A bi-obective genetic appoach fo the selection of sugacane vaieties to comply with envionmental and economic equiements Helenice de Oliveia Floentino and Magaida Vaz Pato CIO Woing Pape 5/0
A bi-obective genetic appoach fo the selection of sugacane vaieties to comply with envionmental and economic equiements Helenice de Oliveia Floentino *, Magaida Vaz Pato,3 Depatamento de Bioestatística do Instituto de Biociências Univesidade Estadual Paulista, Botucatu, SP, Basil addess: Rubião Júnio, P.O. Box 50, CEP 868-000, Botucatu, SP, Basil e-mail: helenice@ibb.unesp.b Instituto Supeio de Economia e Gestão, Univesidade Técnica de Lisboa, Potugal addess: Depatamento de Matemática, ISEG, Rua do Quelhas, 6, 00-78 Lisboa, Potugal e-mail: mpato@iseg.utl.pt 3 Cento de Investigação Opeactional, Faculdade de Ciências, Univesidade de Lisboa, Potugal * coesponding autho: tel: +55 04 38 67, ext. Abstact The selection of sugacane vaieties is an impotant poblem facing companies in Bazil that exploit sugacane havest fo enegy poduction. In the light of cuent concens egading the eduction of envionmental damages and the efficiency of the poduction system, eseach into this poblem is called fo. In this context the authos begin by outlining the sugacane vaiety selection in accodance with technical constaints with the pupose of minimizing collection and tanspot costs and maximizing enegy obtained fom esidues of the sugacane havest. They then pesent a peviously developed modelization fo the poblem within biobective binay linea pogamming and study its had computational complexity. Fundamentally, this pape is devoted to the application of a bi-obective genetic heuistic to the question addessed. A computational expeiment pefomed using a test set including eal and semi-andomly geneated instances is then epoted. The esults pove to be vey good in tems of solution quality spacing and cadinality of the set of potentially nondominated points besides computing time. Fo these easons, this will be an appopiate tool to help sugacane company manages to plan thei poducing activities. Additionally, the bi-obective genetic heuistic can be easily adapted to othe binay bi-obective poblems that shae the assignment and the napsac constaints with this model.
Keywods: Selection of sugacane vaieties, Bi-obective binay linea pogamming poblem, Bi-obective genetic algoithm, Paeto-optimality.. Intoduction The sugacane is among the main Bazilian agicultual poducts (see [5]). Its stals ae used fo poduction of the alcohol, suga and enegy. Sugacane havesting may be mechanized o not. If the havest is not mechanized then the common pactice is to bun esidue (tops, staw, leaves, etc) pio to havest. Such taditional pactice pollutes both the atmosphee and the wate netwos. Moeove, it fequently tuns into uncontolled fies thus buning neighbou popeties, buildings, public distibution netwos and equipments. Envionmental and govenmental oganizations have been acting against this pactice and law aleady establishes deadlines to the end of buning esidue. Sugacane esidue epesents a new and impotant souce of enegy, because it has high caloific value and a low poduction of mico pollutants. Used as fuel, this biomass can educe the envionmental poblem of wold enegy poduction. But availability of this biomass depends on the sugacane havesting being mechanized. Fo the above easons, in the shot futue, the only havesting pocess is the mechanized one thus implying the ecovey of sugacane esidue fom soil. Its use fo powe geneation at a pocessing cente has been advised by scientists and pactitiones. On one hand, the collection and tanspot of sugacane havest biomass fom the planting field to the pocessing cente includes vaious high cost opeations and machiney. On the othe hand, thee is lage diffeence in the amount of esidue geneated and powe heat among sugacane vaieties. Hence, the selection of vaieties to be planted uges. But, to choose the ight vaieties is not easy; it depends on fundamental agonomic infomation, industial factos, and on the inteaction of all the biotic, abiotic, administative and economic factos (see [5, 6]). Recently, Floentino and othe authos have been developing and pesented in [0] and [] bi-obective optimization models fo the selection of sugacane vaieties complying with envionmental, economic and technical equiements. The poblem, hee denoted fo shot by SSVP, calls fo minimizing the cost of the esidual biomass tansfe pocess fom the field to the pocessing cente, maximizing the enegy contained in these esidues, and at the same time addesses sucose poduction and planting aea
estictions. Those authos tacle the SSVP with an exact method and apply the methodology to the case of a small sugacane poduce. One nows that most of the Bazilian sugacane poduces wo at a lage scale, dealing with planting aeas much bigge than the one discussed in the above papes. At these eal dimensions, nowadays any exact methodology is no longe useful. Non-exact methods, namely natue-inspied metaheuistics appea to be appopiate tools fo high dimension combinatoial models. Following successful developments fo othe bi-obective combinatoial poblems (e.g. [3]) we opted in favou of a genetic heuistic fo the SSVP in view of the possibility of geneating at easonable computational expenses many diffeent solutions coesponding to vaying attainment levels of the optimization obectives. A bief eview of the main agonomic technical aspects, along with the SSVP itself ae pesented in Section, wheeas Section 3 is devoted to a bi-obective binay linea pogamming model fo the SSVP and also to the poof of its computational complexity. Solutions fo this poblem ae obtained using a bi-obective genetic algoithm, pesented in Section 4. Section 5 shows the esults obtained fom the computational expeiments and, lastly, Section 6 contains the conclusions.. The poblem Ripoli and Ripoli in [5] discuss elevant technical aspects fo the SSVP. All the data and paametes fo the poblem ae detailed in [0]. Hee, to bette attain the scope of this wo, we pesent the main aspects fo the mathematical study of the poblem and give all the data involved. Let us conside a specific company that owns o ents a field whee sugacane plantation is being planned. The company also contols the poduction at a sugacane pocessing cente o mill. The field is made up of plots which will be planted with this cop. We assume that the necessay distances ae nown, namely that plot is at a distance D (in m) to and fom the mill, fo all =,...,. In this context a single vaiety will be selected fo plot measuing L (in ha). Thus the total aea planted equals L = (in ha). The quality of sugacane is mainly measued by the levels of sucose and fibe. Fo the suga and alcohol industies in the State of São Paulo and the Mid-South of Bazil, the ecommended sucose value (POL) of sugacane should be geate than 4%. This figue allows 3
