Solving Design of Pressure Vessel Engineering Problem Using a Fruit Fly Optimization Algorithm

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1 Solvng Desgn of Pressure Vessel Engneerng Problem Usng a Frut Fly Optmzaton Algorthm Xantng Ke, Yongsen Zhang, Yuchuan L,, * and Tngsong Du College of Scence, Chna Three Gorges Unversty, Ychang 44300, Chna Hube Provnce Key Laboratory of Systems Scence n Metallurgcal Process, Wuhan , Chna Abstract We nvestgate a frut fly optmzaton algorthm (FOA) to solve constraned structural engneerng desgn optmzaton problems. In our work, we compared PSO and QAFSA, and found the FOA to be more vald to search for the optmal soluton of three typcal functons. As an applcaton, optmal results provded by FOA concernng desgn of pressure vessel optmzaton problem are reported, and our result demonstrates that the best soluton yelded by FOA s superor to those of stateof-the-art algorthms n the lterature. Keywords - Frut fly optmzaton algorthm, desgn of pressure vessel, nonlnear constrant. I INTRODUCTION Durng the past several decades there has been a growng nterest n nonlnear and constraned problem solvng systems based on prncples of evoluton and heredtary: such as systems mantan a populaton of potental solutons, they have some selecton process based on ftness and ndvduals. One type of such systems s a class of evoluton strateges algorthms ([]994), whch mtate the prncples of natural evoluton for parameter optmzaton problem. The heurstc technques contan genetc algorthm (GA), smulated annealng (SA), Tabu search (TS), partcle swarm optmzaton (PSO), quantum artfcal fsh swarm algorthm (QAFSA), artfcal bee colony (ABC) etc. [, 5,,, 5, 3]. Frut fly optmzaton algorthm (FOA) s one of the meta-heurstc approaches frstly consdered by Wen-Tsao Pan (0)[6]. Such an optmzaton algorthm has advantages such as smple computatonal process, ease of transformaton of such concept nto program code and ease of understandng, etc. Very recently, Pan (0) [6] and QuanKe Pan et al. (04) [5] have respectvely appled FOA nto some unmodal and multmodal functons to obtan the approxmate optmal results. However, some structural engneerng desgn problems are general large scale, nonlnear and constraned optmzaton problems. So as to settle these desgn optmzaton problems, there are a growng number of concentratons on artfcal ntellgence heurstc algorthm, whch has been developed wth an am to carry out global search wth three purposes: solvng problem faster, solvng large scale problem, and obtanng robust algorthms. In earler papers, Deb and Goyal (986) [7] have presented a method combned bnary and real coded GAs to settle the mxed varables. In 005, Tsa (005) [9] has put forward a method to solve nonlnear fractonal programmng problems nvolved wth engneerng desgn optmzaton. In ths paper, we propose the FOA by ncorporatng the nformaton of the global average maxmum, the maxmum value,standard devaton and best soluton nto the search strategy to solve the structural engneerng desgn problem. Based on ths, not only can FOA be appled to resolve these smple problems but also can be utlzed to handle the structural engneerng desgn optmzaton problems. The present paper shows that the computaton results by FOA are better than those conventonal heurstc methods. An outlne of ths paper s as follows. In Secton, the detaled processes of FOA are descrbed to search for global optmal soluton for general optmzaton problems. Three typcal functons among PSO, QAFSA and FOA are compared by smulaton experment n Secton 3. Secton 4 apples FOA to solve desgn of pressure vessel optmzaton problem and compared wth some other algorthms results whle research conclusons drawn are dscussed n Secton 5. II. FRUIT FLY OPTIMIZATION ALGORITHM The FOA ([6] 0) s a new approach for searchng global optmzaton, whch s based on two man foragng processes: smell the food source by osphress organ and towards the relevant locaton to fnd food by usng senstve vson. The frut fly on sensory percepton s superor to other speces, partcularly n osphress and vson, as