UNinhabited aerial vehicles (UAVs) are becoming increasingly

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1 A Process Algebr Genetic Algorithm Sertc Krmn Tl Shim Emilio Frzzoli Abstrct A genetic lgorithm tht utilizes process lgebr for coding of solution chromosomes nd for defining evolutionry bsed opertors is presented. The lgorithm is pplicble to mission plnning nd optimiztion problems. As n exmple the high level mission plnning for coopertive group of uninhbited eril vehicles is investigted. The mission plnning problem is cst s n ssignment problem, nd solutions to the ssignment problem re given in the form of chromosomes tht re mnipulted by evolutionry opertors. The evolutionry opertors of crossover nd muttion re formlly defined using the process lgebr methodology, long with specific lgorithms needed for their execution. The vibility of the pproch is investigted using simultions nd the effectiveness of the lgorithm is shown in smll, medium, nd lrge scle problems. Index Terms Genetic lgorithms, process lgebr, UAV tsk ssignment. I. INTRODUCTION UNinhbited eril vehicles (UAVs) re becoming incresingly effective in performing missions tht hve previously been performed by mnned irplnes. Their efficcy minly stems from the lck of n on-bord humn opertor. This enbles development of systems with significnt weight svings, lower costs, nd llows performnce of long endurnce tsks. Currently, bsic tsks of UAVs such s flying nd trjectory plnning from wy-point to wy-point cn be utomted. To enble the simultneous coopertive opertion of multiple such systems in complex missions, higher levels of utonomy re constntly sought. Within the lst decde, coopertive control lgorithms hve been proposed to coordinte such multi gent systems in preferbly optiml wy (see for exmple Refs. [], [2], [3], [4], [5]). Recently, coopertive control lgorithms hve been extended to hndle more complex tsks nd constrints, clled the mission specifictions, which re expressed using forml lnguges (see for exmple Refs. [6], [7]). These mission specifictions include, but re not limited to, combintions of temporl ordering mong tsks s well s conjunctive nd disjunctive logicl constrints. Specifiction lnguges with strict dedlines to specify nd solve more complex UAV missions lso hve been considered (see, for exmple, Ref. [8]). In Ref. [7], Process Algebr (PA) is used to specify clss of complex coupling constrints between tsks in UAV missions. This pper lso dopts PA s the specifiction lnguge for resons to be outlined shortly. In computer science, Sertc Krmn is with the Deprtment of Electricl Engineering nd Computer Science, Msschusetts Institute of Technology, Cmbridge, MA, 0239; sertc@mit.edu Tl Shim is with the Fculty of Aerospce Engineering, Technion - Isrel Institute of Technology, Hif, 32000, Isrel; tl.shim@technion.c.il Emilio Frzzoli is with the Deprtment of Aeronutics nd Astronutics, Msschusetts Institute of Technology, Cmbridge, MA, 0239; frzzoli@mit.edu process lgebr is used for resoning bout the time behvior of softwre. For such resoning, the softwre is ssumed to be ble to execute ctions from set A in some order. An ction is very generl bstrction; it my refer to blink of light, writing into file, or moving robotic rm. A behvior of the system is, then, sequence of ctions which re ordered with respect to the order tht they were executed by the softwre. The sequence (, 2,..., n ) for which, 2,..., n A, for instnce, would be behvior of the system. Then, using the process lgebr terms one cn indeed specify the set of behviors tht system cn exhibit. This cn be used s design specifiction for utomtic genertion of softwre; or it cn be used for checking whether given softwre stisfies such criterion. In mny militry multiple-uav missions, individul tsks like re serch, clssifying, or destroying trget, cn be coupled with ech other with temporl nd logicl constrints. Intuitively, these tsks, which we will refer to s tomic objectives, correspond to the ctions, wheres the coupling constrints will be represented by the process lgebr terms. High level tsks will be described using sets of tomic objectives, coupled through process lgebr terms. We will denote such high level tsks s complex objectives. In the end, the entire UAV mission cn be given s single specifiction, i.e., single complex objective. The ssignment of multiple cooperting UAVs to multiple tsks, such s collections of the tomic objectives mentioned bove, requires the solution of combintoril optimiztion problem. The significnt difficulty in solving mny combintoril optimiztion problems is tht they re NP-hrd nd therefore cnnot be solved in polynomil time by deterministic methods. So, due to the prohibitive computtionl complexity of the problem, the trditionl deterministic serch lgorithms provide n optiml solution only for smll-sized problems. For lrge sized problems they my provide fesible solution in given bounded run-time. Approximtion lgorithms cn lso be used for solving such problems. These lgorithms give solution of cost J to the problem with optiml cost J* such tht the rtio J/J* is bounded by known constnt [9]. If optimlity is not sought nd the gol is to obtin good fesible solution quickly, then stochstic serch lgorithms tht employ degree of rndomness s prt of their logic cn be used. An lgorithm of this type uses rndom input to guide its behvior in the hope of chieving good performnce in the verge cse nd converge to good solution in the expected runtime. Evolutionry lgorithms (EA), which re inspired by the muttion selection process witnessed in nture, re common stochstic serch methods used to solve mny combintoril optimiztion problems. These methods involve itertively mnipulting popultions of solutions, termed chromosomes, tht encode cndidte good

