Trip Distribution Meeghat Habibian Transportation Demand Analysis Lecture note Trip Distribution Meeghat Habibian

Demand models of trip distribution Destination choice models Outline Introduction Hitchkock Model Entropy Model Gravity Model Demand models of trip distribution Destination choice models Intervening Opportunity Model Transportation Demand Analysis- Lecture note

Transportation Demand Analysis Lecture note Introduction

Second Step in Urban Transportation Modeling System Socio-economic Forecasts (Population, Employment, …) Trip Generation Trip Distribution Transportation Demand Analysis- Lecture note Mode Split Trip Assignment

Situation Four-step model: Given Ti, Aj  Tij= f(Ti,Aj, …ij) Direct approach: Tij= f(Ti,Aj)=f(g(popi),h(popj))=Ψ(popi, popj) More general: Tij= f( …i, …j, …ij) Example: Tij= f(popi, popj, Number of callsij) Transportation Demand Analysis- Lecture note

Definitions N = Number of zones in the urban area i = Subscript, used to denote origin zones j = Subscript, used to denote destination zones 𝑂 𝑖 = Number of trips originating in zone i 𝐷 𝑗 = Number of trips destined for zone j 𝑇 𝑖𝑗 = Number of trips (“flow”) from origin zone i to destination zone j Transportation Demand Analysis- Lecture note

General Equations T=Total Trips= 𝑖 𝑂 𝑖 = 𝑗 𝐷 𝑗 Logical constraints which any feasible trip matrix must satisfy: 𝑗 𝑇 𝑖𝑗 = 𝑂 𝑖 𝑖=1,…,𝑛 [1] 𝑖 𝑇 𝑖𝑗 = 𝐷 𝑗 𝑗=1,…,𝑛 [2] Transportation Demand Analysis- Lecture note

Elementary Trip Distribution Models Growth factor models Uniform growth factor model Average growth factor model Simple average Fratar model Positive and Negative points? Transportation Demand Analysis- Lecture note

Transportation Demand Analysis Lecture note The Hitchkock Model

Hitchcock Model A minimum-cost flow problem over the network 𝑀𝑖𝑛 𝑧 𝑥 = 𝑖=1 𝐼 𝑗=1 𝐽 𝑐 𝑖𝑗 𝑋 𝑖𝑗 Such to: 𝑗 𝑥 𝑖𝑗 = 𝑂 𝑖 ∀𝑖=1,2,…,𝐼 Transportation Demand Analysis- Lecture note 𝑖 𝑥 𝑖𝑗 = 𝐷 𝑗 ∀𝑗=1,2,…,𝐼 J 𝑥 𝑖𝑗 ≥0 ∀ 𝑖,𝑗

Hitchcock Model 𝑐 𝑖𝑗 and 𝑥 𝑖𝑗 : (fixed) cost (per unit of flow) and the flow, respectively, on the link leading from node i to node j Oi : Total flow supplied by node i Dj : Total flow required at node j. Assume further: ∑ Oi = ∑ Dj Transportation Demand Analysis- Lecture note At the optimal solution: Maximum number of links carrying flow equals minimum number of links that can connect I supply nodes to J demand nodes, that is (I+J-1)

Transportation Demand Analysis Lecture note The Entropy Model

Entropy Model All of the states can be occurred in trip distribution matrix: 𝑇 𝑡 11 𝑇− 𝑡 11 𝑡 12 𝑇− 𝑡 11 − 𝑡 12 𝑡 13 … T = All of the trips It can be simplified as: Transportation Demand Analysis- Lecture note 𝑇! 𝑖𝑗 𝑡 𝑖𝑗 !

Entropy Model This model finds the highest probability of trip distribution matrix with respect to all constraints, so the objective function is: 𝑀𝑎𝑥 𝑇! 𝑖𝑗 𝑡 𝑖𝑗 ! And it equals to: Transportation Demand Analysis- Lecture note 𝑀𝑎𝑥 𝑇! 𝑖𝑗 𝑡 𝑖𝑗 ! = ln 𝑇!− 𝑖𝑗 ln 𝑡 𝑖𝑗 ! Ln𝑇! Is constant and equation can be simplified as: 𝑀𝑎𝑥 − 𝑖𝑗 ln 𝑡 𝑖𝑗 !

