Exercise

1. Given the following data set, develop a trip table with four categories of households. List the problems you face while doing so.

   Household	Monthly	Occupants	Trips
   number	income	between 6	produced
   	(1000 Rs.)	and 60 yrs.	per day
   1	10	2	4
   2	15	5	4
   3	35	5	14
   4	30	6	14
   5	40	3	8
   6	15	5	6
   7	20	3	3
   8	32	3	8
   9	40	3	9
   10	38	2	6
   11	40	6	10
   12	12	6	6
   13	35	2	8
   14	40	2	7
   15	15	2	3

2. Using the trip table obtained in Problem 1, determine the total number of trips produced from a zone with 2000 households in each category.
3. Determine which of the following three trip distribution tables (giving the trips made from Zone A to Zones B, C, and D) is likely to be obtained if one used the maximum entropy model of trip distribution.

   	B	C	D			B	C	D			B	C	D
   A	3	4	5		A	4	4	4		A	4	6	2

4. For the 1000 shopping trips from Zone A, four destinations exist. The destinations are Zones W, X, Y, and Z. The shopping areas available in each of the zones and their distances from Zone A are given in the following table. Assuming a proportionality constant of 0.35 and assuming a thousand square meter of shopping area as one opportunity determine the trip distribution from Zone A.

   Zone	Shopping area	Distance from Zone A
   	(in '000 sq.m.)	(in km)
   W	2.0	5.0
   X	1.0	8.0
   Y	5.0	11.0
   Z	3.0	5.0

5. A Logit model is being developed for mode choice behaviour. The choice is between two modes -- automobile and bus. The utilities derived from using an automobile and a bus are, $U_A$, and $U_B$, respectively. The following models for $V_A$ and $V_B$ are proposed (note that, the notation used is, $U_i = V_i + e_i$, where $e_i$ is a stochastic term): $V_A = a_A + bt_A$ and $V_B = bt_B$; where, $t_k$ is the travel time by mode $k$; $a_A$ and $b$ are constants. Based on the above description, answer the following:
   1. Show that $\pi_A/\pi_B = e^{V_A}/e^{V_B}$; where $\pi_i$ is the probability of choosing Mode $i$.
   2. What is the purpose of including a constant term in $V_A$?
   3. What sign should one expect for $b$ and why?
   4. If it is observed that people are more inclined to use their automobiles (that is, they are biased towards the automobile mode) what sign should one expect for $a_A$?
   5. Suppose a new mode, say rail, is introduced and $V_R = bt_R$, what changes can be observed in $\pi_A$, $\pi_B$, and $\pi_A/\pi_B$?
6. For the incremental assignment example problem given in the text assign the O-D matrix in (a) three increments by dividing the matrix into three parts in the ratio 40:30:30, and (b) in five increments by dividing the matrix into five parts in the ratio 40:30:10:10:10. Compare the results and state what you observe.
7. Use user equilibrium method to assign the trips given in Problem 6.
8. Show that for the example problem on user equilibrium given in the text the objective function value at $n=1$ is 1975. Also show that the value of $\alpha^{*,n}$ for $n=1$ is 0.5964.