132 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND

132 IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, VOL. 2, NO. 2, APRIL 2005

Modeling and Performance Evaluation ofSupply Chains Using Batch Deterministic

and Stochastic Petri NetsHaoxun Chen, Lionel Amodeo, Feng Chu, and Karim Labadi

Abstract—Batch deterministic and stochastic Petri nets are in-troduced as a tool for modeling and performance evaluation ofsupply chains. The new model is developed by enhancing deter-ministic and stochastic Petri nets (DSPNs) with batch places andbatch tokens. By incorporating stochastic Petri nets (SPNs) withthe batch features, inhibitor arcs, and marking-dependent weights,operational policies of supply chains such as inventory policies canbe easily described in the model. Methods for structural and per-formance analysis of the model are developed by extending existingones for DSPNs. As applications, an inventory system and an indus-trial supply chain are modeled and their performances are evalu-ated analytically and by simulation, respectively, using this BSPNmodel. The applications demonstrate that our model and associ-ated methods can solve some important supply chain modeling andanalysis issues.

Note to Practitioners—This paper was motivated by the problemof performance analysis and optimization of supply chains but italso applies to other discrete event systems where materials areprocessed in finite discrete quantities (batches) and operations areperformed in a batch way because of batch inputs and/or in orderto take advantages of the economies of scale. Existing Petri netmodeling and analysis tools for such systems ignore their batch fea-tures, making their modeling complicated. This paper suggests anew model called batch deterministic and stochastic Petri nets (BD-SPNs) by enhancing deterministic and stochastic Petri nets withbatch places and batch tokens. Methods for structural and perfor-mance analysis of the model are developed. We then show how aninventory system and a real-life supply chain can be modeled andtheir performances can be evaluated analytically and by simulationrespectively based on the model. The model and associated analysismethods therefore provide a promising tool for modeling and per-formance evaluation of supply chains.

Index Terms—Modeling, performance evaluation, Petri nets,supply chain management.

I. INTRODUCTION

DRIVEN by fierce global competition, many manufacturershave recognized that operational coordination across their

supply chains is critical for them to further reduce costs whileimproving services to customers [14]. The coordination, how-ever, is quite difficult because of the inherent complexity anduncertainty of the supply chains. The goal of this study is to

Manuscript received January 29, 2004; revised July 28, 2004. This paperwas recommended for publication by Associate Editor M. Zhou and EditorN. Viswanadham upon evaluation of the reviewers’ comments.

The authors are with the Technology University of Troyes, 10010Troyes Cedex, France (e-mail: [email protected]; [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TASE.2005.844537

develop a tool which can help industrial practitioners to model,evaluate performances, and optimize operational policies oftheir supply chains.

In the literature, supply chains are usually described as multi-echelon inventory systems [10]–[14]. However, most existingmodels can only describe a restricted class of supply chainswith simplifications. For instance, most multi-echelon inventorymodels do not explicitly take account of transportation opera-tions and capacity constraints in supply chains by simply as-suming a constant lead time between any two adjacent stockinglocations [10]–[14]. These models lack flexibility and generalityin describing real-life supply chains. Furthermore, no mathe-matical model exists that can describe material, information,and financial flows of a supply chain in an integrated way.

Petri nets are a powerful tool for modeling and analysis ofdiscrete event systems [1]–[17] such as manufacturing systems.Since supply chains are also discrete event systems from a highlevel of abstraction, it is possible to develop a Petri net tool formodeling and analysis of supply chains.

Although the literature of Petri nets is very plentiful, verylittle work applied Petri nets to the modeling of supply chains.Supply chains are modeled by using colored Petri nets, whereeach supply chain entity is modeled as a block with action, re-source, and control, which is a subnet of the colored Petri netmodel [15]. Supply chains are also modeled by using general-ized stochastic Petri nets (GSPNs) [16]. For inventory systemswith independent demand, a basic supply chain entity, they aremodeled by using first-order hybrid Petri nets that combine fluidand discrete event dynamics [2]–[8]. These Petri net models,however, ignore an important feature of supply chain, that is,many operations such as inventory replenishment and distribu-tion are usually performed in a batch way because of the batchnature of customer orders and the requirement for taking advan-tages of the economies of scale.

In this paper, batch deterministic and stochastic petri nets(BDSPNs) are introduced as a tool for modeling and perfor-mance evaluation of supply chains. The new model is developedby enhancing deterministic and stochastic Petri nets (DSPNs)[13] with batch places and batch tokens. Batch tokens, whichhave sizes and reside in batch places, are viewed as differentindividuals. They are used to describe the batch behavior ofa supply chain. By incorporating stochastic Petri nets (SPNs)with the batch features, inhibitor arcs, and marking-dependentweights, operational policies of supply chains such as inventorypolicies can be easily described in the model.

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CHEN et al.: MODELING AND PERFORMANCE EVALUATION OF SUPPLY CHAINS 133

Our model is different from batch Petri nets [6] in the liter-ature. The introduction of batch places and batch transitions inthat model is for describing the system behavior that a batch ofmaterials with a given size is transferred into a material flow andvice versa. The model is a hybrid Petri net with discrete and con-tinuous transitions, and complex transition enabling and firingconditions. Our model, however, is a discrete Petri net model,which keeps most simplicity of classical discrete Petri nets.

