J. Shanghai Jiaotong Univ. (Sci.), 2009, 14(6): 713-719

DOI: 10.1007/s12204-009-0713-z

Solving Resource-constrained Multiple Project Scheduling ProblemUsing Timed Colored Petri Nets

WU Yu (� �), ZHUANG Xin-cun∗ (���), SONG Guo-hui (���)XU Xiao-dong (���), LI Cong-xin (���)

(National Die and Mould CAD Engineering Research Center, Shanghai Jiaotong University, Shanghai 200030, China)

Abstract: To solve the resource-constrained multiple project scheduling problem (RCMPSP) more effectively, amethod based on timed colored Petri net (TCPN) was proposed. In this methodology, firstly a novel mappingmechanism between traditional network diagram such as CPM (critical path method)/PERT (program evaluationand review technique) and TCPN was presented. Then a primary TCPN (PTCPN) for solving RCMPSP wasmodeled based on the proposed mapping mechanism. Meanwhile, the object PTCPN was used to simulate themultiple projects scheduling and to find the approximately optimal value of RCMPSP. Finally, the performanceof the proposed approach for solving RCMPSP was validated by executing a mould manufacturing example.Key words: timed colored Petri nets, resource-constrained multiple project scheduling problem (RCMPSP),mapping mechanismCLC number: TH 164 Document code: A

Introduction

There is no doubt that about 90% projects are car-ried out in multi-project environment in recent businessor industry. That is to say, projects are not isolatedfrom each other, they are connected with certain re-lations such as resource competition and sharing, duedate conflict etc. Because of these mutual relationsespecially the resource conflict in concurrent multipleprojects, project management becomes more and morecomplicated. Therefore, it is important to find “better”ways to solve the resource-constrained multiple projectscheduling problem (RCMPSP) which is a well knownNP-hard problem. Recently, a number of researches onRCMPSP have been reported in literatures[1]. Theirworks have dealt with varieties of situations based onsingle project in which one or both of these types ofconstraints are relaxed, or at least simplified, and a fewstudies on project management have started to explorehow to manage and schedule multiple projects.

Fricke and Shenhar[1] considered the insight into thatthe most important multiple project success factors ofthis environment differed from factors of success of tra-ditional single project management, and were consis-tent with other emerging research in product develop-ment environment. Lova and Tormos[2] analyzed theeffect of the schedule generation schemes such as serialor parallel, and priority rules that minimized latest fin-ish time, total slack, and maximized total work content,

Received date: 2008-04-07∗E-mail: georgezxc@sjtu.edu.cn

shortened activity from the shortest project, or first-come first-served in single project and multiple projectenvironments. Liao et al[3] put forward a heuristic al-gorithm to solve the RCMPSP and discussed the appli-cation in mould manufacturing. Kim et al[4] proposed ahybrid genetic algorithm with fuzzy logic controller tosolve the RCMPSP. The proposed approach was basedon the design of genetic operators with fuzzy logic con-troller through initializing the revised serial methodwhich outperforms the non-preemptive scheduling withprecedence and resources constraints. However, theabove-mentioned researches were all based on the con-ventional project management models such as criticalpath method (CPM) or program evaluation and reviewtechnique (PERT). Although CPM/PERT method canbe perfectly applied in single project management, itcan not be employed to manage the concurrent multi-ple projects effectively since the resource constraint isnot taken into account. In addition, a number of fac-tors such as interdependency, criticality, conflicting pri-orities, and substitution of resources, which affect thecontrol over project execution, can not be representedby traditional CPM/PERT models. The later modelssuch as venture ERT(VERT), graphical ERT(GERT),controlled activity networks, and tools like time/costtradeoff curves and influence matrix approaches alsoare not capable of incorporating these factors[5]. Incontrast, Petri nets can incorporate these factors andprovide a powerful graphical and analytical methodol-ogy in project management.

