Page - 157 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 57 Jp= 2 ∑ i=1 3 ∑ æ=1 ⎡⎣(a(o)iæ −x(o)iæ)2+ ( z(o,1)iæ1 z (o,3) iæ1 + ( a(o)iæ )2 2 −z(o,2)iæ1 )2( z(o,1)iæ1 )2⎤⎦∣∣∣∣∣∣ t=Tf − 2∑ i=1 3 ∑ æ=1 14∫ 0 γiæ(τ)u (o) iæ1(τ)dτ, (26) where γiæ(t) are given time functions denoting the most preferable operation intervals: γ11 =15γ+(6− t), γ12 = 10γ+(9− t), γ13 = 10γ+(11− t), γ21 = 20γ+(8− t), γ22 = 15γ+(8− t), γ23 = 30γ+(11− t). Here, γ+(α) = 1 ifα≥ 0 andγ+(α) = 0 ifα< 0. The integration element in (27) can be interpreted similarly to previous formulas through penalties for operations beyond the bounds of the specified intervals. The scheduling problem can be formulated as follows: facilities-functioningschedule [controlprogram → u(t)] returningaminimalvalue to the functional (27) under theconditions (26)andtheconditionof interruptionprohibitionshouldbedetermined. Theresultingschedule is showninFigure3. Theupperpart of Figure 3 shows the initial feasible schedule implementingFirst-in-first-out processing. Thesecondandthirdcases illustrateaconflictresolutionforD(1)1 interruption. Theoptimal schedule ispresented in the thirdandthe forthrows. Figure3.Optimalscheduleresults. 5.QualitativeandQuantitativeAnalysisof theMSSchedulingProblem Therepresentationof theschedulingproblemintermsofadynamicsystem(1)controlproblem allowsapplicationsofmathematicalanalysis tools frommoderncontrol theory(38,43,44). For example, qualitative analysis based on control theory as applied to the dynamic system (1) provides the results listed in Table 1. This table confirms that our interdisciplinary approach to the description ofMSOPCprocesses provides a fundamental base forMSproblemdecisions (or control andmanagement) thathaveneverbeenpreviously formalizedandhavehighpractical importance. The tablealsopresentspossibledirectionsofpractical implementation(interpretation) for theseresults in thereal schedulingofcommunicationwithMS.Forexample, criteria forcontrollability andattainability inMSOPCproblems canbeused forMScontrol processverification for agiven timeinterval. 157