THE ROBUSTNESS OF GENETIC ALGORITHMS IN SOLVING UNCONSTRAINED BUILDING OPTIMIZATION PROBLEMS

Transcription

1 Nnth Internatonal IBPSA Conference Montréal, Canada August 5-8, 2005 THE ROBUSTNESS OF GENETIC ALGORITHMS IN SOLVING UNCONSTRAINED BUILDING OPTIMIZATION PROBLEMS Jonathan Wrght, and Al Alajm Department of Cvl and Buldng Engneerng, Loughborough Unversty, Loughborough, Lecestershre, LE 3TU UK ABSTRACT Ths paper nvestgates the robustness of a Genetc Algorthm (GA) search method n solvng an unconstraned buldng optmzaton problem, when the number of buldng smulatons used by the optmzaton s restrcted. GA search methods can be classfed as beng probablstc populatons based optmzers. The probablstc nature of the search suggests that GA s may lack robustness n fndng solutons. Further, t s a common percepton that snce GA s terate on a populaton (set) of solutons, they requre many buldng smulatons to converge. It s concluded here that a partcular GA was robust n fndng solutons wth.4% mean dfference n buldng energy use from that for the best soluton found n all tral optmzatons. Ths performance was acheved wth only 300 buldng smulatons (n any one tral optmzaton). INTRODUCTION The beneft of the smulaton-based optmzaton of buldng desgn has been demonstrated through a potental reducton n buldng energy use by as much as 30% of that resultng from a benchmark desgn (Wetter and Wrght, 2003). Recent research examned the performance of several dfferent optmzaton algorthms n solvng a nonsmooth, sngle-crteron, non-lnear, unconstraned buldng optmzaton problem (Wetter and Wrght, 2004). The optmzaton problem ncluded elements of wndow geometry, the setponts for the control of wndow shadng devces, nght-setback setponts for the zone temperature, and the desgn supply ar temperature setpont. The performance of twelve dfferent optmzaton algorthms was examned; the algorthms ncluded, drect search methods (pattern and smplex searches), a gradent based method, probablstc populatons based methods (partcle swarm and genetc algorthms), and hybrd methods (partcle swarm and pattern search). It was concluded that the drect and gradent based methods could converge far from the optmum when the objectve functon was non-smooth (whch was the case for the example buldng optmzaton problem). Conversely, the probablstc populatonbased algorthms were more robust n fndng nearoptmum solutons, wth a hybrd partcle-swarm pattern-search algorthm havng the most consstent performance. A smple genetc algorthm also proved to be robust n fndng near-optmum solutons. Although robust, the hybrd partcle-swarm patternsearch algorthm requred n the order of 765 buldng smulatons before convergng on a soluton. In comparson, the genetc algorthm requred n the order of 585 smulatons before convergence. However, the convergence of the genetc algorthm was determned by arbtrarly lmtng the maxmum number of teratons of the algorthm. In contrast, the number of teratons of the hybrd algorthm was governed, n-part, by the characterstcs of the search space and the formally defned convergence of the pattern search algorthm. In ths respect, the patternsearch s guaranteed to have converged onto a local optmum, whereas the arbtrary convergence crtera of the genetc algorthm, means that convergence onto a local optmum cannot be guaranteed. Although prevous research, ndcates that genetc algorthms have the potental to solve unconstraned buldng optmzaton problems, the number of algorthm teratons requred to fnd a soluton remans unclear. Further, genetc algorthms are probablstc optmzers and as such ther behavor vares probablstcally wth the choce of startng condtons and algorthm control parameters. Although prevous research has examned the range of solutons found durng a partcular search (Coley and Schukat, 2002; Caldas and Norford, 2002), no research has been conducted to examne the effect of the algorthm startng condtons or control parameters on the robustness of the search n fndng near-optmum solutons. Ths paper examnes the robustness of a genetc algorthm n solvng an unconstraned non-smooth buldng optmzaton problem (Wetter and Wrght, 2003). The robustness of the algorthm s examned n relaton to the range of the solutons found for dfferent startng condtons and algorthm control parameters. In partcular, the ablty of the search to fnd a soluton wth a lmted number of tral smulatons s examned (the computatonal overhead assocated wth smulatng the buldng performance

