List of Symbols

The list of symbols is organized as follows. Symbols are sorted by alphabetin a lexicographic order. We have separated the symbols by capital and non­capital letters. As an example, AFTER comes before BEFORE, but afterafter. Greek letters have been treated as alphabet letters, that is, a, {3, I,etc. correspond with a, b, c etc.

The symbols used have been summarized by chapters. We did not includeall symbols, but only the most important ones and only those which areused in different parts of the thesis. If a symbol has not been included thenit is or, at least, should have been introduced near its occurrence. On thecontrary, the list bears some redundancy, i.e. some symbols that are neitherself-explanatory nor standard symbols are listed several times while othersonly have been explained once. Of course, these symbols have been selectedon a quite subjective level.

Finally, according to the famous rule "let x be a number that we call y",sometimes different symbols have been used for the same object. On thecontrary, some symbols have been "overloaded", i.e. have different meaningsin different chapters. As an example, we use A or 0' to denote a subset ofoperations. Another example is the use of S which denotes the search spaceof a local search algorithm and the set of all solutions of a DSP instancewhich in both cases seems to be a natural choice due to the common roots.The reason for this was not to distract the reader, but definitely to improvereadability. Again, however, this is quite a matter of taste.

238

Symbol

a

c

11008(x)8(Xi 1 ,··· ,XiJ

f

nselect

xz

rr(~)

CONS~

'D(x)'DOM:F(P)N

PSo

v

LIST OF SYMBOLS

Descriptionset of all subsets of S.assignment: a = (at, ,an) E 'D(xt} X ... X 'D(xn).constraint: c : 'D(XiJ x X 'D(Xik) -+ {true, false}.(domain) consistency test: 1 : 0 -+ 0.series of consistency tests: 100 = bg )gEN E r N

.

current domain of x: 8(x) ~ 'D(x).cartesian product of current domains:8(Xi 1 l ... ,Xik) ~ 'D(Xi 1 ) x ... x'D(Xik).evaluation function (of a local search algorithm):f: S -+ JR.number of variables.selection function (of a local search algorithm):select: 2s -+ S.variables.objective function of a COP instance: z : 'D(xt} x... x 'D(xn ) -+ Rset of consistency tests.~-fixpoint induced by r: definition 2.9.set of constraints.set of current domains: ~ = {8(x) I x E V}.domain of x.set of domains: 'DOM = {'D(x) I x E V}.set of all feasible assignments (solutions) of P.neighbourhood structure (of a local search algo­rithm): N: S -+ 2s .instance of the CSP or COP: definitions 2.1 or 2.2.search space (of a local search algorithm).set of all sets of current domains: 0 = 2"D(x 1 ) X ... x2"D(x n ) •

set of all variables: V = {Xl, ... , x n }.

Table 8.1: List of symbols for chapter 2.

LIST OF SYMBOLS

Symbol

0, *ii-+ji+7jjob( i)mach( i)n

Pires( i)st;CC*(P)C*(S)C*(ST)

D99s

9(P)oop

PSSTt1'"

239

Description(dummy) operations start and end.operation: i EO.i has to be processed before j: i -+ j E C.i and j cannot be processed in parallel: i +7 JED.job of i.machine used by i.number of operations.processing time of i: Pi EN.~et of resources used by i.start time (variable) of i: st; E No.relation of conjunctions: C ~ 0 x O.optimal makespan of P.makespan of the selection S: lemma 3.1.makespan of the schedule ST = (st 1, ... , st n ):

C*(ST) =maXiEo{sti + p;}.relation of disjunctions: D ~ 0 x O.disjunctive graph: subsection 3.1.2.directed graph associated to the disjunctive graph9 and the selection S: 9s = (O,CuS).disjunctive graph associated to P: definition 3.1.set of operations: 0 = {I, ... ,n}.clique decomposition: definition 3.4.set of processing times: P = {Pi liE O}.instance of the DSP: subsection 3.1.1.partial or complete selection: S ~ D.schedule: ST = (st 1, ... ,stn ) E No.set of weights: W = {w;--+j I i -+ j E CUD}.

Table 8.2: List of symbols for chapter 3.

240

Symbol(h)

fA,;

<5;est;estfeet;let;1st;Ist7np; (tl ,t2)

[it,t2)[tl, t2]AA -+ i, i -+ A

C*(Po(A))

LIST OF SYMBOLS

Description

h ~ 4: output consistency test for the couple (A, i):subsection 4.4.3,h ;::: 5: input negation consistency test for the cou­ple (A, i): subsection 4.5.3.current domain of i: <5; ~ No.earliest start time of i: est; = min<5;.adjusted earliest start time of i.earliest completion time of i: eet; = est; + p;.latest completion time of i: let; = 1st; + p;.lat~st start time of i: 1st; =max <5; .adjusted latest start time of i.number of operations.interval processing time of i in the time interval[tl, t2): see (4.60).time interval: [tl, t2) = {tl' tl + 1 , t2 - I}.time interval: [tl, t2) = {tl, tl + 1 ,t2}'subset of operations: A ~ O.i has to be processed after (before) all operationsin A.optimal makespan if all oper-a-tions in A arescheduled within their current domains withoutpreemption.optimal makespan if all operations in A are sched­uled within their current domains with preemptionallowed.

