製品系列統合化設計における最適性と最適化法に関する Title研究 ( 第 3 報, システム属性とモジュール組合せの同時最適化法 ) Citation 日本機械学会論文集 C 編. 68(668) P.1329-P

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1 製品系列統合化設計における最適性と最適化法に関する Ttle研究 ( 第 3 報, システム属性とモジュール組合せの同時最適化法 ) Author(s) 藤田, 喜久雄 ; 吉田, 寛子 Ctaton 日本機械学会論文集 C 編. 68(668) P.1329-P.1337 Issue Date Text Verson publsher URL DOI Rghts Osaka Unversty

2 (C ) (2002-4) 1329 No ( 3 ) 1, 2 Optmzaton Methodologes for Product Varety Desgn (3rd Report: Smultaneous Optmzaton Method for Module Attrbutes and Module Combnaton) Kkuo FUJITA 3 and Hroko YOSHIDA 3 Department of Computer-Controlled Mechancal Systems, Osaka Unversty, 2-1 Yamadaoka, Suta, Osaka , Japan Ths research dscusses and develops the optmzaton methodologes for product varety desgn, whch means the challenge to smultaneously desgn multple products. Followng the succeedng reports, ths paper proposes a smultaneous optmzaton method for both module combnaton and module attrbutes of multple products, whch s vewed as the thrd class of product varety optmzaton n the frst report. Whle t s the most dffcult one among the classes, t ncludes a herarchcal structure on the desgn optmalty, whch s composed of commonalty and smlarty pattern, smlarty drectons, module attrbutes, under a systematc understandng of product varety representaton. Based on such structure, ths paper confgures an optmzaton method for both module combnaton and module attrbutes across multple products. The optmzaton method hybrdzes a genetc algorthm, a mxed-nteger programmng method wth a branch-and-bound technque, and a constraned nonlnear programmng method,.e., a successve quadratc programmng method. In ts optmzaton process, the frst optmzes the combnatoral pattern, the second optmzes the well-structured drectons, and the thrd optmzes the contnuous module attrbutes under the others. Fnally t s appled to the smultaneous desgn problem of multple arplanes to demonstrate ts valdty and effectveness. Key Words : Product Varety Desgn, Desgn Optmzaton, Desgn Engneerng, Hybrd Genetc Algorthm, Mxed-Integer Programmng 1 (1) ( ) 1 (2) , ( ). 2 ( ) ( ) ( ). Emal: futa@mech.eng.osaka-u.ac.p Class I 2 (3) Class II 1 Class III 1 (2) Class III 2 3

3 ( 3 ) (2) 1 (2) Class I Class II Class III Class I Class II Class III (2) Class I Class III 1 (2) Class III P 1, P 2,, P n P n P P n M M 1, M 2,, M nm P M m ( = 1, 2,, n P ; = 1, 2,, n M ) m [ ] T x = x,1, x,2,, x,n A M n A P x [ ] z = x1t, x T 2,, x T T nm [ ] z 1T, z 2T,, z n T P T ( ) 2 x x,1 M P 1 P 2 m 1 m 2 x 2,1 > x 1,1 (1) x 2,k = x 1,k (k = 2, 3,, n A ) (2) P 1 P 2 2 M x 2,k = x 1,k (k = 1, 2,, n A ) (3) (2) 2

4 ( 3 ) 1331 P z Feasble(s ) ( = 1, 2,, n P ) (4) s P Feasble( ) s z 2.5 z ( = 1, 2,, n P ) (2) (2) Class III 2.3 P C D C D (z ) = C D (x ) = n M =1 C D (x ) (5) α D W ( β D W W Base W Base ) + γ D C Base D 0 α D, β D, γ D W m m m Base ( ) (6) m Base (6) 1 m (6) 2 m m Base m Base (6) 1 m (6) 3 C F W x,1 (6) (6) C F 2.7 Class III z ( = 1, 2,, n P ) (4) (1), (2), (3) Class III m 1 m 2 m 2 m 1 (1) 3 ()

