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Cb32492496

  1. 1. Mullya Satish Anand, Santosh Krishnaji Sindhe / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 2, March -April 2013, pp.492-496492 | P a g eModeling and Simulation of Job Shop Scheduling Using Petri-NetsMullya Satish Anand*, Santosh Krishnaji Sindhe***(Department of Mechanical Engg, IOK COE Pune)** (Department of Mechanical Engg, IOK COE Pune)ABSTRACTIn most of the manufacturing unitsscheduling is a difficult task due to the complexityof the system. Hence powerful tools that canhandle both modeling and optimization arerequired. Most of the research in this area focusesin either developing optimization algorithms, or inmodeling complex production systems. However,few tools are aimed to the integration of both ofthem. In this paper, a Petri Net based integratedapproach, for simultaneously modeling andscheduling manufacturing systems, is proposed.The procedure is illustrated with an exampleproblem.Keywords - Analysis of Petri Nets, INA, Petri Nets,Scheduling.I. INTRODUCTIONThe prime subject of research on schedulingproblems consists of the optimal allocation of scarceresources to tasks over time. Despite the complexityof many scheduling problems, effective algorithmshave been developed. However, most of researchfocused on the effectiveness of the algorithms,neglecting the issue of flexibility. Research on Petrinets addresses the issue of flexibility. To facilitate themodeling of complex systems many extensions havebeen proposed. Typical extensions are the addition of„color‟, „time‟ and „hierarchy‟. These Petri nets haveall the advantages of the classical Petri net, such asthe graphical nature, mathematical foundation andthe various analysis methods. Therefore, it isinteresting to investigate the application of Petri netsto scheduling. In this paper we concentrate on timedPetri nets, i.e. Petri nets extended with a timingconcept.II. LITERATURE REVIEWWhen Petri Nets were introduced, manypapers were published on topics such as resourceutilization, bottlenecks, throughput, cycle times andcapacity estimations. Petri Nets were evaluated inmanufacturing systems using different techniqueslike simulation, queuing theory, probability andstochastic Petri nets. Most of the results in this areafocused on cyclic scheduling problems.Ramamoorthy & Ho (1980) and Hillion & Proth(1989) have used a technique based on a „markedgraphs‟ (a subclass of Petri nets)to analyze the throughput of cyclic (production)processes. Carlier, Chretienne & Girault (1984, 1988,1983), Gao, Wong & Ning (1991) and Watanabe &Ya-mauchi (1993) also focused on minimal cycletimes for repetitive scheduling problems. In thispaper focus is on the traditional non-cyclicscheduling problems such as machine scheduling andjob shop scheduling (Pinedo, 1995). In this papertimed Petri net model is used and time is associatedwith transitions.III. TIMED PETRI NETSPetri nets originate from the early work ofCarl Adam Petri in 1962. Since then there has beenlot of research in study and application of Petri nets.The classical Petri net is a bipartite directed graphwith two node types called places and transitions.The nodes are connected via directed arcs. Twonodes of the same type cannot be connected. Placesare represented by circles and transitions byrectangles. Places may contain zero or more tokens,which are represented by black dots. The number oftokens may change during the execution of the net.A place „p‟ is called an input place of a transition„t‟if there exists a directed arc from p to t, p is called anoutput place of t if there exists a directed arc from t top. The net shown in Fig.I illustrate the classical Petrinet model. These Petri net models a machine whichprocesses jobs and has two states free and busy.There are four places in, free, busy and out and twotransitions start and finish. In the state shown in Fig.I there are five tokens; four in place in and one inplace free. The tokens in place in represent jobs to beprocessed by the machine. The token in place freeindicates that the machine is free and ready to processa job. If the machine is processing a job, then thereare no tokens in free and there is one token in busy.The tokens in place out represent jobs which havebeen processed by the machine. Transition start hastwo input places in and free and one output placebusy. Transition finish has one input place busy andtwo output places out and free. A transition is calledenabled if each of its input places contains at leastone token.