2 2 solutions. The genertionl process is performed by pplying evolutionry opertors like selection, crossover, nd muttion. Cndidte solution selection is performed by evluting the fitness (commonly chosen s inversely proportionl to the cost) of ech chromosome in the popultion. Historiclly, there were three min types of EAs: genetic lgorithms (GAs), evolutionry strtegies, nd evolutionry progrmming, with GA being the most populr one [0]. Much work in pplying GAs to combintoril optimiztion problems is concerned with the encoding of the chromosomes nd the use of specil crossover opertors tht preserve the vlidity of the solution defined by the chromosome. The encoding nd definitions of the evolutionry opertors re problem specific. Recently, GAs hve been proposed for solving UAV coopertive ssignment problems [], [2]. In Ref. [], GA ws proposed for scenrio where homogeneous set of multiple UAVs cooperte in performing multiple tsks (such s clssify, ttck, nd verify) on multiple sttionry ground trgets. Solving such problems required ssigning different tsks to different vehicles nd consequently ssigning ech vehicle with flyble pth tht it must follow. In Ref. [2], GA ws used to solve coopertive UAV ssignment problem where trgets required simultneous ctions from severl UAVs. In both of these studies simultion results showed the effectiveness of GAs in providing in rel-time good qulity suboptiml fesible solutions. Evolutionry lgorithms hve lso been pplied to relted problems, such s the vehicle routing problem [3], [4]. Our work in this pper is mostly relted to our previous work in Refs. [3], [6], [7], [8], [5], [], [2], [6]. In Ref. [6], [8], [5], forml lnguges such s Liner Temporl Logic (LTL) nd Metric Temporl Logic (MTL) were employed to describe complex tsks nd constrints in militry UAV opertions. Although LTL nd MTL re highly expressive specifiction lnguges, the lgorithms presented in Ref. [6], [8] re limited to smll problem sizes due to computtionl intrctbility of checking whether specifiction given in LTL or MTL cn be stisfied. To hndle lrger-scle problems more effectively, in Ref. [7], computtionlly more trctble process lgebr specifictions were incorported into tree serch bsed tsk ssignment lgorithm (see Ref. [3] for the detils of tree serch). The computtionl efficiency of the lgorithms tilored to hndle process lgebr specifictions mde their implementtion on stte-of-the-rt UAV pltforms possible. The lgorithm proposed in [7] ws recently demonstrted on tem of three UAVs in joint U.S.A.-Austrlin militry exercise in Austrli [7]. Process lgebr type specifictions, if not s expressive s the temporl logics, were shown to describe brod clss of complex mission specifictions of prcticl importnce in Ref. [7]. Genetic lgorithms, on the other hnd, were used in Refs. [], [2] to improve the computtionl effectiveness of the tsk ssignment lgorithms; however, their integrtion with complex mission specifictions were never considered. This pper fills this gp by proposing computtionlly effective lgorithm, which cn yet hndle brod clss of complex tsks specified using process lgebr. The min contribution of this pper is genetic lgorithm solution to n ssignment problem with high-level specifictions represented vi process lgebr terms, s well s the PA bsed definition of the evolutionry opertors of crossover nd muttion. The pper is orgnized s follows. Nottion is provided in the next section. In Section III the process lgebr specifiction frmework is introduced. Section IV is devoted to the specifiction of complex multiple UAV missions using PA. Then, the GA-bsed tsk ssignment lgorithm tht cn hndle PA specifictions is given in Section V, followed by the results of Monte-Crlo simultion study which is presented in Section VI. The pper is concluded with remrks in Section VII. Proofs of importnt results re given in the ppendix. II. NOTATION The sets of nturl numbers, positive nturl numbers, rel numbers, nd positive rel numbers re denoted by N, N +, R, nd R +, respectively. A finite sequence (of distinct elements) on set S is n injective mp σ from {, 2,..., K} to S where K N +. A finite sequence will often be written s σ = ( σ(), σ(2),..., σ(k) ). For the ske of brevity, finite sequence will be simply referred to s sequence from this point on. An empty sequence, specil structure used in the pper, is represented by δ. An element s is sid to be n element of sequence σ, denoted by s σ, with slight buse of nottion, if there exists k {, 2,..., K} such tht σ(k) = s. The nottion σ is used to denote the number of elements of sequence σ, i.e., σ = K. Given two sequences σ nd σ 2 both defined on the sme set S, we will denote their conctention by σ σ 2, which itself is lso sequence defined on the set S with domin {, 2,..., σ + σ 2 }. More precisely, (σ σ 2 )(k) = σ (k) for ll k {, 2,..., σ } nd (σ σ 2 )( σ + k) = σ 2 (k) for ll k {, 2,..., σ 2 }. The conctention of sequence σ with the empty string δ is σ itself. For ny two elements s, s 2 S, the ordering reltion < σ defined on the elements of the sequence σ is formlized s follows: s < σ s 2 if there exists i, j {, 2,..., K} with i < j such tht σ(i) = s nd σ(j) = s 2. Given sequence σ defined on set S, n order preserving projection of σ on to set S S is defined s the sequence σ, for which the following hold: (i) for ll s σ, we hve tht s S implies s σ, nd (ii) for ny s, s 2 σ, s < σ s 2 implies s < σ s 2. Given sequence σ defined on set S, its order preserving projection on to set S S will be denoted s [σ] S. Given set S, we denote the set of ll sequences on set S by Σ S. III. PROCESS ALGEBRA Most engineering systems hve set of ctions tht llows them to communicte with the outer world or mnipulte the objects therein to ccomplish high-level tsk. Of course, in this context, the definition of the high-level tsk, s well s the ctions, depend on the grnulrity of the bstrction; but we will ssume tht these notions re such tht the system under considertion will be designed to hndle only one high-level tsk nd the ctions re tomic in the sense tht they cn not be ccomplished by executing sequence of other ctions. Even though systems tht do not terminte nd operte in persistent mnner exist nd they re interesting in their own

3 3 right, most of the rel-world systems execute sequence of ctions, which eventully led to ccomplishment of the highlevel tsk nd termintion of the system. In the rest of the pper, we will ssume tht ech high-level tsk, if it cn be ccomplished t ll, cn be ccomplished by terminting execution of the system, i.e., finite sequence of ctions. Such n execution is clled behvior of the system. It is importnt to note t this point tht, most of the time, such behvior tht leds to successful fulfillment of the requirements is not unique, nd it is crucil nd chllenging to nturlly nd formlly specify the set of desired behvior of system. Moreover, given the specifiction, designing lgorithms tht utomticlly enble the system to fulfill the specifiction is importnt nd chllenging in its own right. Along with mny other formliztions such s temporl logics [8], µ clculus [9], or Petri nets [20], process lgebr [2], [22], [23] is methodology tht cn be used to specify the desired behvior of