Structural analysis methods of the model are developed alongwith state equation and reachability analysis. Performance eval-uation methods are developed by exploring the marking processof the model and by extending existing methods for DSPNs. Thenew model and analysis methods developed in this paper are ap-plied to an inventory system with independent demand and anindustrial supply chain which produces and distributes electricalconnectors for high voltage lines. The performance of the inven-tory system with policy is evaluated analytically, whereasthat of the real-life supply chain is evaluated by simulation basedon its BDSPN model. The applications demonstrate advantagesand potentials of BDSPNs as a modeling and analysis tool ofsupply chains.

The remainder of this paper is organized as follows. BDSPNsare introduced and defined formally in Section II. Structural andperformance analysis methods of the model are developed inSections III and IV, respectively. An inventory system is mod-eled using BDSPNs and its performance is evaluated analyti-cally in Section V. In Section VI, an industrial supply chain ismodeled and its performance is evaluated by simulation. Con-cluding remarks are given in Section VII.

II. BATCH DETERMINISTIC AND STOCHASTIC PETRI NETS

Several SPN models have been proposed in the literature.Among them are SPNs, GSPNs [1], and DSPNs [1]–[13].SPNs are Petri nets with exponential transitions whose firingtime is subject to an exponential distribution. GSPNs extendSPNs by allowing immediate transitions with zero firing time,whereas DSPNs extend GSPNs by allowing deterministic tran-sitions with a constant firing time. In this section, we introducea new class of stochastic Petri nets called BDSPNs, whichextend DSPNs by introducing batch places and batch tokens.A BDSPN has two types of places: discrete places and batchplaces. Tokens in a discrete place are viewed indifferentlyas in classical Petri nets, whereas tokens in a batch place,called batch tokens, may have different sizes and are viewed asdifferent individuals.

Batch features of supply chain operations motivate this ex-tension. In a supply chain, many operations such as inventoryreplenishment and distribution are usually performed in a batchway, where a given number of products involved in the opera-tions are viewed and treated as an inseparable entity [10]–[14].In the past, when we used Petri nets to model manufacturingsystems, it is usually assumed that each product is produced ata given production rate or at a rate subject to a given probabilitydistribution. The time for producing a batch (a given number) ofthe product thus depends on the size of the batch. In this case,each unit of the product can be represented by a token and the

production activity can be described by a deterministic or sto-chastic transition [1]–[13]. However, for operations such as in-ventory replenishment and distribution, their lead times usuallydon’t depend on the size (quantity) of the customer order in-volved, and orders with different sizes usually have the samelead time for replenishment or delivery. For this case, batch to-kens and batch firing of transitions are necessary for modelingthese batch operations in the framework of Petri nets.

As DSPNs, BDSPNs have three types of transitions: imme-diate transitions, deterministic transitions, and exponential tran-sitions. They may also have inhibitor arcs and marking-depen-dent arc weights. The transition enabling conditions and thestate evolution of BDSPNs are similar to those of DSPNs ex-cept in BDSPNs transitions may be fired in a “batch” way whenbatch tokens are involved.

To formally define BDSPNs, two types of markings are in-troduced, called -marking and -marking, respectively. The

-marking, which represents the state of a BDSPN, is denotedby a vector (or mapping) , where is a nonnegative in-teger for a discrete place or a set of nonnegative integers for abatch place . The set may contain identical elements and eachelement in it represents a batch token with a given size. The

-marking, used for defining marking-dependent arc weights,is denoted by a vector (or mapping) , where is the sameas for a discrete place and is the total size of batch tokensfor a batch place .

A. Model

Definition 1: A BDSPN is specified as a nine-tuple

(1)

where the following are true.

• is a finite set of places consisting of thediscrete places in and the batch places in .

• is a finite set of transitions consisting ofthe immediate transitions in , deterministic transitionsin , and exponential transitions in .

• , , and definethe input arcs, the output arcs and the inhibitor arcs of thetransitions, respectively. It is assumed that only immediatetransitions are associated with inhibitor arcs, i.e.,

, and the inhibitor arcs and the input arcs are twodisjoint sets, i.e., .

• : defines the weights of theinput, output and inhibitor arcs that may depend on the

-marking of the net, where is the set of nonnegativeintegers and is the cardinality of set . It is assumedthat the weights of all arcs associated with the determin-istic and exponential transitions are constant.

• : is a firing priority function for the transitions,which assigns a priority level to each transition. It is as-sumed that timed transitions have the lowest priority level,i.e., if . For immediate transitions,

.• : defines the firing delay

for timed transitions which may be a function of the-marking at which a transition is enabled. It specifies a


Fig. 1. Transition enabling and firing of BDSPNs.

constant firing delay for each deterministic transition anda mean firing delay for each exponential transition.

• : is the initial -marking, where de-notes the superset consisting of all subsets of .

if , and if .Note that the above definition is similar to that of DSPNs if

we don’t distinguish between batch places and discrete placesand if each batch token is regarded as a number of discrete to-kens with the number given by its size [13]. For simplicity, thefiring weight function for immediate transitions in DSPNs is notintroduced in our model. Instead, all immediate transitions areassumed to have the same firing weight 1.

In graphical representation, discrete places and batch placesare represented by single circles and squares with an embeddedcircle, respectively. Immediate, deterministic, and exponentialtransitions are represented by thin bars, filled rectangles, andempty rectangles, respectively. Inhibitor arcs are represented byarrows ending with a small circle. Discrete tokens (tokens ina discrete place) are represented by dots, whereas each batchtoken is represented by an Arabic number that indicates its size(see Fig. 1).

