Structural optimization using unconstrained nonlinear goal programming algorithm

compvrers& SIrucrUres Vol. 52, No. 4.

Pergamon

pp. 723-721, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved

00457949(94)E4M&A

004%7949/94 s7.w + 0.00

STRUCTURAL OPTIMIZATION USING ~~ONSTRAINED NONLINEAR GOAL PROGRAMMING ALGORITHM M. E. M. apartment

of ~~hanical

EL-SAYED

and T. S.

Engineering, Florida Inte~ational

JANG

University, Miami, FL 33199, U.S.A.

(Received 21 January 1993)

Altstraet-This paper presents a method for solving structural optimization problems using nonlinear goal programming techniques. The developed method removes the difficulty of having to define an objective function and constraints. It also has the capacity of handling rank ordered design objectives or goals. The

formulation of the structural optimization problem into a goal programming form is discussed. The resulting optimization problem is solved using Powell’s unconstrained conjugate direction search algorithm. To demonstrate the effectiveness of the method, as a design tool, the solutions of some numerical test cases are included.

INTRODUCI‘ION

One of the major difficulties in applying traditional mathematical prorating techniques to design problems is matching the model with reality. Real-life design problems occur with conflicting and multiple objectives rather than with a single one. It is often difficult to define the problems exactly in mathematical programming traditional form. There are, however, techniques that are capable of handling weights and priority factors for conflicting and multiobjective optimization. These techniques are called linear and nonlinear goal programming [ 1,2]. In a goal programming (GP) formulation, the design goals are defined and weights and priority factors are assigned to each. The algorithm then attempts to minimize the sum of the deviational variables with weights and priority factors, starting with the highest priority goal, which makes goal programming an ideal real-life design tool. The initial development of the goal programming concept was in 1961 by Charnes and Cooper [3]. In 1965, Ijiri [4], further developed linear goal programming (LGP). He presented the definition of ‘preemptive priority factors’ to treat the multiple conflicting goals according to their ranked importance. He also assigned weights to the goals of the same priority level. In the late 1960s and early 197Os, LGP began to receive wide acclaim as a decisionmaking tool. Griffith and Stewart 151, developed a nonlinear programming technique in 1961. The application of LGP to many areas was conducted in the 1970s. Lee [6], illustrated the application of LGP and contributed to the continuous growth in the use of LGP. He also developed a computer program using the revised simplex method for the solution of LGP problems. CAS J2/4-I

723

In the late 1970s and mid-1980s the emphasis was more evenly shared between applications for management [7], marketing [8], and industrial engineering [9]. The development of goal pro~amming models basically consisted of improving the solution procedures [lo, 111.There was also some efforts to apply multi-objective criteria to structural optimization problems [ 121. Recently Rao [ 131, applied fuzzy goal programming to structural systems. El-Sayed et al. [14, 151, used the successive linearization method to apply LGP techniques to large scale optimization problems. As indicated in [14, IS]. for large scale structural optimization problems with high nonlinearity, successive LGP may not converge to the optimum design. In this paper the general formulation of the structural optimization problem into a nonlinear goal programming (NLGP) form is presented. The resulting NLGP problem is then solved using Powell’s conjugate direction method. NONL~EAR

GOAL STRUCXXJRAL OPTIMIZATION MODEL

Following [ 161,the standard form of the nonlinear goal pro~amming model is: find to minimize

x=(X,,-$,Xj,.*.,X,) z = (A(&, fkv-,

subject to

d+),

d+),f,(d-, . J&Pr

d+), . . . , d+N

g,(x) f d; - d: = bj (fori=l,2,...,1)

and

d;, d: 30,

(1)

124

M. E. M. EL-SAYED and T. S. JANG

where z represents an achievement vector, structured as an ordered set such that a pre-emptive priority structure is maintained. The dimension of z represents the number of pre-emptive priority levels which is equal to or less than the number of objectives. The value of z will be equal to zero if all the objectives meet their aspiration levels. (for i=l,2

g,(x)+d;-d’=b,

,...,

d+) = 1

(w,d;

xL

The nonlinear goal programming model for the structural optimization problem using eqns (1) and (4), for weight, stress, displacement, and frequency requirements can be expressed as: find

x=(x,,.X?..YJ ,...,

to minimize

z = (S,(d-,

X,)