Entropy Model The problem is changed to a minimization problem: min 𝑖𝑗 ln 𝑡 𝑖𝑗 ! [Stirling Approximation log n!≈nlogn−n] Therefore: min 𝑖𝑗 (𝑡 𝑖𝑗 ln 𝑡 𝑖𝑗 − 𝑡 𝑖𝑗 ) Transportation Demand Analysis- Lecture note 𝑖𝑗 𝑡 𝑖𝑗 =𝑇 Is constant again and equation can be simplified as: min 𝑖𝑗 𝑡 𝑖𝑗 ln 𝑡 𝑖𝑗

Entropy Model with respect to just one constraint max − 𝑖 𝑗 𝑡 𝑖𝑗 ln 𝑡 𝑖𝑗 𝑠.𝑡. 𝑖 𝑗 𝑡 𝑖𝑗 =𝑇 𝑡 𝑖𝑗 ≥0 Transportation Demand Analysis- Lecture note

Entropy Model If 𝜆 is a dual variable, with Lagrange multiplier solution : 𝐿 𝑡 𝑖𝑗 ,𝜆 =− 𝑖 𝑗 𝑡 𝑖𝑗 ln 𝑡 𝑖𝑗 −𝜆 𝑖 𝑗 𝑡 𝑖𝑗 −𝑇 𝜕𝐿 𝜕 𝑡 𝑖𝑗 =0 →−1− ln 𝑡 𝑖𝑗 −𝜆=0 → 𝑡 𝑖𝑗 = 𝑒 −𝜆−1 𝜕𝐿 𝜕𝜆 =0 → 𝑖 𝑗 𝑡 𝑖𝑗 =𝑇 → Transportation Demand Analysis- Lecture note 𝑖 𝑗 𝑒 −𝜆−1 =𝑇 → 𝑒 −𝜆−1 𝑖 𝑗 1 =𝑇 → 𝑒 −𝜆−1 𝑚𝑛 =𝑇 𝑒 −𝜆−1 = 𝑇 𝑚𝑛 → 𝑡 𝑖𝑗 = 𝑇 𝑚𝑛

Example 20000 trips estimated in peak hour for an area contains 5 origins and 5 destinations, make the trip distribution matrix through entropy approach. With the information above, the most likely matrix is: 20000 5×5 =800 Transportation Demand Analysis- Lecture note

Transportation Demand Analysis Lecture note The Gravity Model

Concept F ij =G m i m j r2 m2 r m1 Transportation Demand Analysis- Lecture note

Analogy T ij ≅ O i i=1,…,N T ij ≅ D j j=1,…,N T ij ≅ f ij =f 1 r ij i, j = 1,…,N Therefore: T ij =kij O i D j /r ij =kikj O i D j /𝑓(c 𝑖j ) kij: constant of proportionality for pair i-j ki: constant of proportionality for zone i (simplification) c ij :Impedance (e.g., time, distance, cost) between i and j f ij = Impedance function; 𝜕 f ij 𝜕 c ij <0 Transportation Demand Analysis- Lecture note

Proportional constants T ij =kikj O i D j f ij Remember: 𝑗 𝑇 𝑖𝑗 = 𝑂 𝑖 𝑖=1,…,𝑛 𝑘𝑖= 1 𝑗 𝑘𝑗 𝐷 𝑗 𝑓 𝑖𝑗 𝑖 𝑇 𝑖𝑗 = 𝐷 𝑗 𝑗=1,…,𝑛 𝑘𝑗= 1 𝑖 𝑘𝑖 𝑂 𝑖 𝑓 𝑖𝑗 Transportation Demand Analysis- Lecture note

Formulation Observation trips  Many Tij=0 For simplification: Assume: kj=1 Therefore: 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 Note: Constraint on destinations is relaxed: 𝑘𝑗= 1 𝑖 𝑘𝑖 𝑂 𝑖 𝑓 𝑖𝑗 is not in place! Transportation Demand Analysis- Lecture note 𝑖 𝑇 𝑖𝑗 = 𝐷 𝑗

