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TNK084 Traffic Theory series Vol.4, number.1 May 2008 Combined models for public and private travel modes Magnus Edgren Abstract – This report analyses a combined network model consisting of a user equilibrium car network model, a model for modeling public transport and a nested logit model for modeling the choices between the travel modes. First a derivation is presented that shows that the combined model do consists of the three sub models. Then some results from a simulation using the model is presented and analyzed. It is shown that monetary costs for different transportation modes play a role for all people but most for people with low income. It is also shown that the dispersion in the logit model must be chosen with great accuracy in order to get valid results. Keywords: Combined models, nested logit functions, public transport, user equilibrium I. model complies with reality these things will be shown in the results. Introduction I.4. I.1. Background The network on which the model is applied is small and not valid for a real case. The results shall only be seen as indications on what happens when certain parameters are changed. The model is also simplified in the way that public transport is assumed to have their own network. No pedestrian or similar transportation is modeled either. Modelling a car network or public transport can be done separately in a relatively easy manner. But what is really interesting to know is when the two models are modelled together with a choice model for which travel mode to use. Then we can see the interactions between the models due to their individual attractiveness and how different user groups react on the available choices. A combined model is often quite advanced mathematically and it is generally easier to apply a model and prove that it is correct than to derive it from scratch. I.2. II. Theoretical background An optimization problem is given together with models for the car network, public transport and the choice between travel modes. Here they are presented together with a description of the notations. Scope and objectives The combined model will be analyzed and proved to be made up by a car network model, a model for public transport and a logit model for choosing among the travel modes. The analysis consists of a theoretical part and a implementation in AMPL. The AMPL model will be used to se how the model behaves while changing some of the parameters used as input to the model. For instance monetary cost will be varied in order to see how the model output is varying. I.3. Limitations II.1. Notations Sets C = set of all OD-pairs M = set of user groups L = set of links in the car network Rpqm = set of all routes in OD-pair pq For all notations below we have lL m r R pq m M p, q C Aim When changing the monetary cost, i.e. the actual cost that people perceive in order to choose a certain travel mode, it is expected that the travelling changes accordingly. If the price for a ticket to the public transport decreases the travelling will most certainly increase. Similarily if the fuel price for private cars increases, the travelling with cars will decrease. If the Demand Number of potential travellers 1 m d pq Magnus Edgren Number of bus travellers d bm pq Number of car travellers cm d pq Number of travellers dm d pq Number not making a trip nm d pq User groups Value of time m Car network model Total flow on link l vl Link flow on link l by user group m wlm Route flow, user group m hrm hrm rl Route cost, user group m Route r uses link l With a Wardrop user equilibrium mind we then get the following model for the car network t lc l , vl Travel time on link l l l Link capacity Monetary cost Public transport Travel time from p to q bm t bm pq pq , d pq Bus frequency pq Free flow travel time p to q pq Monetary cost pq Choice model Dispersion logit model, level i i , i 1,2 Other t nm pq Cost of not making a trip II.2. Car network model We assume link costs as follows t lc l , vl l m with tl being convex function for the travel time due to link flows, road capacity and others, τl being the monetary cost, e.g. the cost for petrol while traveling on the link, and βm indicating the value of time for a certain user group, m. We then get the cost for a certain route in OD-pair “pq” crm rl t lc l , vl lL l m 2 Magnus Edgren