Hybrid Petri net modeling and schedulability analysis of high fusion point oil transportation under...

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 40, NO. 2, MARCH 2010 159

Hybrid Petri Net Modeling and SchedulabilityAnalysis of High Fusion Point Oil Transportation

Under Tank Grouping Strategy for Crude OilOperations in Refinery

NaiQi Wu, Senior Member, IEEE, Feng Chu, Chengbin Chu, and MengChu Zhou, Fellow, IEEE

Abstract—There are varieties of constraints for a short-termscheduling problem of crude oil operations in a refinery. Theseconstraints are difficult to model and complicate the short-termscheduling problem. Among them, oil residency time and high fu-sion point crude oil transportation constraints are the challengingones. With high setup cost for high fusion point oil transportation,it is desired that the volume of high fusion point oil can be trans-ported as much as possible by a single setup. This may result in latetransportation of other types of crude oil, leading to the violation ofcrude oil residency time constraint. These constraints are ignoredby existing methods in the literature. To solve this problem, thispaper studies the problem in a control theory perspective by view-ing an operation decision in the schedule as a control. With thisidea, the system is modeled by a hybrid Petri net. With this modeland tank grouping strategy, schedulability analysis is carried outand schedulability conditions are presented with tank charging anddischarging costs being taken into consideration. These conditionsare necessary for determining a refining schedule and can be usedto check whether a target-refining schedule is realizable or not.If so, a feasible detailed schedule for the refining schedule can beeasily obtained by creating the operation decisions one by one.

Index Terms—Crude oil operations, discrete event systems, hy-brid system, oil refinery, Petri net.

I. INTRODUCTION

A S IT can greatly increase profit if a plant is well oper-ated [12], great attention has been paid to the develop-

ment of effective techniques for the operations of refinery in the

Manuscript received December 7, 2008; revised April 12, 2009. Firstpublished November 10, 2009; current version published February 18, 2010.This paper was recommended by Associate Editor M. Jeng. This work wassupported in part by National Natural Science Foundation of China underGrant 60474061, Natural Science Foundation of Guangdong Province underGrant 6104659, le Conseil Regional de la Champagne-Ardenne, France, andthe Chang Jiang Scholars Program, Ministry of Education of the People’s Re-public of China.

N. Q. Wu is with the Department of Industrial Engineering, School of Mecha-tronics Engineering, Guangdong University of Technology, Guangzhou 510006,China, and also with the University of Technology of Troyes, Troyes 10000,France (e-mail: [email protected]).

F. Chu is with the Laboratoire d’IBISC FRE CNRS 3190, Universite d’EvryVal d’Essonne, CE 1455 Courcouronnes 91020 Evry Cedex, France (e-mail:[email protected]).

C. Chu is with the Laboratoire Genie Industriel, Ecole Centrale Paris,Grande voie des Vignes, Chatenay-Malabry Cedex 92295, France (e-mail:[email protected]).

M. C. Zhou is with the Department of Electrical and Computer Engineering,New Jersey Institute of Technology, Newark, NJ 07102-1982 USA, and alsowith the School of Electro-Mechanical Engineering, Xidian University, Xi’an,710071, China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSMCC.2009.2032661

past years. With the installation of advanced control systems forprocess control and the planning technique well developed byusing linear programming-based commercial software [1], [13],there is a gap between the process control level and the planninglevel [7], [20]. Therefore, to effectively operate an oil refineryplant, it is necessary to integrate process control and produc-tion planning by short-term scheduling. A short-term sched-ule for oil refinery should provide all the activities in everydetail for the whole scheduling horizon, it is the detail thatmakes the short-term scheduling problem so difficult as pointedout by Honkomp et al. [7]. Besides, it is subject to variousconstraints, including physical and process constraints. If oneof the constraints is violated, a short-term schedule becomesinfeasible.

However, up to now, it lacks effective techniques and soft-ware tools for short-term scheduling problem in oil refinery,because of its extreme complexity. In fact, this job is still donemanually by a planner in oil refinery plants. Thus, it is mean-ingful to search for effective techniques and tools for short-termscheduling, and effort was made to do so in the recent years. Themathematical programming models that are applied to study thescheduling problem of batch processes are modified for short-term scheduling of oil refinery processes and a variety of mathe-matical programming models are currently available. There aremainly two categories: discrete-time and continuous-time mod-els. The work in [5], [6], [8], [10], [11], [14], and [15] belongs tothe former. The main drawback with such models is that there isa huge number of binary variables and makes the problem verydifficult to solve.

To overcome the drawback in the discrete-time representa-tion, continuous-time representation is adopted by some re-searchers [5], [8], [9], and [11]. Although, with these models, thenumber of discrete variables is reduced significantly, there arenonlinear constraints in it, which makes the problem difficult tosolve. Moreover, because of the complexity for the requirementof providing detailed activity, to make the problem solvable, formost of models in both discrete-time and continuous-time mod-els, special assumptions are made, which generally make the so-lution inefficient or unrealistic for real world cases. Meanwhile,in these models, often some constraints, such as the oil residencytime and high fusion point crude oil transportation constraintsare ignored. This leads their solutions to be infeasible. Thus,although there is significant progress in theory for the short-term scheduling problem of oil refinery, there is a serious gap

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160 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 40, NO. 2, MARCH 2010

Fig. 1. Architecture of short-term scheduling for refinery.

between theory and applications [20]. It is crucial to search foreffective techniques that work well in practical applications.

Short-term scheduling for crude oil operations is one of themost difficult scheduling problems in operating an oil refin-ery plant. To reduce the complexity, a new method is proposedin [21]–[23]. By this method, the problem of short-term schedul-ing for crude oil operations is decomposed into two subproblemsin a hierarchical way. At the upper level, it finds the refiningschedule to optimize the objectives, such as crude oil inventoryand production rate. At the lower level, it provides the detailedschedule to realize a given refining schedule. In doing so, it givesrise to new problems: 1) how can we guarantee that the refiningschedule found is realizable, because, at the upper level, it doesnot decide the detailed schedule; and 2) how can we guaranteethat a detailed schedule can be found when a refining schedule isrealizable. In [23], a Petri net (PN)-based heuristic is presentedto check the realizability of a refining schedule, if realizable, adetailed schedule is found as well. However, if a detailed sched-ule is not found it cannot guarantee that the refining schedule isnot realizable. Feasibility is essential to the short-term schedul-ing problem of crude oil operation in a refinery. If schedulabilityconditions are known, the upper level subproblem can be solvedby taking these conditions as constraints. In this way, the realiz-ability of the obtained refining schedule is guaranteed. Thus, ifgiven a refining schedule, a detailed schedule can be efficientlyobtained with the schedulabilty conditions, the issue of short-term scheduling can be resolved. For this purpose, with the archi-tecture as shown in Fig. 1, schedulability analysis is done in [21]and [22] and schedulability conditions are presented. These con-ditions can be used as constraints for finding realizable refiningschedule. At the same time, with these conditions, it is easy tofind detailed short-term schedule for a given refining schedule.

In viewing each operation decision in a short-term schedule asa control command to the system, its schedulability is analyzedin a control theory perspective in [21] and [22]. As shown inFig. 1, the system is modeled by a hybrid PN. The executionof an operation decision transfers the system from one state toanother. Thus, if a system is schedulable, the initial state must besafe and an operation decision should transfer the system froma safe state to another safe one. Then, a newly reached statecan be seen as an initial state at the time point. This implies

that all the reached states should be safe. Hence, safeness of thesystem is equivalent to schedulability [21], [22]. In this way,with safeness conditions known and the system state given, itis simple to find an operation decision such that its executiontransfers the system from the given safe state to another safe one,and the detailed schedule can be obtained by creating operationdecisions one by one given a refining schedule.

In scheduling crude oil operations, there are various con-straints, including resource and process ones. Resource con-straints include: 1) the limited number of storage and chargingtanks and the capacity of each tank; 2) the limited flow rate ofoil unloading and pipeline; and 3) the volume of various crudeoil types available in storage and charging tanks and in comingtankers. Process constraints include: 1) a distiller should be keptin working all the time uninterruptedly; 2) at least one chargingtank should be dedicated to a distiller at any time for feedingthe latter; 3) a tank cannot be charged and discharged at thesame time; 4) after being charged, the crude oil should stay inthe charged tank for a certain amount of time before it can bedischarged. This is called oil residency time (RT) constraint;and 5) theoretically, crude oil can be kept in pipeline withoutflowing. However, when a type of crude oil with high fusionpoint is being transported via a pipeline, such crude oil must bekept flowing until it goes out of the pipeline; and otherwise thepipeline will be blocked. This requirement is called high fusioncrude oil transportation constraint. Most of the existing methodsignore the fourth and fifth process constraints, and there is nomodel that deals with the fifth process constraint. This paperconducts schedulability analysis with all the constraints takeninto account.

With high fusion point crude oil types to be processed, costis an important issue and should be considered in schedulingthe system. In crude oil operations, when a tank is dischargedsuch that there is no oil in the tank can be discharged any more,this tank is said to be emptied. However, it cannot be reallyemptied. Instead there is some oil remaining (usually about 1/6capacity). Thus, when a tank with oil type i in it is emptied,and is then recharged with oil type j with i �= j, some volumeof high quality oil is likely treated as low quality oil due tothe oil mixture. This leads to a loss, called tank charging anddischarging loss. Therefore, when a tank with oil type i in it isemptied and recharged, it is better to charge the same or similartype of oil that contains similar components.

Often, different distillers process different types of crude oilat a time. A distiller may process different crude oil type atdifferent time. However, these crude oil types processed bythe same distiller at different time are similar. Thus, to reducecharging and discharging loss, for a time, the charging tanks aregrouped such that each group of charging tanks serve a distiller.Crude oil to be processed by distiller DSi is charged into thetanks in the tank group assigned to DSi . We call such strategyas charging tank grouping. In this paper, this strategy is appliedin analyzing the schedulability so as to reduce scheduling costas much as possible.

With the architecture given in Fig. 1, a model that describesthe behavior of the system is necessary. For crude oil oper-ations, the start and end of crude oil movement are discrete

WU et al.: HYBRID PETRI NET MODELING AND SCHEDULABILITY ANALYSIS OF HIGH FUSION POINT OIL TRANSPORTATION 161

events and oil flows continuously. Thus, it is a typical hybridsystem and should be described by a hybrid model. There arehybrid PN models [2], [3], [4], [16], [24]. However, they cannotbe directly used here because they cannot describe the above-mentioned constraints well. The hybrid PN model developedin [21], [22] is modified to model such systems. With it, schedu-lability conditions are presented that describe the relationshipamong the number of charging tanks, number of distillers, ca-pacity of charging tanks, production rates, and amount of highfusion point oil that can be transported by a single setup. Theseconditions provide the base for finding a refining schedule andchecking the realizability of a given one. By using them, onecan find a detailed schedule for a given realizable refining one.

The contribution of this paper includes: 1) it is the first timethat high fusion point crude oil is considered in schedulingcrude oil operations; 2) a hybrid PN model is developed to de-scribe such a system; 3) schedulability conditions are presented;4) with the schedulability conditions, one can derive how muchhigh fusion point crude oil can be transported from storage tanksto charging tanks, which is important in scheduling such sys-tems; and 5) with them, one can easily find a detailed feasibleone given a realizable refining schedule.

Next, we briefly introduce the process of crude oil operationsand its short-term scheduling problem. The hybrid PN modelis developed in Section III. With the PN model, schedulabilityanalysis for systems with one and two distillers is carried outin Section IV. Section V generalizes the results obtained inSection IV into general cases. In Section VI, an industrial casestudy is presented to show the power and applications of theresults obtained. Finally, conclusions are given in Section VII.

II. PROCESS AND SHORT-TERM SCHEDULING

This section briefly introduces the process of crude oil oper-ations and its short-term scheduling problem. The readers arereferred as to [20]–[23] for detail.

In a general refinery process, crude oil is carried to the portby crude oil tankers, where crude oil is unloaded into storagetanks by the port. The crude oil in the storage tanks is trans-ported to charging tanks in the refinery plant through a pipeline.From the charging tanks, oil is fed into distillers for distillation.The middle products from the distillers are then sent to otherproduction units for fractionation and reaction. The productsafter fractionation and reaction are blended to produce the finalproducts that are ready for delivery. This paper addresses onlyshort-term scheduling problem for crude oil operations from atanker to distillers.

Often, in a refinery, various types of crude oil are processed.Some crude oil types have fusion point higher than 30 ◦C, andit is in a solid state under ordinary temperature. To prevent suchcrude oil from being frozen in the pipeline, the pipeline andthe crude oil that is being transported must be kept hot enough.Notice that, generally, a pipeline has tens of kilometers long,it can be heated only by hot crude oil that is flowing throughit. Thus, before high fusion point crude oil is to be transportedfrom storage tanks to charging tanks through the pipeline, acertain amount of hot crude oil with low fusion point should

be transported via the pipeline such that the pipeline can beheated first. After it is heated in this way, the heated high fusionpoint crude oil can be sent to charging tanks through it. After aparcel of such oil is sent to the pipeline, another parcel of crudeoil with low fusion point must be sent to it such that the highfusion point crude oil can go out of the pipeline. This resultsin high set up cost for transporting high fusion point crude oil.Thus, in scheduling crude oil operations, when high fusion pointcrude oil is transported from storage tanks to charging one, itis desired that it can transport as much as possible, or at leastan expected amount by a single setup. Often, in determining arefining schedule, a planner wants to know if a given amount ofhigh fusion crude oil in the storage tanks can be transported tocharging tanks for processing by a single setup.

