Partnering Strategies for Fitness Evaluation in a Pyramidal Evolutionary Algorithm

Partnering S trategies for Fitness E valuation in a Pyramidal Evolutionary Algorithm GECCO 2002: Proceedings o f the Genetic a nd Ev olutionary C omp utation C onference, pp 2 63- 270 , New Yor k, USA, 2002. Uwe Ai ckelin School of Co mputer Science University of Not tingham NG8 1B B UK uxa@cs.nott.ac .uk Larry Bull Intelligent Co mputer Systems Centre University of t he West of England Bristol B S16 1QY, U K Abstract This paper combines the idea of a hierarchical distributed genetic algori thm with differe nt inter - agent par tnering strategies. Ca scading cl usters o f sub-populations are buil t from bo ttom up, with higher-level sub-population s optimising larger parts of the prob lem. He nce higher-level sub- populations search a lar ger s earch space with a lower resolutio n whilst lower-level sub- populations searc h a smaller s earch space with a higher resolution. T he effects of different part ner selection schemes for (sub-)fitness evaluation purposes are examined for two multiple -choice optimisation proble ms. It i s s hown that rando m partnering strategies per form best by p roviding better sa mpling and more d iversity. 1 INTRODUCTION When hierarchically distributed evolutionary algorithms are c ombined with multi-ag ent str uctures a n umber o f new q uestions beco me appare nt. O ne o f the se que stions is addressed in this paper: the issue of assigning a meaningful (sub-) f itness to a n agent. T his paper will loo k at seven differe nt partnering strate gies for fitness evaluation when combi ned with a genetic algorit hm that uses a co-operative s ub-population structure. W e will evaluate t he different strategies accor ding to their optimisation per formance of t wo schedulin g problems. Genetic algori thms are genera lly attributed to Holla nd [1976] and hi s students in the 1970s, although evolutionar y comp utation d ates back further (refer to Fogel [1998 ] for an extensive review of ea rly appro aches). G enetic al gorithms are stochastic meta- heuristics that mimic some features of natural evolutio n. Canonical ge netic al gorithms were not intended for function o ptimisation, as di scus sed by De Jo ng [19 93]. However, slightly modified versions p roved very successful. For an introd uction to gene tic algori thms for function opti misation, see Deb [1996]. The tw i st when ap plying o ur type of distributed genetic algorithm lies in its special hier archical str ucture. All sub- populations follow different ( sub-) fitness functions, so i n effect only searchin g specific par ts of the solution space. Following spec ial c rossover -operat ors these parts are then gradually merged to full solutio ns. The a dvantage of suc h a divide and conquer appr oach i s reduced epi stasis within the lo wer-level sub-pop ulations which makes the optimisation tas k easier for the genetic algor ithm. The p aper is arra nged as fol lo w s : the following sect ion describes the nurse sc hedulin g and tena nt selec tion problems. P yramid al genetic algorithm s and their application to these t wo prob le ms are detailed in sect ion 3. Section 4 explains the seven p artnering s trategies examined in t he p aper and section 5 describes their use and co mputational res ults. T he final sectio n disc usses all findings and dr a w s conc lusions. 2 THE NURSE SCHEDULING PROBLEM Two optimisation probl ems are considered in this paper, the n urse schedulin g pr oblem and the tena nt se lection problem. B oth have a number of charact eristics that make them an ideal testbed for the enhanced ge netic algorith m using partnering strategies. Fir stly, t hey are b oth in the class of NP- complete problems [Jo hnson 199 8, Martel lo & To th 1990]; hence, they ar e challeng ing prob lems. Secondly, t hey have pr oved re sistant to o ptimisation b y a standard genetic al gorithm, with good solutions onl y found b y using a novel strategy of i ndirectly optimising the prob lem with a deco der b ased genetic algorit hm [Aickelin & Do wsland 20 01]. Finall y, both p roblems are similar multiple -choice allocat ion prob lems. F or the nurse scheduling, t he c hoice is to all ocate a shift-pattern to eac h nurse, whilst for the tenant selection it is to allocate an area of the m all to a shop. H o wever, as the f o llowing more detailed explanation of the two will s how, the t wo problems al so ha ve some ver y distinct characterist ics making them d ifferent yet si milar e nough for a n interesting co mparison of result s. The nurse-schedulin g problem is t hat o f c reating weekly