The state of the net is represented by its -marking :. Its corresponding -marking is denoted by :, or simply by . if is a discrete place,

and if is a batch place. In the following,a place connected with a transition by an arc is referred to asinput, output, and inhibitor place, depending on the type of thearc. The set of input places, the set of output places, and the setof inhibitor places of transition are denoted by , , and ,respectively, where ,

, and .

B. Transition Enabling and Firing

We first consider the transition enabling and firing condi-tions for untimed batch Petri nets without priority, i.e., BDSPNswithout taking account of time and transition priorities. The con-ditions depend on whether a transition has batch input places.For simplicity, they are formalized by assuming constant arcweights.

Case 1: : , i.e., a transition has no batchinput place. In this case, the enabling conditions of the transi-tion are the same as those for a transition in a DSPN. That is,transition is enabled if and only if

(2a)

(2b)

The firing of the transition results in a new marking

(3a)

(3b)

(3c)

The operator in (3c) is a “batch” addition. Given any two sets,the batch addition of the two sets will keep all their identicalelements in the resulted set. For any batch output place, the firingof transition generates a batch token with the size given by theweight of the arc connecting the transition to the batch place.

Case 2: : , i.e., a transition has one or morethan one batch input places. In this case, transition is enabledif and only if there is , such that

(4a)

(4b)

(4c)

The firing of the transition results in a new marking

(5a)

(5b)

(5c)

(5d)

The operator in (5d) is the batch addition as explained be-fore. For a transition with one or more than one batch inputplaces, it is enabled if and only if the following are true.

1) Each batch input place of the transition has a batch tokenwith size such that all these batch tokens have the samediscrete firing index defined as for the tran-sition.

2) Each discrete input place of the transition has enough to-kens to simultaneously fire the transition for a number oftimes given by the index.

3) The number of tokens in each inhibitor place of the transi-tion is less than the weight of its associated inhibitor arc.

Note that several discrete firing indexes satisfying thetransition enabling conditions (4a)–(4c) may exist for some

-marking and some transition .For any batch output place, the firing of transition generates

a batch token with the size given by the product of the discretefiring index and the weight of the arc connecting the transitionto the batch place. For any discrete output place, the firing of thetransition generates a number of discrete tokens with the numbergiven by the product of the discrete firing index and the weightof the arc connecting the transition to the discrete place.

Three examples are given in Fig. 1 to illustrate the transitionenabling and firing of BDSPNs. Fig. 1(a) illustrates the enabling


and the firing of a transition without input batch place but withone inhibitor place and one output batch place . Fig. 1(b)illustrates the enabling and the firing of a transition with oneinput batch place . Fig. 1(c) illustrates the enabling and thefiring of a transition with two batch places and .

The above transition enabling and firing conditions don’t takeaccount of transition priorities. When the priorities are consid-ered, only the transitions with the highest priority are enabledand may be fired at any marking.

When time is involved in BDSPNs, we need additional rulesor policies to resolve conflicts when some conflicting transitionsare enabled at the same time.

When both timed and immediate transitions are enabled atany marking, it is assumed that only immediate transitions canfire because timed transitions have the lowest priority level 0according to the definition of BDSPNs. When immediate tran-sitions with different priorities are enabled, only those with thehighest priority may be fired. When some conflicting transitionswith the same highest priority are enabled, they have the sameprobability to be fired. The probability is obtained by dividing1 by the number of the conflicting transitions.

Immediate transitions are fired in zero time whereas timedtransitions are fired after either an exponentially distributed or adeterministic delay. When several timed transitions are enabledin any marking, the transition with the minimum firing delaywill cause the marking change. This is called the race condi-tion [1]–[13]. Furthermore, as in DSPN, it is assumed that aftera marking change each timed transition newly enabled sam-ples a remaining firing delay from its firing delay distribution.Each timed transition which has already been enabled in the pre-vious marking and is still enabled in the current marking keepsits remaining firing delay. This execution policy corresponds tothe race with enabling memory policy of stochastic Petri nets[1]–[13].

The policy for the choice of a batch token in each batch placeto fire its output transition should also be specified when thereare several batch tokens in the place. We assume the choice fol-lows the first-in first-out (FIFO) rule with random breakdownof ties. That is, a batch token which arrived in each batch placefirst will be selected to fire its output transition if it can. In thecase when several batch tokens arrived in the place at the sameearliest time, they are assumed to have the same probability tofire the transition.

III. STRUCTURAL ANALYSIS OF BDSPNs

In this section, linear algebraic techniques based on stateequation and reachability analysis are developed for BDSPNs,which provide a basis for structural analysis of the model.Some important structural properties of the model are definedand their verifications using the two methods are discussed.

A. Structural Properties

Several important properties can be defined for BDSPNs byanalogy to classical Petri nets and with consideration of theirbatch features. When studying structural properties, the timespecification of BDSPNs is ignored. In this case, a BDSPN be-comes an untimed batch Petri net (an untimed Petri net with

batch places and batch tokens), in which transitions are associ-ated with priorities. We called the untimed net a batch Petri netfor simplicity.

Definition 2: For a batch Petri net, if there is a transitionwhose firing causes a change from a -marking to a

-marking , we say that is directly reachable from ,written as or . More generally, if there is asequence of transitions and a sequence ofmarkings such that ,

, we say is reachable from , written asor .

