Optimized solution of a two-bar truss nonlinear problem using ... - ispacs

Journal Nonlinear Analysis and Application 2014 (2014) 1-9

Available online at www.ispacs.com/jnaa Volume 2014, Year 2014 Article ID jnaa-00230, 9 Pages doi:10.5899/2014/jnaa-00230 Research Article

Optimized solution of a two-bar truss nonlinear problem using fuzzy Geometric programming S. H. Nasseri1∗, Z. Alizadeh1 (1) Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran c S. H. Nasseri and Z. Alizadeh. This is an open access article distributed under the Creative Commons Attribution License, Copyright 2014 ⃝ which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract Geometric programming is a methodology for solving algebraic nonlinear optimization problems. It provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by an exponential or power function. Many applications of geometric programming are engineering design problems in which some of the parameters are approximately known. The goal of this paper is to state the problem of a two-bar truss which some parameters are estimates of actual values and suggest a solution to derive the fuzzy objective value of the fuzzy polynomial geometric programming problem. Keywords: Geometric programming, Two-bar Truss, Optimization, Fuzzy set.

1 Introduction In the real world, many applications of geometric programming are engineering design problems and it is useful in the study of a variety of optimization problems. There are a lot of examples such as engineering design [3, 4], project management [5] and inventory management [6, 10, 11, 17] that geometric programming has very applicability. Efficient algorithms have been developed for solving the geometric programming problems when the parameters are known exactly [12, 13, 15, 16]. However, in the real world many design problems have parameters that are estimates of actual values [9]. In such cases fuzzy geometric programming is introduced. If some parameters are uncertain, then the most likely values are usually adopted to make the conventional geometric program workable. Fuzzy set theory has been widely used in system design optimization. To deal quantitatively with imprecise information, Bellman and Zadeh [14] and Zadeh [1] introduce the notion of fuzziness. The remainder of this paper is organized as follows: The fuzzy geometric programming problem is first introduced. In the next section, first we introduce the trapezoidal fuzzy number and then state an approach to transform these numbers to a form that can be useful for the problems. After that we formulate the geometric programming problem in the form of two-level mathematical programs and by the use of transformation trapezoidal numbers, change them into the conventional one-level mathematical programs to solve. The Two-bar truss problem is stated in section four. Finally, a summary of the research is presented. ∗ Corresponding

author. Email address: [email protected]

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2 Mathematical formulation A constrained polynomial geometric program is an optimization problem of the following form: so

n

t=1

j=1

∑ cot ∏ x j aot j

min fo (x) = x

s.t.

si

n

t=1

j=1

∑ cit ∏ x j γit j 6 1,

fi (x) =

x j > 0,

(2.1)

i = 1, ..., m.

j = 1, ..., n.

The polynomial fo (x) containing so terms is the objective function, while the polynomials fi (x) for i = 1, ..., m containing si terms represent m inequality constraints. By the definition of polynomial all the coefficients Cit for i = 0, 1, ..., m and t = 1, ..., sm are positive. If the right hand sides of the constraints in the geometric program (2.1) are modified as so

n

t=1

j=1

min fo (x) = x

s.t.

si

n

t=1

j=1

∑ cot ∏ x j aot j

fi (x) = ∑ cit ∏ x j γit j 6 bi , x j > 0,

(2.2)

i = 1, ..., m.

j = 1, ..., n.

where all bi are positive numbers. If bi = 1 ∀i, then this modified geometric program coincides with the original one. Otherwise, the constraints need some amendment to be consistent with model (2.1). Intuitively, if any of the parameters bi , c0t or cit is fuzzy, the objective value should be fuzzy as well. The conventional geometric programming problem which is defined (2.1) will know change into fuzzy geometric programming problem. suppose that the right-hand side bi , the cost coefficient c0t and the constraint coefficient cit are approximately known. They can be represented by the convex fuzzy sets B˜ i , C˜0t and C˜it , respectively. Suppose µB˜i , µC˜0t and µC˜it denote their membership functions and we have: ( )} { B˜ i = bi , µB˜i (bi ) |bi ∈ S B˜ i , { ( )} C˜0t = c0t , µC˜0t (c0t ) |c0t ∈ S C˜0t , { ( )} C˜it = cit , µC˜it (cit ) |cit ∈ S C˜it , ( ) ( ) ( ) where S B˜ i , S C˜0t and S C˜it are the supports of B˜ i , C˜0t and C˜it which denote their universe sets respectively. The fuzzy geometric programming can be stated as: s0

n

t=1 si

j=1

a Z˜ = ∑ C˜0t ∏ x j 0t j

s.t.

n

γ ∑ C˜it ∏ x jit j ≤ B˜i ,

t=1

x j > 0,

(2.3) i = 1, ..., m,

j=1

j = 1, ..., n.

