Correa Es una banda flexible de caucho que se coloca alrededor de la mano para sujetar la caja que lleva los componentes electrónicos, brindando facilidad en

Time is continuous, but VRP models require discrete time periods. Time is split into finite chunks to enable the VRP model to potentially consider all the possible options. For each interval, τ , the travel time is modelled as one equation or distribution, and the intervals are then combined together create the travel time function. This is of the form:

ˆ

yt = fτ(t); t ∈ τ, (2.2.2)

whereby ˆytis the predicted travel time at time t and fτ is a function that is dependent

upon the corresponding interval τ to time t. The function can also be dependent upon other factors rather than time alone.

CHAPTER 2. EVALUATING SINGLE LINK MODELS 33

can be applied to distributions. For a distribution with a set number K of different values with corresponding probability pk then:

fτ(t) =gτ(k)(t), p = pk; ∀k ∈ K. (2.2.3)

The intervals for distribution estimates are generally wider, as computing accurate distributions is more difficult with less data.

Let N be the number of discrete time intervals. Then the N functions provide the travel inputs to the VRP and hence the function must provide sensible travel time estimates. Carey et al. (2003) study some of the properties that a travel time function requires for a single link in a VRP and Horn (2000) discuss several assumptions that are often made. These were discussed in the context of point forecasts but also apply to distributions. The properties are first summarised below before the assumptions are discussed.

First In First Out. The first in first out property or FIFO ensures that vehicles can’t miss a delay along the same link by setting off later. This means that if vehicle one enters the link before vehicle two, vehicle one will always exit the link before vehicle two. This issue occurs because of the discretization of the problem from continuous time. Hence

t1+ ˆyt1 < t2+ ˆyt2 ∀t1 < t2. (2.2.4)

This is a reasonable requirement for our problem, as despite travelling upon roads with multiple lanes, delivery vehicles are unlikely to overtake due to their size and speed restrictions and most delays affect all lanes of a road. To reduce the model complexity all lanes are modelled as a single link.

CHAPTER 2. EVALUATING SINGLE LINK MODELS 34

Time dependence. The travel time for time t depends upon the traffic up to and including time t but not later. Time is linear and events that happen further into the future won’t impact now. This only refers to the state of the traffic at time t + 1 etc. which will be unknown. Extra information such as bank holidays, for which the impact can be estimated in advance, can be included in the model.

Constant reduction. The last property is that if traffic flows through the link at the same constant rate any model should reduce to the simple constant model with no variation through time. The travel time is thus just a constant value.

Constant speed. The speed on a link should be constant for the entire link. If this is not the case then the link should be split into sections where the speed is constant. Otherwise, if an accident occurs on the link, the effect upon the travel time will be directly linked to which part of the link the accident occurs in.

Maximum speed. All vehicles travel are assumed to travel as fast as they can (with respect to the current maximum link speed and other factors.) Many HGVs are limited to a certain speed, hence the link length may be needed to calculate a maximum travel time which is lower than the free flowing traffic.

Time invariant network. The defined network is time invariant so doesn’t change. All roads remain in the network and none have their speed reduced by roadworks. This is particularly important in complete graphs which are discussed in Section 1.2.

Positivity. All values must obey the rules of the physical world. The distance, speed and travel time are all positive.

CHAPTER 2. EVALUATING SINGLE LINK MODELS 35

Simple travel time profiles

We now look at the different ways that have been suggested for constructing travel time functions in VRP problems. These start from the very simplest models which are then improved to try and fit the assumptions and properties of the ideal travel time function.

Plotting the travel time function over time gives the travel time profile which is much easier to understand. Creating travel time profiles for distributions is consider- ably more difficult so the following are all point forecasts.

Constant travel time. The simplest way that travel time has been modelled is to assume that each link has a constant travel time for the entire time period un- der consideration (Pradhananga et al., 2010; Zografos and Androutsopoulos, 2004; Pradhananga et al., 2014). Hence ˆyt = c for some constant c.

Step function. A logical extension to having a constant travel time is to make the travel time depend upon the time of day. The simplest travel time function that is time dependent is a step profile (Malandraki and Daskin, 1992). The day is divided into a given number of time periods and each time period has a constant travel time. Thus fτ(t) = cτ. A five time period step function can be seen in Figure 2.2.1.

However the step function doesn’t obey the FIFO property if the differences be- tween two steps are great enough. For example, in Figure 2.2.1, if a vehicle leaves at 19:59:00 it will arrive at 20:03:00, however if it leaves at 20:00:30 it will arrive at 20:02:30, 30 seconds before.

Slanted step function. A slanted step function removes the discontinuities. For a short interval around where each interval τi ends and τi+1begins, the function changes

with a constant gradient between the two constant values. Figure 2.2.2 shows how this would look. If the gradient isn’t too steep then the FIFO property will hold. Malandraki and Daskin (1992) suggest this function without any advice on how to

CHAPTER 2. EVALUATING SINGLE LINK MODELS 36

Figure 2.2.1: Step travel time function plot

generate the travel time profile. For a step-function with predefined time intervals the average can be taken for each period to give cτ, but it is unclear how wide the

slanted connecting sections should be.

CHAPTER 2. EVALUATING SINGLE LINK MODELS 37

More complicated functions The previous functions have all been linear within each time interval. Relaxing this permits a much wider range of functions.

We next look at how to create travel time functions that accurately reflect the travel time for a given road segment. We will begin by looking at point forecasts which can be used to create an estimate for each time interval.