Using MILP Tools o Sudy R&D Porfolio Selecion Model for Large Insances in Public and Social Secor Usando Herramienas de MILP para Esudiar el Modelo de Selección de Porafolios R&D para Casos de Grandes Careras de Proyecos en el Secor Social Igor Livinchev, Fernando López Irarragorri, Miguel Maa Pérez and Elisa Schaeffer Posgraduae Program in Sysems Engineering Faculy of Mechanical and Elecrical Engineering, UANL San Nicolás de los Garza, Nuevo León, Mexico {igor,ferny,miguel,elisa}@yalma.fime.uanl.mx Aricle received on March 10, 2008; acceped on Augus 12, 2008 Absrac In his paper a mixed-ineger linear programming (MILP) model is sudied for he bi-obecive public R&D proecs porfolio problem. The proposed approach provides an accepable compromise beween he impac and he number of suppored proecs. Lagrangian relaxaion echniques are considered o ge easy compuable bounds for he obecives. The experimens show ha a soluion can be obained in less han a minue for insances comprising of up o 25,000 proec proposals. This brings significan improvemen o he previous approaches ha efficienly manage insances of a few hundred proecs. Keywords: R&D proecs porfolios, mixed ineger programming, muli-obecive opimizaion. Resumen En ese rabao se presena un modelo de programación lineal enera mixa (MILP) para el problema del porafolio de proyecos públicos R&D bi-obeivo. El enfoque propueso provee un puno medio enre el impaco y el número de los proyecos. Se consideran écnicas de relaación Lagrangiana para obener coas fácilmene calculables para los valores obeivos. La experimenación muesra que puede obenerse una solución en menos de un minuo incluso para casos de careras de más de 25,000 proyecos propuesos. Eso implica una meora significaiva a los enfoques previos que resuelven eficienemene casos con sólo algunos cienos de proyecos. Palabras clave: Porafolios de proyecos, programación enera mixa, opimización muliobeivo. 1 Inroducion Porfolio opimizaion problems are very well known and inensively sudied in capial invesmen, sock marke, and privae-secor R&D proec selecion, o menion a few. Somewha surprisingly, he public secor has no shown a similar ineres on his opic; he researches in he public-secor raher approach heir proec-selecion problems by simple heurisics. Typically, public R&D proecs can be considered as saisically independen wih very small (or zero) correlaion. Moreover, he amoun of resources sufficien o realize he proec is no known exacly and he budge is frequenly overesimaed by he proponen. In recen years various models and soluions o he R&D porfolio opimizaion problem were proposed [Hsu e al., 2003; Ringues e al., 2004] and corresponding decision suppor sysems were considered [Fernández e al., 2006; Summer and Heidenberger, 2003; Tian e al., 2005]. Bu o he bes of our knowledge, only he work of Fernández e al. (2004) and Fernández e al. (2006) and ha of Navarro (2005) propose mehods ha have a heoreical foundaion and robus heurisics. Those approaches are based on mahemaical decision heory, fuzzy logic, rough ses, evoluionary opimizaion, such differen ools incorporaed in a decision suppor sysem. However, he proposed echniques lack scalabiliy and only work wih medium-sized insances (wih a mos 400 proecs). Moreover, he exising approaches are direced owards he porfolio qualiy as he unique crierion. This can be accepable only if he decision maker provides reliable and perfecly consisen informaion on his/her preferences. In pracice, he informaion of preferences provided by he decision maker is generally rough and parially inconsisen. Livinchev and López (2008) and Livinchev e al. (2008) have shown ha by inroducing he quaniy