Table 8.3: List of symbols for chapter 4 (Part 1).

LIST OF SYMBOLS

Symbol

r out (h)ESTmin(A)

LBh(A, i)

LCTmax(A)

P(A)

241

Descriptionset of input negation consistency tests: subsection4.5.3.set of output consistency tests: subsection 4.4.3.minimal earliest start time of operations in A:ESTmin(A) =miniEA{est;}.time bound adjustment for out-put con-sis-ten-cytests: theorem 4.8.time bound adjustment for input negation consis­tency tests: theorem 4.16.maximal latest completion time of operations in A:LCTmax(A) =maXiEA {leti}'subset of operations which must be processed com­pletely or partially in the time interval [tl, t2):0(tl,t,) ={i E 0 IPi(tl, t2) > O}.sum of processing times of operations in A: P(A) =L:iEA Pi·sum of interval processing times of operations inA in the time interval [tl,t2): P(A,tl,t2) =L:iEA Pi(tl, t2)'task set of A: T(A) =T(ESTmin(A), LCTmax(A)).task set: T(it,t2) = {i E 0 Itl::; esti,lcti::; t2}'upper bound: U B > C*(P).

Table 8.3: List of symbols for chapter 4 (Part 2).

242

Symbola(u)j3(u)vqiPriBBC*(P)C*(S)Ea

Ebg

LB(i -+ 0' \ {i})

LB(O' \ {i} -+ i)

M(v)

0'P(O')

Prq(O')

sS(v)

S(v)

LIST OF SYMBOLS

Descriptionson node of after-candidate u.son node of before-candidate u.node.tail of i: qi = UB -leti'root node.head of i: ri = esti'path or block of operations: definition 5.1.list of paths or blocks: B = (B1 , ... , Bh)'optimal makespan of P.'makespan of the selection S: lemma 3.1.set of after-candidates: subsection 5.1.2.set of before-candidates: subsection 5.1.2.preemptive lower bound if i is processed first in 0':see (5.3).preemptive lower bound if i is processed last in 0':see (5.3).set of solutions compatible with S(v): M(v) ={S E S IS(v) ~ S}.subset of operations: 0' ~ O.subproblem instance associated to the subset 0' ofoperations: subsection 5.2.1.1.subproblem instance associated to the subset 0' ofoperations with heads and tails: subsection 5.2.1.2.subproblem instance associated to the subset 0' ofoperations with heads and tails and preemption al­lowed: subsection 5.2.1.3.set of all solutions (complete and acyclic selections).selection associated to node v, see also (5.1) and(5.2).solution compatible with S(v): S(v) C S(v).

Table 8.4: List of symbols for chapter 5.

LIST OF SYMBOLS

Symbol1f

C*(P)C*(S)P(O')

Prq(O')

UB(P', 1r)

243

Descriptionpercent value.optimal makespan of P.makespan of the selection S: lemma 3.1.subproblem instance associated to the subset 0' ofoperations: subsection 5.2.1.1.subproblem instance associated to the subset 0' ofoperations with heads and tails: subsection 5.2.1.2.hypothetical upper bound for the subproblem in­stance P': UB (P' , 1r) =C* (P') . (1 + 1r/100) .

Symbol

0, *e

A(U, v)nqi

ri

C*(5)C*(5)

SS'Succ(i)

Table 8.5: List of symbols for chapter 6.

Description(dummy) operations start and end.maximum number of all immediate predecessorsand successors of all operations.length of the longest path from u to v.number of operations.tail of i with respect to a solution, thus, qi is thelength of the longest path from i to the operationend (*) minus Pi.head of i with respect to a solution, thus, ri is thelength of the longest path from the operation startoto i.makespan of the selection S: lemma 3.1.

approximation of the makespan of 5: C*(5) ~C*(5).directed graph associated to the disjunctive graphg and the selection 5: gs = (0, C U 5).neighbourhood structures: definitions i.I and i.2.immediate predecessors of i in the neighbouring so­lution 5': Pred(i) ={j I j -t i E C U5'}solution (complete and acyclic selection).neighbouring solution.immediate successors of i in the neighbouring solu­tion S': Succ(i) = {j Ii -t j E CU S'}

Table 8.6: List of symbols for chapter 7.