I)

represents the design objectives where d; measures the negative deviation from the aspired level for the objective i and d: measures the positive deviation. fk(d-, d+) is a weighted linear function of the deviation variables, d-, d +. Each function can be represented as the weighted sum of the negative and the positive deviation variables for all the objectives contained in a priority level Pk with weights w, and p” as: fk(d-,

and

hW. subject to

d+),_&(d-,

d+),

...,

d+))

d+), . ,f&-.

W/W,-l+d,-d:=O ui/oa-

l+d,-

-d:

=0

u,/u, -

1 + d; - dI+ = 0

j-x--l+d;-d,f=O

+ pjd;)

(5)

JEPk

and (fork=1,2

,....

K).

The general mathematical programming model for the optimization problem can be expressed as follows: find

x=(X,,X2,Xj,~..,X,)

to minimize

F(x)

subject to

G,(x) f 0

and

xL
(for i = 1,2,

. . , I)

(3)

where F(x) is the objective function and G,(x) represents the inequality constraint function including the equality constraints. The elements of the vector x are the design variable. For structural optimization problems, the constraint function Gi(x) may be a function of stress, displacement, and natural frequency. The elements of the vector x correspond to the size of the structural members. The variables xL and x” represent the lower and upper bound on the design variables, respectively. The minimum weight structural optimization problem with stress, displacement, and frequency requirements can be expressed as: find

x=(x,,x*,xs

to minimize

W(x)

subject to

f&/a,-1

. . . . 9x,)

GO

UjlU, - 1 < 0 f/X-1GO

XL< x
(2)

(4)

d;,d:,d,~,d:,d,-,d:,d,,d:

20,

where W is the total weight of the structure, ei is the ith stress objective, u, is the ith displacement objective, and f is the natural frequency. W,, TV,,u, and f, represent the target weight, allowable stress, allowable displacement, and frequency, respectively. NONLINEAR

GOAL PROGRAMMING POWELL’S METHOD

USING

A variety of methods can be used to solve nonlinear optimization problems. The success of a method over other methods to find a solution may be relative to the particular problem. A straightforward approach for solving the nonlinear goal structural optimization problem of eqn (5) is to use a zero-order optimization method. Powell [ 171, introduced a method of multi-variable optimization based on the concept of conjugate directions. This method is one of the efficient and reliable nongradient methods available today [ 181. The basic concept of Powell’s method is first to search in n orthogonal directions, where each search consists of updating the design vector using the minimum along the previous search direction as the starting point. After performing these successive minimizations, a new search direction is formed between the original starting point and the resulting point of the successive n searches. The first search direction is then dropped and the remaining search directions are kept along with the new direction, which is placed last among the directions. The search is continued until convergence is achieved. A nonlinear goal programming algorithm is developed using Powell’s algorithm as the optimization

725

Structural optimization using nonlinear goal programming Table 2. Results for case I for IO-member truss

360”

1

Optimum cross-sectional area (in*) Member number

Ref. 1191 _ _

NLGP

1 2 3 4 5 6 7

7.9319 0.1 8.0621 3.9379 0.1 0.1 5.7447

1.9392 0.1 8.0616 3.9332 0.1013 0.1 5.7433

5.5690 0.1

5.5699 5.5711 0.1

1593.18

1593.17

X

Fig. 1. Ten-member planar truss.

routine. The NLGP algorithm first minimizes, as nearly as possible, the objectives with the highest priority level. It then proceeds to satisfy the objectives of the next priority level, as nearly as possible, without degrading the achievement of any objective in a higher priority level. This process is continued until all priority levels have been considered. At each priority level the search is terminated when the difference between present and previous achievement Function value becomes sufficiently small. The vahre of z will be equal to zero if all the objectives meet their aspiration levels. The value of z, will be positive if one or more objectives in priority level k are not met.

f 10 Optimum weight (lb) Total CPU (see) on Vax 8650

25.58

case represents the minimum weight design with stress, displacement and frequency constraints with different priority levels and weights. In all cases the maximum positive or negative deviation was limited to be less than 0.01. The stresses, displacements and Frequencies were obtained using a finite element analysis program. A Vax 8650 computer was used to solve the structural optimization problems. In the following we discuss the three cases and the results obtained.