164 Igor Livinchev, Fernando López Irarragorri, Miguel Maa Pérez and Elisa Schaeffer of funded proecs as an addiional obecive in he porfolio-opimizaion model his issue can be resolved. When solving a muli-obecive porfolio-opimizaion model, one needs o opimize repeaedly he porfolio according o every obecive included. Hence he opimizaion mehod has o be highly efficien: even for large insances, he response ime should be very shor o allow he decision maker o be concenraed and o allow him/her o explore a larger fracion of he se of feasible porfolios. In his work we consider an effecive approach o he muli-obecive porfolio opimizaion for large insances arising in real world public secor (wih several housands of proec proposals). For example, housands proecs are considered every year by scienific foundaions such as he NSF in he Unied Saes and he CONACyT in Mexico. In conras o he work of Navarro (2005), presening a geneic algorihm o explore he space of possible porfolios, he proposed mehod is based on he mixed-ineger linear-programming (MILP). This model differs from he one proposed by Livinchev and López (2008) in he way he ineger variables are inroduced. Using binary variables we give here an exremal MILP characerizaion of he original nonlinear disconinuous obecive funcion. Lagrangian relaxaions are considered o obain simple compuable bounds for he opimal values. Bi-obecive formulaion aking ino accoun he quaniy of funded proecs is also sudied using MILP approach. 2 The porfolio-opimizaion problem in he public secor The proposed model is based on he normaive soluion approach proposed by Fernández e al. (2004), where a nonlinear preference model is consruced from he fuzzy generalizaion of he classic scheme of 0-1 programming and from he muli-aribue decision heory. For hundreds of variables, he complexiy of he nonlinear opimizaion problem is no manageable by radiional algorihms. I has been addressed wih geneic algorihms and neural neworks [Navarro, 2005], and more recenly by differenial evoluion [Casro, 2007]. Livinchev and López (2008) reformulae he problem as a muliobecive problem on a mixed-ineger linear-programming model. The original non-linear model can be saed, in marix form, as follows [Fernández e al., 2004; Livinchev and López, 2008; Livinchev e al., 2008]: max ( η, ϕ) such ha w c T T μ( δ) ϕ T δ η T p q d p h d δ m d δ M (1) where η and φ represen he obecives: quaniy of funded proecs and porfolio qualiy respecively. The oal amoun of funds available for disribuing among proecs is. The componens of he vecor δ ake binary values, depending on wheher a proec is funded (δ = 1) or no (δ = 0). The decision variables of he vecor d are he funding assignmens: componen d corresponds o he amoun of funding assigned o proec. The funcion μ is a fuzzy predicae ha models he level of funding of a proec. For he res of his paper, we say ha he funding of he proec is sufficien if d, being he amoun of funds assigned, belongs o a cerain inerval, m d M, where M is he amoun requesed by he proponen and m is he minimal funding wih which i is possible o carry ou proec. See Figure 1 for an illusraion of he fuzzy predicae. The componens of he vecors m and M are m and M, respecively. Each proec mus be assigned o exacly one secor (being an applicaion area, scienific discipline, ec.) The
Using MILP Tools o Sudy R&D Porfolio Selecion Model 165 division ino secors can incorporae organizaional srucure or he aspiraion o balance he porfolio among differen disciplines. The funding consrains for each secor are represened by he vecors p and p ha conain he minimum and he maximum budge (respecively) o be assigned o each of he secors. The impac measures of he proecs are he componens of he weigh vecor w. Informaion on how he proecs have been evaluaed is conained in heir impac measures. Thus, he decision variables are d, δ, while c, w, p, p, m, M, q, h, and are known coefficien vecors. μ 1 m M d Fig. 1. A fuzzy piecewise linear funcion o model sufficien funding of a proec The complexiy of solving his model lies in he non-lineariy and he disconinuous naure of boh he feasible region and he qualiy-characerizing obecive funcion. These unforunae feaures make he model very hard o opimize wih heurisic mehods and direcly rules ou he use of exac mehods. However, considering he problem srucure, i may be possible o divide he obecive funcion and he feasible region o less complex subproblems [Hooker, 2007; Jain and Grossman, 2001]. In he nex secion we presen a linear mixed-ineger represenaion of he non-linear disconinuous qualiy obecive funcion and ransform he original problem o a linear bi-obecive mixed ineger problem. 