combinations
increases to such an extended that the method may become impractical. For each combination one truss
structure is formed and one structural analysis is performed. If this is a feasible structure then its weight is
recorded. Similar calculations are performed for every combination of the truss structure. When all the
combinations are completed then the recorded feasible weights are compared and the least one is determined.
The truss structure which has the least weight is the true/best discrete optimumsolution of the problem using
the related MCM.
In phase two of the method in order to calculate the true/best discrete optimum solution of the truss
optimization problems, MCM with q = 4 is the standard solution method to start with. But sometimes q = 5 or
6 ..etc. must be used in order to calculate the true discrete optimum solution.
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In order to reduce the number of structural analysis calculations in the second phase of the method, there
are some strategies which are used in this study. For a given combination all design variables are assigned a
profile area value which means that the weight of the truss structure for this combination is known. If this
weight is in between a given lower and upper bounds, then, the structural analysis is performed otherwise it is
not performed. By this strategy the number of structural analysis calculations is reduced in the second phase.
In order to apply this strategy the lower and the upper bounds of the discrete solution of the truss optimization
problem needs to be defined.
3.1.2.2.Lower and upper bounds of the discrete solution
As explained above, at the end of the first phase of the method the continuous optimum solution of the
truss structure is calculated. It is known that the discrete optimum of the truss optimization problem can not
be less than this optimum weight. Forthis reason the continuous optimum solution calculated at the end of
the first phase of the method is considered to be the lower bound of the discrete solutions. For a given
combination if the weight of the truss structure is greater than or equal to the lower bound value then the
calculations continue. If it is less than the lower bound value then the calculations terminate and the next
combination calculations start.
For a combination if the calculated truss weight is greater than or equal to the lower bound then, it must be
also tested if it is less than the upper bound. This upper bound is calculated by Rounding-off the Senior
Area(RSA) method(Kaymak, 2011). RSA gives a discrete solution which is based on nonlinear optimization
of the truss problems by SLP method.
For each combination if the weight of the truss structure is equal or greater than the upper bound then the
calculations of the combination terminate and the next combination calculations start. If it is less than the
upper bound then this also means that the weight of the truss structure is greater than or equal to the lower
bound, therefore structural analysis calculations are performed. In order to calculate the total structural
analysis needed for the second phase of the method, these structural analyses are counted. If the solution is
feasible then the weight of this truss structure becomes the new upper bound of the problem for the rest of the
calculations and the next combination calculations start. After completing all the combination calculations the
current upper bound value becomes the true discrete/best discrete optimum solution of the truss problem for
the related MCM.
The main steps of the computer code developed for MCM are given in section 4.
3.2. Bi-combination Methods (BCM)
If the number of combinations(N) for MCM is too many, then for practical purposes, some combinations
may be skipped. This new approximate method is generated from MCM where q=4 and it is called Bi-
combination method(BCM).
Bi-combination Method(BCM) is introduced in the following: At the end of the first phase of the method
the continuous optimum is calculated for the truss optimization problem and at the optimum, the design
variables which are called reference areas are placed in set 2. The two neighbouring profile areas which are
smaller and larger than the reference areas are placed in the related sets given in Eq. 12.
2�34 < 2�3� < 2 ≤ 2�6� < 2�64 (12) In this method the combinations are always calculated from the given sets by forming the couples given in
Eq. 13. Because of the coupling of the sets, the method is called Bi-combination Method(BCM).
8 2�34, 2�6�9, 8 2�34, 2�649, 8 2�3�, 2�6�9, 8 2�3�, 2�649, 8 2�6�, 2�649 (13)
For BCM the number of combinations is equal to/ = 5 × 2�1. These combinations are also included in
the combinations of related MCM, where / = 0�1. For practical purposes the reduction in the number of
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combinations is a necessary operation if 4�1becomes an inconciveable number. In general, for the MCM if
the number of combinations(N) becomes an inconcievable number then BCM is used where the number of
combinations are reduced. When BCM is used then an acceptable discrete optimum solution is calculated.
The total number of structural analysis needed for the first phase as well as for the second phase of the
method are added that gives the total number of structural analysis needed to calculate the best
discrete/acceptable discrete optimum solution by using MCM/BCM.The main steps of the computer codes
developed for BCM are given the following section.
4. Main steps of the developed computer code for MCM and BCM
In this study the sample problems are solved by the developed computer code based on the MCM with q =
4 values and the true discrete optimum solutions are calculated for the related MCM. For the sample
problems BCM with reduced number of combinations are also used to calculate acceptable discrete optimum
solutions. The discrete solutions and the number of structural analysis needed for calculating these solutions
are compared with those found in literature which are solved by various heuristic methods.
