References

[BP] Burgess R.R., Pritsker A.A.B.: The GERTS Simulation Programs: GERTS ill,

GERTS ill Q, GERTS ill C, GERTS III. Electronic Research Centre,

NASA

[BS] Best E., Schmid H.: Towards a Constructive Solution of the Liveness Problem

in Petri Nets. Technical Report 4/76, Institut flir Informatik, Universitiit

Stuttgart, 1976

[CMl] Clayton E.R., Moore L.J.: GERT Network Simulation. Proc. of the 3rd Annual

Meeting of the Southeastern Division of the American Institute of Decision

Sciences, 1973

[CM2] Clayton E.R., Moore L.J.: GERT Modeling and Simulation: Fundamentals and

Applications. Petrocelli Charter, New York 1976

[CS] Colom J.M., Silva M.: Convex Geometry and Semiflows in Pff Nets. A

Comparative Study of Algorithms for Computation of Minimal P-Semiflows. In:

Proceedings ofthe 10th International Conference on Application and Theory of

Petri Nets, Bonn 1989

[De] De Ambrogio W.: Prograrnmazione reticolare, Vou. Etas Libri, Milan 1977

[Fi] Fisz M.: Probability Theory and Mathematical Statistics. John Wiley and Sons,

New York, London 1963

[FJ] Feldbrugge F., Jensen K.: Petri Net Tool Overview 1986. In: W. Brauer et al.

220 References

(eds.): Petri Nets: Applications and Relationships to Other Models of

Concurrency. Lecture Notes in Computer Science, VoL 255, Springer-Verlag,

Berlin, Heidelberg, New York, 1987

[Fo] Fourier J.B.J.: Solution d'une question particuliere du calcul des inegalites. In:

Oeuvres II, pp. 317-328; Gauthier-Villars, Paris

[Gr] Grubbs E.F.: Attempts to Validate Certain PERT Statistics. Operations Research,

10: 6, 1962

[JK] Jaxy M., KrUckeberg F.: Mathematical Methods for Calculating Invariants in

Petri Nets. In: G. Rozenberg (ed.): Advances in Petri Nets 1987. Lecture Notes

in Computer Science, Vol. 266, Springer- Verlag, Springer-Verlag, Berlin,

Heidelberg, New York, 1987

[La] Lautenbach K.: Liveness in Petri Nets. Internal Report GMD-ISF 72-02.1,

sankt Augustin, 1972

[Li] Lipton lR.: The ReachabiIity Problem Requires Exponential Space. Yale Dniv.,

Dept. of Compo Sci., Research Report No. 62, 1976

[Lie] Lien Y.: Termination Properties in Generalized Petri Nets. SIAM Journal of

Computing 5: 2, 1976,251-265

[MM] Mayr E.W., Meyer A.R.: The Complexity of the Finite Containment Problem for

Petri Nets. Journal of the ACM 28: 3, 1981,561-576

[MP] Moder J.J., Phillips C.R.: Project Management with CPM and PERT. Van

Nostrand Reinhold, New York, 1970

[NS] Neumann K., Steihard D.: GERT Networks and the Time-Oriented Evaluation

of Projects. Springer-Verlag, Berlin, Heidelberg, New York, 1979

[MR] MacCrimmon K.R.: Ryavec C.A., An Analytical Study of the PERT

Assumptions. Operations Research 12: 1, 1964

[Ro] Roy B.: Partial Preference Analysis and Decision-Aid: the Fuzzy Outranking

Relation Concept. In: SEMA, Paris, 1976

[RV] Roy B.: Vinke Ph., Multicriteria Analysis: Survey and New Directions. EJOR 8,

207-218

[Sa] Saaty T.L.: Exploring the Interface between Hierarchies, Multiple Objectives and

Fuzzy Sets. Fuzzy Sets and Systems 1, 1978,57-68

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[Vi] Vianelli S.: Prontuari per calcoli statistici. Abbaco, Palenno, Roma, 1959

[Ya] Yager R.R.: Fuzzy Decision Making Including Inequal Objectives. Fuzzy Sets

and Systems 1, 1978, 87-95

[Zi] Zimmennann H.J.: Fuzzy Sets, Decision Making, and Expert Systems. Kluwer

Academic Publishers, Dordrecht, Boston, London, 1987

Further Reading

This section is aimed at helping the reader deepen the understanding of the notions and

techniques presented in this book. The proposed readings are comprehensive works on

general or special topics presented in the book. For the reader's convenience, we have

avoided suggesting conference proceedings or scattered papers. The selection mirrors the

author's personal taste and experience as a reader. It does not pretend to be complete, but

should certainly help the interested traveller along the way.