Objective function vl min z c t lc l , s ds l wlm lL 0 mM m II.4. The choice model is assumed to be a nested logit model as shown in fig 2. subject to h cm d pq m r rR m pq vl pq C , m M wlm lL m l m d pq β1 w mM Choice model for travel modes rlm hrm nm No trip d pq l L, m M ( p , q )C rR m pq dm d pq Make trip β2 Un Ud r R , pq C, m M h 0 m pq m r cm Car d pq Bus d bm pq II.3. Public transport model The public transport is assumed to travel on its own network and also with direct links between the ODpairs, i.e. links and routes are identical. It is also assumed that the general travel time function is t bm pq pq ,d bm pq a pq d bm pq b pq pq Fig. 2. The nested logit model. The utilities for each level is marked Ux, with x indicating travel mode. The probability of selecting one alternative is given by the logit model as n pq e 1U n e 1U n e 1U d e 1U d p d 1U n e e 1U d e 2U c pc pd 2U c e e 2U b e 2U b pb pd 2U c e e 2U b pn where apq and bpq are constant parameters describing the link, and αpq describing the bus frequency. The demand for bus travel is denoted d bm pq . Beside the travel time function we have the term pq Uc Ub pq m with ρpq being a parameter for modelling changes in travel time and τpq being the ticket price. The cost of time for user group m is βm. Then the demands for different travel modes are calculated as nm m m d pq d pq pn d pq The resulting model will then be d pq bm min z b t bm pq pq , s ds d pq pq m pqC mM 0 bm pq 1 1 e 1 U n U d 1 dm m d pq d pq 1 e 1 U d U n 1 cm dm d pq d pq 2 U c U b 1 e 1 dm d bm pq d pq 2 U b U c 1 e The composite utility for making a trip is also known as Ud 3 1 2 ln e 2U c e 2U b Magnus Edgren II.5. Optimization problem The optimization problem is given as follows vl c t l l , s ds l wlm lL 0 m M m bm d pq t bm , s ds d bm pq pq pq pq pq m ( p , q ) C m M 0 d pqnmt nmpq min T ( p , q ) C m M 1 1 d ln d nm pq dm dm 1 d pq ln d pq 1 d ln d dm pq 1 d ln d cm pq bm 1 d bm pq ln d pq 1 ( p , q ) C m M 1 2 ( p , q ) C m M 1 2 ( p , q ) C m M nm pq dm pq cm pq subject to cm nm d pq d bm pq d pq , nm dm m d pq d pq d pq , h m r cm d pq , p, q C , p, q C , p, q C , mM mM mM rR m pq ( p , q )C rR m pq m rl hrm wlm , hrm 0, wlm vl , mM dm d pq nm d pq cm d pq bm d pq 0, 0, 0, 0, l L, m M m r R pq , lL p, q C , p, q C , p, q C , p, q C , p, q C , mM mM mM mM mM 4 Magnus Edgren II.6. L 0 hrm Test network The network used for test of the model is depicted in fig 1 below. The circles in bold denotes nodes in the OD-matrix, whereas the other circles are intersections. The solid lines are links in the car network and the dashed lines for the public transport. The public transport is assumed to not intersect with the car network. hrm Doing that we have t c l l m m pq 0 m rl m 0 hrm tlc l rlm pq m lL lL 1 2 4 L 0 hrm This describes a model with route costs 5 crm tlc l rlm m lL 3 In equilibrium either the route flows hrm=0 or the route costs takes the minimum values πpqm, hence it is a user equilibrium model. 6 In order to show that the solution is unique it has to be shown that the objective function, T, is convex. This can be done by showing that the Hessian matrix, H, is positive definite. 7 2T2 v1 H 2T v v 1 K Fig. 1. The test network. III. Verification of the model Formulate the Lagrangean equation as follows d d d L T ( p ,q )C mM m pq ( p ,q )C mM ( p ,q )C mM where m m pq , pq m pq m pq m pq dm pq T t ac a , va va dm d pq cm d pq hrm mM and m pq 2T 2 vK 2T vK v1 By deriving T once we have cm d pq d bm pq nm pq which we know is convex. Further derivations will not change this, i.e. T is convex. are Lagrangean III.2. Public transport model multipliers for the first three constraints in the optimization problem. The public transport model consist only of a costfunction, hence no proof is needed in order to see that it is a part of the optimization problem. III.1. Car network model It can be shown that the car traffic route choice is made according to a user equilibrium model by using the following conditions 5 Magnus Edgren Un Ud t III.3. Choice model The following equations will be used to show that the used choice model is the nestled logit model as described in II.4. nm pq pq 2 t bm pq pq 2 m m pq ln e e 2 1 From this equation we see that Ud is the composite m cost of car ( p q ) and bus ( t pq L 0 xm d pq d xm pq L 0 xm d pq d dm pq e x n, d , c, b d bm pq e cm d pq