shown n Fg. ([0] 03). The osphress organs of frut fles can fnd all knds of scents floatng n the ar. When t gets close to the food locaton, uses ts senstve vson to fnd food and the companons flockng locaton, after that fly towards that drecton.in ths algorthm, based on the food searchng behavor of frut fly, t conssts of several essental steps as follows: Step. Intalze the frut fly swarm locaton randomly and parameters, ncludng maxmum number of generatons and populaton sze. X _ axs Value rand( ) and Y _ axs Value rand( ) Step. Produce the random drecton and dstance to the search of food dependng on osphress for an ndvdual frut fly. DOI 0.503/IJSSST.a ISSN: x onlne, prnt

2 Fgure. Food fndng teratve process of frut fly swarm. X X _ axs RandomValue and Y Y _ axs RandomValue Step3. Smell-based searchng process: snce the food locaton cannot be known, the orgnal dstance s thus estmated frst (Dst), then the flavor concentraton judgment value (S) whch s the recprocal of dstance s computed. After that, randomly generate several frut fles around the frut fly group to set up a populaton. Dst X Y S Dst Step4. Substtute flavor concentraton judgment value (S) nto the smell concentraton judgment functon (ftness functon) so as to yeld the smell concentraton of ndvdual locaton of frut fly. Smell Functon S Step5. Seek out the maxmal smell concentraton and note down the locaton ndex of the maxmal value among the frut fly swarm. bestsmell bestindex max Smell Step6. Keep the best smell concentraton value and the locaton coordnate, then let the frut fly group fly to the locaton. Smellbest bestsmell X axs X bestindex Y axs Y bestindex Step8. Substtute the X _ axs and Y _ axs from Step6 nto the flavor concentraton judgment value ( S ), and then put S nto the constrants. Afterwards, substtute S whch meets the constrants nto the Smellbestf (or called ftness functon) and lead to the best optmzaton soluton. Smellbest f Smellbestf end X best S end In the present paper, FOA s appled to cope wth desgn of pressure vessel optmzaton problem. III. SIMULATION EXPERIMENT AND ANALYSIS To verfy the feasblty and better convergence precson of the FOA, three typcal test functons are proposed and the results obtaned by PSO, QAFSA, and FOA are compared. The functons are expressed as follows: Ex: max f ( x) sn( Ex: max f Ex3: max f 3 ( x, x ( x, x n x ) / n x s. t. x exp(( n cos(x )) / ).789 x ) ( x x ) cos( x )cos( ) 4000 s. t. 5 x, x 5 sn x x 0.5 ) 0.5 [ 0.00( x x )] s. t. 00 x, x 00 Step7. Intate the teratve optmzaton and repeat the Step-6, and judge the smell concentraton whether s superor to prevous teraton, f so, carry out Step6. Fgure. The dagram of f ( x ). DOI 0.503/IJSSST.a ISSN: x onlne, prnt

3 Fgure 3. The dagram of f ( x). Fgure 4. The dagram of f 3 ( ). Fgure 6. The optmzaton process of f ( x ). We can fnd the optmal values of the three functons are, and 0 through the analyss and the functon dagrams shown n Fg., 3 and 4. The theoretcal maxmum values of these three multdmensonal functons are also, and 0 n the defnton doman. Usng PSO, QAFSA and FOA, we get maxmum value of three dfferent functons n Fg. 5, 6, and 7, respectvely. The maxmum teraton s set to be 500 tmes. All algorthms run 80 tmes from dfferent random ntal populaton respectvely. The local average maxmum value, the global maxmum value and standard devaton are to be as a measurement of the performance of algorthm, the results are shown n Table. It has been turned out that the theoretcal maxmum values of these functons are,, and 0 n the defnton doman respectvely. PSO, QAFSA and FOA are used to fnd the maxmum value of three dfferent functons, severally. The maxmum teraton s set to be 500 tmes. All algorthms are run 80 tmes from dfferent random ntal populaton. The global average maxmum value, the global maxmum value and standard devaton are used as a measurement of the performance of algorthm, the results are shown n Table I: Fgure 5. The optmzaton process of f ( x ). Fgure 7. The optmzaton process of f 3 ( x ). DOI 0.503/IJSSST.a ISSN: x onlne, prnt