system. Initited nd used in Computer Science to reson bout computer softwre, process lgebrs found mny pplictions in severl diverse fields from web pplictions to hybrid systems [24]. Using process lgebr for specifiction of UAV missions ws first considered in Ref. [7]. This section presents n introduction to the process lgebr bsed specifiction frmework of Ref. [7]. An importnt notion in process lgebr is the definition of the set of terms, which is formlly given s follows: Definition III. (Terms of Process Algebr) Given finite set A of ctions, the set T of PA terms (defined on A) is defined inductively s follows: ech ction A is in T; if p, p T holds, then p + p T, p p T, nd p p T lso hold. Ech element of T is clled PA term, or term for short. Terms re relted to ech other s they cn evolve from one to nother. This evolution is mde through trnsition, denoted s p p, where p, p T, nd A. This trnsition is red s Process p cn evolve into process p by executing the ction. There is lso specil process, denoted s, which corresponds to the terminted process. By definition, the process hs no ctions to execute nd cn not evolve into ny other process. Following the definition of the set of terms, ech ction cn be term such s p =, where A. Informlly speking, the system specified s hs only one behvior: it cn execute nd then terminte, which is denoted s. To specify more complex systems, the PA specifiction frmework offers the opertors (+), ( ), nd ( ), which re clled the lterntive, sequentil, nd prllel composition opertors, respectively. Intuitively, process tht is specified with the term p + p behves either like p or p, i.e., either executes behvior of p or executes one of p (but not both). The term p p, on the other hnd, first executes behvior of the process p, nd right fter p termintes, it executes behvior of p. The process The words evolve nd evolution re used here in the context of process lgebr, nd they should not be confused with the sme terms commonly used in describing evolutionry lgorithms. p p is sid to terminte when p termintes. The process p p executes behvior of ech of p nd p concurrently. The process p p is sid to hve terminted, when both p nd p terminte. This informl presenttion of the behvior of processes cn be formlized by the opertionl semntics, which is defined s set of trnsition system specifictions (TSSs). A TSS is composed of set H of premises nd conclusion π, denoted s H π, where π is trnsition nd H is set of trnsitions. A TSS sttes tht if the premisses H re possible trnsitions, then so is the trnsition π. The semntics (mening) of ech PA term is defined using the opertionl semntics of process lgebr given s follows: Definition III.2 (Opertionl Semntics of PA) The opertionl semntics of the process lgebr is given by the following set of trnsition system specifictions: p 2 p p p p 2 p 2 p 2 p + p 2 p + p 2 p p + p 2 p + p 2 p 2 p p p 2 p2 p p p p 2 p p 2 p p p p 2 p 2 p p 2 p2 p p 2 p p 2 p p 2 p p p 2 p p 2 where A nd p, p, p 2, p 2 T Notice tht the first TSS formlly sttes tht ny ction A cn execute nd then evolve to the terminted process, without requiring ny other premisses to hold. The next four TSSs provide the semntics of the lterntive composition opertor. Essentilly, the second TSS sttes tht if process p cn execute n ction nd evolve to the terminted process, then so does the process p + p 2 for ny PA term p 2. The next three TSSs complement the semntics with other cses. The other TSSs in Definition III.2 provide the semntics of the sequentil nd prllel composition opertors similrly. With its recursive definition, the opertionl semntics ssocites ech process with set of trces tht the process cn execute. This set is merely the set of ll behviors of the system. More formlly, ny sequence σ = (, 2,..., k ) of ctions is clled trce of term p 0 if nd only if there i pi exists set of processes p, p 2,..., p k such tht p i for ll i {, 2,..., k} nd p k =. In other words, trce of process is behvior of the process s it is sequence of its vilble ctions which led to successful termintion. Moreover, the set of ll such trces of process p 0, which will be denoted by Γ p0, formlly defines the behvior of the process p 0. Exmple Let A = {, 2, 3, 4, 5 } be set of ctions. Consider the process p := which cn execute the ction nd terminte. Thus, is legitimte trnsition, which yields the trce ( ). Notice tht this is the only trce of p ; hence, the behvior of is the set {( )}, which includes its only trce.

4 4 Let us consider p 2 := + 2. Notice tht, for this process, + 2 is legitimte trnsition, since is lso legitimte trnsition (recll the second TSS in the opertionl semntics). Hence, ( ) is trce of p 2. Furthermore, notice tht ( 2 ) is nother trce. Thus, the behvior of p 2 is the set {( ), ( 2 )} of its trces. Finlly consider lrger exmple: p 3 := ( + 2 ) ( 3 ( 4 5 )). Proceeding s bove, the behvior of p 3 cn be determined s {(, 3, 4, 5 ), (, 4, 3, 5 ), (, 4, 5, 3 ), ( 2, 3, 4, 5 ), ( 2, 4, 3, 5 ), ( 2, 4, 5, 3 )}. Ech process lgebr term cn be represented by specil dt structure clled prse tree. The prse tree of process lgebr term is binry tree composed of nodes, ech of which encode either n opertor or n ction from set A of ctions. More precisely, ech lef node in the tree encodes n ction nd every other node (this includes the root if the prse tree is not single node) encodes n opertor. Given term p T its prse tree is recursively defined s follows: If p A, i.e., p is n ction itself, then the prse tree of p is single node which is lbeled with p. If p = p p 2, where {, +, }, then the prse tree of p is binry tree which is rooted t node lbeled with nd hs the prse tree of p nd p 2 s its left nd right children, respectively. Exmple Consider the process ( + 2 ) ( 3 ( 4 5 )). The prse tree of this process is presented in Figure. Notice tht this prse tree is indeed combintion of the prse tree of the two processes + 2 nd 3 ( ), bound with the sequentil composition opertor. Notice lso tht the former process hs prse tree formed by binding the prse trees of nd 2 with n lterntive composition opertor. The prse tree of 3 ( ) cn lso be investigted with its subtrees, similrly. Fig.. Prse tree of the process ( + 2 ) ( 3 ( 4 5 )). Some of the lgorithms tht will be introduced in the next sections hevily employ the prse tree of the given specifiction. To render these lgorithms more redble, let us present some nottion, which will be used throughout the pper. Let N p denote the set of nodes in the prse tree of process p, nd