8.2.1. Trip matrix

The trip pattern in a study area can be represented by means of a trip matrix or origin-destination (O-D) matrix. This is a two-dimensional array of cells where rows and columns represent each of the zones in the study area. The notation of the trip matrix is given in figure 1.1.

A trip matrix is a table that shows how many trips are made from one zone to another in a study area. Each row in the matrix represents a zone that trips are coming from, while each column represents a zone that trips are going to. The intersection of a row and a column shows the number of trips between those two zones. For example, if you have three zones, the matrix might look like a 3x3 table where the cell at row 1, column 2 indicates the number of trips from zone 1 to zone 2.

Imagine a school cafeteria where each table represents a different zone (for instance, the playground, library, and gym). If you want to track how many students move from the playground to the library and back, you would create a table. Each cell in the table counts how many students go from one table to another, similar to how the trip matrix counts trips between different locations.

The cells of each row i contain the trips originating in that zone which have as destinations the zones in the corresponding columns. T is the number of trips between origin i and destination j. O is the total number of trips emanating from that zone.

Each cell in the trip matrix specifies a certain number of trips from one zone to another. The notation uses 'T' to denote the number of trips from origin i to destination j. Furthermore, 'O' represents the total number of trips that start from zone i. This means that if you want to know how many trips are generated from zone 1, you would look at all the numbers in row 1 and add them up. This helps planners know where trips are starting from.

Think of the trip matrix like a student attendance chart in class. Each cell tells you how many students are going from one class to another. For instance, if class A has 10 students going to class B and 5 students going to class C, the cells in the matrix indicate these movements. By adding up the numbers in a row, you find out how many students are moving from class A in total.

The sum of the trips in a row should be equal to the total number of trips emanating from that zone. The sum of the trips in a column is the number of trips attracted to that zone. These two constraints can be represented as: Σ T = O and Σ T = D.

In a trip matrix, the way the data is organized follows specific rules. The total number of trips originating from a zone (O) must equal the sum of the trips listed in that row of the matrix. Similarly, the total number of trips to a destination zone (D) must equal the sum of the trips in that column. This is crucial for accurate trip estimation and planning because it ensures that the numbers are consistent across the board.

Imagine a pizza shop where each kind of pizza represents a zone. If a customer orders two pepperoni pizzas and three veggie pizzas (total of five), the kitchen must be able to account for those five pizzas being prepared (the row's total). Conversely, if the delivery driver counts the types of pizzas to deliver and adds up to five pizzas (the column's total), both must match to ensure that every pizza ordered is correctly accounted for.

If reliable information is available to estimate both O and D, the model is said to be doubly constrained. In some cases, there will be information about only one of these constraints; the model is called singly constrained.

In trip distribution modeling, sometimes you may have complete information about both trip origins and destinations—this is known as a doubly constrained model because it relies on two sets of constraints to balance the data appropriately. However, if you only know about trips starting from zones or only about trips arriving at zones, you use a singly constrained model where one side remains estimated based on available data.

Consider organizing a school field trip. If you have a list of how many students are going (origins) and a finalized list of permissions from parents for how many students can attend (destinations), you have all the information you need, akin to a doubly constrained model. If you only know how many students want to go but not how many can get permissions, you're working with only half the information, like a singly constrained model.

--