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The fuzzy numbers are assumed to be trapezoidal fuzzy numbers. Denote the α -cuts of B˜ i , C˜0t and C˜it as: ] [ (Bi )α = (Bi )Lα , (Bi )Uα [ ] { ( ) } { ( ) } = min bi ∈ S B˜ i |µB˜i (bi ) ≥ α , max bi ∈ S B˜ i |µB˜i (bi ) ≥ α , bi bi [ ] L U (C0t )α = (C0t )α , (C0t )α [ } { }] { ( ) ( ) ˜ ˜ = min c0t ∈ S C0t |µC˜0t (c0t ) ≥ α , max c0t ∈ S C0t |µC˜0t (c0t ) ≥ α , c0t c0t [ ] (Cit )α = (Cit )Lα , (Cit )Uα [ { } { }] ( ) ( ) = min cit ∈ S C˜it |µC˜it (cit ) ≥ α , max cit ∈ S C˜it |µC˜it (cit ) ≥ α , cit

cit

Z˜ is a fuzzy number for that we apply Zadeh’s extension principle [1, 8] to transform the problem into a family of conventional geometric program to be solved. The membership function µZ˜ after using the extension principle can be defined as: { } µZ˜ (z) = sup min µB˜i (bi ) , µC˜0t (c0t ) , µC˜it (cit ) , ∀i, j,t|z = Z (a, b, c) , a,b,c

where Z (a, b, c) is the function of the conventional geometric program that is defined in model 2.2. As mentioned by Liu [2] for finding the membership function µZ˜ , it is sufficient to find the upper bound of the objective value ZαU and the lower bound of the objective value ZαL at specific α level. 3 Solution approach First of all, the fuzzy trapezoidal numbers should be transformed to a form that can be useful for the problem. Consider a trapezoidal fuzzy number: [ ] a˜ = aL , aU , γ , β

Figure 1: trapezoidal fuzzy number In a trapezoidal fuzzy number, for different α -cuts suppose an upper bound and a lower bound. It can be expressed as: ( ) ( ) γ + α aL − γ ≤ a˜ ≤ β − α β − aU . (3.4)

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For each fuzzy number in model (2.3) we have:  ( ) ( ) ( ) B˜ i , C˜0t , C˜it |γ + α BLi − γ ≤ B˜ i ≤ β − α β − BUi       ( L ) ( )  U γ + α C0t − γ ≤ C˜0t ≤ β − α β −C0t M=   ( ) ( )   γ + α CitL − γ ≤ C˜it ≤ β − α β −CitU   

              

(3.5)

By the use of equations (3.5) the fuzzy model (2.3) can be expressed as conventional geometric program. 3.1 Lower bound The lower bound of the objective value is expressed as: { ( ) ( ) } ZαL = min Z B˜ i , C˜0t , C˜it | B˜ i , C˜0t , C˜it ∈ M .

(3.6)

We have to solve the two level mathematical program for obtaining the lower bound of the objective value: ZαL =

min

(B˜i ,C˜0t ,C˜it )∈M

min Z˜ = s.t.

s0

n

t=1 si

j=1

∑ C˜0t ∏ x j 0t j a

n

(3.7)

γ

∑ C˜it ∏ x jit j ≤ B˜i ,

i = 1, ..., m,

j=1

t=1

x j > 0,

j = 1, ..., n.