schedules for wards of up to 30 nurses at a major UK hospital. These sc hedules have to sat isfy working contracts a nd mee t the d emand for given n umbers of nurses o f dif ferent grade s o n e ach shi ft, whil st at the same time being s ee n to b e fair by t he sta ff concerned. The latter obj ective is achieved by m eeti ng as many of the nurses’ r equests as p ossible and by co nsiderin g hi storical information to ensure that unsatisfied r equests and unpopular shifts are evenly distributed. Du e to v a rious hospital p olicies, a nurse ca n normally o nly work a s ub- set of the 411 theor etically p ossible shift -patterns. For instance, a nurse should wor k either days o r nights i n a given week, b ut no t bo th. The interested r eader is d irected to Aickel in & Dows land [200 0] and Do wsland [1 998] for further details of t his prob lem. For our purposes, the proble m ca n b e m o delled a s follows. Nurses are sc heduled weekly o n a w ard basis such t hat they work a feasible pattern with regard s to their contract and that the demand for all da ys and nights and for all qualification levels is c overed. In tot al t hree qualification le vels with correspo nding dema nd exist. I t is hospital p olic y that more qualified nurses ar e allowed to cover for les s q ualified one . I nfeasible sol utions with respect to cover are not accep table. A solution to the problem w o uld be a string, with the number of elements equal to the number of n urses. E ach eleme nt would then indicate the shift-pattern worked by a p articular nurse. Depending on the nurses’ preference s, the r ecent histor y of patter ns worked and the overall attractiveness of the pattern, a penalty cost is th en allocated to ea ch n urse- shift-pattern p air. These values were set i n close consultation with the ho spital and ran ge fro m 0 (perfect) to 100 (unacceptab le), with a b ias to lower values. The sum of these values gives the quality of the sched ule. 52 data sets are available, with a n average pro blem size o f 30 nurses per ward and up to 411 po ssible shift-patterns per nurse. The prob lem can b e for mulated as an integer linear program as follo ws. Indices: i = 1.. . n nurse index. j = 1.. . m shift pattern index. k = 1. ..7 are days and 8... 14 are nights. s = 1... p grade index. Decision variab les:    = else 0 pattern shift works nurse 1 j i x ij Parameters: n = Nu mber of nurses. m = Number o f shift pattern s. p = Nu mber of grades.    = else 0 night day / covers pattern shift 1 k j a jk    = else 0 higher or grade of is nurse 1 s i q is p ij = Pre ference cost of nurse i working shift p attern j . N i = Shifts per week of n urse i i f night shifts are worked. D i = Shifts per week of nurse i i f day shifts are worked. B i = Shifts per week of nurse i i f both are worked. R ks = Demand of nurse s with grade s o n day or night k . F(i) = Set of feasible shift pat terns for nur se i , defined as i shifts combined j B a or shifts night j N a or shifts day j D a i F i k jk i k jk i k jk ∀                           ∈ ∀ = ∈ ∀ = ∈ ∀ = = ∑ ∑ ∑ = = = 14 1 14 8 7 1 ) ( Target function: ! min 1 ) ( → ∑ ∑ = ∈ n i m i F j ij ij x p Subject to: i x i F j ij ∀ = ∑ ∈ 1 ) ( (1) s k R x a q ks i F j n i ij jk is , ) ( 1 ∀ ≥ ∑ ∑ ∈ = (2) Constraint set (1 ) ensure s that ever y nurse works exact ly one sh ift patter n from his/ her feasible set, and constraint set (2) ensures that the de ma nd for n urses is co vered for every grade on every da y and night. Note t hat the definition of q is is such t hat higher grad ed n urses can substituted t hose at lower gra des if necessar y. T ypical problem d imensions ar e 3 0 nurses of three grades and 411 shift patterns. T hus, the I nteger P rogramming formulation has about 120 00 binary variab les and 100 constraints. Finally for al l de coders, the fitnes s of co mpleted solut ions has t o be calculated. Unfortunatel y, feasibility cannot b e guaranteed, as o therwise an unlimited s upply o f nurses, respectively overti me, would be necessar y. T his is a problem-specif ic is sue a nd cannot be chan ged. T herefor e, we still need a pe nalty function ap proach. Since the chosen encod ing automaticall y satisfies co nstraint set (1) of the integer p rogramming f ormulation, w e can use the following formula, where w demand is the pena lty weight, to calculate the fitness of solutio ns. Hence the penalty is proportio nal to t he number of uncovered shifts and the fitness of a soluti on is calculate d as follo ws. min! 0 ; max 14 1 1 1 1 1 1 →       − + ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = k p s n i m j ij jk is