### What is a Mode Choice Model?

Mode choice models model the travellers choice of which mode of transport to take, eg car, public transport or whatever. They take as input variables about each possible mode of transport that the traveller has available for the trip and gives the proportion of travellers which would use each mode of transport. (eg 70% by car and 30% by public transport).

If a trip matrix is input into the mode choice model, it would split the matrix into separated matrices, one for each mode of transport. (For this reason it is sometimes called a mode split model.) These matrices can be assigned to the highway and public transport networks respectively to give flows on each section of highway or each public transport service.

If individual trips are input into the mode choice model (eg trips obtained from a household interview survey), then the mode choice model will provide the probability of this trip using each mode of transport. If the trip record has an expansion factor which represents the total number of people with this choice, then the mode choice model will apply the probability to the expansion factor to give the expanded number of people who chose each mode of transport.

### Input Data

The modeller must decide which variables are relevant to the decision making process but the most common variables for urban and interurban travel are: in-vehicle time, walk time, wait time, cost and how many interchanges are involved. This data can be 'skimmed' from the highway and public transport networks for each origin-destination pair to form matrices which are called skims because they are obtained from the paths by skimming along the paths accumulating the data. The networks can therefore be skimmed for in-vehicle time, walk time, fare, etc and each skim put into the mode choice model. In this case the mode choice model will split an input trip matrix into a trip matrix which uses each mode of transport, eg car and public transport.

The modeller must also determine which modes are available - or the choice set. This is sometimes not so easy but some software suites will provide this as a skim.

### The Concept of Utility

Economists have a concept called utility which broadly speaking represents the amount of 'use' you get from buying something. The value of the utility can be used to compare different commodities - the higher the utility, the more 'use' or benefit you get from it and the more likely you are to choose it. If you have an amount of money to spend on a set of commodities, the concept of utility says that you would chose the set which gives you the highest utility.

For example if you wanted to buy an electric kettle for boiling water you may have the choice of a small cheaper kettle or a larger more expensive one, which boils water quickly. You may choose to buy the cheaper one because you think the extra features are not worth the extra cost. The concept of utility combines the cost with the other features of (in this case the kettle) and would say that the cheaper kettle gave you the higher utility.

Utility can be used to compare completely different alternatives so for example, you may already have a method of boiling water and you may instead want to buy a packet of tea and a cup so you could drink a hot cup of tea but you could not buy everything. The concept of utility would say that you would form the utility of the alternatives: 1 cheap kettle, 2 expensive kettle, 3 cup and tea and then you would select the alternative with the highest utility.

The concept of utility assumes that you have a method of combining the various features (called attributes) of all the alternatives including their price, to give one measure of utility which is consistent across all the alternatives within the set of choices (or choice set) open to you.

Let us consider utility in the context of mode choice. The concept of utility would say that we would consider the utility of each mode in turn. The utility for each mode would consist of the attributes (or features) of each mode which were relevant to the decision making process. The attributes most commonly used include:

£ = the cost of each mode eg fare, parking cost, petrol cost,

IVT = the amount of time spent travelling on each mode of transport (in the vehicle itself as opposed to wait and walk time - see below)

WAIT = the amount of time spent waiting for the bus train etc to arrive,

WALK = the amount of time spent walking to, from or between buses, trains etc

IC = the number of interchanges needed (eg between buses, trains)

The utility for each mode can be formed from the weighted sum of the attributes of choice so for example the utility for mode m could be given by:

U(m) = b(m) + b1*IVT(m) + b2*£(m) + b3*WALK(m) + b4*WAIT(m) + b5*IC(m)

Where:

U(m) = the utility of travel by mode m

b(m) = the perception of mode m (or mode constant)

b1, b2, b3, b4, b5 = weight of each attribute

IVT(m) = the accumulated time spent in vehicles while travelling by mode m

£(m) = the accumulated fare for travelling by mode m

WALK(m) = the accumulated time spent walking while travelling by mode m

WAIT(m) = the accumulated time spent waiting while travelling by mode m

IC(m) = the number of interchanges needed to make the trip by mode m

Faced with a choice between alternative modes of transport, the traveller would find out the value of each attribute (eg cost, in vehicle time, walk time, wait time and the number of interchanges), weight and add them so as to calculate the utility for each mode of transport. He (or she) would then compare the utility of each mode of transport and (in principle) choose to travel on the mode with the highest utility. For modelling more than one person, we need to consider the mathematical form of the choice model.

Non - mathematicians may like to read this paragraph in order to aid their understanding of Exp(U) used in the Mathematical Form section described below. Please consider raising a number such as 2 to a power such as 3 which would be represented as 2^3 which means 2 multiplied by itself three times ie 2*2*2 which is 8. So if utility was 3 then 2^U would be 2*2*2 or a value of 8. If utility were 4 then 2**U would be 2*2*2*2 which would be 16. In mathematics there is a special number called by the letter 'e' which has a value of about 2.1 or more precisely about 2.17. e is a bit like pi (the ratio of the diameter of a circle to its circumference - about 3.14159) which you can calculate to as many decimal places as you like but you will never get it exactly. Like pi, mathematicians therefore denote it by using the letter 'e'. So let us consider e^U which if U was 3 would be given by finding e on your calculator and multiplying it by itself three times to give e*e*e or a bit more than 8. You can get the same answer a bit easier by pressing 'exp' on your calculator followed by pressing 3 which is better especially if U was not a round number. Exp(U) is therefore the e raised to the power of the utility and Exp(Um) is e raised to the power of the utility for mode m. It's a bit of a mouthful to say, so you may like to think of it as utility after you have pressed the Exp key on your calculator.

