Proceedngs of the 5th WSEAS Int. Conf. on Instrumentaton, Measurement, Crcuts and Systems, Hangzhou, Chna, Aprl 6-8, 26 (pp56-6) Mult-objectve Desgn Optmzaton of MCM Placement Chng-Ma Ko ab, Yu-Jung Huang a, Shen-L Fu a, MeHu Guo c a Department of Electronc Engneerng, I-Shou Unversty, Kaohsung, Tawan, ROC. b Communcaton & Informaton Servce Dvson, Open System Department, SYSWARE Co., Tape, Tawan, ROC c Departmrnt of Appled Mathematc Natonal Sun Yat-sen Unv., Kaohsung, Tawan Abstract :- Placement of multple des on an MCM substrate s a dffcult combnatoral task n whch multple crtera need to be consdered smultaneously to obtan a true mult-objectve optmzaton. Our desgn methodologes consder mult-objectve component placement based on thermal relablty, routng length and chp area crtera for mult-chp module. The purpose of the mult-objectve optmzaton placement s to enhance the system performance, relablty and reduce the substrate area by obtanng an optmal cost durng mult-chp module placement desgn phase. For relablty consderatons, the desgn methodology focuses on the placement of the power dsspatng chps to acheve unform thermal dstrbuton. For route-ablty consderaton, the total wre length mnmzaton s estmated by Stener tree approxmaton method. For substrate area consderaton, the area s estmated by mnmum area contans all chps. The cost functon s formulated by the weght sum calculaton. For desgn flexblty, dfferent weghts can be assgned dependng on desgner s consderatons. Varous methods ncludng teraton, smulated annealng and generc approxmaton are appled to solve the placement solutons. An auto generated optmal placement layout based on the analytcal soluton s presented. Key-Words: -Mult-chp module, placement, mult-objectve, smulate annealng algorthm, generc algorthm, fuzzy thermal placement algorthm. Introducton Optmal electronc component placement studes have tradtonally focused on sngle objectve optmzaton [,2]. It has manly used a sngle objectve of mnmzng the overall wre length or mnmzng the overall heat generaton or mnmzng the overall tme delay n ts functonng. Osteman et al. [3] developed a force drected placement methodology to solve coupled relablty and routablty placement procedure for arrangng electronc components on a convectvely cooled two-dmensonal workspace. Quepo et al. [4] ntroduced a genetc algorthm for the search of optmal or near optmal placement solutons on prnted wrng boards. Deb et al [5] use evolutonary algorthms to solve a two-objectve optmzaton problem ncludng mnmzng the overall wre length and mnmzng the falure rate of the board. A fuzzy analytcal model for the optmal component placement on the multchp module (MCM) substrate s presented n [6, 7] There are many factors to consder n selectng the correct MCM package desgn. In a conventonal layout flow, placement studes have focused on sngle objectve optmzaton. For nstance, placement and routng are optmzed for tmng, wth lttle or no consderaton for power, routablty or sgnal ntegrty. Some of the mportant goals for the MCM placement desgns are: even heat dstrbuton, mnmzaton of the total substrate area, the total routng length, and the number of routng layers. Therefore, the desgn must consder the combned cost of the heat dsspaton, area, routng length etc., not just each cost n solaton. In ths paper, we focus on the mult-objectve placement optmzaton studes. These objectves are routng length, substrate area, and thermal dstrbuton. The man desgn ssue addressed s on the multobjectve optmzaton placement for relablty, routeablty and substrate area. The weghted sum approach s used to formulate the placement cost functon. The optmum solutons of the cost functon are obtaned