Petri net is a powerful graphical and mathematicalmodeling tool that has been successfully applied in the

714 J. Shanghai Jiaotong Univ. (Sci.), 2009, 14(6): 713-719

areas of discrete event dynamics system (DEDS) suchas performance evaluation, communication protocols,legal systems, and decision models[6]. The primary dif-ference between Petri nets and other modeling tools isthe presence of tokens which are used to simulate dy-namic, concurrent and asynchronous activities in a sys-tem. The behavior of a system can be represented in aPetri net by setting up state equations, algebraic equa-tions and other mathematical models. Project manage-ment has also been regarded by some researchers[7] asa prospective area where the modeling power of Petrinets can be used. Recently, application of Petri netsin resource-constrained single project management hasbeen reported. Kim et al[8] introduced an interactiveproject management approach based on the Petri netto overcome the limitations of network diagrams. Ku-mar and Ganesh[9] described the use of Petri nets tofacilitate resource allocation in projects under someconditions commonly encountered in practice. Thena Petri net aided software including genetic algorithmbased search and heuristics was developed to deal witha multi mode, multiple constrained scheduling problemwith pre-emption of activities. But up to now there isstill no reported achievement addressed on the appli-cation of Petri nets in multiple projects environment.According to the advantages of Petri nets mentionedabove and the researches in single project management,the attempt to solve the RCMPSP using Petri nets isshould to be necessary and feasible.

The aim of this paper is to study the application ofPetri nets for solving RCMPCP. At first a novel map-ping mechanism between traditional network diagramsuch as CPM/PERT and Petri nets representation inmultiple projects environment is proposed, because thenetwork diagram applied for single project can be eas-ily obtained by mature commercial software package.So converting the single project represented by the net-work diagram to a relative Petri net can make it conve-nient to combine each single project and establish theRCMPSP model. Then the converted Petri nets modelabout RCMPSP is to be simulated to find the optimalresult.

1 Introduction of Petri nets

1.1 Advantages of Applying Petri nets inProject Management

Petri nets offer a lot of advantages in project man-agement. They are summarized as follows.

(1) A Petri net is capable of modeling a systemwhere many activities take place concurrently and asyn-chronously. It is capable of modeling concurrence, andconflicts. Moreover, System deadlocks can be deter-mined.

(2) It provides information for a project manager tohelp, check, and reason about the tardy progress of the

activity.(3) It is capable of regenerating and rescheduling of

activities. A Petri net is also a dynamic representationof a system, and hence, is suitable for on-line monitor-ing.

(4) Logic expressed in words is transferred into agraphical form and a suitable mathematical form foranalysis.

(5) Analytical models maximize modeling flexibility,whereas simulation can reduce flexibility. Petri netslie between these two approaches, providing analyticalresults with much of the modeling flexibility of simula-tion.

(6) Dynamic simulation of a project can be visualizedgraphically. Using subnets, the parts of the project canbe conveniently modeled. It is possible to simulate theentire project and keep all subnets in the foregroundand the rests in background.

(7) A Petri net is capable of representing resourceinterdependency, partial allocation, substitution of re-sources and mutual exclusivity.

(8) Using behavioral properties such as reachabilityand boundedness, project planning can be improved.

(9) A Petri net can be simplified by combining simi-lar places and transitions. Using higher-level Petri netssuch as colored Petri nets (CPNs) or object Petri nets(OPNs), the graphical size of the network can be re-duced.

(10) Modeling dummy activities in network diagramsof CPM/PERT is always cumbersome. However, rep-resentation of precedence relationships by using Petrinets become easier.1.2 Timed Colored Petri nets (TCPNs)

In multiple projects environment, the number of con-current projects is always to be the level of hundredsand thousands, and the graphical size of Petri nets forRCMPSP will be prodigious when it is modeled by ba-sic Petri nets. Therefore, it is necessary to employ thehigher-level Petri nets to combine similar places andtransitions which are converted from the network dia-gram to Petri nets in different concurrent project. Sothe graphical size can be reduced and make it easyto analyze. Among the various extension to the basicPetri nets, TCPNs[10] is one of the high level Petri netswhich has advantages over basic Petri nets in makingthe model of a DEDS much more compact and concise.This is gained by introducing “colors” to distinguishamong tokens presenting different entities and using ad-ditional logic to control token flows. So according tothese advantages, the TCPN is selected to model theRCMPSP in this study.