Similarly, we can define and for-markings.Definition 3: A batch Petri net is called bounded if the-marking of its any place does not exceed a finite number

for any -marking reachable from the initial marking. That is,, if , and , if

, for any with .Definition 4:

a) A batch place of a batch Petri net is called batch-dead if,for at least one reachable -marking, the place has a batchtoken that cannot be used for firing any of its output tran-sitions. In other words, the discrete firing index of eachoutput transition with respect to the batch token is not aninteger.

b) A batch Petri net is called batch-dead if it has at least onebatch-dead batch place.

c) A transition of a batch Petri net is live for the initial-marking , if for any : , there is :

such that is enabled at .d) A batch Petri net is said to be live for the initial -marking

, if all its transitions are live for .e) A batch Petri net is said to be batch-live if it is live and

has no batch-dead place.If the enabling condition of a transition is not verified because

there is no token whose size is a multiple of the weight of thecorresponding arc, a batch-deadlock will occur. This deadlockmay be caused by a system design and operational problem ora modeling problem. For the latter, we can change the corre-sponding batch place to a discrete place to avoid the deadlock.

B. Linear Algebraic Techniques

Let and . For a batchPetri net, when all its arc weights are constant, we can also de-fine its incidence matrix:

(6)

where is an matrix whose rows correspond to theplaces and whose columns correspond to the transitions ofthe net. In (6) and hereafter, we abuse the symbol to repre-sent both the incidence matrix and the arc weight function ofBDSPNs in case of no confusion.

Suppose that the current -marking of the net is and tran-sition is enabled at the marking. The firing of the transitionwill result in a new marking written as

(7a)


(7b)

(7c)

where is a discrete firing index of transition at markingdefined as

if

if

(8)

The operator in (7b) is the batch addition when it involves the-marking of a batch place.Let be an 1 characteristic vector repre-

senting the firing of transition defined as

ifif (9)

We have

(10)

Equation (10) is called the state equation of the net. This equa-tion can be extended to the sequential firing of multiple transi-tions when is a constant matrix.

For example, the net in Fig. 1(b) has its incidence matrix. In the net, transition is enabled at -marking

since we have a discrete firing indexsuch that

After the firing of transition , the -marking of the net ischanged to one shown in Fig. 1(b). That is,

One important linear algebraic technique for classical Petrinets is -invariants and -invariants. Similarly, we have:

Definition 5: For a batch Petri net with constant incidencematrix , a -dimensional column vector , whose entries arenatural numbers, is called a -invariant of the net if .

If is a -invariant of a batch Petri net, then we have

(11)

If in a batch Petri net all places are covered by a -invariantin the sense that each entry of the invariant is positive, then themaximum number of tokens (the maximum total size of batchtokens) in any place is finite and the net is structurally bounded.

Similarly, we can define -invariants for the batch Petri net.This part is skipped because of space limitation.

C. Reachability Analysis

The reachability analysis plays an important role for struc-tural analysis of untimed Petri nets. This approach can also beapplied to the study of structural properties of batch Petri nets.

Definition 6: ( -Reachability Set): Let be a batch Petrinet defined by (1). The set of all -markings reachable from the

initial marking by firing a sequence of transitions and theinitial marking itself constitute the reachability set of the net,which is denoted by .

Definition 7: ( -Reachability Graph): Let be a batch Petrinet defined by (1). The reachability graph of the net is a directedgraph , where the set of vertices is given by thereachability set , whereas the set of directed arcsis given by all feasible marking changes of the net due to tran-sition firing in all reachable -markings.

The -reachability graph can be constructed by using an im-plicit approach when the underlying net is bounded.

Similarly, we can define -reachability set and-reachability graph . Set and graph

can be obtained from and re-spectively by transforming each -marking in (in

) into its corresponding -marking and by merging dupli-cated -markings (and duplicated arcs).

The absence of batch-dead places and the liveness (batch-liveness) of a batch Petri net can be checked in the -reachabilitygraph by looking for a dead batch token in a batch place in allreachable -markings and a dead transition.

The above definition of -reachability graph does not con-sider transition priorities. However, the construction of thereachability graph in the case of batch Petri nets with priorityis similar to that for Petri nets without priority. The only differ-ence is in the enabling rule that in the case of a net with prioritymay produce a smaller reachability graph (only the transitionswith the highest priority are enabled and may be fired at anymarking).

Furthermore, when performing the performance analysis ofa BDSPN, we are interested only in the markings (states) thatcan stay for a positive time. For this reason, the reduced reach-ability graph is introduced, which is obtained by deleting all“vanishing” markings (nodes) in the reachability graph and theirassociated transitions (arcs) while keeping all “tangible” mark-ings. A marking is called vanishing if at least one immediatetransition is enabled at the marking. Otherwise, the marking istangible.

IV. PERFORMANCE ANALYSIS OF BDSPNS

One ultimate goal for the introduction of the BDSPN modelis to evaluate the performances of its modeled systems. Forthis purpose, the marking process of the model which describesits stochastic behavior is studied by exploring the similaritybetween BDSPNs and DSPNs. Performance analysis methodsfor BDSPNs are then developed by extending existing methodsfor DSPNs. It is shown that BDSPNs have marking processessimilar to those of DSPNs and most performance analysismethods of DSPNs can be extended to BDSPNs after a propermodification.

A. Marking Process of BDSPNs

For a stochastic Petri net with exponential transitions, im-mediate transitions, and deterministic transitions, the character-istics of its (stochastic) marking process only depends on theprobabilistic choice of the transition(s) to be fired next and the


joint probability distributions of the firing times of all possiblecombinations of transitions that can be fired concurrently at anyreachable marking. The simplest stochastic process that can rep-resent the marking process of such a net is only determined bythe probability distributions associated with the transitions.