Graph Theory

L. W. Beinecke, R. J. Wilson, Applications of Graph Theory, Academic Press, New

York,1979

N. Deo, Graph Theory with Applications to Engineering and Computer Science,

Prentice-Hall, Englewood Cliffs, New Jersey, 1974

F. Harary, Graph Theory, Addison-Wesley, Reading, Mass., 1969

F. Harary, R. Z. Norman, D. Cartwright, Structural Models: An Introduction to the

Theory of Directed Graphs, John Wiley and Sons, New York, 1965

R. J. Wilson, Introduction to Graph Theory, Academic Press, New York, 1972, and

Longman Group, Harlow, Essex, 1975

224 Further Reading

Networking Techniques

A. Alan, B. Pritsker, Modeling and Analysis Using Q-GERT Networks, John Wiley and

Sons, New York, 1979

R. D. Archibald, R. L. Villoria, Network-Based Management Systems (PERTICPM) ,

John Wiley and Sons, New York, 1967

D. D. Bedworth, Industrial Systems, Planning, Analysis, Control, Ronald Press, New

York,1973

E. S. Buffa, W. H. Taubert, Production-Inventory Systems: Planning and Control,

Irwin, Homewood, Ill., 1972

E. R. Clayton, L. J. Moore, GERT Modeling and Simulation: Fundamentals and

Applications, Petrocelli Charter, New York, 1976

S. E. Elmaghraby, Activity Networks: Project Planning and Control by Network Models,

John Wiley and Sons, New York, 1977

C. A. Kirkpatrick, R. Levin, Planning and Control with PERTICPM, McGraw-Hill,

New York, 1966

R. F. Gonzales, C. McMillan, Systems Analysis, A Computer Approach to Decision

Models, (3rd ed.), Irwin, Homewood, Ill., 1973

R. W. Miller, Schedule, Cost and Profit Control with PERT, McGraw-Hill, New York,

1963

J. J. Moder, C. R. Phillips, Project Management with CPM and PERT, Van Nostrand

Reinhold, New York, 1970

K. Neumann, U. Steihard, GERT Networks and the Time-Oriented Evaluation of

Projects, Springer-Verlag, Berlin, Heidelberg,1979

G. E. Whitehouse, System Analysis and Design Using Network Techniques, Prentice­

Hall, New York, 1973

Petri Nets

G.W. Brams (nom collectiv), Reseaux de Petri: Theorie et Practique, Tomes 1 et 2,

Editions Masson, Paris, 1982

Further Reading 225

W. Brauer, G. Rozenberg, W. Reisig (eds.), Petri Nets: Central Models and their

Properties, Lecture Notes in Computer Science, Vo1.254, Springer-Verlag,

Berlin, Heidelberg, New York, 1987

W. Brauer, G. Rozenberg, W. Reisig, (eds.), Petri Nets: Applications and Relationships

to Other Models of Concurrency, Lecture Notes in Computer Science, Vo1.255,

Springer-Verlag, Berlin, Heidelberg, New York, 1987

W. Reisig, Petri Nets, An Introduction, Springer-Verlag, Berlin, Heidelberg, New

York, 1985

w. Reisig, Systementwurfmit Netzen, Springer-Verlag, Berlin, Heidelberg, New York,