TCPNs to be defined below are based on semanticsmade by Jensen[11]. The temporization of CPNs can beachieved by attaching time either to places, resultingin timed place CPNs, to transition, resulting in timedtransition CPNs, or to the arc expression function. In

J. Shanghai Jiaotong Univ. (Sci.), 2009, 14(6): 713-719 715

this study, we utilize timed place CPNs. To be exact,in this method, the deterministic time delay of a tokenthat reaches an output place of a transition after beinggenerated by firing the transition is specified by theexpression function of the output arc of the transition.When a token is put into a place by the occurrenceof a step, it remains unavailable for an amount of timeequal to the time delay of a token, after which the tokenbecomes available. The TCPN can be formally definedas

TCPN = (Σ, P, T, A, C, G, E, M0, t0),

where P = {p1, p2, · · · , pn} is the finite set of places,n > 0 is the total number of places; T = {t1, t2, · · · , tm}is the finite set of transitions, m > 0 is the total numberof transitions; P ∩ T = ∅; A is the finite set of arcs,and the elements of A are denoted by the expression“pi − tj” or “tj − pi”, i ∈ n�j ∈ m; Σ is the finiteset of colors, Σ(pi) = {ai,1, ai,2, · · · , ai,ui}, Σ(tj) ={bj,1, bj,2, · · · , bj,vj}, ui = |Σ(pi)| denotes the numberof colors about pi(i ∈ n), vj = |Σ(tj)| denotes thenumber of colors about tj(j ∈ m); C : P → Σ is a setof color functions that map places to the set of colors,they specify attributes of tokens in a specific place; Gis a set of guard functions defined from T into Booleanfunctions, that is, their evaluations are either True orFalse, they prevent tokens of specific colors from flowingthrough a transition; E is a set of input and outputarc expression functions, they are defined from A intocolors of places and time delays that represent the timeduration of activities; M0 is a set of initial markings ofplaces, they are defined from places into colors of placesand time stamps; t0 is the starting model time.

The marking of a place (say p) denoted by M(p), iscomposed of a set of tokens with their individual timestamps. It represents the realized states among thepossible states specified by colors of place p. It may bechanged by firing an enabled transition that the placep as an input or output place.

2 Resource-constrained Multiple Pro-ject Scheduling Model

2.1 Problem DefinitionThe classical RCMPSP with precedence consists of

the activity shop, flow shop, assembly line balancing,and related scheduling problem, so the object problemstudied in this paper is described as follows.

(1) The RCMPSP consists of multiple projects whichhave a number of activities with known processing timeand multiple resources.

(2) The RCMPSP must be finished without changingthe project, when once initiated in a specific project.

(3) The start time of each activity is dependent uponthe completion of some other activities. After finishing

a specific activity, next activity must be also started ina project.

(4) The multiple resources are available in limitedquantities but renewable from period to period.

(5) Activities cannot be interrupted, it means thatthere is only one execution mode for each activity.

(6) The optimal objective is to minimize the totalproject time for all projects.2.2 Mathematical Model of RCMPSP

Mathematical model:

min tf =N∑

i=1

witfiM , (1)

∑

i∈Af

∑

j∈Af

lijr � br, r ∈ R, (2)

tsi(j−1) + tpi(j−1) � tsij , ∀i, j, (3)

where N is total number of multiple projects; M isthe maximum number of activities in each project; Ris the total number of resource type; tpij is the pro-cessing time of activity j in project i; tfij is the finishtime of activity j in project i; tsij is the start timeof activity j in project i; lijr is the scheduling activ-ity j in project i consumes resource units per periodfrom resource r; br is the maximum-limited resourcer only available with the constant period availability;Af is the set of activities being in progress in periodf , Af = {j|tsij � f < tsij + tpij ; i = 1, 2, · · · , N ;j = 1, 2, · · · , M}; wi is the weight of project i.

Equation (1) minimizes the total project time thatis the sum of the product of the completion times andthe weight which indicates the priority of each project.Equation (2) correspond to resource constrains regard-ing nonrenewable resources. Lastly, in Eq. (3), activityj in each project must not be started before all activi-ties of precedence j − 1 are finished.

3 Mapping Mechanism Between Net-work Diagram and Petri nets

Kim et al[8] studied a kind of mapping mechanismbetween CPM and Petri nets based on basic Petri netsin single project environment. Reddy et al[5] proposedanother mapping mechanism between PERT and ba-sic Petri nets in single project situation. The commonground on these two researches is that activities in thenetwork diagram were mapped into timed transitionsand they only discussed the mapping mechanism in sin-gle project environment. However, there is a crypticshortcoming of these two mechanisms. That is to say,after a certain transition which is associated with thetime parameter is fired, tokens will be removed fromthe input place of this transition immediately, but to-kens cannot be deposited into the output place of thistransition correspondingly, because tokens must keep

716 J. Shanghai Jiaotong Univ. (Sci.), 2009, 14(6): 713-719

unavailable until the time interval associated with thetransition passed. Therefore, in this time interval, thestate of system markings is unknown and the additionaltracing mechanism must be established to track themovement of markings, which makes it difficult to an-alyze the system dynamic performance.

