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

22 Chapter 3 General Light Controlled Intersection ModelThe axes show the speeds of transitions T 4 and T 5 . A gray polygon boundsthe area where the possible speed v 4 and v 5 can be placed. This polygonis bounded by v 1 , v 2 and v 3 from one side and by the axes from the otherside. All combinations of the speeds v 4 and v 5 which are in proportion to themaximal speed V 4 and V 5 are located on the line λ. Point A, in Fig. 3.2(a),is where v 4 and v 5 are at a maximum and are located on line λ. The conflictresolution is given by this point A and the speed transitions for T 4 and T 5are v 4 = 30 and v 5 = 10, respectively.Let us consider the modification of speed v 1 from v 1 = 35 to v 1 = 25(see Fig. 3.2(b)). This situation corresponds to point B where v 4 and v 5 areat a maximum and are located closest to line λ. The conflict resolution isgiven by this point and the speed of the transitions T 4 and T 5 is v 4 = 25and v 5 = 15, respectively. Note that the speed is not in proportion to themaximal speed V 4 and V 5 , but it is the maximum firing speed combination.Let us assume v 1 to be equal to 15 (see Fig. 3.2(c)). This situationcorresponds to point C where v 4 and v 5 are at a maximum and are locatedclosest to line λ. The conflict resolution is given by this point and the speedof the transitions T 4 and T 5 is v 4 = 15 and v 5 = 18, respectively. Note thatthere is no actual conflict in this case.We can generalize that the conflict resolution corresponds to the pointlocated on the line segment DE closest to line λ. In the special case whenpoint D is equal to E i.e. |DE| = 0 the transition speed is placed to thispoint, see Fig. 3.2(c) for point C.This geometrical interpretaion of speed computing can be formulated asfollow: consider place P i and the set of transitions Pi◦ having a conflict.The set of transitions Pi◦ indexes is denoted as C i . Variable vi s (t) denotesthe sum of the speeds supplying the place P i at time t. For each transitionT j where j ∈ C i initialize maximum permissible transition speed vjmax (t) asa minimum from the maximal speed V j and separately sum the speed oftransitions which supplied the place from ◦ T j with zero marking. Set thetransition speed v j (t) = 0 and modify it by the iterative Algorithm 3.1.3.2.2 Linear Programming for Conflict ResolutionThis subsection shows how linear programming (LP) can be used for conflictresolution computation. At first, we will use the example in Fig. 3.1 to showhow a conflict resolution can be solved by linear programming at time t. In