Table of activities and their dependencies of studied process.
\r\n\t
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Following post-doc studies at the Division of Neonatology, and the Department of Pediatric Surgery of the University Hospital Graz, he became consultant of Pediatrics in 1997 and consultant of Neonatal and Pediatric Intensive Care Medicine in 2000. Since 2004, he is Professor of Pediatrics and since 2008, Head of the Research Unit of Neonatal Infectious Diseases and Epidemiology of the Medical University Graz. Since 2012 he is Deputy Head of the Division of Neonatology of the Medical University of Graz. His main research fields include neonatal infectious diseases, RSV infection, periventricular leukomalacia of the preterm infant, the neonatal microbiome and the role of probiotics.\nHe is a member and board member of several scientific societies including ESPID, past president of the Austrian Society of Perinatal Medicine, and member of the editorial board of several international journals including "BMC Infectious Diseases" and "Frontiers in Pediatrics"',institutionString:"Medical University of Graz",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"5",totalChapterViews:"0",totalEditedBooks:"3",institution:{name:"Medical University of Graz",institutionURL:null,country:{name:"Austria"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"16",title:"Medicine",slug:"medicine"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"205697",firstName:"Kristina",lastName:"Kardum Cvitan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/205697/images/5186_n.jpg",email:"kristina.k@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"990",title:"Human Respiratory Syncytial Virus Infection",subtitle:null,isOpenForSubmission:!1,hash:"b5d980a120eec6669c7f37a5082f0696",slug:"human-respiratory-syncytial-virus-infection",bookSignature:"Bernhard Resch",coverURL:"https://cdn.intechopen.com/books/images_new/990.jpg",editedByType:"Edited by",editors:[{id:"66173",title:"Prof.",name:"Bernhard",surname:"Resch",slug:"bernhard-resch",fullName:"Bernhard Resch"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"2990",title:"Neonatal Bacterial Infection",subtitle:null,isOpenForSubmission:!1,hash:"093e4b7e0964b0fe0229a4b4cafef28c",slug:"neonatal-bacterial-infection",bookSignature:"Bernhard Resch",coverURL:"https://cdn.intechopen.com/books/images_new/2990.jpg",editedByType:"Edited by",editors:[{id:"66173",title:"Prof.",name:"Bernhard",surname:"Resch",slug:"bernhard-resch",fullName:"Bernhard Resch"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"8250",title:"The Burden of Respiratory Syncytial Virus Infection in the Young",subtitle:null,isOpenForSubmission:!1,hash:"576eea34382b48ecff475d3f267837a2",slug:"the-burden-of-respiratory-syncytial-virus-infection-in-the-young",bookSignature:"Bernhard Resch",coverURL:"https://cdn.intechopen.com/books/images_new/8250.jpg",editedByType:"Edited by",editors:[{id:"66173",title:"Prof.",name:"Bernhard",surname:"Resch",slug:"bernhard-resch",fullName:"Bernhard Resch"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6550",title:"Cohort Studies in Health Sciences",subtitle:null,isOpenForSubmission:!1,hash:"01df5aba4fff1a84b37a2fdafa809660",slug:"cohort-studies-in-health-sciences",bookSignature:"R. 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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.
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;
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 n ∈ N. 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 σ ∈ ASQ, τ ∈ ASQ, n ∈ N, m ∈ N, 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:
length: ASQ → N0, so that: length(σ) := n, length(ε) := 0,
elements: ASQ →
prefix: ASQ × N0 → ASQ, 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,
suffix: ASQ × N0 → ASQ, 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,
create: N × A → ASQ, so that: create(n, a) := < a, a, …, a>, where length(<a, a, …, a>) = n,
sort: (N0)SQ → (N0)SQ, so that: sort(σ) := ρ,
We use the following subsets of the set (N0)SQ:
N# := {σ ∈ (N0)SQ | (σ = ε) ∨ ((σ = <a1, a2, …, an>) ∧ (a1 ≤ a2 ≤ … ≤ an)), n ∈ N},
N0 := {σ ∈ (N0)SQ | (σ = ε) ∨ ((σ = <0, 0, …, 0>) ∧ (length(σ) = n)), n ∈ N}.
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, P ∩ T = ∅, 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 k ∈ N, 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
(P, T, A) is the connected net, ∀t ∈ T: (•t ≠ ∅) ∧ (t• ≠ ∅),
AF: (P × T) ∪ (T × P) → N0 is the arc function,
TP is the transition priority function, TP: T → N,
TI: (T × P) → N0 is the time interval function,
IP is the input place, (IP ∈ P) ∧ (•IP = ∅) ∧ (∀p ∈ (P \\ ({IP} ∪ RP)): •p ≠ ∅),
OP is the output place, (OP ∈ P) ∧ (OP• = ∅) ∧ (∀p ∈ (P \\ ({OP} ∪ RP)): p• ≠ ∅),
RP is the set of resource places, (RP ⊆ (P \\ {IP, OP})) ∧ (∀t ∈ T: ⌐(•t ⊆ RP)).
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:
marking M of the PNTS PNTS is a function M: P → N0,
time marking m of the PNTS PNTS is a function m: P → N#,
where ∀p ∈ P: |M(p)| = length(m(p)),
variable τ ∈ N0 is the net time of the PNTS PNTS,
state S of the PNTS PNTS is an ordered triple S := (M, m, τ),
transition t ∈ T 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)) ∧ (∀n ∈ elements(prefix(m(p), AF(p, t))): n ≤ τ)),
firing of the transition t ∈ T 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 [t〉 S′, where ∀p ∈ P:
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))),
elapsing of time interval δ ∈ N results in changing the state S := (M, m, τ) of PNTS PNTS into its state S′ := (M, m, τ + δ), where ∀t ∈ T: ⌐(t en (M, m, τ)), that is denoted by S [δ〉 S′, so that:
if the transitions t1, t2, …, tn ∈ T 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,
finite nonempty sequence σ := t1 t2 … tn of the transitions t1, t2, …, tn ∈ T 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),
∀t ∈ T: ⌐(t en (Mn + 1, mn + 1, τ1)),
is called step σ in the given state S1 of PNTS PNTS and it is denoted by.