The main steps of the developed computer codes which are based on BCM and MCM are given below:
A. First Phase
1) Assign zjs=0, Tj=0 calculate continous optimum of nonlinear problem Eq.1-11 by SLP
B. Second Phase
1) Determine the lower bound(Wl=WCont.Sol.) and upper bound(W
u). Determine the reference design
variable values. Choose q=4. Calculate number of combination for BCM(NBCM).
2) t=0
3) t=t+1
4) Calculate the weight of combination(Wt)
5) If W? ≤ W@ ≤ WA go to step 6 else step 3
6) Perform structural analysis. If structure is feasible go to step 7 else step 3
7) Wu = Wt
8) If t < NBCM is true go to step 3 else step 9
9) Acceptable discrete optimum for BCM = Wu. Take this W
u as the upper bound for MCM. Choose q ≥
4.
10) Calculate number of combination for MCM(NMCM)
11) If NMCM inconciveable go to step 20 else step 12
12) t=0
13) t=t+1
14) Calculate the weight of combination(Wt)
15) If W? ≤ W@ ≤ WA go to step 16 else step 13
16) Perform structural analysis. If structure is feasible go to step 17 else step 13
17) Wu = Wt
18) If t < NMCM is true go to step 13 else step 19
19) True/Best discrete optimum for MCM = Wu
20) True/best/acceptable discrete optimum calculated. Print results.
5. Sample Problems
5.1. 10-bar planar truss
The 10-bar plane truss problem is solved using different optimization techniques by researchers(Camp and
Bichon, 2004; Capriles et al, 2007) The geometry of the truss structure shown in Fig. 1. The truss structure is
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loaded by 100kips at joints 2 and 3 along negative Y direction. All nodes in verticaldirection are subjected to
displacement limits of ±5.08 cm. The members are subjected to stress limits of ±172.37MPa. The discrete
variables are selected from the set D = {10.452, 11.613, 12.839, 13.742, 15.355, 16.903, 16.968, 18.581,
18.903, 19.935, 20.194, 21.806, 22.387, 22.903, 23.419, 24.774, 24.968, 25.032, 26.968, 27.226, 28.968,
29.613, 30.968, 32.064, 33.032, 37.032, 46.581, 51.419, 74.193, 87.097, 89.677, 91.613, 100.000, 103.226,
109.032, 121.290, 128.387, 141.935, 147.742, 170.967, 193.548,216.129} cm2. Forthis sample problem there
are 10 design variables(nE = 10). The modulus of elasticity is E = 68947.59MPaand the material density
isρj=2767.99kg/m3.
The discrete optimum solution calculated by Camp and Bichon (2004) using Ant Colony
Optimization(ACO) is equal to 2492.80 kg. Capriles et al (2007) using Rank-based Ant System(RBAS)
calculates the same weight as seen in Table 1.
Using BCM an acceptable discrete optimum solution is found to be 2492.80 kg which is calculated after
149 structural analysis. For this case the number of combinations is equal to (N=5,120). Initial upper bound
value of the sample is 2512.28 kg, which is calculated by RSA.
Using MCM with q=4 the true discrete optimum solution is calculated to be 2492.80 kg. For this case the
number of combinations is equal to (N=1,048,576). When the developed computer code for MCM is used
then this true discrete optimum solution is calculated after 11,414 structural analysis. For this sample problem
BCM can be preferred because it gives the true discrete optimum solution with few structural analysis, which
can be seen from Table 1. The reference areas which are calculated in the First Phase of the method are also
given in Table 1 which is the continuous optimum solution of the sample problem.
5.2. 25-bar space truss
The 25-bar space truss with discrete variables is solved by many researchers(Wu and Chow, 1995;
Capriles et al, 2007; Camp and Bichon, 2004; Lee et al, 2005). The modulus of elasticity is E = 68947.59
MPa and the material density is ρ = 2767.99 kg/m3 . The members are subjected to stress limits of ±275.79
MPa and all nodes are subjected to displacement limits of ±0.889 cm. The profile set includes cross-
sectional areas ranging from 0.645 cm2 to 21.945 cm2 increasing by 0.645 cm2 . The geometry of the truss
structure is shown in Fig.2. The truss members are collected in 8 groups: group 1: 1, group 2: 2 to 5, group
3: 6 to 9, group 4: 10 and 11, group 5: 12 and 13, group 6: 14 to 17, group 7: 18 to 21 and group 8: 22 to 25.
to 52, group 12: 53 and 54, group 13: 55 to 58, group 14: 59 to 66, group 15: 67 to 70 and group 16: 71 and
72. Which means that forthis sample problem there are 8 design variables(�. = 8). The twenty five bar
space truss structure is subjected to a single load case where: Px = 4.448 kN, Py = −44.482 kN, Pz =
−44.482 kN at joint 1, Py = −44.482 kN, Pz = −44.482 kN at joint 2, Px = 2.224 kN at joint 3 and Px =
2.669 kN at joint 6.