3 The mixed-ineger linear model Consider firs he problem (1) aking ino accoun only qualiy obecive. Bearing in mind Figure 1 we will consider he qualiy of he proec as a non-negaive funcion z(x) defined for all 0 x M, monoonously increasing for x [ mm, ] wih z(m) = 1, 0 < zm ( ) <1, and z(x) = 0 for 0 x < m. Respecively, expeced uiliy of he proec is defined as wz(x), where w > 0 is a known consan. The obecive is o maximize he sum of individual uiliies (porfolio qualiy) subec o limied funds. To formalize he problem we firs give an exremal presenaion of he nonlinear funcion zx ( ) 0 defined for x [0, M ] as follows: 0 for x< m zx ( ) =. (2) α + γx for m x M Here γ > 0, while α is free. In wha follows we se zm ( ) = 0.5 Combining his wih zm ( ) = 1 yields 1 γ = 2( M m) and M α = 1 2( M m ). Proposiion 1. For any fixed x [0, M ] he funcion z(x) defined in (2) coincides wih he opimal value of he
166 Igor Livinchev, Fernando López Irarragorri, Miguel Maa Pérez and Elisa Schaeffer following problem: z ( x) = maxz z αy+ γx, z y, x my, 0 z, y {0,1}. (3) Proof. a) Suppose ha 0 x < m. Then by x my we need o se y = 0. Hence by z y we have z 0, while z α y+ γ x yields z γ x wih γ x 0. Combining wih he las consrain in (3), we ge z = 0. b) Suppose ha m x M. By definiion of α, γ, for hese values of x we have 0.5 α + γ x 1. Consrain x my is fulfilled for any y {0,1}. If y = 0, hen z = 0 as before. For y = 1 he problem (3) reduces o max{ z z α + γ x} giving z = α + γ x > 0. Hence, for m x M, he opimal y = 1 and z ( x) = α + γ x as desired. Suppose we have J proecs, each characerized by is own vecor of parameers ( w, αγ,, m, M), = 1, 2, K, J. Based on Proposiion 1, maximizing he overall expeced uiliy subec o linear consrains Ax b on funding x = ( x, K, x, K, x ) can be saed as 1 J F = wz (4) max z α y + γ x, z y, x my, 0 z, y {0,1}, 0 x M = 1, 2, K, J (5) Ax b (6) By linear consrains (6) we can represen, for example, consrains for he overall funding of he proecs. If he proecs are grouped in cerain areas of specializaions, hen similarly we can sae consrains for he oal funding in a paricular area. Typically, he number of applican proecs is very large. Meanwhile, he number of proec areas (i.e., he secors ino which he proposals are divided) is relaively small. Tha is, he mixed-ineger problem defined by (4)- (6) has a large number of variables and relaively few linear consrains 6. In real decision-making processes he problem of Equaions 4-6 has o be solved repeaedly for differen values of original daa. So i is very desirable o have a fas mehod a leas o esimae he opimal value F. 4 Lagrangian Relaxaion and Bounds Mos large scale opimizaion problems exhibi a srucure ha can be exploied o consruc efficien soluion echniques. In one of he mos general and common forms of he srucure he consrains of he problem can be divided ino easy and complicaed. In oher words, he problem would be an easy problem if he complicaing consrains could be removed. One ypical example is a block-separable problem decomposing ino a number of smaller independen subproblems if he binding consrains could be relaxed. A well-known way o exploi his srucure is o form he Lagrangian relaxaion wih respec o complicaing consrains. Tha is, he complicaing consrains are relaxed and a penaly erm is added o he obecive funcion o discourage heir violaion. Typically, he penaly is a linear combinaion of corresponding slacks wih coefficiens called Lagrange mulipliers. The opimal value of he Lagrangian problem, considered for fixed mulipliers, provides
Using MILP Tools o Sudy R&D Porfolio Selecion Model 167 an upper bound (for maximizaion problem) for he original opimal obecive. The problem of finding he bes, i.e. bound minimizing Lagrange mulipliers, is called he Lagrangian dual. The lieraure on Lagrangian relaxaion is quie exensive. We refer here only o a few survey papers [Fisher, 1985; Rangioni, 2005; Guignard, 2003; Lemaréchal, 2001]. To derive an upper bound for F we will use he Lagrangian relaxaion, dualizing he consrains (6) wih mulipliers u 0, where he vecors u and b are dimensioned correspondingly. The Lagrangian problem, considered for fixed mulipliers, has he form { } Lu ( ) = ub+ max ( wz cx ) (7) subec o he consrains (5), where c c ( u) is h componen of he vecor A u. We have F Lu ( ) for all u 0. This Lagrangian problem decomposes ino J independen subproblems in variables ( x, yz, ) subproblems have he form l = max wz cx z αy+ γx, z y, x my, 0 z, y {0,1}, 0 x M,. The laer where we have omied he indices o simplify he noaion. Problem (8) is solved by inspecion. If y = 0, hen z = 0, and consrain z α y+ γ x is hen saisfied for all x 0. So he problem (8) becomes l y= 0 = max{ cx 0 x a} = max{0, ca}. For y = 1, he problem (8) resuls in l y= 1 = max wz cx z α + γx, 0 z 1 m x M, By he definiions of α and γ we have α + γ x 1 for m x M. Hence we may relax condiion z 1 since i follows from z α + γ x. Remaining consrains of he problem (9) form he polyhedron P = { x z α + γ x, z 0, m x M}, (8) (9) having four verices Vk, k = 1, K, 4, wih he following componens ( x, z ) k and obecive values : V 1 V 2 V 3 V 4 ( x, z ) k ( m,0) ( m, α + γ m) ( M,0) ( a, α + γ a) k l cm w( α + γ m) cm cm w( α + γ M) cm y= 1 k l y= 1
168 Igor Livinchev, Fernando López Irarragorri, Miguel Maa Pérez and Elisa Schaeffer 2 1 Noe ha by definiion of α, γ boh α + γm and α + γ M are posiive, such ha l y= 1 > l y= 1 and Thus he opimal obecive value of he problem (9) is 4 3 y= 1 > y= 1 l l. l y= 1 = max{ w( α + γm) cm, w( α + γm ) cm}. Since l = max{ l, l } we obain for he opimal value of he problem (8): y= 0 y= 1 l = max{0, w( α + γm) cm, w( α + γm ) cm}. (10) Consider now he Lagrangian dual problem o find he bes (he smalles) Lagrangian bound UB = min l ( u) + u b, u 0 where variables l ( u) u, l : is defined similar o (8) for each proec. This dual problem can be saed as a linear program in UB = min l + u b l w( α + γ m) cm, l w ( α + γ M ) c M, c = ( A u), u, l 0, wih F UB. Here ( A u ) is he h componen of he vecor A u and variables l are used for max in (10). Suppose now ha a number of sufficienly funded proecs is limied from below. As i was shown in Proposiion 1, y = 1 for x [ mm, ] and y = 0 for 0 x < m. So we consider he problem (4)-(6) wih he addiional consrain y p, where p is he minimal number of sufficienly funded proecs. Dualizing his consrain wih muliplier θ 0, we ge he corresponding Lagrangian problem { } Lu (, θ) = ub θ p+ max ( wz cx + θy ) subec o he consrains (5). This problem also decomposes ino J independen subproblems differen from he problem (8) only in he obecive: l = max wz cx +θ y. I can also be analyzed by inspecion, resuling in he following expression for is opimal obecive: l = max{0, w( α + γm) cm + θ, w( α + γ M ) cm + θ}.
Using MILP Tools o Sudy R&D Porfolio Selecion Model 169 Respecively, he dual Lagrangian problem becomes l l UB = min l + u b θ p w ( α + γ m ) c m + θ, w ( α + γ M ) c M + θ, c = ( A u), u, l, θ 0. As was shown above, he number of sufficienly funded proecs subec o consrains (5) can be presened by. Thus he bi-obecive problem o maximize he number of funded proecs and he porfolio qualiy can be y saed as a bi-obecive MILP: { wz y } max, subec o he consrains (5) and (6). Due o he non-convexiy of he feasible se o his problem, is efficien soluion se in general can no be fully deermined by parameerizing on [0,1] π he weighed-sum problem max π wz + (1 π) y subec o he consrains (5) and (6). ha is, here may exis efficien soluions ha can no be reached even if he complee parameerizaion in π is aemped (see, e.g., [Alves and Clímaco, 2007] and he references herein). Meanwhile, he parameerizaion of his weighed-sum problem provides efficien poins and can be useful o ge an iniial rough represenaion of he efficien se. Noe, ha he proposed Lagrangian echniques can also be applied o ge easily compuable bounds for he weighed-sum problem. The characerizaion of all efficien poins ypically consiss in inroducing addiional consrains ino he weighed-sum problem. Generally, hese consrains impose bounds on he obecive funcion values, which can be regarded as a paricularizaion of he general characerizaion provided by Soland (1979). The inroducion of bounds on he obecive funcion values enables he weighed-sum problem o compue all efficien soluions. Oher characerizaions based on reference poins can be defined, using, for example, he augmened weighed Tchebycheff program (see, e.g., [Alves and Clímaco, 2007] and he references herein). We do no consider here he full characerizaion of efficien poins for our bi-obecive MILP leaving his ineresing opic for our fuure research. In he nex secion we presen numerical