let n be node from the set N p. Then, the function Prent p (n) : N p N p {δ} returns the prent node of given node. If n hs no prent, then we hve Prent p (n) = δ. The function Lef p (n) : N p {flse, true} returns True if node n is lef, i.e., it hs no children, nd Flse otherwise. As mentioned before, ech node in the tree encodes either n opertor or n ction. The functions Opertor p (n) : N p {+,, } nd Action p (n) : N p A return the encoded opertor nd the ction, respectively. While the function Action is defined only for the lef nodes, Opertor is defined for ll the other nodes in N p. Finlly, the function Children p (n) mps ech node to n ordered sequence of its children such tht the left child is the first element of the sequence, wheres the right child is the lst one. IV. HIGH-LEVEL SPECIFICATION OF GENERIC OBJECTIVES In this section, the process lgebr frmework is used to specify clss of vehicle routing problems. Although process lgebr is not s expressive s other formlisms to express temporl logic constrints (e.g., Liner Temporl Logic), in this pper we choose process lgebr for two resons. First, process lgebr offers computtionlly efficient lgorithms, e.g., for checking whether given string stisfies given specifiction. This property llows us to design computtionlly efficient vlid GA opertors. Second, the hierrchicl specifiction methodology of the process lgebr llows building more complex specifictions from simpler ones, which is illustrted with exmples throughout this section. Most of the mteril in this section is derived from tht in [7], where the reder is referred to for more thorough discussion in the context of UAV mission plnning. We consider vehicle routing problem, in which set O of tomic objectives re ssigned to set V of vehicles, so s to optimize given cost function, while stisfying specifiction given in the PA lnguge. First, we define tomic objectives nd employ process lgebr to represent more generic objectives in terms of tomic ones. Then, we proceed with some preliminry definitions, followed by formliztion of the problem definition. A. Objectives Intuitively speking, n tomic objective is tsk tht cn not be represented by combintion of ny others. In essence, tomic objectives re bstrctions of individul tsks in vehicle routing problem. In the context of, for instnce, UAV mission plnning, the first such bstrctions were presented recently in Ref. [25] nd further developed nd employed in Ref. [7]. Definition IV. An tomic objective o is tuple (x i o, x f o, To e, v o ) where x i o R 2 is the entry point x f o R 2 is the exit point T0 e R + is the execution time v 0 V is cpble vehicle Intuitively, only v o cn execute the tomic objective o in exctly T e o mount of time; moreover, v o moves to the coordinte x i o to strt executing o nd ends up t coordinte x f o fter the execution. We will ssume, without ny loss of generlity, tht o cn not be executed by ny vehicle other thn v 0. Lter in the

5 5 pper, we will show tht the tsks tht cn be executed by one of severl vehicles cn be represented s combintion of different tomic objectives. The definition of tomic objectives is quite generl nd cn cpture mny different types of individul tsks for vehicle routing problems. In [25], Rsmussen nd Kingston show tht similr bstrction cn model, e.g., sector or re serch, clssifiction, ttck, rescue, trget trcking, reconnissnce, et ceter, in the UAV mission plnning context. Using the process lgebr specifiction frmework, more generic objectives, which impose temporl nd logicl constrints, cn be composed from simpler ones. Given set O of tomic objectives, (generic) objective is represented by process lgebr term p defined on O s the set of ctions. Exmple Let us present exmples of tomic nd generic objectives in the context of vehicle routing. Let us consider scenrio, in which heterogenous set {v, v 2, v 3, v 4 } of four vehicles re bound to visit four cities, nmed A, B, C, nd D, respecting the following constrints. The first nd the second vehicles cn only visit A, the third vehicle cn visit only B, nd the fourth vehicle cn visit the remining two cities, C, nd D. The mission is to first visit A by either v or v 2, then obtining certin intelligence, e.g., surveillnce dt, from A, visit B, C, nd D, in ny order such tht C is visited before D s some crgo hs to be moved from C to D by the fourth vehicle. This high-level specifiction of the vehicle-routing problem t hnd cn be described within the process lgebr frmework s follows. Let O = {o, o 2, o 3, o 4, o 5 } denote the set of tomic objectives. The tomic objectives o nd o 2 visit the city A, wheres the tomic objectives o 3, o 4, nd o 5 visit the cities B, C, nd D, respectively; the tomic objective o cn be executed by v, o 2 by v 2, o 3 by v 3, nd o 4 nd o 5 by v 4. Notice tht the constrint tht the city A must be visited either by v or by v 2 cn be represented by the process o +o 2. Similrly, the whole mission cn be specified by the process lgebr string (o + o 2 ) (o 3 (o 4 o 5 )). For more exmples of specifictions of vehicle routing problems using the process lgebr frmework, we refer the reder to [7]. B. Schedules, Observtions, nd Specifictions A single vehicle schedule is sequence of distinct pirs of tomic objectives nd time instnts. Intuitively speking, single vehicle schedule σ v for vehicle v is list of tomic objectives nd their execution times to be executed by v. More precisely, if (o, t) σ v for some o O nd t R +, then the tomic objective o is sid to be scheduled to be executed t time t by vehicle v. Note tht the tomic objective nd time pirs in single vehicle schedule σ v re ordered ccording to their time component, i.e,, (o i, t i ) < σv (o j, t j ) only if t i t j for ll (o i, t i ), (o j, t j ) σ v. A complete schedule, or schedule for short, is set S of single vehicle schedules tht contins exctly one single vehicle schedule for ech vehicle in V. Given two tomic objectives o i nd o j, let us denote the time it tkes for vehicle v to trvel from the exit point of o i to the entry point of o j s Tv,o t i,o j. Then, complete schedule P is sid to be vlid if for ech vehicle v V nd for ll pirs (o i, t i ) σ v the tomic objective o i cn indeed be executed t time t i by vehicle v. More precisely, for ny σ v = {(o, t ), (o 2, t 2 ),..., (o k, t k )} in P, the following holds: t i + To e i + Tv,o t i,o i t i for ll i {2,..., k}. A sequence π = (o, o 2,..., o k ) of tomic objectives is n observtion of the schedule S if the following holds: (i) ny