To characterize the marking process of a BDSPN, one ques-tion we should answer is whether the batch places, the batchtokens and the policy for choosing batch tokens to fire the nexttransition defined in Section II have an effect on the character-istics of the stochastic process. To answer this question, we ob-serve that for a BDSPN with FIFO batch token choice policyand random breakdown of ties of arrival times of batch tokens:

1) At any -marking, if the net does not have any two batchtokens in a tie of arrival times that can fire an enabledtransition, the transition to be fired next can only befired with a unique discrete firing index determined bythe -marking. In this case, the probabilistic choice ofthe transition to be fired next when several transitionsare enabled at the same marking only depends on theprobability distributions associated with these transitions.

2) At any -marking, when the transition to be fired nextcan be fired with several discrete firing indexes becauseof an arrival time tie of multiple batch tokens with dif-ferent sizes in an input batch place of the transition, thechoice of the discrete firing index to fire the transition ismade randomly with an equal probability for all possibleindexes. The probability is determined by the -marking.In this case, if we view the firing of a transition with dif-ferent discrete firing indexes as the firing of different tran-sitions and the choice of the discrete firing index is madeby introducing auxiliary immediate transitions precedingthe transitions as shown in Fig. 2, the probabilistic choiceof the transition to be fired next also depends only on theprobability distributions associated with all enabled tran-sitions at the -marking. In the figure, timed transitions

and in (b) correspond to the timed transition withdiscrete firing indexes 1 and 2, respectively in (a). Imme-diate transitions and are introduced to model thechoice between the two possible firing indexes.

3) The probability distribution of the sojourn time in any-marking only depends on the probability distributions

associated with all enabled transitions at the -marking.The former distribution can be expressed as a compositionof the later probability distributions.

With the above observations and the results about the sto-chastic process of DSPNs, we have the following classificationof the marking processes underlying BDSPNs.

a) If a BDSPN does not contain any deterministic transition,its marking process is a continuous-time Markov chain.

b) If a BDSPN has no deterministic transition concurrentlyenabled with other deterministic transitions or exponen-tial transitions at any marking, its marking process is asemi-Markov chain.

c) If a BDSPN has a deterministic transition concurrently en-abled with other deterministic transitions or exponentialtransitions at some marking with restrictions on the usage

Fig. 2. Representation of batch token choice by immediate transitions.

of deterministic transitions ([13]), its marking process isa Markov regenerative process.

d) The marking process of a BDSPN without any restrictionon the usage of deterministic transitions is a generalizedsemi-Markov process.

B. Performance Evaluation

The characterization of the marking process of BDSPNs inlast subsection allows us to analytically and efficiently evaluatethe performance of a subclass of BDSPNs—BDSPNs withoutdeterministic transitions called BSPNs. For a BSPN, its steady-state distribution is the basis for analysis of its behavior ex-pressed in terms of performance indexes. These indexes can becomputed using a unified approach in which proper index func-tions are defined over the -marking of the net and an averageindex is derived using the steady-state distribution. Assumingthat is such an index function and that is the steady-stateprobability for state , the average index can becomputed using the following weighted sum:

(12)

Some important performance indexes of BDSPNs are formal-ized in the following.

1) The mean number of tokens in a given place

for discrete place (13a)

for batch place (13b)

where is the cardinality of set .2) The mean total size of tokens in a given place

for discrete place (14a)

for batch place

(14b)

3) Let be the set of -markings at which transitionis fired, the firing frequency of the transition can be

computed by

(15)


where is the firing rate of transition .4) Let be the set of -markings at which transition

is fired with discrete firing index , the firing frequencyof the transition with this index can be computed by

(16)

where is the firing rate of transition with index .The general procedure for performance evaluation of BSPNs

is summarized in the following.

1) Construct the -reachability graph. The nodes of thegraph represent all -markings reachable from the initialmarking. Each arc is labeled by its corresponding firedtransition. The transition is marked by its discrete firingindex for batch firing.

2) Eliminate all vanishing markings (nodes) and its associ-ated immediate transitions (arcs) to generate the reduced

-reachability graph.3) The stochastic marking process is obtained from the re-

duced -reachability graph by labeling each arc with thefiring rate of its corresponding fired transition. The statespace of the process is composed of all nodes of the graph.The state transition process from one state to another isspecified by the distribution of its corresponding exponen-tial transition with a given firing rate.

4) Determine the steady-state distribution of the stochasticprocess.

5) Evaluate the performance indexes under considerationusing the steady-state distribution obtained in Step 4.

The performance evaluation of general BDSPNs becomesmore complicated analytically or has to rely on simulation.This is a topic for further research.

V. MODELING AND PERFORMANCE EVALUATION OF

INVENTORY SYSTEMS USING BDSPNS

In this section, we discuss how BDSPNs can be used to modeland evaluate the performance of inventory systems with inde-pendent demand, whereas the capability of BDSPNs for mod-eling and performance evaluation of multi-echelon inventorysystems (supply chains) will be demonstrated using an indus-trial case in the next section.

Inventory systems are an important component of supplychains and inventory management plays a very important rolein supply chain management. Inventory management addressestwo fundamental issues: when should a stock replenish itsinventory and how much should it order from its supplier(s) foreach replenishment [10].