the companies of the egion to calculate the minimum of POL fo the planting field, hee epesented by Slo (in t). Let us conside the n sugacane vaieties adapted to local soil and climate conditions. If S i epesents the estimated sucose poduction fom one hectae planted with sugacane vaiety i (in t ha - ) then the total amount of suga poduced fom vaiety i in plot (in t) is given by si = S i L () whee, as mentioned above, L is the aea (in ha) of plot. With espect to sugacane fibe, the ecommended value is between % and 3%, because high fibe pecentage educes the efficiency to extact the uice and low fibe levels favou mechanical damage of the cane when cutting and tanspoting. Again, on the basis of these values, the lowe and uppe limits fo the quantity of sugacane fibe poduced fom the field unde study ae thus calculated and hee epesented by Flo and Fup, espectively (both in t). If we conside F i to be the estimated fibe content of sugacane vaiety i planted in one ha (in t ha - ) we can calculate fi = F i L () which epesents the total amount of fibe (in t) poduced fom vaiety i in plot. Finally, to contol sugacane vaieties (e.g. genetically modified ones) let us suppose that this company, lie some othes, impose additional uppe bound constaints on the aea to be planted with each vaiety, epesented by Lup i (in ha) fo all i. To tae pofit of the esidual biomass poduced fom the sugacane havest fo enegy poduction, the poblem involves othe equests fo vaiety selection, mainly, minimization of esidue collection cost and maximization of esidue enegy balance. These ae tacled as soft constaints, in opposition to the pevious had constaints. The cost and enegy consumed fo the use of the esidual biomass depend on the necessay opeations: the staw of the sugacane is aed into piles, is compacted with a special machine, is loaded into tucs, and tanspoted to the mill. Hence, by taing all the agonomical and economic data in Table, the total cost of tansfeing the esidual biomass fom sugacane vaiety i planted in plot (in R$) is calculated though: c = ( Cccl + Ct ) L (3) i i i 4
whee Cccli = PbiCl is the cost to collect, compact and load esidual biomass fom each hectae of sugacane vaiety i (in R$ ha - Ct ( V Pb Vc) D Co P ) and is the cost to tanspot the i = esidual biomass fom vaiety i poduced in one hectae of plot (in R$ ha - ). The enegy balance equates to the diffeence between the enegy available fom the esidual biomass and the enegy consumed to tansfe this esidue fom the planting aea to the mill. Then, again taing data of Table, the esidual biomass enegy balance fo sugacane vaiety i planted in plot is detemined (in MJ) in accodance to: ei = Ea i Et i (4) whee Ea = Ec Pb L is the enegy available fom the esidual biomass i i i sugacane vaiety i planted in plot (in MJ) and Et = + i i i ( Elm Etm D Vi Vc) Pbi L obtained fom is the enegy used in the pocess of tansfeing collect/compact/load/tanspot the biomass fom sugacane vaiety i planted in plot to the pocessing cente (in MJ). Table. Data paametes fo the poblem. L D (ha) aea of plot (m) total distance fom plot to the mill including the way bac... F i (t ha - ) poductivity of fibe fom vaiety i Si (t ha - ) poductivity of suga of vaiety i Lup i (ha) maximal aea fo vaiety i Pbi (t ha - ) estimated mass of esidue geneated pe hectae of vaiety i Vi (m 3 t - ) volume of one ton of compacted esidue fom vaiety i Eci (MJ t - ) estimated caloific enegy of one ton of esidue fom vaiety i... Cl (R$ t - ) cost to collect, compact and load one ton of esidue 3 Vc (m ) load capacity of the tuc Co (L m P (R$ L - - ) fuel consumption of the tuc pe m ) pice of the fuel pe L Elm (MJ t - ) enegy consumed by machiney to collect, compact and load pe ton of esidue Etm (MJ m - ) enegy consumed by the tuc to tanspot the esidue pe m Slo (t) minimal ecommended total POL quantity 5
Flo (t), Fup (t) lowe and uppe ecommended limits fo the total fibe content Finally, the selection of sugacane vaieties complying with envionmental, economic and technical equiements poblem studied in this pape, the SSVP fo shot, equies choosing one single vaiety pe plot of the field such that the aea planted with each vaiety is uppe bounded, the minimum and maximum total amounts of fibe ae espected, as well as the minimum total amount of suga to be poduced. The soft constaints ae consideed as optimization obectives: minimization of total costs associated with tansfeing the sugacane esidue fom field to mill and maximization of enegy balance of the esidue. 3. Mathematical fomulation and computational complexity This section pesents a mathematical model fo the SSVP defined in the pevious one. Consideing: n numbe of sugacane vaieties; numbe of plots that compose the aea to be planted with sugacane; i index fo the sugacane vaieties (i=,...,n); index fo the plots (=,...,); the data in Table ae used to calculate, though the expessions () to (4), the essential paametes fo the model: s i f c e i i i (t) total amount of suga poduced fom sugacane vaiety i in plot ; (t) total amount of fibe poduced fom sugacane vaiety i in plot ; (R$) total cost of collecting the esidual biomass fom sugacane vaiety i in plot ; (MJ) total enegy balance fo esidual biomass fom sugacane vaiety i in plot. Taing the decision vaiables x = [xi] i=,...,n; =,..., such that x i is equal to if sugacane vaiety i is selected fo plot and 0 othewise (i=,...,n; =,...,), a bi-obective binay linea pogamming model fo the SSVP studied in this pape follows: minimize n ( ) = i f x c i x (5a) i= = subect to maximize f ( ) n x = e i x (5b) i i= = 6