d 1 mpq t nm pq d bm pq d 1 1 e pq 2 mpq t bm pq pq m 1 1 e 2 U c U b p q m U c U b p q t bpmq p q m m which is the minimum cost for car, p q , minus the pq 2 mpq t bm pq pq m nm d pq d ). πpqm is the and from this we see that 2 mpq mpq cm pq cost for the public transport alternative. IV. Analysis Constructing the quota on the first logit level nm pq pq m On the second level in the choice model we have 2 1 m pq mpq 2 1 cm d pq e pq minimum route cost for car trips from p to q as shown next. This will result in the following nm d pq e bm 1 dm pq 1 e 1 mpq t nm pq 2 1 2 m m pq pq IV.1. Method The analysis is made with respect to variations in some of the input parameters. The parameters analyzed is as follows 1 and comparing it with the following from the logit theory l Monetary cost for bus, pq Monetary cost for car, d d nm pq nm pq d dm pq 1 1 e 1 U n U d Bus frequency, pq Dispersion in the choice model, it is clear that Un Ud t nm pq m pq t nm pq 2 2 1 m pq m pq For each parameter change, the output travel patterns, d pqxm , x n, d , c, b, will be analysed. We will look at the following m m 1 pq 2 pq 1 2 How the decisions on making the trip or not changes for each user group. How the travel mode is changed for those making the trip To clear this expression up we look at the sum of trip makers dm cm d pq d pq d bm pq The user groups here used are “poor” (low income) and “rich” high income. m m 1 pq 2 pq 1 ln e e 2 1 2 2 i , i 1,2 m pq bm t pq pq pq 2 m 6 Magnus Edgren TABLE I CHANGED NUMBER OF TRAVELERS DUE TO PARAMETER VARIATIONS IV.2. Results and discussion Parameter In fig 3 the total flow in the network before changing parameters is indicated. The main flow in the car network goes in OD-pairs (1,5), (1,7), (5,1) and (7,1) via node 3 and in OD-pairs (5,7) and (7,5) via node 6. The biggest movement in the public transport mode is between nodes 7 and 1. Mon. cost car Mon. cost bus Bus frequency Dispersion choice model level 1 Dispersion choice model level 2 1 5 “Poor” “Rich” +20% -20% +20% -20% +20% -20% Mult 5 Div 5 Mult 5 Div 5 -2.0% +2.0% -1.4% +1.4% +0.2% -0.3% +40% -22% -14% +40% -0.8% +0.8% -0.6% +0.6% +0.2% -0.2% +32% -26% -12% +30% By examining a change in the monetary cost for cars in a wider range we get the result in fig 4. The y-axis shows the ratio between making the trip and the total demand for each user group and in total. We see that in order to reduce the number of travelers from 98% to 95% we need to increase the cost to 16 kr per link for poor people. But for the rich people to reduce the traveling as much as the poor the cost needs to be increased to 40 kr, which is 250% more. 2 4 Variation 3 1 0,98 6 0,96 0,94 0,92 Ratio 7 Fig. 3. Total flow in the network. Poor 0,9 Rich Total 0,88 0,86 IV.3. Change in no of travelers due to param changes 0,84 The change in number of travelers due to parameter changes are presented in table I. It can be seen that “Rich” people are less sensitive to changes in the monetary cost for both the car and the public transport travel mode. However, a change in the bus frequency generates the same result for the two user groups, which is reasonable due to the priceless nature of this parameter. The increased dispersion on level 1 yields a bigger raise in number of poor travelers than the rich ones. As a increased dispersion makes people more tend to follow their true wish, the interpretation of this is that the number of poor people wanting to do a trip but don’t actually do it is bigger than the corresponding subset among the rich people. The change in dispersion on level 2 is not intuitively easy to interpret. Raised dispersion yields more tendency to choose the true wish alternative and maybe it is so that people in this model would want to go by car, but as they don’t afford it they rather stay at home than go with the alternative they can afford (i.e. public transport). 0,82 47 42 37 32 27 22 17 12 7 2 0,8 Mon. cost car [kr/link] Fig. 4. Change of the monetary cost for car. We continue by looking at the dispersion in the choice model (level 1), as shown in fig 5. With a very low dispersion all groups have the same travel habits but for high dispersion the values differ a lot. We also see that for high dispersion the values are stable. This complies with the theory that says that people tend to select more and more according to their utility as the dispersion increases. 