Inventory systems with independent demand may use fixedor variable order quantity policies based on either continuousor periodic review of inventory position. The inventory posi-tion is defined as on-hand inventory plus outstanding ordersminus backorders. Fixed order quantity policies place an orderof fixed size whenever the inventory position of a stock fallsto a pre-specified level, whereas variable order quantity poli-cies place an order of variable size at regular intervals to raisethe inventory position to a pre-specified value. The most fre-quently used inventory policies include order-up-to-level policy,

Fig. 3. BDSPN models of an inventory system.

batch ordering policy , and policy. Inventory poli-cies also differ in their treatment of unmet customer demands.Some policies record unfilled customer demands as backordersand fill the demands later when stock is available, whereas un-filled demands are lost with the other policies.

A. Modeling

As an example, we consider an inventory system with con-tinuous review policy. For this system, when the inven-tory position IP drops below a given reorder point , an orderwith quantity – IP will be placed to raise the inventory posi-tion to a given order-up-to-level . Fig. 3(a) and (b) show theBDSPN models of the inventory system with backorders andwithout backorder, respectively. In the model (a), place rep-resents on-hand (physical) inventory of the stock considered.Batch place represents outstanding orders – the orders thatare placed by the stock but not received from its supplier yet.Discrete place represents backorders – unfilled customer de-mands. Place represents on-hand inventory of the stock plusits outstanding orders, that is, .The inventory position IP equals

. Inventory replenishment decisions are basedon the position.

Customer demand is assumed to be a Poisson process, whichis specified by transition whose firing time is subject to an ex-ponential distribution. Customer demand will be filled if thereis sufficient on-hand inventory. Otherwise, the demand will bebackordered. The fulfillment of customer demand will decreaseon-hand inventory as well as the size of backorders. This isdescribed by the arcs from places , and to transition

. When the inventory position drops belowthe reorder point , i.e., or equivalently

, an order with quantitywill be placed to the supplier(s). The place-

ment of the order will increase the outstanding orders by thisquantity. This is described by immediate transition , its asso-ciated arcs to places and , and the weights of the arcs. Whenthe -marking of place , is less than , tran-sition will be fired, which generates a batch token with size

in place anddiscrete tokens in place . If there is a batch token in place ,a replenishment event represented by the firing of transitionwill occur, which delivers a corresponding order to the stock.


Fig. 4. Reduced �-reachability graph and its underlying CTMC.

For modeling the system without backorder, the place usedfor representing backorders is removed, exponential transition

and immediate transition are merged to a new exponentialtransition , the term in the weights of the arcs associ-ated with transition is removed accordingly.

BDSPNs can also model an inventory system with periodicreview policy. For this purpose, an additional subnet usedfor describing the periodic review feature of the policy has to beintroduced. The subnet is coupled with the existing net in Fig. 3through a common transition ([4]). Our model can also de-scribe inventory systems with order-up-to level policy and batchordering policy.


To illustrate the performance evaluation approach of BDSPNsproposed in last section, we consider the BDSPN model of theinventory system without backorder in Fig. 3(b). For simplicity,we assume that the inventory replenishment time is subject toan exponential distribution and that the inventory policy param-eters of the system are taken as and , respectively.

The reduced -reachability graph of the BDSPN model andthe state transition graph of its corresponding continuous-timeMarkov chain (CTMC) are shown in Fig. 4(a) and (b), respec-tively, where and are the parameters of the exponentialfiring time distributions of transitions and , respectively.

The infinitesimal generator (transition rate matrix)of the CTMC is

Fig. 5. Performance curves of an inventory system.

where is the transition rate from state (marking) to stateand so that .

The steady-state probability vectoris the solution of the

linear equation system: and , wheredenotes the steady-state probability that the stochastic

process is in state . For this linear system, the steady-stateprobabilities can be explicitly obtained as functions ofparameters and as


Fig. 6. Structure of the supply chain.

Fig. 7. Manufacturing process.

With the steady-state probability, the average inventory levelof the system (the mean number of tokens in discrete placecan be obtained by

The stockout rate of the system (the probability that placeis empty) is given by

Given the demand parameter , the average inventorylevel and the stockout rate as functions of inventory replenish-ment parameter are depicted in Fig. 5(a) and (b), respectively.

VI. MODELING AND PERFORMANCE EVALUATION OF AN

INDUSTRIAL SUPPLY CHAIN

In this section, we model and evaluate the performance of anindustrial supply chain by using BDSPN. This supply chain iscomposed of three suppliers, three transporters for the suppliers,a manufacturer, a transporter for the manufacturer and severalcustomers (see Fig. 6). The manufacturer produces an electricalconnector for high voltage lines. There are three types of flowsin the supply chain: material flow, information flow and financialflow.

For material flow (see Fig. 7), the manufacturer needs threeraw materials to produce the connector: flat, rod, and screw.They are purchased from supplier 1, 2 and 3, respectively. These

materials are delivered to the manufacturer by the correspondingtransporters. At the manufacturer site, rods of aluminum are cutinto shafts with a given length, flats are bored and ground. Eachfinished product is then produced by assembling a shaft, a flat,and two screws. The products will be packaged and delivered tocustomers by a transporter.

For information flow, the manufacturer receives customer de-mands in orders. When the inventory position of the finishedproduct (Stock 4) is below a pre-specified reorder point, an as-sembly order with a given batch size will be released to the as-sembly line of the manufacturer. For each of the raw materialstocks S1, S2 and S3, when its inventory position is below itsreorder point, a purchase order with a given batch size will beplaced to its corresponding supplier. Each purchase or assemblyorder contains the information such as ordering time and orderquantity, which are determined by an inventory policy involved.The inventory policies of all four stocks are periodic reviewbatch ordering policy.

For financial flow, customers pay the manufacturer within agiven time period after receiving finished products. The manu-facturer pays its suppliers similarly. The suppliers and the man-ufacturer pay their transporters within a given time period afterthe completion of a delivery.