4 fun TABLE I. THE RESULTS OF SIMULATION WITH PSO, QAFSA AND FOA. PSO QAFSA FOA A max G max A max G max A max G max f ( ) x f ( ) x f ( ) 3 x where fun denotes functon, Amax s global average maxmum, G max s global maxmum value, s standard devaton, respectvely. Ths paper uses PSO, QAFSA and FOA for 80 tmes, gans the average of the evolutonary curve and draws the Fg 5-7. In the Table I, the global average maxmum value, the global maxmum value of the three functons have been acheved and number of teratons by usng PSO, QAFSA and FOA. In each Fgure, the ordnate s denoted as optmal results of the functon, and the abscssa s expressed by evoluton generaton, three lnes wth dfferent colors demonstrate the changng tendency of maxmum value obtaned by PSO, QAFSA and FOA wth the ncrease of the teraton tmes. From Table I, t s clearly ndcated that each functons' average optmzaton result and the best optmal result of FOA are much better than those solved by PSO and QAFSA. Furthermore, the precson of the optmzaton results are also greater than PSO and QAFSA, most of standard devatons of FOA are nearly 0. Moreover, the stablty of FOA n the fgures s better than those of PSO and QAFSA by comparng the standard devaton. IV. DESIGN OF PRESSURE VESSEL OPTIMIZATION PROBLEM A pressure vessel desgn model s as shown n Fg. 8, whch nvolves four decson varables: x s defned thckness of the pressure vessel T s, x stands for thckness of the head T H, x3 represents nner radus of the vessel R, and s on behalf of length of the vessel barrng head L, the total varables descrbed as x ( x, x, x3, ). The objectve functon of the problem s to mnmze the total cost, ncludng the cost of materal, formng, and weldng. So the general pressure vessel desgn optmzaton model can be expressed as: Mnmze f x 0.64x x3x 4.778x x3 3.66x 9.84x x3 st.. gxx0.093x 30 gxx x g3xx3 x g x x Fgure 8. Desgn of pressure vessel problem. Frst of all, let us consder the range of the four decson varables as follows: Regon I : x, x ; 0 x3, 00 Many researchers have obtaned dfferent results by varous algorthms n ths regon. For example, n early papers, an augmented Lagrangan multpler method ([6] 994), genetc adaptve search ([6] 997), and a branch and bound approach ([8] 988) have been come up wth to conduct that problem. In recent years, n order to dspose of pressure vessel desgn problem fundamentally, many methods were rapdly blossomed, ncludng a GA-based coevoluton model ([4] 000), a feasblty-based tournament selecton scheme ([5] 00), cuckoo search ([0] 003), a co-evolutonary partcle swarm optmzaton ([] 007), and an evoluton strategy ([3] 008) and so on. Very recently, hybrd algorthm based on partcle swarm optmzaton wth passve congregaton ([7] 009), mproved ant colony optmzaton ([8] 00), and quantum behaved PSO ([3]00) etc. have been employed nto that problem. The best soluton acqured by FOA n the present paper s f x , the relevant decson varable x ( x, x, x3, ) ( , , , ), and the correspondng constrans g x), g ( ) ( 3 4 x ( , , , ).The all solutons from dfferent approaches n Regon I about ths problem have been compared n Table II, whch can be concluded that FOA s of superor searchng method for ths problem. DOI 0.503/IJSSST.a ISSN: x onlne, prnt

5 TABLE II. COMPARISON OF THE BEST SOLUTION FOR PRESSURE VESSEL IN REGION I BY DIFFERENT METHODS. Method Desgn Varables x x x3 f ( x ) Sandgren ([8] 988) C.Zhang et al. ([30] 993) Kannan et al.([6] 994) K.Deb et al.([6] 997) Coello ([4] 000) Coello et al.([5] 00) X.H.Hu et al. ([4] 003) Gandom et al. ([0] 003) S.He et al. ([] 004) K.S.Lee et al. ([9] 005) Montes et al. ([4] 007) Q.He et al. ([] 007) Montes et al. ([3] 008) Cagnna et al.