In this problem one has decided to use the following utility function: U(Cost, TVTT, CO) = 0.5[5] + 0.2[20] +0.3[10]

Construction experts have determined the marginal distribution on Cost to be that shown in Table 1-1.

Table 1-1. Marginal PMF for Cost Alternative Cost Pr(Cost)

Alt. Cost. Pr(Cost)

A1 : 1.65 0.8

3.00 0.2

A2 : 1.5 0.6

3.5 0.4

Travel demand experts have determined the marginal distribution on annual total vehicle travel time TVTT to be that shown in Table 1-2. (Note: This distribution is one in which the peak period travel times have been converted to annual peak period travel times.)

Table 1-2. Marginal PMF for TVTT

Alt. TVTT. Pr(TVTT)

A1: 7.5 0.8

19.0 0.2

A2: 8.0. 1.0

Environmental experts are predicting the CO level from the TVTT value. In this simplified problem, they believe that this prediction can be made according to the following equation:

CO = k TVTT

They are uncertain about the value of k. They believe that this value does not depend on the alternative or on the value of TVTT and express their uncertainty about the value by the probability distribution presented in Table 1-3.

Table 1-3. Probability distribution on the value of k

Value of k. Pr(k) 0.5 0.3

0.2 0.7

As a first approximation, it is assumed that the value of TVTT and CO are independent of the values of Cost. That is, Pr(Cost, TVTT, CO) = P(Cost) x P(TVTT,CO).

One wishes to choose the alternative, A1 or A2, that maximizes the expected utility associated with (Cost, TVTT, CO) outcomes.

1. (5 pts) Draw the branch of the decision tree associated with choosing Alternative A1. Show values of Cost, TVTT, k, CO, associated probabilities, and outcomes (COST, TVTT, CO) in the appropriate place. Do not clutter the diagram with utilities or with calculations.

2. (3 pts) Show that EU(A1) = 0.62474