finite nonempty sequence ρ of steps and time intervals elapsing that represents the following finite sequence
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,
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 δn〉 S′; 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,
the set of all the states S := (M, m, τ) of PNTS PNTS is denoted by S,
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)) ∧ (S ∈ S)},
static state Ss := (Ms, ms, τs) of PNTS PNTS is every of its states where
the set of all the static states Ss := (Ms, ms, τs) of PNTS PNTS is denoted by Ss,
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)) ∧ (Ss ∈ Ss)},
the function ξ: M → Ms which assigns to each marking M ∈ M of a given PNTS PNTS the associated static marking Ms ∈ Ms is defined as follows:
∀ p ∈ RP: ξ(M(p)) := M(p),
∀ p ∈ P \\ RP: ξ(M(p)) := 0,
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),
the set of all the entry states Se := (Me, me, τe) of PNTS PNTS is denoted by Se,
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) = <>),
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 [Se〉x,
the set of all the exit states Sx of PNTS PNTS that are reachable from all its entry states Se ∈ Se 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}.
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 Ms ∈ Ms 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) ∧ (∀n ∈ elements(prefix(m(IP), AF(IP, T1))) = elements(prefix(<0, 2>, 1)) = elements(<0>) = {0}: 0 ≤ 0) ∧ (∀n ∈ elements(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) ∧ (∀n ∈ elements(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, n ∈ N 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 [T1〉 S1—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 [T1〉 S3), 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 [T3〉 S5 [2〉 S6 [T3〉 Sx, 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 AMs ⊆ Ms 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 ASe ∈ ASe, where ASe ⊆ Se. For instance, the set AMs of the PNTS PNTS1 (see Figure 1) can be defined as AMs := {(0, 0, k, 0) | k ∈ N}, 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 ASe ∈ ASe.
Definition 5. Let PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) be a PNTS, AMs ⊆ Ms 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:
PNTS is k-bounded PNTS if
PNTS is proper-formed PNTS if
proper-formed PNTS is well-formed PNTS if
well-formed PNTS is pure-formed PNTS if
PNTS is proper-formed PNTS if for any of its state S that is reachable from any allowed entry state ASe ∈ ASe there exists its output state Sx that is also reachable from its allowed entry state ASe ∈ ASe such that the output state Sx is also reachable from the state S (i.e., ∀S ∈ [ASe〉 ∃Sx ∈ [ASe〉x: 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., (∃n ∈ N: |[ASe〉〉| = n).
Proper-formed PNTS is well-formed PNTS if for any of its allowed entry state ASe ∈ ASe and for any of its exit state Sx ∈ [ASe〉x, 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 ASe ∈ ASe (i.e., ∀ASe ∈ ASe ∀Sx ∈ [ASe〉x, Sx := (Mx, mx, τx): ξ(Mx) ∈ AMs).
Well-formed PNTS is pure-formed PNTS if for any of its allowed entry state ASe ∈ ASe, where ASe := (AMe, ame, τe), and for any of its exit state Sx ∈ [ASe〉x, 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., ∀ASe ∈ ASe ∀Sx ∈ [ASe〉x, Sx := (Mx, mx, τx): ξ(AMe) = ξ(Mx)).
For instance, if the set AMs of the PNTS PNTS1 (see Figure 1) is defined as:
AMs := {(0, 0, k, 0) | k ∈ N} (i.e., there must be at least one token in the resource place R1 in any allowed entry state ASe ∈ ASe), then it can be shown that PNTS PNTS1 is k-bounded, proper-formed, well-formed and pure-formed PNTS,
AMs := {(0, 0, 0, 0)} (i.e., there may not be any token in the resource place R1 in any allowed entry state ASe ∈ ASe), 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
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 p ∈ P by firing each of the transitions t ∈ T. The number of states S ∈ [ASe〉 for any allowed entry state ASe ∈ ASe must be also finite because PNTS is proper-formed PNTS (i.e., ∃n ∈ N: |[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 p ∈ P, 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
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 Se ∈ Se is the entry state of PNTS PNTS. The class of all PPNTSs will be denoted by PPNTS.□
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 (n ∈ N) and whose application requires the specification of values of k formal parameters (k ∈ N) par1, par2, … park, will be denoted by the expression.
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}, n ∈ N, m ∈ N} 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}, k ∈ N, h ∈ N} be the set of all the allowed static markings of PNTS2.
Cartesian product AMs1 ⊗ AMs2 is then the following set:
PNTS PNTS1 and PNTS2 are disjoint and we denote this fact by PNTS1 ∠ PNTS2 if.
