Wardrop's First Principle and its Dynamic Extensions
Traditionally, the travel choice component of DTA is developed based on Wardrop's (1952) first principle of static traffic assignment. War dr op's first principle or the user equilibrium principle states that the journey times on all routes actually used are equal and are not greater than those which would be experienced by a single vehicle on any unused route. In other words, the travel times of all used routes between the same origin-destination (OD) pair are equal and minimal. This principle assumes that each traveler is identical, non-cooperative and rational in selecting the shortest route, and knows the exact travel time he/she will encounter. If all travelers select routes according to this principle the road network will be at equilibrium, such that no one can reduce their travel times by unilaterally choosing another route of the same OD pair. This principle has been extended to consider generalised travel cost instead of travel time, where generalised travel cost can include the monetary cost of in-vehicle travel time, tolls, parking charges and fuel consumption costs.
User equilibrium can also be explained from the economic concept of utility maximisation (Oppenheim, 1995), where utility measures the degree of satisfaction travelers derive from their choices. In the simplest case, a traveler's utility equals his/her budget or income minus the travel time. In this sense, the assumption of Wardrop's first principle can be viewed as travelers selecting routes to maximise their individual utility and, at equilibrium, no traveler can change his/her route to obtain a higher utility.
The following equilibrium principles for the travel choice component of DTA can be considered as simple dynamic extensions of the travel choice principles adopted in static traffic assignment:
• the dynamic user equilibrium (DUE) or dynamic user optimal (DUO) route choice principle (Friesz et al., 1989);
• the DUE departure time choice principle (Vickrey, 1969) and
• the DUE route/departure time choice principle (Mahmassani & Herman, 1984).
The dynamic user equilibrium (DUE) route choice principle, which is the simplest dynamic extension of Wardrop's (1952) first principle, states that for each origin- destination pair, any routes used by travelers departing at the same time must have equal and minimal travel time. This principle is used when the demand at each departure time is known. That is, the principle is often used in the pure dynamic route choice problem.
The DUE departure time choice principle considers departure time choice instead of route choice. This principle requires that the generalised travel costs for travelers between the same OD pair departing at any time are equal and minimal. The principle is adopted when the route choice is pre-determined or there is no route choice for travelers (i.e. the pure departure time choice problem) and the generalised travel cost normally includes the penalty cost due to early and/or late arrivals in addition to the travel time cost.
The DUE route/departure time choice principle considers departure time choice in addition to route choice, and considers generalised travel cost instead of travel time. This principle states that for each OD pair, the generalised travel costs incurred by travelers departing at any time using any route are equal and minimal. This principle is essentially a generalisation of the DUE route choice principle and the DUE departure time choice principle, and can be used in the simultaneous route and departure time choice problem. In fact, the example discussed in the introduction is constructed using this principle. Assuming that 1 minute of travel time is equivalent to 1 dollar and the cost of early arrival is 0.5 dollar per minute, the minimum generalised travel cost is equal to 90 dollars for all drivers. The travel time cost for the last driver is 90 dollars as the travel time is 90 minutes. The penalty cost for the last driver is 0 dollar, as he/she arrived at B at 9:00 am sharp. The travel time cost for the first driver is 30 dollars and the penalty cost is 60 dollars (i.e. 120 minutes multiplied by 0.5 dollar per minute). If this driver chose Link 1 instead, he/she would arrive at B at 7:30 am, and his/her travel time cost and penalty cost would become 60 dollars and 45 dollars, respectively, leading to a general travel cost of 105 dollars. Therefore, the choice of departing at 6:30 am and going to B via Link 1 is not optimal and no driver picks this choice at equilibrium.
To sum up, this section briefly reviews the common types of travel choice principles used in DTA. For a more detailed review on this component, please refer to Szeto and Wong (2011).