In this study, a novel mapping mechanism is pro-posed to solve this problem. To be exact, an activity ina project is converted into a timed place, and an eventis converted into an instant transition, so that the to-ken tracing mechanism can be omitted. Otherwise anetwork diagram of a project just maps one Petri netin single project environment, but for multiple projectswhich always consists of hundreds of projects whichexecuted concurrently, so it will be extremely compli-cated and time-consumed to convert network diagramsof projects into Petri nets one by one, and the resultPetri net model is always cumbersome and big graph-ical sized. Therefore, the concept of colors is adoptedin this paper to convert network diagrams of multipleprojects into a kind of TCPN in which the differentproject will be represented by different colors, so thatthe complexity and modeling time can be reduced con-sumedly.3.1 Network Diagram Definition in Multi-

project EnvironmentThe primary elements of the network diagram in

multi-projects environment are defined as Fig. 1(a),where Ee,i denotes event i in project e (e =1, 2, · · · , N ′; i = 1, 2, · · · , M ′), where N ′ is totalnumber of projects and M ′ is the total events inproject e; Ak (Eu, Ev) denotes the activity connectedby event Ek,u and Ek,v in project k, Ak (Eu, Ev) =A(Ek,u, Ek,v); A′

k(Eu, Ew) denotes the dummy activ-ity.3.2 Mapping Mechanism

In this section, the mapping mechanism between net-work diagrams and TCPN which time parameter asso-ciated with the transition is discussed in detail, and themapped TCPN is defined as primary TCPN (PTCPN).Figure 1 shows the contrast of network diagram andPTCPN.

Ek,u Ek,v tu tv

tw

p(u, v)

p(u, w)Ek,w

Ak(Eu, Ev)

(a) Network diagram (b) PTCPN

Fig. 1 Conversion of a network diagram to a PTCPN

The PTCPN is generated as follows.(1) An event Ee,i will be mapped to an instant

transition ti (that is a transition without time delay,

where i = 1, 2, · · · , M ′). For example, the mould or-ders during mould manufacturing process can be re-gard as multiple projects and the network diagrams ofthose projects have the same topological structure, thatis to say, the number of activities in each project issame. So the color of transition ti is the project num-ber e(e = 1, 2, · · · , N ′) and the colors should be sort as-cending according to the project number. For instance,E1,i and E4,i are mapped to ti simultaneously, so thecolor of it denotes as C(ti) = {1, 4}.

(2) An activity Ak (Eu, Ev) will be mapped to a col-ored place p (Eu, Ev), of which the colors also can bethe state of activities, resource constraint, and etc be-sides the project number e.

(3) The activity time constraint will be mapped tothe input arc expression function of place p, which canbe denoted as e@ + tm(e), where e is the project num-ber and tm(e) is the time constraint of this activity inproject e.

(4) The mapping mechanism of dummy activityA′

k (Eu, Ew) is the same as activity, but there is no timeconstraint in its relative input arc expression function.

(5) The arrow direction of PTCPN is consistent withit in the network diagram.

(6) ps and pf will be added at the start and end of thePTCPN to represent the start and final of the model,and C (ps) and C (pf) are also the project numbers.

Deduction 1 The early start time (EST) of activ-ity Ak (Eu, Ev) is the time when a token is depositedto the place p (Eu, Ev) of PTCPN.

Deduction 2 The late finish time (LFT) of activityAk (Eu, Ev) is the time when a token is removed fromplace p (Eu, Ev).3.3 Resource-constrained PTCPN

The resource-constrained PTCPN which is named asRC PTCPN is defined as follows based on PTCPN.