5 ( 3 ) 1332 alty and smlarty pattern ty drectons Module attrbutes Genetc algorthm Branch-and-bound technque Successve quaduratc Start Start programmng No Generatng an ntal set of solutons Generatng a new set of solutons Selectng pars of solutons Crossover operatons Mutaton operatons Evaluatng new solutons, respectvely Converged? Yes End Translator of a GA strng to a commonalty and smlarty pattern for consstency Intalzng nodes Selectng a node Solvng a relaxed optmzaton problem on the selected node Boundng nodes Optmal soluton? No Branchng nodes End Yes No Start Assumng an ntal soluton Solvng an assocated quaduratc programmng problem Solvng a lne search problem Converged? Yes End Fg. 1 Outlne of optmzaton method () () () () () () () 0-1 () () () () () (GA) (4) () (5) () 2 (SQP) (6) GA SQP GA SQP GA 3 Class III 3.3 GA () () () 0 m 1 m 2 ξ 1 2 = 1 m 1 m 2 2 m 1 m 2 ( 1 = 1, 2,, n P ; 2 = 1 + 1, 1 + 2,, n P ; = 1, 2,, n M ) (7) ξ 1 2 n MnP C 2

6 ( 3 ) 1333 () ξ 1 2 = 2 ξ 1 3 = 2 ξ ξ 2 3 = 2 ξ 2 3 = 2 ξ 1 2 = 2 ξ ξ 1 3 = 2 ξ 1 3 = 2 ξ 2 3 = 2 ξ ξ 1 2 = 2 (( ) ( )) ξ ξ 1 3 = 1 ξ 1 2 = 1 ξ ξ 2 3 = 0 ξ 2 3 = 1 (( ) ( )) ξ ξ 1 2 = 1 ξ 2 3 = 1 ξ ξ 1 3 = 0 ξ 1 3 = 1 (( ) ( )) ξ ξ 2 3 = 1 ξ 1 3 = 1 ξ ξ 1 2 = 0 ξ 1 2 = 1 1 < 2 < 3. M P 1, P 2,, P P α, β { 1, 2,, P }, α < β ξ α β = 2 P 1, P 2,, P P α, β { 1, 2,, P }, α < β ξ α β = 1 P α β ; α < β n P ξ α β = 0 γ ;1 γ < α ξ γ α = GA ξ 1 2, ( 1 = 1, 2,, n P ; 2 = 1 + 1, 1 + 2,, n P ; = 1, 2,, n M ) GA ξ ξ 1 2 ξ 1 2 ξ 1 2 {0, 1, 2, 3} 2 ξ 1 2 ξ 1 2 ξ 1 2 = 0 ξ 1 2 = 0 = 1 ξ 1 2 = 2 ξ 1 2 = 1 ξ 1 2 = 3 ξ 1 2 = 2 M ( = 1, 2,, n M ) ξ 1 2 ( 1 = 1, 2,, n P ; 2 = 1 + 1, 1 + 2,, n P ) ξ Ξ 2 n MnP C 2 1, 2 ξ GA ξ 1 2 ξ 1 2 = 0 m 1 m 2 ξ 1 2 = 1 m 1 m 2 ξ 1 2 ξ 1 2 Ξ Smple-GA (4) σ- 3.5 GA (1), (2), (3) (1) 2.5 (6) P 1, P 2,, P n M Base { 1, 2,, n } (1) x,1 > x Base,1 for { 1, 2,, n }, Base (8) (6)