Camp and Bichon (2004) used ACO and obtain this solution after 7700 structural analyses which is
the least number of structural analysis needed amongst the heuristic methods for this sample problem.
Where as Lee et al. (2005) obtain the same solution after 13523 structural analyses by using HS. Capriles et
al. (2007) obtain 220.21 kg after 7700 structural analyses by using rank-based ant system(RBAS). Wu and
Chow (1995) obtain 220.78 kg after 40000 structural analyses by using steady-state genetic
algorithm(SSGA)
Using BCM an acceptable discrete optimum solution is found to be 220.21 kg which is calculated
after 26 structural analysis. For this case the number of combinations is equal to (N=1,280).Initial upper
bound value of the sample is 220.45 kg, which is calculated by RSA.
Using MCM with q=4 the true discrete optimum is calculated to be 220.12 kg. For this case the
number of combinations is equal to (N=65,536). When the developed computer code for MCM is used then
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this true discrete optimum solution is calculated after 1,476 structural analysis. Therefore for practical
purposes approximate discrete optimum solution needs to be calculated for this sample problem and BCM
calculates an acceptable discrete optimum which is 0.04% heavier than the true discrete optimum.
For this sample problem BCM must be preferred because it calculates almost the true discrete
optimum solution with relatively few structural analysis, which can be seen from Table2. The reference
areas which are calculated in the First Phase of the method are also given in Table 2 which is the continuous
optimum solution of the sample problem.
5.3. 72-bar space truss
The 72-bar space truss shown in Fig.3 is solved by Wu and Chow(1995) using Steady-State Genetic
Algorithm(SSGA) and Lee et al (2005) using Harmony Search(HS) method. The members are subjected to
stress limits of ±172.37MPa and the joints 17, 18, 19 and 20 are subjected to displacement limits of
±0.635cm. The profile set includes 32 profile cross-sections ranging from 0.645cm2 to 20.645cm
2 increasing
by 0.645cm2.
The modulus of elasticity is E = 68947.59MPa and the material density is ρj=2767.99kg/m3. The truss
structure is under two load cases. In the first load case, the joints 17, 18, 19 and 20 are loaded by 22.241kN
in the negative Y direction and in the second load case the joint 17 is loaded by 22.241kN in the positive X
direction, 22.241kN in the negative Y direction and 22.241kN in the negative Z direction.
The truss members are collected in 16 groups: group 1: 1 to 4, group 2: 5 to 12, group 3: 13 to 16, group 4:
17 and 18, group 5: 19 to 22, group 6: 23 to 30, group 7: 31 to 34,group 8: 35 and 36, group 9: 37 to 40,
group 10: 41 to 48, group 11: 49 to 52, group 12: 53 and 54, group 13: 55 to 58, group 14: 59 to 66, group 15:
67 to 70 and group 16: 71 and 72. Which means that forthis sample problem there are 16 design variables
(�. = 16).
The discrete optimum solution calculated by Wu and Chow using SSGA is equal to 181.90 kg and byLee
et.al. using HS is equal to176.12 kg which are given in Table 3.
Using BCM an acceptable discrete optimum solution is found to be 175.67 kg which is calculated after
20,217 structural analysis. For this case the number of combinations is equal to (N=327,680). Initial upper
bound value of the sample is 177.21 kg, which is calculated by RSA.
Using MCM with q=4 the true discrete optimum is calculated to be 175.04 kg. For this case the number of
combinations is equal to (N=4,294,967,296), which is too many. When the developed computer code for
MCM is used then this true discrete optimum solution is calculated after 122,497,180 structural analysis,
which is also too many. Therefore for practical purposes approximate discrete optimum solution needs to be
calculated forthis sample problem and BCM calculates an acceptable discrete optimum which is 0.36%
heavier than the true discrete optimum. For this sample problem BCM must be preferred because it calculates
almost the true discrete optimum solution with relatively few structural analysis, which can be seen from
Table3. The reference areas which are calculated in the First Phase of the method are also given in Table 3
which is the continuous optimum solution of the sample problem.
6. Discussions and Conclusions
In this study multi-combination method(MCM) is introduced for solving discrete truss optimization
problems. It is a two phase method where in the first phase a nonlinear continuous truss optimization problem
is solved by SLP method. In the second phase of the method, in the vicinity of the calculated continuous
optimum solution point, the neighbouring profile area values are used in calculating the combinations for
searching the true/best discrete optimum solution. Forthis appropriate (0 ≥ 4) must be chosen.
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In this study three sample problems which are taken from literature, are solved by MCM and the true
discrete optimum solutions are calculated. For sample problems 1 and 2 the same true optimum solutions are
calculated but for sample problem 3 the true discrete optimum calculated by MCM is a better optimum then
the ones found in literature. For sample problem 3 too many combinations are tried, which is not practical.