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Vol. 423: K. Marti. P. Kall (Eds.), Stochastic Programming.Numerical Techniques and Engineering Applications. VIll,351 pages. 1995.

Vol. 424: G. A. Heuer. U. Leopold-Wildburger. Silverman'sGame. X, 283 pages. 1995.

Vol. 425: J. Kohlas, P.-A. Monney, A Mathematical Theoryof Hints. XIll, 419 pages, 1995.

Vol. 426: B. Finkensttidt, Nonlinear Dynamics in Eco­nomics. IX, 156 pages. 1995.

Vol. 427: F. W. van Tongeren, Microsimulation Modellingof the Corporate Firm. XVII, 275 pages. 1995.

Vol. 428: A. A. Powell, Ch. W. Murphy, Inside a ModernMacroeconometric Model. XVIII. 424 pages. 1995.

Vol. 429: R. Durier, C. Michelot. Recent Developments inOptimization. VIll, 356 pages. 1995.

Vol. 430: J. R. Daduna, I. Branco, J. M. Pinto Paixiio (Eds.),Computer-Aided Transit Scheduling. XIV, 374 pages. 1995.

Vol. 431: A. Aulin, Causal and Stochastic Elements in Busi­ness Cycles. XI, 116 pages. 1996.

Vol. 432: M. Tamiz (Ed.), Multi-Objective Programmingand Goal Programming. VI, 359 pages. 1996.

Vol. 433: J. Menon, Exch~c;ge Rates and Prices. XIV, 313pages. 1996.

Vol. 434: M. W. J. Biok. Dynamic Models of the Firm. Vll,193 pages. 1996.

Vol. 435: L. Chen, Interest Rate Dynamics, DerivativesPricing, and Risk Management. XII, 149 pages. 1996.

Vol. 436: M. Klemisch-Ahlert. Bargaining in Economic andEthical Environments. IX, 155 pages. 1996.

Vol. 437: C. Jordan, Botching and Scheduling. IX, 178 pages.1996.

Vol. 438: A. Villar. General Equilibrium with IncreasingReturns. XIll, 164 pages. 1996.

Vol. 439: M. Zenner, Learning to Become Rational. VII,201 pages. 1996.

Vol. 440: W. Ryll, Litigation and Settlement in a Game withIncomplete Information. VIII. 174 pages. 1996.

Vol. 441: H. Dawid, Adaptive Learning by GeneticAlgorithms. IX, 166 pages. 1996.

Vol. 442: L. Corch6n, Theories of Imperfectly CompetitiveMarkets. XIll, 163 pages. 1996.

Vol. 443: G. Lang, On Overlapping Generations Models withProductive Capital. X, 98 pages. 1996.

Vol. 444: S. !l'lrgensen, G. Zaccour (Eds.). DynamicCompetitive Analysis in Marketing. X, 285 pages. 1996.

Vol. 445: A. H. Christer, S. Osaki, L. C. Thomas (Eds.).Stochastic Modelling in Innovative Manufactoring. X, 361pages. 1997.

Vol. 446: G. Dhaene, Encompassing. X, 160 pages. j ~97.

Vol. 447: A. Artale, Rings in Auctions. X, 172 pages. 1997.

Vol. 448: G. Fandel, T. Gal (Eds.), MUltiple Criteria DecisionMaking. XII, 678 pages. 1997.

Vol. 449: F. Fang, M. Sanglier (Eds.), Complexity and Self­Organization in Social and Economic Systems. IX, 317pages, 1997.

Vol. 450: P. M. Pardalos, D. W. Hearn, W. W. Hager, (Eds.),Network Optimization. VIII. 485 pages. 1997.

Vol. 451: M. Salge, Rational Bubbles. Theoretical Basis,Economic Relevance, and Empirical Evidence with a SpecialEmphasis on the German Stock Market. IX, 265 pages. 1997.

Vol. 452: P. Gritzmann, R. Horst, E. Sachs, R. Tichatschke(Eds.), Recent Advances in Optimization. VIII, 379 pages.1997.

Vol. 453: A. S. Tangian, J. Gruber (Eds.), ConstructingScalar-Valued Objective Functions. VIII, 298 pages. 1997.

Vol. 454: H.-M. Krolzig, Markov-Switching Vector Auto­regressions. XIV, 358 pages. 1997.

Vol. 455: R. Caballero, F. Ruiz, R. E. Steuer (Eds.), Advancesin Multiple Objective and Goal Programming. VIII, 391pages. 1997.

Vol. 456: R. Conte, R. Hegselmann, P. Terna (Eds.), Simu­lating Social Phenomena. VIII, 536 pages. 1997.

Vol. 457: C. Hsu, Volume and the Nonlinear Dynamics ofStock Returns. VIII. 133 pages. 1998.

Vol. 458: K. Marti, P. Kall (Eds.). Stochastic ProgrammingMethods and Technical Applications. X, 437 pages. 1998.

Vol. 459: H. K. Ryu, D. J. Sloltje. Measuring Trends in U.S.Income Inequality. Xl, 195 pages. 1998.

Vol. 460: B. Fleischmann, J. A. E. E. van Nunen, M. G.Speranza, P. Stahly, Advances in Distribution Logistic. XI,535 pages. 1998.

Vol. 461: U. Schmidt, Axiomatic Utility Theory under Risk.XV, 201 pages. 1998.

Vol. 462: L. von Auer, Dynamic Preferences, ChoiceMechanisms, and Welfare. XII, 226 pages. 1998.

Vol. 463: G. Abraham-Frois (Ed.). Non-Linear Dynamicsand Endogenous Cycles. VI, 204 pages. 1998.

Vol. 464: A. Aulin, The Impact of Science on EconomicGrowth and its Cycles. IX, 204 pages. 1998.

Vol. 465: T. J. Stewart, R. C. van den Honert (Eds.), Trendsin Multicriteria Decision Making. X. 448 pages. 1998.

Vol. 466: A. Sadrieh, The Alternating Double AuctionMarket. VII, 350 pages. 1998.

Vol. 467: H. Hennig-Schmidt, Bargaining in a Video Ex­periment. Determinants of Boundedly Rational Behavior.XII, 221 pages. 1999.

Vol. 468: A. Ziegler, A Game Theory Analysis of Options.XIV. 145 pages. 1999.

Vol. 469: M. P. Vogel, Environmental Kuznets Curves. XIII,197 pages. 1999.

Vol. 470: M. Ammann, Pricing Derivative Credit Risk. XII,228 pages. 1999.

Vol. 471: N. H. M. Wilson (Ed.). Computer-Aided TransitScheduling. XI, 444 pages. 1999.

Vol. 472: J.-R. Tyran. Money Illusion and StrategicComplementarity as Causes of Monetary Non-Neutrality.X, 228 pages. 1999.

Vol. 473: S. Helber, Performance Analysis of Flow Lineswith Non-Linear Flow of Material. IX, 280 pages. 1999.

Vol. 474: U. Schwalbe, The Core of Economies withAsymmetric Information. IX. 141 pages. 1999.

Vol. 475: L. Kaas, Dynamic Macroeconomics with ImperfectCompetition. XI, 155 pages. 1999.

Vol. 476: R. Demel, Fiscal Policy. Public Debt and theTerm Structure of Interest Rates. X, 279 pages. 1999.

Vol. 477: M. Thera, R. Tichatschke (Eds.), III-posedVariational Problems and Regularization Techniques. VIII,274 pages. 1999.

Vol. 478: S. Hartmann. Project Scheduling under LimitedResources. XII, 221 pages. 1999.

Vol. 479: L. v. Thadden, Money, Inflation. and CapitalFormation. IX, 192 pages. 1999.

Vol. 480: M. Grazia Speranza, P. Stahly (Eds.), New Trendsin Distribution Logistics. X. 336 pages. 1999.

Vol. 481: V. H. Nguyen, J. J. Strodiot, P. Tossings (Eds.).Optimation. IX, 498 pages. 2000.

Vol. 482: W. B. Zhang, A Theory of International Trade.Xl, 192 pages. 2000.

Vol. 483: M. Ktinigstein. Equity. Efficiency and EvolutionaryStability in Bargaining Games with Joint Production. XII.197 pages. 2000.

Vol. 484: D. D. Gatti. M. Gallegati. A. Kirman, Interactionand Market Structure. VI, 298 pages. 2000.

Vol. 485: A. Garnaev, Searc~ Games and Other Applicationsof Game Theory. VIII. '.5 pages. 2000.

Vol. 486: M. Neur,.(t. Nonlinear Labor Market Dynamics.X, 175 pages. 20'j().

Vol. 487: Y. Y. Haimes. R. E. Steuer (Eds.), Research andPractice in Multiple Criteria Decision Making. XVII.553 pages. 2000.

Vol. 488: B. Schmolck. Ommitted Variable Tests andDynamic Specification. X, 144 pages. 2000.

Vol. 489: T. Steger. Transitional Dynamics and EconomicGrowth in Developing Countrie,. VIIT. 151 page,. 2000.

Vol. 490: S. Minner, Strategic Safety Stocks in SupplyChain,. XI, 214 page,. 2000.

Vol. 491: M. Ehrgott, Multicriteria Optimization. VIII. 242page,. 2000.

Vol. 492: T. Phan Huy, Constraint Propagation in FlexibleManufacturing. IX, 258 pages. 2000.