ks demand n i m j ij ij x a q R w x p Here we use an encoding that follows directly from t he Integer Progra mming formulatio n. Eac h indi vidual represents a full o ne-week sched ule, i.e. it i s a string of n elements with n being t he number of nurses. T he ith element of the stri ng is the index of t he shi ft p attern worked b y nurse i . For examp le, if we have 5 nurses, t he string (1 ,17,56 ,67,3) represents the schedul e i n which nurse 1 w o rks pa ttern 1, nurse 2 pattern 17 etc. For comparison, all data se ts were attempted u sing a standard Integer P rogram ming p ackage [Fu ller 19 98]. However, some re mained unsolved aft er each being allowed 15 ho urs run-time o n a Pentium II 20 0. Experiments with a number o f descent methods using different neighbour hoods, and a standar d simulated annealing i mplementation, were even less success ful and frequently fai led to find feasible solutio ns. A straightfor ward ge netic algorithm a ppr oach failed to solve the pro blem [ Aickelin & Do wsland 2000]. T he bes t evolutionar y result s to date have been achieved with an indirect ge netic ap proach employing a decod er function [Aickelin & Dowsland 2001]. However, we believe that there is further l everage in dir ect evolu tionary appr oaches to this pro blem. H ence, we propose to use an enhanced pyramidal genetic algorithm i n this paper. 3 TENANT SELECTION PROBLEM The second p roblem is a mall l ayout a nd tena nt se lection problem; ter med the m all proble m here. T he mall pr oblem arises both in the plan ning phase of a ne w shopping ce ntre and on completio n when the type and number o f shops occup ying the mall has to b e decided . To maximise revenue a good mix ture of shops that i s both heterogeneous a nd hom ogeneous ha s to b e achie ved. Due to the difficulty o f o btaining real-li fe data because of confidentiality, the pro blem and data used in this r esearch are constructed a rtificially, but closely modelled after the actual real -life pro ble m as d escribed for instance in Bean et al. [1 988]. In the follo wing, we will br iefly outline o ur model. The o bjective o f the mall pr ob lem is to maxi mise the r ent revenue of the mall. Although there i s a s mall fi xed rent per shop, a large p art of a shop’s rent dep ends on the sales revenue genera ted by it. T he refore, it i s import ant to select the r ight number, size and type o f tenants and to place them i nto the right l ocations to maximise revenue. As outlined in Bean et al. [1 988], the rent o f a shop depends on the following fact ors: • The attractiveness of the are a in which the shop is located. • The total number of shops o f the sa me t ype in the mall. • The size o f the shop. • Possible s ynergy ef fects wit h neighbo uring si milar shops, i.e. sho ps i n the sa me group ( not u sed by Bean et al.). • A fixed amount o f re nt based on the type o f t he shop and the area in which it is loca ted. This pr oblem c an be modelled as follo ws: B efore placing shops, t he mall is divid ed i nto a discrete number of locations, each bi g enough to hold t he smallest shop si ze. Larger sizes ca n be crea ted b y p lacing a s hop o f the same type in ad jacent locatio ns. Hence, the proble m i s that o f placing i shop-t ypes ( e.g. menswear) i nto j locatio ns, where each shop-type can belong to o ne or more of l groups ( e.g. c lothes s hops) and each l ocation is sit uated in one of k areas. For each type o f s hop there will be a minimum, ideal and maxim um n umber allo wed in the mall, as co nsumers are d rawn to a mall b y a balance of variety and ho mogeneity of s hops. The size of shops is dete rmined by how many lo cations they occ upy within the same a rea. For the p urpose of this study, shop s are grouped into three s ize classes, namely small, medium a nd lar ge, occupying one, two a nd t hree locations i n o ne ar ea of the mall re spectively. For instance, if there are two locations to be filled with the same shop- type withi n one area, then this will be a shop of medium size. If t here are five location s with the sa me shop-type assig ned in the same area, then t hey w il l for m one lar ge a nd one medium shop etc. Usua lly, there will also be a m a ximum to tal num b er o f small, mediu m and large shops allo wed in the mal l. To test the rob ustness a nd perfor mance of o ur algo rithms thoroughly o n t his pro blem, 50 problem in stances were created. All proble m instances have 10 0 locat ions grouped into five areas. Ho wever, t he sets d iffer in the number of shop-types a vailable (b etween 50 and 20) and in the ti ghtness of t he co nstraints r egarding the minimum and maximum nu mber of shop s o f a certain type or siz e. Full d etails o f the mod el and ho w the data was created, its dimensions and t he differences b etween the set s can be found in [Aickeli n 1999 ]. 