Open access peer-reviewed chapter

Process Petri Nets with Time Stamps and Their Using in Project Management

By Ivo Martiník

Submitted: December 5th 2017Reviewed: March 26th 2018Published: September 19th 2018

DOI: 10.5772/intechopen.76769

Downloaded: 174

Abstract

Process Petri nets with time stamps (PPNTS) are the newly introduced class of low-level Petri nets, whose definition and the properties are the main topic of this chapter; they generalize the properties of Petri net processes in the area of design, modeling and verification of generally parallel systems with the discrete time. Property-preserving Petri net process algebras (PPPAs) were originally designed for the specification and verification of manufacturing systems. PPPA does not need to verify composition of Petri net processes because all their algebraic operators preserve the specified set of the properties. These original PPPAs are generalized for the class of the PPNTSs in this chapter. The new COMP, SYNC and JOIN algebraic operators are defined for the class of PPNTS and their chosen properties are proved. With the support of these operators, the PPNTSs can be extended also to the areas of project management and the determination of the project critical path with the support of the critical path method (CPM). The new CPNET subclass of PPNTS class is defined in this chapter. It is specially designed for the generalization of the CPM activity charts and their properties. This fact is then demonstrated on the simple project example and its critical path and other property specifications.

Keywords

  • process Petri nets with time stamps
  • property-preserving Petri net process algebras
  • critical path method
  • discrete time
  • property preservation
  • parallel systems modeling

1. Introduction

There are currently a number of formally defined classes of Petri nets [1, 2] available for modeling of generally parallel systems. When studying distributed parallel programming systems, real-time systems, economic systems and many other types of systems, it plays a role modeling of the time variables associated with individual system events, the duration of the studied activities, the time history of the modeled system and many other time characteristics. Special classes of Petri nets were introduced for the modeling of these types of systems with discrete time and their properties were studied in detail. Time Petri nets and timed Petri nets [3, 4] are currently the two most important classes of low-level Petri nets that use the concept of discrete time in their definition. Other classes of low-level Petri nets with discrete time are introduced and discussed for instance in [5, 6, 7]. It can be stated that most of the currently studied classes of Petri nets with discrete time use only the relative time variables usually related to the specific marking of the given Petri net. This fact can then cause difficulties, for example, in modeling complex time-synchronized distributed systems in which an external time source is usually available and individual components of this system must be synchronized with this external time source.

Process Petri nets (PPN) [8] were primarily introduced as a special subclass of classic low-level Petri nets for their using in the area of workflow management. PPN is a continuous Petri net that include within the set of all its places the unique input place, the unique output place and a finite set of so-called resource places which may contain, along with the input place, the tokens in the entry marking of the given PPN. These tokens located in the entry marking of PPN at the resource places usually represent the permanent resources of the modeled system. The given PPN can pass into its exit marking that is reachable from its entry marking by performing the final sequence of the transition firings. The tokens of the PPN’s exit marking may be then located only at its single output place and also at its resource places.

Process Petri nets with time stamps (PPNTS) are the newly introduced class of low-level Petri nets whose definition and the properties are the main topics of this chapter. PPNTS generalize the properties of PPNs in the area of design, modeling and verification of generally parallel systems with the discrete time.

Property-preserving Petri net process algebras (PPPA) [9] were originally designed for the specification and verification of manufacturing systems. PPPA has four types of operators: extensions, compositions, refinements and reductions. All operators can preserve about 20 PPN’s properties (some of them under additional conditions), such as liveness, boundedness, reversibility, RC-property, traps, siphons, proper termination, and so on. PPPA does not need to verify composition of PPNs because all their algebraic operators preserve the specified set of the properties. Hence, if the source PPNs satisfy the desirable properties, each of the composite PPN, including the PPN that models the resulting system itself, also satisfies these properties. These original PPPA are generalized for the class of the PPNTS in this chapter and their properties of proper-formed, well-formed and pure-formed PPNTS are then newly introduced. The new COMP, SYNC and JOIN algebraic operators are defined for the class of PPNTS and their chosen properties are proved.

With the support of these operators, the PPNTS can be extended also to the areas of the project management and the determination of the project critical path with the support of the critical path method (CPM) [10]. The new CPPNET subclass of PPNTS class is then defined in this chapter to represent pure-formed time-dependent processes. It is specially designed for the generalization of the CPM activities charts and their properties. This fact is then demonstrated on the simple project example and its critical path and other properties specification.

This chapter is arranged into the following sections: Section 2 explains the base term of this chapter, that is, process Petri nets with time stamps and introduces the terms of proper-formed, well-formed and pure-formed PPNTS; Section 3 discusses algebraic operators COMP and SYNC defined over the class of PPNTS and their main properties; Section 4 then introduces the special subclass CPPNET of the PPNTS class and explains its use in the area of the project management to represent pure-formed time-dependent processes with using of PPNTSs and to find critical paths for these processes similarly as in the case of the well-known critical path method (CPM). Finally, Section 5 gives the conclusions of the research to conclude the chapter.

2. Process Petri nets with time stamps and their properties

Let N denote the set of all natural numbers, N := {1, 2, …}; N0 the set of all non-negative integer numbers, N0 := {0, 1, 2, …}; ∅ the empty set; |A| the cardinality of the given set A; P(A) denotes the family of all the subsets of the given set A; f: AB a function on a domain A to a codomain B; ⌐ the logical negation operator. Let (AN0) ∧ (∃nN: |A| = n) ∧ (A ≠ ∅); then max(A) := x, where (xA) ∧ (∀yA: x ≥ y). Multiset M over a nonempty set S is a function M: SN0. The non-negative number M(a) ∈ N0, where aS, denotes the number of occurrences of the element a in the multiset M. The multiset M over a nonempty set S will be represented by the notation M := [aM(a), bM(b), cM(c), …] = [a, …, a, b, …, b, c, …, c, …], where S := {a, b, c, …}. Notation SMS then denotes the class of all the multisets over the set S.