n s i i= = x i Slo (6) n Flo f i= = i x i Fup (7) n x = =,..., (8) i= i L = x i Lup i i=,...,n (9) x i = 0 o i=,...,n; =,...,. (0) The goals of the SSVP efe to tansfeing the esidual biomass fom field to mill and use it fo enegy puposes. They ae hee expessed by the two poblem obectives: in (5a) cost minimization and in (5b) enegy balance maximization. The inequalities (6) and (7) guaantee, espectively, the needs of fementable suga and the constaints on the fibe content. The semiassignment equations (8) impose that each plot be planted with one and only one vaiety. Constaints (9) impose the maximal aea pe vaiety and, finally, (0) define the vaiables. This mathematical model is a bi-obective binay linea pogamming poblem simila to the models aleady developed by Floentino et. al [0, ]. Moe specifically, it diffes fom the one pesented in [0] by the constaints (9) which ae not consideed thee. Respecting the computational complexity, the SSVP is difficult in view of its combinatoial natue. In fact, as shown below it is an NP-had poblem. Fist, let us conside a paticula case of SSVP chaacteized by Slo = si ; min i =,..., n ; =,..., Flo = f ; Fup = f min i =,..., n ; =,..., i n i= = i. Then, constaints (6) and (7) ae edundant and the poblem given by (5a), (5b), (8), (9) and (0) is a special case of a bi-obective genealized assignment poblem (GAP) insofa as the coefficients of the vaiables in constaints (9) do not depend on the two indexes as in GAP (see [9]). Now, let us conside the poblem of detemining a feasible solution fo the above special case of SSVP, that is, a solution x satisfying to (8), (9) and (0). L = By taing n = and Lup = Lup = one obtains a -patition poblem which is a well nown NP-complete poblem (see [3]). As seen, the -patition poblem is polynomially educible to the feasibility SSVP, hence the feasibility SSVP is an NP-complete poblem. 7
Consequently, the (optimization) SSVP itself, fomalized though (5a) to (0), is an NP-had poblem. Moeove, the SSVP has a multiple obective natue because it is a paticula multiobective optimization poblem (MOP). Accoding to Cohon (978) [] the study of MOPs was, until the seventies, developed in thee domains: opeations eseach, economics and philosophy. Within the opeations eseach, Kuhn and Tuce in 95 [7] laid the foundation fo seveal algoithms, wheeas Gass and Saaty [4] pesented the fist MOP application 955. In ecent decades multi-obective optimization has eceived inceased attention within opeations eseach, both in the fields of application and theoy (e.g., [4, 7]). The Paeto optimality was intoduced by the economist Vilfedo Paeto in 896 [] as a quantitative definition of optimality fo contadictoy obective economic poblems. Koopmans in 95 [6] was a pionee in the use of the Paeto-optimum concept fo a esouce assignment poblem. Within the philosophical domain the initial contibution to multi-obective optimization came fom Togeson in 958 [8] when studying altenative decision poblems. A MOP can be fomulated, with all minimization obectives, as follows: minimize f(x) subect to x Γ whee f(x) = (f (x), f (x),..., f T (x)) is the obective function vecto with each component f q (x) R and Γ is the feasible egion o the solutions space. Given x, x Γ, the solution x dominates x (o x is dominated by x ) if f q (x ) f q (x ), q =,...,T and thee is at least an s such that f s (x ) < f s (x ). This means that x is bette than o equal to x fo all the obectives and stictly bette fo at least one of the obectives. The solution x Γ is an efficient solution if it is not dominated by any othe of Γ. In the solutions space evey efficient solution, x, coesponds to a point in the obectives space, f(x), which is denoted by nondominated point. The set of all nondominated points is the Paeto optimal set o the Paeto fontie. In most cases, the esolution methods fo MOPs ae designed to obtain o appoximate the Paeto fontie o subsets of this and the coespondent efficient solutions. Theefoe the SSVP is a MOP with two linea obective functions (T = ): one minimization and one maximization function. Vectoial optimization within this MOP is subect 8
to linea constaints and binay conditions on the vaiables, all them defining the solutions space Γ. In addition, the two obectives of the SSVP ae, in pactice, contadictoy as seen though the computational expeiments. In fact, the moe the esidual biomass fom sugacane, the moe is the net enegy poduced (the enegy balance), which is theefoe bette fo obective. Howeve, the costs of tansfeing this esidual biomass ae highe, which maes the solution wose fo obective. As a esult of the combinatoial featues and, mainly, the high computational complexity of the poblem, exact techniques can offe good esults only fo small/medium instances of the poblem. In fact, when the dimension scale is lage such techniques become difficult to apply, with pohibitively lage consumption of computational esouces, even when detemining a single efficient solution. Othe methods, of a non-exact natue, ae then needed to cope with these difficulties. The following section discusses a bi-obective heuistic technique to tacle the SSVP which we have biefly pesented in [4]. 4. Bi-obective genetic appoach Fom among the many types of multi-obective heuistic methods, the genetic o evolutionay heuistics have poved successful in obtaining solutions fo difficult MOPs as they deal with a population of solutions with diffeent chaacteistics as to the optimization obectives. Deb in [6] gives a compehensive synthesis of these multi-obective methods, wheeas the boo by Coello, Lamont and Velduizen [3] coves the cuent eseach and application in the field. Specifically fo multi-obective poblems with napsac and semi-assignment type constaints, lie the SSVP, genetic heuistics have also been successfully applied, as shown in the ecent aticles [, 8,, 5, 8, 0]. Hee the aim is to obtain a set of feasible solutions fo SSVP whose coesponding points in the obectives space ae faily spead and ae not fa fom the exact Paeto fontie. The genetic algoithm we developed, algoithm GenSuga, contains the main components of an evolutionay appoach fo combinatoial optimization. It involves binay tounament selection and the genetic opeatos of cossove, mutation and epai. An elitist stategy is also applied. The heuistic solutions ae the output of GenSuga, pecisely the solutions that ae not dominated in the population of the last geneation. 9