Although there can be more detailed types of re-sources, at the top level of the hierarchy we classify theresources into two categories: consumable and reusable.In Figs. 2(a) and 3(a) represent the single and multipleconsumable resource constraint, and Figs. 2(b) and 3(b)represent the single and multiple reusable resource con-straint, respectively. Resources can be shared by differ-ent activities in different projects or them in a same

Mr

p(u,v)

(a) Single reusable resource (b) Multiple reusable resourcep(u,v)

nr nrtv tu tv

nr1

nr2nr2nr1

Mr2

Mr1

tu

Fig. 2 Reusable resource constraint in multiple projectsenvironment

J. Shanghai Jiaotong Univ. (Sci.), 2009, 14(6): 713-719 717

(a) Single consumable resource (b) Multiple consumable resource

p(u,v)

p(u,v)

Mcnc

tu tv

nc2nc1 Mc1

Mc2

tvtu

Fig. 3 Consumable resource constraint in multiple projectsenvironment

project. Where Mr represents the reusable resource,Mc represents the consumable resource, p(u, v) denotesthe colored place, ti represents the time constraint ofith event, nr represents the number of reusable resourceand nc represents the number of consumable resource.

4 Case Study

This case study concerns a 3-projects concurrentscheduling problem for mould manufacturing process.Because the design and manufacturing processes ofmould projects are homology, the topology structuresof them are proofed to be the same. In this case study,a mould manufacturing example studied in Ref. [3] isemployed to validate the effectiveness of the proposedmapping mechanism. To be exact, the basic idea of theproposed approach is to map the given network dia-gram to the PTCPN by the proposed mapping mecha-nism firstly, then use the simulation ability of TCPN tosimulate the scheduling process and find the partiallyoptimal schedule from the simulation result. Figure4 shows the network diagram of this multiple mouldprojects.

1 2

5

83 6 9 12

74

10 111 2

4

10 13

3

6

8

5

7 11

129

14 15 16

Fig. 4 Network diagram of mould projects schedulingproblem

In Fig. 4, there are 16 activities and 12 events ineach mould project, and the dashed arrow indicates thedummy activity. There are 9 different resources sharedby them, which the details are listed in Tables 1 and 2.The resource demand (RD) and the planning date (T)without resource constraint of each activity are listedin Table 3. The weights of each project are w1 = 0.40,w2 = 0.28, w3 = 0.32. Suppose that the initial startdates of each project are identical, and the finish dateswithout resource constraint of each project are 33, 38,35 days.

According to the proposed mapping mechanism, thenetwork diagram of RCMPSP for mould manufactur-

ing was mapped to PTCPN. In Fig. 5, the places weredivided into 3 types and the transitions were dividedinto 2 types. They are listed as follows.

(1) P = Pa ∪ Pr ∪ Pe, where Pa is the finite set ofactivity places, Pr is the set of resource places, and Pe

Table 1 The type and gross of resources for mouldmanufacturing process

Resource ID Resource name Gross

1 Designers 10

2 Buyers 6

3 Processing technicians 4

4 Programmers 5

5 Heat processing center 8

6 Electrical machining 14

7 NC machining 18

8 Assembly 7

9 Die trying 5

Table 2 Matching relation of activities andresources

ActivityID

Start (Stop)events

Activityname

Neededresource type

1 1(2) Conceptdesigning

1

2 2(3) Die structuredesigning

1

3 3(4) NCprogramming

4

4 3(5) Detaildesigning

1

5 3(6) Processdesigning

3

6 3(10) Die carrier producingor buying

2

7 5(6) CAE 1

8 5(10) Standard partspurchasing

2

9 6(7) Moving and soliddie NC rough

machining

7

10 6(8) Electric poleprocessing

6

11 6(10) Other partsmanufacturing

7

12 7(8) Heatprocessing

5

13 8(9) Moving and soliddie electricalmachining

6

14 9(10) Moving and soliddie NC finemachining

7

15 10(11) Die assembly 8

16 11(12) Die trying 9

718 J. Shanghai Jiaotong Univ. (Sci.), 2009, 14(6): 713-719

is set of status places.(2) Pa = {Ps, P 1, P 2, · · · , P 16, Pf}, Pr =

{R 1, R 2, · · · , R 9}, Pe = {S1, S2, · · · , S9, Sa, Sb, Sc,Sd, Se, Sf, Sg, Sh, Si, Sj, Sk, Sl, Sm, Sn, So, Sp, Sq, Sr,Ss, St}.

(3) T = Te ∪ Tu, where Te is the set of events transi-tion, and Tu is the set of assistant transitions.