Definition 8. The function COMP: PNTS × PNTS → PNTS 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, PNTS1 ∠ PNTS2, t be an arbitrary transition, where (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, then PNTS := [PNTS1, PNTS2].COMP(t, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:
P := P1 ∪ P2,
T := T1 ∪ T2 ∪ {t},
A := A1 ∪ A2 ∪ {(OP1, t), (t, IP2)},
AF := AF1 ∪ AF2 ∪ {((OP1, t), 1), ((t, IP2), 1)},
TP := TP1 ∪ TP2 ∪ {(t, 1)},
TI := TI1 ∪ TI2 ∪ {(t, IP2), ti)},
IP := IP1,
OP := OP2,
RP := RP1 ∪ RP2. □
Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti) can be seen in 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, PNTS1 ∠ PNTS2, t be an arbitrary transition, (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, 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 = AMs1 ⊗ AMs2 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 × … × PNTS → PNTS 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 ≠ j ⇒ PNTSi ∠ PNTSj, where n ∈ N, pi and po be the arbitrary places, (pi ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (po ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (pi ≠ po), ti and to be the arbitrary transitions, (ti ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (to ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (ti ≠ to), af1 ∈ N, af2 ∈ N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0, then
where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:
P := P1 ∪ P2 ∪ … ∪ Pn ∪ {pi, po},
T := T1 ∪ T2 ∪ … ∪ Tn ∪ {ti, to},
A := A1 ∪ A2 ∪ … ∪ An ∪ {(pi, ti), (ti, IP1), …, (ti, IPn), (OP1, to), …, (OPn, to), (to, po)},
AF := AF1 ∪ AF2 ∪ … ∪ AFn ∪ {((pi, ti), 1), ((ti, IP1), af1), …, ((ti, IPn), afn), ((OP1, to), af1), …, ((OPn, to), afn), ((to, po), 1)},
TP := TP1 ∪ TP2 ∪ … ∪ TPn ∪ {(ti, 1), (to, 1)},
TI := TI1 ∪ TI2 ∪ … ∪ TIn ∪ {((ti, IP1), ti1), …, ((ti, IPn), tin), ((to, po), tio)},
IP := pi,
OP := po,
RP := RP1 ∪ RP2 ∪ … ∪ RPn. □
Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio) can be seen in 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 ≠ j ⇒ PNTSi ∠ PNTSj, where n ∈ N, pi and po be arbitrary places, (pi ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (po ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (pi ≠ po), ti and to be arbitrary transitions, (ti ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (to ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (ti ≠ to), af1 ∈ N, af2 ∈ N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0 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 = AMs1 ⊗ AMs2 ⊗ … ⊗ 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. □
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 CPNET ⊂ PNTS 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: PNTS → PNTS of net transition joining is defined as follows: if PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) be the arbitrary PNTS, p ∉ P1 be the arbitrary place, t1 and t2 be the arbitrary transitions, (t1 ≠ t2) ∧ (t1 ∈ T1) ∧ (t2 ∈ T1), ti ∈ N0, then PNTS := PNTS1.JOIN(p, t1, t2, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:
P := P1 ∪ {p},
T := T1,
A := A1 ∪ {(t1, p), (p, t2)},
AF := AF1 ∪ {((t1, p), 1), ((p, t2), 1)},
TP := TP1 ∪ {(t, 1)},
TI := TI1 ∪ {(t1, p), ti)},
IP := IP1,
OP := OP1,
RP := RP1. □
Symbolic representation of the unary operator JOIN application over the PNTS PNTS1 can be seen in 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 n ∈ N, be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i ≠ j ⇒ PTSNi ∠ PTSNj. The class CPNET ⊂ PNTS then contains the following PNTSs:
if p be an arbitrary place, p ∉ (P1 ∪ P2 ∪ … ∪ Pn), then PNTS BASEp ∈ CPNET, where BASEp := ({p}, ∅, ∅, ∅, ∅, ∅, p, p, ∅},
if PNTS1 ∈ CPNET, PNTS2 ∈ CPNET, t be an arbitrary transition, where (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, then also [PNTS1, PNTS2].COMP(t, ti) ∈ CPNET,
if PNTS1, PNTS2, …, PNTSn ∈ CPNET, pi and po be the arbitrary places, (pi ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (po ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (pi ≠ po), ti and to be the arbitrary transitions, (ti ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (to ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (ti ≠ to), ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0, then also PNTS PNTS ∈ CPNET, where
if PNTS1 ∈ CPNET, p ∉ P1 be an arbitrary place, t1 and t2 be the arbitrary transitions, (t1 ≠ t2) ∧ (t1 ∈ T1) ∧ (t2 ∈ T1), ti ∈ N0 and af ∈ N, then also PNTS ∈ CPNET, where PNTS := (P, T, A, AF, TP, TI, IP, OP, RP), such that:
PNTS := PNTS1.JOIN(p, t1, t2, ti),
⌐(∃ x1 x2 … xn ∈ CIRCUITSPNTS: (x1 ∈ T) ∧ (xn ∈ P) ∧ (n ∈ N)). □
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).
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 PNTS ∈ CPNET 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 PNTS ∈ CPNET 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.
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 CPPNET ⊂ PPNTS contains all the PPNTSs PPNTS := (PNTS, Se) where PNTS ∈ CPNET 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):
Activity | Duration | Previous activities |
---|---|---|
A | 2 | — |
B | 3 | — |
C | 4 | — |
D | 2 | C |
E | 6 | B, D |
F | 5 | A |
G | 5 | C |
H | 3 | E, F |
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.
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)]
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), PPNTS ∈ CPPNET. The critical path function CP: (P ∪ T) → N0 is defined as follows:
CP(IP) := 0,
∀p ∈ (P \\ IP): CP(p) := CP(t) + TI(t, p), where t = •p,
∀t ∈ T: CP(t) := max({CP(p1), CP(p2), …, CP(pn)}), where •t = {p1, p2, …, pn}, n ∈ N. □
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.).
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 t ∈ T 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}, n ∈ N;
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.
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.
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.
Multicriteria decision-making (MCDM) is now considered as one discipline of knowledge, which has been expanding very fast in its own domain. Basically, it is about how to make decision when the undertaken issue is surrounded with a multiple number of criteria. The MC problem consists of two main components, alternatives and criteria. In real-life situations, the alternatives are options, organizations, people, or units to be analyzed which are prescribed under a set of finite criteria or attributes. If the number of alternatives is finite and known, the task is to select the best or the optimal alternative, to rank the alternatives according to their overall quality or performance, or to sort or group the alternatives based on certain measurements or values. In this case, the MC problem is usually called as a multiattribute decision-making (MADM) problem, and the alternatives are prescribed under a finite number of criteria or attributes [1]. The MADM methods are utilized to handle discrete MCDM problems [2]. This chapter focuses on MADM problems or more generally MCDM problems, where this type of problem has a finite number of predetermined alternatives, which is described by several criteria or attributes. MCDM problems can be found in various sectors.