2 beng the domnant computatonal element of the optmzaton). GENETIC ALGORITHM DESIGN A large volume of lterature on the form and operaton of Genetc and other Evoluton Algorthms exsts (among others; Bäck, 996; Bäck et al, 2000; Deb, 2000). In bref, Genetc Algorthms (GA s), terate on a populaton (set), of solutons. Havng frst, randomly ntalzed the populaton wth solutons (each varable beng randomly assgned a value wthn ts bounds), there are fve man operatons n the teraton:. Ftness assgnment: the evaluaton of the solutons objectve functon (often followed by some scalng or rank orderng). 2. Selecton: of ndvduals (solutons), from the exstng populaton for further operaton. The selecton s normally probablstc wth the probablty of selecton ncreasng wth ftness (n ths case of a mnmzaton problem, ncreasng as the objectve functon values decreases). 3. Recombnaton: the mxng of solutons to produce new solutons. Recombnaton s performed probablstcally, and as such may not take place. 4. Mutaton: the probablstc perturbaton of varable values. 5. Replacement: of old solutons n the populaton wth the new solutons resultng from selecton, recombnaton, and mutaton. Further, GA s operate wth an encodng of the problem varables know as a chromosome. The encodng can be n the form of a concatenated strng of bnary numbers, or a vector of real (or nteger), values. Recombnaton and mutaton operate on the chromosome gene values. In the case of the bnary encodng, a gene s represented by a sngle bt n the bnary strng (the value of a problem varable beng represented by several bts). In contrast, real encoded GA s, operate drectly on the real value of the problem varable. Form of Chromosome In contrast to a real vector chromosome, a bnary encodng has potentally greater exploratory power than a real vector chromosome, and naturally lends tself operatng wth both dscrete and contnuous varables. Buldng optmzaton problems are mxednteger problems. For example, alternatve wall constructons mght be dentfed by an nteger ndex that ponts to a partcular combnaton of constructon materals, whereas a supply ar temperature setpont may be treated as beng contnuous. Both contnuous and dscrete varables can be encoded n a bnary chromosome through controllng the number of bts assgned to a gven varable (a three bt encodng resultng 8 dscrete values for the varable). In practce the number of bts can be set to gve a real value precson up to the lmt of the machne precson, although n practce, the number of bts s set to provde a practcable resoluton n varable precson. The nherent encodng of mxednteger problems and the assocated control of varable precson, lends a bnary encodng to the soluton of buldng optmzaton problems. A Gray bnary encodng has been adopted for use n ths research. Ths encodng mproves the contnuty of the encoded search space, snce t lmts the occurrence of Hammng clffs (these occur when a sequental change bt values results n large, dscontnuous, changes n the value of the decoded problem varable). Ftness In ths study, we seek to mnmze the buldng energy use and therefore, the lower the energy use, the hgher the ftness of an ndvdual. In ths research, the solutons have been rank-ordered on the energy use objectve functon, the ftness of a soluton beng gven by ts rank. Selecton A characterstc of GA s s that, through the choce of algorthm operators, and ther control parameter values, t s possble to nfluence the behavor of the search. A common consderaton n choosng the algorthm operators and parameters, s the balance between the convergence relablty and the convergence velocty (or exploraton versus explotaton ; Bäck, 996). One of the prncpal operators governng ths balance s the selecton mechansm. The selecton operator s used to select solutons from the current populaton that wll be used to form the next populaton of solutons (ths beng the bass for the next teraton of the algorthm). If for example, the selecton method smply selected the best soluton of the populaton to form the bass of the next populaton, then the search would soon converge towards the current best soluton. However, f the selecton method randomly selected solutons from the populaton, then t s lkely that no one soluton would domnate the search drecton, and therefore, the search would be more random n nature. In ths research, we seek robust convergence wth as few buldng smulatons as possble (that s, relable convergence wth a hgh convergence velocty). In order to examne ths, the tournament selecton