TEST CASES

To demonstrate the efficiency of the method and its application to design problems some numerical test cases are conducted using the 10 member planar truss structural optimization problem. Figure 1 shows the geometry and dimensions of the lo-member truss. The design data For the truss elements and load data are given Table 1. Three different cases of nonlinear goal structural optimization problems are considered. The first case represents the minimum weight design with stress constraints only. The second case represents the minimum weight design with stress and displacement constraints with different priority levels. The third Table 1. Design and load data for IO-member planar truss Modulus of elasticity = 104ksi Material specific weight = 0.10 lb/in’ Lower limit on cross-sectional areas = 0.10 in2 Initial value of design variable = 10 in* Stress limit = +25 ksi Number of loading conditions = 1 Load data Load component (kips) in direction Loading condition

Node

X

1

2 4

0.0 0.0

- 100.0 - 100.0

0.0 0.0

2

1

0.0

50.0

0.0

: 4

0.0 0.0

- -50.0 150.0 - 150.0

0.0 0.0

,?

Y

Case I In this case, the minimum weight design with stress constraints only is performed using the developed NLGP technique. Two loads of 100 kips in the negative y direction are applied at nodes 2 and 4 as in loading condition 1. The structural optimization problem was solved using the NLGP technique with deviational variables only. The results for the optimum cross-sectional area of each member, and the optimum weight are compared with the results obtained From a traditional mathemati~l pro~amming algorithm 1191,in Table 2. Case II In this case, the structural optim~tion is performed For loading condition 2 with stress and displacement constraints of +2.0 in. at nodal points 1 and 2. The NLGP algorithm was used to solve two structural optimization problems. The first problem was solved with deviational variables only. The second problem was solved with priority level 1 for the weight and displacement constraints and priority level 2 For the stress constraints. The results for the two optimization problems are compared with the results obtained From a traditional mathematical programming algorithm [19], in Table 3. Cu.re ZIZ This case is the same as Case II except that a natural frequency constraint of 22 Hz was imposed on the structure. The NLGP algorithm was used to

726

M. E. M. EL-SAYEDand T. S. Table 3. Results for Case II for 10 member truss

Optimum cross-sectional area (in*) Member number 2 3 4 5 6 7 8 9 10 Optimum weight (lb)

Ref. [19]

NLGP

NLGP (P)

30.03 1 0.1 23.274 15.286 0.1 0.5565 7.4683 21.198 21.618 0.1

28.7878 0.1 22.3483 16.7847 0.1 0.5516 7.5633 21.9861 21.0175 0.1

32.9657 0.1 22.7988 14.1463 0.1 0.7393 6.3807 20.9121 20.9779 0.1

5061.6

5051.75

5013.24

64.35

108.18

Total CPU (set) on Vax 8650

solve three structural optimization problems. The first problem was solved with deviational variables only. The second problem was solved with priority level 1 for the weight, displacement and frequency constraints and priority level 2 for the stress constraints. For the third problem priority level 1 was given to the weight, displacement and frequency constraints and a weighting factor of 3 was given to the frequency constraint. Priority level 2 was given to the stress constraints. The results for the three optimization problems are compared with the results obtained from a traditional mathematical programming algorithm [20], in Table 4.

JANG

solution to the problem is always obtained. This is due to the unconstrained search method which does not require feasibility, and the ordered set of preemptive priority levels. At the end of the optimization process the achievement vector indicates the degree to which the desired aspiration levels are met. As in the case of general nonlinear optimization problems, there is no guarantee that the solution will be a global optimum. The test cases performed, using the developed algorithm, demonstrated that a classical structural optimization problem can be solved with the same efficiency using the nonlinear goal programming formulation. By using different priority levels and weighting factors a further reduction in the structural weight can be achieved. The resulting optimum design, however, depends on the priority levels and the weighting factors used. Therefore, careful selection of the priority levels and the weighting factors is of great importance for achieving the proper design.