7 Magnus Edgren tell us that all groups are similarly tend to use the car. 0,9 0,6 0,8 0,55 0,7 0,5 Poor 0,5 0,45 Rich 0,4 Ratio Ratio 0,6 Total Poor 0,4 Rich Total 0,3 0,35 0,2 0,3 0,1 0,25 45 47 42 37 32 27 22 17 7 Dispersion, logit level 1 12 0,2 2 40 35 30 25 20 15 10 5 0 0 Mon. cost car [kr/link] Fig. 5. Change of the dispersion, choice model level 1. Fig. 6. Change of the monetary cost for car. IV.4. Change in no of car users due to param changes With the dispersion for “no trip” or ”make trip” set low we have equally car usage for the user groups. With a raised dispersion we get quite low shares of car usage for all groups. The values go to stable values as the dispersion increases. Table II presents the change in number of car users due to changes in the parameters. We see that raised fuel price for car gives less car users as they switch to public transport, while a raised fee for the public transport gives more car users. As before the rich group is less sensitive to changes in the monetary cost. A change in the bus frequency is independent of user groups. Changes in the dispersion on level 1 in the logit model do not make a big difference on the ratios that is presented here. The reason is that the number of people actually doing the trip (which is changed due to the dispersion change) still chooses the travel mode as they did before. The dispersion on level 2 makes a big difference however. A raised dispersion makes people choose the car alternative in a bigger extent which also reduces the number of travelers in total as discussed above. 0,9 0,8 0,7 Ratio 0,6 0,2 0,1 45 40 35 30 25 -1.9% +1.9% +2.4% -2.5% -0.6% +0.9% +0.1% +0.2% +42% -16% 20 -3.7% +3.6% +4.6% -4.8% -0.6% +0.9% 0.0% +0.2% +42% -18% 15 +20% -20% +20% -20% +20% -20% Mult 5 Div 5 Mult 5 Div 5 0 10 “Rich” 5 “Poor” Total 0 Variation Rich 0,4 0,3 TABLE II CHANGED NUMBER OF CAR USERS DUE TO PARAMETER VARIATIONS Parameter Poor 0,5 Dispersion, logit level 1 Mon. cost car Mon. cost bus Bus frequency Dispersion choice model level 1 Dispersion choice model level 2 Fig. 7. Change of the dispersion for choice mode level 1. V. Conclusion The given model is shown to be a combination of the given models for the car network, public transport and choice. It is also, due to the tests performed within this work, likely that the model is valid in reality. However the parameters must be chosen carefully as discussed below. The results indicate that people are more sensitive to variations in monetary cost than for variations in other costs such as bus frequency. However there are important parameters that have not been analyzed here, such as changes in travel time. Between user groups we With a look on the monetary cost we get the graph in fig 6, which has the share of car users among all travelers for each user group on the y-axis. High cost yields that poor people uses the bus instead so the ratio decreases rapidly for this group. A low cost, however, 8 Magnus Edgren see that rich people are less sensitive to changes in cost than poor people are. We also see that the results are very sensitive to the dispersion parameters. With the dispersions being wrong it doesn’t matter how good the model and parameter is in other respects. Time spent on the project The time spent on the project is shown in table III. TABLE III TIME SPENT ON THE PROJECT Parameter Theory Implementation Tests Analysis Documentation 45 h 20 h 14 h 9h 35 h Appendix The AMPL code is attached last in the report. The following source files are attached: prob_def.txt network.txt prob_solve.txt defines the optimization problem defines all parameters AMPL script for running a calculation References Journal Papers: [1] C. Rydergren, Short text on the combined mode and route choice model, (2005) Books: [2] Y. Sheffi, Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods, (1985), Prentice-Hall, Englewood Cliffs, NJ. 9 Magnus Edgren prob_def.txt var var var var var var var var var var var var #-----------------------------------------------------------# TNK084 Traffic theory # Project - Combined models for private and public transport # Optimization problem # Magnus Edgren # 2008-04-27 #------------------------------------------------------------ #-------------------# Sets #-------------------set set set set set set set set set LINKS; ODPAIR; GROUPS; R15; R17; R51; R57; R71; R75; v {LINKS}; # link flows w {LINKS, GROUPS}; # link flows per group h15 {R15, GROUPS}; # route flows OD15 h17 {R17, GROUPS}; # route flows OD17 h51 {R51, GROUPS}; # route flows OD51 h57 {R57, GROUPS}; # route flows OD57 h71 {R71, GROUPS}; # route flows OD71 h75 {R75, GROUPS}; # route flows OD75 dn {ODPAIR, GROUPS}; # demand no trip dd {ODPAIR, GROUPS}; # demand trip db {ODPAIR, GROUPS}; # demand bus dc {ODPAIR, GROUPS}; # demand car #-------------------# Objective function #-------------------- # # # # # # # # # minimize T: sum{l in LINKS} (ffc[l]*v[l] + A[l]*((1/crho[l])^n[l])*((v[l])^(n[l]+1))/(n[l]+1) + sum{m in GROUPS} (ctao[l]*w[l,m]/beta[m])) + sum{pq in ODPAIR, m in GROUPS} (db[pq,m]/alfa[pq] + db[pq,m]*(brho[pq]+(btao[pq]/beta[m]))) + sum{pq in ODPAIR, m in GROUPS} (dn[pq,m]*tn[pq,m]) + (1/beta1) * (sum{pq in ODPAIR, m in GROUPS} (dn[pq,m]*(log(dn[pq,m])-1) + dd[pq,m]*(log(dd[pq,m])-1))) (1/beta2) * (sum{pq in ODPAIR, m in GROUPS} (dd[pq,m]*(log(dd[pq,m])-1) )) + (1/beta2) * (sum{pq in ODPAIR, m in GROUPS} (dc[pq,m]*(log(dc[pq,m])-1) + db[pq,m]*(log(db[pq,m])-1))); #-------------------# Parameters #-------------------param ffc {LINKS}; # Free flow param A {LINKS}; # param crho {LINKS}; # param n {LINKS}; # param ctao {LINKS}; # param alfa {ODPAIR}; # param d {ODPAIR, GROUPS}; # param brho {ODPAIR}; # param btao {ODPAIR}; # param beta {GROUPS}; # param beta1; # param beta2; # param delta15 {R15, LINKS}; # param delta17 {R17, LINKS}; # param delta51 {R51, LINKS}; # param delta57 {R57, LINKS}; # param delta71 {R71, LINKS}; # param delta75 {R75, LINKS}; # param tn {ODPAIR, GROUPS}; # #-------------------# Constraints #-------------------subject to Constraint1 {pq in ODPAIR, m in GROUPS}: dc[pq,m] + db[pq,m] = dd[pq,m]; subject to Constraint2 {pq in ODPAIR, m in GROUPS}: dn[pq,m] + dd[pq,m] = d[pq,m]; subject to Constraint3a {m in GROUPS}: sum{r in R15} h15[r,m] = dc['OD15',m]; #-------------------# Variables #-------------------- subject to Constraint3b 10 Magnus Edgren {m in GROUPS}: sum{r in R17} h17[r,m] = dc['OD17',m]; subject to Constraint7 {pq in ODPAIR, m in GROUPS}: dn[pq,m] >= 1; subject to Constraint3c {m in GROUPS}: sum{r in R51} h51[r,m] = dc['OD51',m]; subject to Constraint8 {pq in ODPAIR, m in GROUPS}: dd[pq,m] >= 1; subject to Constraint3d {m in GROUPS}: sum{r in R57} h57[r,m] = dc['OD57',m]; subject to Constraint9 {pq in ODPAIR, m in GROUPS}: dc[pq,m] >= 1; subject to Constraint3e {m in GROUPS}: sum{r in R71} h71[r,m] = dc['OD71',m]; subject to Constraint10 {pq in ODPAIR, m in GROUPS}: db[pq,m] >= 1; subject to Constraint3f {m in GROUPS}: sum{r in R75} h75[r,m] = dc['OD75',m]; subject to Constraint11 {l in LINKS}: v[l] >= 0; subject to Constraint4 {l in LINKS, m in GROUPS}: sum{r in R15} (delta15[r,l]*h15[r,m]) + sum{r in R17} (delta17[r,l]*h17[r,m]) + sum{r in R51} (delta51[r,l]*h51[r,m]) + sum{r in R57} (delta57[r,l]*h57[r,m]) + sum{r in R71} (delta71[r,l]*h71[r,m]) + sum{r in R75} (delta75[r,l]*h75[r,m]) = w[l,m]; subject to Constraint12 {l in LINKS, m in GROUPS}: w[l,m] >= 0; network.txt subject to Constraint5a {m in GROUPS, r in R15}: h15[r,m] >= 0; #-----------------------------------------------------------# TNK084 Traffic theory # Project - Combined models for private and public transport # Input data # Magnus Edgren # 2008-04-25 #------------------------------------------------------------ subject to Constraint5b {m in GROUPS, r in R17}: h17[r,m] >= 0; subject to Constraint5c {m in GROUPS, r in R51}: h51[r,m] >= 0; set LINKS := 1 4 5 6 7 8 9 10 11 12 14 15 16 18 19 20 21 24; set ODPAIR := OD15 OD17 OD51 OD57 OD71 OD75; set GROUPS := POOR RICH; set R15 := R1 R2 R3 R4 R5 R6; set R17 := R7 R8 R9 R10 R11 R12; set R51 := R13 R14 R15 R16 R17 R18; set R57 := R19 R20 R21 R22 R23; set R71 := R24 R25 R26 R27 R28 R29; set R75 := R30 R31 R32 R33 R34; subject to Constraint5d {m in GROUPS, r in R57}: h57[r,m] >= 0; subject to Constraint5e {m in GROUPS, r in R71}: h71[r,m] >= 0; subject to Constraint5f {m in GROUPS, r in R75}: h75[r,m] >= 0; #------------------------# Car model parameters #------------------------param ffc:= 1 0.0125 4 0.0125 subject to Constraint6 {l in LINKS}: sum{m in GROUPS} w[l,m] = v[l]; 11 Magnus Edgren 5 0.03 6 0.033333 7 0.03 8 0.025 9 0.075 10 0.033333 11 0.026667 12 0.07625 14 0.025 15 0.026667 16 0.02 18 0.075 19 0.07625 20 0.02 21 0.0125 24 0.0125; 24 1800; param n:= 1 