  2. 2. Mullya Satish Anand, Santosh Krishnaji Sindhe / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 2, March -April 2013, pp.492-496493 | P a g eFigure I. A Petri net which represents a machineAn enabled transition can fire. Firing a transition tmeans consuming tokens from the input places andproducing tokens for the output places, i.e. t „occurs‟.Transition start is enabled in the state shown inFig. I, because each of the input places in and freecontains a token. Transition finish is not enabledbecause there are no tokens in place busy . Therefore,transition start is the only transition that can fire.Firing transition start means consuming two tokens,one from in and one from free, and producing onetoken for busy. The resulting state is shown in Fig.II.In this state only transition finish is enabled. Hence,transition finish fires and the token in place busy isconsumed and two tokens are produced, one for outand one for free. Now transition start is enabled, etc.As long as there are jobs waiting to be processed, thetwo transitions fire alternately, i.e. the machinemodeled by this net can only process one job at atime.Timing Concept:To model the real systems it is often important todescribe the behaviour of the system with the help ofdurations and delays. Since the classical Petri net isnot easily capable of handling quantitative time, weadd a timing concept. In this paper a timing conceptis used where time is associated with transitionswhich determine delays. Firing is instantaneous, i.e.the moment a transition consumes tokens from theinput places the produced tokens appear in the outputplaces. However, because of the firing delay it takessome time before the produced tokens becomeavailable for consumption. This results in thefollowing definition of a timed Petri net .Figure II. Transition start has firedDefinition 1A timed Petri net is a six tuple TPN = (P, T, I, O, TS,D) satisfying the following requirements:(i) P is a finite set of places.(ii) T is a finite set of transitions.(iii) I ∈ T →P (P) is a function which defines the setof input places of each transition.(iv) O ∈ T →P (P) is a function which defines the setof output places of each transition.(v) TS is the time set.(vi) D ∈ T → TS is a function which defines thefiring delay of each transition.The state of a timed Petri net is given by thedistribution of tokens over the places. Firing atransition results in a new state. This way a sequenceof states M0, M1, .... Mn is generated such that M0 isthe initial state and Mi+1 is the state reachable fromMi by firing a transition. Transitions are eager, i.e.they fire as soon as possible. If several transitions areenabled at the same time, then any of these transitionsmay be the next to fire. Therefore, in general, manyfiring sequences are possible. Let M0 be the initialstate of a timed Petri net. A state is called a reachablestate if and only if there exists a firing sequence M0,M1, ...Mn which „visits‟ this state. A terminal state isa state where none of the transitions is enabled, i.e. astate without successors.IV. THE GENERAL SCHEDULINGPROBLEMScheduling is concerned with the optimalallocation of scarce resources to tasks over time.Scheduling techniques are used in productionplanning, project planning, computer control,manpower planning etc.Resources can be called as „machines‟ or„processors‟ and tasks as „operations‟ or „steps of ajob‟. Resources are used to process tasks. However, itis possible that the execution of a task requires morethan one resource, i.e. a task is processed by aresource set. There may be multiple resource sets thatare capable of processing a specific task. Theprocessing time of a task is the time required toexecute the task given a specific resource set. Byadding precedence constraints it is possible toformulate requirements about the order in which thetasks have to be processed. It is assumed thatresources are always available, but shall notnecessarily assume the same for tasks. Each task hasa release time, i.e. the time at which the task becomesavailable for processing. This leads to the followingdefinition.Definition 2A scheduling problem is a six-tupleSP = (T, R, PRE, TS, RT, PT) satisfying thefollowing requirements.(i) T is a finite set of tasks.(ii) R is a finite set of resources.
  3. 3. Mullya Satish Anand, Santosh Krishnaji Sindhe / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 2, March -April 2013, pp.492-496494 | P a g e(iii) PRE ⊆ T × T is a partial order, the precedencerelation.(iv) TS is the time set.(a) the resource sets capable of processing task tand(b) the processing time required to process t by aspecific resource set.This definition specifies the data required toformulate a scheduling problem. The tasks aredenoted by T and the resources are denoted by R. Theprecedence relation PRE is used to specifyprecedence constraints. If task t has to be processedbefore task t‟, then (t, t‟) ∈ PRE, i.e. the execution oftask t has to be completed before the execution oftask t‟ may start. TS is the time set. IN and IR+ ∪ {0}are typical choices for TS. The release time RT (t) ofa task t specifies the time at which the task becomesavailable for processing, i.e. the execution of t maynot start before time RT (t).Scheduling Problem: 3 jobs 4 machines problemTable I. Path of operations Table II. Time ofoperationsAbove, TABLE I. shows path for operations which isto be followed by 3 jobs and the TABLE II. showsthe corresponding processing time of each job on aparticular machine.Assumptions1. No resource may process more than one task at atime.2. Each resource is continuously available forprocessing.3. Each operation, once started, must be completedwithout interruptions.4. The processing times are independent of theschedule. Moreover, the processing times are fixedand known in advance.Fig. V shows Petri Net model of 3 job 4 machines.The typical characteristics exhibited by the activitiesin a dynamic event-driven system, such asconcurrency, decision making, conflict, sequentiality,mutual exclusion, synchronization and priorities, canbe modeled effectively by Petri nets. Here mutualexclusion is used to build the model. This is due tothe fact that 3 jobs may be ready to be processed witha single machine, but as per our assumption machinecan process 1 job at a time. Mutually exclusive- Twoprocesses are mutually exclusive if they cannot beperformed at the same time due to constraints on theusage of shared resources. Fig.III shows thisstructure. Here either transition t1 or t3 can be firedfirst. For example, a robot may be shared by twomachines for loading and unloading.Figure III. Mutual ExclusiveIn Fig. V places containing tokens indicate that jobsand machines are available initially. Processing timesare given to the transitions. TABLE III shows thetransitions and its time. Other transitions given in themodel apart from the transitions given in the table has0 unit time. These transitions indicate start of theprocess. T25 transition is a switch transition used tocomplete the cycle; its time is 0 units. Initial stateconsists of places P0, P1, P2 containing 1 token eachindicating start of job 1, 2 & 3. And final stateconsists of places P38, P39, P40 containing 1 tokeneach indicating completion of job 1, 2, 3. To achievethe final state transitions are fired according tosequence of operations. The sequence of transitionswith minimum time is the optimum schedule.Analysis of the Petri Net model is done with the helpof INA tool.Table III. Transitions with time unitsV. INTEGRATED NET ANALYZERWith the help of INA, Petri nets of verydifferent kinds can be investigated under differentfiring rules (in particular timing) with regard to theirgeneral properties. Properties which can be verifiedthrough the analysis are boundedness of places,liveness of transitions, and reachability of markingsor states.INA combines the following: a textual editor for nets,a by-hand simulation part, a reduction part forPlace/Transition-job 1 2 3 41 1 2 3 42 1 2 4 33 3 1 2 4job 1 2 3 41 9 8 4 42 5 6 3 63 10 4 9 2T1 9 T6 4 T12 9 T18 2T3 5 T8 8 T14 3 T20 4T4 10 T10 6 T16 4 T22 6
  4. 4. Mullya Satish Anand, Santosh Krishnaji Sindhe / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 2, March -April 2013, pp.492-496495 | P a g eFigure V. Petri Net model of 3 job 4 machines.Figure VI. A Schedule for 3 jobs and 4 machines.