### Mathematical Model Form

There are various forms of mode choice model (eg probit or logit) but by far the most common form is the logit model (pronounced lodge it). The logit model has a theoretic pedigree from random utility theory, has been found to fit mode choice making behaviour quite well and is computationally tractable. The probit model has a better theoretical pedigree (athough this is arguable) but is computationally much more intractable. We will concentrate on the logit model. The generalised extreme value (gev) model is a generic form of model from which can be derived various forms of model including the logit 'family' of models. It has important properties which are used to estimate model coefficients using both revealed and stated preference techniques and which form the basis of disaggregate modelling. The mixed logit form of the model provides greater flexibility in the model.

The logit model comes in two main forms: the multinomial logit model or the nested (sometimes called hierarchical or tree) logit model. The difference is best demonstrated by means of an example where there are three alternative modes of transport to chose from: car, bus, train. The multinomial logit model would say that the traveller would look at all three modes and choose one of these three. The nested logit model would say that the traveller would choose first whether to go by car or public transport and if he chose public transport he would then consider whether to choose bus or train. Note that in the nested logit, the bus versus train choice is not even considered if he chooses car. The binomial logit model is a special case of the multinomial logit model where there are only two alternatives.

Let us first consider framing our mode choice model as a multinomial logit model which gives the proportion of travellers who would use mode m (Pm) as the following:

Pm = [Exp(Um)] / [Sum of Exp(Um) over all modes available]

Where:

Pm = the proportion of trips (or the probability of) travelling on mode m

Um = the utility of travel by mode m (calculated as shown above)

Exp(Um) = e, about 2.17, raised to the power of Um as described above.

Sum of Exp(Um) over all modes available is simply the exponentiated utility (obtained by entering the utility into your calculator and pressing the Exp key) added-up for all the modes - so if there are two modes you add the two exponentiated utilities together. The logit model is simply the ratio of the exponentiated utility to their sum. So if we have two modes, car and public transport which have exponentiated utilities of 3 and 2 respectively, then car will have 3/(3+2) or three fifths or 60% of them and public transport will have 2(3+2) or two fifths or 40% of them. In terms of probability this give car a probability of three fifths or 0.6 and public transport a probability of two fifths or 0.4. Note that the proportions always will add up to 1 or 100% and the probabilities will always add-up to 1.

If there are three modes the sum is the sum of all three modes and the above equation can be used to find the proportion using each of the three modes so for example if the choice also included train with an exponentiated utility of 5 then the proportion by car would be 3/(3+2+5) or three tenths or 0.3 or 30%; the proportion by bus would be 2/(3+2+5) or two tenths or 0.2 or 20% and the proportion by train would be 5/(3+2+5) or five tenths or 0.5 or 50%.

It is worth pointing out that by introducing the third mode, the ratio of car to bus choice remains the same at 3:2 (ie for two modes it is 60:40 and for three modes is 30:20 both coming down to the same ratio: 3:2). This property means that if another mode is introduced it extracts travel from both modes so as to preserve their relative mode shares. At first sight, this seems to be intuitively correct because by introducing train we have done nothing to change car or bus and have not changed the relative attractiveness of car with respect to bus (for a further discussion see the Independence from Irrelevant Alternatives or IIA property).

### Disaggregate (or Individual) Mode Choice

If we are considering an individual trip, then to forecast their mode choice we identify the choice set (ie the set of alternative mode they have available to them). We find the cost, in-vehicle time, walk time, wait time, number of interchanges (and any other attribute of choice we have in our utility function) by each mode of transport in their choice set. We use the weights (more about the weights later) to calculate the utility which we exponentiate. We sum the exponentiated utility over all the modes in the choice set and the probability of choosing each mode is given by the ratio of their exponentiated utility to the sum of all exponentiated utilities. We can use the probability to identify the best mode and allocate that individual trip to the mode with the highest probability.

If we have a set of individual trips we might consider forecasting the probability of each trip in the set. If we are interested in the overall mode shares we might add-up the number of trips allocated to each mode. However a more accurate estimate of the overall mode shares might be to sum the probabilities for each individual trip. If our individual trips had an expansion factor to expand our sample to represent all travel, the mode shares of all travel is best obtained by multiplying each expansion factor by the forecast probability and summed over the sample. This is called disaggregate mode choice modelling.

### Matrix Mode Choice

If we have a trip matrix then we can split it into trip matrices for each mode of transport by considering each movement (or matrix cell) from one origin to one destination in turn. The calculation is the same as above: calculate the utility, exponentiate it, sum over all modes and calculate the proportion by each mode, apply these proportions to the trip matrix to give the number of trips for each mode and write these out to the matrix for each mode. This calculation is then repeated for each origin to destination zone pair in the trip matrix to give the trip matrix for each mode.

For matrix mode choice the attributes (cost, in-vehicle time, walk time, wait time and number of interchanges) need to be provided for each origin to destination zone pair for each alternative mode. This is usually done in the form of a 'skim' matrix. The values of ivt, £, walk, wait, interchange can be skimmed from the networks for car and public transport. The coefficients b0 to b5 can be taken from a stated preference survey with b0 being the public transport mode specific constant relative to car (ie b0 for car is zero).

### The Composite Utility or Logsum

The sum of the exponentiated utility for each mode of transport has a special significance - it is a measure of the closeness of the origin and the destination of the trip.(The closeness is the inverse of separation between the origin and destination.) It is usual to take the natural logarithm of it when it is called the logsum or composite utility.

It has some important properties in particular that if any mode of transport is improved then it will be bigger. It also retains the mode proportions so if a mode of transport is improved then it will attract mode trips and these trips will be allocated by the mode choice model to the improved mode with the trips on the (unchanged) other modes of transport remaining the same. These are important properties and the logsum or composite utility is usually passed on to the distribution model to form the measure of separation between each pair of zones.

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