Transportation Planning Asian Institute of Technology Traffic Assignment Transportation Planning Asian Institute of Technology

Contents Basic Concepts Traffic Assignment Methods All-or-Nothing Assignment Stochastic Methods Congested Assignment

Basic Concepts Introduction The equilibrium point between demand and supply defines the price at which goods will be exchanged. Supply side is made up of road (transport) network, S(L,C) Cost C is a function of distance, free-flow speed, capacity, and speed-flow relationship Demand side is made up of the number of trips by O-D pair and mode under a given level of service.

Basic Concepts Introduction In private transport mode, a main element defining LOS is travel time. In public transport mode, addition consideration should be given to routes, capacity, frequency and other random characteristics. At a higher level resulting flow may affect choices of mode, destination and time of day for travel. (System Equilibrium)

Basic Concepts Definitions and Notation Tijr = number of trips from i and j via route r Va = the flow on link a C(Va) = the cost-flow relationship for link a c(Va) = the actual cost for a particular level of flow Va c(Va) = 0 is free-flow cost cijr = cost of traveling from i and j via route r dijra = 1 if link a is on path r from i to j; 0 otherwise superscript n shows iteration #, and superscript * indicates optimum value.

Basic Concepts Speed-Flow and Cost-Flow Curve Speed, S (km/h) Travel Time (minutes/km) Vmax Vmax Flow, V Flow, V

Basic Concepts Speed-Flow and Cost-Flow Relationship The cost on a link a is a function of all the flows V in the network. Ca = Ca({V}) Considering long links to simplify the problem (separable): Ca = Ca(Va) The function should be Realistic - Allowing overloading Non-decreasing - Easy to transfer Continuous and differentiable

Basic Concepts Total operating cost = VaCa(Va) Marginal cost contributed by the marginal addition of a vehicle to the stream (10.3) BPR Function (10.6) UK Department of Transport divides speed-flow into three regions: free-flow, moderate flow, and over-capacity.

Traffic Assignment Methods Introduction Primary objectives To obtain aggregate network measures To estimate zone-to-zone travel costs To obtain reasonable link flow & identify congested links Congestion will be studied using peak-hour trip matrices

Traffic Assignment Methods Route Choice Most common approximation considers only time and monetary costs (proportional to travel distance) Different drivers may choose different routes when traveling between the same two points Different individual perceptions Congestion effects affect shorter route first and attract more traffic to initially less attractive routes

Traffic Assignment Methods Example Demand 3,500 from A to B Capacity (through)= 1,000 vph Capacity (bypass) = 3,000 vph A B Stochastic Effects? No Yes No All-or-nothing pure stochastic Capacity Restraint? Wardop’s equilibrium Stochastic user equilibrium Yes

Traffic Assignment Methods Tree Building Identifies a set of routes which might be considered attractive to drivers. All dA = ∞ except dS = 0 set L = loose-end containing all nodes Set origin S as current node A then Check each link (A, B). If dA + dA,B < dB then dB = dA + dA,B Remove A from L then select another node until L is empty.

All-or-Nothing Assignment Example A 5 6 2 10 4 8 4 3 6 4 C 3 4 8 10 5 2 3 2 D B A-C = 400 B-C = 300 A-D = 200 B-D = 100

All-or-Nothing Assignment Example A 5 6 2 10 4 8 4 3 6 4 C 3 4 8 10 5 2 3 2 D B

All-or-Nothing Assignment Example A 600 400 400 200 400 200 C 200 200 D B 200

All-or-Nothing Assignment Example A 600 400 400 200 400 300 500 C 300 300 200 100 300 D B 300 Does not reflect real situation Act like desired lines

Stochastic Methods Varied perceptions of costs and other attributes. Second-best routes must be identified. The number of second-best routes may be very large. Simulation-based method Proportion-based method

Stochastic Method Simulation-based Method Relies on Monte Carlo Simulation Distribution of perceived costs are assumed independent. Drivers choose the route that minimizes their perceived route costs. Proportion of drivers Mean link cost

Stochastic Method Simulation-based Method Split population traveling between each OD pair into N segments. (Tij/N trips) n = 0 n = n+1 For each O-D compute perceived cost for each link using random numbers to sample link cost. Build a minimum path from i to j and assign Tij/N If n = N stop, otherwise go to 2.