Table 3 Resource and planning date constraint ofactivities

ActivityProject 1 Project 2 Project 3

RD T RD T RD T

1 24 4 40 5 40 4

2 24 3 42 6 48 6

3 8 2 10 2 4 1

4 40 4 27 3 28 4

5 6 3 6 2 12 3

6 6 2 24 6 30 6

7 20 4 36 6 27 3

8 4 1 12 3 6 2

9 20 2 14 1 11 1

10 36 4 22 2 16 2

11 36 3 27 3 30 3

12 28 4 20 4 21 3

13 39 3 20 2 48 4

14 32 2 52 4 28 2

15 18 3 15 3 24 4

16 12 4 16 4 20 4

(4) Te = {T1, T2, · · · , T12}; Tu = { Ta, Tb, Tc, Td,Te, Tf, Tg, Th, Ti, Tj, Tk, Tl, Tm, Tn, To, Tp, Tq,Tr, Ts, Tt, Tu, Tv, Tw, Tx}.

In the experiment, the target PTCPN was simulatedfor 100 times randomly. Actually we can use the reach-able graph of PTCPN to find the general optimiza-tion, but it may consume more CPU time. Therefore,in order to find a middle course with an acceptableCPU time, we employ the partial reachable graph tosolve this problem. The simulating result of resource-constrained finish dates (in Table 4) which are partiallyoptimal are 43, 63, 57 days. The comparison of thescheduling result between the proposed approach andthe method proposed by Liao et al[3] is listed in Table5.

According to Tables 4 and 5, the conclusion is drawnthat the proposed approach based on the mappingmechanism obtained a better result than laser quantumwell intermixing (LQWI) method, which optimized the

Table 4 Result comparison

Approach Project 1 Project 2 Project 3Object function

value/d

PTCPN 43 63 57 53.08

LQWIheuristic

59 65 69 63.72

S3 S7 Sa Sb

Se

P_4 P_7TS4 TS5 Ta

S4 S8P_5TS6S2 T3 TS7 Ta

Sf SrP_11

R_7 R_5

R_6

TiSd T6 Tn Tb

Sh SiP_9Tk Tm P_12 TdT7

Sg Sj SiP_10Tj Ti Tp Tr

P_13 P_14So Sp StT9Sn TsT8 Tt Tu Tx

S5 P_d

P_6

P_3

R_3

R_4

R_2

TS3

TS5S6 S9TS8

T4 Tb

Ty

T5 Tc

Th

Te

Sc SqP_8Td Tf Tvs@+(case s of(O_1, 1)=>4|(O_2, 1)=>3|(O_3, 1)=>4)

t@+(case t of(O_1, 2)=>3|(O_2, 2)=>2|(O_3, 2)=>3)

r@+(case r of(O_1, 3)=>2|(O_2, 3)=>2|(O_3, 3)=>1)

w@+(case w of(O_1, 4)=>2|(O_2, 4)=>6|(O_3, 4)=>6)

case w of(O_1, 4)=>6’e|(O_2, 4)=>24’e|(O_3, 4)=>30’e

case w of(O_1, 4)=>6’e|(O_2, 4)=>24’e|(O_3, 4)=>30’e

case t of(O_1, 2)=>32’e|(O_2, 2)=>52’e|(O_3, 2)=>28’e

case t of(O_1, 2)=>32’e|(O_2, 2)=>52’e|(O_3, 2)=>28’e

case t of(O_1, 2)=>36’e|(O_2, 2)=>22’e|(O_3, 2)=>16’e

w@+0x@+0

P

case t of(O_1, 2)=>6’e|(O_2, 2)=>6’e|(O_3, 2)=>12’e)

case r of(O_1, 3)=>8’e|(O_2, 3)=>10’e|(O_3, 3)=>4’e

case r of(O_1, 3)=>8’e|(O_2, 3)=>10’e|(O_3, 3)=>4’e

case z of(O_1, 5)=>20’e|(O_2, 5)=>14’e|(O_3, 5)=>11’e

case r of(O_1, 3)=>4’e|(O_2, 3)=>12’e|(O_3, 3)=>16’e)

s@+(case s of(O_1, 1)=>4|(O_2, 1)=>6|(O_3, 1)=>3)

r@+(case r of(O_1, 3)=>1|(O_2, 3)=>3|(O_3, 3)=>2)