3 method has been adopted for use n ths study, snce t allows control of the selecton pressure on the best solutons. The tournament operator randomly selects n, solutons from the populaton, the wnner of the tournament and soluton carred forward for recombnaton, beng the soluton havng the hghest ftness of all solutons n the tournament. The probablty of a soluton havng a hgh ftness beng selected, depends on the number of ndvduals n the tournament, and the sze of the populaton (that s, the percentage of the populaton selected for the tournament). One measure of the selecton pressure s the takeover tme (Bäck, 996). Ths s the number of generatons (algorthm teratons), for the populaton to fll wth the best soluton found n the ntal generaton, n the absence of recombnaton and mutaton (n the absence of recombnaton or mutaton, no new solutons are produced, the search only then samplng solutons from the randomly generated ntal populaton). The takeover tme for a tournament selecton s approxmated by (Bäck, 996): τ = ( ln( q) + ln(ln( q)) ) () ln( n) where, τ s the takeover tme, q the number of ndvduals n the populaton, and n, the number of ndvduals n the tournament. In ths research, we examne the effect of selecton pressure on the performance of the search by varyng the populaton sze (q), for a tournament sze (n). Table, gves the takeover tmes for a bnary tournament (n=2) and three populatons szes. The table also ncludes the number of tournaments necessary to fll the populaton wth the best of the ntal solutons (taken as, τ ' = τ q q ). Table Takeover tmes for a bnary tournament Populato n Sze, q (-) Takeover Tme τ (-) ' τ (-) The effect of populaton sze on the selecton pressure s clear from Table, wth a populaton sze of 5 ndvduals convergng n under half the number of teratons requred for a populaton of 30. Perhaps what s more sgnfcant s the dfference n number ' τ of tournament selectons ( ), snce, ths reflects the mpact of the populaton sze and selecton pressure on the lkely number of buldng smulatons requred to reach convergence (wth larger populatons requrng sgnfcantly more smulatons before the search begns to converge on a soluton). Recombnaton The recombnaton operator controls the mxng of genetc nformaton from pared ndvduals through a process know as crossover (each ndvdual n the par resultng from a separate tournament selecton). In the case of a bnary chromosome, crossover take place by swappng bt values between the two ndvduals. In the unform crossover operator used here, each par of bts are swapped wth a 50% probablty (an average of 50% of the bts wll be swapped). A 50% probablty of bt crossover gves the greatest mxng of genetc nformaton between pared ndvduals. However, n ths research, a further probablty parameter has been used to control the occurrence of chromosome crossover. If the probablty of chromosome crossover s such that crossover takes place, the bt values are swapped between chromosomes wth a 50% probablty. The effect of chromosome crossover probablty on the performance of the search, s examned here by experment. Mutaton A probablstc bt-wse mutaton, n whch a gven gene value f flpped from 0 to, or vsa versa, has been adopted n ths study. The effect of mutaton probablty on the performance of the search s examned here by experment. Replacement and Eltsm It s common to control the proporton of solutons replaced n the populaton at each teraton of the algorthm. In ths case, we replace all solutons, except the soluton havng the hghest ftness (the elte soluton). Ths guarantees that the search does not dverge to a soluton havng a hgher objectve functon than that already found by the search. The replacement ndvduals are a result of selecton, recombnaton, and mutaton (although, snce recombnaton and mutaton are probablstc, t s possble that some selected solutons reman unchanged when replaced n the populaton). Convergence and Automatc Restart GA s are conventonally stopped after a fxed number of algorthm teratons (generatons). However, snce the am of ths research s to study the convergence behavor of the GA n relaton to the number of buldng performance smulatons, the search s stopped here after a fxed number of buldng smulatons. Snce, the recombnaton and