First we turn the inner program of model (3.7) to the following standard geometric program form: ZαL =

min

(B˜i ,C˜0t ,C˜it )∈M

min Z˜ = s.t.

s0

n

t=1 si

j=1

∑ C˜0t ∏ x j 0t j C˜it

a

n

γ

∑ B˜i ∏ x jit j ≤ 1,

i = 1, ..., m,

j=1

t=1

x j > 0, (

(3.8)

j = 1, ..., n.

)

In model (3.8) we want to find a set of B˜ i , C˜0t , C˜it that derive the smallest objective value. Hence, model (3.8) can be rewritten as the following mathematical program: s0 ( L ) n a − γ ∏ x j ot j ZαL = min ∑ γ + α C0t t=1

si

s.t.

∑

t=1

(

(3.9)

j=1

( )) n γ + α CitL − γ γ ( ) ∏ x jit j ≤ 1, U β − α β − Bi j=1

i = 1, ..., m,

x j > 0, 0 ≤ α ≤ 1, j = 1, ..., n. Now we can derive the lower bound of the objective value simply by solving model (3.9). 3.2 Upper bound For obtaining the upper bound of the objective value we have the following two-level program: { ( ) ( ) } ZαU = max Z B˜ i , C˜0t , C˜it | B˜ i , C˜0t , C˜it ∈ M .

(3.10)

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By repeating the same steps expressed for obtaining the lower bound of the objective value, the following standard geometric program should be solved: s0 ( ) n aot j U ZαU = min ∑ β − α β −C0t ∏ xj

(

si

s.t.

∑

t=1

(3.11)

j=1

t=1

( )) n β − α β −CitU γ ( ) ∏ x jit j ≤ 1, γ + α BLi − γ j=1

i = 1, ..., m,

x j > 0, 0 ≤ α ≤ 1, j = 1, ..., n. The upper bound of the objective value can be obtained easily by solving model (3.11). 4 Design of a Two-bar Truss The two-bar truss [7] shown in Fig.2 is subjected to a vertical load 2P and is to be designed for minimum weight. The members have a tubular section with mean diameter d and wall thickness t and the maximum permissible stress in each member (σ0 ) is approximately equal to 60,000 psi. Determine the values of h and d using geometric programming for the following data: P = (32.500, 33.250, 32.000, 33.750) lb,t = 0.1in., b = 30in., σ0 = (58.000, 61.000, 57.500, 61.750) psi. and ρ (density) = (0.2, 0.5, 0.1, 0.7) lb/in3 . Solution:

Figure 2: Two-bar truss under load

The objective function is given by f (d, h) = 2ρπ dt

√ b2 + h2

and the stress constraint can be expressed as

σ=

√ P b2 + h2 ≤ σ0 π dth

by replacing the given amounts we have: √ f (d, h) = 2 (0.2, 0.5, 0.1, 0.7) (3.14) (0.1) d 900 + h2 √ (32.500, 33.250, 32.000, 33.750) 900 + h2 s.t. ≤ (58.000, 61.000, 57.500, 61.750) (3.14) (0.1) dh

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The functions can be converted to polynomials by introducing a new variable y as √ y = 900 + h2 and the variable gives the new constraint as 900 + h2 ≤1 y2 Thus the optimization problem can be stated as min s.t.

f = 0.628 (0.2, 0.5, 0.1, 0.7) yd (32.500, 33.250, 32.000, 33.750) d −1 h−1 y ≤ 1 (0.314) (58.000, 61.000, 57.500, 61.750) 900y−2 + y−2 h2 ≤ 1 y, h, d ≥ 0

According to the equation (3.5) the trapezoidal fuzzy numbers can be written in the following form: 57.500 + α (58.000 − 57.500) ≤ B˜ 1 ≤ 61.750 − α (61.750 − 61.000) 0.1 + α (0.2 − 0.1) ≤ C˜01 ≤ 0.7 − α (0.7 − 0.5) 57.500 + α (58.000 − 57.500) ≤ C˜11 ≤ 61.750 − α (61.750 − 61.000) According to models (3.9) and (3.11), the lower bound and the upper bound of the objective value at possibility level α can be solved as: 0.628 (0.1 + 0.1α ) yd 57.500 + 0.500α s.t. d −1 h−1 y ≤ 1 (61.750 − 0.750α ) (3.14)

ZαL = min

(4.12)

