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

4.3 Implemented Algorithm 51P 2P 1 T 6 T 17 T 12 T 1 T 14 T 9 T 5 T 16T 15 T 8 T 7 T 11 T 2 T 3 T 13 T 10 T 40 5 10 15 tFig. 4.10: The optimal schedule of the Jaumann filterTORSCHE includes a multicommodityflow function.The MMCF problem is defined by a directed flow network graph G(V , E),where the edge (u, v) ∈ E from node u ∈ V to node v ∈ V has a capacitycap uv and a cost a uv . There are ψ commodities K 1 , K 2 , . . . , K ψ definedby K i = (source i , sink i , b i ) where source i and sink i stand for sourceand sink node of commodity i, and b i is the volume of the demand. Theflow of commodity i along the edge (u, v) is f i (u, v). The objective isto find an assignment of the flow f i (u, v) which minimizes the total costJ = ∑ (∀(u,v)∈Ea uv · ∑ψ )i=1 f i(u, v) and satisfies the following constraints:∑ ψi=1 f i(u, v) ≤ cap uv ∀(u, v) ∈ E,∑u∈V f i(u, w) = ∑ v∈V f i(w, v) w ∈ V \ {source i , sink i },∀i = 1 . . . ψ,∑w∈V f i(source i , w) = ∑ w∈V f i(w, sink i ) = b i ∀i = 1 . . . ψ.(4.1)The function multicommodityflow solves MMCF problem by the transformationto the linear programming problem [Kort 06]. Let P be the set ofall paths from the source node source i to the sink node sink i for all commoditiesi = 1, 2, 3 . . . ψ. Let M be a 0-1-matrix whose columns correspondto the elements p of P and whose rows correspond to the edges of G, whereM (u,v),p = 1 iff (u, v) ∈ p. Similarly, let N be a 0-1-matrix whose columnscorrespond to the elements of P and whose rows correspond to the commodities,where N i,p = 1 iff source i and sink i are the start and end nodes of pathp. Then the LP problem using the variables defined above is:min (a P · y) , (4.2)