Example Determine the trip distribution matrix for following four cities assuming f(cij)= tij-2 Travel-time Matrix (minutes) Time 1 2 3 4 7 35 45 40 5 20 12 8 Transportation Demand Analysis- Lecture note Trip Generation data City 1 2 3 4 Product 4724 901 193 108 Attract 4909 774 174 69

Example Travel-time Matrix (minutes) Calculate f(cij)= tij-2 Time 1 2 3 4 7 35 45 40 5 20 12 8 Transportation Demand Analysis- Lecture note Calculate f(cij)= tij-2 *0.001 1 2 3 4 20.408 0.816 0.494 0.625 40.000 2.500 6.944 111.111 15.625 250.000

Example Trip distribution matrix 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 The sum of attracted trips are not satisfied! Reason: The attraction constraint has been ignored: 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 1 2 3 4 4688 30 4724 101 777 11 12 901 19 15 151 8 193 20 10 66 108 Aj 4820 842 176 88 Transportation Demand Analysis- Lecture note Observed 4909 774 174 69 Nj 1.019 0.919 0.989 0.784 𝑖 𝑇 𝑖𝑗 = 𝐷 𝑗

Row - Column factor correction method Concept to proportionally adjust the trip matrix until it approximately (e.g., ±5%) matches the forecast year row and column sums An iterative procedure is required to balance rows and columns Let: O i n = j T ij n for the n th iteration T ij 0 = base year O−D flow O i new = forecast year row sum Similar definitions for D j n 𝑎𝑛𝑑 𝐷 𝑗 𝑛𝑒𝑤 Transportation Demand Analysis- Lecture note

Row - Column factor correction algorithm 𝑘=1 Nj 𝑇 𝑖𝑗 𝑘 = 𝑇 𝑖𝑗 𝑘−1 ( 𝐷 𝑗 𝑛𝑒𝑤 𝐷 𝑗 𝑘−1 ) k=k+1 Yes 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒? No k=k+1 Mi Transportation Demand Analysis- Lecture note 𝑇 𝑖𝑗 𝑘 = 𝑇 𝑖𝑗 𝑘−1 ( 𝑂 𝑖 𝑛𝑒𝑤 𝑂 𝑖 𝑘−1 ) 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒? No Yes Yes Stop

Row-Column correction example Transportation Demand Analysis- Lecture note

Row-Column correction example Transportation Demand Analysis- Lecture note

Row-Column correction example Transportation Demand Analysis- Lecture note

Row-Column correction example Transportation Demand Analysis- Lecture note

Impedance Function Common impedance functions (transportation system effect): Hyperbolic: Exponential: 𝑓 𝑖𝑗 = 𝑐 𝑖𝑗 𝜃 𝜃<0 𝑓 𝑖𝑗 =𝜃1𝑒𝑥𝑝 𝜃2 𝑐 𝑖𝑗 𝜃1>0, 𝜃2<0 Transportation Demand Analysis- Lecture note θ, θ1, θ2= parameters which must be estimated from observed data

Hyperbolic Impedance Function The simplest form: θ = -2  Original gravity hypothesis Overestimation of shorter trips it increases quickly as c decreases and approaches infinity when c approaches zero 𝑓 𝑖𝑗 = 𝑐 𝑖𝑗 θ θ<0 Transportation Demand Analysis- Lecture note 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗

Exponential Impedance Function To correct the infinity problem 𝑓 𝑖𝑗 =𝜃1𝑒𝑥𝑝 𝜃2 𝑐 𝑖𝑗 𝜃1>0, 𝜃2<0 This function approaches θ1 when c approaches zero. Transportation Demand Analysis- Lecture note

Comparison (hyperbolic vs. exponential) Both are monotonically decreasing functions of c θ1 and θ2 are related to the total (or average) trip cost in a system, although this relationship is usually not expressed explicitly Transportation Demand Analysis- Lecture note