tomic objective o tht is scheduled in S is n element of π, nd (ii) for ech tomic objective o π there exists time instnce t such tht t t t + To e, where t is such tht (o, t) σ v for some v V, To e is the execution time of o, nd (iii) for ll we hve o i, o j π, o i < π o j if nd only if t i t j, where t i nd t j re the time instnces corresponding to o i nd o j, respectively. Intuitively, n observtion is sequence π of tomic objectives such tht corresponding to ech o i tht pper in π one cn find time instnce t i within the execution intervl of the tomic objective o i so tht the ordering of these time instnces is the sme s the ordering of their corresponding tomic objectives in π. From here on, we will denote the set of ll observtions of vlid complete schedule S by Π S. Following the definition of observtions, specifiction nd its stisfction is formlized s follows. Definition IV.2 (Specifiction) A specifiction is process lgebr term defined on the set O of tomic objectives. A vlid schedule S is sid to stisfy specifiction p if nd only if ny observtion of S is trce of p, i.e., Π S Γ p holds. Exmple Consider the scenrio in the previous exmple. Recll tht the specifiction ws p spec = (o +o 2 ) (o 3 (o 4 o 5 )). Consider the schedule tht ssigns S = {σ v, σ v2, σ v3, σ v4 }, where σ v = ((o, t )), σ v2 = δ, σ v3 = ((o 3, t 3 )), nd σ v4 = ((o 4, t 4 ), (o 5, t 5 )). The time instnces t i re depicted in Figure 2. Notice tht this schedule hs exctly three observtions: π = (o, o 3, o 4, o 5 ), π 2 = (o, o 4, o 3, o 5 ), nd π 3 = (o, o 4, o 5, o 3 ) (see Figure 3 for depictions of the time instnces t i tht led to these observtions). Notice tht ll these observtions re indeed trces of p spec. Hence, S stisfies p spec. Fig. 2. A timeline of the exmple schedule. C. Problem Definition Given schedule S = {σ v v V}, ech single vehicle schedule σ v in S cn be nturlly ssocited with rel number τ v, which represents the time tht vehicle v is finished with the execution of its lst tomic objective. More formlly,

6 6 () (b) (c) Fig. 3. The time instnces t i tht led to the three observtions of the exmple schedule re shown in (), (b), nd (c) s dots. τ v = t + T e o, where (o, t ) = σ v ( σ v ). The rel number τ v will be referred to s the completion time of σ v. Using the completion times {τ v } v V, it is possible to define the cost of the schedule S in mny wys. Two of the common cost functions include the totl completion time J nd mximum completion time J 2, which re defined, respectively, s J (S) = v V τ v, The problem definition is given s follows. J 2 (S) = mx v V τ v. () Problem IV.3 Given set V of vehicles, set O of tomic objectives, trveling times Tv,o t i,o j for ll v V nd ll o i, o j O, nd process lgebr specifiction p spec defined on O, the optiml plnning problem with PA specifictions is to find vlid schedule S such tht (i) S stisfies the specifiction p spec, nd (ii) the cost function J (S) (or J 2 (S)) is minimized. Recently, tree serch bsed solution to similr problem ws given in [7], which extends the lgorithm in [3] to hndle process lgebr specifictions. The tree serch lgorithm presented in these references effectively serches the stte spce of ll solutions nd returns fesible solution to the problem in time polynomil with respect to the size of the specifiction p spec s well s the number of vehicles. Moreover, given extr time, the lgorithm improves the existing solution with the gurntee of termintion with n optimum solution in finite time. In the next section, we provide genetic lgorithm heuristic solution to Problem IV.3. V. GENETIC ALGORITHM Given specifiction p spec, ny trce of p spec is chromosome, usully denoted by X, X, X 2 et ceter. The genetic lgorithm (GA) mintins set X of chromosomes clled the genertion. The GA is initilized with rndomly creted genertion of chromosomes. At ech itertion, (i) prent chromosomes re selected stochsticlly from X ccording to their fitness, (ii) new chromosomes re generted from their prents using the crossover opertion, (iii) some of the chromosomes re generted by mutting the existing ones rndomly, nd (iv) the chromosomes tht re more fit thn others re crried to the next genertion. In this section, we first present the detils of these four evolutionry opertors, fter discussing the reltionship between the schedules nd the chromosomes. We lso present the genetic lgorithm solution s whole nd discuss its correctness. A. The Reltionship Between Chromosomes nd Schedules Ech chromosome X i corresponds nturlly to vlid complete schedule denoted s S(X i ). Before formlizing the construction of S(X i ), let us introduce the following definition. An tomic objective ō is sid to be predecessor of nother tomic objective o in specifiction p if the following two conditions hold: (i) there exists trce γ of p, in which ō ppers before o in γ, i.e., ō < γ o, (ii) there is no trce of p, in which o ppers before ō. Equivlently, it cn be shown tht ō is predecessor of o in p if nd only if the prse tree of p includes node tht binds two terms p nd p 2 with sequentil composition opertor s in p p 2, where p includes ō nd p 2 includes o s n tomic objective (see Ref. [7]). We will denote the set of ll predecessors of given tomic objective o in specifiction p by Pred p (o). Notice tht the sets Pred p (o) for ll o O cn be formed efficiently by observing the prse tree of p. This process cn be executed (only once before strting the lgorithm) in time tht is bounded by polynomil in the size of p. Given chromosome X, the complete schedule S(X) is generted recursively s follows: For chromosome of the form X = (o), where o = (x i o, x f o, T o, v o ), we define S(X) to be schedule tht ssigns the tomic objective o to vehicle v o to be executed t time t, where t is the time required for v o to trvel from its initil position to the entry point x i o of the tomic objective o i. More precisely, S(X) := {σ v, σ v2,..., σ vn }, where σ vo = ( (o, t) ), with t being the time it tkes v i to trvel from its initil position to x i o, nd for ll v j v o nd v j V, σ vj is n empty sequence. Note tht N represents the number of vehicles, i.e. N = V. Given chromosome X with X = K >, let o = (x i o, x j o, T o, v o ) be the lst tomic objective tht ppers in X, i.e., o = X( X ), nd X be the sequence tht is the sme s X, except tht it does not contin o; more formlly, X is such tht X = X (o) holds. Let us denote S(X ) s {σ v, σ v 2,..., σ v N }. Then, the schedule S(X) is defined s {σ v, σ v2,..., σ vn }, where σ vi = σ v i for ll v i v o nd σ vo is such tht σ vo (k) = σ v o (k) for ll k {, 2,..., σ v o } nd σ vo ( σ v o + ) = (o, t). Tht is, S(X) is the sme s S(X ) except tht, in ddition, in S(X) tomic objective o is ssigned to vehicle v o to be executed t time t. The execution time t is computed s follows. Recll tht τ σ vo denotes the completion time of the schedule σ v o. The execution time, t i, is the smllest time tht is greter thn both τ σ vo nd the mximum