It should be pointed out that when there is crude oil with highfusion point in the pipeline, the oil in it must be kept in flowing;and otherwise the pipeline and oil in it will cool down and theoil will be frozen. This implies that once crude oil with highfusion point is sent to the pipeline there must be enough spacein charging tanks to hold the coming crude oil, otherwise, theoil in the pipeline cannot be kept in flowing. With the numberof charging tanks limited and there are multiple distillers thatprocess different types of crude oil concurrently, it is a greatchallenge to schedule the system.

A crude oil operation process is composed of a series ofoperations. For each operation to take place, a decision shouldbe made. To describe a short-term schedule, we first define anoperation decision.

Definition 2.1: OD = (COT, ζ, S,D, INT = [a, b]) is definedas an operation decision, where COT is the crude oil type; ζ isthe volume of crude oil to be unloaded from a tanker to a storagetank, or transported from a storage tank to a charging tank, or fedfrom a charging tank to a distiller; S is the source from whichthe crude oil is to be delivered; D is the destination to which thecrude oil is to be delivered; and INT is a time interval in whicha and b are the start and end time points of the operation.

The flow rate in delivering crude oil in [a, b] can be vari-able. However, in reality, it is kept as a constant for a singleoperation. Thus, given volume ζ and time interval [a, b] in anOD, ζ/(b − a) is determined and used as its flow rate. EachOD is a command that transfers the system from a state toanother.

There are three types of ODs: crude oil unloading, trans-portation, and feeding, denoted by ODU, ODT, and ODF, re-spectively, and their time interval is denoted as [α, β], [λ, µ],and [ω, π], respectively. For ODU, S is a tanker and D is astorage tank. For ODT, S is a storage tank, D is a charging tank,and the transportation must be conducted through a pipeline. ForODF, S is a charging tank and D is a distiller. We use ODFki

to denote the ith OD for feeding distiller k during the schedulehorizon. Let Γ = [τs, τe ] be the schedule horizon that often lastsfor a week or 10 days and g = ζ/(β − α), f = ζ/(µ − λ), andh = ζ/(π − ω) denote flow rates for a tanker unloading, pipelinetransportation, and distiller feeding decided by ODs, and K de-note index set for the distillers. Given the system state at τs ,i.e., the inventory of crude oil and state of all the devices, andinformation of tanker arrival, the short-term scheduling problem


is to find a series of ODs described as follows:

SCHD = {ODU1 , . . . ,ODUw ,ODT1 , . . . ,

ODTz ,ODF1 , . . . ,ODFK }. (2.1)

Subjected to

ωk1 = τs, πk1 = ωk2 , . . . , πk(i−1) = ωki, . . . , and πkn = τe ,

for ∀k ∈ K (2.2)

the resource and process constraints given in Section I.

Constraint (2.2) requires that the schedule should cover theentire scheduling horizon and a distiller cannot be stopped.

A schedule for crude oil operations may not be feasible. Itsfeasibility is strongly related to the safeness of the system in acontrol theory perspective. The execution of an infeasible ODresults in an unsafe state. On the other hand, a safe state guaran-tees the existence of a feasible schedule. This can be explainedas follows. An operation decision OD in a schedule is a controlthat transfers the system from one state to another. If a stateis reached such that one or more constraints are violated, thisstate is said to be an infeasible state. If a state is not infeasible,it must be feasible. Assume that, at time τ , after the action ofOD1 , OD2 , . . ., and ODi , the system is transferred from stateM1 to Mi , and Mi itself is feasible. However, no matter whatODs are applied after time τ , the system will enter an infeasiblestate. Then, state Mi is called an unsafe state. If a state is notunsafe, it must be safe. If a short-term schedule SCHD = (OD1 ,OD2 , . . ., ODn ) is obtained such that OD1 transfer the systemfrom M1 to M2 , OD2 from M2 to M3 , . . ., and ODn from Mn

to Mn+1 , and all the states M1 , M2 , . . ., Mn+1 are safe, theschedule is feasible. On the other hand, at any time, if the systemis in a safe state, there must exist a short-term schedule such thatthe system is always in a safe state.

III. HYBRID PN MODELING

It follows from the short-term schedule defined in the lastsection that a hybrid model suitable for short-term schedulingis needed. A hybrid PN is developed for crude oil operationsfor schedulability analysis in [21] and [22], which consider onlythe situations without high point crude oil being processed. Bymodifying it, we present a new hybrid PN model here. We firstpresent the models for devices, e.g., tanks and pipeline. Then,based on the models for devices, the model for the whole systemis developed. A reader is referred to [3], [17], [25]–[29] for thebasic knowledge of PN.

The PN model used here is a kind of colored-timed PN(CTPN) defined as CTPN = (P = PD ∪ PC ∪ PE , T = TD ∪TT ∪ TC , I,O,Φ,M0 ), where PD , PC , and PE are sets of dis-crete, continuous, and enforcing places; TD , TT , and TC aresets of discrete, timed, and continuous transitions; I and O areinput and output functions; Φ(p) and Φ(t) represent the colorsets of place p ∈ P and transition t ∈ T ; and M0 is the initialmarking. The icons are shown in Fig. 2.

Fig. 2. Icons in the model.

Fig. 3. Petri net for a tank.

A. Device Modeling

The devices include tanks (storage and charging tanks) andpipeline. A tank is modeled by a PN shown in Fig. 3. A token in adiscrete place is a discrete token and acts just as that in ordinaryPN. A token in a continuous place represents a type of materialswith a real number of volume, called token volume for short.A discrete transition acts just as that in ordinary PN. When atimed transition fires, it delivers a token from a place to anotherwith a constant time delay. When a continuous transition fires,some type of material represented by a token is delivered froma place to another with a known flow rate determined by an OD.Whether the token is completely removed from its input placeis determined by its firing duration.

In Fig. 3, continuous places ps and pc that can hold at mostone token at a time model the state of a tank. A token in eitherps or pc or both represents that there is crude oil in the tank.However, a token in ps represents that the oil in the tank is notready for discharging, and it is ready only if pc has a token and ps

is empty. Continuous transitions t1 and t3 model the processesof filling to and discharging from a tank. The time delay in t2models the RT constraint. Thus, when charging oil to a tankstops, a token representing a type of crude oil must stay in ps

for some time. After the time delay specified by t2 , a token witha real number of volume moves from ps to pc , and is ready tobe discharged from a tank. Because there is only one token indiscrete place p4 , only one of transitions t1 and t3 can fire at atime. This guarantees that a tank cannot be filled and dischargedat the same time. The self-loop between p4 and t2 together withthe inhibitor arc (ps , t3) guarantees the RT requirement of the oilin the tank before it can be discharged. The volume associatedwith the token in continuous place p3 models the capacity of


Fig. 4. PN model for pipeline.

the tank available at the current marking. Thus, the behaviorof the charging and discharging of a tank is exactly described.Thereafter, by Ki(ps), Ki(pc ), and Ki(ps , pc ) (or Ki in short)we mean places ps , pc , and the model for tank i.

The PN model in Fig. 4 describes the behavior of a pipeline.It is modeled by a macro transition tpipeline in which thereare continuous places, such as p1 , p2 , and p3 , connected bydiscrete transitions in a serial way. A token in a continuousplace in tpipeline represents a segment of crude oil in the pipeline.The tokens in different continuous places represent segments ofdifferent crude oil types. When a discrete transition in tpipeline ,say t1 , is enabled and fires, a token in p2 is removed immediately,and when t2 is enabled and fires, a token enters p2 immediately.Thus, the behavior of a segment in the pipeline for holding onetype of crude oil is described by the PN.

The PN model shown in Fig. 4 describes the behavior ofthe pipeline that can hold three different types of crude oilwith continuous places p1 , p2 , and p3 in tpipeline . Continuoustransitions tI1 to tIk and tO1 to tOk in tpipeline are used tomodel the behavior that crude oil flows into and out of thepipeline with a given flow rate. Let TI = {tI1 , . . . , tIk} andTO = {tO1 , . . . , tOk}. It should be noticed that, at a time, onlyone type of crude oil can flow into (out of) the pipeline. Thisimplies that only one transition in TI and one transition in TO canfire at a time. When one of transitions in TO is firing, the tokenvolume in p1 is continuously decreasing until p1 is emptied.When p1 is emptied, t1 is enabled and fires, and the token inp2 is moved into p1 immediately. In this way, one transition inTO can fire continuously. When one transition in TI , say tI1 ,fires, a token goes into p3 immediately. With the continuousfiring of tI1 , the token volume in p3 increases continuouslyuntil p3 is full. The number of continuous places in the modelmeans the number of segments of crude oil of different typesin the pipeline. It can be set as the largest number of crude oilsegments that may occur. Since the pipeline is modeled as asingle macro transition tpipeline , a transition in TI and anotherin TO fire concurrently with the same flow rate. However, thecrude oil type flowing into a pipeline may be different from thatflowing out of it.

Fig. 5. PN model for the whole system.

There is also an enforcing place p4 in tpipeline . Every time,a token enters p3 by firing a transition in TI , a token with thesame color and volume goes into p4 too. When there is a tokenrepresenting a high fusion point crude oil type in p4 , it enforcesone of transition in TO to fire, which requires firing of onetransition in TI too. When the token representing a high fusionpoint crude oil in a continuous place in tpipeline goes out oftpipeline by firing a transition in TO , the token in p4 is removedtoo. For the sake of simplicity, transition y as shown in Fig. 4is used to represent this process pictorially. Then, a place p iny can be denoted as p(y). When y fires, it implies that onetransition in TI and one transition in TO fire with the same ratesimultaneously, or crude oil is delivered from a storage tank tothe pipeline (a place in y) by a transition in TI , and at the sametime crude oil in p1(y) is being delivered into a charging tankby a transition in TO .

B. PN Model for the Whole System

The PN in Fig. 5 gives the illustration for the whole systemwith two storage and charging tanks, respectively. For the reasonof simplicity, we omit the discrete place and its associated arcs,and the inhibitor arc in a tank PN model. Later, we also omit pi3 .Crude oil in a storage tank is discharged through the pipeline,and then transition y for the pipeline is the discharging transitionfor every storage tank. Similarly, y is the charging transition forevery charging tank. Thus, {t11 , t12 , y, p1s , p1c , p13} and {t21 ,t22 , y, p2s , p2c , p23} are for two storage tanks, respectively, and{y, t31 , t32 , p3s , p3c , p33} and {y, t41 , t42 , p4s , p4c , p43} for twocharging tanks, respectively. Transition t1 creates a token whena tanker arrives. If a tanker carries k types of crude oil, thenk tokens are created with their respective volumes. However, atoken can be moved into p1 from t1 only if p1 is empty. Placesp3 and p4 represent two distillers. From the definition of a short-term schedule, we know that firing a continuous transition in thePN model must be triggered by an OD. Thus, the dynamics ofcrude oil flow in the PN model should be governed by the flowrate given in OD. This can be modeled by the transition enablingand firing rules that are defined below.

We use •t (•p) to denote the set of input places of transition t(input transitions of place p) and t• (p•) the set of output placesof t (output transitions of p). Let V (M(p)) denote the volumeof material in p at marking M .

Definition 3.1: A discrete transition (including discrete onesin y) t is said to be enabled at marking M , if M(p) ≥ 1,∀p ∈ •t and M(p) = 0,∀p ∈ t•. When t fires, M is changed into


M ′ with M ′(p) = M(p) − 1,∀p ∈ •t and M ′(p) = M(p) +1,∀p ∈ t•.

Compared with the transition enabling and firing rules forordinary PN, it requires that t’s output places are empty, simi-larly to that for finite capacity PN. Notice that, in ordinary PN,tokens are discrete, but the tokens in the PN model here may bediscrete or continuous. However, to fire a discrete transition, alltokens in the model are treated as discrete one. By Definition3.1, when some part of oil in p1 in Fig. 4 is discharged, thereis still a token in p1 and t1 cannot fire, or a token in p2 cannotgo to p1 . Only when all the oil in p1 is discharged such thatp1 is empty, then t1 can fire and a token in p2 can go to p1 . Inthis way, it guarantees that each continuous place in Fig. 4 canhold one type of crude oil at a time. A timed transition is usedonly in the PN for a tank to guarantee the RT before oil in atank can be discharged. Assume that the time associated with atimed transition is Ψ. Its enabling and firing rules are defined asfollows:

Definition 3.2: A timed transition t is said to be enabled atmarking M if M(pi) ≥ 1, ∀pi ∈ •t. When t starts to fire at timeτ , M is changed into M ′ such that: 1) at τ , M ′(pi) = M(pi) −1, if pi ∈ •t and pi ∈ PD and M ′(pi) = M(pi) if pi ∈ •t andpi ∈ PC ; 2) at τ + Ψ, M ′(pi) = M(pi) − 1 and V (M ′(pi)) =0, pi ∈ •t and pi ∈ PC ; and 3) at τ + Ψ, M ′(pj ) = 1 for ∀pj ∈t• and V (M ′(pj )) = V (M(pj )) + V (M(pi)), for pj ∈ t• andpj ∈ PC .

