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In this question, we are going to learn about the gravity model. As discussed in class, an alternative and frequent way to model the demand is doing it in 4 steps: Generation/Attraction, Distribution, Mode choice, and Route choice. In the generation/attraction step, we estimate the number of trips that depart and arrive from each zone, in a similar way as we studied generation in the lectures. The idea of the distribution step is, taking the generation and attraction numbers as known, estimate the number of people going from each zone to each zone. Let's be precise: Consider that you know O₂ for every zone i, representing the number of users that have i as their origin. Similarly, you know Dj, representing the number of users that have j as their destination. We want to estimate Mij, i.e., the number of users going from i to j. The matrix M is known as the Origin-Destination matrix. Let us denote Tij the travel time required to go from i to j. The gravity model consists of estimating Mij = Oi Dj. f (Tij) Σk Dk ⚫ f(Tik) where we sum over all zones k except for i, and ƒ is a given decreasing function, e.g. f (x) : or f(x) = 1 x2 a. Discuss intuitively the properties of the formula given above. b. Consider a city with 4 zones. They generate and attract as follows: Zone 1 2 3 4 Generation 100 50 30 80 Attraction 60 50 60 90 The travel times between each pair of zones are: Origin/Destination 1 1 3 4 5 4 2 2 3 8 5 3 4 4 4 4 LC 10 x Considering f(x) = χ Estimate the Origin-Destination (OD) matrix. (Note: this is the single-constrained gravity model. As you might see with your results, the origin-destination matrix correctly reproduces the number of trips generated by each zone, BUT this is not true for attraction. The alternative is a doubly-constrained gravity model, where it is imposed that the origin-destination matrix reproduces generation and attraction. In that case, the matrix is solved following an iterative process.

In this question, we are going to learn about the gravity model. As discussed in class, an alternative and frequent way to model the demand is doing it in 4 steps: Generation/Attraction, Distribution, Mode choice, and Route choice. In the generation/attraction step, we estimate the number of trips that depart and arrive from each zone, in a similar way as we studied generation in the lectures. The idea of the distribution step is, taking the generation and attraction numbers as known, estimate the number of people going from each zone to each zone. Let's be precise: Consider that you know O₂ for every zone i, representing the number of users that have i as their origin. Similarly, you know Dj, representing the number of users that have j as their destination. We want to estimate Mij, i.e., the number of users going from i to j. The matrix M is known as the Origin-Destination matrix. Let us denote Tij the travel time required to go from i to j. The gravity model consists of estimating Mij = Oi Dj. f (Tij) Σk Dk ⚫ f(Tik) where we sum over all zones k except for i, and ƒ is a given decreasing function, e.g. f (x) : or f(x) = 1 x2 a. Discuss intuitively the properties of the formula given above. b. Consider a city with 4 zones. They generate and attract as follows: Zone 1 2 3 4 Generation 100 50 30 80 Attraction 60 50 60 90 The travel times between each pair of zones are: Origin/Destination 1 1 3 4 5 4 2 2 3 8 5 3 4 4 4 4 LC 10 x Considering f(x) = χ Estimate the Origin-Destination (OD) matrix. (Note: this is the single-constrained gravity model. As you might see with your results, the origin-destination matrix correctly reproduces the number of trips generated by each zone, BUT this is not true for attraction. The alternative is a doubly-constrained gravity model, where it is imposed that the origin-destination matrix reproduces generation and attraction. In that case, the matrix is solved following an iterative process.

Traffic and Highway Engineering
Traffic and Highway Engineering
5th Edition
ISBN:9781305156241
Author:Garber, Nicholas J.
Publisher:Garber, Nicholas J.
Chapter13: Evaluating Transportation Alternatives
Section: Chapter Questions
Problem 9P
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In this question, we are going to learn about the gravity model. As discussed in class, an alternative and frequent way to model the demand is doing it in 4 steps: Generation/Attraction, Distribution, Mode choice, and Route choice. In the generation/attraction step, we estimate the number of trips that depart and arrive from each zone, in a similar way as we studied generation in the lectures. The idea of the distribution step is, taking the generation and attraction numbers as known, estimate the number of people going from each zone to each zone. Let's be precise: Consider that you know O₂ for every zone i, representing the number of users that have i as their origin. Similarly, you know Dj, representing the number of users that have j as their destination. We want to estimate Mij, i.e., the number of users going from i to j. The matrix M
Transcribed Image Text:In this question, we are going to learn about the gravity model. As discussed in class, an alternative and frequent way to model the demand is doing it in 4 steps: Generation/Attraction, Distribution, Mode choice, and Route choice. In the generation/attraction step, we estimate the number of trips that depart and arrive from each zone, in a similar way as we studied generation in the lectures. The idea of the distribution step is, taking the generation and attraction numbers as known, estimate the number of people going from each zone to each zone. Let's be precise: Consider that you know O₂ for every zone i, representing the number of users that have i as their origin. Similarly, you know Dj, representing the number of users that have j as their destination. We want to estimate Mij, i.e., the number of users going from i to j. The matrix M
is known as the Origin-Destination matrix. Let us denote Tij the travel time required to go from i to j. The gravity model consists of estimating Mij = Oi Dj. f (Tij) Σk Dk ⚫ f(Tik) where we sum over all zones k except for i, and ƒ is a given decreasing function, e.g. f (x) : or f(x) = 1 x2 a. Discuss intuitively the properties of the formula given above. b. Consider a city with 4 zones. They generate and attract as follows: Zone 1 2 3 4 Generation 100 50 30 80 Attraction 60 50 60 90 The travel times between each pair of zones are: Origin/Destination 1 1 3 4 5 4 2 2 3 8 5 3 4 4 4 4 LC 10 x Considering f(x) = χ Estimate the Origin-Destination (OD) matrix. (Note: this is the single-constrained gravity model. As you might see with your results, the origin-destination matrix correctly reproduces the number of trips generated by each zone, BUT this is not true for attraction. The alternative is a doubly-constrained gravity model, where it is imposed that the origin-destination matrix reproduces generation and attraction. In that case, the matrix is solved following an iterative process.
Transcribed Image Text:is known as the Origin-Destination matrix. Let us denote Tij the travel time required to go from i to j. The gravity model consists of estimating Mij = Oi Dj. f (Tij) Σk Dk ⚫ f(Tik) where we sum over all zones k except for i, and ƒ is a given decreasing function, e.g. f (x) : or f(x) = 1 x2 a. Discuss intuitively the properties of the formula given above. b. Consider a city with 4 zones. They generate and attract as follows: Zone 1 2 3 4 Generation 100 50 30 80 Attraction 60 50 60 90 The travel times between each pair of zones are: Origin/Destination 1 1 3 4 5 4 2 2 3 8 5 3 4 4 4 4 LC 10 x Considering f(x) = χ Estimate the Origin-Destination (OD) matrix. (Note: this is the single-constrained gravity model. As you might see with your results, the origin-destination matrix correctly reproduces the number of trips generated by each zone, BUT this is not true for attraction. The alternative is a doubly-constrained gravity model, where it is imposed that the origin-destination matrix reproduces generation and attraction. In that case, the matrix is solved following an iterative process.
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