7 7 execution time of ny predecessor of o tht is scheduled in S(X ), i.e., t i = mx { τ σ ui, mx{tõ (õ, tõ) σ u j, σ u j S(X ), õ Pred p (o)} } Given chromosome X with X = K, let X, X 2,..., X K be the sequences defined s follows: for ll k {, 2,..., K }, we hve X k = k nd X k (i) = X(i) for ll i {, 2,..., k}. Notice tht, lgorithmiclly, S(X ) cn be computed esily, nd S(X k ) cn be computed using S(X k ). Hence, S(X) cn be constructed recursively strting from S(X ). Exmple Consider the scenrio in the previous exmple. Recll tht the specifiction ws (o + o 2 ) (o 3 (o 4 o 5 )). It cn be shown tht o nd o 2 hve no predecessors. However, o 3 nd o 4 both hve o nd o 2 s their predecessors. The tomic objective o 5, on the other hnd, hs o, o 2, nd o 4 s its predecessor. Trces of this process lgebr term were presented in n erlier exmple. One of these trces ws (o, o 4, o 3, o 5 ). Algorithmiclly, the schedule corresponding to this chromosome is formed s follows. First, the tomic objective o is hndled: it is ssigned to v, since v is the cpble vehicle for o ; the time of execution t is selected such tht t is the lest time tht, strting from its initil position, v gets to A. Next, o 4 is scheduled to be executed by its cpble vehicle, v 4, t time t 4. Recll tht o is predecessor of o 4. Hence, t 4 set fter both the rrivl time of v 4 to the city C s well s the completion time of o. Continuing this procedure similrly, o 3 is scheduled next to be executed by v 3 t time t 3, where t 3 is fter both the rrivl of v 3 to the city B nd the completion time of o. Finlly, o 5 is ssigned to be executed t time t 5 such tht t 5 is fter the rrivl of v 4 to D coming from C nd fter the end execution of ll its predecessor tomic objectives, o, o 3, nd o 4. Hence, we hve generted the complete schedule S = {σ v, σ v2, σ v3, σ v4 }, where σ v = ((o, t )), σ v2 = δ, σ v3 = ((o 3, t 3 )), nd σ v4 = (o 4, t 4 ), (o 5, t 5 ) (Recll Figure 2 for its representtion). B. Evolutionry Opertors In this section detiled discussions of rndom chromosome genertion s well s the other four phses of the GA re provided. ) Rndom Chromosome Genertion: Notice tht generting chromosomes t rndom cn not be ccomplished by solely picking rndom sequence of tomic objectives, since ech chromosome must be trce of the given specifiction. In this section, we provide n lgorithm, which rndomly genertes chromosome, i.e., trce of the specifiction, such tht there is nonzero probbility for ny trce of the specifiction to be chosen. The lgorithm hevily employs procedure denoted s Next, which mps given term p to the set of ll pirs (p, o ) of terms nd tomic objectives such tht for ny (p, o ) in Next(p) we hve tht p cn evolve into p by executing o, i.e., the trnsition p o p holds. An lgorithmic procedure to compute Next(p) is provided in Algorithm, the correctness of which follows esily from the opertionl semntics of process lgebr. The Next lgorithm runs recursively. Its execution is best visulized with the prse tree of the process lgebr term, p, tht it tkes s prmeter. By the semntics of lterntive composition opertor, the tomic objectives tht cn be executed next by p = p + p 2 is exctly those tht cn be executed either by p or p 2. Hence, if the root node of the prse tree of p is + opertor, i.e., p is of the form p = p + p 2, then the lgorithm clls itself recursively with prmeters p nd p 2, nd returns the union of tht returned fter these two clls (Lines 2-3). The semntics of the sequentil composition opertor is such tht the tomic objectives tht cn be executed next by p = p p 2 is exctly those tht cn be executed by p ; However, if p evolves to p fter executing n tomic objective, p evolves to p p 2 fter executing the sme tomic objective. Hence, if p is of the form p = p p 2, then the lgorithm clls itself recursively with prmeter p, conctentes p 2 to the end of p in ll pirs (p, t ) returned by this recursive cll, nd returns ll the resulting pirs (Lines 4-5). The semntics of the prllel composition opertor is such tht p = p p 2 cn execute the tomic objectives tht either p or p 2 cn execute. If, p is of the form p p 2, then the lgorithm recursively clls itself twice with prmeters p nd p 2. The pirs returned fter these clls re ppropritely conctented by p nd p 2, nd ll resulting pirs re returned (Lines 6-7). Finlly, if the p = o, where o is n tomic objective, then the lgorithm returns (, o) (Lines 8-9), since p cn only execute the tomic objective o nd evolve to the terminted process,. Algorithm : Next(p) Procedure switch p do 2 cse p = p + p 2 3 return Next(p ) Next(p 2 ) 4 cse p = p p 2 5 return {(p p 2, t ) (p, t ) = Next(p 2 )} 6 cse p = p p 2 7 return {(p p 2, t ) (p, t ) = Next(p )} {(p p 2, t ) (p 2, t ) = Next(p 2 )} 8 cse p = o O 9 return {(, o)} 0 endsw endsw Algorithm recursively explores the prse tree of p in this mnner, nd extrcts ll the tomic objectives tht cn be executed next. Note tht ech node in the prse tree is explored t most once in the lgorithm, which implies tht the running time of the lgorithm is liner with respect to the size of the specifiction even in the worst cse. Given finite set S of elements, let Rnd(S) be procedure tht returns n element of S uniformly t rndom. The lgorithm tht genertes rndom chromosome, denoted s RndomGenerte, is given in Algorithm 2. The RndomGenerte(p) procedure runs the Next(p) procedure