Gamma function fij is expected to approach zero as c does in following cases: Models for vehicular trips (excluding walking trips) Models for work trips or specialty shopping trips The most commonly used function with these characteristics is a Gamma function fij = θ1cijexp-(θ2cij) Transportation Demand Analysis- Lecture note

Real cases Transportation Demand Analysis- Lecture note

Gravity model calibration Aim: Calibrate the impedance function (based on current trip matrix) A try and error approach is suggested fij cij Transportation Demand Analysis- Lecture note Initial value f =1.0 1.0

Example Calibrate the impedance function in 5min intervals for a 4-zone city such to: Current Travel-time matrix (minutes) Time 1 2 3 4 5 16 13 18 7 20 12 9 Transportation Demand Analysis- Lecture note Current Trip interchange matrix (trips) Trips 1 2 3 4 Pi 250 125 375 75 825 100 400 50 225 775 205 60 420 910 155 215 320 175 865 Aj 710 800 970 895 3375

Example Select the O-D pairs according to the requested intervals: Time 1 2 3 4 5 16 13 18 7 20 12 9 Travel Time Zones 0.1-5.0 5.1-10.0 22,34,43 10.1-15 13,31,24,42 15.1-20 12,21,14,41,23,32 Transportation Demand Analysis- Lecture note 11,33,44

Example Determine the observed trips for each of the requested intervals: Trips 1 2 3 4 Pi 125 375 75 825 100 400 50 225 775 205 60 420 910 155 215 320 865 Aj 710 800 970 895 3375 250 225 175 = T11+T33+T44 250+225+175= 650 Transportation Demand Analysis- Lecture note Travel Time Zones Observed F1 (assumed for 1st iteration) 0.1-5.0 650 1 5.1-10.0 22,34,43 1140 10.1-15 13,31,24,42 1020 15.1-20 12,21,14,41,23,32 565 11,33,44

Example 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 Trips 1 2 3 4 Pi 173.56 195.56 237.11 218.78 825 163.04 183.70 222.74 205.52 775 191.44 215.70 261.54 241.32 910 181.97 205.04 248.61 229.39 865 Aj 710 800 970 895 3375 Transportation Demand Analysis- Lecture note 𝑇11= 825∗710∗𝐹1 710∗𝐹1+800∗𝐹1+970∗𝐹1+895∗𝐹1 = 825∗710∗1 710∗1+800∗1+970∗1+895∗1 =173.56

Example Trips 1 2 3 4 Pi 173.56 195.56 237.11 218.78 825 163.04 183.70 222.74 205.52 775 191.44 215.70 261.54 241.32 910 181.97 205.04 248.61 229.39 865 Aj 710 800 970 895 3375 173.56+261.54+229.39=664.48 Travel Time Zones Calculated F2 ∆F 0.1-5.0 11,33,44 664.48 0.978 0.022 5.1-10.0 22,34,43 673.62 1.692 0.692 10.1-15 13,31,24,42 839.10 1.215 0.216 15.1-20 12,21,14,41,23,32 1197.78 0.471 0.528 Transportation Demand Analysis- Lecture note F2 = 𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 ×𝐹1= 650 664.48 ×1=0.978 Convergence criterion: ∆F≤0.01

Example 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 =214.34 Trips 1 2 3 4 Pi 214.34 Travel Time Zones Calculated F2 0.1-5.0 11,33,44 664.48 5.1-10.0 22,34,43 673.62 1.692 10.1-15 13,31,24,42 839.10 15.1-20 12,21,14,41,23,32 1197.78 Trips 1 2 3 4 Pi 173.56 195.56 237.11 218.78 825 163.04 183.70 222.74 205.52 775 191.44 215.70 261.54 241.32 910 181.97 205.04 248.61 229.39 865 Aj 710 800 970 895 3375 0.978 1.216 0.472 𝑇 𝑖𝑗 = 𝑂 𝑖 𝐷 𝑗 𝑓 𝑖𝑗 𝑗 𝐷 𝑗 𝑓 𝑖𝑗 = 0.978×710+ + + 825×710×0.978 =214.34 0.472×800 1.216×970 0.472×895 Trips 1 2 3 4 Pi 214.34 116.46 363.9 130.29 825 80.25 324.41 109.63 260.69 775 212.04 92.71 233.12 372.12 910 75.749 219.95 371.28 198.01 865 Aj 582.379 753.53 1077.93 961.11 3375 Transportation Demand Analysis- Lecture note