4 mutaton operators are probablstc, t s possble that a selected soluton s smply coped from one generaton to the next (ths also occurs for the elte ndvdual). When ths occurs, the objectve functon value s taken from memory so that the need to resmulate the buldng performance s avoded. Therefore, all of the buldng smulatons performed are guaranteed to be unque. Further, n order to guarantee that the search s able to contnue untl the specfed number of smulatons s reached, the search s automatcally re-ntalzed f the populaton collapses onto a sngle soluton. Such convergence can be measured n terms of the problem varables (the genotype ), or the objectve functon (the phenotype ). In ths case we choose to dentfy the collapse of the populaton n terms of the objectve functon, as should the objectve functon have a low gradent n the regon of the optmum, convergence of the objectve functon may not be reflected by the same degree of convergence n the problem varables. Collapse of the populaton has been defned n terms of: where, fmax ( ) fmn ( ) α = 00 mn ( ) (2) f f max ( ), s the maxmum objectve functon value found n the current populaton; f mn( ), the mnmum (best) objectve functon value n the current populaton; α the convergence parameter. Should the current populaton have collapsed, then the next populaton s frst re-seeded wth the elte (best) soluton, wth the remanng ndvduals beng randomly ntalzed wthn ther bounds. Ths strategy s normally appled to a mcro-ga (for example; Caldas and Norford, 2002); mcro-ga use small populaton szes, that due to the hgh selecton pressure have a tendency to converge prematurely. In ths case, the re-ntalzaton s appled regardless of populaton sze. Note that, to some extent, the rentalzaton of the populaton can mask the apparent convergence and therefore the effect of the selecton pressure. A value of α <=% has been used n ths study to ndcate convergence and trgger rentalzaton of the populaton. EXPERIMENTS There are two elements to the experments performed here, the example buldng optmzaton problem, and the choce of GA control parameter sets. Example Buldng and Optmzaton Varables The mnmzaton of annual energy use of a mdfloor fve zone offce buldng (Wetter and Wrght, 2004) has been used as a bass for ths study. All exteror zones have daylght control, and the zone condtons are mantaned by a VAV system. The performance of the buldng s smulated by EnergyPlus, whch also auto-szes the capacty of the HVAC system. All experments have been performed usng verson of EnergyPlus, on a MS Wndows platform (Pentum 4 chp set). The buldng locaton (and assocated weather data, s taken as beng Chcago, IL, USA. The optmzaton problem varables are gven n Table 2. The normalzed wndow geometry varables determne the wndow wdth and heght (wth a value of 0.0 correspondng to wndow area of 20.4% of the façade, and.0 for 7.3% of the façade). The normalzed wndow overhang varables govern the depth of overhang above each wndow (wth a value of 0.0 correspondng to an overhang of 0.05m and.0 to.05m). The wndow shadng setponts control the use of an external shadng devce (the shadng devce beng actvated when the total nsolaton on the wndow exceeds the setpont). The zone ar temperature setponts govern nght operaton of the HVAC system, whle the supply ar temperature setpont s used to n the auto-szng of the HVAC system capacty. Note that the ncrement n each varable, together wth the lower and upper bounds, are used to dscretze the search space durng the bnary encodng of the varables. The dscrete ncrement n each varable has been set based on reasonable engneerng tolerance for each class of varable. The total dscretzed search space has a sze of.94 x Genetc Algorthm Control Parameters In ths research, we am to examne the robustness of a GA n solvng