8 8 (Line 4), rndomly picks one of the tomic objectives returned by Next, sy o (Line 5), nd runs itself recursively with the process p tht p evolves to fter executing o (Line 6). The recursion ends when the lgorithm is run with the terminted process (Line 2). The chromosome tht is returned is essentilly conctention of the tomic objectives tht were picked rndomly long the wy during the recursion. Algorithm 2: RndomGenerte(p) Procedure if p = then 2 return δ 3 else 4 S Next(p) 5 (p, o ) Rnd(S) 6 γ Rndomgenerte(p ) 7 return o γ 8 end Note tht Algorithm 2 returns exctly one rndom trce of p. Using Algorithm 2 repetedly, however, set X of chromosomes cn be generted to initilize the GA. After this initiliztion, ech itertion of the GA proceeds with the forementioned five phses, detiled fter the next short exmple. Exmple To illustrte the rndom chromosome genertion, consider the running exmple specifiction p spec = (o + o 2 ) (o 3 (o 4 o 5 )). The lgorithm RndomGenerte(p spec ) first clls the Next procedure with p spec, which returns S = {(p, o ), (p 2, o 2 )}, where p = p 2 = o 3 (o 4 o 5 ). Sy the Rnd(S) procedure returns the pir (p 2, o 2 ) mong the two elements of S. Then, the RndomGenerte lgorithm clls itself with p 2, which genertes second cll to Next procedure, this time with p 2, which returns {(p 3, o 3 ), (p 4, o 4 )}, where p 3 = o 4 o 5 nd p 4 = o 3 o 5. Assume tht, in this itertion, the Rnd(S) procedure returns (p 4, o 4 ). The RndomGenerte lgorithm, this time, runs itself with p 4. The Next(p 4 ) cll returns S = {(p 5, o 3 ), (p 6, o 5 )}, where p 5 = o 5 nd p 6 = o 3. Assume tht Rnd(S) returns (p 5, o 3 ). Finlly, the RndomGenerte procedure clls itself with p 5, which results in Next(p 5 ) returning S = (, o 5 ) nd Rnd(S) returning o 5. The finl cll of RndonGenerte with, then, returns δ, nd the procedure termintes. After termintion the rndom chromosome obtined is X = (o 2, o 4, o 3, o 5 ). 2) Selection: In the selection phse, pirs of chromosomes re selected rndomly from the set X of ll chromosomes to be the prents of the next genertion. The rndomiztion is bised, however, so tht those chromosomes tht re more fit thn others re selected to be prents with higher probbility. Throughout this pper, the fitness of chromosome is evluted using the cost of its corresponding schedule s follows: f Xi = J i (S(X i )), where i =, 2 (see Eqution ()). Tht is, the chromosomes with lower-cost corresponding schedules re rted s more fit ones. After the selection phse, child chromosome is produced from these two prent chromosomes vi the crossover opertion. 3) Crossover: The crossover opertion genertes child chromosome X from given pir X nd X 2 of prent chromosomes. Note tht merely picking cutting point nd joining prts of two vlid chromosomes does not necessrily produce vlid chromosome in this cse. In this section, cut nd splice crossover opertor tht lwys produces vlid chromosome is provided. Informlly speking, the crossover opertion first prtitions the set O of tomic objectives into two sets of tomic objectives denoted s S nd S 2. Then, two different sequences, σ nd σ 2, re formed such tht σ i is the order preserving projection of X i onto the set S i for i =, 2. In the end, the child chromosome is the conctention of σ nd σ 2. Let us first identify four primitive procedures, which help clrify the presenttion of the crossover lgorithm. Let p be process lgebr term. The procedure ChildrenAO p (n) tkes node n of the prse tree of p nd returns the set of ll ctions (equivlently, tomic objectives) tht re lbels of the lef nodes of the tree rooted t n. An lgorithmic procedure for computing ChidlrenAO p (n) is given in recursive form in Algorithm 3. The procedure RightMostChild p (n) returns the rightmost lef of the tree rooted by node n. This function is presented in n lgorithmic form in Algorithm 4. Given n tomic objective o O, let the functions Left p (o) nd Right p (o) return the set of tomic objectives tht re, intuitively speking, to the left of n nd right of n, respectively. More precisely, we hve õ Left p (o) if nd only if there exists n, m left, m right N p such tht (m left, m right ) = Children p (n), õ ChildrenAO p (m left ) but õ / ChildrenAO p (m right ), o ChildrenAO p (m right ) but o / ChildrenAO p (m left ). The procedure Right p (o) is defined symmetriclly. As mentioned erlier, the crossover lgorithm first cretes two disjoint sets S nd S 2 of tomic objectives, such tht S S 2 = O. The sets S nd S 2 re, indeed, formed using nturl ordering of the tomic objectives, which comes from the prse tree itself. Intuitively speking, the tomic objectives in prse tree cn be ordered such tht o p o 2 if nd only if o is to the left of o 2 in the tree. More precisely, first cutting point tomic objective, sy o, is chosen ccording to some procedure to be outlined shortly, nd S nd S 2 re defined s S := Left p (o) {o} nd S 2 := Right p (o). It cn be shown rther esily tht this selection of S nd S 2 stisfies S S 2 = nd S S 2 = O. Then, the child chromosome cn be generted using S nd S 2 s outlined bove. Note, however, tht not ll choices of o would yield vlid chromosome, i.e., trce of the specifiction. Yet, it is possible to select the cutting point tomic objective so tht the resulting chromosome will be vlid. Such procedure runs s follows. Informlly speking, first, the prse tree of the specifiction is rndomly rerrnged, while preserving the behvior (set of trces tht cn be generted) of the specifiction. The rerrngement is done by rndomly choosing to either switch the left nd right children

9 9 of ech lterntive nd prllel composition opertor in the prse tree or keep them s is. This procedure is denoted s RndomRerrnge, which tkes process p nd returns the rerrnged one. RndomRerrnge llows vriety of different schemes for cutting the prent chromosomes, s will be cler lter. Then, n tomic objective o rnd is selected t rndom mong O, nd used s cutting point if the resulting child chromosome yields fesible ssignment. If not, the nerest tomic proposition, to the right of o rnd, tht would yield fesible ssignment. This procedure is given in Algorithm 5. Algorithm 3: ChildrenAO p (n) Procedure if Lef p (n) = True then 2 S {n} 3 else 4 σ Children p (n) 5 S 6 for i to σ do 7 S S ChildrenAO p (σ(i)) 8 end 9 end 0 return S Algorithm 4: RightMostChild p (n) Procedure while Lef p (n) = Flse do 2 σ Children p (n) 3 n σ( σ ) 4 end 5 return n Algorithm 5: CutAtomicObjective p (X, X 2 ) Procedure o rnd Rnd(O) 2 S {o rnd } 3 while (o / X for o S) or (o / X 2 for o S) do 4 while Opertor p (n) = + do 5 n Prent p (n) 6 end 7 S ChildrenAO p (n) 8 end 9 if Lef p (n) = Flse then 0 n RightMostChild p (n) end 2 o cut Action p (n) 3 return o cut S := Left(o cut ) {o cut } nd S 2 := Right(o cut ), which represent the set of tomic objectives to the left of o cut nd the ones to the right of o cut, respectively. Using these two sets, two sequences, σ nd σ 2, re generted from chromosomes X nd X 2. Finlly, the resulting child chromosome X is the conctention of the two sequences σ nd σ 2. Algorithm 6: Crossover p (X, X 2 ) Procedure p RndomRerrnge(p) 2 o cut := CutAtomicObjective p (X, X 2 ) 3 S := Left