8.4. Gravity model

This model originally generated from an analogy with Newton’s gravitational law. Newton’s gravitational law says, F = GM1M2/d2. Analogous to this, T_ij = CO_iD_j/c_ij.

The gravity model in trip distribution is inspired by Newton's law of gravitation, which states that two objects attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them. In transport modeling, this translates to trips (T_ij) being proportional to the product of trip productions (O_i) at the origin and trip attractions (D_j) at the destination, while inversely related to a measure of cost or distance between these locations (c_ij). Essentially, this means that the likelihood of trips occurring between two locations increases with the number of attractions at the destination and decreases with the distance or cost associated with traveling to that destination.

Imagine planning a vacation. You’re more likely to choose a destination that has a lot of attractions (like beaches, parks, and museums) if they're close by, compared to a destination that has few attractions and is far away. The gravity model captures this intuition of choosing destinations based on attractiveness and distance.

Introducing some balancing factors, T_ij = A O_i B D_j f(c_ij) where A and B are the balancing factors, f(c_ij) is the generalized function of the travel cost.

In the gravity model, balancing factors A and B are used to adjust outputs so that they align with actual trip productions and attractions. A scaling factor (A) applies to the origins and another (B) affects the destinations. This adjustment ensures that the total number of trips emanating from origins and those attracted to destinations match the data available, allowing the model to reflect observed patterns accurately. The function f(c_ij), which represents the cost or distance factor, further modifies how trips are distributed by imposing a deterrent effect based on the cost of travel.

Think of a salesperson trying to visit clients in various districts. While she would like to see as many clients as possible, she must also consider how much time and money each meeting will take. Balancing factors in the model help her decide which clients to prioritize based on both their importance (attraction) and the travel cost involved.

This function is called deterrence function because it represents the disincentive to travel as distance (time) or cost increases. Some of the versions of this function are: f(c_ij)=e^(-βc_ij) f(c_ij)=c_ij^(-n) f(c_ij)=c_ij^(-n) × e^(-βc_ij)

The deterrence function quantifies how the likelihood of trip-making diminishes as distance, travel time, or cost increases. The variations of the deterrence function account for different sensitivity to cost/distance when predicting trips. The exponential function illustrates a rapid decline in trips as costs rise, while power functions might suggest a more gradually decreasing trend. Essentially, the model aims to capture the diminishing returns of attractiveness as the disincentives of distance and cost become more significant.

Consider a person deciding whether to attend a concert. If the concert is nearby and cheap, they are very likely to go. However, if it’s far away and expensive, their inclination may drop significantly. The deterrence function in the gravity model is like measuring that decrease in enthusiasm based on how inconvenient the concert becomes.

As in the growth factor model, here also we have singly and doubly constrained models. The expression T = A O B D f(c_ij) is the classical version of the doubly constrained model.

In trip distribution models, singly constrained models make assumptions based on one side of trip generation or attraction, while doubly constrained models account for both sides. In a doubly constrained model, both the total number of trip productions (origins) and attractions (destinations) must match the generated trips according to the costs associated with traveling. This approach adds robustness to the model, as it should resonate more closely with real-situation conditions.

When planning a delivery route, if you only account for the number of packages to be delivered (productions) and ignore the destinations (attractions), your approach may fall short. This is where doubly constrained models shine, ensuring that both the starting point and endpoints are considered in distribution planning.

From this, we can find the balancing factor B as B = 1/Σ A O f(c_ij) and A = 1/Σ B D f(c_ij).

The calculation of balancing factors requires iterating between the two factors until the model stabilizes. Initially, you might set one factor equal to one and calculate the other based on the current estimates. You recursively apply this process to refine both factors until the trip distribution estimates converge, meaning that they reliably reflect the actual observed trips. This iterative method helps in fine-tuning the model to enhance accuracy.

It’s like a craftsperson adjusting the tension in a string instrument. One adjustment might make the sound better, but the perfect tone often requires several tweaks to different strings to reach that harmonious sound. In trip distribution, we tweak balancing factors until the total trips for origins and destinations match.

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