1985

M. Silva, Las Redes de Petri: en la Automatica y la Informatica, Editorial AC, Madrid,

1985

G. Rozenberg, (ed.), Advances in Petri Nets 1987, Lecture Notes in Computer Science,

Vo1.266, Springer-Verlag, Berlin, Heidelberg, New York, 1987

G. Rozenberg, (ed.), Advances in Petri Nets 1988, Lecture Notes in Computer Science,

Vo1.340, Springer-Verlag, Berlin, Heidelberg, New York, 1988

Decision Support Systems

S. S. Mitra, Decision Support Systems - Tools and Techniques, John Wiley and Sons, .

New York, 1986

R. J. Thierauf, Decision Support Systems for Effective Planning and Control, Prentice­

Hall, Englewood Cliffs, New Jersey, 1982

Multicriteria Decision Making

K. J. Arrow, Individual Choice under Certainty and Uncertainty. In: Collected Papers of

K. J. Arrow, Blackwell, Oxford, 1984

D. Bell, R. Keeney, H. Raiffa (eds.), Conflicting Objectives in Decisions, John Wiley

and Sons, New York, 1977

A. H. Cornell, The Decision-Maker's Handbook, Prentice-Hall, Englewood Cliffs, 1980

226 Further Reading

G. Fandel, J. Spronk, Multiple Criteria Decision Metlwds and Applications, Springer­

Verlag, Berlin, Heidelberg, New York, 1985

G. Fandel, Optimale Entscheidungen in Organisationen, Springer-Verlag, Berlin,

Heidelberg, New York, 1979

N. A. J. Hastings, J. M. C. Mello, Decision Networks, John Wiley and Sons, New

York, 1979

R. Keeney, H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Trade­

offs, John Wiley and Sons, New York, 1976

D. W. Miller, M. K. Starr, Executive Decisions and Operations Research, Prentice-Hall,

Englewood Cliffs, 1960

J. de Montgolfier, P. Bertier, Approche multicritere des problemes des decision, Editions

Hommes et Techniques, Paris, 1978

Saaty T. L., The Analytic Hierarchy Process. Planning, Priority Setting and Resource