Notice that when a timed transition t fires, there may be atoken in its output place pj . When the token in the input place pi

moves into pj these two tokens merge into one with the volumebeing the sum. This definition is used to describe such a factthat when a tank is neither full nor empty, this tank can still becharged. When it is charged, the volume of oil should be added.Meanwhile, the time delay associated with the timed transitionguarantees the RT constraint. In this way, the behavior of a tankis precisely modeled by the PN.

Because there are multiple types of crude oil, it is necessaryto distinguish them. To do so, colors are introduced into the PNmodel. We use ϕ ∈ Φ to denote the color of crude oil and say thata token in place p representing crude oil of type i has color ϕi andthe number of tokens in p with color ϕi at marking M is denotedby M (p, ϕi), and the volume for this token at M is denotedby V (M(p, ϕi)). When V (M(p, ϕi)) = 0, it implies that thenumber of tokens with color ϕi in p is zero. If a continuoustransition t is firing to move crude oil type i from a place toanother, we say t is firing with color ϕi . A continuous transitionmust fire with a color. As discussed above, the volume of atoken in p3 in Fig. 3 models the capacity of a tank available.Let Θ denote the color for such a token. In other words, p3 ineach tank’s PN model has the same color as the token in ps

and/or pc . Further, let Φ1 ⊂ Φ be the set of colors for crude oiltypes with low fusion point and Φ2 ⊂ Φ be the set of colors forcrude oil types with high fusion point with Φ1 ∩ Φ2 = Ø andΦ1 ∪ Φ2 = Φ.

Definition 3.3: A continuous transition (including transitionsin TI and TO in y) t is said to be enabled with color ϕi atmarking M , if the following conditions are satisfied:

1) M(p, ϕi) ≥ 1 or M(p,Θ) ≥ 1, ∀p ∈ •t; and

2) Kj (ps) ∈ t• for some j, then M(Kj (ps), ϕi) ≥ 1 orM(Kj (pc), ϕi) ≥ 1, or M(Kj (ps)) = M(Kj (pc)) = 0.

By condition 1), it says that, if p ∈ PC , there must be crudeoil with a right color or the tank to be charged must not be full.If p ∈ PE , it implies that firing t will consume a token in p. By2), it says that the oil in Kj (ps) has color ϕi that is same as thatin p ∈•t. This implies that if there is crude oil in a tank, onlythe same type of crude oil can be filled into it. However, if atank is empty, any type of crude oil can be filled into it. When acontinuous transition t is enabled and triggered by an OD, it canthen fire. This firing must be associated with a flow rate givenin OD, i.e., the flow of crude oil is governed by a flow rate. Thetransition firing rules below describe this dynamics. We assumethat t’s firing with color ϕi begins at time τ1 , ends at time τ2 ,τ ∈ [τ1 , τ2 ], and the flow rate is f . Then, the marking changesas follows. At τ1 , if p ∈•t and p is a discrete place

M ′(p) = M(p) − 1. (3.1)

At τ2 , if p ∈ t• and p is a discrete place

M ′(p) = M(p) + 1. (3.2)

At τ ∈ [τ1 , τ2 ], if p ∈•t and p ∈ PC or p ∈ PE , and there isa token with color ϕi in p

V (M ′(p, ϕi)) = V (M(p, ϕi)) − (τ − τ1)f. (3.3)

If p ∈•t and p ∈ PC , and there is a token with color Θ in p

V (M ′(p, Θ)) = V (M(p, Θ)) − (τ − τ1)f. (3.4)

If p ∈ t• and p ∈ PC or p ∈ PE , and there is a token withcolor ϕi in p

V (M ′(p, ϕi)) = V (M(p, ϕi)) + (τ − τ1)f. (3.5)

If p ∈ t• and p ∈ PC or p ∈ PE , and there is no token withcolor ϕi in p

M ′(p, ϕi) = 1 and

V (M ′(p, ϕi)) = V (M(p, ϕi)) + (τ − τ1)f. (3.6)

Because any firing of a continuous transition leads to a crudeoil operation determined by an OD, expressions (3.1) through(3.6) describe the dynamics of crude oil flow in the PN model forthe system. Notice that, with (3.5) and (3.6), when a transition tin TI in y fires with color ϕi , a token with color ϕi and volumethat flows through t is moved into t’s output enforcing place.From (3.3), such a token can be removed by firing transitions inTO .

Definition 3.4: The PN model for the crude oil operations islive if and only if the following conditions are satisfied.

1) Let Pdsl denote the set of places representing the distillersand |Pdsl | = h. Then, at any time t (or at any marking M )for any pi ∈ Pdsl , there exists a ti ∈ •pi such that ti is infiring or enabled and t1 ∩ t2 ∩ . . . ∩ th = Ø.

2) If M(p4(y), ϕi) > 0, ϕi ∈ Φ2 , and p4(y) ∈ PE , at mark-ing M , there exists at least a transition t ∈ p4(y)• is infiring or enabled.


By Definition 3.4, a nonlive state is an infeasible state result-ing from an infeasible schedule. Thus, it is necessary to schedulecrude oil operations such that the PN model is live. If a schedulecan be found such that the PN is live, the system is schedulable.It should be pointed out that the conditions in Definition 3.4 arevery easy to check. Thus, the PN model developed here is aneffective tool for scheduling the system.

IV. SCHEDULABILITY WITH ONE AND TWO DISTILLERS

With the PN model developed in the last section, this sectionanalyzes the schedulability and the amount of high fusion pointoil that can be transported by a single setup based on chargingtank grouping. It assumes that there is always enough crude oilin the storage tanks to be processed, or schedule feasibility isindependent of the number of storage tanks and their capacity.Notice that the pipeline always holds some oil in it and weuse Cpipe to denote its capacity. Hence, when a volume V1 >Cpipe of oil is transported from storage tanks to charging tankCTK1 , Cpipe of it will remain in the pipeline if there is no otherparcel of oil that is transported via the pipeline immediately afterV1 volume of oil is transported. Therefore, after transporting aparcel of oil with high fusion point, a parcel of oil with low fusionpoint and volume V2 ≥ Cpipe must be transported immediately.This implies that, after charging CTK1 with V1 of high fusionpoint oil, there must be a charging tank that can receive at leastvolume Cpipe of oil. Without loss of generality, we assume thatthe capacity of any charging tanks is greater than Cpipe . Toreduce charging and discharging cost, it is also desired that atank is charged to capacity when it is charged. Thus, we use thenumber of tanks to discuss the volume of oil transported.

Definition 4.1: A system of crude oil operations with initialstate S0 is said to be schedulable if there exists a feasible short-term schedule for a horizon Γ = [0,∞).

Notice that the key here is the schedule horizon Γ = [0,∞),because the operation of distillers cannot be interrupted.The existence of a feasible short-term schedule SCHD1 ={OD1 , OD2 , . . . , ODn} for a schedule horizon [0, a] doesnot guarantee that a feasible short-term schedule SCHD2 ={OD1 , OD2 , . . . , ODn , ODn+1 , . . . , ODn+k} with SCHD1 ⊂SCHD2 for schedule horizon [0, b], with b > a, can be found.

Definition 4.2: A state M of a system of crude oil opera-tions is said to be safe if, with M as initial state, the system isschedulable.

Therefore, with Definitions 4.1 and 4.2, to analyze the schedu-lability of crude oil operations is to analyze the safeness of thesystem. In [21] and [22], schedulability conditions are presentedby considering oil residency time constraint only. Here, both oilresidency time constraint and crude oil types with high fusionpoint are considered. Thus, the problem addressed here is muchmore complex than that in [21] and [22].

A. Case 1: System with One Distiller

Let fp max , fds , and Ψ be the maximal crude oil transportationrate of the pipeline, the feeding rate to the single distiller, andthe oil residency time, respectively. It is shown in [21] that ifthere is only a single charging tank and Ψ > 0 the system is notschedulable even if fp max = ∞. Obviously, such a system is

Fig. 6. PN model for the proof of Theorem 4.1.

not schedulable for the situation considered in this paper. Nev-ertheless, it is shown in [21] that if there are two charging tanksand fp max is large enough the system is schedulable if only oilresidency time is considered. However, the following theoremshows that it is not schedulable for the system considered inthis paper. Let ξ = (Ψ × fp max × fds)/(fp max − fds), CTKi

denote charging tanks i, α = fds × Ψ, and ϕ be the color ofcrude oil type being processed, respectively.

Theorem 4.1: Assume that: 1) there is a single distiller withfeeding rate fds ; 2) there are two charging tanks CTK1 andCTK2 with capacity greater than or equal to ξ; 3) the crude oil be-ing processed is a type with high fusion point and its color is ϕ ∈Φ2 ; 4) fp max > fds ; and 5) initially there is crude oil ready forfeeding with volume ζ ≥ (Ψ × fp max × fds)/(fp max − fds)in tank CTK1 and CTK2 is empty. Then, the system is notschedulable.

Proof: This situation can be modeled by the PN shownin Fig. 6. At the beginning (time τ0), we have M0(p1c) =1, M0(p1s) = M0(p2c) = M0(p2s) = 0, V (M0(p1c , ϕ)) = ζ,and M0(p4(y), ϕ) = 1 as shown in Fig. 6(a). Then, t12 canfire with rate fds for feeding p1 (the distiller) and y can firewith rate fp max to charge p2s . At time τ1 = τ0 + (ζ/fp max),M1 is reached such that M1(p2s) = M1(p1c) = M1(p4(y),ϕ) = 1, V (M1(p2s , ϕ)) = ζ, and V (M1(p1c , ϕ)) = ζ − (τ1 −τ0)fds > 0 as shown in Fig. 6(b). At τ1 , if y stops firing, then attime τ2 = τ0 + (ζ/fds), M2 is reached such that M2(p1c) = 0and V (M2(p1c)) = 0. This time, we have

τ2 − τ1 =ς

fds− ς

fp max=

fp max − fds

fp max × fds× ς

≥ fp max − fds

fp max × fds× Ψ × fp max × fds

fp max − fds= Ψ

or we have M2(p2c) = 1 and V (M2(p2c , ϕ)) = ζ, or t22 can firewith rate fds and the distiller can continue its operation withoutinterruption. However, because M1(p4(y), ϕ) = 1 with ϕ ∈ Φ2 ,at τ1 , y cannot stop firing. Thus, at τ2 , it reaches M3 such thatM3(p1c) = M3(p2c) = 0 as shown in Fig. 6(c). At this mark-ing, both t11 and t22 cannot fire, or the PN is not live, or this isan infeasible state. Therefore, the system is not schedulable. �


Fig. 7. PN model for a system with three charging tanks and one distiller.

Theorem 4.1 shows that with high fusion point types of crudeoil to be processed more charging tanks are required.

Theorem 4.2: Assume that: 1) there is a single distiller withfeeding rate fds ; 2) there are three charging tanks CTK1−3 withcapacity Cj ≥ α, j ∈ {1, 2, 3}; 3) the crude oil being processedis a type with high fusion point and its color is ϕ ∈ Φ2 ; and4) fp max ≥ fds ; and 5) initially there is crude oil in CTK1 andCTK2 with amount ζ1 and ζ2 , respectively, where ζ1 = ζ2 =ζ = min{C1 , C2 , C3}. The oil in CTK1 is ready for feedingand CTK3 is empty. Then, the system is schedulable and anynumber of tanks of high fusion oil can be transported by a singlesetup.

Proof: The PN model is shown in Fig. 7. Initially (attime τ0 , or at marking M0) we have M0(p1c) = M0(p2s) =M1(p4(y), ϕ) = 1 with ϕ ∈ Φ2 and M0(p1s) = M0(p2c) =M0(p3s)= M0(p3c) = 0, V (M0(p1c , ϕ))= V (M0(p2s , ϕ))= ζ,and V (M0(p1s))= V (M0(p2c))= V (M0(p3s))= V (M0(p3c))= 0. This time we fire t12 with rate fds to feed p1 , meanwhiley can fire with rate fds to charge p3s , for fp max ≥ fds .At time τ1 = τ0 + (ζ/fds) ≥ τ0 + Ψ, M1 is reachedsuch that M1(p1c) = M1(p2s) = 0, M1(p2c) = M1(p3s) =M1(p4(y), ϕ) = 1, and V (M1(p2c , ϕ)) = V (M1(p3s , ϕ)) = ζ.At this time, t22 can fire with rate fds to feed p1 and y can firewith rate fds to charge p1s . In fact, marking M1 is equivalentto M0 , hence, the previous process can be repeated unlimitedly.By doing so, there is always a transition firing to feed p1 andat the same time y keeps firing without interruption. Thus, thePN is live and it is schedulable. At the same time, y can firewith color ϕ ∈ Φ2 unlimitedly, or any number of tanks of highfusion oil can be transported. This can be done by a singlesetup. �

Obviously, if there are more than three charging tanks, thesystem is schedulable and any amount of crude oil with highfusion point can be transported by a single setup.