Example Travel Time Zones Calculated F3 ∆F 0.1-5.0 11,33,44 645 0.985 0.007 5.1-10.0 22,34,43 1068 1.807 0.114 10.1-15 13,31,24,42 1057 1.173 0.042 15.1-20 12,21,14,41,23,32 605 0.440 0.031 Trips 1 2 3 4 Pi 223.276 112.48 363.39 125.84 825 74.9019 346.20 102.33 251.56 775 201.7518 85.32 231.37 391.55 910 69.614 208.98 390.13 196.26 865 Aj 569.543 752.99 1087.23 965.22 3375 Transportation Demand Analysis- Lecture note

Example: Travel Time Zones Calculated F4 ∆F 0.1-5.0 11,33,44 651 0.984 0.001 5.1-10.0 22,34,43 1128 1.826 0.019 10.1-15 13,31,24,42 1026 1.167 0.007 15.1-20 12,21,14,41,23,32 570 0.436 0.004 Trips 1 2 3 4 Pi 224.22 112.03 363.41 125.33 825 74.12 349.64 101.26 249.97 775 200.20 84.31 230.55 394.91 910 68.77 207.32 393.37 195.51 865 Aj 567.32 753.32 1088.61 966.74 3375 Transportation Demand Analysis- Lecture note

Example: ∆F≤0.01 √ Transportation Demand Analysis- Lecture note Travel Time Zones Calculated F5 ∆F 0.1-5.0 11,33,44 650 0.9832 0.000 5.1-10.0 22,34,43 1138 1.82942 0.003 10.1-15 13,31,24,42 1021 1.165921 0.001 15.1-20 12,21,14,41,23,32 566 0.435539 ∆F≤0.01 √ Transportation Demand Analysis- Lecture note

Notes f reflects the effect of transportation system in Gravity model Modifications in transportation system could be captured by f Distance-based calibration of f results in Fratar model (why?) f is usually adjusted for generalized cost (e.g., time, cost,…) f may represent by Gamma or Negative-binomial distribution Transportation Demand Analysis- Lecture note

Demand models of trip distribution Transportation Demand Analysis Lecture note Demand models of trip distribution

Destination choice models Transportation Demand Analysis Lecture note Destination choice models

Determine the percentage distribution of trips Concept Determine the percentage distribution of trips from a given origin to available destinations not directly the flow of traffic It follows the general structure of transportation choice model based on the principle of individual utility maximization Transportation Demand Analysis- Lecture note

Definitions i: individual (or a homogenous group of individual trip makers) J: destinations available for a particular trip purpose Pi(j/J): Probability of choice of destination j among J destinations Aij is a vector of attributes of destination j for traveler i attractiveness, travel cost, socioeconomic attributes Transportation Demand Analysis- Lecture note

Multinomial logit structure Formulation Multinomial logit structure Vi(j): Generalized travel cost of destination j for traveler i a function of Attributes of destination j tij: trip distribution ai: number of trip makers Transportation Demand Analysis- Lecture note

Intervening opportunity models Transportation Demand Analysis Lecture note Intervening opportunity models

Concept According to Stouffer, 1940: 𝑇 𝑖𝑗 =𝑘 𝑎 𝑗 𝑉 𝑗 The probability of choice of a particular destination is proportional to the opportunity for trip purpose satisfaction at that destination, (aj) inversely proportional to all such opportunities that are closer to the trip maker’s origin, (Vj) (i.e., intervening opportunities) Transportation Demand Analysis- Lecture note 𝑇 𝑖𝑗 =𝑘 𝑎 𝑗 𝑉 𝑗