4. Encoding Each individual of the population is chaacteized by a single chomosome which diectly epesents a solution fo the SSVP. The chomosome is encoded though an intege valued vecto whose components give the sugacane vaieties selected fo the espective plot, as illustated in Figue. Thus, in this epesentation each gene is a vaiety, the one that is poposed fo the plot. vaieties fom {,,...,n}... plots Figue. Encoding of the chomosomes (solutions). Note that, as opposed to a staightfowad binay encoding that pais the genes with the binay vaiables, this non-binay encoding natually fulfills the semi-assignment constaints of the SSVP equations (8) of the fomulation wheeas the minimum levels fo POL and fibe, the maximum of fibe, as well as the maximal aea pe vaiety can be violated constaints (6), (7) and (9) and, consequently, the solution is unfeasible. 4. Initial population A constuctive heuistic involving two diffeent geneatos is applied to geneate the N individuals fo the initial population, Pop(). The fist geneato is devised to poduce individuals andomly, thus ceating a highly divese population, wheeas the second poduces individuals with less divesity but achieves a easonable quality of the optimization obectives. In both cases, if bounding constaints ae not satisfied then a disuptive/constuctive phase endeavos to enfoce feasibility. The pseudo-code of the algoithm Build, which poduces chomosome co, chaacteizing the -th individual of the population, that is, an SSVP solution, is pesented in Figue., algoithm Build(co, f f ) step //initialize// β = 0 ; β = 0; β = 0 s f L step //detemine co // if = o o o N Nc //andom geneato // 0
fo = to do andomly select i fom {,,,n} = co = i ; cs = S i L ; cf F i L endfo else //geedy geneato // fo = to do i = ag ci ei endif step 3 sum = cf if if = min i=,..., n = co = i ; cs = S i L ; cf F i L endfo //disuptive/constuctive phase ty to impose feasibility on co // sum > Fup then call FeasUp(F,S,Fup,cf,cs ) sum < Flo then call FeasLow(F,S,Flo,cf,cs ) sum = cs if = sum < Slo then call FeasLow(S,F,Slo,cs,cf ) step 4 //calculate penalizations, should it be the case// = max Slo cs =, 0 β s ; = max{ Flo Fup } f cf, cf, 0 f = cost of co f f + β + β + β f = = = f f stop s enegy balance of co β β β s f f L L β ; = = n β L = max i= : co = i L Lupi, 0 Figue. Algoithm to geneate an individual and its fitness. In step the algoithm stats by constucting co, which is epesented by an intege vecto whose components identify sugacane vaieties: this is a andom based pocess fo (N Nc) chomosomes and a geedy one fo the emaining Nc chomosomes. The andom geneato simply builds each chomosome by andomly assigning vaieties to the plots, following a unifom discete distibution. The second is used to detemine Nc chomosomes by following a geedy citeion. Hee, the vaiety fo each plot is chosen by minimizing the atio ci ei. Then, in step 3, the thee main SSVP constaints uppe bound on fibe, and lowe bounds on fibe and suga ae sequentially tested and, if necessay, enfoced on the solution following
the pocedues FeasUp and FeasLow pocedues, as shown in the pseudo-codes of Figues 3 and 4, espectively. Each of these geedy pocedues identifies the most favoable sugacane vaiety with espect to the specific estaint and assigns it to plots until the constaint is imposed. The numbe of sugacane vaiety changes is bounded by the algoithm paamete Ncha ( Ncha n ) while the algoithm paamete M epesents a big positive eal numbe. When, afte calling the thee pocedues, co violates the fibe, POL o aea constaints of SSVP, the espective fitness value is penalized in step 4 on the stength of the poximity to feasibility. It is woth noting that the suga constaint is most fequently satisfied because the pocedue enfocing suga constaint is the last one to be called within the algoithm of Figue see step 3. This option was taen in view of the impotance attibuted by the companies to the minimal POL content of sugacane, as opposed to the fibe equiements. The case of maximal aeas is not that elevant, hence no pocedue enfocing these constaints was defined, apat fom the penalization steps. pocedue FeasUp(F,S,Fup,cf,cs ) step //initialize// i = ag min F i //detemine the smallest component of F// i=,..., n fo = to do SU p = step while = sum > Fup and p Ncha do = ag max cf SU //detemine the geatest hitheto unchanged component of cf // =,..., sum = sum cf co = i ; p = p + endwhile stop + F i L cf = F i L ; cs = Si L ; SU M = Figue 3. Algoithm to enfoce uppe bound feasibility fo fibe. pocedue FeasLow(G,H,Glo,cg,ch )
step //initialize// i = ag maxgi //detemine the geatest component of G// i=,..., n fo = to do SU p = step while = sum < Glo and p Ncha = ag min cg =,..., sum = sum cg co = i ; p = p + endwhile stop SU + G L i do //detemine the smallest hitheto unchanged component of cg// cg = Gi L ; ch = H i L ; SU M = Figue 4. Algoithm to enfoce lowe bound feasibility fo fibe o POL. 4.3 Fitness To evaluate the individual s fitness, the simple an concept is used thus giving elevance to the dominance elations. Note that an is the basic element of many evolutionay algoithms (see [3]). The values of the two obective functions of SSVP fom (5a) and (5b), updated with unfeasibility penalizations, ae detemined fo all the N population s individuals, as seen in the algoithm Build. The obective values ae then used to apply the dominance concepts to the N individuals of the cuent population. In this way, all nondominated individuals, o potentially efficient solutions, within the cuent population, denoted as Paeto level solutions, ae assigned a an equal to. Hence, the fitness of each of these individuals, that is, of the espective chomosome co, epesented by Fit(co ), is set to. In the sequence, the emaining individuals in the cuent population ae once again classified as to the dominance elation and the nondominated ones, the Paeto level solutions eceive fitness equal to ½, that is, all such co have Fit(co ) = ½. While thee is an individual in the emaining population whose espective fitness has not been calculated, the pocess continues by assigning to each individual a fitness value equal to the invese of its level an. 4.4 Othe genetic featues 3