Definition 1. Let A be a nonempty set. By the (nonempty finite) sequence σ over the set A we understand a function σ: {1, 2, …, n} → A, where nN. Function ε: ∅ → A is called the empty sequence on the set A. We usually represent the sequence σ: {1, 2, …, n} → A by the notation σ = <a1, a2, …, an > of the elements of the set A, where ai = σ(i) for 1 ≤ i ≤ n. Empty sequence ε: ∅ → A on the set A we usually represent by the notation ε = <>. We denote the set of all finite (and possible empty) sequences over the set A by the notation ASQ.

If σ = <a1, a2, …, an > and τ = <b1, b2, …, bm > are the finite sequences, where σASQ, τASQ, nN, mN, then by the concatenation of the sequences σ and τ, denoted by σ++τ, we understand the finite sequence σ++τ := < a1, a2, …, an, b1, b2, …, bm>. The following functions are defined:

  1. length: ASQN0, so that: length(σ) := n, length(ε) := 0,

  2. elements: ASQP(A), so that: elements(σ) := {a | ∃i, 1 ≤ i ≤ n: a = σ(i)}, elements(ε) := ∅,

  3. prefix: ASQ × N0ASQ, so that:

    prefix(<a1, a2, …, an>, m) := < a1, a2, …, am>, if m ≤ n,

    prefix(<a1, a2, …, an>, m) := < a1, a2, …, an>, if m > n,

    prefix(ε, m) := ε,

  4. suffix: ASQ × N0ASQ, so that:

    suffix(<a1, a2, …, an>, m) := < am + 1, am + 2, …, an>, if m < n,

    suffix(<a1, a2, …, an>, m) := ε, if m ≥ n,

    suffix(ε, m) := ε,

  5. create: N × AASQ, so that: create(n, a) := < a, a, …, a>, where length(<a, a, …, a>) = n,

  6. sort: (N0)SQ → (N0)SQ, so that: sort(σ) := ρ,

where (ρ = <b1, b2, …, bn>) ∧ (b1 ≤ b2 ≤ … ≤ bn) ∧ ([a1, a2, …, an] = [b1, b2, …, bn]).

We use the following subsets of the set (N0)SQ:

  • N# := {σ ∈ (N0)SQ | (σ = ε) ∨ ((σ = <a1, a2, …, an>) ∧ (a1 ≤ a2 ≤ … ≤ an)), nN},

  • N0 := {σ ∈ (N0)SQ | (σ = ε) ∨ ((σ = <0, 0, …, 0>) ∧ (length(σ) = n)), nN}.

Thus, the elements of the set N# constitute the empty sequence ε and all the finite ascending ordered sequences σ consisting of non-negative integer numbers. Similarly, the elements of the set N0 then form an empty sequence ε and all the sequences in the form <0, 0, …, 0 > of any finite length.

Definition 2. Net NET is an ordered triple NET := (P, T, A), where P is finite nonempty set of places, T is finite set of transitions, PT = ∅, and A is finite set of arcs, A ⊆ (P × T) ∪ (T × P). □

The given net NET is then described with a bipartite graph containing a finite nonempty set P of places used for expressing of the conditions of a modeled process (we usually use circles for their representation), a finite set T of transitions describing the changes in the modeled process (we usually draw them in the form of rectangles) and a finite set A of arcs being principally oriented while connecting the place with the transition or the transition with the place and we usually draw them as lines with arrows.

Some commonly used notations for the nets are •y = {x | (x, y) ∈ A} for the preset and y• = {x | (y, x) ∈ A} for the postset of a net node y (i.e., place or transition). A path of a net NET := (P, T, A) is a nonempty sequence <x1, …, xk> of net nodes, where kN, which satisfies (x1, x2), (x2, x3), …, (xk-1, xk) ∈ A. A path of the net NET := (P, T, A) leading from its node x to its node y is a circuit if (y, x) ∈ A. We denote the set of all the circuits of the net NET by CIRCUITSNET. Net NET’ := (P′, T’, A’) is a subnet of the net NET := (P, T, A) if (P′P) ∧ (T’T) ∧ (A’ = A ∩ ((P′ × T’) ∪ (T’ × P′))). Net NET is connected if and only if it is not composed of two or more disjoint and nonempty subnets.

Definition 3. Process net with time stamps (PNTS) PNTS is an ordered tuple PNTS := (P, T, A, AF, TP, TI, IP, OP, RP), where

  1. (P, T, A) is the connected net, ∀tT: (•t ≠ ∅) ∧ (t• ≠ ∅),

  2. AF: (P × T) ∪ (T × P) → N0 is the arc function,

    AFxy>0xyA,AFxy=0xyA,wherex,yPT,

  3. TP is the transition priority function, TP: TN,

  4. TI: (T × P) → N0 is the time interval function,

  5. IP is the input place, (IPP) ∧ (•IP = ∅) ∧ (∀p ∈ (P \ ({IP} ∪ RP)): •p ≠ ∅),

  6. OP is the output place, (OPP) ∧ (OP• = ∅) ∧ (∀p ∈ (P \ ({OP} ∪ RP)): p• ≠ ∅),

  7. RP is the set of resource places, (RP ⊆ (P \ {IP, OP})) ∧ (∀tT: ⌐(•tRP)).

The class of all the PNTS will be denoted by PNTS. □

The given PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is represented by the connected net (P, T, A) with nonempty preset and postset of each of its transitions t; the arc function AF assigning each arc with a natural number (such number has the default value of 1, if not explicitly indicated in the PNTS diagram) expressing the number of removed or added tokens from or to the place associated with that arc when firing a particular transition; transition priority function TP assigns with each transition the natural number value expressing its priority (with the default value of 1); the time interval function TI assigns to each arc of the type (transition, place) a non-negative integer d expressing the minimum time interval during which the token has to remain in the place instead of being able to participate in the next firing of some transition and it thus determines the so-called time marking of the given PNTS (the value d associated with the respective arc is given in the format +d in the PNTS diagram); the input place IP is the only one nonresource place of PNTS PNTS with no input arc(s); the output place OP is the only one nonresource place of PNTS PNTS with no output arc(s); the finite set RP of resource places is used for expressing conditions of a modeled process containing some initial resources and we use circles with the double line for their representation.