A. Modeling

Fig. 8 shows a BDSPN model of the industrial supply chain.The interpretations of the places and the transitions in the modelare given by Tables I and II, respectively, where in Table Idenotes the mean firing delay (in days) of transition .

In the model, the material flow is represented by timed transi-tions t1, t2 and t3 (inventory replenishments of the suppliers), t7,t8 and t9 (transportation from the suppliers to the manufacturer),t11 (assembly), t13 (delivery preparation), t14 (transportation


Fig. 8. BDSPN model of the supply chain.

TABLE IINTERPRETATION OF THE TRANSITIONS IN FIG. 8

from the manufacturer to customers), and their associated placesand arcs.

The information flow is represented by immediate transitionst23 to t26, their associated places and arcs, and the weights of thearcs. The parameters of the inventory policies are representedby the weights of the arcs as we model inventory systems inSection III.

The financial flow is represented by transitions t15 to t22, dis-crete places p16 to p19, and their associated arcs. These transi-tions represent payment operations from one company to an-other whereas the markings of the discrete places represent themoney available in the companies.

After establishing the BDSPN model, we can use structuralanalysis methods developed in Section IV to verify whether themodel is structurally bounded and/or batch-live. If not, it maybe caused by modeling itself or caused by a design problem ofthe industry supply chain. Our analysis shows that the BDSPNmodel has no structural problem. Because of page limitation, wewill not present the analysis.


The performance of the industrial supply chain may be eval-uated analytically if its BDSPN model is bounded in the case of


TABLE IIINTERPRETATION OF THE PLACES IN FIG. 8

no backorder of final customers. However, in this study, the per-formance is evaluated by the simulation using a BDSPN simu-lator (a C++ program) developed by us, because currently com-puterized BDSPN analytic performance evaluation tool is notavailable.

For the supply chain, the mean replenishment lead times forstocks S1, S2, S3, S4 are 40, 20, 70, and 20 days, respectively.The mean delivery lead times for all transporters are 2 days. Be-cause the deviations of these lead times are not known and neg-ligible, we treat the firing delays of their corresponding tran-sitions as constant numbers. Customer orders arrive randomlywith the inter-arrival times subject to an exponential distributionwith mean value 0.0355. The reorder points of stocks S1, S2, S3,S4 are 2300, 590, 2000, 400, respectively, whereas the orderquantities of these stocks are 5000, 3000, 9300, 2000, respec-tively. The annual holding costs of the raw materials and finalproduct in stocks S1, S2, S3, S4 are 12% of their prices whichare 0.3 Euro, 0.6 Euro, 0.16 Euro, and 3 Euro, respectively.

The performance criteria of the supply chain we want to ob-tain include average inventory level and service level for eachstock, where the service level is defined as the probability thatcustomer orders are filled on time. The first criterion is easy toobtain since it corresponds to the average number of tokens inthe discrete place representing the stock. For the evaluation ofthe service level, we need to know the total time that the discreteplace has no token while the batch place representing customerorders is not empty in each simulation. This can be done by ob-serving the markings of the two places during the simulation.

TABLE IIIPERFORMANCE EVALUATION OF A SUPPLY CHAIN

Our BDSPN simulator provides functions for calculating av-erage number of tokens (respectively, average total size of batchtokens) for discrete places (respectively, batch places) and thetotal time that a BDSPN is at a specified state (or in a specifiedstate set) in a simulation.

Because of stochastic nature of the model, multiple repli-cations of simulation over a long time horizon should be per-formed to obtain a reliable estimation of the performance in-dexes. For the industrial case, the number of replications is takenas and the simulation horizon is taken astime units (days) with (i.e., ). The reason fortaking 25 replications is because Kelton [12] has demonstratedthat there is little value in dividing the available computationtime into more than 25 replications. The formulais suggested by Banks, Carson, Nelson and Nicol in their simu-lation textbook [3], where the first observations are deletedin each replication to reduce the point-estimation bias caused byinitial conditions in a steady-state simulation. The time horizonof each replication, beyond the deletion point, is taken as tentimes of the amount of observations deleted. Sincecorresponds to more than half a year and the observed period

corresponds to more than five years, we believethat these two durations are long enough to reduce the initializa-tion bias to a negligible level and to obtain a reliable estimationof steady-state performance. All performance indexes are eval-uated (estimated) as their mean values over the 25 replications.

The point estimation of each performance index and thestandard error of the estimation obtained by the simulation areshown in Table III, where each entry no bracketed is the pointestimation value of a performance index, whereas the bracketedentry just below the value is the standard error of the estimation.

The standard error is quite small, usually within 1% ofits corresponding point estimation value. This implies thatthe obtained results are trustful. Given the point estima-tion and the standard error, a confidence interval for eachperformance index can be calculated for any given levelof significance under the condition of the independence ofreplications. A confidence interval for per-formance index , based on the -distribution, is given by

, whereand are the point estimation and the standard error of theestimation of , respectively, is the number of independentreplications, and is the percentage


TABLE IVCONFIDENCE INTERVALS OF PERFORMANCE EVALUATION

point of a -distribution with degrees of freedom ([3]).Using this formula, the 99% confidence interval for eachperformance index of the supply chain is given in Table IV,where the bracketed entry just below the point estimation of theindex indicates the half length of the confidence interval, i.e.,the value . We have also verified our results bycomparing them with their industrial values. It is shown thatour results well approximate the values.

