Utilities

Basic idea

These trade-offs are operationalized via giving utitilies to the different aspects of the situation. The utilities in this chapter will be negative, which is why they are sometimes called disutilities. Let us assume that we have the following utilities:

• The (dis)utility of the trip time, $U_{trip}(T_{trip})$. It depends on the trip time, $T_{trip}$.

• The (dis)utility of being early, $U_{early}(T_{early})$. It depends on how early the traveler is. If the traveler is late, this contribution is zero.

• The (dis)utility of being late, $U_{late}(T_{late})$. It depends on how late the traveler is. If the traveler is early, this contribution is zero.

Let us further assume that these utilities are additive (see Fig. 14.1):

$$U_{dep} = U_{trip}(T_{trip}) + U_{early}(T_{early}) + U_{late}(T_{late}) \ .$$

An example is:

$$U_{dep} = - \frac{0.4}{60~sec} \, T_{trip} - \frac{0.25}{60~sec} \, T_{early} - \frac{1.5}{60~sec} \, T_{late}\ .$$ (14.1)

The results of this come out in arbitrary utility units, sometimes called ``utils''.

Dependence on departure time

Fig. 14.1 gives the function of the different utilities as a function of the arrival time. For the calculation that we will do later, we need them as a function of departure time. For example, if $t_{des}$ is the desired arrival time, then

$$T_{early}(t_{dep}) = \max\Big( 0, t_{des} - t_{early} \Big) = \max\Big( 0, t_{des} - (t_{dep} + T_{trip}) \Big) \ .$$

Here, $T_{trip}$ again depends on $t_{dep}$, and therefore

$$T_{early}(t_{dep}) = \max\Big( 0, t_{des} - (t_{dep} + T_{trip}(t_{dep})) \Big) \ .$$

As we will see later, we will essentially need a table of the values of $T_{early}$ as a function of $t_{dep}$ where $t_{dep}$ increases in 5-min time steps. Because of this simplification, the problem can be solved as a sequence of look-ups, resulting in a table similar to the following (where $t_{des} = 8:00$)

$t_{dep}$	$T_{trip}(t_{dep})$	$T_{early}(t_{dep})$
6:00	0:15	1:45
$\vdots$
7:00	0:15	0:45
7:05	0:19	0:36
7:10	0:30	0:20
$\vdots$

Figure 14.1: Utility contributions