The dimension of the population, N, is equal in evey geneation of the evolutionay pocess. The maximum numbe of geneations was set equal to Nmax, this being the only stopping citeion used in the genetic algoithm GenSuga, whose pseudo-code is shown in Figue 5. This algoithm detemines a set S* of potentially efficient solutions fo the SSVP given by the feasible individuals that ae not dominated at the last geneation. As fo the opeatos, five basic opeatos ae applied on each cuent population, to ceate the population of the next geneation: selection, cossove, mutation, epai and elitism. The selection opeato acts N times on the population of the cuent geneation, Pop(nge), with the pupose of choosing a set of copy individuals, the so-called Pool, upon which the genetic opeatos of cossove, mutation and epai will be applied. The selection opeato is a standad binay tounament whose pioity is given to the individual beaing the lowest atio f f, that is, a feasible solution with a low cost and high enegy balance levels. The cossove acts N/ times, each involving a pai of andomly chosen individuals fom the Pool. The pocedue is a one-point cossove which poduces two childen fom the two paents. When a child is not feasible, it is epaied though the action of the epai opeato, a pocedue coesponding to steps 3 and 4 of the algoithm Build. Aftewads, each child eplaces one of the paents in the Pool, but only if it is significantly bette in tems of dominance elation than the paent is. In fact, the eplacement of a paent by a child occus should the child be feasible and its cost, f child, be at least Rep% lowe than the cost of the paent in question and, at the same time, its enegy balance, f child, should be Rep% o moe geate than the enegy balance of that paent. In case a child is Rep% bette than both paents it eplaces the wost paent and if each child is Rep% bette than at least one of the paents, both paents ae eplaced by childen. Then mutation applies with pobability p m Finally, within the elitist opeato pocedue, all the feasible individuals that ae not dominated in the pevious geneation s population, hee epesented by S*, ae included in the Pool and the S* less fitted individuals ae eliminated fom the Pool. The population fo the next geneation, Pop(nge+), is then set equal to Pool. on each gene of all the chomosomes of the Pool. In the event of a gene mutating, the sugacane vaiety fo the espective plot is andomly chosen by giving equal pobability to all the n vaieties. Again, if the mutant individual is not feasible, then the epai opeato is applied. 4
algoithm GenSuga step //initialize// nge = Pop() = φ fo = to N do //build population fo geneation // call Build(co, f, f ) Pop() = Pop() {co } endfo fo = to N do calculate Fit(co ) // fo all the individuals of Pop() // S* = set of feasible individuals fom Pop() that ae not dominated step epeat until nge = Nmax //geneation nge// select fom Pop(nge) N individuals fo the Pool apply cossove on the Pool and the epai opeato when necessay apply mutation on the Pool and the epai opeato when necessay Pool = Pool S* //apply elitism// detemine Pop(nge + ) fom Pool update S* nge = nge + endepeat stop //S* contains the potentially efficient solutions// Figue 5. Genetic algoithm fo the SSVP. 5. Computational expeiments In Sections 5. and 5. all data fo the SSVP instances used in the computational expeiments ae detailed: a eal case instance coesponding to a small company studied in [0], as well as othes geneated to simulate eal companies in the State of São Paulo, in Bazil. Fist, the bi-obective genetic algoithm GenSuga, pesented in Section 4, was computationally tested with the eal case along with an exact method, thus geneating the fist esults shown in Section 5.3. This section follows with the esults of the expeiments undetaen, whee algoithm GenSuga uns with all the othe instances. 5. Geneal data and the eal case The main data fo the eseach came fom an agicultual study of Lima [9] fo the abovementioned Bazilian egion and ae shown in the following tables. Table displays some data elated to the sugacane quality bounds, the cost, machiney fuel consumption, fuel pice, 5
tuc capacity, besides consumption of enegy, wheeas Table 3 contains the sugacane vaieties that ae suitable to the local soil and weathe conditions and all data elative to these vaieties. Table. Some data needed to apply the model. Slo L = Flo L = - Fup L = - Cl Vc Co P Elm Etm - 3 (t ha - ) (t ha ) (t ha ) (R$ t ) (m ) (L m ) (R$ L ) (MJ t ) (MJ m ) 8.4.00 5.00 3.69 60.00 0.37.5 65.0 5.5 Table 3. Vaieties data. - - - - vaiety i specification S F i i Pb i V i Ec i (t ha - ) - (t ha ) - (t ha ) (m t - ) - (MJ t ) SP80-86 6.4 3.94 33.36 7.964 67.99 RB7454 0.40.90 37.58 8.60 649.95 3 SP80-380 8.46.63 36.7 9.369 60.4 4 SP8-350 8.38.3 34.5 0.69 947.85 5 RB855536 7.05.5 6.43 9.78.95 6 RB8553 7.54 0.9 9.38 0.87 30.37 7 SP790 5.80 0.33 4.09 8.9 977.47 8 RB835486.84 9.8.53 9.56 444.0 9 RB7406 0.77 6. 33.0.3 008.83 0 SP70-43 5.0.59.4 7.05 94.80 SP70-84 3. 0.04 8.37 7.74 56.45 SP7-663.74 9.65 3.57 8.7 0.00 3 SP7-346.86 0.33 0.5 9.56 963.06 4 NA56-79.84 9.8 7.4.3 3.4 5 RB806043 0.77 4..53 9.08 408.96 6 RB835089 4.48.5 33.30 8.60 98. 7 SP87-365 7.03 3. 0.96 9.00 57.80 8 SP80-84 6.5.7.90 9.60 749.85 9 RB943365 7.80.99 3.40 9.80 50.4 0 RB95 5.64.3 7.3 0.90 47.9 The eal case unde study involves the fist 0 vaieties of Table 3. Moeove, Table 4 includes the espective plot aeas and the distances between each plot and the mill. In this study, the maximal aeas pe vaiety ae not elevant, which is why values fo Lup i ae not povided. We may assume that Lup i = L. Table 4. Plot data. = 6
plot D (m) L (ha) 3.49 8.49.49 4.5 3 6.08 58.8 4 3.49 4. 5.59 5.74 6.59 6.6 7 5.33 30.4 8 8.30 5.08 9 9.4.0 0.63 54.95 6.43 38.66 8.5 3.78 3 7.80 0.43 4 8.59 6.5 5.5 8.79 6 7.0 57.79 total 35.8 5. Othe test instances A test set of 80 instances was designed to cope with the eal life cases aising in the State of São Paulo. Eight sets of 0 instances each wee built: fou sets of instances with 0 sugacane vaieties, pecisely those consideed fo the eal case, and the othe fou sets with instances containing all the 0 vaieties found in Table 3. Fo all such instances, the paametes Cl, Vc, Co, P, Elm and Etm assume the values indicated in Table, as well as the POL and fibe equiements pe hectae, also given in this table. Fo each instance, the total aea of the field to be planted, the numbe, shapes and aeas of the espective plots and the location of the mill, as well as the distances D wee detemined to simulate eal situations in the egion. Table 5 summaizes some of these options and the semiandom pocess is descibed though the following paagaphs. Table 5. Main chaacteistics of all the test instances. instances nb field aea n instances (ha) I0 eal case 35.8 0 6 I to I0 0 405.0 0 0 I to I0 0 405.0 0 0 I to I30 0 5.0 0 60 7