Congested Assignment Wardrop’s Equilibrium No user can improve the trip travel time (cost) by unilaterally change changing the routes Traffic arranges itself in congested networks such that all used routes between O-D pair have equal and minimum costs while all unused routes have greater of equal costs.

Congested Assignment Example Time Ct = 10 + 0.02Vt Cb = 15 + 0.005Vb Town Bypass Time Time Flow, V Flow, V Ct = 10 + 0.02Vt Cb = 15 + 0.005Vb Vb + Vt = V; Vb = 0.8V – 200 = 1,400

Congested Assignment Wardrop’s Equilibrium Second Principle: under social equilibrium conditions, traffic should be arranged in congested networks in such a way that the average or total travel cost is minimized. This become known as social equilibrium or system optimum.

Congested Assignment Wardrop’s Equilibrium Complexity City suburb Route 1: x1 = 3 km v1f = 45 km/hr t1f = (3x45)/60 = 4 min t1 = 4 + (q1/1000)2 min Route 2: x2 = 6 km v2f = 60 km/hr t2f = (6x60)/60 = 6 min t2 = 6 + 2(q2/500) min

Congested Assignment Wardrop’s Equilibrium Complexity

Congested Assignment Incremental Assignment Traffic does not all come at the same time, but gradually spread over a given time period. Incremental loading traffic into the network is more realistic. Easy to program and solve. Represents a build up of traffic congestion during a short peak period.

Congested Assignment Incremental Assignment Select an initial set of current link costs. Set Va = 0 and select fraction pn such that (Typical value 0.4, 0.3, 0.2, 0.1) Build the set of minimum cost trees using current costs. Set n = n + 1 Load Tn = pnT (all-or-nothing) with a set of auxiliary flow Fa. Calculate flow on each link as: Calculate a new set of current link cost based on the flows Van Continue until all fractions of T is assigned.

Congested Assignment Example Consider again the problem of the two routes, town centre and bypass, of previous example. We split the demand of 2000 trips into four increments of 0.4, 0.3, 0.2 and 0.1 of this demand, i.e. 800, 600, 400 and 200 trips. At each increment we calculate the new travel costs using equations The following table summarizes the results of this algorithm

Congested Assignment Example V = 2000 Ct = 10 + 0.02Vt Cb = 15 + 0.005Vb N Increment Flow Town Cost Town Flow Bypass Cost Bypass 10 15 1 800 26 2 600 18 3 400 1000 20 4 200 1200 21

Congested Assignment Method of Successive Averages (MSA) Iterative algorithm developed to overcome the problem of overloading on low- capacity links. Heuristics approach. Provide close solution to real equilibrium.

Congested Assignment Method of Successive Averages (MSA) Initialize all Va = 0 and set n = 0 Build a set of minimum cost tree with the current costs; set n = n + 1 Load the whole matrix T (all-or-nothing) then obtain auxiliary flow Fa. Calculated the current flows where 0 ≤ f ≤ 1 Calculate a new set of current link costs based on flows Van. Stop when converged.

Congested Assignment Method of Successive Averages (MSA) Iteration f Flow town Cost town Flow bypass Cost bypass 1 F 2000 Vn 50 15 2 1/2 1000 40 20 3 1/3 667 23.3 1333 21.7 4 1/4 500 1500 22.5 5 1/5 800 26 1200 21

Congested Assignment Method of Successive Averages (MSA) Iteration f Flow town Cost town Flow bypass Cost bypass 6 F 2000 Vn 1/6 667 23.3 1333 21.7 7 1/7 572 21.4 1428 22.1 8 1/8 750 25 1250 21.2 9 1/9 10 1/10 600 22 1400

Assignment #9 Providing two origins, A and B, and two destinations, L and M, with the following trip interchanges A − L = 600 A − M = 400 B − L = 300 B − M = 400 Find the shortest path and the link volumes by all-or-nothing assignment 5 11 13 2 A C F I L 15 9 12 7 7 9 9 11 7 3 B D G J 12 11 7 9 9 7 3 2 E H K M

Assignment #9 Consider the following network where there are 100 vehicles per hour travelling from A to D and 500 from B to D. The travel time versus flow relationships are depicted in the figure in minutes and the flow q in vehicles per hour. Use an incremental loading technique with fractions 40, 30, 20 and 10% of the total demand to obtain an approximation to equilibrium assignment. t = 1.0+0.001q t = 2.0+0.01q A C D t = 3.4+0.001q t = 12.0+0.001q t = 1.0+0.001q A C