4 PYRAMIDAL GENETIC ALGORITHMS Both problems failed to be optimised with a standard genetic algorith m [Aickelin & Dows land 2000, 2001]. Our p revious resear ch s howed that the d ifficulties were attributable t o epistas is cr eated b y the constrained nature of the opt imisation. B riefly, epistasis r efers to the ‘ non- linearity’ of the solution st ring [Davidor 1991], i.e. individual vari able values which w ere good in their own right, e. g. a particul ar shif t / l ocati on for a pa rticular nurse / shop for med low qualit y sol utions o nce co mbined. This effect was creat ed b y t hose c onstraints t hat c ould onl y be incorporat ed into the genetic algorithm via a penalt y function a pproach. For instance, most nurses preferred working da ys; thus, par tial sol utions with many ‘d ay’ shift-patterns have a higher fi tness. Ho wever, combining these shift -patterns lead s t o shortage s at ni ght a nd therefore infeasible solutions. The situation for the mall problem is similar yet more complex, as two types of constraints have to be d ealt wi t h: size con straints a nd number constrai nts. In [ Aickelin & Dowsland 2000 ] we prese nted a simple, and o n i ts o wn unsuccessful , pyramidal genet ic algorithm for the nurse- scheduling p roble m. A pyramidal approach can b est be described as a hier archical coevolutionar y genetic algorithm where casc ading clusters of sub- populations are built fro m b ottom up. Higher-level sub- populations have i ndividual s with loner stri ngs and optimise lar ger pa rts of the p r oblem. Thus, the hierarch y is not within one string but rather between sub- populations which opti mise d ifferent prob lem porti ons. Hence, higher-level sub-pop ulations s earch a larger search space wit h a lower r eso lution whilst lower-le vel sub-populations searc h a smaller s earch space with a higher resolutio n. A related hierarchical framework was presented using G enetic Programming [Koza 1991] whereby main p rogram trees co evolve with succes sively lower l evel f unctions [e.g. Ahluwalia & Bull 1998]. T he pyramidal G A can b e app lied to the nurse-sched uling problem in the following wa y: • Solutions in sub-pop ulations 1, 2 and 3 hav e their fitness based on co ver a nd r equests only for grade 1, 2 and 3 respectively. • Solutions in sub-pop ulations 4, 5 and 6 hav e their fitness based on co ver and re quests fo r grades (1+2), (2+3) and (3+1). • Solutions i n sub -population 7 optimise cover and requests for (1+2+3). • Solutions in s ub-populatio n 8 s olve the or iginal (all) proble m, i.e. cover for 1 , for (1+2) and for (1+2+ 3). The full structure is illustrated in figure 1. Sub-solution strings fro m lower populatio ns are cascaded upward s using suitable crossover and selection m ec hanisms. For instance, fixed cro ssover points are used such that a solution fro m sub -population (1) combined with one f rom (1+2) fo rms a new sol ution in sub-population (1+2). Eac h sub-population p erfor ms 50% of c rossovers unifor m with two p arents from itself. T he o ther 50% are done by taking one p arent fro m itsel f a nd the other from a suitable lower level p opulatio n and then performing a fixed-point crossover. B ottom le vel sub-pop ulations use onl y uni form crossover. The top level ( all) po pulation rando mly chooses the second par ent fro m all other pop ulations. Although the full proble m is as epistatic as be fore, the sub-proble ms ar e less so as the interactio n bet ween nurse grades is (p artially) ignored . Co mpatibility p roble ms o f combining the p arts ar e r educed by t he p yramidal structure with its hierarchical and gradual co mbining. This can be seen as similar to the “Island Inje ction” parallel GA syst em [Eby et al. 1999]. Using this appr oach i mproved solution quality in comparison to a sta ndard ge netic algorithm was r ecorded . Initially r oulette wheel se lectio n ba sed on fitness rank had been used to choose pare nts. The fitness of each sub- string is calculated using a substit ute fitness measure based o n the req uests a nd cov er as detailed above, i.e. the possibility of more qualified nurses co vering for less- qualified ones is partia lly igno red. Unsatisfied constrai nts are still i ncluded v ia a penal ty functio n. T his paper will investigate various pa rtnering strategies be tween the agents o f the sub-pop ulations t o i mprove upo