B. Case 2: System with Two Distillers

When there are multiple distillers in a refinery, often eachdistiller processes different type of crude oil with different crudeoil feeding rate. Thus, we always assume that different distillerprocesses different type of crude oil at a time. Without loss ofgenerality, we also assume that at any time there is no more thanone type of high fusion point crude oil to be processed at a time.Let fds1 and fds2 denote the feeding rate to distiller #1 (DS1)and distiller #2 (DS2), and α1 = Ψ × fds1 and α2 = Ψ × fds2 ,respectively.

In crude oil operations, when fp max = fds1 + fds2 the sys-tem reaches its maximal productivity, for fds1 + fds2 cannot be

Fig. 8. PN model with five charging tanks for feeding two distillers.

greater than fp max . In this paper, schedulability is discussed un-der the assumption that fp max = fds1 + fds2 . It is shown in [22]that if there are four charging tanks for feeding two distillers andit requires fp max = fds1 + fds2 with fds1 �= fds2 , the systemis not schedulable. This conclusion must hold here too. We con-sider the situation that there are five charging tanks. Let Π be areal number with Π ≥ 1 and the following conditions are given.

1) Production condition: there are two distillers DS1 and DS2with feeding rates being fds1 and fds2 , fds1 > fds2 , α1 =Ψ × fds1 , α2 = Ψ × fds2 , and fp max = fds1 + fds2 .

2) Tank capacity: there are five charging tanks CTK1−5 withcapacity C1 ≥ 2Πα1 , C2 ≥ 2Πα1 , C3 ≥ 2Πα1 , C4 ≥2Πα2 , and C5 ≥ 2Πα2 , and CTK1 , CTK2 , and CTK3are for DS1 , CTK4 and CTK5 for DS2 .

3) Initial Condition: the volume of oil type 1 (color ϕ1) inCTK1 and CTK2 is ζ1 = ζ2 = 2Πα1 , and the volume ofoil type 2 (color ϕ2) in CTK4 is ζ4 = 2Πα2 , CTK3 andCTK5 are empty, and the oil in CTK1 and CTK4 is readyfor feeding.

4) Crude oil type switching: after processing oil in CTK4 byDS2 , it should process a high fusion point oil type (type#3).

Theorem 4.3: If conditions (1)–(4) are satisfied, the system isschedulable and one tank of crude oil of type 3 can be transportedby a single setup.

Proof: With the PN model shown in Fig. 8, the system canbe scheduled as follows. Initially, at marking M0 (time τ0), wehave M0(p1c) = M0(p2s) = M0(p4c) = 1, V (M0(p1c , ϕ1)) =V (M0(p2s , ϕ1)) = 2Πα1 , V (M0(p4c , ϕ2)) = 2Πα2 , and otherplaces are empty. At this marking, we fire t12 and t42 withcolors ϕ1 and ϕ2 to feed p1 and p2 , respectively. At the sametime, we fire y with color ϕ3 ∈ Φ2 and rate fds1 + fds2 duringtime interval [τ0 , τ0 + 2Πα2/(fds1 + fds2)] to charge p5s andfire y with color ϕ1 and rate fds1 + fds2 during time interval[τ0 + Πα2/(fds1 + fds2), τ0 + 2ΠΨ] to charge p3s . In this way,at τ1 = τ0 + 2Πα2/(fds1 + fds2), we have M(p4(y)) = 1, butwith M(p3s) = 0, y can fire. It is a live state. Also, at τ1 , t51 canstart to fire and at τ0 + 2ΠΨ, the firing of t51 must have ended,because 2Πα2/(fds1 + fds2) < Π(α1 + α2)/(fds1 + fds2) =


Π(fds1 + fds2)Ψ/(fds1 + fds2) = ΠΨ. Thus, after ΠΨ ≥ Ψtime units, at time τ2 = τ0 + 2ΠΨ, marking M1 is reached suchthat M1(p2c) = M1(p5c) = M1(p3s) = 1, V (M1(p2c , ϕ1)) =V (M1(p3s , ϕ1)) = 2Πα1 , and V (M1(p5c , ϕ3)) = 2Πα2 , andother places are empty. This marking is equivalent to the ini-tial marking M0 or the previous process can be repeated andy can be kept firing in all the time. In other words, the systemis schedulable. In this process, one tank of high fusion pointoil is transported by firing y uninterruptedly, or by a singlesetup. �

Theorem 4.3’s conditions require that fds1 > fds2 . Thiscondition guarantees that when M1 is reached such thatM1(p4c) = 0, M1(p5s) = 0 and V (M1(p5s , ϕ2)) = 2Πα2must hold, such that the oil residency time constraint can bemet. If fds2 > fds1 , and when a marking M is reached suchthat V (M(p4c , ϕ2)) = α2 and V (M(p5s , ϕ2)) = 2Πα2 , theoil residency time constraint must also be met. In this case,it requires that 2Πα2/(fds1 + fds2) ≤ (2Π − 1)α2/fds2 ⇒2ΠΨ × fds2/(fds1 + fds2) ≤ (2Π − 1)Ψ × fds2/fds2 ⇒ Π≥(fds1 + fds2)/2fds1 . In general, Ψ is just several hour longand Π can be more than 2 for most charging tanks. Thus, iffds2 ≤ 3fds1 the condition can be met, or the theorem holdsfor fds2 > fds1 too. It follows from the proof of Theorem 4.3that it is impossible to transport two tanks of high fusion pointoil by a single setup. Thus, with five charging tanks for feedingtwo distillers, at most one tank of high fusion point oil can betransported by a single setup. Now we consider the situationwith six charging tanks and the following conditions.

5) Production condition: there are two distillers DS1 andDS2 with fds1 �= fds2 , α1 = Ψ × fds1 , α2 = Ψ × fds2and fp max = fds1 + fds2 .

6) Tank capacity: there are six charging tanks CTK1−6 withcapacity C1 ≥ Πα1 , C2 ≥ Πα1 , C3 ≥ Πα1 , C4 ≥ Πα2 ,C5 ≥ Πα2 , and C6 ≥ Πα2 , and Π ≥ (fds1 + fds2)/fds1 ;CTK1 , CTK2 , and CTK3 are for DS1 ; and CTK4 , CTK5 ,and CTK5 for DS2 .

7) Initial condition: the volume of oil type 1 (color ϕ1) inCTK1 and CTK2 is ζ1 = ζ2 = Πα1 , the volume of oiltype 2 (color ϕ2) in CTK4 is ζ4 = Πα2 , CTK3 , CTK5 ,and CTK6 are empty, and the oil in CTK1 and CTK4 isready for feeding.

8) Crude oil type switching: after finishing the oil of type 2in CTK4 , DS2 should process oil type 3 (color ϕ3) withhigh fusion point.

Theorem 4.4: If conditions (5)–(8) are satisfied, the systemis schedulable and two tanks of crude oil of type 3 can betransported via the pipeline by a single setup.

Proof: The PN model for this case is shown inFig. 9. We first show that when fds1 > fds2 the the-orem holds. Initially, at marking M0 (time τ0), wehave M0(p1c) = M0(p2s) = M0(p4c) = 1, V (M0(p1c , ϕ1)) =V (M0(p2s , ϕ1)) = Πα1 , V (M0(p4c , ϕ2)) = Πα2 , and otherplaces are empty. At this marking, we fire t12 with colorϕ1 and rate fds1 to feed p1 , and t42 with color ϕ2and rate fds2 to feed p2 , respectively. At the same time,we fire y with color ϕ3 ∈ Φ2 and rate fds1 + fds2 tocharge p5s . At time τ1 = τ0 + Πα2/(fds1 + fds2), marking

Fig. 9. PN model with six charging tanks for feeding two distillers.

M1 is reached such that M1(p1c) = M1(p2c) = M1(p4c) =M1(p5s) = M1(p4(y), ϕ3) = 1 and V (M1(p5s , ϕ3)) = Πα2 .At this marking, we can continue the firing of t12 andt42 to feed p1 and p2 , respectively. Meanwhile, M1(p6s) =M1(p6c) = 0. Hence, we can fire y with color ϕ3 ∈ Φ2 andrate fds1 + fds2 to charge p6s continuously. Because fds1 >fds2 , at time τ2 = τ0 + 2Πα2/(fds1 + fds2), marking M2 isreached such that V (M2(p6s , ϕ3)) = Πα2 , but M2(p1c) =M2(p4c) = M1(p4(y), ϕ3) = 1 still holds. At this marking,with M2(p3s) = M1(p6c) = 0, we can fire y with color ϕ1and rate fds1 + fds2 to charge p3s , and at the same time con-tinue the firing of t12 and t42 to feed p1 and p2 . At timeτ3 = τ0 + ΠΨ, M3 is reached such that M3(p5c) = M3(p2c) =1, M3(p4c) = M3(p1c) = 0, and V (M3(p3s , ϕ1)) = Π(α1 −α2). The question is if M3(p4(y), ϕ3) = 1, which dependson whether Π(α1 − α2) ≥ Cpipe or not. However, we can as-sume that M3(p4(y), ϕ3) = 1. At this marking, we can firet22 with color ϕ1 to feed p1 , t52 with color ϕ3 to feedp2 , and fire y with color ϕ1 to charge p3s and p1s withvolume Πα2 and Πα1 , respectively. Let τ4 = τ0 + 2ΠΨ, wehave τ4 − Πα2/(fds1 + fds2) − τ3=ΠΨ−ΠΨ×fds2/(fds1 +fds2) ≥ Ψ((fds1+fds2)/fds1 − fds2/fds1) = Ψ. This impliesthat t31’s firing has ended at τ4 . Thus, at timeτ4 , marking M4 is reached such that M4(p6c) =M4(p3c) = M4(p1s) = 1, M4(p6s) = M4(p3s) = M4(p4(y),ϕ3) = 0, V (M4(p3c , ϕ1)) = V (M4(p1s , ϕ1)) = Πα1 , andV (M4(p6c , ϕ3)) = Πα2 . This marking is equivalent to M0 , theprevious process can be repeated in a cycle way.

Now we discuss the case with fds2 > fds1 . At M0 , we firet12 with color ϕ1 and rate fds1 to feed p1 , and t42 with colorϕ2 and rate fds2 to feed p2 , respectively. At the same time,we fire y with color ϕ3 ∈ Φ2 and rate fds1 + fds2 to chargep5s . At time τ1 = τ0 + Πα2/(fds1 + fds2), M1 is reachedsuch that M1(p1c) = M1(p2c) = M1(p4c) = M1(p5s) =M1(p4(y), ϕ3) = 1 and V (M1(p5s , ϕ3)) = Πα2 . At thismarking, we can continue the firing of t12 and t42 to feed p1and p2 , respectively. Meanwhile, M1(p6s) = M1(p6c) = 0,


the PN is in a live state. Hence we can fire y with colorϕ3 ∈ Φ2 and rate fds1 + fds2 to charge p6s . Becausefds2 > fds1 , at time τ2 = τ0 + ΠΨ, M2 is reached such thatV (M2(p6s , ϕ3)) = Πα1 , and p1c and p4c are just emptiedor M2(p1c) = M2(p4c) = 0 and M2(p4(y), ϕ3) = 1. At thismarking, we have τ2 − τ1 = ΠΨ − Πα2/(fds1 + fds2) =ΠΨ − ΠΨ × fds2/(fds1 + fds2) ≥ Ψ((fds1 + fds2)/fds1 −fds2/fds1) = Ψ. This implies that, at τ2 , t51’s firing has endedor M2(p5s) = 0, M2(p5c) = 1, and V (M2(p5s , ϕ3)) = Πα2 .Thus, at τ2 , we can fire t22 and t52 to feed p1 and p2 , respectively,and at the same time fire y with color ϕ3 and rate fds1 + fds2to charge p6s . At τ3 = τ2 + Π(α2 − α1)/(fds1 + fds2), M3 isreached such that M3(p2c) = M3(p5c) = M3(p4(y), ϕ3) = 1and V (M3(p6s , ϕ3)) = Πα2 . At this marking, with p3s , p3c ,p1s , and p1c being empty, we can continue the firing of t22and t52 to feed p1 and p2 , respectively, and at the sametime fire y with color ϕ1 and rate fds1 + fds2 to charge p3s

and then p1s . At τ4 = τ0 + 2ΠΨ, M4 is reached such thatM4(p2c) = M4(p5c) = M4(p4(y), ϕ3) = 0. Notice that thevolume that is charged to p6s and p35 after τ2 is Π(α2 −α1) + Πα1 = Πα2 . Thus, τ4 − (τ2 + Πα2/(fds1 + fds2)) =ΠΨ − Πα2/(fds1 + fds2) = ΠΨ − ΠΨ × fds2/(fds1 + fds2)≥ Ψ((fds1 + fds2)/fds1 − fds2/fds1) = Ψ. This impliesthat, at τ4 , the firing of t61 and t31 has ended or M4(p6s) =M4(p3s) = 0, M4(p6c) = M4(p3c) = M4(p1s) = 1, V (M4(p3c , ϕ1)) = V (M4(p1s , ϕ1)) = Πα1 , and V (M4(p6c , ϕ3)) =Πα2 . This marking is equivalent to M0 , or the previous processcan be repeated in a cycle way, or the system is schedulable. Inboth cases, two tanks of crude oil with high fusion point aretransported via the pipeline without being interrupted. Thus,such a parcel of oil can be transported by a single setup. �

By making Π as large as possible, the charging tanks can becharged as full as possible. It follows from Theorem 4.4 thattwo tanks of oil with high fusion point can be transported fromstorage tanks to charging tanks by a single setup. At the sametime, maximal productivity can be reached. From the proof ofTheorem 4.4, it is obvious that it is impossible to transport threetanks of oil with high fusion point if there are three chargingtanks for each distiller. From the proof, it is known that CTK3and CTK1 are charged interruptedly. This implies that two tanksof high fusion point crude oil can be transported by a single setupfor any one of the two distillers.