Definition 4. Let PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) be the PNTS. Then:

  1. marking M of the PNTS PNTS is a function M: PN0,

  2. time marking m of the PNTS PNTS is a function m: PN#,

    where ∀pP: |M(p)| = length(m(p)),

  3. variable τN0 is the net time of the PNTS PNTS,

  4. state S of the PNTS PNTS is an ordered triple S := (M, m, τ),

  5. transition tT is enabled in the state S := (M, m, τ) of the PNTS PNTS that is denoted by t en S, if ∀p ∈ •t: (M(p) ≥ AF(p, t)) ∧ (∀nelements(prefix(m(p), AF(p, t))): n ≤ τ)),

  6. firing of the transition tT results in changing the state S := (M, m, τ) of the PNTS PNTS into its state S′ := (M’, m’, τ) that is denoted by S [tS′, where ∀pP:

    • M’(p) := M(p) − AF(p, t) + AF(t, p),

    • m’(p) := sort(suffix(m(p), AF(p, t))++create(τ + TI(t, p), AF(t, p))),

  7. elapsing of time interval δN results in changing the state S := (M, m, τ) of PNTS PNTS into its state S′ := (M, m, τ + δ), where ∀tT: ⌐(t en (M, m, τ)), that is denoted by S [δS′, so that:

    tTnN1n<δ:tenMmτ+ntT:tenMmτ+δ,

  8. if the transitions t1, t2, …, tnT are enabled in the state S := (M, m, τ) of PNTS PNTS (i.e., (t1 en S) ∧ (t2 en S) ∧ … ∧ (tn en S)) we say that these transitions are enabled in parallel in the state S that is denoted by {t1, t2, …, tn} en S,

  9. finite nonempty sequence σ := t1 t2 tn of the transitions t1, t2, …, tnT for which the following is valid in the state S1 := (M1, m1, τ1) of PNTS PNTS:

    • (M1, m1, τ1) [t1⟩ (M2, m2, τ1) [t2⟩ … [tn⟩ (Mn + 1, mn + 1, τ1),

    • tT: ⌐(t en (Mn + 1, mn + 1, τ1)),

    is called step σ in the given state S1 of PNTS PNTS and it is denoted by.

    M1m1τ1σMn+1mn+1τ1,

  10. finite nonempty sequence ρ of steps and time intervals elapsing that represents the following finite sequence

    M1m1τ1σ1M2m2τ1δ1Mn+1mn+1τnδnMn+1mn+1τn+1

    of the state changes of PNTS PNTS is the sequence ρ := σ1 δ1 σ2 δ2 σn δn of steps σ1, σ2, , σn and time intervals elapsing δ1, δ2, , δn,

  11. we say the state S′ of PNTS PNTS is reachable from its state S if there exists the finite sequence ρ := σ1 δ1 σ2 δ2 σn δn of steps σ1, σ2, , σn and time intervals elapsing δ1, δ2, , δn such that S [σ1 δ1 σ2 δ2 σn δnS′; the set of all the reachable states of PNTS PNTS from its state S is denoted by [S⟩; the set of all the finite sequences ρ := σ1 δ1 σ2 δ2 σn δn associated with all the reachable states S′ ∈ [S⟩ is denoted by [S⟩⟩, that is,

    S{σ1δ1σ2δ2σnδnS[S:S[σ1δ1σ2δ2σnδnS,nN},

  12. the set of all the states S := (M, m, τ) of PNTS PNTS is denoted by S,

  13. the set of all the markings M associated with the set S of all the states of PNTS PNTS is denoted by M, that is, M := {M | (S = (M, m, τ)) ∧ (SS)},

  14. static state Ss := (Ms, ms, τs) of PNTS PNTS is every of its states where

    pP\RP:Msp=0msp=<>,

  15. the set of all the static states Ss := (Ms, ms, τs) of PNTS PNTS is denoted by Ss,

  16. the set of all the static markings Ms associated with the set Ss of all the static states of PNTS PNTS is denoted by Ms, that is, Ms := {Ms | (Ss := (Ms, ms, τs)) ∧ (SsSs)},

  17. the function ξ: MMs which assigns to each marking MM of a given PNTS PNTS the associated static marking MsMs is defined as follows:

    • ∀ p ∈ RP: ξ(M(p)) := M(p),

    • ∀ p ∈ P \ RP: ξ(M(p)) := 0,

  18. entry state Se := (Me, me, τe) of PNTS PNTS is every of its states where

    • ∃k ∈ N: (Me(IP) = k) ∧ (length(me(IP)) = k),

    • ∀ p ∈ P \ (RP ∪ {IP}): (Me(p) = 0) ∧ (me(p) = <>),

    • ∀ p ∈ RP: (Me(p) ≥ 0) ∧ (me(p) ∈ N0),

  19. the set of all the entry states Se := (Me, me, τe) of PNTS PNTS is denoted by Se,

  20. exit state Sx := (Mx, mx, τx) of PNTS PNTS that is reachable from its entry state Se := (Me, me, τe) is every of its states where

    • Sx ∈ [Se⟩,

    • Mx(OP) = Me(IP),

    • ∀ p ∈ P \ (RP ∪ {OP}): (Me(p) = 0) ∧ (me(p) = <>),

  21. the set of all the exit states Sx := (Me, me, τe) of PNTS PNTS that are reachable from its entry state Se := (Me, me, τe) is denoted by [Sex,

  22. the set of all the exit states Sx of PNTS PNTS that are reachable from all its entry states SeSe is denoted by Sx. □

The above established concepts are demonstrated in a simple example of the PNTS PNTS1 := (P, T, A, AF, TP, TI, IP, OP, RP) that is shown in Figure 1, where P := {IP, P1, R1, OP}, T := {T1, T2, T3}, A := {(IP, T1), (IP, T2), (T1, P1), (T2, P1), (R1, T1), (P1, T3), (T3, R1), (T3, OP)}, AF := {((IP, T1), 1), ((IP, T2), 1), ((T1, P1), 1), ((T2, P1), 2), ((R1, T1), 1), ((P1, T3), 1), ((T3, R1), 1), ((T3, OP), 1)}, TP := {(T1, 2), (T2, 1), (T3, 1)}, TI := {((T1, P1), 3), ((T2, P1), 3), ((T3, R1), 1), ((T3, OP), 4)}, IP := IP, OP := OP, RP := {R1}.