resuls obained by parameerizing on π [0,1] he weighed-sum problem. For all problem insances he linear consrains Ax b represen he limi for he overall funding of he proecs, as well as he bounds for he oal funding in a paricular area. 5 Numerical experimens Five insances were considered wih 40, 400, 1,200, 10,000, and 25,000 proecs, while π was moved from zero o one wih he sep-size 0.01. The small insances were included for comparabiliy and were consruced based on he previous works of Navarro (2001) and Fernández e al. (2004). The larger insances wih 1,200, 10,000 and 25,000 proecs, were generaed wih an insance-generaion ool developed by Casro (2007). The ILOG CPLEX, version 9.0 was used as opimizaion ool. For all insances and values of π used, he run ime of CPLEX was below 20
170 Igor Livinchev, Fernando López Irarragorri, Miguel Maa Pérez and Elisa Schaeffer seconds on a four-processor SunFire server running he Solaris operaing sysem, also version 9.0. We compare our soluions wih previously repored resuls only for he insances wih 40 and 400 proecs, since hose mehods canno solve larger problems. For he 40-proec insance, using π = 1 for comparabiliy, we obained a 28-proec porfolio wih qualiy indicaor a 156.574, being very similar o ha obained by Navarro [2001]. Meanwhile, using π = 0.43 in our approach, he resuling porfolio qualiy was almos he same (154.704), bu wo more proecs were suppored as a 30-proec porfolio was obained. For he case of 400 proecs, we did no generae he insance idenical o ha used by Fernández e al. (2004) and hus no compared wih heir mehod in erms of porfolio qualiy. For larger insances we did no find references presening a mehod capable o handle such a number of proecs. Opimizing he weighed-sum problem for a fixed π akes less han 20 seconds for all problem insances. Thus, for our bi-obecive problem we could generae he Pareo fron of 40 non-dominaed soluions in a reasonable ime using 40 differen values of π. In Figure 2, he Pareo fron for he 10,000-proec insance is shown. For oher insances he shape of he curve behaved similarly. Such a shape usifies he use of he muli-obecive model as a ool o search for a compromise beween porfolio qualiy and he number of suppored proecs. 6 Conclusions Fig. 2. The Pareo fron of he 10,000-proec insance This paper presens a MILP model for he nonlinear muli-obecive porfolio opimizaion for public R&D proecs. The Lagrangian relaxaion is sudied o ge simple compuable bounds for he opimal obecive. The weighed-sum scalarizaion of he bi-obecive model is numerically esed for large and very large problem insances. In numerical ess only simple resricions for funding were considered. Meanwhile, Lagrangian bounds derived in he paper are valid for he general form of funding consrains. Sudying real problems wih more complicaed funding consrains is an ineresing area for a fuure research. The proposed model allows a decision maker o find a compromise beween he qualiy and he size of he R&D proecs porfolio. The experimens show ha opimizing insances wih up o 25,000 proecs akes less han a minue, which superiors significanly he exising soluions echniques capable o handle in a reasonable ime only up o 400 proecs. The fas opimizaion is very imporan for an ineracive decision-suppor sysem. This gives he decision maker an opporuniy o explore differen Pareo-opimal soluions and choose an accepable compromise beween he porfolio qualiy and he number of proecs suppored.
Using MILP Tools o Sudy R&D Porfolio Selecion Model 171 Acknowledgmens The work of he firs auhor was parially funded by CONACyT (gran number 61343) while F. López and E. Schaeffer were suppored by PROMEP (gran number 103,5/07/2523). References 1. Alves, M. and J. Clímaco, A review of ineracive mehods for muliobecive ineger and mixed-ineger programming, European Journal of Operaions Research, 180: 99-115 (2007). 2. Casro, M., Diseño de un sisema de sopore a la decision para la opimización de careras en organizaciones públicas, Maser's Thesis, Universidad Auónoma de Nuevo León, San Nicolás de los Garza, Mexico, 2007. 3. Fernández, E., F. López and J. Navarro, Decision suppor ools for R&D proec selecion in public organizaions, IAMOT 2004, Washingon, DC, USA, 2004. Available online a 4. Fernández, E., F. López, J. Navarro and A. Duare, Inelligen echniques for R&D proecs selecion in large social organizaions, Compuación y Sisemas, 10(1): 28-56 (2006). 