\nSelection problems are really of an MCDM type, a simple problem that we are facing almost every day, for example, when we want to select a dress or a shirt to wear. A decision to choose which dress or shirt is based on certain attributes or factors, such as for what function (office, leisure, and business), color preference, and style or fashion. Here, the types of dress/cloth are the alternatives, while all factors that become the basis of evaluation are the attributes. Another example is when we want to choose the best location to set up projects such as housing, industrial, agricultural activities, recreation center, hoteling, and so on. Many factors or criteria that may be conflicting with each other should be considered by the decision-makers. Selecting the best candidate for various positions that can be conducted in many settings such as face-to-face interviews or online test is also an MCDM problem since the selection will be based on certain requirements. Selecting employees in different organizations with different scope of jobs with different requirements imposed by the related organization can also be categorized as an MCDM problem.
\nAnother example is about selection of the best supplier of a manufacturing firm [3, 4], selection of the best personal computer [5], and selection of a suitable e-learning system [6] to be implemented in an educational institution. These studies focused on selecting the best alternative from a finite number of alternatives that were prescribed under a few evaluation criteria. These studies have the same main issue that is the relative importance or the weights of the evaluation criteria toward the overall performance of the alternatives under study. The studies provide ways to find weights subjectively and how to aggregate the weights when a group of decision-makers were involved in judging the importance of the criteria.
\nIn addition, conducting an evaluation of a program, for example, is usually done after identifying the aspects of the program to evaluate. We may have many programs to be evaluated under several aspects of evaluations with the involvement of one evaluator or a group of evaluators. In a different situation, it may be only one program to be evaluated under several aspects and may be evaluated by one or many evaluators. Besides, many other evaluation situations are usually performed with the presence of many criteria such as evaluation of students, evaluation of employees’ performance, evaluation of learning approaches [7], and evaluation of students’ performance [8]. In relation to the study about the evaluation of students’ academic performance in primary schools, five academic subjects were assumed to have different contribution toward the overall performance of the students. A few experienced teachers were asked to evaluate the degree of importance of the subjects. The resulting weights of the academic subjects were incorporated in finding the overall academic performance of the students in year six in one selected primary school in the northern part of Malaysia. For the purpose of illustrating the practical consideration of the credibility of the evaluators, the problem of evaluation of students’ academic performance is extended by considering the credibility of the teachers who participated in weighting the academic subjects. The detailed discussion is available in the Appendix at the end of the chapter.
\nReferring to those examples of MCDM scenarios, decision-maker(s) or evaluator(s) are involved in many stages of the evaluation process in searching for the optimal solution. As all MCDM problems have two main components, the alternatives and the criteria or attributes, the decision-maker(s) or the evaluator(s) would involve in at least two situations: deciding the quality of each alternative based on each of the criteria and also finding the relative importance of the criteria toward the overall performance of the alternatives. As what is usually arose in solving MCDM problems, criteria are contributing at different level of importance and should become a concern to the decision-maker(s) or evaluator(s). The criteria or attributes of the units to be analyzed should not be assumed to have same contribution toward the overall quality of the alternatives.
\nBesides having a challenge in finding the suitable evaluator(s) or decision-maker(s), since they might come with different background and experience, they also come with different levels of superiority or credibility that should be taken into consideration. This issue should be thought seriously because the results may be misleading if those who are involved in doing the evaluation or judgment do not have enough experience or less credible to give judgment regarding the MCDM problem under study. Moreover, the results may differ among the evaluators if the evaluators are at different levels of superiority [9]. Therefore, the credibility of expert(s) or evaluator(s) or decision-maker(s) who are involved in assessing quality of the alternatives or relative importance of attributes should be taken into consideration.
\nWebster’s New World College Dictionary defines credibility as the quality of being trustworthy or believable. Credibility is also interpreted by good reputation, reputation, honor, and the presence of someone who stands out in the professional community [10]. Meanwhile, professionalism refers to competence or skill expected of a professional. In other words, a professional is someone who is skilled, reliable, and entirely responsible for carrying out their duties and profession [11]. This definition of professionalism has a resemblance to the term of credibility so that the two are like two sides of a coin that cannot be separated. For the purpose of assessment or evaluation, professionalism and credibility are the competencies of assessors in carrying out their functions and roles well, full of commitment, trustworthiness, and accountability.
\nIt is normal that the assessors have different levels of credibility, and their credibility should be considered together with their assessments or evaluations. This chapter provides an overview of the current work on how the credibility of the decision-maker(s) or evaluator(s) could be considered especially on evaluating the importance of the criteria or attributes of any MCDM under investigation, how to quantify the credibility of those people, and how that quantitative values could be incorporated in finding the overall score of the alternatives. This issue falls under the concept of group decision-making and extends it with the consideration of the degree of superiority or credibility of the decision-maker(s) or evaluator(s). By deliberation of different relative importance of the attributes plus the different level of credibility or superiority of those who are involved in finding the optimal solution of the MCDM problem, the solution of the problem would be more realistic, accurate, and representative of the true setting of the problem.
\nIn achieving the objective of the writing, the chapter is organized as follows. The next section describes the basic notations for this chapter. Section 3 discusses the concept of weights and the related methods, particularly the rank-based weighting method. Section 4 discusses on the aggregation of criteria weights and the values of criteria. Section 5 explains how to aggregate the credibility of the evaluators who are involved in weighting or finding weights or relative importance of the criteria. Furthermore, Section 5 also illustrates two approaches to aggregate the degree of credibility of evaluators in finding the relative importance in order to find the overall performance of the alternatives and their rankings. Section 6 suggests a few ways to quantify the credibility of the evaluators. The conclusion of the chapter is in Section 7, which is followed by a list of all references of the chapter. A numerical example is provided in the Appendix at the end of the chapter.
\nLet \n
A multiattribute problem as a decision matrix.
