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# 1 - KTH Kungliga Tekniska hgskolan

By Harold Gray,2014-02-26 16:52
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Formulate an entropy maximising model for the number of shopping trips by modeConsider then the extended network including also link 5 dotted line

Written exam in the course Optimisation Models in Transport and Location

Analysis

Monday, March 10, 2003, 14.00-18.00

1. Assume an individual n is choosing alternative i from a finite set {1,2,…,I} with probability

VinenP. iIVjne?

1j

Formulate a random utility maximisation problem for individual n that leads to this

nVaprobability (no derivation). How will change if a common constant is added to all . Pjnni

Va;b;tAssume that for all j, , where t is an observed variable. Formulate a jnjnnjjn

a,b and maximum likelihood estimation problem for the parameters . What data do you nj

tneed in addition to ? Can all parameters be estimated? jn

m2. Consider the problem of finding the number of shopping trips from residential areas i to Sij

Sshopping centres j by mode m. The total number of shopping trips from zone i is given as. i

mWThe net travel cost between zone i and zone j by mode m is where is the (cW)jijj

benefit obtained through shopping in centre j. The total net travel cost for shopping trips in the region is NC. Formulate an entropy maximising model for the number of shopping trips by mode! Derive the solution and interpret the components of the model! What model for the

choice of travel mode between zone i and shopping centre j is implied?

3. Consider a region consisting of N population sites indexed by i = 1, 2,…, N. Let the number

hof inhabitants in site i be . K mobile telephone masts are to be located in the region. There i

are M potential location sites indexed by j =1, 2,…, M, where M > K. Let d be the distance ij

between population site i and potential mobile telephone mast j. Formulate an optimisation

problem that minimises the number of inhabitants in the region that will have a mobile telephone mast within the distance of . d

4. Consider the following network:

B 4

c(f)5;10f 111

2 c(f)55;f 222

c(f)55;f 5 3 333

c(f)5;10f 444

1 Six vehicles should travel between A and B.

f A is the vehicle flow on link i. i

Find the user equilibrium assignment of vehicles to the original (continuous line) network of

four links (1-4) and compute the total travel cost.

Consider then the extended network including also link 5 (dotted line) with cost function

c(f)10;f. Find the new user equilibrium assignment after the investment and compute 555

the total travel cost.

Explain the effects of the investment! How can the paradox be resolved?

5. The following indicator for accessibility to work has been suggested:

w1iwa where = number of working places in zone i iipi

p = working population residing in zone i i

Discuss the properties (strengths, weaknesses) of this indicator!

A more elaborate indicator is the following:

wj2aq(d) where q and f are non-increasing functions of distance ?iijf(d)pj?kjkk

(q may be decided by planners and f estimated on commuting

data)

Discuss the properties of this indicator!

Give other examples of how accessibility can be measured?

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6. Assume that you have been hired as an advisor to the mayor in a city that is contemplating to introduce some form of road pricing. There is a ring road immediately outside the city centre. Indicate how you would design a road pricing system if the mayor wants to a) maximise revenue

b) reduce congestion in the city centre

emissions in the city as a whole. c) reduce CO2

7. In a standard monocentric urban economics model, let every individual be identical with income Y and transport costs T(r), where r is distance to Central Business District (CBD). Individuals have preference for lot size s and consumption z, given by the utility function

The price of consumption good z is set to unity. U(z,s).

a) State the utility maximisation problem for the individual!

b) Define the bid rent function formally, and explain in word what it is! c) Motivate why the bid rent function is decreasing in r!

d) Suppose that the utility function of the individual is given by U(z,s)logz;logs where 0,0, and ;1. Then the optimum consumption,

ugiven lot size s and utility level u, is given by Derive the bid rent function Z(s,u)se.

(r,u)!

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Solution to

Written exam in the course Optimisation Models in Transport and Location Analysis

Monday, March 10, 2003, 14.00-18.00

UV;X1. Let the utility of alternative I for individual n be , where Xis independent in ininin

nGumbel (0,). Then . Adding the same constant ato both PPr{UU for all j}n iinjn

nPsides of the previous inequality will not change . The parameters b and can be ji

nmaxPestimated by solving the following ML-problem:, where z is the alternative n~zn,...,bb1,In

chosen by individual n and which needs to observed (in fact it is only necessary to observe the number of individuals choosing each alternative). aand one of the bcannot n j nPbe estimated since they cancel out in . i

3. The following minimisation problem solves the stated problem

N

hzmin?iiy,...,y1Mi1

subject to

M

yK?jj1

zyaij for all and ijij

(；y0,1j

(；z0,1i

and where

?1 if ddija ?ij0 otherwise?

6. a) Place several toll rings outside and perhaps also inside the ring road and collect a (high) toll fee night and day. Allow season tickets.

b) Place a toll ring immediately inside the ring road and collect a reasonable high toll fee during congested hours. In addition there could also be some toll lines inside the city centre. Do not allow season tickets.

c) The simplest solution would perhaps be to introduce an additional local citywide petrol tax. Another possibility, at least in the future, would be to have a distance-based toll fee (implemented by some GPS-based system). Presently a number of toll rings throughout the city would be a way of implementing a CO-reducing system. 2

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