Seite - 155 - in Algorithms for Scheduling Problems
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Algorithms 2018,11, 57
• modelsM<ν>ofauxiliaryoperationsprogramcontrol.
Thesemodelsweredescribed in [13,19,33–38].Alongwith the initialproblemofprogramcontrol
(marked by the symbolГ), we consider a relaxed one (marked by P). The latter problemhas no
constraintsof interruptiondisabilityoperationsD(i)æ (whereD (i)
æ isanoperationæwithobject i. See,
forexample, [36,37]). Let thegoal functionofproblemГbe:
Jp= ⎧⎨⎩Job+ n∑i=1 si
∑
æ=1 m
∑
j=1 l
∑
λ=1 [
z(o,1)iæjλz (o,3)
iæjλ+ (aiæ) 2
2 −z(o,2)iæjλ ]2(
z(o,1)iæjλ )2⎫⎬⎭ ∣∣∣∣∣∣
t=Tf (23)
where the auxiliary variables z(o,1)iæjλ(t), z (o,2)
iæjλ(t), z (o,3)
iæjλ(t) are used for operationswith interruption
prohibition. More detailed information about these auxiliary variables and themethods of their
utilizationcanbefoundin[20].
The following theorem expresses the characteristics of the relaxed problem of MS
OPCconstruction.
Theorem1.LetPbea relaxedproblemfor theproblemГ.Then
(a) If theproblemPdoesnothaveallowable solutions, then this is true for theproblemГaswell.
(b) Theminimalvalueof thegoal function in theproblemPisnotgreater than theone in theproblemГ.
(c) If theoptimalprogramcontrolof theproblemPisallowable for theproblemГ, then it is theoptimalsolution
for theproblemГaswell.
Theproofof the theoremfollowsfromsimpleconsiderations.
Proof.
(a) If theproblemPdoesnothaveallowable solutions, thena controlu(t) transferringdynamic
system(1) fromagiveninitial statex(T0) toagivenfinalstatex(Tf)doesnotexist. Thesameend
conditionsareviolated in theproblemГ.
(b) It canbeseen that thedifferencebetween the functional Jp in (23)andthe functional Job in the
problemPisequal to lossescausedbyinterruptionofoperationexecution.
(c) Letu∗p(t), ∀ t∈ (T0, Tf] be anMS optimal program control in P and an allowable program
control inГ; letx∗p(t)beasolutionofdifferential equationsof themodelsM<o>,M<ν> subject
tou(t)=u∗p(t). If so, thenu∗p(t)meets therequirementsof the local sectionmethod(maximizes
Hamilton’s function) for theproblemГ. In thiscase, thevectorsu∗p(t),x∗p(t) return theminimum
tothe functional (1).
Theschemeofcomputationcanbestatedas follows.
Step 1. An initial solutionug(t), t∈ (T0,Tf] (an arbitrary allowable control, in otherwords,
allowableschedule) is selected. Thevariantug(t)≡0 isalsopossible.
Step2. ThemainsystemofEquation (1) is integratedsubject toh0(x(T0))≤ 0andu(t)=ug(t).
Thevectorx(o)(t) is receivedasa result. Inaddition, if t=Tf, then the recordvalue Jp= J (o)
p canbe
calculated,andthe transversalityconditions (5)areevaluated.
Step3. Theadjoint system(2) is integratedsubject tou(t)=ug(t)and(5)over the interval from
t=Tf to t=T0. For time t=T0, the first approximationψ (o)
i (T0) is received as a result. Here the
iterationnumber r=0iscompleted.
Step4. Fromthepoint t=T0 onwards, thecontrolu(r+1)(t) isdetermined(r=0,1, 2, . . . is the
numberof iteration) throughtheconditions (19). Inparallelwith themaximizationof theHamiltonian,
themainsystemofequationsandtheadjointsystemare integrated. Themaximization involves the
solvingofseveralmathematicalprogrammingproblemsateachtimepoint.
155
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Buch Algorithms for Scheduling Problems"
Algorithms for Scheduling Problems
- Titel
- Algorithms for Scheduling Problems
- Autoren
- Frank Werner
- Larysa Burtseva
- Yuri Sotskov
- Herausgeber
- MDPI
- Ort
- Basel
- Datum
- 2018
- Sprache
- englisch
- Lizenz
- CC BY 4.0
- ISBN
- 978-3-03897-120-7
- Abmessungen
- 17.0 x 24.4 cm
- Seiten
- 212
- Schlagwörter
- Scheduling Problems in Logistics, Transport, Timetabling, Sports, Healthcare, Engineering, Energy Management
- Kategorien
- Informatik
- Technik