From Theorem 4.2, we notice that when there are three charg-ing tanks for feeding one distiller, unlimited amount of highfusion point oil can be transported by a single setup. However,only two tanks of high fusion point oil can be transported bya single setup when there are two distillers with each distillerhaving three charging tanks. This implies that, with more dis-tillers in the system, it is more difficult to schedule it. Now weconsider the situation with more charging tanks and the follow-ing conditions are given.

9) Tank capacity: there are eight charging tanks CTK1–8with capacity Ci ≥ Πα1 , i = 1, 2, 3, and 4, and Cj ≥Πα2 , j = 5, 6, 7, and 8, and Π ≥ (fds1 + fds2)/fds1 ,CTK1–4 are for DS1 , and CTK5–8 for DS2 .

10) Initial condition: the volume of oil type 1 (color ϕ1) inCTK1–3 is ζ1 = ζ2 = ζ3 = Πα1 , the volume of oil type 2

Fig. 10. PN model with eight charging tanks for feeding two distillers.

(color ϕ2) in CTK5 is ζ5 = Πα2 , CTK4 , and CTK6–8 areempty, and the oil in CTK1 and CTK5 is ready for feeding.

11) Crude oil type switching: after finishing the oil of type 2in CTK5 , DS2 should process oil type 3 (color ϕ3 ∈ Φ2)with high fusion point.

Theorem 4.5: If conditions (5), (9)–(11) are satisfied the sys-tem is schedulable and three tanks of crude oil of type 3 can betransported via the pipeline by a single setup.

Proof: The PN model for this case is shown inFig. 10. We first show that when fds1 > fds2 the theo-rem holds. Initially, at marking M0 (time τ0), we haveM0(p1c) = M0(p2s) = M0(p3s) = M0(p5c) = 1, V (M0(p1c ,ϕ1))=V (M0(p2s , ϕ1))=V (M0(p3s , ϕ1))=Πα1 , V (M0(p5c ,ϕ2)) = Πα2 , and other places are empty. At this marking, wefire t12 with color ϕ1 and rate fds1 to feed p1 , and t52 withcolor ϕ2 and rate fds2 to feed p2 , respectively. For y’s firing,there are three cases: 1) fds1 < 2fds2 ; 2) fds1 = 2fds2 , and3) fds1 > 2fds2 . For Case 1), we fire y with color ϕ3 ∈ Φ2and rate fds1 + fds2 to charge p6s with volume Πα2 , thencharge p7s with volume Πα2 , and then charge p8s with volumeΠ(α1 − α2). For Case 2), we fire y with color ϕ3 ∈ Φ2 andrate fds1 + fds2 to charge p6s , then p7s , and then p8s , all withvolume Πα2 . For Case (3), we fire y with color ϕ3 ∈ Φ2 andrate fds1 + fds2 to charge p6s , then p7s , and then p8s , all withvolume Πα2 , and then continue to fire y with color ϕ1 tocharge p4s with volume Π(α1 − 2α2). At τ1 = τ0 + ΠΨ,M1 is reached such that M1(p1c) = M1(p5c) = 0,M1(p2c) = M1(p3c) = M1(p6c) = M1(p4(y), ϕ3) = 1 for allthe three cases; V (M1(p6c , ϕ3)) = V (M1(p7s , ϕ3)) = Πα2 ,and V (M1(p8s , ϕ3)) = Π(α1 − α2) for Case (1);V (M1(p6c , ϕ3)) = V (M1(p7s , ϕ3)) = V (M1(p8s , ϕ3)) =Πα2 for Case (2); and V (M1(p6c , ϕ3)) = V (M1(p7s , ϕ3)) =V (M1(p8s , ϕ3)) = Πα2 , V (M1(p4s , ϕ1)) = Π(α1 − 2α2) forCase (3). At this marking, we can fire t22 with color ϕ1 and ratefds1 to feed p1 , and t62 with color ϕ3 and rate fds2 to feed p2 ,respectively. Meanwhile, there is enough space in p4s and p1s ,the PN is in a live marking. We fire y with color ϕ3 ∈ Φ2 tocharge p8s with volume Π(2α2 − α1), then with ϕ1 to chargep4s with volume Πα1 , and then with ϕ1 to charge p1s withvolume Π(α1 − α2) for Case (1); fire y with ϕ1 to chargep4s with volume Πα1 , and then with ϕ1 to charge p1s withvolume Π(α1 − α2) for Case (2); and fire y with ϕ1 to charge


p4s with volume 2Πα2 , and then with ϕ1 to charge p1s withvolume Π(α1 − α2) for Case (3). At τ2 = τ0 + 2ΠΨ, M2 isreached such that M2(p2c) = M2(p6c) = M2(p4(y), ϕ3) = 0,M2(p3c) = M2(p7c) = M2(p8s) = M2(p4s) = M2(p1s) = 1,V (M2(p7c , ϕ3))= V (M2(p8s , ϕ3))= Πα2 , V (M2(p4s , ϕ1))=Πα1 , and V (M2(p1s , ϕ1)) = Π(α1 − α2). At this marking, wefire t32 with color ϕ1 and rate fds1 to feed p1 , and t72 with colorϕ3 and rate fds2 to and p2 , respectively. At the same time, wecontinue to fire y with color ϕ1 to charge p1s with volume Πα2and then charge p2s with volume Πα1 . Thus, at τ3 = τ0 + 3ΠΨ,M3 is reached such that M3(p3c) = M3(p7c) = M3(p4(y),ϕ3) = 0, M3(p4c) = M3(p8c) = M3(p1s) = M3(p2s) = 1,V (M3(p1s , ϕ1)) = V (M3(p2s , ϕ1)) = V (M3(p4c , ϕ1)) =Πα1 , and V (M3(p8c , ϕ3)) = Πα2 . This marking is equivalentto M0 , or the previous process can be repeated in a cycle way.

Now we discuss the case with fds2 > fds1 . There arealso three cases: 1) fds2 < 2fds1 ; 2) fds2 = 2fds1 ; and 3)fds2 > 2fds1 . At M0 , we fire t12 with color ϕ1 and rate fds1to feed p1 , and t52 with color ϕ2 and rate fds2 to feed p2 ,respectively. At the same time, we fire y with color ϕ3 ∈ Φ2and rate fds1 + fds2 to charge p6s with volume Πα2 and thento charge p7s with volume Πα1 . Let τ1 = τ0 + ΠΨ, we haveτ1 − Πα2/(fds1 + fds2) − τ0 = ΠΨ − ΠΨ × (fds2/(fds1 +fds2)) ≥ Ψ((fds1 + fds2)/fds1 − fds2/fds1) = Ψ. This im-plies that t61’s firing has ended at τ1 . Thus, at time τ1 , M1is reached such that M1(p1c) = M1(p5c) = 0, M1(p2c) =M1(p6c) = M1(p7s) = M1(p4(y)) = 1, V (M1(p6c , ϕ3)) =Πα2 , and V (M1(p7s , ϕ3)) = Πα1 . At this marking, we firet22 with color ϕ1 and rate fds1 to feed p1 , and t62 with colorϕ3 and rate fds2 to feed p2 , respectively. At the same time,we fire y with color ϕ3 ∈ Φ2 to charge p7s with volumeΠ(α2 − α1), then with color ϕ3 to charge p8s with volumeΠα2 , and then with color ϕ1 to charge p4s with volumeΠ(2α1 − α2) for Case 1); with color ϕ3 ∈ Φ2 to chargep7s with volume Π(α2 − α1), then with color ϕ3 ∈ Φ2 tocharge p8s with volume Πα2 for Case 2); and with colorϕ3 ∈ Φ2 to charge p7s with volume Π(α2 − α1), then withcolor ϕ3 ∈ Φ2 to charge p8s with volume 2Πα1 for Case3). At τ2 = τ0 + 2ΠΨ,M2 is reached such that M2(p2c) =M2(p6c) = 0 and M2(p3c) = M2(p7c) = M2(p4(y), ϕ3) = 1for all the three cases; V (M2(p7c , ϕ3)) = V (M2(p8s , ϕ3)) =Πα2 , and V (M2(p4s , ϕ1)) = Π(2α1 − α2) for Case (1);V (M2(p7c , ϕ3)) = V (M2(p8s , ϕ3)) = Πα2 for Case (2);and V (M2(p7c , ϕ3)) = Πα2 and V (M2(p8s , ϕ3)) = 2Πα1 forCase (3). At this marking, we fire t32 with color ϕ1 and t72with color ϕ3 to feed p1 and p2 , respectively. At the sametime, there is space in p4s , p1s , and p2s , we fire y with colorϕ1 to charge p4s with volume Π(α2 – α1), then to chargep1s with volume Πα1 , and then to charge p2s with volumeΠα1 for Case (1); fire y with color ϕ1 to charge p4s , p1s ,and p2s , sequentially, all with volume Πα1 for Case (2);and fire y with color ϕ3 ∈ Φ2 to charge p8s with volumeΠ(α2 − 2α1), then with color ϕ1 to charge p4s , p1s , andp2s , sequentially, all with volume Πα1 for Case (3). At τ3 =τ0 + 3ΠΨ,M3 is reached such that M3(p3c) = M3(p7c) =M3(p4(y), ϕ3) = 0,M3(p4c) = M3(p8c) = M3(p1s) = M3(p2s) = 1, V (M3(p1c , ϕ1)) = V (M3(p2s , ϕ1)) = V (M3(p4c ,

ϕ1)) = Πα1 , and V (M3(p8c , ϕ3)) = Πα2 . This marking isequivalent to M0 , or the previous process can be repeated in acycle way.

In both cases, three tanks of high fusion point crude oil aretransported via the pipeline without being interrupted. Thus,such a parcel of oil can be transported by a single setup. �

From Theorems 4.4 and 4.5, it seems that, with one morecharging tank adding to each distiller, one more tank of highfusion point crude oil can be transported via the pipeline by asingle setup for a distiller to process. It should be noticed that,for Case (2), after charging CTK6 and CTK7 to Πα2 by firingy, CTK5 has been emptied. Thus, while p1 and p2 are being fedby CTK2 and CTK6 , CTK8 and CTK5 can be charged by firingy. Then, while p1 and p2 are being fed by CTK3 and CTK7 ,CTK4 and CTK1 can be charged by firing y. In this way, the PNis live. This implies, in some case, that four tanks of high fusionoil can be transported by a single setup.

V. SCHEDULABILITY FOR SYSTEMS WITH

MULTIPLE DISTILLERS

This section discusses schedulability for systems with morethan two distillers and generalizes the results obtained in thelast section. We assume that there are K ≥ 3 distillers for thesystem discussed and the following conditions are given.12) Production condition: there are K distillers DS1–K

with feeding rates fdsi > fds(i+1) , i = 1, . . . , K −1, αi = Ψ × fdsi , i = 1, . . . ,K, and fp max = fds1 +fds2 + · · · + fdsK .

13) Tank capacity: there are 2K + 1 charging tanksCTK1−(2K +1) with capacity C1 ≥ Kα1 , C2 ≥ Kα1 ,C3 ≥ Kα1 ; C4 ≥ Kα2 , C5 ≥ Kα2 ; . . . ,C2i ≥ Kαi,C2i+1 ≥ Kαi ; . . . ;C2K ≥ KαK , C2K +1 ≥KαK . CTK1–3 serve DS1 while CTK2i−CTK2i+1 serve DSi , 2 ≤ i ≤ K.

14) Initial condition: the volume of oil type 1 in CTK1and CTK2 is ζ1 = ζ2 = Kα1 , the volume of type 2 inCTK4 is ζ4 = Kα2 , . . ., the volume of type i in CTK2 i

is ζ2i = Kαi, . . ., the volume of type K in CTK2K isζ2K = KαK , the other tanks are empty, and the oil inCTK1 , and CTK2i , 2 ≤ i ≤ K is ready for feeding.