Proceedngs of the 5th WSEAS Int. Conf. on Instrumentaton, Measurement, Crcuts and Systems, Hangzhou, Chna, Aprl 6-8, 26 (pp56-6) based on smulated annealng, and generc approxmaton algorthms. The rest of ths paper s organzed as follows. Secton 2 provdes problem formulaton. Secton 3 explans dfferent soluton technques. Secton 4 presents expermental results. Secton 5 concludes our paper. 2. Problem Formulaton The placement problem can be stated as follows: Gven a set of modules (cells) M = m, m, Lm }, a { 2 n set of sgnals S = { s, s2, Lsk}, and a set of power P = { p, p2, L pn}. Each module m M s assocated wth a set of sgnals S, where S. Also each sgnal s S s assocated wth a set of modules M, where M m s S }. s s called a sgnal net. The power set ndcates the assocate power value for the correspondng module n module set. Placement conssts of assgnng each module m M to a unque locaton such that a gven cost functon s optmzed and constrants are satsfed. The objectve of an mult-chp module (MCM) placement s to mnmze a weghted sum of some optmzaton crtera subject to constrants on others. E.g., f k crtera are consdered, the objectve s to mnmze the sngle-valued cost functon C j = = w f s. t. c Sc s = { j c M j s = j +, K, k : f F () for some j, j k. Here f s the cost of the soluton wth respect to the th crteron. w and F s are user-defned weghts and bounds, respectvely. Three objectves representng the general performance of a placement system are consdered n ths study. There are mnmzng substrate area, mnmzng routng length, and mnmzng thermal gradent. The cost due to these objectves can be defned as follows: a. Substrate Area= area that contans all the des b. Routng length=entre Stener tree lengths connect all the de c. Power cost=thermal dstrbuton on the entre substrate The weght sum approach s appled n order to combne these three objectves. The optmzed soluton f s to obtan a mnmum sum of weghts of the form gven as: k mn w f (2) = where w are the weghtng coeffcents representng the relatve mportance of the objectves. It s usually assumed that k w = (3) = The fuzzy thermal placement algorthm [8] s used to estmate the power cost. The average repulsve force among the des can be expressed as the followng formula: F j P = (4) N N ( ) Where F j s the repulsve force based on fuzzy Z functon [7, 8], N s the total number of des. 3. Soluton Methodology The basc consderaton to approach the multobjectve approxmaton optmze soluton s to make the fnal cost as low as possble. Due to dfferent purposes and requrements, varous weght settngs can be assgned to the three objectves to calculate the total combned cost of thermal, power and area. Varous methods ncludng teraton, smulated annealng, convergence perturbaton and genetc algorthm are appled to solve the placement solutons. Smulated annealng method [9] could escape from the local mnmum soluton to approach the global mnmum soluton. Durng teraton process, the new readng wll be always accepted as long as the new energy value E becomes smaller. If the value ncreased, then t could be accepted n a certan probablty. The accepted probablty P can be determned from the followng equaton. P = exp (E next E current ) Temperature f (5) The temperature s set to hgh value at the begnnng such that t could escape from the local mnmum soluton. It then contnues to cool down the temperature to approach the best soluton. The pseudo code of the smulated annealng algorthm s lsted n Table.
Proceedngs of the 5th WSEAS Int. Conf. on Instrumentaton, Measurement, Crcuts and Systems, Hangzhou, Chna, Aprl 6-8, 26 (pp56-6) Table Smulated annealng algorthm for MCM Placement Algorthm Smulated_annealng(S, T, T,, M); (*S s the ntal soluton*) (*T s the ntal temperature *) (*T s the fnal temperature *) (* s the coolng rate *) (*M represents the tme untl the next parameter update *) Begn T=T; S=S; Iteraton=M; NewS=RandomPoston(S); ρh=cost(news)-cost(s); f ((ρh<) or (random<e-ρh/t) then S=NewS; Iteraton=Iteraton+; untl (Iteraton=) T= x T; untl (T<T) End. For teraton technque, the soluton methodology start to randomly exchange the de poston and set a loop to approach the optmze soluton. Wth random choce of the new poston, t s dffcult to converge to the global mnmum soluton. If we could randomly change the de poston at the begnnng and then lmt the new poston to the area around prevous poston, then t could be easer to get the better global soluton. To acheve ths goal, we apply small perturbaton approxmaton method to lmt the area of random poston. The advantage of the smulated annealng method s that t can escape from the local mnmum. But t would not be easy to get the global mnmum soluton f we randomly set the new poston. To rapdly get the global mnmum soluton, we set the convergng poston nstead of random poston. To combne