I3 to I40 0 5.0 0 60 I4 to I50 0 3 645.0 0 80 I5 to I60 0 3 645.0 0 80 I6 to I70 0 6 075.0 0 300 I7 to I80 0 6 075.0 0 300 All cases wee geneated on the basis of small fields of 405 ha, each consisting of 0 plots, aggegated in fou tacts, and built though a sequential pocess descibed by the pseudo-code in Figue 6. algoithm SimulSmall step //initialize// choose the patten step choose the mill location calculate the distance lining the mill to the neaest point at the edge of the field, point X step 3 //design the plots// define the 0 field s plots by splitting each tact into a cetain numbe of plots fo each plot calculate the distance fom the cental point of the espective tact to X complete the distances calculation fo the plot endfo stop Figue 6. Algoithm to build a semi-andom small instance of SSVP. Thus, when building a semi-andom instance of 405 ha (cases I to I0), the algoithm SimulSmall stats up by selecting a patten fo the plantation aea, step. The patten is andomly chosen with a unifom discete distibution fom the fou pattens shown in Figue 7. patten patten 500 m 700 m 3.80 m 3 085 m tact 9 plots 3 tact plots tact plot 4 tact 0 plots tact 3 4 plots tact 3 3 plots tact 4 6 plots tact 4 5 plots 3 4 patten 3 patten 4 3 4 8
90 m 0.4 m 00 m 840.9 m tact 7 plots 3 tact 3 plots tact 3 plots tact 3 plots tact 3 3 plots 4 tact 3 5 plots tact 4 7 plots tact 4 9 plots Figue 7. Field pattens fo the small test instances. The mill s location step of the algoithm is then andomly chosen (unifom discete distibution) among fou possibilities, as illustated in Figue 8. In this figue, the ectangle epesents the entie small field to be planted. The hypothetical distances of the possible locations of the mill to the neaest bode of the field ae calculated. 5 m 0 m 4 m 3 Figue 8. Mill s possible location and distances fo the small instances. 4 0 m In step 3 of the algoithm, each tact of the patten is split into a cetain numbe of equal aea plots as indicated in Figue 7, thus defining the 0 plots fo the instance. The lining distances to the mill, fo all the plots to be found in a tact, ae calculated fom the cental point of the tact to the edge of the field (point X detemined in step ), thus assuming that each tuc goes staight though the field to that bode point. We ae dealing with Euclidian distances. Figue 9 illustates how the distances ae calculated. Conside that patten and location have aleady been selected, as well as the Euclidean distance between the mill and the bode point X. The distances fo the thee plots of tact 3 (see Figue 7) emain to be detemined. Fist, the Euclidean distance fom the cental point of tact 3 to bode point X is calculated. The two distances ae then added and finally doubled, beaing in mind that the tuc is supposed to go fom the plot to the mill and bac again. The esulting numbe is the simulated distance fo all the thee plots of tact 3. 3 4 9
X Figue 9. Illustation of the distances calculation fo the small fields. All the small instances I to I0 and I to I0 wee obtained by esoting to the pocedue descibed above. The thee othe goups coespond to medium-sized dimension fields I to I40 and I4 to I60 and high dimension fields I6 to I80. All wee built by gouping a specific numbe of small fields consisting of 405 ha. Simila sizes and shapes ae commonly found in the companies in the egion addessed. Figue 0 illustates a medium dimension case of 5 ha. All instances of this same aea ae built fom thee small fields andomly chosen fom the fou possible pattens in Figue 7. Let us suppose that by following andom selection, patten was chosen once and patten chosen twice. The thee small fields ae assumed to be oined in sequence, as seen in the figue. One stats by andomly geneating the position of the mill. In the sequence, the distances fo each plot ae calculated, as in the case of the small instances, on the assumption that the tucs coss the small field whee the plot is located taing the diection of the point on its edge that is closest to the mill and then continue bodeing the small fields up to the point of the 5 ha field that is closest to the mill. 3 4 3 4 3 4 3 3 Figue 0. Illustation of a medium size instance of 5 ha. Each of the fields of 3 645 ha coesponds to nine small fields and each of the 6 075 ha ones is made up of 5 small fields. All ae semi-andomly built following a pocess simila to the one above descibed fo the 5 ha cases. 5.3 Computational esults A computational expeiment was undetaen to study the behavio of the genetic algoithm GenSuga with the instances we descibed above. The feasible individuals/solutions found at the last geneation that ae not dominated by any othe of the population, above epesented by S*, ae the so-called potentially efficient o, simply, heuistic solutions. The coesponding points in the obectives space build the last geneation fontie. 0
All the pogams wee coded in MATLAB [] and an on CORE QUAD computes with.83 GHz and G RAM at the Depatment of Biostatistics of UNESP in Botucatu, Bazil. On the basis of peliminay expeiments, the algoithm paametes wee set equal to N = 300, Nc = 4, Ncha = 30, Nmax = 000, p m = 0.05 and Rep = 5. As mentioned in Section 5., the maximal aea constaints wee not consideed in this computational study. Multi-obective non-exact algoithms ae developed to find a set of solutions which points in the obectives space ae as close as possible to the (tue) Paeto fontie of the multi-obective poblem, besides being as divese as possible within the set. The pefomance of these appoaches has been studied by many authos who have poposed measues fo the quality of the solutions obtained, namely Schott [7], Van Veldhuizen [9] and Zitzle, Deb and Thiele [30]. In [4] and [6] one can find detailed desciptions and comments on the subect. In this study, to evaluate the degee of achievement of the algoithm GenSuga thee pefomance measues ae used: the geneal distance (GD), the spead (Sp) and the coefficient of vaiation (CV). Fist, GD measues the poximity between the heuistic points and the nondominated points taen fom an exact appoach, by calculating the aveage distance: GD = N g d = N g sq () whee d sq = h= min,..., N e m= * h sq ( f f ) ( =,..., N g), i.e., d is the Euclidean distance between m m the -th heuistic and the neaest nondominated point; f m is the m-th obective function value especting the -th heuistic point; f *h m is the m-th obective function value especting the h-th nondominated point; N g and N e epesent, the numbe of heuistic and nondominated points, espectively. A good quality heuistic poduces esults with a small value fo GD. The spead, Sp, measues the extent of spead achieved among the heuistic points. It shows how well the Paeto fontie is coveed by the last geneation fontie: Sp = d m= e m d m= N g + d e m = + N su g d d su su ()
