**Lecture notes in Transportation Systems Engineering**

**August 23, 2011**

- To estimate the volume of traffic on the links of the network and obtain aggregate network measures.
- To estimate inter zonal travel cost.
- To analyze the travel pattern of each origin to destination(O-D) pair.
- To identify congested links and to collect traffic data useful for the design of future junctions.

The types of traffic assignment models are all-or-nothing assignment (AON), incremental assignment, capacity restraint assignment, user equilibrium assignment (UE), stochastic user equilibrium assignment (SUE), system optimum assignment (SO), etc. The frequently used models all-or-nothing, user equilibrium, and system optimum will be discussed in detail here.

and total flows from 1 to 2 is given by.

Since the shortest path is Link 1 all flows are assigned to it making =12 and = 0.

where is the flow on path , is the travel cost on path , and is the minimum cost. Equation 3 can have two states.

- If = 0, from equation 2 0. This means that all used paths will have same travel time.
- If 0, then from equation 2 = 0.

- The user has perfect knowledge of the path cost.
- Travel time on a given link is a function of the flow on that link only.
- Travel time functions are positive and increasing.

where is the path, equilibrium flows in link a, travel time on link a, flow on path k connecting O-D pair r-s, trip rate between r and sand is a definitional constraint and is given by

(5) |

The equations above are simply flow conservation equations and non negativity constraints, respectively. These constraints naturally hold the point that minimizes the objective function. These equations state user equilibrium principle.The path connecting O-D pair can be divided into two categories : those carrying the flow and those not carrying the flow on which the travel time is greater than (or equal to)the minimum O-D travel time. If the flow pattern satisfies these equations no motorist can better off by unilaterally changing routes. All other routes have either equal or heavy travel times. The user equilibrium criteria is thus met for every O-D pair. The UE problem is convex because the link travel time functions are monotonically increasing function, and the link travel time a particular link is independent of the flow and other links of the networks. To solve such convex problem Frank Wolfe algorithm is useful.

subject to

Substituting , in the above formulation will yield the unconstrained formulation as below :

Differentiate the above equation and equate to zero, and solving for and then leads to the solution = 5.8, = 6.2.

subject to

(7) | |||

(8) | |||

(9) | |||

(10) |

equilibrium flows in link a, travel time on link a, flow on path k connecting O-D pair r-s, trip rate between r and s.

(11) | |||

(12) |

Substituting

(13) |

Differentiate the above equation to zero, and solving for and then leads to the solution = 5.3,= 6.7 which gives Z(x) = 327.55

subject to

Substituting , in the above formulation will yield the unconstrained formulation as below :

Differentiate the above equation and equating to zero,

Travel times are

It follows that the travel times are at user equilibrium.

Prof. Tom V. Mathew 2011-08-23