900y−2 + y−2 h2 ≤ 1 y, h, d ≥ 0 0.628 (0.7 − 0.2α ) yd 61.750 − 0.750α s.t. d −1 h−1 y ≤ 1 (57.500 + 0.500α ) (3.14)

ZαU = min

(4.13)

900y−2 + y−2 h2 ≤ 1 y, h, d ≥ 0 Since the problem has a zero degree of difficulty, the minimum of the primal problem can be obtained by maximizing the corresponding dual function [7]. The dual function can be stated as: [

57.500 + 0.500α = 0/628 (0.1 + 0.1α ) 3.14 (61.750 − 0.750α ) [ ]λ22 1 (λ22 + λ21 ) λ22 s.t. λ01 = 1 λ01 + λ11 − 2λ21 − 2λ22 = 0 λ01 − λ11 = 0 −λ11 + 2λ22 = 0

VαL

]λ11 [

]λ21 900 (λ22 + λ21 λ ) λ21

(4.14)

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[

61.750 − 0.750α = 0/628 (0.7 − 0.2α ) 3.14 (57.500 + 0.500α ) ]λ22 [ 1 (λ22 + λ21 ) λ22 s.t. λ01 = 1 λ01 + λ11 − 2λ21 − 2λ22 = 0 λ01 − λ11 = 0 −λ11 + 2λ22 = 0

VαU

]λ11 [

]λ21 900 (λ22 + λ21 λ ) λ21

(4.15)

The equality constraints can be used to express λ01 , λ11 , λ21 and λ22 so we have:

λ01 = 1 λ11 = 1 λ21 = 1/2 λ22 = 1/2 The upper bound and lower bound of the objective value are determined by the value of possibility level α . By using the method which is described in [9] λ ∗ can be transformed to the corresponding primal solution and the values of decision variables y, d and h are obtained. Tables 1 and 2 lists the α -cuts of the objective value at 11 distinct α values and the corresponding primal solution.

Table (1): The upper bound of the objective value for Two- bar truss problem

α ZαU y d h

0.0 9.020 42.45 0.419 30.04

α ZαL y d h

0.0 1.117 42.45 0.419 30.04

0.1 8.744 42.45 0.419 30.04

0.2 8.470 42.45 0.420 30.04

0.3 8.196 42.42 0.480 30.00

0.4 7.923 42.42 0.479 30.00

0.5 7.651 42.42 0.478 30.00

0.6 7.381 42.42 0.477 30.00

0.7 7.112 42.42 0.476 30.00

0.8 6.843 42.42 0.475 30.00

0.9 6.576 42.42 0.474 30.00

1.0 6.310 42.42 0.428 30.00

0.9 2.163 42.42 0.427 30.00

1.0 2.281 42.42 0.428 30.00

Table (2): The lower bound of the objective value for Two- bar truss problem

0.1 1.231 42.45 0.419 30.04

0.2 1.346 42.45 0.420 30.04

0.3 1.461 42.42 0.422 30.00

0.4 1.577 42.42 0.422 30.00

0.5 1.693 42.42 0.423 30.00

0.6 1.810 42.45 0.424 30.04

0.7 1.927 42.42 0.425 30.00

0.8 2.045 42.42 0.426 30.00

The α value indicates the level of possibility and the degree of uncertainty of the obtained information. When the α value is greater, the level of possibility is greater too and the degree of uncertainty becomes less. In particular, α = 0 has the widest interval indicating that the objective value will absolutely fall into this range. On the other side, the possibility level α = 1 is the most possible value of the objective value. In this example the objective value is impossible to fall below 1.117 or exceed 9.020 and the most possible value is to lie within 2.281 and 6.310. 5

Conclusions

Geometric programming is a known method of solving a class of nonlinear programming problems. In real world applications, the parameters in the geometric program may not be known precisely. In particular, we have mentioned an optimization problem which the parameters are supposed to be fuzzy trapezoidal numbers. The idea is based on Zadeh’s extension principle to transform the fuzzy geometric programming problem to a pair of twolevel mathematical programs and then by the solution proposed solve it. The illustrated example which is a fuzzy engineering optimization problem with geometric programming form, show that the solution is in reality able to solve such problems in this type.

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