([] 008) A.Kaveh et al. ([7] 009) A.Kaveh et al. ([8] 00) L.S.Coelho ([3] 00) B.Akay et al. ([] 0) Present study InRegon I, due to the upper bound of s 00, at ths pont the fourth constrant condton s satsfed automatcally. In order to take all constrants nto account, the upper bound of s adjusted to 40 ([] 007). In the lght of ths, the regon s denoted as Regon II. Regon II : x, x ; 0 x3 00; 0 40 TABLE III. COMPARISON OF THE BEST SOLUTION FOR PRESSURE VESSEL IN REGION II BY DIFFERENT METHODS. Method Desgn Varables x x x3 f ( x ) Hedar et al. ([3] 006) Mahdav et al. ([] 007) Dmopoulos ([8] 007) Gandom et al. ([9] 0) Present study In ths scope, Hedar and Fukushma (006) [0], Dmopoulos (007) [], Mahdav (007) [8], and Gandom et al. (0) [4] have nvestgated the problem by multfarous methods. The best outcome found by FOA s f ( x) , where x x, x, x, ) ( 3 ( , , , ), and constrants g x), g ( ) ( 3 4 x ( , , , 5.660). Table III shows a comparson between the tradtonal methods and FOA, whch demonstrates that the solutons ganed by FOA are better than those of algorthms. DOI 0.503/IJSSST.a ISSN: x onlne, prnt

6 TABLE IV. STATISTICAL RESULTS OF DIFFERENT APPROACHES FOR PRESSURE VESSEL (NA MEANS NOT AVAILABLE). Method Best Mean Worst Std-dev Sandgren([8] 988) NA NA NA Kannan et al.([6] 994) NA NA NA K.Deb et al([6] 997) NA NA NA Coello ([4] 000) Coello et al ([5] 00) Gandom et al.([0] 003) S.He et al.([] 004) NA B.Akay et al.([] 0) NA 05 Montes et al.([3] 008) Kaveh et al.([8] 00) AKaveh et al.([7] 009) Coello ([3] 00) Q.He et al.([] 007) Cagnna et al.([] 008) NA.75 Present study Hedar et al.([3] 006) Mahdav et al([] 007) NA NA NA Dmopoulos ([8] 007) NA NA NA Gandom et al.([9] 0) Present study Accordng to the statstcal smulaton results, summed up n Table IV, whch can be seen that the average searchng ablty of FOA s better than those of other algorthms appled n the lterature from [8, 9, 3, ]. Moreover, the standard devaton of the outcome acqured by FOA s smaller comparatvely after runnng 50 trals wth MATLAB ndependently. We can see that the stablty of FOA s better than those of other approaches by comparng the standard devaton. V. CONCLUSION Ths paper ntroduces FOA, whch ndcates the better searchng ablty of FOA such as effectveness and robustness than PSO and QAFSA va three typcal functons smulaton experment. Then FOA s appled to solve a desgn of pressure vessel optmzaton problem. Snce the procedure of FOA s relatvely uncomplcated, t s effortless to understand. Furthermore, FOA s more convenent and sutable to deal wth some engneerng desgn optmzaton problems. However, there are some shortcomngs of that algorthm, for example, the decson value of the taste concentraton s the recprocal of the dstance, whch has lmted that varables should be postve values, the scope of FOA s narrowed by ths lmtaton, and that may be mproved n further study. ACKNOWLEDGEMENTS The research was supported by the Natonal Natural Scence foundaton of Chna (No ), Hube Provnce Key Laboratory of Systems Scence n Metallurgcal Process of Chna (No. Z040), and the Natural Scence Foundaton of Hube Provnce, Chna under Grants 03CFA3. REFERENCES [] B. Akay, and D. Karaboga, Artfcal bee colony algorthm for large-scale problems and engneerng desgn optmzaton, Journal of Intellgent Manufacturng, vol. 3, pp.00 04, 0. [] L. C. Cagnna, S. C. Esquvel, and C. A. C. Coello, Solvng engneerng optmzaton problems wth the smple constraned partcle swarm optmzer, Informatca, vol.3, pp.39 36, 008. [3] L. S. Coelho, Gaussan quantum-behaved partcle swarm optmzaton approaches for constraned engneerng desgn problems, Expert Systems wth Applcatons, vol.37, pp , 00. [4] C. A. C. Coello, Use of a self -adaptve penalty approach for engneerng optmzaton problems, Computers n Industry, vol.4, pp.3 7, 000. [5] C. A. C. Coello, and E. M. Montes, Constrant- handlng n genetc algorthms through the use of domnance-based tournament selecton, Advanced Engneerng Informatcs, vol.6, pp.93 03, 00. [6] K. Deb, and A. S. Gene, A robust optmal desgn technque for mechancal component desgn, (Eds. D. Dasgupta, Z. Mchalewcz), Evolutonary Algorthms n Engneerng Applcatons, Sprnger, Berln, pp , 997. [7] K. Deb, and M. Goyal, A combned genetc adaptve search (GeneAS) for engneerng desgn, Computer Scence and Informatcs, vol.6, pp , 986. [8] G. G. Dmopoulos, Mxed-varable engneerng optmzaton based on evolutonary and socalmetaphors, Computer Methods n Appled Mechancs and Engneerng, vol.96, pp , 007. [9] A. H. Gandom, X. S. Yang, and A. H. Alav, Mxed varable structural optmzaton usng frefly algorthm, Computers & Structures, vol.89, pp , 0. [0] A. H. Gandom, X. S. Yang, and A. Alav, Cuckoo search algorthm: a meta heurstc approach to solve structural optmzaton problems, Engneerng wth Computers, vol.9, pp.7 35, 003. [] Q. He, and L. Wang, An effectve co - evolutonary partcle swarm optmzaton for constraned engneerng desgn problems, Engneerng Applcatons of Artfcal Intellgence, vol.0, pp.89 99, 007. DOI 0.503/IJSSST.a ISSN: x onlne, prnt