From the above modeling and performance evaluation of anindustrial supply chain, we can see our proposed model wellmeet the modeling needs of supply chains by providing a goodvisibility of orders and batch operations in the supply chain.

VII. CONCLUSION

In this paper, BDSPs are introduced as a tool for modelingand performance evaluation of supply chains. The new Petrinet model is developed by enhancing deterministic and sto-chastic Petri nets with batch places and batch tokens. Structuralanalysis and performance evaluation methods of the model aredeveloped by extending existing ones for deterministic andstochastic Petri nets. Applications to inventory systems and areal-life supply chain demonstrate that our model well meetsthe modeling needs of supply chains where inventory replenish-ment and distribution are usually performed in a batch way. Theapplications also show that our model and associated methodscan solve important supply chain issues such as evaluation ofinventory policies. Compared with batch Petri nets and DSPNsin the literature, our model keeps most simplicity of classicaldiscrete Petri nets and all good features of DSPNs, allowingus to develop its analysis methods. The model and associatedapproaches therefore provide a promising tool for modelingand performance evaluation of supply chains. Further workincludes the development of performance analysis methods formore general BDSPNs and the theoretical study of the proposedsimulation-based approach.

REFERENCES

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[5] H. Chen, L. Amodeo, and L. Boudjeloud, “Supply chain optimizationwith Petri nets and genetic algorithms,” in Proc. Int. Conf. IndustrialEngineering and Production Management, Porto, Portugal, Jun. 2003.

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[7] C. Fleurent and G. A. Ferland, “Algorithmes génétiques hybrides pourl’optimization combinatoire,” Rech. Operat., vol. 30, no. 4, pp. 373–398,1996.

[8] R. Furcas et al., “Hybrid Petri net modeling of inventory managementsystems,” J. Eur. Syst. Autom., vol. 35, pp. 417–434, 2001.

[9] D. Goldberg, Genetic Algorithms in Search, Optimization and MachineLearning. Reading, MA: Addisson-Wesley, 1989.

[10] S. C. Graves et al., Logistics of Production and Inventory. New York:Elsevier, 1993.

[11] P. J. Haas, Stochastic Petri Nets: Modeling, Stability, Simulation. NewYork: Springer-Verlag, 2002.

[12] W. D. Kelton, “Replication splitting and variance for simulating discreteparameter stochastic processes,” Oper. Res. Lett., vol. 4, pp. 275–279,1986.

[13] C. Lindemann, Performance Modeling with Deterministic and Sto-chastic Petri Nets. New York: Wiley, 1998.

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Haoxun Chen received the B. S. degree in appliedmathematics from Fudan University, Shanghai,China, the Master’s degree in systems engineeringfrom Shanghai Jiaotong University, Shanghai, China,and the Ph.D. degree in systems engineering fromXi’an Jiaotong University, Xi’an, China, in 1984,1987, and 1990, respectively.

He was a Lecturer with Xi’an Jiaotong Universityfrom 1990 to 1992 and an Associate Professor from1993 to 1996. He visited the National Research Insti-tute in Computer Science and Automation (INRIA),

Lorraine, France, as a Visiting Professor, in 1994, the University of Magdeburg,Magdeburg, Germany, as a Research Fellow of the Alexander von HumboldtFoundation, in 1997 and 1998, and the University of Connecticut, Storrs, as aResearch Assistant Professor in 1999 and 2000. Since 2004, he has been a Pro-fessor at the Technology University of Troyes, Troyes, France. His research in-terests include supply chain management, production planning and scheduling,and discrete event systems. He has published more than 70 papers in technicaljournals and Conference proceedings.

Prof. Chen received the King-Sun Fu Memorial Best Transactions PaperAward from IEEE Robotics and Control Society in 1998.


Lionel Amodeo received the Ph.D. degree inautomatic and production management from theTechnology University of Belfort-Montbéliard,Belfort, France, in 1999.

Since 2000, he has been with the TechnologyUniversity of Troyes, Troyes, France, where he iscurrently an Associate Professor in the Departmentof Industrial Engineering. His research interestsinclude Petri nets, discrete event systems, modeling,analysis and optimization of production systems,and supply chains. He has published over 30 papers

and one book.Dr. Amodeo was a finalist of the Kayamori Best Paper Award of the IEEE

International Conference on Robotics and Automation in 2002.

Feng Chu received the B.S. degree in electricalengineering from Hefei University of Technology,Hefei, China, the Master’s degree in electricalengineering from Institut National Polytechnique deLorraine, Lorraine, France, and the Ph.D. degree incomputer science from University of Metz, Metz,France, in 1986, 1991, and 1995, respectively.

She worked at the Technology University ofJiangsu, Jiangsu, China, for two years and at theNational Research Institute in Computer Scienceand Automation (INRIA), Lorraine, France, for four

years. She is currently an Associate Professor at the Technology Universityof Troyes, Troyes, France. She is mainly interested in modeling, analysis andoptimization of supply chains or production systems, transportation planning,production planning, and scheduling.

Karim Labadi received the Bachelor’s degree inelectronic and control engineering from the Institutd’Electronique, Université de Tizi, Ouzou, Algeria,and the Master’s degree in automation and appliedcomputer science from the Ecole Centrale de Nantes(ECN), Nantes, France, in 1998 and 2001, respec-tively. He is currently working toward the Ph.Ddegree in industrial engineering at the TechnologyUniversity of Troyes, Troyes, France.

His research interests include Petri nets theory,modeling, analysis, performance evaluation, and

optimization of industrial systems.

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