15) Crude oil type switching: after finishing the oil of typeK in CTK2K , distiller K should process another type ofcrude oil (type K + 1) that has high fusion point.

Theorem 5.1: If conditions (12)–(15) are satisfied the systemis schedulable and one tank of crude oil of type K + 1 can betransported via the pipeline by a single setup.

Proof: Let a charging tank CTKi be modeled by {pis , pic , ti1 ,ti2} with ti1 and ti2 being the timed and discharging transitions,respectively, and distiller DSi be modeled by pi in the PNmodel. Then, this system can be scheduled as follows. Initially,at M0 (time τ0), M0(p1c) = M0(p2s) = 1,M0(p(2i)c) =1, i = 2, 3, . . . ,K, V (M0(p1c , ϕ1)) = V (M0(p2s , ϕ1)) =Kα1 , V (M0(p(2i)c , ϕi)) = Kαi, i = 2, 3, . . . ,K, and theother places are empty. At this marking, we fire t12 withcolor ϕ1 to feed p1 , and t(2i)2 with color ϕi to feedpi, i = 2, 3, . . . ,K, respectively. Meanwhile, we fire y in the


following order: with color ϕK +1 ∈ Φ2 to charge p(2K +1)swith volume KαK → with color ϕK−1 to charge p(2K−1)s withvolume KαK−1 → · · · → with color ϕi to charge p(2i+1)s withvolume Kαi → · · · → with color ϕ2 to charge p5s with volumeKα2 → with color ϕ1 to charge p3s with volume Kα1 . Itshould be pointed out that after firing y for charging p(2K +1)s ,we have M(p4(y), ϕK +1) = 1. However, at that marking,there are K − 1 empty places: p(2K−1)s , . . . , p(2i+1)s , . . . , p5s ,and p3s . Thus, y can continue to fire and after sometime M(p4(y), ϕK +1) = 0 must hold. Notice also thatK α1

fp m a x= K×fd s 1 ×Ψ

fd s 1 + ···+fd s K≥ (fd s 1 + ···+fd s K )×Ψ

fd s 1 + ···+fd s K= Ψ.

This implies that charging p5s stops before timeτ0 + (K − 1)Ψ. Hence, the firing of t51 can start beforeτ0 + (K − 1)Ψ and must end before τ1 = τ0 + KΨ, or oilresidency time constraint is met. Thus, at τ1 , marking M1 isreached such that M1(p2c) = M1(p3s) = 1,M1(p(2i+1)c) =1, i = 2, 3, . . . ,K, V (M1(p2c , ϕ1))=V (M1(p3s , ϕ1))=Kα1 ,V (M1(p(2i+1)c , ϕi)) = Kαi, i = 2, 3, . . . ,K, and the otherplaces are empty. This marking is equivalent to M0 . Hence, theprocess can be repeated in a cyclic way and a feasible schedulecan be obtained. It can be seen that, in this process, a tank of oiltype K + 1 is transported by a single setup. �

Theorem 5.1’s conditions require that the capacity of a charg-ing tank and the volume of crude oil in a charging tank are greateror equal to Kαi . It seems that it is difficult to be satisfied. How-ever, in practice, it is not a serious restriction. In practice, ingeneral, there are no more than four distillers in most refineries.Similarly, feeding rate for a distiller is no more than 650 t/h. Gen-erally, Ψ is from 4–6 h. Thus, we have 4 × 650 × 6 = 15 500 tthat is less than the capacity of a general charging tank. In thetheorem, it states that a tank of a high fusion oil type can betransported for DSK . It follows from the proof of the theoremthat one tank of high fusion crude oil can be transported for anydistiller if the conditions in the theorem are satisfied. However,the cost may be very high if only one tank of high fusion crudeoil is transported with one setup, unless the capacity of tank isvery large. Now we consider more charging tank situations withthe following conditions.16) Production condition: there are K distillers DS1–K

with feeding rates fdsi �= fdsj , i �= j, αi = Ψ × fdsi , i =1, . . . ,K, and fp max = fds1 + fds2 + · · · + fdsK .

17) Tank capacity: there are 3K charging tanks CTK1−3K

with capacity C3i−2 ≥ Παi , C3i−1 ≥ Παi , andC3i ≥ Παi, i = 1, . . . ,K, and Π ≥ (fds1 + . . . +fdsK )/(fds1 + . . . + fds(K−1)), and CTK3(i−1)+1 ,CTK3(i−1)+2 , and CTK3(i−1)+3 are for DSi , 1 ≤ i ≤ K.

18) Initial condition: the volume of oil type 1 in CTK1 andCTK2 is ζ1 = ζ2 = Πα1 , . . . , the volume of oil type i inCTK3i+1 and CTK3i+2 is ζ3i+1 = ζ3i+2 = Παi+1 , 1 <i ≤ K − 2, the volume of oil type K in CTK3(K−1)+1 isζ3(K−1)+1 = ΠαK , the other tanks are empty, and the oilin CTK1 , CTK3i+1 , 1 < i ≤ K − 1, is ready for feeding;

19) Crude oil type switching: after finishing the oil of type Kin CTK3(K−1)+1 , distiller K should process another typeof crude oil (type K + 1) that has high fusion point.

Theorem 5.2: Assume that conditions (16) to (19) given aboveare satisfied, then the system is schedulable and two tanks of

crude oil of type K + 1 can be transported via the pipeline by asingle setup.

Proof: There are two cases: 1) fds1 > fdsK ; and 2)fds1 < fdsK . We first show that the theorem holds forCase (1). Initially, at M0 (time τ0), we have M0(p1c) =M0(p2s) = 1, M0(p(3i+1)c) = M0(p(3i+2)s) = 1, 1 ≤ i ≤K − 2,M0(p(3(K−1)+1)c) = 1, V (M0(p1c , ϕ1))=V (M0(p2s ,ϕ1)) = Πα1 , V (M0(p(3i+1)c , ϕi+1))=V (M0(p(3i+2)s , ϕi+1))= Παi+1 , 1 ≤ i ≤ K − 2, V (M0(p(3(K−1)+1)c , ϕK ))=ΠαK ,and other places are empty. At this marking, we fire t(3i−2)2with color ϕi to feed pi, 1 ≤ i ≤ K. At the same time, we firey with the following order: with color ϕK +1 ∈ Φ2 and ratefp max to charge p(3K−1)s with volume ΠαK → with colorϕK +1 ∈ Φ2 and rate fp max to charge p(3K )s with volumeΠαK → · · · → with color ϕi to charge p(3i)s with volumeΠαi,K − 1 ≥ i ≥ 2,→ · · · →with color ϕ1 to charge p3s withvolume Π(α1 − αK ). It should be pointed out that, after charg-ing p(3K )s with volume ΠαK , we have M(p4(y), ϕK +1) = 1.However, with p(3i)s , 1 ≤ i ≤ K − 1, being empty, y canfire uninterruptedly. Hence, the PN is in a live marking andthe previous process is feasible. Then, at τ1 = τ0 + ΠΨ,M1is reached such that M1(p4(y), ϕK +1) = 0,M1(p2c) = 1,M1(p(3i+2)c)= M1(p(3i+3)s) = 1, 1 ≤ i ≤ K − 1, V (M1(p2c ,ϕ1))= Πα1 , V (M1(p(3i+2)c , ϕi+1))= V (M1(p(3i+3)s , ϕi+1))= Παi+1 , 1 ≤ i ≤ K − 2, V (M1(p(3K−1)c , ϕK +1)) =V (M1(p(3K )s , ϕK +1)) = ΠαK , V (M1(p3s , ϕ1)) = Π(α1 −αK ), and other places are empty. At this marking, we fire t22with color ϕ1 to feed p1 , t(3i−1)2 with color ϕi to feed pi ,2 ≤ i ≤ K − 1, and t(3K−1)2 with color ϕK +1 to feed pK . Atthe same time, we fire y with the following order: with color ϕ1and rate fp max to charge p3s with volume ΠαK → with colorϕ1 and rate fp max to charge p1s with volume Πα1 → · · · →with color ϕi+1 to charge p(3i+1)s with volume Παi+1 ,1 ≤ i ≤ K − 2. Let τ2 = τ0 + 2ΠΨ, we have (τ2 − ΠαK )/(fds1 + · · · + fdsK ) − τ1 = ΠΨ − ΠΨfdsK /(fds1 + · · · +fdsK ) = ΠΨ(1 − fdsK /(fds1 + · · · + fdsK )) ≥ Ψ. This im-plies that t31’s firing has ended at τ2 . Thus, at τ2 , M2 is reachedsuch that M2(p3c)= M2(p1s) = 1,M2(p(3i)c)= M2(p(3i−2)s)= 1, 2 ≤ i ≤ K − 1, M2(p(3K )c) = 1, V (M2(p3c , ϕ1)) =V (M2(p1s , ϕ1))= Πα1 , V (M2(p(3i)c , ϕi))= V (M2(p(3i−2)s ,ϕi)) = Παi, 2 ≤ i ≤ K − 1, V (M2(p(3K )c , ϕK +1)) = ΠαK ,and other places are empty. This marking is equivalent to M0 ,and thus the process can be repeated in a cyclic way.

Now we show that the theorem still holds for Case (2).Without loss of generality, we assume that α1 + α2 − αK > 0.At M0 , we fire t12 with color ϕ1 to feed p1 , t(3i−2)2 withcolor ϕi to feed pi, 2 ≤ i ≤ K. At the same time, we firey with the following order: with color ϕK +1 ∈ Φ2 andrate fp max to charge p(3K−1)s with volume ΠαK → withcolor ϕK +1 ∈ Φ2 and rate fp max to charge p(3K )s withvolume ΠαK → with color ϕK−1 and rate fp max to chargep(3K−3)s with volume ΠαK−1 · · · → with color ϕi to chargep(3i)s with volume Παi, 3 ≤ i ≤ K − 2,→ · · · → withcolor ϕ2 to charge p6s with volume Π(α1 + α2 − αK ).It should be pointed out that, after charging p(3K )s withvolume ΠαK , we have M(p4(y), ϕK +1) = 1. How-ever, with p(3i)s , 1 ≤ i ≤ K − 1, being empty, y can fire


uninterruptedly. Hence, the previous process is feasible.Let τ1 = τ0 + ΠΨ, we have τ1 − ΠαK /(fds1 + · · · +fdsK ) − τ0 = ΠΨ − ΠΨfdsK /(fds1 + · · · + fdsK ) = ΠΨ(1 − fdsK /(fds1 + · · · + fdsK )) ≥ Ψ. This implies thatt(3K−1)1’s firing has ended at τ1 . Thus, at τ1 , M1 isreached such that M1(p4(y), ϕK +1) = 0,M1(p2c) = 1,M1(p(3i+2)c) = M1(p(3i+3)s) = 1, 1 ≤ i ≤ K − 1, V (M1(p2c , ϕ1)) = Πα1 , V (M1(p(3i+2)c , ϕi+1)) = V (M1(p(3i+3)s ,ϕi+1)) = Παi+1 , 2 ≤ i ≤ K − 2, V (M1(p5c , ϕ2)) = Πα2 ,V (M1(p6s , ϕ2)) = Π(α1 + α2 − αK ), V (M1(p(3K−1)c ,ϕK +1)) = V (M1(p(3K )s , ϕK +1)) = ΠαK , and other placesare empty. At this marking, we fire t22 with color ϕ1 to feed p1 ,t(3i−1)2 with color ϕi to feed pi, 2 ≤ i ≤ K − 1, and t(3K−1)2with color ϕK +1 to feed pK . At the same time, we fire y with thefollowing order: with color ϕ2 and rate fp max to charge p6s withvolume Π(αK − α1) → with color ϕ1 and rate fp max to chargep3s with volume Πα1 → with color ϕ1 and rate fp max to chargep1s with volume Πα1 → · · · → with color ϕi+1 to chargep(3i+1)s with volume Παi+1 , 1 ≤ i ≤ K − 2. Notice that,between τ1 and the time point when it stops charging p3s , thetotal volume of oil charged by firing y is ΠαK . Hence, let τ2 =τ0 + 2ΠΨ, we have (τ2 − ΠαK )/(fds1 + · · · + fdsK ) − τ1 =ΠΨ − ΠΨfdsK /(fds1 + · · · + fdsK ) = ΠΨ(1 − fdsK /(fds1 + · · · + fdsK )) ≥ Ψ. This implies that t31’s firing hasended at τ2 . Thus, at τ2 , M2 is reached such that M2(p3c) =M2(p1s) = 1,M2(p(3i)c) = M2(p(3i−2)s) = 1, 2 ≤ i ≤ K −1,M2(p(3K )c) = 1, V (M2(p3c , ϕ1)) = V (M2(p1s , ϕ1)) =Πα1 , V (M2(p(3i)c , ϕi)) = V (M2(p(3i−2)s , ϕi)) = Παi, 2 ≤i ≤ K − 1, V (M2(p(3K )c , ϕK +1)) = ΠαK , and other placesare empty. This marking is equivalent to M0 , and thus theprocess can be repeated in a cyclic way. �

In both cases, a feasible schedule can be obtained and twotanks of crude oil of high fusion point type by firing y contin-uously. This implies that this amount of oil can be transportedby just a single setup.