7 [] S. He, E. Prempan, and Q. H. Wu, An mproved partcle swarm optmzer for mechancal desgn optmzaton problems, Engneerng Optmzaton, vol.36, pp , 004. [3] A. R. Hedar, and M. Fukushma, Dervatve - free flter smulated annealng method for constraned contnuous global optmzaton, Journal of Global Optmzaton, vol.35, pp.5 549, 006. [4] X. H. Hu, R. C. Eberhart, and Y. H. Sh, Engneerng optmzaton wth partcle swarm, Proceedngs of the 003 IEEE Swarm Intellgence Symposum, pp.53 57, 003. [5] Y. Hu, T. S. Du, J. H. Wu, X. H. Lu, D. Y. L, and W. W. LI. Solvng Hgh Dmensonal and Complex Non-convex Programmng Based on Improved Quantum Artfcal Fsh Algorthm. Open Automaton and Control Systems Journal, vol.6, pp.9-37, 04. [6] B. K. Kannan, and S. N. Kramer, An augmented lagrange multpler based method for mxed nteger dscrete contnuous optmzaton and ts applcatons to mechancal desgn, Transactons of the ASME, Journal of Mechancal Desgn, vol.6, pp.405 4, 994. [7] A. Kaveh, and S. Talatahar, Engneerng optmzaton wth hybrd partcle swarm and ant colony optmzaton, Asan journal of cvl engneerng (buldng and housng), vol.0, pp.6 68, 009. [8] A. Kaveh, and S. Talatahar, An mproved ant colony optmzaton for constraned engneerng desgn problems, Engneerng Computatons, vol.7, pp.55 8, 00. [9] K. S. Lee, and Z. W. Geem, A new meta-heurstc algorthm for contnuous engneerng optmzaton: harmony search theory and practce, Computer Methods n Appled Mechancs and Engneerng, vol.94, pp , 005. [0] Hong-ze L, and Sen Guo, A hybrd annual power load forecastng model based on generalzed regresson neural network wth frut fly optmzaton algorthm, Knowledge-Based System, vol.37, pp , 03. [] M. Mahdav, M. Fesanghary, and E. Damangr, An mproved harmony search algorthm for solvng optmzaton problems, Appled Mathematcs and Computaton, vol.88, pp , 007. [] Z. Mchalewcz, Genetc Algorthms + Data Structures = Evoluton Programs, Sprnger -Verlag, Berln, 994. [3] E. M. Montes, and C. A. C. Coello, An emprcal study about the usefulness of evoluton strateges to solve constraned optmzaton problems, Internatonal Journal of General Systems, vol.37, pp , 008. [4] E. M. Montes, C. A. C. Coello, J. V. Reyes, and L. M. Davla, Multple tral vectors n dfferental evoluton for engneerng desgn, Engneerng Optmzaton, 39, , 007. [5] Quan-Ke Pan, Hong-Yan Sang, An mproved frut fly optmzaton algorthm for contnuous functon optmzaton problems, Knowledge-Based System, vol.6, pp.69 83, 04. [6] Wen-Tsao Pan, A new Frut Fly Optmzaton Algorthm: Takng the fnancal dstress model as an example, Knowledge- Based System, vol.6, pp.69 74, 0. [7] K. H. Raj, R. S. Sharma, G. S. Mshra, A. Dua, and C. Patvardhan, Anevolutonary computatonal technque for constraned optmsaton n engneerng desgn, Journal of the Insttuton of Engneers Inda Part Me Mechancal Engneerng Dvson, vol.86, pp. 8, 005. [8] E. Sandgren, Nonlnear nteger and dscrete programmng n mechancal desgn, Proceedngs of the ASME Desgn Technology Conference, F.L. Kssmne, vol.95 05, 988. [9] J. Tsa, Global optmzaton of nonlnear fractonal programmng problems n engneerng desgn, Engneerng Optmzaton, vol.37, pp , 005. [30] C. Zhang, and H. P. Wang, Mxed-dscrete nonlnear optmzaton wth smulated annealng, Engneerng Optmzaton, vol., pp.77 9, 993. [3] K. Zhang, T. S. Du, T. B. Wang, W. Q. Lu, Optmzaton model of power system unt commtment allocaton problem consderng the value-pont effect and ts smulaton analyss, Computer Modellng and New Technologes, vol.8, pp.6-3, 04. DOI 0.503/IJSSST.a ISSN: x onlne, prnt