n these results. Figure 1: Nurse P roblem P yramidal Structure. Similar to the nurse prob lem, a solution to the mall problem can be repre sented by a string with as many elements as locatio ns in the mall. Each ele ment then indicates what s hop-type i s to b e locate d there. The m al l is geograp hically split i nto di fferent reg ions, for i nstance north, east, sout h, west and central. So me of the objectives are reg ional; e. g. the size of a shop, the s ynergy effects, the attract iveness of an area to a shop-type, whereas others are global, e. g. the tot al number of shop s of a certain t ype or size. The app lication of the pyramidal struct ure to t he mall problem follows along simila r lines to t hat o f the nurse problem. In line with deco mposing partitio ns into those with nurses o f the same grad e, the pro blem is now split into the area s of the mall. T hus, we will have s ub-strings with a ll t he shop s in one a rea in t hem. T hese can then b e combined to create larger ‘p arts’ of t he mall and finally full solutions. However, the que stion arises ho w to calculate the substitute fitness measure o f t he partial stri ngs. T he solution chosen here will be a pseudo measure based on area dep endant components only, i.e. glob al aspects are not taken into account when a substit ute fit ness for a partial string is calculated. T hus, sub-fitne ss will b e a measure of the rent re venue create d by p arts of the mall, taking i nto acco unt those con straints that ar e area based . All ot her constraints are ig nored . A penalt y functi on is used to account for unsatisfied constraints. Due to the complexit y o f t he fitnes s calculations and the limited overall population size, we ref rained from using several le vels in the hierarc hical d esign as we d id with the nurse scheduling. Instead a simpler two-level hierarch y is used as sho wn in figure 2: Five sub-pop ulations optimising t he five ar eas sep aratel y (1 ,2,3 ,4,5) and one main pop ulation o ptimising the ori ginal p roblem ( all). Within t he s ub-populatio ns 1-5 unifor m cro ssover is used. The top -level pop ulation u ses uni form crossover between two member s o f the p opulation h a lf t he ti me and for the remainder a special cro ssover that selects one sol ution from a r andom sub-populatio n that then p erforms a fixed- point crossover with a mem b er of the top po pulation. Figure 2: Mall Probl em Pyramidal Structure . The remainder o f t his p aper will investigate ways to try to improve on p reviously fo und poor results by suggesting ways of combini ng partial str ings more intellige ntly. An alternative, par ticularly for the m all p roblem, would be a more gradual b uild-up of sub-pop ulations. W ithout increasing t he overall pop ulation size, t his w o uld lead to more and hence s maller sub-po pulations. However, this more gradual app roach might have enabled the algor ithm to find good f eas ible sol utions b y m ore slo wly j oining together pro mising b uilding b locks. This is in contra st to the relativel y harsh two-level a nd three-le vel d esign where building bl ocks had to ‘succeed’ immediately. Exploring the exact bene fits o f a gradual bui ld-up o f s ub- solutions would make for a nother c hallenging area o f possible future r esearch. 5 PARTNERING STRATEGIES The problem o f how to p ic k par tners has bee n noted in both competitive and co-operat ive coevolutionary algorithms. Ma ny strategies have b een presented in the literature as summaris ed for instance i n [B ull 1997 ]. In this p aper, the following st rategies are co mpared for their effectiveness i n fightin g epistasis b y giving m ea nin gful (sub-) fitness values i n the pyramida l gene tic algorit hm optimising the nurse scheduli ng and the mall p roble ms . • Rank-Selection (S): T his is the method used so far in our al gorithms. Solutions are assigned a sub-fitness 1+2+3 1+2 all 3+1 2+3 1 3 2 2 1 all 3 4 5 score based as closely a s p ossible o n t he co ntribution of t heir partial string to full sol utions. All solutions are then ran ked within each sub-population and selection follo ws a roulette wheel sche me based on the ranks [e.g. Aickelin & Do wsland 200 0]. • Random (R): Solutions choose their mating partners rando mly f ro m amongst all those in the sub- population their sub-population is paired w i th [e.g. Bull & Fogart y 1993]. • Best (B ): In this str ategy, each age nt is paired w i th the curr ent best solution o f the other sub- population(s). In case o f a tie, the solution with the lower pop ulation index is c hosen [e.g. Potter & De Jong 199 4]. • Distributed (D): The idea behind t his approach is to match sol utions with si milar ones to t hose pair ed with previously [e.g. Ackley & Litt man 19 94]. T o achieve this each s ub-populatio n is spac ed o ut e venly across a single toroid al grid. Subsequently, solutions are paired with others on t he sa me grid location i n t he appropri ate o ther sub-pop ulations. Children created by t his a re inserted in a n ad jacent grid location. This is said to be b eneficial to the sear ch proc ess because a consistent co evolutio nary pre ssure emerges since all offspring ap pear in their parents’ nei ghbourhoo ds [Husbands 199 4]. In our algorith ms, we use local mating wit h the neighbourhood s et t o the eight age nts surroundin g the chosen locat ion. • Best / Random (B R): A sol ution is p aired twi ce : with the best o f the o ther sub-po pulation(s) a nd with a rando m part ner(s). The bet ter o f the t wo fitness values is record ed. • Rank-based / Ra ndo m (SR): A solution i s paired twice: with ro ulette wheel s elected solution(s) and with (a) random par tner(s). T he b etter of the t wo fitness values i s record ed. • Random / Random (RR): A solution is pai red twice with rando m partner(s). T he better of t he two fitness values is record ed. 6 EXPERIMENTAL RESULTS 6.1 THE MO DEL To allow for fair co mparison, the para meters and strategies used for bo th proble ms are kept as similar as possible. Bot h have a to tal population of 1000 agents. These are split into sub-pop ulations o f size 10 0 for the lower-levels and a m ai n p opulation of size 300 for the nurse sched uling and respecti vely of size 500 for t he mall problem. In principle , t wo ty pes of crossover take pla ce: within sub-populatio ns a t wo-parent-two-children parameterised uniform cross over w ith p= 0.66 for genes coming fro m one parent takes place. Each new solution created u ndergoes mut at ion with a 1% bit mutation probab ility, where a mutation would re- initialise t he bit in the feasible range. T he al gorithm is ru n in gener ational m o de to acco mmodate the sub-popul ation structure bett er. In every ge neration the wor st 90% of parents of all s ub-populations ar e r eplaced . For all fit ness and sub-fitness function calc ulations a fitness score as described before is used. Co nstraint violatio ns are penalised with a d ynamic p enalty para meter, which adjusts itself dep ending on t he ( sub)-fitness d ifference between t he b est and the be st fe asible agent in e ach ( sub-) population. Full detai ls on this type of weight a nd how it was calc ulated can b e found in Smith & Tate [1993 ] and Aickelin & Do wsland [20 00]. The s top ping criterio n i s the top sub-po pulation sho wing no improvem ent for 50 generations. To o btain statistically so und results a ll experime nts were conducted as 20 runs over all pr oblem instances. All experiments were started with the same set of random seeds, i.e. with the same initial pop ulations. The results are presented in feas ibility and cost re spectively rent format. Feasibility denotes the pro bability of finding a feasible sol ution a veraged over all pr oblem insta nces. Cost / Re nt r efer to the ob jective function val ue of the best feasible sol ution for each p roblem instance averaged over the number of instances for which at least o ne feasible solutio n was found. Should the algorith m f ail to find a single feasible sol ution for al l 20 runs on one proble m instance, a censored observation of one hundred in the nurse cas e and zer o for the mall p roblem is mad e instead . As we are minimising the cost for the nurses and maximising the rent of the mall, this is equi valent to a ver y poor solution. For the nurse-scheduli ng problem, th e cost represents the sum of unfulfilled nurses’ requests and unfavourable shift- patterns worked. F or the mall , the values for the rent are in thousands of po unds per year. 6.2 RESULTS Tab le 1 sho w s the res ults for a variety of fit ness evaluation strategies used a nd compares these to the theoretic bounds (Bound) and the sta ndard genetic algorithm ap proac h (SGA). For the Nurse Schedulin g Problem all strategies used give better results than those found b y t he SG A. However, as explained abo ve, most credit for this is attrib uted to the p yramidal str ucture reducing ep istasis. On closer examina tion, rank-based (S), random (R) and distributed ( D) perform al most equall y well, with the rank-based method being sli ghtly better t han the other two. All three m e thods have in com mon that they contain a stochastic element in the choice of part ner. T he benefit of