In relation to the numerical example in the Appendix, the students are the alternatives, while the academic subjects are the criteria. So, \n
In finding the relative importance of the criteria or simply the weights of the criteria, \n
This subsection focuses on the discussion of rank-based weighting methods [18, 19] as these methods are used in this chapter in the illustration of practical consideration of evaluators’ credibility in evaluating relative importance of criteria for some real-life multicriteria problems. These methods are very easy to use but have good impact [20]. Three popular rank-based methods are rank-sum (RS), rank reciprocal (RR), and rank order centroid (ROC). The mathematical representations of the three methods are as follows.
\nSuppose \n
where \n
Referring to the numerical example in the Appendix, there are five criteria representing five academic subjects; \n
Many studies were conducted to study the performance of these rank-based methods as criteria weighting methods. For example, a simulation experiment was conducted on investigating the performance of the three rank-based weighting methods (RS, RR, RS) and equal weights (EW) where the data was generated on a random basis [16]. Three performance measures of the methods were “hit rate,” “average value loss,” and “average proportion of maximum value range achieved.” The results show that the ROC was found to be the best technique in most cases an in every measure. Another study on these three rank-based weighting techniques and EW concludes that the rank-based methods have higher correlations with the so-called true weights than EW [21].
\nA study is also done where EW, RS, and ROC methods were compared to direct rating and ratio weight methods [22]. Basically, the direct rating method is a simple type of weighting approach in which the decision-maker or the evaluator must rate all the criteria according to their importance. The evaluator can directly quantify their preference of the criteria. The rating does not constrain the decision-maker’s responses since it is possible for the evaluator to alter the importance of one criterion without adjusting the weight of another [23]. The comparison was conducted under a condition that the evaluators’ judgments of the criteria weights are not certain and subject to random errors. The results show that the direct rating tends to give better quality of decision results when the uncertainty is set as small, while ROC provides comparable results to the ratio weights when a large degree of error is placed. Please note that the ratio weight method requires the evaluators firstly rank the related criteria based on their importance. The evaluators should allocate certain value such as 10 for the least important attribute, and the rest of attributes are judged as multiples of 10. The weight of a criterion is obtained by dividing the criterion’s weight with the sum of all attributes’ weights.
\nThe superiority of ROC over other rank-based methods is also subsequently confirmed in different simulation conditions [24]. An investigation on RS, RR, and ROC weighting methods was also carried out by changing the number of criteria from two to seven [25]. It is found that ROC gives the largest gap between the weights of the most important criterion and the least. RS provides the flattest weight function in the linear form. For RR, the weight of the most important one descends most aggressively to that of the second highest weight value, and then, the function continues to move flatter. In relation to rank-based weighting methods, another rank-based method was proposed [26]. This new rank-based method is called as generalized sum of ranks (GRS). Further investigation was carried out where the performance of GRS was compared to RS, RR, and ROC using a simulation experiment. The result of the investigation shows that GRS has a similar performance to ROC.
\nBased on the previous discussion, it can be concluded that the three rank-based weighting methods, RS, RR, and ROC, are having good features especially the ROC method. Therefore, these rank-based methods are used in the current study to illustrate how to include the degree of credibility of the evaluators who are involved in ranking the importance of the criteria. Furthermore, converting the ranks into weight values is not difficult, and the related formula is given as in Equations (1), (2), and (3).
\nOther subjective weighting methods are analytic hierarchy process (AHP) [4, 27, 28], swing methods [29, 30], graphical weighting (GW) method [31], and Delphi method [32]. The AHP technique was introduced in 1980 [33]. It is a very popular MC approach, and it is done by conducting pairwise comparison of the importance of each pair of criteria. A prioritization procedure is implemented to draw a corresponding priority vector, where this priority vector represents the criteria weights. Thus, if the judgments are consistent, all prioritization procedures would give the same results. At the same time, if the judgments are inconsistent, prioritization procedures will provide different priority vectors [34]. Nevertheless, AHP is widely criticized for being such a tedious process, especially when there are a significant number of criteria or alternatives.
\nFor the swing method, the evaluator must identify an alternative with the worst consequences on all attribute. The evaluator(s) can change one of the criteria from the worst consequence to the best. Then, the evaluator(s) is asked to choose the criteria that he/she would most prefer to modify from its worst to its best level, the criterion with the most chosen swing is the most important, and 100 points is allocated to the most important criterion.
\nThe GW method begins with a horizontal line that is marked with a series of number, such as (9-7-5-3-1-3-5-7-9). The evaluator is expected to place a mark that represents the relative importance of a criterion on the horizontal line with the basis that a criterion is either more, equally, or less important than another criterion by a factor of 1–9. Then, a decision matrix is built as a pairwise comparison matrix. A quantitative weight for a criterion can be calculated by taking the sum of each row, and then the scores are normalized to obtain an overall weight vector. The GW method enables the evaluators to express preferences in a purely visual way. However, GW is sometimes criticized, since it allows evaluator(s) to assign weights in a more relaxed manner.
\nA Delphi subjective weighting method [35] requires one focus group of evaluators to evaluate the relative importance of the criteria. Each evaluator remains nameless to each other that can reduce the risk of personal effects or individual bias. The evaluation is conducted in more than one round until the group ends with a consensus of opinions on the relative importance of the criteria under study. The main advantage of this method is that the method avoids confrontation of the experts [36]. However, to pool up such a focus group is quite costly and timely.
\nFinding the final score of each alternative is very important since the final scores of the alternatives are required to rank the alternatives. Basically, those alternatives with higher scores should be positioned at higher rankings and vice versa. In order to find the overall or composite or final values of each alternative, the criteria weights should be aggregated with each alternative’s values of the corresponding criteria. There are many aggregation methods available in literature. The section focuses on simple additive weighted average (SAW) method as the chapter uses SAW in the numerical example (in the Appendix at the end of the chapter). Furthermore, SAW method is a very well-established method and very easy to use [16].
