Discrete optimization truss structures by Multi ... · The discrete nonlinear truss optimization problem presented in relations(1 - 11), is a nonlinear problem due to Eq. 4. In the

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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

European International Journal of Science and Technology Vol. 3 No. 7 September, 2014

201

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

Arora, J.S., Huang, M.W., Hsieh, C.C. (1994) Methods for optimization of nonlinear problems with discrete

variables: a review. Structural Optimization 8:69–85

Baugh, Jr J., Caldwell, S., Brill, Jr E. (1997) A mathematical programming approach for generating

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

Cella, A., Logher, R. (1971) Automated optimum design from discrete components. Journal of

Structural Engineering 97:175–189

Chai, S., Sun, H.C. (1997) A two-level delimitative and combinatorial algortihm for discrete optimization of

structures. Structural Optimization 13:250–257

Dakin, R.J. (1965) A tree search algorithm for mixed integer programming problems. Computer Journal

8:250–255

Huang, M.W., Arora, J.S. (1997) Optimal design with discrete variables: some numerical experiments.

Internatinal Journal for Numerical Methods in Engineering 40:165–188

Kaymak, B. (2011) Least weight design of truss structures with discrete variables. PhD thesis

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Econometrica 28:497–520

Lee, K.S., Geem, Z.W., Lee, S., Bae, K. (2005) The harmony search heuristic alhorithm for discrete

structural optimization. Engineering Optimization 37(No. 7):663–684

Olsen, G.R., Vanderplaats, G.N. (1989) Method of nonlinear optimization with discrete design variables.

American Institute of Aeronautics and Astronautics 27:1584–1589

Souza, R.P., Fonseca. J.S.O. (2008) Optimum truss design under failure constraints combining continuous

and integer programming. International Conference on Engineering Optimizaiton

Toakley, R. (1968) Optimum design using available sections. Journal of Structural Engineering 94:1219–

1241


Tong, W., Liu, G. (2001) An optimization procedure for truss structures with discrete design variables and

dynamic constraints. Computers and Structures 79:155–162

Wu, S., Chow, P. (1995) Steady-state genetic algorithms for discrete optimization of trusses. Computers and

Structures 56:979–991

Fig. 1. 10-bar planar truss

Fig. 2. 25-bar planar truss

User

Typewritten Text

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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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209


210

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

Documents

Discrete optimization truss structures by Multi ... · The discrete nonlinear truss optimization problem presented in relations(1 - 11), is a nonlinear problem due to Eq. 4. In the