OPTIMIZED SOLUTION OF PRESSURE VESSEL DESIGN USING GEOMETRIC PROGRAMMING

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1 OPTIMIZED SOLUTION OF PRESSURE VESSEL DESIGN USING GEOMETRIC PROGRAMMING S.H. NASSERI, Z. ALIZADEH AND F. TALESHIAN ABSTRACT. Geometric programmig is a methodology for solvig algebraic oliear optimizatio problems. It provides a powerful tool for solvig oliear problems where oliear relatios ca be well preseted by a expoetial or power fuctio. This feature is especially advatageous i situatios where the optimal value of the objective fuctio may be all that is of iterest. I such cases, calculatio of the optimum desig vectors ca be omitted. The goal of this paper is to state the problem of Pressure vessel desig ad after that fidig a better optimized solutio usig geometric programme. 1. INTRODUCTION I the real world, may applicatios of geometric programmig are egieerig desig problems. Oe of the remarkable properties of geometric programmig is that a problem with highly oliear costraits ca be stated equivaletly as oe with oly liear costraits. This is because there is a strog duality theorem for geometric programmig problems. The dual costrais are liear ad liearly costraied programs are geerally easier to solve tha oes with oliear costraits. Its attractive structural properties as well as its elegat theoretical basis have led to a umber of iterestig Key words ad phrases. Geometric programmig; pressure vessel desig; optimizatio. Correspodig author: asseri@umz.ac.ir. 344

2 applicatios ad the developmet of umerous useful results. The optimazatio problem which is itroduced i the followig has bee solved before by Deb ad Gee [2] usig Geetic Adaptive Search, by Kaa ad Kramer [3] usig a augmeted Lagragia Multiplier approach, ad by Coello [4] usig Geetic Algorithm ad the by M. Mahdavi et al. [1] usig a improved harmoy search algorithm. I this paper, we first use the duality theorem [6] ad the fid the optimal solutio which is better optimized tha ay other earlier solutios reported before. The remaider of this paper is orgaized as follows: The fuzzy geometric programmig problem is first itroduced. Next, the problem is stated. By the duality theorem, we write the dual of the problem ad fially the compariso of results are show i a table. 2. MATHEMATICAL FORMULATION A costraied posyomial geometric program is a optimizatio problem of the followig form: mi x f o x) = s o C ot a otj 2.1) s i s.t. f i x) = C it γ itj 1, i = 1,..., m. > 0, j = 1,...,. The posyomial f o x) cotaiig s o terms is the objective fuctio, while the posyomials f i x) for i = 1,..., m cotaiig s i terms represet m iequality costraits. By the defiitio of posyomial all the coefficiets C it for i = 0, 1,..., m ad t = 1,..., s m are positive. If the right had sides of the costraits i the geometric program 2.1) are modified as mi x f o x) = s o C ot a otj 2.2) s i s.t. f i x) = C it γ itj b i, i = 1,..., m. > 0, j = 1,...,. where all b i are positive umbers. If b i = 1 i, the this modified geometric program coicides with the origial oe. Otherwise, the costraits eed some amedmet to be cosistet with model 2.1). 345

3 FIGURE 1. Schematic of pressure vessel 3. PRESSURE VESSEL DESIGN A cylidrical vessel is capped at both eds by hemispherical heads as show i Fig1). The objective is to miimize the total cost, icludig the cost of material, formig ad weldig. There are four desig variables: T s thickess of the shell, x 1 ), T h thickess of the head, x 2 ), R ier radius, x 3 ) ad L legth of cylidrical sectio of the vessel, ot icludig the head, x 4 ). T s ad T h are iteger multiples of ich, which are the available thickess of rolled steel plates, ad R ad L are cotiuous. By usig the same otatio give by Coello [5], the problem is stated as follows: mi f x ) = x 1 x 3 x x 2 x x 2 1x x 2 1x 3 s.t. g 1 x ) = x x 3 0 g 2 x ) = x x 3 0 g 3 x ) = πx 2 3x π x g 4 x ) = x The comparisos of results are show i Table 1). 346