\nThe mathematical equation for SAW is given as follows:
\n\n\n
SAW is an old method, and MacCrimmon is one of the first researchers that summarized this method in 1968 [37]. As a well-established method, it is used widely [38] in solving MC problems, particularly for the evaluation of alternatives. Basically, this method is the same as the simple arithmetic average method, but instead of having the same weight values for the criteria, SAW method uses mostly distinct weights values of the criteria. As given in Eq. (4), the overall performance of each alternative is obtained by multiplying the rating of each alternative on each criterion by the weight assigned to the criterion and then summing these products over all criteria [15]. The best alternative is the one that obtained the highest score and will be selected or ranked at the first position. Many recent studies used the SAW method, for example, in [39, 40, 41], and a review on its applications is also available [42].
\nBesides SAW or also known as weighted sum method (WSM), there is another average technique, called weighted product model (WPM) or simple geometric weighted (SGW) or simple geometric average method. In WPM, the overall performance of each alternative is determined by raising the rating of the alternative to the power of the criterion weight and then multiplying these products over all criteria [15]. However, WPM is a little bit complex as compared to SAW since WPM involves power and multiplications.
\nAHP [14], technique for order preference by similarity to ideal solution (TOPSIS), and VlseKriterijumska Optimizacija Kompromisno Resenje (VIKOR) [43] are also popular aggregation methods in solving MC problems. As previously mentioned in Section 3.2, AHP is built under the concept of pairwise comparison either in finding the criteria weights or criteria values of the alternatives. The aggregation of criteria weights and the criteria values obtained by AHP is sometimes done by using the SAW or SGW methods.
\nAHP and TOPSIS are two different aggregation methods. TOPSIS assigns the best alternative that relies on the concepts of compromise solution, where the best alternative is the one that has the shortest distance from the ideal solution and the farthest distance from the negative ideal solution [44]. In other words, alternatives are prioritized according to their distances from positive ideal solutions and negative ideal solutions, and the Euclidean distance approach is utilized to evaluate the relative closeness of the alternatives to the ideal solutions. There is a series of steps of TOPSIS, but this method starts with the weighted normalization of all performance values against each criterion. Some recent applications of the TOPSIS method are available [45, 46, 47, 48].
\nVIKOR method [49] is quite similar to TOPSIS method, but there are some important differences, and one of the differences is about the normalization process. TOPSIS uses the vector linearization where the normalized value could be different for different evaluation unit of a certain criterion, while VIKOR uses linear normalization where the normalized value does not depend on the evaluation unit of a criterion. VIKOR has also been used in many real-world MCDM problems such as mobile banking services [50] digital music service platforms [51], military airport location selection [52], concrete bridge projects [53], risk evaluation of construction projects [54], maritime transportation [55], and energy management [56].
\nThis section discusses how credibility can be included practically in solving MC problems. Suppose the evaluators are requested to evaluate the relative importance of the criteria based on rank-based weighting methods as explained in Section 3.1. Suppose there is a panel of \n
Approach 1.
For the first approach as portrayed in Figure 2, the degree of credibility of the evaluators is attached to the resulted weights from the ranks of criteria by using any of the equations, Eq. (1), Eq. (2), or Eq. (3). So, here there are p sets of weights of the criteria, and the average of that p weights for each criterion is calculated by summing up all weights for that criteria and divide the sum with the total number of evaluators. So now, there is only one set of weights that can be aggregated with the values of alternatives for each corresponding criterion as given in Eq. (4). There is only one set of overall performance of all n alternatives.
\nFor the second approach, the criteria weights obtained from each evaluator are kept, and then each set of weights is aggregated with the quality values of each alternative. So, here there are p sets of overall values of the alternatives. In order to get the final overall score of the alternatives, the average of the p scores for each alternative should be calculated. The ranking or sorting of the alternatives or selecting the best alternative is done based on the average of that p overall scores of each individual alternative. The following section provides some suggestion on how to quantify the credibility of the evaluators.
\nReferring to the numerical example in the Appendix, there were three evaluators involved in ranking the importance of the five academic subjects, and the number of students is 10. So, \n
Credibility is synonym to professionalism, integrity, trustworthiness, authority, and believability. A study focuses on how to assess the credibility of expert witnesses [58]. A 41-item measure was constructed based on the ratings by a panel of judges, and a factor analysis yielded that credibility is a product of four factors: likeability, trustworthiness, believability, and intelligence. Another study concerns about the credibility of information in digital era [59]. Credibility is said to have two main components: trustworthiness and expertise. However, the authors conclude that the relation among youth, digital media, and credibility today is sufficiently complex to resist simple explanations, and their study represents a first step toward mapping that complexity and providing a basis for future work that seeks to find explanations.
\nIt can be argued that the degree of credibility of evaluators or judges or decision-makers can be determined subjectively or objectively, where the former one can be done by using certain construct as proposed in [58] or can be determined based on certain objective or exact measures such as years of experience, salary scale, or amount of salary. The quantification of the degree of credibility opens a new potential area of research as there are very few researches done especially on finding the suitable objective proxy measures of the degree of credibility.
\nFinding the degree of credibility subjectively requires more time and much harder as it involves a construct or an instrument which would be used as a rating mechanism to obtain the degree of credibility. Meanwhile, finding the degree of credibility based on objective information is simpler and easier to do. As an illustration on how to quantify the credibility objectively, suppose there are three experts with their basic salaries in a simple ratio of 1:2:3. So, this ratio can be converted as 0.167:0.333:0.500, so that the sum of credibility of the evaluators is equal to 1. These values can be used to represent the degree of credibility of the evaluators or experts 1, 2, and 3, respectively. It should be noted that the sum of the degrees of credibility of the three evaluators is equal to one to make the future calculation simple while easier for interpretation of the values. Here, evaluator 3 is the most credible one since he/she has the highest salary among the three, and it is a usual practice that those who are higher in terms of expertise usually are paid higher. The same computation can be used for the years of experience or salary scale.