whee d m e epesents the distance between the exteme point, fo obective m, of the last geneation fontie and the espective lexicogaphic point in the (tue) Paeto fontie (m=,). Wheneve the Sp value is close to zeo, the spead is low. Finally, the spacing measue S povides the divesity among the genetic heuistic points. It is calculated fom the elative distance between consecutive points in the last geneation fontie, as follows: S = N N g g = su su ( d d ) N g su su s su d whee d = min fm fm ( =,..., N g) and d =. As s=,..., N ; s g m= = N g S epesents the standad deviation of the distances, it is highly dependent of the values of the cost and enegy balance themselves. Hence, we use the coefficient of vaiation, CV, which gives the popotion of this vaiation elatively to the aveage: S CV =. su d Let us stat by consideing the detailed esults fom the application of GenSuga to the eal instance descibed in Section 5., denoted by I0, with a total aea of 35.8 ha. Figue shows the effect of the genetic evolution on the initial population. It depicts the points coesponding to the initial population of GenSuga, epesented by (4), and the heuistic points, that is, the ones which ae associated with the potentially efficient solutions of the last geneation, epesented by small cicles. In accod to the definition, all these last points coespond to feasible solutions of SSVP. But the initial population s points may coespond to feasible o unfeasible solutions. Actually, both the initial individuals/solutions, built by the algoithm Build, and the ones geneated duing the genetic evolution ae subect to the disuptive/constuctive pocedues (Figues 3 and 4) to foce satisfaction of all constaints of the model howeve, in many cases, these pocedues alone ae not effective. (3) Fo puposes of compaison we an the ε-constaint method (see, fo instance [6]) to obtain efficient solutions which coespond to points of the (tue) Paeto fontie of the bi-obective poblem (5a) to (0). The ε-constaint method stats by minimizing cost subect to constaints on the enegy balance. Howeve, as this method taes consideable CPU time, we divided the enegy balance ange into 0 intevals to obtain a maximum of 0 efficient solutions and also
imposed a CPU time limit of 5 minutes to solve each (single obective) binay linea pogamming poblem with MATLAB ILP solve []. If the CPU time bound becomes estictive in attaining o poving optimality of the cuent best solution, even if it is feasible, that solution might not be efficient and is discaded. Consequently, the maximum numbe of efficient solutions obtained by unning the appoach is 0, signaling that the maximum cadinality of the Paeto fontie appoximation by the ε-constaint method in this expeiment is 0 as well. Figue shows the appoximation of the Paeto fontie fom the ε-constaint algoithm epesenting such nondominated points with the symbol +. The two exteme points of this set ae appoximate lexicogaphic points (identified in this gaphic by the symbol ). Finally, the gey point coesponds to a solution that was detemined and actually applied at the company [9]. x 0 7 3.8 Enegy Balance (M).6.4..8.6 3.5 4 4.5 5 5.5 6 6.5 Cost (R$) x 0 4 Figue. Results fom GenSuga fo the eal instance I0. 3
This figue displays points with a positive enegy balance, anging fom moe than.9 0 7 MJ to ove 3 0 7 MJ, values valid fo the heuistic points fom the genetic algoithm and fo the nondominated points obtained with the ε-constaint algoithm. This situation complies with the chaacteistics of the sugacane vaieties that ae ich in tems of enegy poduced fom sugacane esidue, wheeas the espective tansfe cost is not that high, anging fom about 4. 0 4 R$ to about 5.6 0 4 R$. As fo the optimization behaviou of the genetic algoithm, its esults loo vey good indeed, with an almost pefect visual ovelapping of the heuistic points with the nondominated points, at least fo the eal instance. Notice that each one of the points detemined eithe by GenSuga o by the ε-constaint algoithm fo the instance I0 coesponds to a diffeent solution. In othe wods, thee is a twoway coespondence between the set of solutions (in the solution space) and the set of points (in the obective space). This fact, not common within multi-obective optimization, esults fom the data of this specific SSVP eal instance. Table 6 pesents futhe esults of the genetic and the ε-constaint algoithms fo the instance I0. GenSuga was independently applied 0 times and the espective esults ae aveages pe un. The table shows computing times (in seconds) in column (5) the total un time of the ε-constaint algoithm and in (6) the aveage CPU time pe un of GenSuga as well as some figues to access the quality of the best solutions obtained at the last geneation of GenSuga. Fistly, the table contains the anges the diffeence between the maximum and minimum values obtained fo cost and enegy balance at the fontie points fom the ε-constaint and the genetic algoithms, in columns ()-() and (6)-(7), espectively. Ranges fo cost and enegy balance obtained fom unning the GenSuga and ε-constaint algoithms have aleady been seen in Figue above. Then the aveage numbe of sugacane vaieties found in the solutions coesponding to the fontie points is pesented in columns (3) and (8), espectively. This table also includes the following cadinality measues: the numbe of nondominated points obtained with the ε-constaint method, N e, in column (4); the numbe of feasible solutions (o points) at the initial and final populations of GenSuga, in columns (9) and (0), espectively; and the cadinality of the last geneation fontie, that is, numbe of heuistic points, N g, in column (). Column () indicates the numbe ND epesenting the heuistic points that ae not dominated by any of the nondominated points fom the ε-constaint algoithm. Columns (3) to (5) show the 4