the advantages of smulated annealng method and small perturbaton method, we develop the smulated annealng wth small perturbaton method. It could not only escape the local mnmum soluton n the begnnng, but also can fast to get closer to the soluton. The pseudo code of the smulated annealng wth small perturbaton program s lsted n Table 2. Table 2 Smulated annealng wth small perturbaton algorthm for MCM Placement Smulated_Annealng_wth_Small_Perturbaton(S, T, T,, M); (*S s the ntal soluton*) (*T s the ntal temperature *) (*T s the fnal temperature *) (* s the coolng rate *) (*M represents the tme untl the next parameter update *) Begn T=T; S=S; Iteraton=M; NewS=LmtePoston(M, T); ρh=cost(news)-cost(s); f ((ρh<) or (random<e-ρh/t) then S=NewS; Iteraton=Iteraton+; untl (Iteraton=) T= x T; untl (T<T) End. The smulaton results for random choce of the swtch poston and small perturbaton of the chosen poston s shown on Fgure, respectvely. The blue damonds are random choce of the swtch poston and the red rectangles are small perturbaton of the chosen poston. Fgure Comparson wth random choce and small perturbaton method Generc Algorthm s used to exchange the sequence of the de postons ncludng crossover, mutaton and reproducton []. Crossover s an operaton where two parent sequences exchange parts of ther correspondng chromosomes. We set two knds of ntal postons to be parent chromosomes as shown on Fgs 2, 3. The
Proceedngs of the 5th WSEAS Int. Conf. on Instrumentaton, Measurement, Crcuts and Systems, Hangzhou, Chna, Aprl 6-8, 26 (pp56-6) parent chromosome poston descrbes the best power dsspaton. And the parent chromosome poston 2 descrbes the smallest area requrement. 5 7 8 3 2 25 2 6 9 23 5 22 8 7 3 24 2 4 9 6 4 2 Fgure 2 Generc parent chromosome poston 3 6 8 2 4 8 3 2 5 7 9 5 22 2 4 6 24 7 9 2 23 25. Table 3 Generc Algorthm for MCM Placement Algorthm Generc(S,M, N); (*S s the ntal soluton*) (*N represents the tme for chromosome exchange*) (*M represents the tme for rearrange de poston*) Intalze de poston; Begn For N= to N S=S; For M= to M NewS[] = Crossover sequence; NewS[2] = Crossover sequence; NewS[3] = Crossover sequence; NewS[4] = Crossover sequence; NewS[5] = Crossover sequence; NewS[6] = Crossover sequence; NewS[7] = Reproducton sequence; NewS[8] = Mutaton sequence; For L= to 8 f (NewS[L] < S) then S=NewS[L]; next L next M Rearrange de poston; next N; End. Fgure 3 Generc parent chromosome poston 2 For example, two parent sequences (23456) and (65432) are selected accordng to pre-defned crossover probabltes. A crossover pont s randomly selected, and then a new sequence (65423) s created. Mutaton s an operaton that randomly changes the sequence. We also use the reproducton to reverse the sequence of parent one. The pseudo code of the generc algorthm program s lsted n Table 3. 4. Experment Results In ths secton, the smulaton results of the multobjectve component placement are presented. Varous weghtng factors are used to observe the multobjectve component placement. Dependng on the values of the selected weghtng factor, the placement procedure allows the desgner to place components for optmal relablty, area, or routablty. In addton, the placement procedure can be used to observe the tradeoffs relatonshp among relablty, area, and routablty. We set the nput condton as shown on Table 4. In ths case study, there are 2 des wth dfferent length, wdth and power, and three dfferent nets to nterconnect these chps. Fgs 4, 5 shows the results obtaned by smulated annealng and smulated annealng wth small perturbaton methods for the weghtng factors of Power : Routng : Area=::2, respectvely. In the fgures, the blue damonds are the power costs curve, the purple rectangles are the routng costs curve, the green trangles are the area costs curve and the red Xs denote the total costs curve. The horzontal axs represents the smulaton tme, and the vertcal axs represents the normalzed total cost value. The tradtonal loop teraton method appled n ths case study cannot always reach the global mnmum solutons. However, wth smulate annealng method, t can obtan the approxmaton global mnmum soluton. The soluton wth loop teraton wth small perturbaton method wll be more close to the mnmum soluton.