4 4. SOLUTION APPROACH First of all we modify the problem as a origial geometric program as follow: mi f x ) = x 1 x 3 x x 2 x x 2 1x x 2 1x 3 s.t. g 1 x ) = x 1x g 2 1 x ) = x 2x g 3 π x ) = x2 3x 4 + π x3 3 1 g 4 x ) = x 4 1 Solutio: I this problem m = 4, N o = 4, N 1 = 1, N 2 = 1, N 3 = 2, N 4 = 1, = 4. The sigum fuctios are σ o = 1, σ 1 = 1, σ 2 = 1, σ 3 = 1 ad σ 4 = 1. The dual objective fuctio ca be stated as follows: max vλ) = Π 4 k=0 ΠN k j=1 C kj λ kj Σ N k l=1 λ kl) σ kλ kj The costraits are give by: Σ N o j=1 λ oj = 1 Σ m k=0 ΣN k j=1 σ ka kij λ kj = 0, i = 1,..., Σ N k j=1 λ kj 0, k = 1,..., m. where C kj are the coefficiets, a kij are the expoets, m idicates the total umber of costraits, N 0 deotes the umber of terms i the objective fuctio ad N k represets the umber of terms i the kth costrait. That is: max vλ) = λ 01 ) λ λ 02 ) λ λ 03 ) λ λ 04 ) λ ) λ ) λ 21 1 π 240 )λ λ 31 λ

5 λ 32 )) λ 31 π λ 32 λ 31 + λ 32 )) λ 32 s.t. λ 01 + λ 02 + λ 03 + λ 04 = 1 λ λ λ 04 λ 11 = 0 λ 02 λ 21 = 0 λ λ 02 + λ 04 + λ 11 + λ 21 2λ 31 3λ 32 = 0 λ 01 + λ 03 λ 31 + λ 41 = 0 λ 31 + λ 32 0 λ 11 0, λ 21 0, λ The dual problem has the desirable features of beig liearly costraied. After solvig the dual problem, the optimum value of the objective fuctio v = f = is kow ad the values of the desig variables x i are as follow: x 1 = , x 2 = , x 3 = 37.70, x 4 = The results obtaied usig geometric programmig were better optimized tha ay other earlier solutios which has bee reported before. Table 1 Optimal results for pressure vessel desig methods M. Mahdavi[1] Deb ad Gee[2] Kaa ad Kramer[3] Coello[4] proposed method results CONCLUSIONS Geometric programmig is a kow method of solvig a class of oliear programmig problems. I particular, we have metioed a optimizatio problem which was bee solved by differet methods ad each method had a result but the results obtaied usig Geometric programmig were better optimized. REFERENCES [1] M. MAHDAVI, M. FESANGHARY, E. DAMANGIR, A improved harmoy search algorithm for solvig optimizatio problems, Appl. Math. Comput., ), [2] K. DEB, A.S. GENE, A robust optimal desig techique for mechaical compoet desig, i: D. Dasgupta, Z. MichalewiczEds.), Evolutioary Algorithms i Egieerig Applicatios, Spriger, Berli., 1997),

6 [3] B.K. KANNAN, S.N. KRAMER, A augmeted Lagrage multiplier based method for mixed iteger discrete cotiuous optimizatio ad its applicatios to mechaical desig, J. Mech. Des. Tras. ASME., ), [4] C.A.C. COELLO, Costrait-hadlig usig a evolutioary multiobjective optimizatio techique, Civ. Eg. Eviro. Syst., ), [5] C.A.C. COELLO, Theoretical ad umerical costrait-hadlig techiques used with evolutioary algorithms: a survey of the state of the art, Comput. Meth. Appl. Mech. Eg., ), [6] SINGIRESU S. RAO, Egieerig Optimizatio: Theory ad Practice, Joh Wiley ad Sos, Ic., Fourth Editio 2009). S.H. NASSERI, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MAZANDARAN, BABOLSAR, IRAN address: asseri@umz.ac.ir Z. ALIZADEH, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MAZANDARAN, BABOLSAR, IRAN address: z.afrouzy@yahoo.com F. TALESHIAN, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MAZANDARAN, BABOLSAR, IRAN address: f.taleshia@yahoo.com 349