Allocation, McGraw-Hill, New York, 1980

H. A. Simon, Models of Bounded Rationality, The MIT Press, Cambridge, Mass., 1982

H. Thiriez, S. Zionts (eds.), Multiple Criteria Decision Making, Lecture Notes in

Economics and Mathematical Systems, Vo1.30, Springer-Verlag, Berlin,

Heidelberg, New York, 1976

M. Zeleny, Multiple Criteria Decision Making, McGraw Hill, New York, 1981

Fuzzy Sets

T. T. Ballmer, M. Pinkal, Approaching Vagueness, North-Holland, New York, 1983

D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic

Press, New York, 1980

T. A. Folger, G. J. Klir, Fuzzy Sets, Uncertainty, and Information, Prentice Hall,

Englewood Cliffs, New Jersey, 1988

A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Academic Press, New

York,1975

M. M. Gupta, A. Kaufmann, Introduction to Fuzzy Arithmetic: Theory and Applications,

Van Nostrand Reinhold, New York, 1985

Further Reading 227

M. M. Gupta, R. K. Ragade, R. R. Yager (eds.), Advances in Fuzzy Set Theory and

Applications, North-Holland, New York, 1979

H. J. Zimmennann, Fuzzy Set Theory - and its Applications, Kluwer-Nijhoff, Boston,

1985

Fuzzy Decision Making

K. Fu, M. Shirnura, K. Tanaka, L. A. Zadeh (eds.), Fuzzy Sets and Their Application to

Cognitive and Decision Processes, Academic Press, New York, 1975

B. R. Gaines, H. J. Zimmermann, L. A. Zadeh (eds.), Fuzzy Sets and Decision

Analysis, North-Holland, New York, 1984

W. J. M. Kickert, Fuzzy Theories on Decision Making, Martinus-Nijhoff, Boston, 1978

H. J. Zimmennann, Fuzzy Sets, Decision Making, and Expert Systems, Kluwer

Academic Publishers, Boston, Dodrecht, Lancaster, 1987

Miscellanea

B. De Finetti, Theory of Probability, John Wiley and Sons, New York, 1974.

M. Fisz, Probability and Mathematical Statistics, John Wiley and Sons, New York, 1963

W. Feller, An Introduction to Probability Theory and Its Applications, Vols. 1 and 2,

John Wiley and Sons, New York, 1950, 1966

R. A. Howard, Dynamic Probabilistic Systems, Vols. 1 and 2, John Wiley and Sons,

New York, 1971

Index

activated 136, 160

activation distribution of sinks 110, 113

activity 44, 45, 52, 53, 66, 85

activity duration 81,85

activity beginning, ending 44

activity cost 86

activity cost function 86

activity network 6,44, 45, 66, 86

actual ca~e 130

actual cost 76

actual plan execution 67

actual project duration 94

actual state 131

adjacent 36

adjacency matrix 43

algorithms for computing generator sets of elementary S-invariants 152

algorithms for constructing reachability graphs 142

a-cut 187

alternative 30, 182, 186, 190

alternatives concurrently enabled at a given marking 182

analytical cost optimization 82

AND/ AND logic 44

AND type node 104

230 Index

arc 38

arc labeling 159

arity 159

best executions 216

beta probability distribution 89

bipartite graph 38

bottom-up construction of control nets 178

capacity function 135

capacity of a place 135

case 17,30

case class 130

causally dependent 119

causally independent 119

causal relationship 122

central limit theorem 95

choice 58,59, 171, 172

choice schema 30, 182

choice schema enabled at a given marking 183

circuit 41

closed walk 41

color 158

color function 159

color set 27,159

composition module 171

composition rules 179

composition schema 58

concession 125

concordance matrix 209

concurrent 119

concurrent composition 58, 59

concurrenteven~ 21

concurrently enabled 125, 136, 160

condition 17, 130

conditional distribution function of durations 108

conditional probability that event s is realized not later than time t 113

condition/event net (CE net) 17, 129, 130

conflict 20, 124, 125, 137

connected graph 41

connected digraph 41

consistent 204

consistent weighting 203

continuous non-negative random variable 87

control net 29, 163, 170, 186

control place 29, 182

copies of resources l35

correct plan representations 60

cost center 75

cost control 86

cost optimization 81

cost plan 86

cost planning 75

CPM (Critical Path Method) 9, 65, 66

CPM-PERT cost planning 76

CPM scheduling planning technique 85, 86

critical activities 69

critical events 69

criticality indicator 99

critical path 11, 69, 93

cycles 41, 108

decision 170

degree of outranking of an execution 210

deterministic type node 105

dichotomy 216

digraph 38

direct costs 75

directed tree 42

disciplined planning 52

disciplined planning syntax (DPS) 52, 58

disconnected digraph 41

Index 231

232 Index

discordance matrix 209

discordance multiplier 209

DPS-correct modules 58

dummy activity 45, 63, 66

dummy arc 41,45

earliest activity start 73

earliest activity tennination 73

earliesteventtime 10,67,68

earliest expected time 93

edge 35

elementary resources 135

elementary S-invariants 147

elementary T-invariants 148

empty multiset XIII

enabled 19, 125, 136

enabled for color k 160

end 36

end vertex 36

event 17,44,45,104,130

event realization 104

event slack 10, 69

exclusive-Or (EOR) type node 104

execution net 163,164,165, 170

execution of a choice schema 183

execution of an alternative 182

execution of a Petri net plan 163

execution supervisor 2, 31, 185

expected activity duration 93

expected project completion date 93, 94

expected project duration 94

first level project plan 86

frrst level project specification 85

follower case 131

free choice net 128

free delay 74, 75

free slack 71

flow relation 121

fuzzy attribute 29,31,185,186,208

fuzzy attribute over alternatives 190

fuzzy attribute over executions 191

fuzzy attribute over operations 189, 190

fuzzy graphs 188

fuzzy outranking 33, 208