Figure 1.

Firing of transition T1 in PNTS PNTS1.

PNTS PNTS1 is in its entry state Se := (Me, me, τe), where marking Me := (Me(IP), Me(P1), Me(R1), Me(OP)) = (2, 0, 2, 0), time marking me := (me(IP), me(P1), me(R1), me(OP)) = (<0, 2>, <>, <0, 0>, <>) and net time τe = 0 (i.e., τe = τ). Static marking MsMs associated with the entry marking Me (see (xiv) and (xvii) of Definition 4) has the value Ms := ξ(Me) = (0, 0, 2, 0).

Time marking m of any PNTS expresses the current time state of the modeled system using the final (or empty) ascending ordered sequences of non-negative integers (i.e., elements of the set N#) associated with each of its places. The individual values of the time marking m associated with the arbitrary place p of the given PNTS in its state S, informally said, represent the values of the net time τ at which the respective token can first participate in the firing of selected enabled transition t of the given PNTS.

The transitions T1 and T2 are enabled in the entry state Se because (see (v) of Definition 4):

  • p ∈ •T1: (2 = M(IP) ≥ AF(IP, T1) = 1) ∧ (2 = M(R1) ≥ AF(R1, T1) = 1) ∧ (∀nelements(prefix(m(IP), AF(IP, T1))) = elements(prefix(<0, 2>, 1)) = elements(<0>) = {0}: 0 ≤ 0) ∧ (∀nelements(prefix(m(R1), AF(R1, T1))) = elements(prefix(<0, 0>, 1)) = elements(<0>) = {0}: 0 ≤ 0),

  • p ∈ •T2: (2 = M(IP) ≥ AF(IP, T1) = 1) ∧ (∀nelements(prefix(m(IP), AF(IP, T2))) = elements(prefix(<0, 2>, 1)) = elements(<0>) = {0}: 0 ≤ 0).

When enabling individual transitions of the given PNTS so-called conflicts can originate in its certain markings (or conflict transitions). At the enabling of the transitions t1 and t2 of the given PNTS in its state S the conflict occurs, if both transitions t1 and t2 have at least one input place, each of the transitions t1 and t2 is individually enabled in the state S, but the transitions t1 and t2 are not enabled in parallel in the state S (see (viii) of Definition 4) and enabling of one of them will prevent enabling of the other (i.e., (•t1 ∩ •t2 ≠ ∅) ∧ (t1 en S) ∧ (t2 en S) ∧ ⌐({t1, t2} en S)). The term of conflict transitions can be obviously easily generalized for the case of a finite set t1, t2, …, tn, nN of the transitions of the given PNTS.

The transitions T1 and T2 in the entry state Se of PNTS PNTS1 are conflict transitions because the time marking me(IP) = <0, 2 > (i.e., only one token of the entry marking Me(IP) may participate in the firing of the transition T1 or T2 in the net time τe = 0). When solving such transitions conflict we therefore follow the rule which determines, informally said, that from the set of conflict transitions the one will be enabled, whose value of the transition priority function TP is the highest. If such transition from the set of conflict transitions does not exist, the given conflict would have to be solved by other means. The transition T1 is then enabled in the entry state Se on the basis of that rule in our studied example (because TP(T1) = 2 and TP(T2) = 1).

Firing of the transition T1 changes the entry state Se := (Me, me, τe) of the PNTS PNTS1 into its state S1 := (M1, m1, τe) (i.e., Se [T1S1—see Figure 1), where (see (vi) of Definition 4):

  • M1(IP) := Me(IP) - AF(IP, T1) = 2 − 1 = 1,

  • m1(IP) := sort(suffix(me(IP), AF(IP, T1))) = sort(suffix(<0, 2>, 1)) = sort(<2>) = <2>,

  • M1(P1) := Me(P1) + AF(T1, P1) = 0 + 1 = 1,

  • m1(P1) := sort(create(τ + TI(T1, P1), AF(T1, P1))) = sort(create(0 + 3, 1)) = sort(create(3, 1)) = sort(<3>) = <3>,

  • M1(R1) := Me(R1) - AF(R1, T1) = 2 − 1 = 1,

  • m1(R1) := sort(suffix(me(R1), AF(R1, T1))) = sort(suffix(<0, 0>, 1)) = sort(<0>) = <0> .

There is no enabled transition in the state S1 := (M1, m1, τ1) and it is necessary to perform the time interval elapsing with the value of δ = 2. This will change the state S1 := (M1, m1, τe) into the state S2 := (M1, m1, τ1), where τ1 := τe + δ = 2 (i.e., S1 [2⟩ S2). It can be easily shown that transition T1 in the state S2 is enabled and firing of this transition changes the state S2 := (M1, m1, τ1) of the PNTS PNTS1 into its state S3 := (M2, m2, τ1) (i.e., S2 [T1S3), where M2 := (M2(IP), M2(P1), M2(R1), M2(OP)) = (0, 2, 0, 0) and m2 := (m2(IP), m2(P1), m2(R1), m2(OP)) = (<>, <3, 5>, <>, <>).

It can then be easily verified that S3 [1⟩ S4 [T3S5 [2⟩ S6 [T3Sx, where:

  • S4 = (M2, m2, τ2) = ((0, 2, 0, 0), (<>, <3, 5>, <>, <>), 3),

  • S5 = (M3, m3, τ2) = ((0, 1, 1, 1), (<>, <5>, <4>, <7>), 3),

  • S6 = (M3, m3, τ3) = ((0, 1, 1, 1), (<>, <5>, <4>, <7>), 5),

  • Sx = (M4, m4, τ3) = ((0, 0, 2, 2), (<>, <>, <4, 6>, <7, 9>), 5).