CONCLUSION

A method for solving structural optimization problems using nonlinear goal programming is developed. The method uses Powell’s algorithm to minimize the achievement function at each priority level. The greatest advantage of the developed nonlinear goal structural optimization method is that the method allows the design engineer to model the problem realistically. Another major advantage of the NLGP over conventional optimization methods is that the

REFERENCES

1. M. J. Schniederjans and N. K. Kwak, An alternative solution method for goal programming: a tutorial. J. Operar. Res. Sot. 33, 247-251 (1982). 2. J. P. Ignizio, Generalized goal-programming: an overview. Comput. Operat. Res. 10, 217-289 (1983). 3. A. Chames and W. Cooper, Management Models and the Industrial Applications of Linear Programming. John Wiley, New York (1961). 4. Y. Ijiri, Management Goals and Accounting for Control. Rand McNally, Chicago (1965). 5. R. E. Griffith and R. A. Stewart, A nonlinear programming technique for the optimization of continuous processing systems. Managem. Sci. 7, 379-392 (1961). 6. S. M. Lee, Goal Programming for Decision Analysis. Auerback Publishers, Philadelphia (1972). 7. J. C. Fisk, A goal programming model for output planning. Decis. Sci. 10, 593-603 (1979). 8. J. M. Wilson, The handling of goals in marketing problems. Managem. Decis. 13, 175-180 (1975). 9. N. K. Kwak and M. J. Schniederjans, A goal programming model for improved transportation problem solutions. Omega 7, 367-370 (1979).

Table 4. Results for Case III for lo-member truss Optimum cross-sectional area (in*) Member number

Optimum weight (lb) Total CPU (set) on Vax 8650

NLGP

NLGP (P)

NLGP (P and W)

24.8602 0.1 25.9162 13.1350 0.1 1.9127 13.1924 15.3214 17.5268 0.1

25.2706 0.1 25. 13.0167 0.1 2.0984 13.5444 15.4454 17.5334 0.1

25.7545 0.1 21.9696 13.9559 0.1 0.6181 9.1645 15.8525 19.3600 0.1

7.8601 0.1 10.2442 4.3975 0.1 0.3489 7.6610 4.6563 6.1863 0.1

4730.29

4734.75 78.02

4514.33 201.43

1176.97 100.34

GRGA [20]

Structural optimization using nonlinear goal programming 10. J. P. Ignizio, Linear Programming in Single and Multiple Objective Sysrems. Prentice-Hall, Englewood Cliffs, NJ (1982). 11. M. J. Schniederjans, Linear Goal Programming. Petrocelli Books, Princeton, NJ (1984). 12. S. S. Rao, Multiobjective optimization in structural design with uncertaih parameters and stochastic processes. AZAA J. 22. 1670-1678 (1984). 13. S. S. Rao, Multi-objective optim~zatidns of fuzzy structural systems. Int. J. Numer. Meth. Engng 24,1157-l 171 (1987). 14. M. E. El-Sayed, B. J. Ridgely and E. Sandgren, Nonlinear structural optimizations using linear goal programming. Comput. Struct. 32, 69-73 (1989). 15. M. E. El-Sayed and T. S. Jang, Large scale structural optimizations using linear goal programming.

121

Proc. ASME Design Automation Conference, Chicago (1990). 16. J. P. Ignizio, Goal Programming and Extensions. Lexington Books, Lexington (1976). 17. M. J. D. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput. J. 7, 155-162 (1964). 18. G. N. Vanderplaats, Numerical Optimization Techniques for Engineering Design: with Applications. McGraw-Hill, New York (1984). 19. E. J. Haug and J. S. Arora, Applied Optimal Design. John Wiley, New York (1979). 20. J. Abadie and J. Carpenter, The Generalization of the Worfe Reduced Gradient Method to the Case of Nonlinear Constraints in Optimization. Academic Press, New York (1969).