\nThe numerical example in the Appendix extends the problem of evaluating students’ academic performance which is discussed earlier in the Introduction. Here, the credibility of the teachers who were asked to assess the relative importance of the five subjects was considered. In order to incorporate the degree of credibility of the teachers, a new set of values is introduced to represent these different degrees of credibility. The example shows two ways of calculations on how the credibility values could be included in finding the overall scores of the alternatives. As expected, the overall scores and the overall ranking are different as compared to overall scores of not considering the different credibility of the teachers. The details and the step-by-step methodology are also included in the Appendix.
\nThis chapter provides an overview on the practical consideration of evaluators’ credibility in evaluating relative importance of criteria for some real-life multicriteria problems. Credibility of the evaluators who are involved in solving any multicriteria problem should be included in calculating the overall scores of the alternatives or the units of analysis. This chapter demonstrates how the credibility of evaluators who participated in finding the criteria weights can be combined with the criteria weights and the quality of the criteria of the alternatives. Rank-based criteria weighting methods are used as an illustration in a numerical example of evaluation of students’ academic performance problem at the end of the chapter. However, other criteria subjective weighting methods are also possible to be used but with caution especially at the stage of aggregation of criteria weights and criteria values. It may exist only one approach to do the aggregation due to the underpinning concepts of the aggregation methods. The chapter uses simple additive weighted average method as the aggregation method since the method is very well established. The use of other aggregation techniques is also plausible. The chapter also suggests a few practical proxy measures of the credibility but is still very limited. More researches should be conducted to find ways of measuring the credibility of evaluators or experts either subjectively or objectively. Inclusion of the credibility of evaluators in solving multicriteria problems is realistic since the evaluators come from different backgrounds and levels of experience. Quantification of the evaluators’ credibility subjectively or objectively opens a new insight in group decision-making field. Furthermore, the credibility of the evaluators should also be considered in other multicriteria problems in other areas, so that the results are more practical and accurate.
\nMr. Zachariah is a class teacher of 10 excellent students in one of the best primary schools of a country. The 10 students were already given the final marks of five main academic subjects by their respective teachers as in Table 1. Mr. Zachariah must rank the students according to their performance because these students will be given awards and recognition on their graduation day.
\n\n | Native language | \nEnglish language | \nMathematics | \nScience | \nHistory | \n
---|---|---|---|---|---|
Student 1, \n | \n0.25 | \n0.34 | \n0.12 | \n0.36 | \n0.45 | \n
\n\n | \n0.33 | \n0.54 | \n0.22 | \n0.44 | \n0.76 | \n
\n\n | \n0.43 | \n0.65 | \n0.57 | \n0.42 | \n0.91 | \n
\n\n | \n0.55 | \n0.32 | \n0.37 | \n0.67 | \n0.53 | \n
\n\n | \n0.27 | \n0.66 | \n0.57 | \n0.82 | \n0.61 | \n
\n\n | \n0.67 | \n0.56 | \n0.46 | \n0.46 | \n0.31 | \n
\n\n | \n0.58 | \n0.87 | \n0.39 | \n0.27 | \n0.43 | \n
\n\n | \n0.32 | \n0.76 | \n0.41 | \n0.37 | \n0.51 | \n
\n\n | \n0.91 | \n0.36 | \n0.47 | \n0.45 | \n0.45 | \n
\n\n | \n0.12 | \n0.33 | \n0.81 | \n0.75 | \n0.32 | \n
Ten students assessed under five academic subjects.
Suppose three experienced teachers, Edward, Mary, and Foong, were asked to evaluate the relative importance of the five academic subjects with their degree of credibility as discussed in previous section, that is, the salary ratio of the three teachers is 0167: 0.333: 0.500. The rank-based technique is used to analyze the ranking of importance of the academic subjects given by these three teachers by using Eq. (1).
\nThe results are given in Table 2. Column 2 displays the ranking of the criteria evaluated by teacher 1, and column 3 shows the corresponding criteria weights as analyzed by Eq. (1), while columns 4 and 5 and columns 6 and 7 show the respective results by teachers 2 and 3, respectively. The second last column of the table summarizes the criteria weights when the teachers are of same credibility. The values were computed as the simple arithmetic average of the corresponding criterion, while the last column has the final weights that were calculated as the simple arithmetic average as well but with consideration of the different degree of credibility according to Approach 1 as given in Figure 2. Please note that the both sets of final weights are already summed to one. So, the normalization process to guarantee the sum of weights is one and is not necessary.
\n\n | Teacher 1 (0.167) | \nTeacher 2 (0.333) | \nTeacher 3 (0.500) | \nFinal weight same credibility (SC) | \nFinal weight different credibility (DF) | \n|||
---|---|---|---|---|---|---|---|---|
\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n|||
Native language | \n1 | \n0.333 | \n2 | \n0.267 | \n2 | \n0.267 | \n0.289 | \n0.278 | \n
English language | \n3 | \n0.200 | \n3 | \n0.200 | \n3 | \n0.200 | \n0.200 | \n0.200 | \n
Mathematics | \n4 | \n0.133 | \n1 | \n0.333 | \n1 | \n0.333 | \n0.267 | \n0.300 | \n
Science | \n5 | \n0.067 | \n5 | \n0.067 | \n5 | \n0.067 | \n0.067 | \n0.067 | \n
History | \n2 | \n0.267 | \n4 | \n0.133 | \n4 | \n0.133 | \n0.178 | \n0.156 | \n
Criteria weights of five academic subjects evaluated by three teachers with the same and different credibility by using rank-sum weighting technique.
Now, in order to find the overall performance of each student, for example, the overall performance of student 1 without consideration of credibility of teachers in evaluating the relative importance of the academic subjects, it is simply done by multiplying row 2 of Table 1 with its corresponding criteria weights in the second last column of Table 2 by using Eq. (4) as follows:
\nThe same process is performed to find the overall scores of student 1, if the credibility of the teachers in finding weights of the criteria is considered but the weights in last column of Table 2 is used, instead.