There are no enabled transitions in the exit state Sx := (M4, m4, τ3) of PNTS PNTS1 that is reachable from the entry state Se := (Me, me, τe) (see (xx) of Definition 4) and there is also no time interval elapsing value δ in this state that enables any of the transitions.

The set AMsMs of all the allowed static markings of the given PNTS PNTS, informally said, expresses how many tokens may be located in its individual resource places if PNTS PNTS be in its (now no longer arbitrary) allowed entry state ASeASe, where ASeSe. For instance, the set AMs of the PNTS PNTS1 (see Figure 1) can be defined as AMs := {(0, 0, k, 0) | kN}, that is, there must be at least one token in the resource place R1 (and of course at least one token in the input place IP) in any allowed entry state ASeASe.

Definition 5. Let PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) be a PNTS, AMsMs be the set of all of its allowed static markings and ASe := {ASe | (ASe = (AMe, ame, 0)) ∧ (ξ(AMe) ∈ AMs)} be the set of all of its allowed entry states. Then:

  1. PNTS is k-bounded PNTS if

    ASeASekN0pPSASe,SMmτ:Mpk,

  2. PNTS is proper-formed PNTS if

    ASeASe:(S[ASeSx[ASex:Sx[S)(nN:[ASe=n),

  3. proper-formed PNTS is well-formed PNTS if

    ASeASeSxASex,SxMxmxτx:ξMxAMs,

  4. well-formed PNTS is pure-formed PNTS if

    ASeASeSxASex,SxMxmxτx:ξAMe=ξMx.

PNTS is proper-formed PNTS if for any of its state S that is reachable from any allowed entry state ASeASe there exists its output state Sx that is also reachable from its allowed entry state ASeASe such that the output state Sx is also reachable from the state S (i.e., ∀S ∈ [ASe⟩ ∃Sx ∈ [ASex: Sx ∈ [S⟩). Furthermore, the cardinality of the set [ASe⟩⟩ of all the sequences ρ := σ1 δ1 σ2 δ2 σn δn associated with all the reachable states S ∈ [ASe⟩⟩ must be finite (i.e., (∃nN: |[ASe⟩⟩| = n).

Proper-formed PNTS is well-formed PNTS if for any of its allowed entry state ASeASe and for any of its exit state Sx ∈ [ASex, where Sx := (Mx, mx, τx), it is true that the exit static marking ξ(Mx) of all its resource places is an element of the set AMs of all its allowed static markings if PNTS PNTS be in its allowed entry state ASeASe (i.e., ∀ASeASeSx ∈ [ASex, Sx := (Mx, mx, τx): ξ(Mx) ∈ AMs).

Well-formed PNTS is pure-formed PNTS if for any of its allowed entry state ASeASe, where ASe := (AMe, ame, τe), and for any of its exit state Sx ∈ [ASex, where Sx := (Mx, mx, τx), it is true that the exit static marking ξ(Mx) of all its resource places is equal to the entry static marking ξ(AMe) of all its resource places that is associated with the allowed entry state ASe (i.e., ∀ASeASeSx ∈ [ASex, Sx := (Mx, mx, τx): ξ(AMe) = ξ(Mx)).

For instance, if the set AMs of the PNTS PNTS1 (see Figure 1) is defined as:

  1. AMs := {(0, 0, k, 0) | kN} (i.e., there must be at least one token in the resource place R1 in any allowed entry state ASeASe), then it can be shown that PNTS PNTS1 is k-bounded, proper-formed, well-formed and pure-formed PNTS,

  2. AMs := {(0, 0, 0, 0)} (i.e., there may not be any token in the resource place R1 in any allowed entry state ASeASe), then it can be shown that PNTS PNTS1 is k-bounded, proper-formed, but not well-formed or pure-formed PNTS (see for instance the sequence

    1000<0><><><>0T20200<><33><><>03.0200<><33><><>3T3 T30022<><><44><77>3,whereξMx=ξ0022=00200000=AMs).

Lemma 1. If PNTS is proper-formed PTNS then PNTS is k-bounded PNTS.

Proof. Clear. PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is a connected net that contains the finite set T of the transitions. Then the finite number of tokens will be added to each of the places pP by firing each of the transitions tT. The number of states S ∈ [ASe⟩ for any allowed entry state ASeASe must be also finite because PNTS is proper-formed PNTS (i.e., ∃nN: |[ASe⟩⟩| = n). From these facts then immediately follows that in any state S ∈ [ASe⟩ the finite number of tokens must be placed in any place pP, where any final number of tokens is placed in the input place IP in the entry state ASe. From these facts then immediately follows that ASeASekN0pPSASe,SMmτ:Mpk.

Definition 6. Process Petri net with time stamps (PPNTS) PPNTS is an ordered couple PPNTS := (PNTS, Se), where PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is the PNTS and SeSe is the entry state of PNTS PNTS. The class of all PPNTSs will be denoted by PPNTS.□

3. Algebraic operators COMP and SYNC and their properties

We study the issue of transforming PNTS through precisely defined binary operator COMP and n-ary operator SYNC over the class PNTS and we also examine the preservation of individual PNTS’s properties when applying each of these operators. Formal enrollment of an application of generally n-ary operator OP whose operands are the PNTS PNTS1, PNTS2, …, PNTSn (nN) and whose application requires the specification of values of k formal parameters (kN) par1, par2, … park, will be denoted by the expression.

PNTSPNTS1PNTS2PNTSn.OPpar1par2park,

where PNTS is the resulting PNTS.

Definition 7. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be the PNTSs. Let AMs1 := {(AMs1(IP1), AMs1(p11), …, AMs1(p1n), AMs1(r11), …, AMs1(r1m), AMs1(OP1)) | P1 := {p11, …, p1n, r11, …, r1m}, RP1 := {r11, …, r1m}, nN, mN} be the set of all the allowed static markings of PNTS1, AMs2 := {(AMs2(IP2), AMs2(p21), …, AMs2(p2k), AMs2(r21), …, AMs2(r2h), AMs2(OP2)) | P2 := {p21, …, p2k, r21, …, r2h}, RP2 := {r21, …, r2h}, kN, hN} be the set of all the allowed static markings of PNTS2.

