1. GEOG 111/211A Transportation Planning Trip Distribution Additional suggested reading: Chapter 5 of Ortuzar & Willumsen, third edition November 2004

2. GEOG 111/211A Transportation Planning Trip Distribution Objectives Replicate spatial pattern of trip making Account for spatial separation among origins and destinations (proximity in terms of time, cost, & other factors) Account for attractiveness among TAZs Reflect human behavior

3. GEOG 111/211A Transportation Planning Trip Distribution Convert Production and Attraction Tables into Origin - Destination (O - D) Matrices Destinations 1 2 3 4 5 6 Sum Origins1 2 3 4 5 6 Sum T 11 T 12 T 13 T 14 T 15 T 16 O 1 T 21 T 22 T 23 T 24 T 25 T 26 O 2 T 31 T 32 T 33 T 34 T 35 T 36 O 3 T 41 T 42 T 43 T 44 T 45 T 46 O 4 T 51 T 52 T 53 T 54 T 55 T 56 O 5 T 61 T 62 T 63 T 64 T 65 T 66 O 6 D 1 D 2 D 3 D 4 D 5 D 6 TAZP 15 13 26 18 8 13 93 A 22 6 5 52 2 6 93 1 2 3 4 5 6 Sum

4. GEOG 111/211A Transportation Planning Crude approximation for HBW

5. GEOG 111/211A Transportation Planning Trip Distribution, Methodology General Equation: –T ij = T i P(T j ) T ij = calculated trips from zone i to zone j T i = total trips originating at zone i P(T j ) = probability measure that trips will be attracted to zone j Constraints: Singly Constrained –Sum i T ij = D j OR Sum j T ij = O i Doubly Constrained –Sum i T ij = D j AND Sum j T ij = O i

6. GEOG 111/211A Transportation Planning Trip Distribution Models Growth Factor / Fratar Model –T ij = T i (T j / T) T ij = present trips from zone i to zone j T i = total trips originating at zone i T j = total trips ending at zone j T = total trips in the entire study –T ij * = T ij (F i F j ) / F F i = T i * / T i F j = T j * / T j F = T* / T * = estimated future trips

7. GEOG 111/211A Transportation Planning Trip Distribution Models Gravity Model –T ij = T i T ij = trips from zone i to zone j T i = total trips originating at zone i A j = attraction factor at j A x = attraction factor at any zone x C ij = travel friction from i to j expressed as a generalized cost function C ix = travel friction from i to any zone x expressed as a generalized cost function a = friction exponent or restraining influence Sum (A x / C ix ) a A j / C ij a You can consider this as the probability spatial distribution P(Tj)

8. GEOG 111/211A Transportation Planning Trip Distribution Models Intervening - Opportunities Model –T ij = T i (e - e ) T ij = trips from zone i to zone j T = trip destination opportunities closer in time to zone i than those in zone j T i = trip end opportunities in zone i T j = trip end opportunities in zone j L = probability that any destination opportunity will be chosen -LT-LT(T + T j )

9. GEOG 111/211A Transportation Planning Model Comparison Growth Factor Gravity Intervening - Opportunities Simple Easy to balance origin and destination trips at any zone Specific account of friction and interaction between zones Does not require origin - destination data Claimed to bear a better “fit” to actual traffic Does not reflect changes in the frictions between zones Does not reflect changes in the network Requires extensive calibration Long iterative process Accounts for only relative changes in time - distance relationship between zones Arbitrary choice of probability factor Model Advantages Disadvantages New: Destination choice models build on intervening opportunities

10. GEOG 111/211A Transportation Planning Gravity Model Process 1.Create Shortest Path Matrix - Minimize Link Cost between Centroids 2.Estimate Friction Factor Parameters - Function of Trip Length Characteristics by Trip Purpose 3.Calculate Friction Factor Matrix 4.Convert Productions and Attractions to Origins and Destinations 5.Calculate Origin - Destination Matrix 6.Enforce Constraints on O - D Matrix - Iterate Between Enforcing Total Origins and Destinations

11. GEOG 111/211A Transportation Planning Shortest Path Matrix Matrix of Minimum Generalized Cost from any Zone i to any Zone j (see OW p. 153) –Distance, Time, Monetary Cost, Waiting Time, Transfer Time, etc.. may be used in Generalized Cost –Time or Distance Often Used –Matrix Not Necessarily Symmetric (Effect of One - Way Streets) TAZ ID 123456123456 TAZ ID 1 2 3 4 5 6 C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36 C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66