p (o cut ) {o cut } 4 S 2 := Right p (o cut ) 5 σ := [X ] S 6 σ 2 := [X 2 ] S2 7 return σ σ 2 Exmple Consider the running exmple with the specifiction p spec = (o + o 2 ) (o 3 (o 4 + o 5 )). Consider the two chromosomes X = (o, o 3, o 4, o 5 ) nd X 2 = (o 2, o 4, o 3, o 5 ). First, the crossover opertion clls the rndom rerrnge procedure, which switches the left nd right children of lterntive nd prllel composition opertors or keeps them s is with equl probbility. The prse tree of p spec ws given in Figure. Notice tht there is exctly one lterntive nd one prllel opertor, ech of which cn hve their children switched with probbility /2. Let us ssume tht the RndomRerrnge procedure switches the children of the prllel composition opertor, while keeping unchnged tht of the lterntive composition opertor. The new prse tree is shown in Figure 4. Next, the CutAtomicObjective p procedure is run with X nd X 2. Assume tht the o rnd turns out to be o 4, which is included in both of the chromosomes. Hence, CutAtomicObjective procedure returns o cut = o 4. From the prse tree presented in Figure 4, notice tht we hve S = Left p (o cut ) {o cut } = {o, o 2, o 4 } nd S 2 = Right p (o cut ) = {o 3, o 5 }. Hence, we get σ = [X ] S = (o, o 4 ) nd σ 2 = [X 2 ] S2 = (o 3, o 5 ). Thus, we find tht X = (o, o 4, o 3, o 5 ), which is lso trce of p spec, thus vlid chromosome. Notice tht this crossover procedure combined the chrcteristics of the prent chromosomes, X nd X 2, in the child chromosome X. The choice of executing o rther thn o 2 is inherited from X, wheres the choice of executing o 3 fter o 4 is inherited from X 2. The crossover opertion is summrized in Algorithm 6. Given two chromosomes X nd X 2, the crossover lgorithm first genertes cut point tomic objective o cut vi Algorithm 5. In the second step, it genertes the two sets Fig. 4. Prse tree of the process ( + 2 ) ( 3 ( 4 5 )). It is worthwhile to mention the role of the RndomRerrnge procedure t this point. Clerly,

10 0 prse tree rooted with n lterntive or prllel composition opertor cn be represented in two different wys. Consider, for instnce, the process p +p 2, which hs the sme behvior exhibited by p 2 + p. Independent of which representtion of the (sme) specifiction the lgorithm is strted with, the RndomRerrnge procedure removes the bised cutting point decisions tht my come out of the crossover procedure by rndomly picking one of the two representtions. Finlly, let us note tht it is not cler t this stge whether Crossover p (X, X 2 ) lwys returns child chromosome tht is indeed trce of p. We postpone this discussion until Section V-C. 4) Muttion: The muttion opertion is used for mking rndom chnges in some smll portion of the genertion so s to void locl minim during optimiztion. In the muttion phse of the lgorithm, set of chromosomes re selected from the current genertion; ech selected chromosome is modified rndomly to nother chromosome nd crried over to the next genertion. However, this opertion, gin, is not trivil, since the modified chromosome must lso be vlid chromosome, i.e., trce of the specifiction p spec. The muttion phse, given in Algorithm 7, proceeds s follows. Firstly, number of chromosomes re picked t rndom from the set X of chromosomes. Secondly, for ech such chromosome X, n tomic objective o mid X is picked uniformly t rndom. Then, X is prtitioned into two sequences, σ nd σ 2, such tht σ σ = X nd σ () = o mid. Finlly, the mutted chromosome X is computed by employing the RndomGenerte procedure (Algorithm 2): X = σ σ rnd, where σ rnd = RndomGenerte(p ) nd p T is such tht p σ p. Algorithm 7: Mutte p (X) Procedure o mid Rnd({o O o X}) 2 σ, σ Σ O re such tht p = σ σ nd σ () = o mid 3 p is such tht p σ p 4 σ rnd RndomGenerte(p ) 5 return σ σ rnd In essence, fter the chromosome is prtitioned into two, the first prt of the chromosome is kept wheres the second prt is re-generted rndomly. Exmple Consider the chromosome X = (o 2, o 4, o 3, o 5 ) from the previous exmple. Let us illustrte the muttion lgorithm on this exmple. The muttion opertor first picks n element of X using uniformly rndom distribution. Let us ssume tht this element turns out to be o 3. Then, the lgorithm prtitions the chromosome into two s in X = σ σ 2, where σ = (o 2, o 4 ) nd σ 2 = (o 3, o 5 ). Notice tht p σ p holds, where p = o 3 o 5. The muttion opertor, then, runs the RndomGenerte procedure with p, which might either return σ 3 = (o 3, o 5 ) or σ 4 = (o 5, o 3 ). Assume tht it returns the ltter. Then, the resulting mutted chromosome is the conctention of σ nd σ 4, i.e., (o 2, o 4, o 5, o 3 ). 5) Elitism: In the elitism phse, the chromosomes with high fitness re selected to move into the next genertion. The selection is mde rndomly, even though bised towrds chromosomes with high fitness vlues. However, we deterministiclly choose smll set of chromosomes, clled the elite members, of the current genertion with the highest fitness vlues, in order to rule out the possibility of loosing ll the good solutions in given genertion. This provides us with solution tht is monotoniclly improving. C. Algorithm nd Correctness Let us extend the primitive procedure Rnd s follows. Let Rnd(S, φ, k) be primitive procedure, where S is finite set, φ : S R + is function, nd k is number such tht k < S. The function Rnd(S, vlue, k) returns set of k distinct elements from S by rndomly picking n element s S with probbility φ(s)/ s S φ(s ) repetedly, until k distinct elements re selected. Let lso SelectBest(S, φ, k) be nother procedure tht returns the k elements with highest vlues of the function φ. More precisely, SelectBest(S, φ, k) is set of k elements from set S such tht for ll s SelectBest(S, φ, k) nd for ll s S\SelectBest(S, φ, k) we hve tht φ(s) φ(s ). The GA is formlized in Algorithm 8, where the initiliztion phse (Lines 2-4) s well s the selection (Lines 6-8), crossover (Lines 9-6), muttion (Lines 7-23), nd elitism (Line 24) opertions re shown explicitly. Algorithm 8: GA(p spec, K totl, K children, K elite, K mutte, N) X 2 for i to K totl do 3 X X RndomGenerte(p spec ) 4 end 5 for j to N do 6 for ll X k X do 7 fitness(x k ) /J(PC(X k )) 8 end 9 C 0 for i to K children do X Rnd(X, fitness, ) 2 X 2 Rnd(X, fitness, ) 3 C Crossover(X, X 2 ) 4 fitness(c) /J(PC(C)) 5 C C C 6 end 7 M 8 for i to K mutte do 9 X Rnd(X, fitness, ) 20 M Mutte(X) 2 fitness(m) /J(PC(C)) 22 M M M 23 end 24 X SelectBest(X, K elite ) Rnd(X C M, fitness, K totl K elite ) 25 end 26 return SelectBest(X, )