the example optmzaton problem for a fxed number of buldng smulatons. In a prevous study, a soluton to the example optmzaton problem, that had an objectve functon value wthn 0.4% of the best soluton, was found by a partcle swarm algorthm wth only 37 tral smulatons; all other algorthms n the study that found a soluton wth an objectve functon value wthn % of the best requred more functon evaluatons (Wetter and Wrght, 2004). It would therefore seem that n the order of 300 tral smulatons s requred to solve the example problem. Therefore, n ths study, each optmzaton s performed for 300 tral smulatons. The behavor of the GA has been examned for 2 dfferent sets of control parameters, consstng of 3 alternatve populaton szes, 2 crossover probabltes, and 2 mutaton probabltes (Table 3)

5 Table 2 Optmzaton problem varables Index Varable Lower Bound Upper Bound Increment Best Soluton North Wndow Geometry (-) West Wndow Geometry (-) West Wndow Overhang (-) East Wndow Geometry (-) East Wndow Overhang (-) South Wndow Geometry (-) South Wndow Overhang (-) West Wndow Shadng Setpont (W/m 2 ) East Wndow Shadng Setpont (W/m 2 ) South Wndow Shadng Setpont (W/m 2 ) Zone Nght Temperature Setpont, Wnter ( o C) Zone Nght Temperature Setpont, Summer ( o C) Supply Ar Temperature Setpont ( o C) Table 3 GA Control parameter sets Control Parameter Values Populaton Sze (-) [5,5,30] Probablty of Chromosome Crossover [0.7,.0] Probablty of Mutaton [0.0, 0.02] Populaton szes n the order of 80 ndvduals are commonly used n GA s. However, snce we have a lmt of 300 smulatons, ths s lkely to result n less than 5 teratons of the GA (assumng that, say, more than 60 of the 80 ndvduals are new n each generaton). Consderng that the larger the populaton, the lower the selecton pressure and convergence velocty, t s unlkely that 5 teratons s suffcent to fnd a soluton. For ths reason, we adopt small populaton szes that allow 0 or more teratons of the algorthm. Further, the alternatve populaton szes have been chosen to provde a range of selecton pressures, as ndcated by ther respectve takeover tmes n Table. In the unlkely event that each populaton was flled wth new solutons, a populaton sze of 30 would result n the mnmum of 0 teratons. Prevous research (Wetter and Wrght, 2004), also ndcated that a populaton n the order of 5 ndvduals was able to fnd a soluton to the optmzaton problem, and fnally, a populaton of 5 ndvduals s typcal of that used n a mcro-ga (for whch, the automatc re-ntalzaton of the algorthm s requred). The probablty of chromosome crossover controls the mxng of genetc nformaton between selected ndvduals. In general, crossover rates n the order of 0.7 or hgher, mprove the chance of good solutons beng found. Gven the hgh selecton pressures resultng from the small populaton szes, the hgher degree of mxng of solutons s also lkely to mantan the exploratory power of the search. In ths respect, we examne the behavor of the search for two hgh, chromosome crossover rates of 0.7 and.0 (70% and 00% probablty). Mutaton perturbs the gene values, and as such has a hgh probablty of ntroducng new solutons to the search. However, too hgh a mutaton rate can result n a random search. It s common for bnary mutaton rates to be set n terms of the number of bt mutatons per chromosome. The encoded bnary chromosomes

6 used n ths problem are 78 bts long, so that a 0.0 probablty of mutaton s lkely to result n less than one bt mutaton per chromosome, and a probablty of 0.02, of greater than one mutaton per chromosome. A probablty of one mutaton per chromosome s commonly used, so that here, we examne the effect of a mutaton rates that are slghtly hgher and lower than normally used. It s expected that the hgher mutaton rate s mportant n terms of mantanng the exploratory power of the search, gven the hgh selecton pressure due to the small populaton szes. Fnally, snce GA s are probablstc optmzers, for each of the 2 GA parameters sets, we conduct 9 separate experments, wth each experment started wth a dfferent randomly generated populaton of solutons. Gven 9 experments for each of the 2 parameter sets and each experment havng 300 tral smulatons, we perform a total of 32,400 smulatons n ths study. RESULTS AND DISCUSSION In