12. GEOG 111/211A Transportation Planning Example: travel time matrix for 6 TAZs

13. GEOG 111/211A Transportation Planning Friction Factor Models Exponential: –f(c ij ) = e c > 0 Inverse Power: –f(c ij ) = c ij b > 0 Gamma: –f(c ij ) = a c ij e a > 0, b > 0, c > 0 - c (c ij ) - b Trip Purpose a b c HBW 285070.0200.123 HBP1391731.2850.094 NHB2191131.3320.010 ref. NCHRP 365 / TransCAD UTPS Manual pg. 80

14. GEOG 111/211A Transportation Planning Example friction factors using travel times alone Friction ij = 1/exp(-0.03* Time ij )

15. GEOG 111/211A Transportation Planning Friction Factor Matrices Matrix of Friction from any Zone i to any Zone j, by Trip Purpose –Each Cell of a Friction Factor Matrix is a Function of the Corresponding Cell of the Shortest Path Matrix –Each Trip Purpose has a separate Friction Factor Matrix Because Trip Making Behavior Changes for Each Trip Purpose TAZ ID 123456123456 TAZ ID 1 2 3 4 5 6 F 11 F 12 F 13 F 14 F 15 F 16 F 21 F 22 F 23 F 24 F 25 F 26 F 31 F 32 F 33 F 34 F 35 F 36 F 41 F 42 F 43 F 44 F 45 F 46 F 51 F 52 F 53 F 54 F 55 F 56 F 61 F 62 F 63 F 64 F 65 F 66

16. GEOG 111/211A Transportation Planning Trip Conversion(Approximate) Home Based Trips:Non - Home Based Trips: –O i = (P i + A i ) / 2 O i = P i –D i = (P i + A i ) / 2 D i = A i O i = origins in zone i (by trip purpose) D i = destinations in zone i (by trip purpose) P i = productions in zone i (by trip purpose) A i = attractions in zone i (by trip purpose) Note: This Only Works for a 24 Hour Time Period If our models are for one period in a day we prefer to work directly with Origins-Destinations

17. GEOG 111/211A Transportation Planning O - D Matrix Calculation Calculate Initial Matrix By Gravity Equation, by Trip Purpose –Each Cell has a Different Friction, Found in the Corresponding Cell of the Friction Factor Matrix Enforce Constraints in Iterative Process –Sum of Trips in Row i Must Equal Origins of TAZ i If Not Equal, Trips are Adjusted Proportionally –Sum of Trips in Column j Must Equal Destinations of TAZ j If Not Equal, Trips are Adjusted Proportionally –Iterate Until No Adjustments Required

18. GEOG 111/211A Transportation Planning O - D Matrix Example: Destinations 1 2 3 4 5 6 Sum Origins1 2 3 4 5 6 Sum T 11 T 12 T 13 T 14 T 15 T 16 O 1 T 21 T 22 T 23 T 24 T 25 T 26 O 2 T 31 T 32 T 33 T 34 T 35 T 36 O 3 T 41 T 42 T 43 T 44 T 45 T 46 O 4 T 51 T 52 T 53 T 54 T 55 T 56 O 5 T 61 T 62 T 63 T 64 T 65 T 66 O 6 D 1 D 2 D 3 D 4 D 5 D 6

19. GEOG 111/211A Transportation Planning Attraction/friction matrix

20. GEOG 111/211A Transportation Planning Gravity model probability Sum (A x / friction ix ) A j / friction ij

21. GEOG 111/211A Transportation Planning Trip Interchange - iteration 1 For each cell value we apply the gravity equation once - in this iteration - after this we use the ratio to adjust the values in the cells - until row and column targets are satisfied - see also OW - chapter 5

22. GEOG 111/211A Transportation Planning Trip interchange - iteration 2 Using the ratios from before we succeed in getting the targets for the sums of cells for each column - look at the other ratios

23. GEOG 111/211A Transportation Planning Trip interchange - iteration 3

24. GEOG 111/211A Transportation Planning Trip interchange - iteration 4 We get both rows and columns to produce the sums we want!

25. GEOG 111/211A Transportation Planning Multiple Matrices For each trip purpose obtain different Origin- Destination Tij matrices Usually these are 24 hour Matrices (number of trips from one zone to another in a 24 hour period) In assignment we will need a matrix of vehicles moving from a zone to another during a specific period (peak usually) in a typical day

26. GEOG 111/211A Transportation Planning Final O - D Matrix (simplified) Combine (Add) O - D Matrices for Various Trip Purposes Scale Matrix for Peak Hour –Scale by Percent of Daily Trips Made in the Peak Hour –0.1 Often Used (10% of daily trips) Scale Matrix for Vehicle Trips –Scale by Inverse of Ridership Ratio to Convert Person Trips to Vehicle Trips –0.95 to 1 Often Used Note: Mode Split Process / Models More Accurate, - we will explore them in class