Therefore for practical purposes for sample problem 3, BCM must be use to calculate an acceptable discrete
optimum solution. For sample problem 3, BCM calculates an acceptable discrete optimum solution which is
only 0.36% heavier than the true discrete optimum solution. In most engineering applications it can be easily
stated that this difference is negligable. As seen from Table 3 BCM gives an acceptable discrete optimum
solution which is better than the optimum solutions found in literature.
In this study it is concluded that the proposed Bi-combination Method(BCM), which is an approximate
method, can always be used in solving the discrete truss optimization problems. But multi-combination
method(MCM) can only be used when the number of combinations is concievable. On the other hand these
methods are very powerfull because they are quite suitable for parallel computing techniques.
For future work the developed MCM and BCM are considered to be used in solving the discrete truss
optimization problems with buckling constraints.
References
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variables: a review. Structural Optimization 8:69–85
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alternatives in discrete structural optimization. Engineering Optimization 28:1–31
Camp, C., Bichon, B. (2004) Design of space trusses using ant colony optimization. Journal of Structural
Engineering 130(No 5)
Capriles, P.V.S.Z., Fonseca, L.G., Barbosa, H.J.C., Lemonge, A.C.C. (2007) Rank-based ant colony
algortihms in discrete structural optimization. Communications in Numerical Methods in Engineering
23:553–575
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structures. Structural Optimization 13:250–257
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Souza, R.P., Fonseca. J.S.O. (2008) Optimum truss design under failure constraints combining continuous
and integer programming. International Conference on Engineering Optimizaiton
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Tong, W., Liu, G. (2001) An optimization procedure for truss structures with discrete design variables and
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Structures 56:979–991
Fig. 1. 10-bar planar truss
Fig. 2. 25-bar planar truss
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Fig. 3. 72-bar space truss
Table 1. Comparison of discrete optimal design solutions for 10 bar planar truss
Design
Variab.
(cm2)
This Study
ACO RBAS BCM MCM Reference
areas
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
147.742
91.613
10.452
10.452
216.129
10.452
147.742
51.419
10.452
141.935
147.742
91.613
10.452
10.452
216.129
10.452
147.742
51.419
10.452
141.935
147.742
91.613
10.452
10.452
216.129
10.452
147.742
51.419
10.452
141.935
147.742
91.613
10.452
10.452
216.129
10.452
147.742
51.419
10.452
141.935
150.322
98.516
10.452
10.452
208.000
10.452
146.387
53.613
10.452
139.290
W(kg) 2492.80 2492.80 2492.80 2492.80 2489.26
# of
Struc.
Analy.
10000 10000 149 11414 326
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Table 2. Comparison of discrete optimal design solutions for 25 bar planar truss
Design
Variab.
(cm2)
This Study
ACO RBAS HS SSGA BCM MCM Reference
areas
A1
A2
A3
A4
A5
A6
A7
A8
0.645
1.935
21.935
0.645
13.548
6.452
3.226
21.935
0.645
3.226
21.935
0.645
12.258
5.806
3.226
21.935
0.645
1.935
21.935
0.645
13.548
6.452
3.226
21.935
0.645
3.226
21.935
0.645
9.677
5.806
3.871
21.935
0.645
3.226
21.935
0.645
12.258
5.806
3.226
21.935
0.645
1.935
21.935
0.645
13.548
6.452
3.226
21.935
0.645
2.716
21.935
0.645
12.414
6.244
3.035
21.935
W(kg) 220.12 220.21 220.12 220.78 220.21 220.12 219.76
# of
Struc.
Analy.
7700 7700 13523 40000 26 1476 399
Table 3. Comparison of discrete optimal design solutions for 72 bar Space truss
Design
Variab.
(cm2)
This Study
HS SSGA BCM MCM Reference
areas
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
A12
A13
A14
A15
A16
12.258
3.226
0.645
0.645
9.032
3.871
0.645
0.645
3.871
3.226
0.645
0.645
1.290
3.226
2.581
3.871
9.677
4.516
0.645
0.645
8.387
3.226
1.290
0.645
3.226
3.226
0.645
1.290
1.290
3.226
3.226
4.516
12.258
3.226
0.645
0.645
8.387
3.226
0.645
0.645
3.871
3.226
0.645
0.645
1.290
3.871
3.226
3.226
12.258
3.226
0.645
0.645
9.032
3.226
0.645
0.645
3.226
3.226
0.645
0.645
1.290
3.871
2.581
3.871
12.129
3.310
0.645
0.645
8.142
3.329
0.645
0.645
3.348
3.335
0.645
0.645
1.006
3.542
2.587
3.671
W(kg) 176.12 181.90 175.67 175.04 172.35
# of
Struc.
Analy.
16044 60000 20217 122497180 406