5. Fisher, M. L., An aplicaion oriened guide o lagrangian relaxaion, Inerfaces, 15: 10--21 (1985). 6. Frangioni, A., Abou lagrangian mehods in ineger opimizaion, Annals of Operaions Research, 139: 163-- 169 (2005). 7. Guignard, M., Lagrangean relaxaion, TOP, 11(2): 151--228 (2003). 8. Hooker, J., Inegraed Mehods for Opimizaion. Inernaional Series in Operaions Research & Managemen Science, Vol. 100. Springer, 2007. 9. Hsu, Y-G., G-H. Tzeng and J.Z. Shyu, Fuzzy muliple crieria selecion of governmen-sponsored fronier echnology R&D proecs, R&D Managemen, (33)5: 539-551, 2003. 10. Jain, V. and I. Grossmann, Algorihms for Hybrid MILP/CP Models for a Class of Opimizaion Problems, INFORMS Journal on Compuing, 13(4): 258-276, 2001. 11. Klapka, J. and P. Pinos, Decision suppor sysem for mulicrierial R&D and informaion sysems proecs selecion, European Journal of Operaion Research, 140(2): 434-446, 2002. 12. Lemaréchal, C., Lagrangian relaxaion. In Compuaional Combinaorial Opimizaion, edied by M., Junger and D. Naddef, pp.115-160. Springer Verlag, 2001. 13. Livinchev, I., and F. López, An ineracive algorihm for porfolio bi-crieria opimizaion of R&D proecs in public organizaions, Journal of Compuer and Sysems Sciences Inernaional, (47)1: 25--32, 2008. 14. Livinchev, I., F. López, A. Alvarez and E. Fernández, Large scale public R&D porfolio selecion by maximizing a biobecive impac measure, Technical Repor PISIS-2008, Graduae Program in Sysems Engineering, UANL, San Nicolás de los Garza, Mexico, 2008. Submied for publicaion. 15. Navarro, J., Modelo difuso de preferencias para resolver problemas de carera en organizaciones públicas, Maser's. Thesis, Universidad Auónoma de Sinaloa, Sinaloa, Mexico, 2001. 16. Navarro, J., Herramienas ineligenes para la evaluación y selección de proyecos de invesigación-desarrollo en el secor público, Docoral Thesis, Universidad Auónoma de Sinaloa, Sinaloa, Mexico, 2005. 17. Ringues, J.L., S.B. Graves and R.H. Caseb, Mean--Gini analysis in R&D porfolio selecion, European Journal of Operaional Research, (154)1: 157-169, 2004. 18. Soland, R.M., Mulicrieria opimizaion: A general characerizaion of efficien soluions, Decision Sciences, (10)1: 26--38, 1979. 19. Summer, C. and K. Heidenberger, Ineracive R/&D porfolio analysis wih proec inerdependencies and ime profiles of muliple obecives, IEEE Transacions on Engineering Managemen, (50)2: 175-183, 2003. 20. Tian, Q., J. Ma, J. Liang, R. Kwok and O. Liu, An organizaional decision suppor sysem for effecive R&D proec selecion, Decision Suppor Sysems, 39(3): 403-413, 2005.
172 Igor Livinchev, Fernando López Irarragorri, Miguel Maa Pérez and Elisa Schaeffer Igor Livinchev received his D.Sc. degree in Sysems Modeling and Opimizaion from he Compuing Cener Russian Academy of Sciences in 1995. Currenly he is wih he Faculy of Mechanical and Elecrical Engineering of he UANL, SNI 2. His main research ineress are large-scale sysem modeling, opimizaion and conrol, as well as decomposiion, aggregaion, and coordinaion in mulilevel and hierarchical sysems. Fernando López Irarragorri go his docoral degree on Technical Sciences a he Poliechnical Superior Insiue Jose Anonio Echeverria from Havana, Cuba, in 1998. Currenly he is an Associae Professor of Operaions Research a he Faculy of Mechanical and Elecrical Engineering of he UANL. His main research ineress are developmen and applicaion of decision suppor mehods and echniques as well as he developmen of Decision Suppor Sysems. Miguel Maa Pérez received his docoral degree in Sysems Engineering from he UANL in 2008. Currenly he is wih he Faculy of Mechanical and Elecrical Engineering of he UANL. His main research ineress are complex sysems modeling and large-scale sysem opimizaion. Sau Elisa Schaeffer go her docoral degree on Compuer Science and Engineering a he Helsinki Universiy of Technology TKK in 2006. She is an associae professor a he Faculy of Mechanical and Elecrical Engineering and research coordinaor of IT & Sofware a he CIIDIT research cener a he UANL, Mexico. Her specialy are complex sysems and graph heory.