Cartesian product AMs1AMs2 is then the following set:

AMs1AMs2AMs1IP1AMs1p11AMs1p1nAMs1r11AMs1r1mAMs1OP1,AMs2IP2AMs2p21AMs2p2kAMs2r21AMs2r2hAMs2OP2.AMs1IP1AMs1p11AMs1p1nAMs1r11AMs1r1mAMs1OP1AMs1.AMs2IP2AMs2p21AMs2p2kAMs2r21AMs2r2hAMs2OP2AMs2.

PNTS PNTS1 and PNTS2 are disjoint and we denote this fact by PNTS1PNTS2 if.

P1P2=T1T2=.

Definition 8. The function COMP: PNTS × PNTSPNTS of nets composition is defined as follows: if PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be the arbitrary PNTSs, PNTS1PNTS2, t be an arbitrary transition, where (tT1) ∧ (tT2), tiN0, then PNTS := [PNTS1, PNTS2].COMP(t, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:

  1. P := P1P2,

  2. T := T1T2 ∪ {t},

  3. A := A1A2 ∪ {(OP1, t), (t, IP2)},

  4. AF := AF1AF2 ∪ {((OP1, t), 1), ((t, IP2), 1)},

  5. TP := TP1TP2 ∪ {(t, 1)},

  6. TI := TI1TI2 ∪ {(t, IP2), ti)},

  7. IP := IP1,

  8. OP := OP2,

  9. RP := RP1RP2. □

Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti) can be seen in Figure 2.

Figure 2.

Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti).

Lemma 2. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be two arbitrary PNTS, PNTS1PNTS2, t be an arbitrary transition, (tT1) ∧ (tT2), tiN0, AMs1 and AMs2 be the sets of all the allowed static markings of PNTS1 and PNTS2. Let PNTS := [PNTS1, PNTS2].COMP(t, ti).

If PNTS1 and PNTS2 are proper-formed, resp. well-formed, resp. pure-formed, PNTS and AMs = AMs1AMs2 be the set of all the allowed static markings of PNTS PNTS, then also resulting PNTS is proper-formed, resp. well-formed, resp. pure-formed, PNTS.

Proof. Clear, it directly follows from Definition 5, Definition 7 and Definition 8. □

Definition 9. The function SYNC: PNTS × PNTS × … × PNTSPNTS of synchronous nets composition is defined as follows: if PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn), be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i ≠ jPNTSiPNTSj, where nN, pi and po be the arbitrary places, (piP1P2 ∪ … ∪ Pn) ∧ (poP1P2 ∪ … ∪ Pn) ∧ (pipo), ti and to be the arbitrary transitions, (tiT1T2 ∪ … ∪ Tn) ∧ (toT1T2 ∪ … ∪ Tn) ∧ (tito), af1 ∈ N, af2 ∈ N, …, afnN, ti1 ∈ N0, ti2 ∈ N0, …, tinN0, tioN0, then

PNTSPNTS1PNTS2PNTSn.SYNCpipotitoaf1afnti1tintio,

where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:

  1. P := P1P2 ∪ … ∪ Pn ∪ {pi, po},

  2. T := T1T2 ∪ … ∪ Tn ∪ {ti, to},

  3. A := A1A2 ∪ … ∪ An ∪ {(pi, ti), (ti, IP1), …, (ti, IPn), (OP1, to), …, (OPn, to), (to, po)},

  4. AF := AF1AF2 ∪ … ∪ AFn ∪ {((pi, ti), 1), ((ti, IP1), af1), …, ((ti, IPn), afn), ((OP1, to), af1), …, ((OPn, to), afn), ((to, po), 1)},

  5. TP := TP1TP2 ∪ … ∪ TPn ∪ {(ti, 1), (to, 1)},

  6. TI := TI1TI2 ∪ … ∪ TIn ∪ {((ti, IP1), ti1), …, ((ti, IPn), tin), ((to, po), tio)},

  7. IP := pi,

  8. OP := po,

  9. RP := RP1RP2 ∪ … ∪ RPn. □

Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio) can be seen in Figure 3.

Figure 3.

Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio).

Lemma 3. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn) be arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i ≠ jPNTSiPNTSj, where nN, pi and po be arbitrary places, (piP1P2 ∪ … ∪ Pn) ∧ (poP1P2 ∪ … ∪ Pn) ∧ (pipo), ti and to be arbitrary transitions, (tiT1T2 ∪ … ∪ Tn) ∧ (toT1T2 ∪ … ∪ Tn) ∧ (tito), af1 ∈ N, af2 ∈ N, …, afnN, ti1 ∈ N0, ti2 ∈ N0, …, tinN0, tioN0 and AMs1, AMs2, …, AMsn be the sets of all the allowed static markings of PNTS1, PNTS2, …, PNTSn. Let PNTS := [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio).

If PNTS1, PNTS2, …, PNTSn are proper-formed, resp. well-formed, resp. pure-formed, PNTS and AMs = AMs1AMs2 ⊗ … ⊗ AMsn is the set of all the allowed static markings of PNTS PNTS, then also PNTS is proper-formed, resp. well-formed, resp. pure-formed, PNTS.

Proof. Clear, it directly follows from Definition 5, Definition 7 and Definition 9. □

4. Process Petri nets with time stamps and their applications in project management area

Critical Path Method (CPM) is a method used in modeling and project management that was developed at the end of 1950s and that is commonly used for all the types of projects including software development [10]. The CPM is the most widely used method of so-called network analysis, even though it is designed to analyze the time consumption of only deterministic projects, that is, projects where the duration of each of their activities is exactly known, including all their sub-activities.