fuzzy outranking relation 210

fuzzy relation 188,208

fuzzy set 187

GERT (Graphical Evaluation and Review Technique) 14,65,103

GERT network 104

global clock 119

graph 35

Heuristic Speeding Up Algorithm 83

hierarchy of plans 53,186

hierarchy of specifications 53

home state 143

homogeneous Markov renewal process 110, 111

incidence matrix 25,43, 139, 150, 151

incident 36

inclusive-Or (lOR) type node 104

incomparable 208

independent delay 75

independent slack 71

indifferent 207

indifference classe 33

indifference threshold 208

initial marking 135, 160

initial, terminal vertex 41

input side of a GERT vertex 104

Index 233

234 Index

invariant 149

intersection of fuzzy sets 187

in-tree 42

isolated 36

isomorphic graphs 37

i-th preference class 217

label 159

latest activity start 73

latest activity tennination 73

latest allowable time 93

latest event time 69

live marking 143

liveness of general place/transition nets 143

live net 143

live transition 143

longest or most pessimistic duration 88

loop 36,39

macro, macro-activity 52, 53, 66, 85

marked graph 126

marking of a net 22, 135, 160

marking of a place 22, 135

marking of a predicate 27, 160

Markov renewal process 111

matrix of relative weights 203

maximal execution 166

max-min composition 188

membership degree 187, 190, 192

milestone 44, 104

modal or most likely duration 88

Monte Carlo method 99, 117

MRP (Markov Renewal Process) method 112

multiple edge 36

multiset xn

nested choice schemata 182,186

net 120, 121

net elements 121

net invariants 25

network 39

nominal cost 76

occur (to) 136, 160

occur concurrently (to) 136

one-step reachability relation 125

operation 22

Operational Decision Making (ODM) 29, 185

optimal execution 200,207

optimal executions 32

output side of a GERT node 105

outranking degree 208

outranking relation 216

outranking techniques 185

out-tree 42

parallel arcs 39

parallel composition 179

partial ordering of activities 66

path 41

PERT (Program Evaluation and Review Technique) 9,65, 87

PERT cost planning and optimization 101

PERT scheduling planning 93

pessimistic, optimistic and modal activity costs 101

Petri net 119

place 22, 135

place/transition net (PT net) 21, 134, 135

. plan 2

planned cost 76

planned project execution 67

plan supervisor 133

polychotomy 216

Index 235

236 Index

possible shortening 84

post-condition 17

post-set 124

precedence relation 44

pre-condition 17

predicate 27, 159

predicate/transition net (PrT net) 25, 157,159

preference categories 33,208

preference classes 216

preference degree 33, 199

preference threshold 208

pre-set 124

primary choice schema 182

probabilistic branching 108

probability density function 87

probability distribution function 87

probability of completing the project not after a certain date 95

project 1, 2, 5

project completion date 67

project design 2

project plan 66

project specification 1,46,52,66

project verification 2

pure 126, 136

reachability graph 23, 141

reachability matrix 43

reachability relation 125

reachable by step x 125

reachable from 125

reachable marking 136

reachable vertex 41

realizable T -count 148

realization date 67

re-assignment of durations to activities 83

refmement 53

renewal 111

renewal function 112

reproducible marking 144

requested project completion date 83

resource 124

resource consumption 135

resource flow 153

resource production 135

root 36

r-step transition function 112

S-element 121

S-graph 126

S-invariant 146, 147, 149, 150, 151

S-sequence 172

S-simple 126

S-subnet 129, 146

S-vector 129

S-weights 146

Saaty's method 203

sequence 58,59,179

shortest or most optimistic duration 88

side-condition 126

simulation of GERT Networks 117

simple 126

simple digraph 39

sink 44, 107, 178

slack 67,70,72

source 44, 107, 178

speeding up cost 83, 86

split function of the subdivision 49

standard deviation of expected project completion time 95

standard deviation of unimodal distributions 90

state element 121

state machine 126

states of plan execution 120

Index 237

238 Index

step 131, 136

stochastic type node 105

stop state 143

strongly connected digraph 41

subdivision 49

subgraph 38

subgraph induced by 38

subproject 52, 66, 85

subnet 128

supervision of plan executions 67

support of a fuzzy attribute 187

support of an S-invariant 147

support of aT-invariant 148

synchronization transition 137

system state 131

T-count 147

T-element 121

T-graph 126

T-invariant 146, 148, 149, 150, 151

T-simple 126

T-subnet 129, 147

T-vector 129

three-parameter beta distribution 88

token 17, 135

top-down development of control net plans 171, 172

top-down development of project plans 52, 86

total concordance matrix 209

total delay 74, 75

transition 27, 135, 159

transition element 121

transition enabled for a constant tuple 27

transition function 111

transition matrix of a Markov renewal process 111

transition rule 19, 136, 160

transitions in conflict 137

tree 42

unconditional distribution function of project duration 110, 113

underlying graph 39

underlying net 135, 159

unfolding 49, 53

unimodal 88

unitary speeding up cost 84

union of fuzzy sets 188

updating coefficient 79

valency 36

variance, expectation of a beta distribution 91

vertex 35

vertex adjacency function 36

veto threshold 208

waiting state 124

walk 41

weak interpretation 121, 123

weakly connected graph 41

weight function 135

weighted attributes 200,201,202

weighted graph 36

weight of an arc 135

weight of an edge 36

Index 239