Theory

Introduction

As the name suggests Doubly Constraint Gravity Model is a model where both the Trip Production and Trip Attraction are constrained.

$$\nonumber T_{ij} = A_{i}O_{i}B_{j}D_{j}f(C_{ij}) $$ Where,

$ A_{i} = $Trip production balancing factor

$ B_{j} = $Trip attraction balancing factor

$ f(C_{ij}) = $Impedance Function

$$\nonumber A_{i} = \dfrac{1}{\sum {B_{j}D_{j}f(C_{ij})}}$$ $$\nonumber B_{j} = \dfrac{1}{\sum {A_{i}O_{i}f(C_{ij})}}$$

Example

Table1
Zone	1	2	3
$O_{i}$	200	400	400
$D_{j}$	500	400	400

$ Impedance\ Matrix(Cost) = \begin{bmatrix}8 & 1 & 4\\3 & 6 & 5\\2 & 7 & 4\end{bmatrix}$

Obtain a trip matrix that satisfy origin & destination constraint Iteration-1

$X_{i}=1$and substitute in equation (c)

α=-2

The value of $Y_{i}$ is obtained

Substitute these values of Yj in equation (b) The results obtain is given below

This Iteration is continued till the satisfying convergence criteria is obtained.
References Books

J. De D. Ortuzar and L.G. Willumsen (1996), Modelling Transport. Wiley Publications, London.
C. S. Papacostas and P. D. Prevedouros (2001), Transportation Engineering & planning. Prentice-Hall of India, New Delhi.

Doubly constrained Gravity ModelTrip Distribution