4.5 4 4.5 5 3 6 3.1 7 3 8 3.2 9 3.5 10 3.1 11 3.1 12 3 14 3.2 15 3.1 16 3.1 18 3.5 19 3 20 3.1 21 4.5 24 4.5; param A:= 1 0.0026515 4 0.0026515 5 0.03 6 0.033333 7 0.03 8 0.025 9 0.015909 10 0.033333 11 0.026667 12 0.016174 14 0.025 15 0.026667 16 0.02 18 0.015909 19 0.016174 20 0.02 21 0.0026515 24 0.0026515; param ctao:= # Car fee / link [kr] 1 5 4 5 5 5 6 5 7 5 8 5 9 5 10 5 11 5 12 5 14 5 15 5 16 5 18 5 19 5 20 5 21 5 24 5; param crho:= 1 1800 4 1800 5 1100 6 1100 7 1100 8 1100 9 1100 10 1100 11 1100 12 1100 14 1100 15 1100 16 1100 18 1100 19 1100 20 1100 21 1800 param delta15: 1 4 5 6 7 8 9 10 11 12 14 15 16 18 19 20 21 24 := R1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 R2 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 R3 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 R4 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 R5 0 1 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 R6 0 1 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0; param delta17: 1 4 5 6 7 8 9 10 11 12 14 15 16 18 19 20 21 24 := R7 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 R8 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 12 Magnus Edgren R9 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 R10 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 R11 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 R12 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1; param delta51: 1 4 5 6 7 8 9 10 11 12 14 15 16 18 19 20 21 24 := R13 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 R14 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 R15 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 R16 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 R17 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 R18 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0; param delta57: 1 4 5 6 7 8 9 10 11 12 14 15 16 18 19 20 21 24 := R19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 R20 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 R21 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 R22 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 R23 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1; param delta71: 1 4 5 6 7 8 9 10 11 12 14 15 16 18 19 20 21 24 := R24 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 R25 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 R26 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 R27 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 R28 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 R29 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0; param delta75: 1 4 5 6 7 8 9 10 11 12 14 15 16 18 19 20 21 24 := R30 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 R31 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 R32 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 R33 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 R34 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0; #------------------------# Other parameters #------------------------param d(tr): OD15 OD17 OD51 OD57 OD71 OD75 := POOR 562.5 525.0 337.5 300.0 525.0 425.0 RICH 562.5 525.0 337.5 300.0 525.0 425.0; param tn(tr): OD15 OD17 OD51 OD57 OD75 := POOR 1 1 1 1 1 1 RICH 1 1 1 1 1 1; param beta:= POOR 30 RICH 60; OD71 # Value of time [kr/hour] prob_solve.txt #-----------------------------------------------------------# TNK084 Traffic theory # Project - Combined models for private and public transport # Problem solver script # Magnus Edgren # 2008-04-27 #-----------------------------------------------------------reset; option solver minos; model prob_def.txt; data network.txt solve > prob_result.txt; display v > prob_result.txt; display w > prob_result.txt; display h15 > prob_result.txt; display h17 > prob_result.txt; display h51 > prob_result.txt; display h57 > prob_result.txt; display h71 > prob_result.txt; display h75 > prob_result.txt; #------------------------# Bus model parameters #------------------------param brho := OD15 0.8 OD17 0.8 OD51 0.16 OD57 0.2 OD71 0.25 OD75 0.2; param alfa := OD15 10 OD17 10 OD51 10 OD57 10 OD71 10 OD75 10; # Bus frequency exit; param btao := OD15 20 OD17 20 OD51 20 OD57 20 OD71 20 OD75 20; # Bus fee [kr] #------------------------# Choice model parameters #------------------------param beta1:= 1; # Dispersion logit model level 1 param beta2:= 0.2; # Dispersion logit model level 1 13