The basis for using CPM is to create a project model that includes:

  • the list of all activities needed to complete the project,

  • the time duration of each activity that is constant,

  • the dependencies between the project activities,

A critical path is then a designation for a sequence of activities whose time duration directly affects the time duration of the entire project. The activities that make up the critical path are then referred to as critical activities. There may be several critical paths in the project. When managing the project, a sequence of activities within a given network chart describing this project that increases the longest total time duration of a project is called its critical path. The critical path within the network chart can be used to determine the shortest time required to complete the project. The application of the CPM method can therefore determine which activities within the studied project are “critical” (i.e., activities on the longest path in the network chart describing the project) and which activities may be delayed in the execution of the project without increasing its total time.

The special class CPNETPNTS of PNTS is introduced in the following paragraphs to represent network chart used in the CPM method through PNTS. Special unary operator JOIN that is required in the definition of the class CPNET is introduced first.

Definition 10. The function JOIN: PNTSPNTS of net transition joining is defined as follows: if PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) be the arbitrary PNTS, pP1 be the arbitrary place, t1 and t2 be the arbitrary transitions, (t1 ≠ t2) ∧ (t1 ∈ T1) ∧ (t2 ∈ T1), tiN0, then PNTS := PNTS1.JOIN(p, t1, t2, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:

  1. P := P1 ∪ {p},

  2. T := T1,

  3. A := A1 ∪ {(t1, p), (p, t2)},

  4. AF := AF1 ∪ {((t1, p), 1), ((p, t2), 1)},

  5. TP := TP1 ∪ {(t, 1)},

  6. TI := TI1 ∪ {(t1, p), ti)},

  7. IP := IP1,

  8. OP := OP1,

  9. RP := RP1. □

Symbolic representation of the unary operator JOIN application over the PNTS PNTS1 can be seen in Figure 4.

Figure 4.

Symbolic representation of PNTS PNTS := PNTS1.JOIN(p, t1, t2, ti).

Definition 11. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn), where nN, be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i ≠ jPTSNiPTSNj. The class CPNETPNTS then contains the following PNTSs:

  1. if p be an arbitrary place, p ∉ (P1P2 ∪ … ∪ Pn), then PNTS BASEpCPNET, where BASEp := ({p}, ∅, ∅, ∅, ∅, ∅, p, p, ∅},

  2. if PNTS1CPNET, PNTS2CPNET, t be an arbitrary transition, where (tT1) ∧ (tT2), tiN0, then also [PNTS1, PNTS2].COMP(t, ti) ∈ CPNET,

  3. if PNTS1, PNTS2, …, PNTSnCPNET, pi and po be the arbitrary places, (piP1P2 ∪ … ∪ Pn) ∧ (poP1P2 ∪ … ∪ Pn) ∧ (pipo), ti and to be the arbitrary transitions, (tiT1T2 ∪ … ∪ Tn) ∧ (toT1T2 ∪ … ∪ Tn) ∧ (tito), ti1 ∈ N0, ti2 ∈ N0, …, tinN0, tioN0, then also PNTS PNTSCPNET, where

    PNTSPNTS1PNTS2PNTSn.SYNCpipotito11ti1ti2tintio,

  4. if PNTS1CPNET, pP1 be an arbitrary place, t1 and t2 be the arbitrary transitions, (t1 ≠ t2) ∧ (t1 ∈ T1) ∧ (t2 ∈ T1), tiN0 and afN, then also PNTSCPNET, where PNTS := (P, T, A, AF, TP, TI, IP, OP, RP), such that:

    • PNTS := PNTS1.JOIN(p, t1, t2, ti),

    • ⌐(x1 x2 … xnCIRCUITSPNTS: (x1 ∈ T) ∧ (xnP) ∧ (nN)). □

Four simple PNTSs BASEP1, CPNET1, CPNET2 and CPNET3 that are the members of the class CPNET can be seen in Figure 5, where:

  • CPNET1 := [[BASEP2, BASEP3].COMP(T2, 2), BASEP4].COMP(T3, 5),

  • CPNET2 := [[BASEP6, BASEP8].COMP(T6, 2), BASEP7]

    .SYNC(P5, P9, T5, T7, 1, 1, 1, 6, 3),

  • CPNET3 := [ANET2, ANET3, BASEP1].SYNC(IP, OP, T1, T8, 1, 1, 1, 4, 1, 8, 2)

    .JOIN(P10, T3, T5, 4).

    .JOIN(P11, T3, T6, 5).

Figure 5.

PNTSs BASEP1, CPNET1, CPNET2 and CPNET3.

Note also that the PNTS CPNET3 does not contain any circuit as it is required in the (4) of Definition 11.

Lemma 4. Let PNTSCPNET be an arbitrary PNTS. Then PNTS is pure-formed PNTS.

Proof. Clear, it directly follows from Definition 5, Definition 7, Definition 10 and Definition 11 and from the fact that any PNTSCPNET does not contain any resource place (i.e., if the given PNTS is proper-formed PNTS, then it is also immediately well-formed and pure-formed PNTS). Furthermore, it is also clear that if we allow the existence of a circuit within the PNTS PNTS (see (iv) of Definition 11), there is always the danger of a deadlock in such a PNTS and PNTS is in this case neither a proper-formed PTSN. See, for instance, PNTS CPNET4 in its entry state Se in Figure 6, where CPNET4 := CPNET3.JOIN(P12, T7, T2, 6). It is true that CPNET4 ∉ CPNET because there exists for instance the circuit.

T2 P3 T3P10T5 P6 T6 P8 T7P12CIRCUITSCPNET4.

Figure 6.

PNTS CPNET4 in its entry state Se.

It is also clear that after the firing of the transition T1 in the entry state Se of the PNTS CPNET4 no one transition will be enabled for any value of the net time τ in this PNTS (i.e., there exists the deadlock marking in this PNTS) and thus the CPNET4 is not even proper-formed PNTS. □

Definition 12. The class CPPNETPPNTS contains all the PPNTSs PPNTS := (PNTS, Se) where PNTSCPNET and Se := ((1, 0, …, 0), (<0>, <>, …, <>), 0). □

An example of a simple process is presented in the following paragraphs the characteristics of which will be studied with using of the PPNTS from the CPPNET class. The studied process is described in the following table of activities (see Table 1):

ActivityDurationPrevious activities
A2
B3
C4
D2C
E6B, D
F5A
G5C
H3E, F

Table 1.