ths study, we seek to obtan good algorthm performance n two potentally competng crtera; robustness n fndng solutons, and mnmzaton of the number of buldng smulatons. Algorthm Robustness The robustness of the algorthm s examned n terms of the varablty of the fnal solutons from each set of experments. Table 4, gves the results for the tral optmzatons (each statstc relatng 9 separate optmzatons). There s no statstcal dfference between any of the solutons, although subjectvely, the smaller populatons appear to have a greater chance of fndng a better soluton. Further, although not statstcally measurable, for the small populaton szes (5 and 5), the better solutons result from the hgher probablty of crossover and mutaton. The varablty (spread) n the solutons s llustrated n Fgure. The spread s llustrated n terms of the dstance from the best soluton found n all tral optmzatons, the horzontal axs beng the dstance n the problem varable doman, and the vertcal axs the dstance of the objectve functon. The normalzed Eucldean dstance d( X, X ), n the problem varables s gven by: where d( X, X ) = (3) nv = x x x = u l 2 x (4) and ( ) nv X = x,..., R, s a soluton and x nv nv ( x ) R X =,..., x nv, the best soluton found n all experments; nv s the number of problem varables, and l, and u are the varable bounds such, ( nv) ( nv) that l x u,..., where and l < u,,...,. l, u R The dfference n the objectve functon values s gven by: ( ( ) ( )) ( ) ( ), f X f X f X f X = f ( X ) d 00 (5) Fgure llustrates that the majorty of the solutons have an objectve functon value wthn 2.5% of the best soluton, the mean dfference beng.4%. The extent to whch ths s consdered suffcently robust, should be judged n relaton to dfference n buldng energy use to that of the energy use for a standard desgn soluton, as well as the nherent uncertanty n the smulaton and desgn process. Unfortunately, such an analyss s beyond the scope of ths paper. Objectve Functon Dfference (%) Best Solutons: 5 Best Solutons: 5 Best Solutons: 30 Random Search Fgure Soluton spread Normalzed Eucldean Dstance ( ) Fgure, also ndcates that there s some overlap n the optmzed solutons wth those from a randomly generated set (these random solutons resultng from the GA ntalzaton procedure). However, gven that the mean objectve functon dfference, for all 270 random solutons, was 28.4%, the overlap s nsgnfcant. Fnally, n terms of the consstency of the solutons, Fgure, ndcates that the smlar objectve functon values can be obtaned from dfferent solutons. Ths can occur for two reasons, frst, the optmzaton problem may be nsenstve to the problem varables (that s, the objectve functon gradent s low), nv

7 Table 4 Best solutons Populaton Probabltes (-) Objectve Functon (kwh/m 2 ) Sze, q (-) Crossover Mutaton Mnmum Maxmum Mean Standard Devaton and/or, the optmzaton problem s hghly multmodal. Fgure 2, llustrates the spread n varable values for solutons that have an objectve functon dfference (Equaton 5), of 0.5% or less. The decson varable ndex n Fgure 2 relates to that gven n Table 2, and the normalzed dstance by Equaton 4. The varable groups G.N, G.W, G.E and G.S relate to the wndow area and overhang sze for the north, west, east, and south wndow respectvely; the group Sp.Sh the wndow shadng setponts; Sp.Z the zone ar temperature nght setponts; and Sp.S the desgn supply ar temperature. It s postulated here that a large scatter n a varable ndcates a low senstvty, or hgh mult-modalty of the objectve functon to the problem varable. The least spread varables relate to the wndow areas (ndexes, 2, 4 and 6), the zone nght setponts (ndexes and 2), and the desgn supply ar temperature (ndex 3). The varables havng the most spread relate to the sze of the wndow overhangs (ndexes, 3, 5, and 7), and the shadng setponts (ndexes 8, 9, and 0). It s probable that the large spread n sze of wndow overhang ndcates a low senstvty of the objectve functon (energy use), to overhang sze for the example buldng. Ths may also be the case for the shadng setpont. In both cases however, further research s requred to confrm the senstvty of the objectve functon to these varables, and n partcular ther mpact on the mult-modalty of the objectve functon. Normalzed Dstance ( ) Fgure 2 Soluton spread n the regon of the optmum 0.8 G.N G.W G.E G.S Sp.Sh Sp.Z Sp.S Decson Varable Index ( ) Convergence Velocty Fgure 3, llustrates the convergence for three of the tral optmzatons (each optmzaton used the same random number sequence, and had a.0 probablty of crossover and 0.02 probablty of mutaton). In order to obtan a true comparson between the convergence of each populaton, the convergence s gven as a reducton rato: r ( f ( U ), f ( U )) f ( U ) = (6) f ( U ) where: a set of np solutons, s gven by U = X ( X,..., X )} and the f U ) s the mean { np ( objectve functon value of the set. Snce the largest