In the proof of the theorem, we assume that α1 + α2 − αK >0. It seems a constraint, in fact it is not. If α1 + α2 − αK ≤ 0,we need only to leave p6s empty and charge p9s with volumeΠ(α1 + α2 + α3 − αK ) during the time interval [τ0 , τ1], thenthe result is unchanged. It follows from the proof of the theoremthat only two tanks of high fusion point oil can be transportedby a single setup. For Case (1), when CTK3K−1 and CTK3K arecharged to ΠαK , CTK3K−2 has not been emptied yet. To chargeCTK3K−2 after charging CTK3K , y should stop firing for sometime. This violates the liveness conditions given in Definition3.4. For Case (2), when CTK3K−1 and CTK3K are chargedto ΠαK , CTK3K−2 has been emptied. However, by chargingCTK3K−2 immediately after charging CTK3K , oil residencytime constraint will be violated. Theorem 5.2 shows that if eachdistiller has three charging tanks, two tanks of high fusion pointoil can be transported for a distiller by a single setup. Thefollowing theorem shows that with one more charging tankadding to each distiller, one more tank of high fusion pointcrude oil can be transported via the pipeline by a single setup.Let H >3 be an integer and the following conditions are given.20) Tank capacity: there are HK charging tanks numbered

CTK(i−1)H +1 , CTK(i−1)H +2 , . . . ., CTKiH , 1 ≤ i ≤ K,

with CTK(i−1)H +1 , CTK(i−1)H +2 , . . . ., CTKiH for DSi ,the capacities of CTK(i−1)H +1 , CTK(i−1)H +2 , . . . .,CTKiH , 1 ≤ i ≤ K, are all greater than or equal to Παi ,and Π ≥ (fds1 + · · · + fdsK )/(fds1 + · · · + fds(K−1)).

21) Initial condition: the volume of oil type i inCTK(i−1)H +1 , CTK(i−1)H +2 , . . . . , CTK(i−1)H +(H−1) ,1 ≤ i ≤ K − 1, is Παi , the volume of oil type K inCTK(K−1)H +1 is Παk , the other tanks are empty, andthe oil in CTK(i−1)H +1 , 1 < i ≤ K, is ready for feeding.

22) Crude oil type switching: after finishing the oil of type Kin CTK(K−1)H +1 , distiller K should process another typeof crude oil (type K + 1) that has high fusion point.

Theorem 5.3: Assume that conditions (16) and (20)–(22)given above are satisfied, then the system is schedulable andH − 1 tanks of crude oil of type K + 1 can be transported viathe pipeline by a single setup.

Proof: It depends on (H − 1)αK /(α1 + · · · + αK ).We have 0 < (H − 1)αK /(α1 + · · · + αK ) < H − 1.We first show that the theorem holds when 0 <(H − 1)αK /(α1 + · · · + αK ) < 1. Initially, at M0 (timeτ0), we have M0(p(i1)c) = 1 and M0(p(ij )s) = 1, 1 ≤i ≤ K − 1 and 2 ≤ j ≤ H − 1,M0(p(K 1)c) = 1; andV (M0(p(i1)c , ϕi)) = V (M0(p(ij )s , ϕi)) = Παi, 1 ≤ i ≤K − 1 and 2 ≤ j ≤ H − 1, V (M0(p(K 1)c , ϕK )) = ΠαK ,and other places are empty. At this marking, the to-tal capacity in p(K j )s , 2 ≤ j ≤ H , and p(iH )s , 1 ≤ i ≤K − 1, is Π(α1 + · · · + αK−1 + (H − 1)αK ), or there isenough capacity to fire y. Hence, at this marking, we firet(i1)2 with color ϕi to feed pi, 1 ≤ i ≤ K, respectively. Atthe same time, we fire y with the following order: with colorϕK +1 ∈ Φ2 and rate fp max to charge p(K 2)s with volumeΠαK → · · · → with color ϕK +1 ∈ Φ2 and rate fp max tocharge p(K H )s with volume ΠαK → with color ϕi and ratefp max to charge p(iH )s with volume Παi,K − 1 ≥ i > Aand 0 ≤ α1 + · · · + αA − (H − 1)αK ≤ αA → with colorϕA to charge p(AH )s with volume Π((H − 1)αK − (α1 +· · · + αA−1)). It should be pointed out that, after chargingp(K H )s with volume ΠαK , we have M(p4(y), ϕK +1) = 1.However, with p(iH )s , 1 ≤ i ≤ K − 1, being empty, y canfire uninterruptedly. Hence, the previous process is feasi-ble. Let τ1 = τ0 + ΠΨ, we have τ1 − ΠαK /(fds1 + · · · +fdsK ) − τ0 = ΠΨ − (ΠΨ × fdsK )/(fds1 + · · · + fdsK ) =ΠΨ(1 − fdsK /(fds1 + · · · + fdsK )) ≥ Ψ. This impliesthat t(K 2)1’s firing has ended at τ1 . Thus, at τ1 , M1 isreached such that M1(p4(y), ϕK +1) = 0,M1(p(i2)c) = 1,1 ≤ i ≤ K,V (M1(p(i2)c , ϕi)) = Παi, 1 ≤ i ≤ K − 1, V (M1(p(iH )s , ϕi))= Παi,A+ 1≤ i ≤ K − 1, V (M1(p(AH )s , ϕA ))= Π((H − 1)αK − (α1 + · · ·+αA−1)), V (M1(p(K 2)c , ϕK +1))= V (M1(p(K j )s , ϕK +1)) = ΠαK , 3 ≤ j ≤ H , and otherplaces are empty. At this marking, we fire t(i2)2 with color ϕi

to feed pi, 1 ≤ i ≤ K − 1, and t(K 2)2 with color ϕK +1 ∈ Φ2to feed pK , respectively. At the same time, we fire y withthe following order: with color ϕA and rate fp max to chargep(AH )s with volume Π(α1 + · · · + αA − (H − 1)αK ) →with color ϕi and rate fp max to charge p(iH )s with vol-ume Παi,A − 1 ≥ i ≥ 1 → with color ϕi and rate fp maxto charge p(i1)s with volume Παi,K − 1 ≥ i > B and


0 ≤ α1 + · · · + αB − (H − 2)αK ≤ αB → with color ϕB

to charge p(B 1)s with volume Π((H − 2)αK − (α1 +· · · + αB−1)). Then, at τ2 = τ0 + 2ΠΨ, M2 is reachedsuch that M2(p4(y), ϕK +1) = 0,M2(p(i3)c) = 1, 1 ≤ i ≤K,V (M2(p(i3)c , ϕi)) = V (M2(p(iH )s , ϕi)) = Παi, 1 ≤ i ≤K − 1, V (M1(p(K 3)c , ϕK +1)) = ΠαK , the remaining ca-pacity in p(i1)s , 1 ≤ i ≤ B, is Π(H − 2)αK . By doing so,at τF = τ0 + FΠΨ, 2 < F < H − 1,MF is reached suchthat MF (p4(y), ϕK +1) = 0,MF (p(i(F +1))c) = 1, 1 ≤ i≤ K,V (MF (p(i(F +1))c , ϕi)) = Παi, 1 ≤ i ≤ K − 1, V (MF

(p(K (F +1))c , ϕK +1)) = ΠαK , and ∃G such that0 ≤ α1 + · · · + αG − (H − F )αK ≤ αG and the remain-ing capacity in p(i(F −1))s , 1 ≤ i ≤ G, is Π(H − F )αK .Finally, at τH−1 = τ0 + (H − 1)ΠΨ,MH−1 is reachedsuch that MH−1(p(iH )c) = 1 and MH−1(p(ij )s) = 1, 1 ≤i ≤ K − 1 and 2 ≤ j ≤ H − 2,MH−1(p(K H )c) = 1; andV (MH−1(p(iH )c , ϕi)) = V (MH−1(p(ij )s , ϕi)) = Παi, 1 ≤i ≤ K − 1 and 2 ≤ j ≤ H − 2, V (MH−1(p(K H )c , ϕK +1)) =ΠαK , and other places are empty. This marking is equivalent toM0 and the previous process can be repeated in a cycle way. Inthis way, a feasible schedule is created and, by such schedule,H − 1 tanks of high fusion point crude oil are transported forfeeding DSK . Because y can fire uninterruptedly wheneverM(p4(y), ϕK +1) = 1, or the transportation of the H − 1 tanksof high fusion point crude oil can be completed by just a singlesetup.

When (H − 1)αK /(α1 + · · · + αK ) ≥ 1 it can be shown inthe similar way and it is omitted. �

Notice that when there are two distillers and four chargingtanks, for each distiller four tanks of high fusion crude oil canbe transported by just a single setup. In that case, we haveH − 1 > K. From the proof of Theorems 5.2 and 5.3, it is easyto show that when H − 1 ≤ K only H − 1 tanks of high fusioncrude oil can be transported by just a single setup. In the abovetheorems, it is required that Π ≥ (fds1 + · · · + fdsK )/(fds1 +· · · + fds(K−1)). This requirement is to guarantee the satisfac-tion of oil residency time constraint. In practice, if Π is largeenough, a charging tank can be charged with volume more thanΠαi , possibly as full as possible.

In the proof of Theorem 5.3, a streamlined procedure withall the parameters analytically given is presented to obtain afeasible detailed schedule such that H − 1 tanks of high fusioncrude oil can be transported from storage tanks to chargingtanks by a single setup. Thus, to do so, almost no computationis necessary. In practice, often Π is large enough and a chargingtank CTKj can be charged with volume more than Παi . Thus,we need to determine the volume of oil to be charged into CTKj

such that oil residency time constraint can be satisfied. To doso, it needs to calculate the time when the oil for feeding DSi isused up. With the state of the system and the feeding rate to DSi

known, this calculation is very simple. In other words, by usingthe proposed method, no complex computation is required.

VI. INDUSTRIAL CASE STUDY

This section presents a case study to show the application andpower of the results obtained in this paper. This case problem

TABLE IINITIAL STATE FOR THE CHARGING TANKS

arises from a practical application scenario. The refinery hasthree distillers DS1 , DS2 , and DS3 with nine charging tanksavailable. Three times each month, a short-term schedule shouldbe created for the next 10 days. The initial state for nine chargingtanks is shown in Table I. The maximal flow rate of the pipelineis fp max = 1250 t/h. There is 62 000 t of crude oil of type #2, atype of high fusion point oil, in the storage tanks. Such crude oilis to be processed by distiller DS3 . A refining schedule is givenas shown in Fig. 11 for the next 10 days with refining rates fordistillers DS1 , DS2 , and DS3 being 333.3, 291.7, and 625.0 t/h,respectively, where #i denotes crude oil type # i, i = 1, 2, . . . , 7.The oil residency time for charging tanks Ψ is 6 h and thecapacity of pipeline is Cpipe = 12 000 t. The question is if therefining schedule is realizable and 62 000 t crude oil of type #2can be transported to charging tanks by a single setup.

According to the schedulability analysis presented above, ex-amine the refining schedule and the initial state of the chargingtanks, we find that: 1) DS1 processes 38 000 t of oil type #1 heldin charging tanks CT122 and CT129 first and then switchesto process crude oil type #5. This implies that charging tanksCT122, CT129, and CT180 can be assigned to DS1 ; 2) DS2first processes the crude oil held in tanks CT181 and CT124,then switches to process crude oil type #7, or charging tanksCT124 and CT181 are usable for DS2 ; 3) DS3 process theoil held in tank CT115 first and then switches to process oiltype #2 and then #4, or charging tanks CT115 is usable forDS3 ; 4) charging tanks CT116, CT127, and CT125, they arefree to be used; and 5) the total production rate is equal tothe maximal transportation rate of the pipeline. Because fds3> fds2 , we assign CT116 and CT127 to DS3 , and CT125to DS2 . Thus, each of the three distillers has three chargingtanks with two tanks being empty for DS3 , and one for DS2 .With CCT125 = 2000 t > Cpipe = 12 000 t. This implies thattwo tanks of #2 oil can be transported for DS3 by a single setup.Furthermore, we have CCT116 + CCT127 = 68 000 t > 62000 t.Thus, from the results obtained in this paper, it concludes thatthe refining schedule is realizable and 62 000 t of #2 oil can betransported for DS3 by a single setup.

With the above observation and following the scheduling ideapresented in this paper, we find the detailed schedule for thiscase study as shown in Figs. 12 and 13, where #1 stands for


Fig. 11. Refining schedule for the case study.

Fig. 12. Detailed charging tank schedule for feeding distillers.

Fig. 13. Detailed schedule for filling the charging tanks.

crude oil type #1 and 122 for charging tanks CT122. This isconsistent to the analysis given in the last section.

Notice that, in Fig. 13, we only present the charging tankfilling schedule up to time point 200H. To this time point, theamount of crude oil of all types filled into the charging tanksis enough for completing the refining schedule. Then, differenttypes of oil may need to fill to the charging tanks for the nextscheduling horizon. It is extremely difficult, if not impossible,for any existing mathematical programming methods to obtaina short-term schedule for this presented industrial case.