7 ( 3 ) 1334 C D (x ) = α D W ( W β W Base D W Base for = Base + γ D ) C D Base (9) for { 1, 2,, n }, Base (8), (9) m, { 1, 2,, n } for = Base δ = (10) 0 for { 1, 2,, n }, Base δ = 1 (11) { 1, 2,, n } (8) ( x α,1 + 1 δ ) β x β,1 (12) for α { 1, 2,, n }, β { 1, 2,, n }, α β (8) x,1 x,1 x,1 x,1 x,1 (9) C D (x ) = α D W ( + k { 1, 2,, n } k δ β D W W k W k ) + γ D α D W k δ k for { 1, 2,, n } (13) (10) 0-1 (12), (2), (3) 0-1 δ δ {0, 1} (11) 2.5 (6) (13) 3.6 (10) δ {0, 1} 0 δ 1 SQP δ δ = 0 δ = 1 δ = 1 (11) δ = 1 ĩ { 1, 2,, n }, ĩ δĩ = (2) 2 5 Class III Boeng 777 (7) 1 (2) Torenbeek (9) Raymer (8) (6) α D = [Y/kg], β D = 0.5, γ D = 0.1, α F = [Y/m] ( α F = [Y/kgf] )

8 ( 3 ) 1335 Fuselage Man wng Tal wng Engne Case Case Case Fg. 2 Desgn condtons and optmal desgn patterns , 1600, 1800, 2100, 2400, 2600, 2800 [km], [ ] [ ] 2 P 1, P 2, P 3, P 4, P 5 1 P 1 P 2 P 3 P 4 P P 1 P P 5

9 ( 3 ) 1336 Table 1 Optmzaton result for Case 2 Totally ndependent desgn Optmzed desgn P 1 P 2 P 3 P 4 P 5 P 1 P 2 P 3 P 4 P 5 Wdth of man wng [ m ] Wdth of horzontal tal wng [ m ] Heght of vertal tal wng [ m ] Engne power [ kgf ] 5,564 6,257 6,934 7,323 8,201 7,269 7,269 7,269 7,269 8,139 Length of fuselage [ m ] Desgn and development cost [ 10 6 Y] 147, , , , , ,992 12,250 19,246 39,618 Faclty cost [ 10 6 Y] 106, , , , , ,788 10,199 7,164 21,115 Materal cost per unt [ 10 6 Y] 1,600 1,728 1,860 2,106 2,271 1,787 1,823 1,917 2, 2,276 Process cost for the frst unt (wthout learnng effect) [ 10 6 Y] 2,588 2,787 3,016 3,400 3,661 2,975 2,981 3,160 3,487 3,709 Process cost for the last unt n the 1st year [ 10 6 Y] 1,423 1,532 1,658 1,869 2,013 1,193 1,195 1,273 1,418 1,609 Process cost for the last unt n the 5th year [ 10 6 Y] 969 1,043 1,129 1,272 1, ,096 Process cost for the last unt n the 10th year [ 10 6 Y] ,081 1, Process cost for the last unt n the 15th year [ 10 6 Y] , Total process cost per model [ 10 6 Y] 515, , , , , , , , , ,027 Total manufacturng cost [ 10 6 Y] 4,551,478 3,300,099 Prce per unt [ 10 6 Y] 2,923 3,4 4,811 4,148 5,544 2,860 3,0 4,793 4,195 5,584 Total proft (obectve) [ 10 9 Y] , Class III 4 5C 2 = 40 ξ 1 2 GA 70 (2 4 5C 2 = C C 2 = 70 ) GA Sun Ultra 10 Workstaton (440MHz UltraSPARC-II) 14.5 GA 5 Class III 2 12 [ 10 Yen ] Obectve Functon best value mean value 5.0 worst value Generatons Fg. 3 Optmzaton hstory (2)(3) 3 (1),, C, Vol. 65, No. 629, (1999), pp (2), ( 1 ),

10 ( 3 ) 1337 C, Vol. 68, No. 666, (2002), pp (3), ( 2 ), C, Vol. 68, No. 666, (2002), pp (4) Goldberg, D. E., Genetc Algorthms n Search, Optmzaton and Machne Learnng, (1989), Addson- Wesley. (5), FORTRAN77, (1991),, pp (6) (5) pp (7) Sabbagh, K, Twenty Frst Century Jet The Makng and Marketng of The Boeng 777, (1996), Scrbner. (8) Raymer, D. P., Arcraft Desgn: A Conceptual Approach, (1989), AIAA. (9) Torenbeek, E., Synthess of Subsonc Arplane Desgn, (1976), Delft Unversty Press.

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