thee pefomance indicatos given at the beginning of this section though the expessions (), () and (4): the geneational distance using nomalized data, GD, the spead metic, Sp, and the coefficient of vaiation, CV, all calculated fo the eal case. Table 6. Computational esults fo the eal case, I0 instance. cost (R$) () ε-constaint algoithm anges measue CPU time enegy balance nb N e (s) (MJ) vaieties () (3) (4) (5) 448.9 45090.7.9 9 6 000.6 GenSuga algoithm (aveages fom 0 uns) cost (R$) (6) anges measues CPU time enegy balance nb nb initial nb final N g ND GD Sp CV (s) (MJ) vaieties feasible feasible (7) (8) (9) (0) () () (3) (4) (5) (6) 4 37.69 33 84.96 3.7 93.6 300.0 0.7 63.8 0.0004 0.39 0.4 0.3 In this table we note that the ε-constaint algoithm too an excessive time to un, that is to geneate the Paeto fontie appoximation which amounts to 9 efficient solutions, as seen in column (5). Hee, one can see that each efficient solution ased fo an aveage of about 85.9 seconds (.4 minutes), figue calculated fom (6 000.6 9 900) / 9. Moeove this algoithm used about 35 minutes (5 minutes pe each of the 9 failues of the ILP solve) without poducing efficient solutions, figue aleady included in the time of column (5). Column () shows that the GenSuga attained on aveage about 63 potentially efficient (feasible) solutions. And all these wee geneated in about 0 seconds CPU time, on aveage, which is fa less time than the ε-constaint algoithm too. Results egading pefomance measues fo the genetic algoithm match the gaphical esults found in Figue. In fact, GenSuga poduced a set of good quality solutions, measued by the insignificant distance value GD = 0.0004, on aveage, along with a well spead last geneation fontie measued by an aveage value of Sp=0.39. It also povided divese points in the last geneation fontie, measued by CV=0.4 on aveage. It should be noted that the eal solution the company actually adopts, and which was poduced by the technicians at the mill, involves 4 sugacane vaieties, not that diffeent fom the aveage numbe of vaieties at the heuistic solutions (3.6 vaieties on aveage). Its values fo the cost and enegy balance ae 49 69.09 R$ and 9 956 05.4 MJ, espectively, coesponding to 5
the gey point aleady plotted in Figue. This solution is indeed highly dominated by heuistic solutions. Next, Table 7 shows the esults fom the expeiment with the 80 simulated SSVP instances. The main featues of the test set numbe of instances pe subset, the total aea, numbe of vaieties and numbe of plots have aleady been summaized in Table 5. Column () identifies the instances, the next eight columns, () to (9), eveal the same type of infomation as columns (6) to (), (5) and (6) in Table 6 fo the eal case. All these figues found in Table 7 ae aveages pe instance fo the ones identified in column (). Due to the unacceptable computing expenses necessay to exactly solve these instances of SSVP the ε-constaint method was applied only fo a single enegy balance subinteval. This was andomly chosen among 0 equal amplitude subintevals of the enegy balance ange. Also the CPU time limit of 5 minutes was again imposed to solve each (single obective) binay linea pogamming poblem with MATLAB ILP solve. With such options the algoithm could detemine a maximum of one efficient solution pe instance. Column (0) shows the aveage CPU time pe efficient solution along with the numbe of efficient solutions obtained fo the espective goup of instances. Table 7. Computational esults fo the 80 SSVP simulated instances. instances GenSuga algoithm ε-constaint algoithm anges enegy balance (MJ) measues nb final feasible cost (R$) nb vaieties nb initial feasible N g CV CPU time (s) CPU time pe efficient sol. (s) / nb efficient sol. () () (3) (4) (5) (6) (7) (8) (9) (0) I to I0 3 358.6 50 384.65 3. 300.0 300.0 85.5 0.4 04.7 40.7 / 3 I to I0 9 635. 0 940 795.89 3.3 300.0 300.0 70.9 0.3.5 57.4 / I to I30 60 657.56 35 347 83.0 7. 83.6 300.0 08.3 0.09 66.6 - I3 to I40 84 406.79 4 7 76.87. 8.6 300.0 06.6 0..0 - I4 to I50 70 634.94 09 30 0.56 9.5 3.6 300.0 07.7 0.0 590.3 - I5 to I60 60 90.75 3 60 969.9 4.6 07.5 300.0 09.9 0.0 36.6 - I6 to I70 36 70.98 75 75 965.6 9.8 6. 300.0 08.7 0.09 0.4 - I7 to I80 340 50.75 3 39 87.9 4.7. 300.0 03.3 0.09 85.0 - The cost and enegy balance of the heuistic solutions (see columns () and (3)) define wide anges, thus poviding the mill management with options involving diffeent economic impacts. The numbe of vaieties selected to cop, column (4), is in eeping with the pactical figues in the egion. The following esults in this table show that the genetic algoithm 6
significantly impoved the feasibility of the populations fom the fist (column (5)) to the last geneation (column (6)) whee all the individuals ae feasible fo all instances. As seen in column (7), fo the smallest instances moe than half of the last population points ae not dominated within the population. Fo all the othe cases the ones with 60, 80 o 300 plots this figue is a little above the thid of the population dimension. In addition, the spacing of the points at the last geneation fontie is easonable. Effectively, the coefficient of vaiation in column (8) is, on aveage, always less than 4 % fo all the cases. These esults ae faily consistent with those obtained fom the eal instance I0 (see Table 6). Fom Table 7 we may also see that the ε-constaint algoithm equies excessive computing time, mainly in case of the big instances of SSVP, those consisting of 60 to 300 plots with 5 o moe ha, as shown in column (0). In fact, the algoithm did not obtain a single efficient solution fo any of these instances within the CPU time limit of 5 minutes. Moeove, it only found a feasible solution fo fou out of the 0 smalle instances: thee fom the set I to I0 in 40.7 seconds of CPU time on aveage and one fom the set I to I0 in 57.4 seconds. Fo all the othe cases the algoithm attained the CPU time limit without success. We also tied a highe time limit, 0 minutes, but only managed to solve moe thee instances (figues not given in the table). Note that the unning time equied to obtain all the potentially efficient solutions with the genetic algoithm GenSuga always too less than 03 seconds on aveage pe instance (see column (9)). 6. Final comments This pape poposes a new bi-obective genetic heuistic fo the the selection of sugacane vaieties complying with economic and poduction equiements poblem. In addition to the basic genetic components the algoithm embeds constuctive and feasibility heuistic pocedues that evealed a stong effect in the computational esults. To the authos nowledge, all these heuistics have neve been developed fo the poblem and can be used fo any eal dimension instances at low computing expenses. 7