this is apparent when compar ed to the best (B) method. Here the r esults are far worse which we attributed to the inherently rest ricted sampling. Int erestingly, using t he double schemes (S R, BR and RR) im proves results across the board , which again stren gthens our hy po thesis ho w important good sa mpling is. The overall b est results ar e found b y the double rando m (RR) method. These results correspond to those rep orted in [Bull 1997 ]. The results for the Mall probl em are simi lar to those found for the nurse p roblem: Doubl e str ategies work better than sin gle ones an d the Best strategy d oes particularl y poorly. Ho wever, unlike for the n urse scheduling none of the sin gle strategies s ignificantl y improves results o ver t he SGA app roach. Reasons fo r t his have a lread y been out lined in the p revious sections, i.e. mainly the nat ure o f splitting the problem into sub- problems being contrary to many of t he prob lem’s constraints. O n t he other hand, even for the simple strategies resu lts are far im p roved over t hose found b y using the p artnering strategi es f o r mating, whilst those found b y t he double strategies even o utperfor m t he SGA. We believe that this can be expl ained as follo ws: The main do wnfall of the partnering for mati ng strategies for the mall prob lem was outside contro l of these strategies. I t lies i n the fact t hat the sub -fitness scores are not a good predicto r for the success of sub-solutions. However, as these results show, if the original (s ub-)fitness m easures are substituted b y full fitness scores based on good partnering method s the p yramidal structure do es work. This confirms our suspicio n that the previous ‘failure’ of the pyramidal i dea for the mal l prob lem was rooted within our choice of sub-fitness measures rat her than i n the hierarchical sub -population id ea itself. Method N Cost N Feasibility M Rent M Feasibility Bound 8.8 100% 2640 100% SGA 54.2 33% 1850 94% S 13.3 7 9% 1860 90% R 14.5 7 7% 1915 94% B 35.9 4 4% 1550 72% D 14.6 7 7% 1820 88% SR 12.7 8 4% 1950 99% BR 14.2 8 1% 1897 86% RR 12.1 83 % 1955 99% Tab le 1: Partnering Strategie s for Fitne ss Evaluation Results (N = N urse, M = Ma ll). 6.3 NURSE SCHEDULI NG WITH A HILLCLIM BER The results pr esented so far show that e ven with the best algorithm for the nurse sc heduling pro blem so me data instances were unsolvab le. In o rder to o vercome this, a special hillclimbe r has been developed w h ich is fully described in [Aickelin & D o ws l and 2001]. The u se of local search to refine solutions pro duced via the GA for complex p roble m do mains is w e ll established – often termed memetic algori thms [e.g. Moscato 1999 ]. Briefly, the hill -climber is local sear ch based al gorith m that iteratively tries to i mprove solutions b y (chai n-) s wapping shift patterns between nurses or alternativel y assigns a strictly solut ion i mproving pa ttern to a nurse. As the hill climber is c omputationall y e xpensive, it i s only used on those solutio ns sho wing favourab le characteristic s for it to exploit. Those solutions are referred to as ‘balanced’ and one example i s a nurse surplus on one da y shift a nd a shortage on another day shift. The l ast se t of experi ments presented in t ab le 2 shows what impact the b est p artnering schemes for evaluation (RR) has o nce the previously excluded hill climber is attached to the genetic algorithm. The results reveal tha t the SG A is outperfor med by the doubl e rand om fit ness evaluation appro ach coupled with the hill climber. One possible explanation for this effect ca n be found b y having a closer look at the R R operator. Gains ar e most likely made d ue to b etter sa mpling. Ho wever, as mentioned be fore there is a large stocha stic element involved in thi s case. J udging from t hese results it see ms that this is beneficial as it l eads t o a bigger variety o f solutions in turn lea ving m ore for the hill cli mber to exploit. Algorithm Short N Cost N Feasibility SGA & Hillclim ber SGA &H 10.8 91% RR & Hillcl imber RR&H 9 .9 95% Table 2: Results for Algorithms co mbined with a Hillclimber for the Nurse Sc heduling Pro blem. 7 CONCLUSIONS Using the p artnering strategies for eval uation purpo ses yields results i n accor dance with those r eported in [Bull 1997]. For both problems the simple st rategies w or ked equally well ap art fr om t he re stricting ‘best ’ choice. Combining two p artnering schemes im proved results further with the overall b est solutions found by t he do uble rando m strate gy. 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