Definitions dv v P(dv): probability of satisfaction after considering dv opportunity P(v): Probability of satisfaction after considering v opportunities L : proportionality constant of accepting a destination opportunity 𝑃 𝑑𝑣 =𝐿 1−𝑃 𝑣 𝑑 𝑣 Assuming uniformity for the probability of satisfaction at destinations: Therefore: 𝑃 𝑑𝑣 =𝑑𝑃 𝑣 Transportation Demand Analysis- Lecture note 𝑑𝑃(𝑣) 1−𝑃(𝑣) =𝐿𝑑 𝑣 ⇒ − ln 1−𝑃 𝑣 =𝑙𝑣+𝑐⇒ 1−𝑃 𝑣 = 𝑘𝑒 −𝑙𝑣 𝑉 𝑗 = total destination opportunities from origin zone i to the jth destination 𝑃 𝑣 𝑗 =1− 𝑘𝑒 −𝑙 𝑣 𝑗

Definitions 𝑃 𝑣 𝑗 =1−𝑘 𝑒 −𝑙 𝑣 𝑗 𝑃 𝑣 𝑗 =1−𝑘 𝑒 −𝑙 𝑣 𝑗 U(vj): Probability of satisfaction after considering vj opportunities and continue the trip to destination j+1 P(vj)+U(vj)=1  U(vj)= ke-lvj Opportunity of destination j: vj -vj-1 Probability of staying at destination j: U(vj-1)-U(vj) Transportation Demand Analysis- Lecture note

Equation k= 1 𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) Tij=Oi*(U(vj-1)-U(vj)) = Oi* 𝑘 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 )  Oi* 𝑘 𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) =𝑂i k= 1 𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) 𝑗 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) = (𝑒 −𝑙 𝑣 −1 − 𝑒 −𝑙 𝑣 0 ) + (𝑒 −𝑙 𝑣 0 − 𝑒 −𝑙 𝑣 1 ) +…+ (𝑒 −𝑙 𝑣 𝐽−1 − 𝑒 −𝑙 𝑣 𝐽 ) = (𝑒 −𝑙 𝑣 −1 − 𝑒 −𝑙 𝑣 𝐽 ) 𝑗 𝑇 𝑖𝑗 = 𝑂 𝑖 Transportation Demand Analysis- Lecture note v-1: opportunity before first destination (j=0): 0 𝑣 𝐽 : total destination opportunities for all J destinations  Tij= Oi∗ 𝑘 (𝑒 −𝑙 𝑣 𝑗−1 − 𝑒 −𝑙 𝑣 𝑗 ) 𝑗 (1 − 𝑒 −𝑙 𝑣 𝐽 )

Example Example travel-time Matrix (minutes) 1 2 3 4 5 16 13 18 7 20 12 9 Example travel-interchange Matrix (trips) Transportation Demand Analysis- Lecture note Trips 1 2 3 4 Pi ? 825 775 910 865 Aj 710 800 970 895 3375

Example Order zones by travel time and subtended volumes 1 2 3 4 5 16 13 18 7 20 12 9 →5<13<16<18 → 𝑇11<𝑇13<𝑇12<𝑇14 𝑂𝑟𝑑𝑒𝑟𝑒𝑑 𝑍𝑜𝑛𝑒:1,3,2,4 Origin Zone   Order 1 3 2 4 Vj 710 1680=710+970 2480=1680+800 3375=2480+895 800 1695 2405 3375 970 1865 2575 895 2665 Transportation Demand Analysis- Lecture note

Example Sample Calculations for “Calibrated” intervening opportunity Model Origin Zone  Destination Zone   𝑂 𝑖  1− 𝑒 −𝐿 𝑉 J   𝑒 −𝐿 𝑉 𝑗−1   𝑒 −𝐿 𝑉 𝑗   𝑇 𝑖𝑗 1 825 0.63212 0.81028 248 3 0.60788 264 2 775 0.78896 259 0.60518 0.49032 141 4 910 0.75021 0.57546 252 865 0.76786 262 Transportation Demand Analysis- Lecture note L= 1 Number of current trips = 1 3375 =2.963× 10 −4

Example Estimated Trip-Interchange for intervening opportunity Model Transportation Demand Analysis- Lecture note Final estimated Trip-Interchange Matrix (using row-column factors)

Transportation Demand Analysis- Lecture note Finish