  5. 5. Mullya Satish Anand, Santosh Krishnaji Sindhe / International Journal of EngineeringResearch and Applications (IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 2, March -April 2013, pp.492-496496 | P a g enets, an analysis part to compute: structuralinformation, place invariants, transition invariants,reachability graphs.Figure IV. Reachability GraphFig. IV shows Reachability Graph. The reachabilitygraph of a timed Petri net is constructed as follows.We start with an initial state M. Then we calculateall states reachable from M by firing a transition.Each node in the reachability graph corresponds toa reachable state and each arc corresponds to thefiring of a transition. firing sequences. Bycomputing the reachability graph, it is possible toanalyze all possible For a net representing ascheduling problem, each of these firing sequencescorresponds to a feasible schedule. Therefore,reachability graph is used to generate many feasibleschedules.VI. RESULTSINA is a textual tool whereas petri net is agraphical tool so the input to the INA tool is a.PNT file which is in text form. INA gave all thestructural properties of the petri net model. INAcomputed reachability graph which consists of 34states and the optimum schedule was having amake-span of 33 units. For this schedule thesequence of firing of transitions is T4 T3 T1 T10T16 T6 T8 T22 T12 T20 T14 T18. The Gantt chartof the schedule is given in Fig. VI.VII. CONCLUSIONThe approach presented in this papershows that it is possible to model many schedulingproblems in terms of a timed Petri net. In fact, arecipe has been formulated for mapping schedulingproblems onto timed Petri nets. This recipe showsthat the Petri net formalism can be used to modeltasks, resources and precedence constraints. Bymapping a scheduling problem onto a timed Petrinet, we are able to use Petri net theory to analyzethe scheduling problem. We can use Petri net basedanalysis techniques to detect conflictingprecedence‟s, determine lower and upper boundsfor the minimal make-span, etc. By inspecting thereachability graph, we can generate many feasibleschedules.REFERENCES[1] Jiacun Wang. “Petri Nets for Dynamic Event– Driven System modeling.[2] W.M.P van der Aalst. “Petri Net basedScheduling” Department of Mathematics andComputing Science, Eindhoven University ofTechnology.[3] Gonzalo Mejia, Nicholas G. Odrey “Job ShopScheduling in Manufacturing: An approachusing Petri Nets and Heuristic Search”.[4] Tadao Murata “Petri Nets: Properties,Analysis and Applications” Proceedings ofthe IEEE, vol 77, no 4, 1989.[5] Raida El Mansouri, Elhillali Kerkouche, andAllaoua Chaoui, “A Graphical Environmentfor Petri Nets INA Tool Based on Meta-Modelling and Graph Grammars”. WorldAcademy of Science, Engineering andTechnology 44 2008.[6] Ramamoorthy, C.V. And G.S. H O (1980),Performance Evaluation of AsynchronousConcurrent Systems Using Petri Nets, IEEETransactions on Software engineering 6,440–449.[7] Carlier , J., P. Chretienne , And C. Girault(1984), Modelling scheduling problems withTimed Petri Nets, in: G. Rozenberg (ed.),Advances in Petri Nets 1984,Lecture Notes inComputer Science 188, Springer-Verlag,Berlin, 62–82.[8] Hillion , H.P. And J.P Proth (1989),Performance Evaluation of Job-Shop SystemsUsing Timed Event Graphs, IEEETransactions on Automatic Control 34, 3–9.[9] Pinedo,M. (1995), Scheduling: theory,algorithms, and systems, Prentice-Hall,Engle-wood Cliffs.

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