8 populaton sze used here contans 30 solutons, the objectve functon means have been calculated for every 30 new tral smulatons. Reducton Rato ( ) Fgure 3, Rate of convergence Populaton: 5 Populaton: 5 Populaton: Number of Smulatons ( ) In ths respect, the smaller populatons would have already completed several teratons (generatons), before reachng 30 smulatons. Even though ths s the case, the ntal mean objectve functon value s lower for the 30 random solutons of the 30 ndvdual populaton (Table 5). Ths n part explans the hgher rate of convergence of the small populatons (the dvsor n Equaton 6 s hgher). Further, the populaton of 30 converged onto a worse soluton, than that found for ether of the two smaller populaton szes, whch had the effect of ncreasng the fnal reducton rato. It can be concluded, that there s some evdence that the populaton sze of 5 ndvduals has a hgher convergence velocty than the larger two populatons, although ths requres further nvestgaton. Table 5 Mean solutons after 30 smulatons Populaton sze f ( U) CONCLUSIONS The am of ths research was to examne the robustness of a GA n fndng solutons to an unconstraned buldng optmzaton problem, gven a lmted number of tral smulatons. Experments where performed for twelve dfferent sets of GA control parameters, wth nne dfferent tral optmzaton completed for each parameter set. It was concluded, that the GA was nsenstve to the choce of GA control parameters, there beng no statstcally sgnfcant dfference n solutons found between any of the parameter sets. However, the better solutons where obtaned usng small populaton szes (5 and 5 ndvdual), wth hgh probabltes of crossover and mutaton (00% and 2% respectvely). The mean dfference n objectve functon values, from that for the best soluton found, was.4%, wth the majorty of solutons beng wthn 2.5% of the best soluton found. Gven that there s no sgnfcant dfference between the sets of solutons, t can be concluded that the GA was robust n fndng solutons. It was also concluded that t s possble to fnd nearoptmum solutons wth a compettve (low) number buldng performance smulatons (n ths case 300 smulatons). Ths may, n part, be due to the hgh convergence veloctes resultng from the low populaton szes, although ths requres further research. Further research s also requred to examne the robustness of the GA, partcularly n relaton to the characterstcs of the problem and search that result n poor solutons beng found. Research s also requred to nvestgate the soluton accuracy that s expected n relaton to that achevable gven normal desgn tolerances and the uncertanty n results from buldng performance smulaton. REFERENCES Bäck.,T, 996, Evolutonary Algorthms n Theory and Practce, Oxford Unversty Press, New York, ISBN Bäck,T., Fogel, D.B., Mchalewcz, Z., 2000, Evolutonary Computaton 2: Advanced Algorthms and Operators (Evolutonary Computaton), Insttute of Physcs, ISBN Caldas., L.G., Norford., L.K., 2002, A Desgn Optmzaton Tool Based on a Genetc Algorthm, Automaton n Constructon,, 73-84, ISSN Coley, D.A., Schukat. S., 2002, Low-Energy Desgn: Combnng Computer-Based Optmsaton and Human Judgement, Buldng and Envronment, 37, , ISSN Deb, K. D., 200, Mult-Objectve Optmzaton usng Evolutonary Algorthms, John Wley and Sons Ltd, ISBN X. Wetter, M., Wrght, J.A., 2003, Comparson of a Generalzed Pattern Search and a Genetc Algorthm Optmzaton Method, Proceedngs of the 8 th Internatonal IBPSA Conference, Endhoven, Netherlands, Wetter, M., Wrght, J.A., 2004, A Comparson of Determnstc and Probablstc Optmzaton Algorthms for Nonsmooth Smulaton-based optmzaton, Buldng and Envronment, 39, ISSN: