Modeling and Optimization of Traffic Flow in Urban Areas - Czech ...

50 Chapter 4 TORSCHE Scheduling Toolbox for Matlab+T 62+T 162+T 102+T 172+T 132*T 143* T +83 T 12*T 93*T 123+T 42+T 152+T 72+T 22+T 112+T 32+T 52processingtime pFig. 4.8: Jaumann wave digital filterprecedence constraints. The nodes are labeled by the operation type (“+” or“∗”) and processing time p i . The example in Fig. 4.8 considers two parallelidentical processors, i.e. two general arithmetic units.Fig. 4.9 shows the consecutive steps performed in the toolbox. The firststep defines the set of the tasks with the precedence constraints for thescheduling algorithm satsch. The resulting schedule is displayed by theplot command. The optimal schedule is depicted in Fig. 4.10.>> procTime = [2,2,2,2,2,2,2,3,3,2,2,3,2,3,2,2,2];>> prec = sparse([6,7,1,11,11,17,3,13,13,15,8,6,2, 9,11,12,17,14,15,2 ,10],...[1,1,2, 2, 3, 3,4, 4, 5, 5,7,8,9,10,10,11,12,13,14,16,16],...[1,1,1, 1, 1, 1,1, 1, 1, 1,1,1,1, 1, 1, 1, 1, 1, 1, 1, 1],17,17);>> TS = taskset(procTime,prec);>> TS = satsch(TS,problem(P|prec|Cmax),2)Set of 17 tasksThere are precedence constraintsThere is schedule: SAT solverSUM solving time: 0.06sMAX solving time: 0.04sNumber of iterations: 2>> plot(TS)Fig. 4.9: Solution of the scheduling problem P |prec|C max in the toolbox4.3.3 Minimum Cost Multi-commodity Flow ProblemVarious optimization problems (e.g. routing) from the graph and networkflow theory can be reformulated on the minimum cost multi-commodity flow(MMCF) problem. The objective of the MMCF is to find the cheapest possibleways of sending a certain amount of flows through the network. Therefore,