Proceedngs of the 5th WSEAS Int. Conf. on Instrumentaton, Measurement, Crcuts and Systems, Hangzhou, Chna, Aprl 6-8, 26 (pp56-6) Table 4 Input condton for program smulaton Fgs 6, 7, 8, 9 show the results of auto generated optmal placements layout based on the analytcal solutons for weghtng factors to ::, ::, :: and ::. Fgure 6 Auto generated optmal placement layout wth power:routng:area=:: Fgure 7 Auto generated optmal placement wth power:routng:area=:: Fgure 4 Results of smulated annealng method Fgure 8 Auto generated optmal placement layout wth power:routng:area=:: Fgure 5 Results of smulated annealng wth small purturbaton Fgure 9 Auto generated optmal placement layout wth power:routng:area=::
Proceedngs of the 5th WSEAS Int. Conf. on Instrumentaton, Measurement, Crcuts and Systems, Hangzhou, Chna, Aprl 6-8, 26 (pp56-6) The fnal costs for these four dfferent weghtng factors are summarzed on Table 5. Table 5 Summary of four dfferent weght settngs The smulaton results obtaned from generc algorthm for power, wre, area, and total cost under varous weghtng assgnment are shown n Fgure. As shown n the power cost fgure, when the power weghtng factor s hgher, then the fnal power cost becomes lower. The area and wre cost fgures have the smlar effect. Note that power and area weght settngs have the opposte effects on cost evaluaton..5.5 Power Cost Area Cost.5.5 Wre Cost Total Cost Fgure Generc method wth dfferent weght settngs 5. Conclusons In ths paper, we formulated the mult-object cost functon for MCM evaluaton. The objectve functon of the optmzaton problem conssts of cost estmates for power dsspaton, routng length, and substrate area. Dependng on the selected weghtng factor, dfferent placement confguratons based on smulated annealng approach and generc algorthm can be obtaned. An auto generated optmal placement layout based on the analytcal solutons can help desgners to make tradeoffs among mult-objectve selectons for MCM desgn consderatons. Acknowledgments Ths research was partally supported by Natonal Scence Councl, R. O. C., under Grant NSC94-225-E- 24-6. References [.] D. Dancer and M. Pecht, Component placement optmzaton for convectvely cooled electronc, IEEE Trans. S st., Man, Cybern., vol. 8, pp. 49-55,989 [2.] R. Elas, T. Elpern and A. Bar-Cohen, Monte Carlo thermal optmzaton of populated prnted crcut board, IEEE Trans. Conp., Hybrds, Manufact. Technol., vol. 3, pp 953-96, Dec. 99 [3.] M. Osterman and M. Pecht, placement for relablty and routablty of convectvely cooled PWBs, IEEE Trans. Computer-Aded Desgn, vol. 9, No. 7, pp. 734-744, July 99 [4.] Quepo, N.V., Humphrey, A.C., and Ortega, A., Multobjectve Optmal Placement of Convectvely Cooled Electronc Components on Prnted Wrng Boards, IEEE Transactons on Components, Packagng and Manufacturng Technology -Part (A)` v.2, No.. March, pp. 42 53, 998. [5.] Deb, K.; Jan, P.; Gupta, N.K.; Maj, H.K., Multobjectve placement of electronc components usng evolutonary algorthms, IEEE Transactons on Components and Packagng Technologes, Volume 27, Issue 3, Sept., pp. 8 492, 24 [6.] Mehu Guo and Yu-Jung Huang, Multobjectve optmal MCM placement based on fuzzy approach, 6th World Congress of the Bernoull Socety for Mathematcal Statstcs and Probablty and 67th Annual Meetng of the Insttute of Mathematcal Statstcs, Barcelona, Spansh, July 26-3, pp. 7 8, 24 [7.] Huang, Yu-Jun, Guo, Mehu, and Fu, Shen-L, Relablty and routablty consderaton for MCM placement, Mcroelectroncs Relablty 42, pp. 83 9, 22 [8.] Huang, Yu-Jung, Fu, Shen-L, Jen, Sun-Lon, and Guo, Mehu, Fuzzy thermal modelng for MCM placement, Mcroelectroncs Journal 32, pp. 863 868, 2 [9.] S. Krkpatrck, C. D. Gelatt, Jr., and M. P. Vecch, Optmzaton by smulated annealng, Scence, Vol. 22, pp. 67-68, 983 [.] Jeffery K. Cochran., Shwu-Mn Horng. and John W. Fowler, A Mult-Populaton Genetc Algorthm to Solve Mult-Objectve Schedulng Problems for Parallel Machnes, Computers and Operatons Research, v.3 n.7, pp.87-2, June 23.