VII. CONCLUSION

A short-term schedule for oil refinery should provide all theactivities in every detail for the whole scheduling horizon. Itis the detail that makes the short-term scheduling problem ex-tremely difficult [7]. There are various constraints includingresource and process ones. Some of them are very difficult tomodel by mathematical programming models. Thus, it is hardto find a feasible schedule that is essential to the short-termscheduling problem for crude oil operations. Moreover, a good

schedule should make operational cost as low as possible, suchas minimizing oil inventory cost. Among various costs, highsetup cost for high fusion point crude oil transportation viapipeline and the cost resulting from mixing different types ofoil in a tank play an important role. Because of the complexity,there is no model in the literature that can be used to reducesuch kinds of costs. In viewing an operation decision, whichforms a short-term schedule, as a control, we propose to studythe short-term scheduling problem for crude oil operations in acontrol theory perspective. In this way, the short-term schedul-ing problem can be solved in a hierarchical way. At the upperlevel, it determines a refining schedule by minimizing oil in-ventory cost and maximizing productivity, while, at the lowerlevel, a detailed schedule is found to realize it and reduce oiltransportation cost and tank charging and discharging cost.

To solve the lower level problem, this paper studies the de-tailed scheduling problem by taking the setup cost of high fusionpoint oil transportation and tank charging and discharging costinto consideration. The system is modeled by a hybrid Petrinet model. The liveness of the model, which is easy to check,


describes the feasibility of a short-term schedule for the sys-tem. To reduce the tank charging and discharging cost, a controlstrategy is used. By this control strategy, the charging tanks aregrouped such that a group of them serve a distiller. With thePN model and control strategy, this paper conducts schedulabil-ity analysis and schedulability conditions are presented. Theseconditions reveal the relationship among the number of charg-ing tanks, number of distillers, capacity of charging tanks, flowrate of the pipeline, productivity, and volume of high fusionpoint crude oil able to be transported by a single setup. They arenecessary for determining a refining schedule at the upper leveland can be used to check its realizability. They are presentedin a structural way, and can be easily used to create its corre-sponding detailed schedule with setup cost of high fusion pointoil transportation and tank charging and discharging cost beingminimized.

It should be noticed that the volume of high fusion pointcrude oil that is able to be transported by a single setup isobtained under the special control strategy. It is ideal that arequired volume of such oil can be transported by a single setupunder this strategy. However, if not, a control policy and itscorresponding scheduling analysis are necessary because of thehigh setup cost for high fusion crude oil transportation. This isour future work.

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NaiQi Wu (M’04–SM’05) received the M.S. andPh.D. degrees in systems engineering both fromXi’an Jiaotong University, Xi’an, China, in 1985 and1988, respectively.

From 1988 to 1995, he was with the ChineseAcademy of Sciences, Shenyang Institute of Automa-tion, Shenyang, and from 1995 to 1998, with ShantouUniversity, Shantou. From 1991 to 1992, he was aVisiting Scholar in the School of Industrial Engineer-ing, Purdue University, West Lafayette, IN. In 1999,2004, and from 2007 to 2009, he was a Visiting Pro-

fessor in the Department of Industrial Engineering, Arizona State University,Tempe, the Department of Electrical and Computer Engineering, New JerseyInstitute of Technology, Newark, and Industrial Systems Engineering Depart-ment, Industrial Systems Optimization Laboratory, University of Technologyof Troyes, Troyes, France, respectively. He is currently a Professor of indus-trial and systems engineering in the Department of Industrial Engineering,School of Mechatronics Engineering, Guangdong University of Technology,Guangzhou. His research interests include production planning and schedul-ing, manufacturing system modeling and control, discrete event systems, Petrinet theory and applications, and information assurance. He is the author orcoauthor of many papers published in International Journal of Production Re-search, IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, IEEETRANSACTIONS ON ROBOTICS AND AUTOMATION, IEEE TRANSACTIONS ON

AUTOMATION SCIENCE AND ENGINEERING, IEEE/ASME TRANSACTIONS ON

MECHATRONICS, IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING,Journal of Intelligent Manufacturing, Production Planning and Control, andRobotics and Computer Integrated Manufacturing.


Dr. Wu is an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS,MAN, AND CYBERNETICS, PART C and IEEE TRANSACTIONS ON AUTOMATION

SCIENCE AND ENGINEERING and Editor-in-Chief of Industrial Engineering Jour-nal. He was a Program Committee Member of the 2003–2008 IEEE InternationalConference on Systems, Man, & Cybernetics, a Program Committee Memberof the 2005–2009 IEEE International Conference on Networking, Sensing andControl, a Program Committee Member of the 2006, 2008, and 2009 IEEEInternational Conference on Automation Science and Engineering, a ProgramCommittee Member of the 2006 IEEE International Conference on servicesystems and service management, a Program Committee Member of the 2007International Conference on Engineering and Systems Management, and re-viewer for many international journals.

Feng Chu received the B.S. degree from Hefei Uni-versity of Technology, Hefei, China, in 1986, the M.S.degree from the Institut National Polytechnique deLorraine, Nancy, France, in 1991, both in electricalengineering, and the Ph.D. degree in computer sci-ence from the University of Metz, Metz, in 1995.

She was with Jiangsu University of Technology,Zhenjiang, for 2 years and at National Research In-stitute in Computer Science and Automation, Nancy,for 4 years. From 1999 to 2009, she was an Asso-ciate Professor with the University of Technology of

Troyes, Troyes. She joined the Laboratoire d’IBISC FRE CNRS 3190, Univer-site d’Evry Val d’Essonne, Evry Cedex, in 2009, where she is currently a FullProfessor. Her research interests include modeling, analysis and optimization ofcomplex systems including intelligent transportation systems, logistic, and pro-duction systems based on Petri nets and operations research. She is the authoror coauthor of more than 30 papers in international journals such as EuropeanJournal of Operational Research, Computers & Operations Research, AppliedMathematics and Computation, Computers & Industrial Engineering, Inter-national Journal of Computer Integrated Manufacturing, International Jour-nal of Production Research, Transportation Research, IEEE TRANSACTIONS

ON SYSTEMS, MAN AND CYBERNETICS, IEEE TRANSACTIONS ON ROBOTICS

AND AUTOMATION, and IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND

ENGINEERING, and over 40 papers in academic conferences.

Chengbin Chu received the B.Sc. degree in electri-cal engineering from Hefei University of Technology,Hefei, China, in 1985, and the Ph.D. degree in com-puter science from Metz University, Metz, France, in1990.

From 1987 to 1996, he was with the National Re-search Institute in Computer Science and Automa-tion, Nancy. He was a Professor with the Univer-sity of Technology of Troyes, Troyes from 1996 to2008, where he was also Founding Director of theIndustrial Systems Optimization Laboratory. He is

currently a Senior Professor in the Laboratoire Genie Industriel, Ecole Cen-trale Paris, Grande voie des Vignes, Chatenay-Malabry Cedex, France. He isalso an Overseas Visiting Professor and Overseas Director of the Departmentof Industrial Engineering, Xi’an Jiaotong University, Xi’an. His research in-terests include operations research and modeling, analysis, and optimizationof supply chain and production systems. He is the author or coauthor of threebooks and more than 100 papers in international journals such as OperationsResearch, SIAM Journal of Computing, IEEE TRANSACTIONS ON ROBOTICS

AND AUTOMATION, and is currently Associate Editor of IEEE TRANSACTIONS

ON AUTOMATION SCIENCE AND ENGINEERING, International Journal of Produc-tion Research, and Naval Research Logistics. He also published many papers inconference proceedings.

Dr. Chu was nominated “Chang Jiang Scholars Programme” Chair Professorby the Chinese Ministry of Education, in 2005. For his research and appli-cation activities, he received the First Prize of Robert Faure Award in 1996.He also received the “1998 Best Transactions Paper Award” from the IEEERobotics and Automation Society. He serve as an Associate Editor of the IEEETRANSACTIONS ROBOTICS AND AUTOMATION from 2001 to 2004. Currently heis an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE

AND ENGINEERING and the IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS.

MengChu Zhou (S’88–M’90–SM’93–F’03) re-ceived the B.S. degree in electrical engineering fromNanjing University of Science and Technology, Nan-jing, China, in 1983, the M.S. degree in automaticcontrol from Beijing Institute of Technology, Beijing,in 1986, and Ph.D. degree in computer and systemsengineering from Rensselaer Polytechnic Institute,Troy, NY, in 1990.

He joined the Department of Electrical andComputer Engineering, New Jersey Institute of Tech-nology (NJIT), Newark, in 1990, and is currently a

Professor of electrical and computer engineering and the Director of Discrete-Event Systems Laboratory. His research interests include Petri nets, computer-integrated systems, wireless ad hoc and sensor networks, system security, semi-conductor manufacturing, life-cycle engineering, sustainable systems, and em-bedded control. He has more than 300 publications including 9 books, more than140 journal papers, and 17 book chapters. He coauthored with F. DiCesare, PetriNet Synthesis for Discrete Event Control of Manufacturing Systems (Boston,MA: Kluwer Academic, 1993), edited Petri Nets in Flexible and Agile Automa-tion (Boston, MA: Kluwer Academic, 1995), coauthored with K. Venkatesh,Modeling, Simulation, and Control of Flexible Manufacturing Systems: A PetriNet Approach (Singapore, Singapore: World Scientific, 1998), coedited withM. P. Fanti, Deadlock Resolution in Computer-Integrated Systems (New York,NY: Marcel Dekker, 2005), coauthored with H. Zhu, Object-Oriented Program-ming in C++ : A Project-based Approach (Beijing, China: Tsinghua UniversityPress, 2005), coauthored with B. Hruz, Modeling and Control of Discrete EventDynamic Systems (New York, NY: Springer, 2007), and co-authored with Z.Li, Deadlock Resolution in Automated Manufacturing Systems: A Novel PetriNet Approach (New York, NY: Springer, 2009). From Scopus database, in thearea of Petri nets, he ranks the top one in terms of the number of publications;and top two in terms of the number of citations. He ranks the top one in the areaof automated manufacturing systems in both indices.

Dr. Zhou is a Life Member of Chinese Association for Science andTechnology-USA and served as its President during 1999. He was invited tolecture in Australia, Canada, China, France, Germany, Hong Kong, Italy, Japan,Korea, Mexico, Taiwan, and United States and served as a Plenary Speakerfor several conferences. He served as Associate Editor of IEEE TRANSACTIONS

ON ROBOTICS AND AUTOMATION from 1997 to 2000, IEEE TRANSACTIONS ON

AUTOMATION SCIENCE AND ENGINEERING from 2004 to 2007, and ManagingEditor of IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS: PART

C from 2005–2008, and currently Editor of IEEE TRANSACTIONS ON AUTOMA-TION SCIENCE AND ENGINEERING, Associate Editor of IEEE TRANSACTIONS

ON SYSTEMS, MAN AND CYBERNETICS: Part A and IEEE TRANSACTIONS ON

INDUSTRIAL INFORMATICS, and Editor-in-Chief of International Journal ofIntelligent Control and Systems. He served as Guest-Editor for many jour-nals including IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and IEEETRANSACTIONS ON SEMICONDUCTOR MANUFACTURING. He was General Co-Chair of 2003 IEEE International Conference on System, Man and Cybernetics,Washington, DC, October 5–8, 2003, Founding General Co-Chair of 2004 IEEEInternational Conference on Networking, Sensing and Control, Taipei, March21–23, 2004, General Chair of 2006 IEEE International Conference on Net-working, Sensing and Control, Ft. Lauderdale, FL, April 23–25, 2006, and Gen-eral Chair of 2008 IEEE Conference on Automation Science and Engineering,Washington D.C., Aug. 23–26, 2008. He was Program Chair of 1998 and 2001IEEE International Conference on SMC and 1997 IEEE International Confer-ence on Emerging Technologies and Factory Automation. He also served otherchair positions including Local Arrangement Chair of 2007 American ControlConference. He organized and chaired over 80 technical sessions and servedon program committees for many conferences. He has led or participated in36 research and education projects with total budget over $10 million, fundedby National Science Foundation, Department of Defense, Engineering Founda-tion, New Jersey Science and Technology Commission, and industry. He wasthe recipient of National Science Foundation’s Research Initiation Award, CIMUniversity-LEAD Award by Society of Manufacturing Engineers, Perlis Re-search Award by NJIT, Humboldt Research Award for US Senior Scientists,Leadership Award and Academic Achievement Award by Chinese Associationfor Science and Technology-USA, Asian American Achievement Award byAsian American Heritage Council of New Jersey, and Distinguished Lecture-ship of IEEE Systems, Man, and Cybernetics (SMC) Society. He was foundingchair of Discrete Event Systems Technical Committee of IEEE SMC Society,and Co-Chair (founding) of Semiconductor Manufacturing Automation Tech-nical Committee of IEEE Robotics and Automation Society.

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