\n\nTable 3 gives the overall scores and the corresponding final rankings of all students based on average criteria weights with the same (SC) and different (DC) credibility of the teachers. The overall scores are all different, while the rankings are different especially for ranks 8 and 9 and 4 and 5.
\n\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n|
---|---|---|---|---|---|---|---|---|---|---|---|
SC | \nScore | \n0.277 | \n0.427 | \n0.597 | \n0.461 | \n0.526 | \n0.514 | \n0.540 | \n0.470 | \n0.571 | \n0.424 | \n
Rank | \n10 | \n8 | \n1 | \n7 | \n4 | \n5 | \n3 | \n6 | \n2 | \n9 | \n|
DC | \nScore | \n0.244 | \n0.408 | \n0.592 | \n0.439 | \n0.518 | \n0.540 | \n0.550 | \n0.462 | \n0.561 | \n0.422 | \n
Rank | \n10 | \n9 | \n1 | \n7 | \n5 | \n4 | \n3 | \n6 | \n2 | \n8 | \n
Overall scores and ranking of students with average criteria weights evaluated by teachers of the same and different credibility based on Approach 1.
\nTable 4 summarizes three individual overall score of the three different teachers without consideration of their credibility, while the second last column and the last column are the average overall scores of the three overall scores and its corresponding rankings, respectively.
\n\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \nRanking | \n
---|---|---|---|---|---|
\n\n | \n0.311 | \n0.259 | \n0.259 | \n0.276 | \n10 | \n
\n\n | \n0.479 | \n0.400 | \n0.400 | \n0.426 | \n8 | \n
\n\n | \n0.620 | \n0.584 | \n0.584 | \n0.596 | \n1 | \n
\n\n | \n0.483 | \n0.449 | \n0.449 | \n0.460 | \n7 | \n
\n\n | \n0.515 | \n0.530 | \n0.530 | \n0.525 | \n4 | \n
\n\n | \n0.510 | \n0.516 | \n0.516 | \n0.514 | \n5 | \n
\n\n | \n0.552 | \n0.534 | \n0.534 | \n0.540 | \n3 | \n
\n\n | \n0.474 | \n0.467 | \n0.467 | \n0.469 | \n6 | \n
\n\n | \n0.588 | \n0.561 | \n0.561 | \n0.570 | \n2 | \n
\n\n | \n0.349 | \n0.461 | \n0.461 | \n0.424 | \n9 | \n
Same credibility: four different sets of overall scores and final ranking of the 10 students based on average overall scores.
\nTable 5 shows the three overall scores by consideration of the credibility of teachers in finding the academic subjects’ weights, and the average overall scores of the three overall scores. The ranking of the students is based on the average overall scores in column 5 of the table. Here, Approach 2 as in Figure 3 is used to find the final overall scores of the students.
\n\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \nRanking | \n
---|---|---|---|---|---|
\n\n | \n0.052 | \n0.086 | \n0.129 | \n0.089 | \n10 | \n
\n\n | \n0.080 | \n0.133 | \n0.200 | \n0.138 | \n9 | \n
\n\n | \n0.104 | \n0.194 | \n0.292 | \n0.197 | \n1 | \n
\n\n | \n0.081 | \n0.150 | \n0.225 | \n0.152 | \n7 | \n
\n\n | \n0.086 | \n0.176 | \n0.265 | \n0.176 | \n4 | \n
\n\n | \n0.085 | \n0.172 | \n0.258 | \n0.172 | \n5 | \n
\n\n | \n0.092 | \n0.178 | \n0.267 | \n0.179 | \n3 | \n
\n\n | \n0.079 | \n0.155 | \n0.233 | \n0.156 | \n6 | \n
\n\n | \n0.098 | \n0.187 | \n0.281 | \n0.189 | \n2 | \n
\n\n | \n0.058 | \n0.153 | \n0.230 | \n0.147 | \n8 | \n
Different credibility: four different sets of overall scores and final ranking of the 10 students based on average overall scores.
Approach 2.
To make the comparison easier, Table 6 summarizes the overall scores and their corresponding rankings of the students with SC and DC of the teachers when calculating the academic subjects’ weights based on Approach 2.
\n\n | \n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n\n\n | \n|
---|---|---|---|---|---|---|---|---|---|---|---|
SC | \nScore | \n0.276 | \n0.426 | \n0.596 | \n0.460 | \n0.525 | \n0.514 | \n0.540 | \n0.469 | \n0.570 | \n0.424 | \n
Rank | \n10 | \n8 | \n1 | \n7 | \n4 | \n5 | \n3 | \n6 | \n2 | \n9 | \n|
DC | \nScore | \n0.089 | \n0.138 | \n0.197 | \n0.152 | \n0.176 | \n0.172 | \n0.179 | \n0.156 | \n0.189 | \n0.147 | \n
Rank | \n10 | \n9 | \n1 | \n7 | \n4 | \n5 | \n3 | \n6 | \n2 | \n8 | \n
Two different set of overall scores of the students by averaging overall performance of the students and their corresponding rankings based on Approach 2.
As the two sets of the overall scores are different, all rankings based on both sets of the overall scores are the same except for ranks 8 and 9. There is not much different in the overall rankings since the MC problem that is considered here is only a small scale problem with only 10 alternatives and 5 criteria. However, the two sets of overall values are totally different. There may be much more differences in terms of rankings if a bigger MC problem with more alternatives and more criteria is considered. The final ranking of the students obtained by consideration of the different credibility of the teachers should be selected as the practical and valid results.
\n"I work with IntechOpen for a number of reasons: their professionalism, their mission in support of Open Access publishing, and the quality of their peer-reviewed publications, but also because they believe in equality. Throughout the world, we are seeing progress in attracting, retaining, and promoting women in STEMM. IntechOpen are certainly supporting this work globally by empowering all scientists and ensuring that women are encouraged and enabled to publish and take leading roles within the scientific community." Dr. Catrin Rutland, University of Nottingham, UK
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