9.6. Multinomial Logit Model

The binary model can easily be extended to multiple modes. The equation for such a model can be written as:

P $i1$ = \frac{e^{-\beta c_1}}{\Sigma e^{\beta c_m}} (9.3)

The multinomial logit model is a generalization of the binary logit model, allowing for more than two modes of transportation. In this model, we calculate the probability of choosing a certain mode by comparing its utility to that of all available modes using a specific equation. This equation incorporates the costs associated with each mode and uses the exponential function to relate these costs to the probability of choosing each mode.

Imagine you're considering dinner options at a restaurant—let’s say pizza, sushi, or burgers. You think about how much each meal will cost (the cost in our equation) and how you feel about each option. The multinomial logit model works similarly, calculating the probability that you'll choose pizza over sushi or burgers based on these costs (and personal preferences), not just in terms of two options, but three or more!

Example Let the number of trips from i to j is 5000, and three modes are available which has the following characteristics:

tv twalk tt F φ ij ij ij ij ij coefficient 0.03 0.04 0.06 0.1 0.1 car 20 - - 18 4 bus 30 5 3 6 - train 12 10 2 4 -

In this example, we have 5000 trips distributed among three modes: car, bus, and train. The table provides the travel times, walking times, waiting times, fares, and coefficients for these modes. We start by calculating the cost for each mode using the formula developed earlier in the context of the model. After determining the cost for each mode, we convert these costs into probabilities of selecting each mode using the multinomial logit formula.

Think of planning a family trip to a nearby city. Your family can drive (the car), take a bus, or take a train. You gather information about the driving time, bus schedule, train schedule, and ticket prices. By calculating the total 'cost' (in terms of time, money, and convenience) for each option, you can decide how likely your family is to choose each mode of transportation. More loads of trips mean more consideration for which service meets your family’s needs the best—a common scenario for families planning travel!

Cost of travel by car (refer to the equation9.1) = c =0.03 20+18 0.1+4 0.1 = 2.8

Cost of travel by bus (refer to the equation9.1) = c =0.03 30+0.04 5+0.06 3+0.1 6 = 1.88

Cost of travel by train (refer to the equation9.1)= c =0.03 12+0.04 10+0.06 2+0.1 4 = 1.28

Each mode's costs are calculated by summing the product of coefficients with their respective times and fares. The calculated costs are: $2.80 for car,$1.88 for bus, and $1.28 for train. After we have these costs, we use them in the multinomial logit equation to find the probabilities of choosing each mode based on the costs.

Continuing with the family trip example, suppose you calculate that driving will cost you $30 in gas and parking, the bus ticket is$15, and train tickets total $12. Just like you would compare which option fits your budget while also considering travel time, similarly, the mathematical calculations help determine the likelihood of choosing each transport method based on their overall costs.

Probability of choosing mode car (refer to the equation9.3) pcar = e^{-2.8}/(e^{-2.8}+e^{-1.88}+e^{-1.28})

Probability of choosing mode bus (refer to the equation9.3) pbus = e^{-1.88}/(e^{-2.8}+e^{-1.88}+e^{-1.28})

Probability of choosing mode train (refer to the equation9.3) = ptrain = e^{-1.28}/(e^{-2.8}+e^{-1.88}+e^{-1.28})

With the costs calculated, we plug them into the multinomial logit equation to find out the probabilities of each transport mode being chosen. The results yield different probabilities based on the relative costs calculated previously, enabling us to see which mode is preferred when all options are considered together.

Think of this step as finalizing your dinner plans after comparing the prices of all three meal options. If pizzas are expensive, sushi is moderately priced, and burgers are on a sale with a discount, you’ll likely rank your choices—favoring the burger over others. The same logical reasoning is applied through calculations to reach the most probable choice for travelers.

Tcar = 5000 * 0.1237 = 618.5 Tbus = 5000 * 0.3105 = 1552.5 Ttrain = 5000 * 0.5657 = 2828.5 We can put all this in the form of a table as shown below:

tv twalk tt F φ C eC p T ij ij ij ij ij ij ij coefficient 0.03 0.04 0.06 0.1 0.1 - - - - car 20 - - 18 4 2.8 0.06 0.1237 618.5 bus 30 5 3 6 - 1.88 0.15 0.3105 1552.5 train 12 10 2 4 - 1.28 0.28 0.5657 2828.5

After obtaining the probabilities, we multiply each mode's probability by the total number of trips (5000) to assign how many trips will use each mode. Our calculations yield approximately 619 trips by car, 1553 by bus, and 2829 by train. This illustrates how the probabilities directly determine the distribution of trips across the available modes of transport.

Returning to our family trips, once you decide on how many of you might prefer each meal based on the price sensitivity and preferences, you can forecast how many pizzas you will order versus sushi or burgers. Using the probabilities calculated in the transport model helps forecast the real-world behavior of travelers among different options.

Fare collected from the mode bus = Tbus * F = 1552.5 * 6 = 9315 Fare collected from mode train = Ttrain * F = 2828.5 * 4 = 11314

Now, we calculate the total fare collected from each mode based on the number of trips assigned to each mode multiplied by the respective fare. For the bus, the total fare collected is calculated to be $9315, while for the train, it amounts to$11314. This step emphasizes the effectiveness of the multinomial logit model in not only predicting usage but also estimating potential revenue from each mode.

If your family orders meals, tracking how much you’ll spend helps set a budget beforehand. If 20 pizzas cost your family $200, then hypothetical collections from each meal option (where probability meets choice) help plan meals and potential spending later effectively, just like public transport systems plan their finances based on expected riders!

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