Bruton,  M. J. 1970. Chapter 4: Trip Distribution. Introduction to Transportation Planning. Hutchinson Technical Education: London. pp. 97- 129.

Part 2:  Gravity Models

Synthetic Methods of estimating Trip Distribution

1.  Gravity Models
2.  Electrostatic Method
3.  Multiple Regression
4.  Opportunities Method

Will concentrate on Gravity Model

• Actually a family of models

• Will provide background examples then a doubly constrained model -- basic method for transportation modeling

Two Basic Assumptions of all Sythetic Methods

1.  Before future (or present) travel patterns can be determined, causes of movement must be understood.

2.  Causal relationship creating movement patterns are considered similar to laws of physics

Gravity Models

Physical Model

F_ij = G [(M_i*M_j )/D_ij²]

Notation

F_ij =  Force of attraction between i and j

G = Universal Gravity Constant

M_i = Mass of object i

D_ij  = distance between i and j

 

Inter-Urban Model

A.  Population based

T_ij = K [(P_i*P_j )/P]

Notation

T_ij =  Trips between i and j

K = Model Constant

P_i = Popultion at TAZ i

P  = Total Population inside model cordon

Example

• Assume homogenous area

• Everyone moves in the same pattern

• To simplify assume K = 1

T_ij = K [(P_i*P_j )/P] = [(P_i*P_j )/P] = P_i(P_j /P)

Where

(P_j /P) = Proportion of trips ending at TAZ j

Note this is a constant, or TAZ j gets a constant proportion of all trips in the system

Spokane	Trip Ends
2002	North Hill	South Hill	Hillyard	Total
Pj	100	300	600	1000
Pj/P	10%	30%	60%

 

Note also how distance does not effect this solution

 

B.  Trip based

T_ij = K [(T_i*T_j )/T]

Notation

T_ij =  Trips between i and j

K = Model Constant

T_i = Trips originating at TAZ i

T_j = Trips destined to TAZ j

T  = All Trips inside model cordon

Example

• All trip ends balance (origin and destination totals)

• Include trips that remain within a TAZ

• To simplify assume K = 1

T_ij = K [(T_i*T_j )/T] = [(T_i*T_j )/T] = T_i(T_j /T)

Where

(T_j /T) = Proportion of trips ending at TAZ j

Note from the example below that the model doesnot balance properly (can't reproduce the intitial flow table).

Initial Flows
	North	South	Totals
North	2	1	3
South	1	1	2
Totals	3	2	5

Our simple model would:

• over predict north-south and south-north flow but

• under predict north-north and south-south

Predicted Flows k=1
	North	South	Totals
North	1.8	1.2	3
South	1.2	0.8	2
Totals	3	2	5

To balance our model we would need different constants for practically each cell

Konstants
	North	South
North	1.11	0.83
South	0.83	1.25

This demonstrates the need for special conditions for each origin-destination pair

Possible solution include DISTANCE since this varies by each pair (shortcoming is that it may be the same i to j as j to i, so symetry might be required).

 

C.  Doubly Constrained Gravity Model: Based on Taylor, Peter J. 1977. Quantitative Methods in Geography. Waveland Press: Prospect Height, IL. pp. 285-304.

The model presented below is  variation on the "gravity model today" presented in Bruton.  By using Taylor's notation and equations it may be a bit easier to understand.

T_ij = A_iB_j * [(O_iD_i) /d_ij^b]

Notation

T_ij = Trips from i to j in a specified amount of time (usually rush hour)

A_i = Constant to balance trips originating from TAZ i

B_j = Constant to balance trips destined for TAZ j

O_i = Total number of trips Originating in TAZ i

D_i = Total number of trips Destined for TAZ j

d_ij = distance from i to j

b = Power factor for discounting distance

Givens

O_i = From Trip Generation Model or Survey

D_i = From Trip Generation Model of Survey

d_ij = From field measurements

b =  Estimated from data or from existing studies

a₁, a₂  = Tolerances lower (1) and upper (2)

Additional Formulas

A_i^k =1/ S (B_j^k ( D_j /d_ij^b))       (iterative estimate of A_i)

B_j^k =1/ S (A_i^k-1 ( O_i /d_ij^b))     (iterative estimate of B_j)

a₁ < A_i^k/ A_i^k-1 < a₂              (tolerances)

a₁ <  B_j^k/ B_j^k-1 < a₂             (tolerances)

ST_ij = O_i                                        (Balance of flows from Origins)

ST_ij = D_j                                        (Balance of flows to Destinations)

SST_ij = SO_i = SD_j                      (Balance of all internal flows in model)

Note:  Since A_i is calculated based on B_j and A_i is calculated based on   B_j a circular logic exists, meaning that we can calculate one value only if we already know the other.  The solution is as simple as it is surprising.  Assume you know  one of theses two factors and determine the other.  Through an iterative procedure continue to estimate each factor based on the other until the estimates converge to a single set of values. 

Don't Believe it?  Just try and discover that it WORKS!!!     

Solution Steps

Step 1:  Start iterative procedure by assuming A_i⁰ = 1.0 and solve for B_j¹

Step 2:  Using B_j¹ solve for A_i¹

Step 3: Move to next iteration and using  A_i¹ solve for B_j²

Step 4: Using B_j² solve for A_i²

Step 5: Check Tolerances, if met continue to Step 6, otherwise return to Step 3

Step 6: Calculate T_ij Then check for balance between flows and origin, destination, and grand totals of trips.

Advantages of Gravity Model

1.  Based on a causal logic: stresses value of trip attraction (pull) and resistance (friction)

2.  Recognizes that trip purpose effects trip pattern -- can be solved independently for different trip types at same time of day (home to work, home to recreation, commercial...)

3.  Changes in land use can be easily introduced (change pull)

4.  Improvements in transportation facilities can be included (change friction)

5.  Deals with flows in both directions

6.  Includes intra- as well as inter- TAZ trips

7.  Easily solved

Disadvantages of Gravity Model

1.  Use of inverse power of distance for resistance not always suitable, varies with purpose and time of day (congestion); fails to give valid results for very long or very short trips

2.  In doubly constrained approach, constants A_i and B_j  cause the model to fit existing set of Trip Generation factors excellently, but due to the fact these are CONSTANTS they might create great distortions in predicting the future (for example, growing congestion on routes not factored in).

3.  In past, iteration procedures were cumbersome, no longer with modern software and computers