case r of(O_1, 3)=>4’e|(O_2, 3)=>12’e|(O_3, 3)=>16’e

case s of(O_1, 1)=>20’e|(O_2, 1)=>36’e|(O_3, 1)=>27’e

s@+(case s of(O_1, 1)=>3|(O_2, 1)=>3|(O_3, 1)=>3)

case s of(O_1, 1)=>36’e|(O_2, 1)=>27’e|(O_3, 1)=>30’e case z of

(O_1, 5)=>28’e|(O_2, 5)=>20’e|(O_3, 5)=>21’e

case z of(O_1, 5)=>28’e|(O_2, 5)=>20’e|(O_3, 5)=>21’ecase z of

(O_1, 5)=>20’e|(O_2, 5)=>14’e|(O_3, 5)=>11’e

case s of(O_1, 1)=>36’e|(O_2, 1)=>27’e|(O_3, 1)=>30’e

z@+(case z of(O_1, 5)=>2|(O_2, 5)=>1|(O_3, 5)=>1

t@+(case t(O_1, 2)=>4|(O_2, 2)=>2|(O_3, 2)=>4)

z@+(case z of(O_1, 5)=>4|(O_2, 5)=>4|(O_3, 5)=>3)case t of

(O_1, 2)=>36’e|(O_2, 2)=>22’e|(O_3, 2)=>16’e

case s of(O_1, 1)=>20’e|(O_2, 1)=>36’e|(O_3, 1)=>27’e

case t of(O_1, 2)=>6’e|(O_2, 2)=>6’e|(O_3, 2)=>12’e

case s of(O_1, 1)=>40’e|(O_2, 1)=>27’e|(O_3, 1)=>28’e

case s of(O_1, 1)=>40’e|(O_2, 1)=>27’e|(O_3, 1)=>28’e

s s

PP P u

2’x (x,1)

(x,1)

(x,2)

(x,3)

(x,4)

12’e

10’e

E

P

P

(x,2)

P P P P U

t t t t (x,2)

P

E36

36’e

w w

10

12

E

4’x

U P P P

t t

(x,3)

P P

ss

P P P

r

s@+0

s@+0

(x,1)

(x,3)s@+0

s@+0

x@+0

t@+0

x@+0

x@+0

x@+0

s@+0 x@+0

z@+0

x@+0x@+0

t@+0

P

E

5656’e

t

z

P

7272’e

E

P

P P

t

z

(x,2)

(x,5) P

z@+0

32’e32

E

z

PP

z

(x,2)

2’x

(x,1)

(x,3)

(x,4)

PPP

r r r@+0

(x,1)

(x,1)

(x,5)

(x,2)

U P

P

P P

ss

Fig. 5 PTCPN of mould projects scheduling problem

J. Shanghai Jiaotong Univ. (Sci.), 2009, 14(6): 713-719 719

Table 5 Start and finish date of activities in 3projects

ActivityProject 1 Project 2 Project 3

ST FT ST FT ST FT

1 0 4 21 26 4 8

2 8 11 30 36 11 17

3 11 13 36 38 17 18

4 17 21 36 39 26 30

5 11 14 36 38 17 20

6 11 12 36 42 17 23

7 21 25 39 45 36 42

8 21 22 39 42 30 32

9 25 27 45 46 39 40

10 25 29 47 49 39 41

11 25 28 45 48 39 42

12 27 31 46 50 40 43

13 31 34 50 52 43 47

14 34 36 52 56 47 49

15 36 39 56 59 49 53

16 39 43 59 63 53 57

activity finish date of 3 projects effectively and mini-mized the total project time for all projects.

5 Conclusion

In this paper, Petri nets approach was introducedto solve the RCMPSP. In order to establish the Petrinet more quickly and easily, a novel mapping mech-anism between traditional network diagram such asCPM/PERT and timed CPNs was proposed, thenbased on this mapping mechanism, a PTCPN for solv-ing RCMPSP was modeled. At last, the object PTCPNcan be used to simulate the project scheduling and tofind the approximately optimal value of RCMPSP. Inthe case study, a mould manufacturing example wasexecuted to validate the proposed approach, and theexperiment results shows that applying Petri nets tosolve RCMPSP is not only feasible but also effective.But there are still issues for the author to research as fu-ture work such as using the reachable graph of PTCPNto find the general optimization, and how to reducingthe CPU time consuming more efficiently. Above all,this methodology provides a new approach to solve the

resource-constrained multiple project scheduling prob-lems.

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