Table of activities and their dependencies of studied process.

The CPM chart of the abovementioned process comprising the activities listed in Table 1 is shown in Figure 7 where:

  • selected activity of the studied process and its duration is associated with each edge of the CPM chart,

  • each node of the CPM chart is associated with its serial number (indicated in the upper half of the node), the earliest possible activation time of the given node (shown in the lower left quarter of the node) and the possible latest activation time of the given node (shown in the lower right quarter circle of the node),

  • the desired links of the individual activities are expressed through the oriented edges and their associated nodes in the CPM chart,

  • the critical path of the CPM chart passes through its nodes where the earliest possible activation time is equal to the latest possible activation time, that is, through nodes 1, 2, 3, 5 and 6, and it is thus formed by A, D, E and H activities (these activities are represented by dashed line graphs in the CPM chart) with a total duration of 15 time units.

Figure 7.

CPM chart of process activities listed in Table 1.

The pure-formed PPNTS PROC that represents the process comprising the activities listed in Table 1 can be seen in Figure 8, where

  • PROC := [PROC1, [BASEC, BASEG].COMP(T5, 5)].SYNC(IP, OP, T1, T7, 1, 1, 0, 4, 0),

  • PROC1 := [[BASEA, BASEF].COMP(T3, 5), [BASEB, BASEE].COMP(T4, 6)]

Figure 8.

PPNTS PROC of process activities listed in Table 1.

.SYNC(P1, H, T2, T6, 1, 1, 2, 3, 3).

Places A, B, C, D, E, F, G and H of PNTS PROC represent individual activities of the studied process and the appropriate values of the time interval function TI then express the time durations of relevant activities (i.e., for instance, the time duration of the activity A is represented by the value of TI(T2, A) = 2, etc.).

In order to find the critical path of the process represented by PPNTS PROC, we first perform the association of each place and transition of the PPNTS PROC with the value of the critical path function CP that is introduced in the following Definition 13.

Definition 13. Let PPNTS PPNTS := (P, T, A, AF, TP, TI, IP, OP, RP, Se), PPNTSCPPNET. The critical path function CP: (PT) → N0 is defined as follows:

  1. CP(IP) := 0,

  2. p ∈ (P \ IP): CP(p) := CP(t) + TI(t, p), where t = •p,

  3. tT: CP(t) := max({CP(p1), CP(p2), …, CP(pn)}), where •t = {p1, p2, …, pn}, nN. □

The PPNTS PROC whose each place and transition is associated with the value of the critical path function CP can be seen in Figure 9 (where, for instance, CP(E) = CP(T4) + TI(T4, E) = 6 + 6 = 12, where T4 = •E; CP(T4) = max({CP(B), CP(D)}) = max({3, 6}) = 6, where •T4 = {B, D}, etc.).

Figure 9.

PPNTS PROC with the associated values of critical path function CP.

It follows directly from the Definition 4 that the value of the critical path function CP associated with any transition tT of the arbitrary PPNTS := (P, T, A, AF, TP, TI, IP, OP, RP, Se), where Se := ((1, 0, …, 0), (<0>, <>, …, <>), 0), then represents the net time τ value when the given transition t will be fired. The value of the critical path function CP associated with the output place OP (i.e., CP(OP)) then immediately indicates the total duration of the process critical path (i.e., the net time τ value when the transition t = •OP will be fired). The algorithm for finding the set of PPNTS nodes of which the project critical path is formed is then obvious and it is expressed by the following pseudocode in PASCAL (the set of nodes forming the critical path of the project is then contained in the CriticalPath variable):

Node := OP; CriticalPath := {OP};

WHILE (Node <> IP) DO

BEGIN

MaxValue := max({CP(X1), CP(X2), …, CP(Xn)}), where •Node = {X1, X2, …, Xn}, nN;

Node := Xi, where (Xi ∈ {X1, X2, …, Xn}) ∧ (CP(Xi) = MaxValue);

CriticalPath := CriticalPath ∪ {Node};

END;

The critical path of PPNTS PROC is after applying of the above algorithm represented by the set CriticalPath := {IP, T1, C, T5, D, T4, E, T6, H, T7, OP} (see Figure 9). It is also clear that the given PPNTS may contain more critical paths with the same total time duration.

5. Conclusions

Further research in the field of PNTSs is mainly focused on the definition of additional unary, binary and n-ary PPPA operators preserving their specified properties, for instance, the binary SUBST operator that performs the substitution of the given PNTS for the selected place of another PNTS, and so on. In the field of the project management the research is focused on modeling complex processes, which individual activities can additionally share in parallel a selected set of the resources. These resources are then represented in the given PNTS by individual tokens located in the resource places of its selected net marking. Finding the time-optimal critical path of such a process as well as verifying the properties of the given PNTS that models such a process is generally a nontrivial problem and the use of PPPAs plays a crucial role here.

Another class currently being studied is the class of multiprocess Petri nets with time stamps that represents the generalization of the class of PNTS. The given multiprocess Petri net with time stamps then represents the finite set of processes each of which is modeled by a separate PNTS that share a common set of resources modeled by its individual resource places and their tokens. Many of the studied properties of the multiprocess Petri nets with time stamps are similar or identical to those of the PNTS class and they allow for a further generalization of the concept of a critical path formed by a sequence of activities of the process modeled by the given multiprocess Petri nets with time stamps.

Acknowledgments

This chapter was supported by the SGS at the Faculty of Economics, VŠB-TU Ostrava, under Grant Evaluation of comparative applications using cognitive analysis and method of data envelopment analysis, number SP2018/146.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ivo Martiník (September 19th 2018). Process Petri Nets with Time Stamps and Their Using in Project Management, Petri Nets in Science and Engineering, Raul Campos-Rodriguez and Mildreth Alcaraz-Mejia, IntechOpen, DOI: 10.5772/intechopen.76769. Available from:

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