10.6. Other assignment methods

Incremental assignment is a process in which fractions of traffic volumes are assigned in steps. In each step, a fixed proportion of total demand is assigned, based on all-or-nothing assignment. After each step, link travel times are recalculated based on link volumes. When there are many increments used, the flows may resemble an equilibrium assignment; however, this method does not yield an equilibrium solution. Consequently, there will be inconsistencies between link volumes and travel times that can lead to errors in evaluation measures. Also, incremental assignment is influenced by the order in which volumes for O-D pairs are assigned, raising the possibility of additional bias in results.

Incremental assignment involves assigning traffic volumes in separate steps rather than all at once. Each step assigns a portion of the total predicted traffic, and after each assignment, the travel times along the links of the network are recalculated based on the new volumes. This process is repeated multiple times. The final result can appear similar to a perfect equilibrium, but it does not truly reach that state. Errors may arise due to mismatches between volumes and calculated travel times, and the assignment order can introduce bias.

Imagine a teacher assigning homework in stages, giving groups of students a few problems at a time. After each group completes their problems, the teacher checks their answers and adjusts future assignments based on the current answers. However, if the teacher gives different groups varying numbers of problems (some get 5, others get 10), the final result may not accurately reflect the students' abilities. Similarly, in incremental assignment, the traffic predictions can be skewed if the assignments are inconsistent.

Capacity restraint assignment attempts to approximate an equilibrium solution by iterating between all-or-nothing traffic loadings and recalculating link travel times based on a congestion function that reflects link capacity. Unfortunately, this method does not converge and can flip-flop back and forth in loadings on some links.

Capacity restraint assignment is designed to imitate an equilibrium by alternating between assigning traffic using the all-or-nothing method and recalculating travel times based on how congested the links become under the current load. However, a significant flaw in this method is that it may not stabilize; the traffic volumes on different links may keep changing back and forth, making it unreliable for consistent outcomes.

Think about trying to pour water into several cups at once. If one cup fills quickly, you might decide to pour more water into it. Then, if another cup fills up faster, you shift your pouring again. This back-and-forth action can make it hard to know how much water each cup will really hold at the end. Similarly, with capacity restraint assignment, the constant adjustments often lead to unstable traffic volumes that don’t truly reflect a stable state.

User equilibrium assignment procedures based on Wardrop’s principle assume that all drivers perceive costs in an identical manner. A solution to the assignment problem on this basis is an assignment such that no driver can reduce his journey cost by unilaterally changing route. Van Vilet considered as stochastic assignment models, all those models which explicitly allow non-minimum cost routes to be selected. Virtually all such models assume that drivers' perception of costs on any given route are not identical and that the trips between each O-D pair are divided among the routes with the most cheapest route attracting most trips. They have important advantages over other models because they load many routes between individual pairs of network nodes in a single pass through the tree-building process, the assignments are more stable and less sensitive to slight variations in network definitions or link costs to be independent of flows and are thus most appropriate for use in uncongested traffic conditions such as in off-peak periods or lightly trafficked rural areas.

Stochastic user equilibrium acknowledges that different drivers have varied perceptions of travel costs, not just a single fixed cost. This model allows for the selection of routes that may not be the cheapest but could still be chosen by drivers based on their individual evaluations. Consequently, the model distributes traffic across multiple routes, which leads to a more stable allocation. This method is particularly effective in scenarios with light traffic where there is minimal congestion.

Imagine a group of friends deciding where to dine out. Some prefer inexpensive places and may argue strongly for a relatively cheap restaurant, while others might prioritize atmosphere or location. Even if a few suggest more expensive places, the overall group will split among varied options, not all choosing the cheapest restaurant. This is similar to stochastic user equilibrium, where traffic is spread across various routes as drivers choose based on differing views on travel costs.

Dynamic user equilibrium, expressed as an extension of Wardrop’s user equilibrium principle, may be defined as the state of equilibrium which arises when no driver can reduce his disutility of travel by choosing a new route or departure time, where disutility includes schedule delay in addition to costs generally considered. Dynamic stochastic equilibrium may be similarly defined in terms of perceived utility of travel. The existence of such equilibria in complex networks has not been proven theoretically and even if they exist the question of uniqueness remains open.

Dynamic assignment considers not just the route a driver takes but also when they travel. It presents a scenario where no driver can reduce their overall discomfort or disutility by changing either their route or their time of departure. This addition of time of travel makes the model more complex and realistic. However, proving that such dynamic equilibria exist in traffic networks is challenging; researchers have yet to establish theoretical groundwork for these concepts.

Imagine planning a road trip. You may not only think about which road to take but also whether to leave early in the morning or later in the day to avoid traffic. If every driver evaluates their unique circumstances and makes the journey best for themselves, no one could benefit by simply changing their route or time. Dynamic assignment captures this complexity but remains difficult to accurately represent in actual traffic networks.

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