Mycity Final Report: Paradise City

Eileen Lee

Brett Leibowitz

ORF 467

MyCity Final Report: Paradise City

Welcome to Paradise City, where residents are happy to live self-sufficiently in their rectangular shaped zones. The river that flows through the city separates a peaceful residential community from the bustling industrial atmosphere. With a retirement community, golf course, and plenty of five star restaurants and retail stores, the west side of the river hosts a myriad of opportunities. Zones east of the river house the centrally located airport, the university, and the amusement park among various other land uses.

The city is divided into 58 Traffic Assignment Zones (TAZ) including residential, work, school, and others to keep the population active and happy. Approximately 30% of the land is designated for water and open space, such that it is not inhabited by any citizens and offers natural beauty throughout the city. Residential areas account for about 20% of the city, while industrial, professional, and government offices that support a majority of employment is another 20% of the land. Schools contribute to about 5% of the city’s area and the remaining “other” zones primarily for recreation, restaurants, and shopping take up the final 25% of the city.

Demographics:

According to the U.S. Census Bureau, the average household size in 2010 was 2.59. With a population of 250,000 people in Paradise City, the total number of households is approximately 96,505.

The resident population is broken down by age distribution and occupation classification. The relationship between these breakdowns is very strong. All of the children under 5 years old are also not yet in school. The number of primary, secondary, and university students is equivalent to the 5-20 year old age category. The 21-66 year old workers include all the workers in the population. There is a minimal 4% unemployment rate in Paradise City, and 14% of the population is senior citizens, which establishes the 18% of non-workers and over 66 age distribution category.

With a majority of the population productively working to contribute to the city, employment breakdown is important to the citizens. A large majority of workers are located in heavy industrial regions, where production facilities and large plants dominate. Medium and light industrial areas consequently take up another significant portion of the labor force along with professional offices for doctors, lawyers, and businesses. The remaining occupations include government officials, teachers and professors, and finally workers at various attractions.

Students also take up a significant portion of the population. Primary students include grades K-5, while secondary students include grades 6-12. Upon graduation, a large number of students are accepted to the prestigious university complete with a convenient golf course and sports stadium nearby.

To maintain a balanced lifestyle, residents of Paradise City also enjoy going out to eat, shop, and play. One of the most popular attractions includes the amusement park located in the southeast corner of the city. Many citizens enjoy shopping at the various retailers and eating at the broad array of diverse restaurants. Being healthy and active is important to all members of the community, so recreational areas are very attractive. Paradise City sponsors various concerts in the park, benefit runs, and several events throughout the year to promote recreation.

Trip Generation:

Because residents must make trips to various zones in order to fulfill their daily activities such as going to work, school, or other locations, trip production and attraction vectors are used to determine the number of people going to each zone for a particular purpose. There are nine sets of production and attraction vectors for the various trip purposes: HB-work, HB-school, HB-Other, Work-Home, School-Home, Other-Home, Work-Other, School-Other, and Other-Other. The sum of the production vectors is equal to the sum of attraction vectors because it is assumed that no inhabitants disappear or leave the city.

HB-work trips are those that go from home directly to work. The production vector values are nonzero in residential areas only and attraction vector values are nonzero in all employment zones. Though the university also has residents who live in the zone, it is assumed that everyone that resides there is a student. Therefore, there is no production from the university to work. However, since professors, management and administrative workers are employed at the university, there is a non-negative attraction for work. About 55% of the residential zone populations are workers, and the production vector includes the number of workers in each zone. The attraction vector consequently is the number of jobs available at each zone in which the workers will distribute accordingly. There is also no structural unemployment, so the skillset of individuals in the city correspond to the jobs available. Because citizens of Paradise City are responsible workers, all 137,500 workers in the population go to work every day. Therefore, the total number of trips for HB-work is 137,500.

HB-school trips similarly include only those that go from home directly to school. About 20% of the population living in residential areas is a primary, secondary, or university student, and these students represent the production vector in each residential zone. The attraction vector is non-negative in the educational zones. Each educational zone has a primary and secondary school, such that all students regardless of grade are attracted to the same zone up to the zone’s capacity. The exception includes Zone 49, which is the university, where only university students attend. The total number of students is 52,500, and the total number of HB-school trips is 46,800. The discrepancy of 5,700 students is caused by the university students who live on campus and also attend school there. Because they continue to stay inside the same zone for home and school, this intrazonal trip is not counted in the number of trips. The attraction to the university includes the commuting students who live in various residential zones in the city and do not live on campus.

HB-other trips include all trips that begin at home and go to retail, dining, recreational, or other miscellaneous zones that are not school or work. The number of trips is determined using the following guideline:

1.5*(number of 21-66 non-workers + number of people over 66)

+ 0.3*(number of students) + 0.5*(number of 21-66 workers)

This arbitrary determination for the number of HB-other trips is used to help generate an approximate 4 trips per person. In this case, the university zone is included as a production vector because the college students living at the school presumably like to travel to other zones to eat out, go shopping, and take excursions. The attraction to an “other” zone is determined proportionally by area. For example, the large recreational areas attract a significant number of people, which is beneficial for this city that values health and wellbeing initiatives.

For Work-home and School-home trips, the number of trips produced is determined using a certain percentage of the population who make HB-Work and HB-School trips. It is assumed that 50% of people who go to work will return directly home, while the remaining 50% will make Work-other trips to run errands, grab a beer, etc. at another location before returning home. 60% of children who go to school go directly home in a School-home trip. This percentage is greater than that of Work-home because students typically have less access to drive to other locations immediately after school. The number of students who make School-other trips is therefore 40% because those that do not go directly home go to another zone.

The Other-home trips include the people who go from retail, restaurants, recreational, and other zones back home. The attraction vector is calculated by determining the remaining people who have not yet returned home. The production vector assumes that the number of trips generated for each “other” zone is proportional to the area of the zone, as in all cases that include “other” trips.

Finally, the production and attraction vectors for Other-other trips are the same because it is assumed that the influx of people remains the same. Other-other trips include trips that do not include home or work, such as getting a haircut and then going to the park to run a workout.

The summary trip production and attraction table below includes the number of trips from each residential zone for a particular trip purpose and the percentage of trips per trip purpose in each zone. Zone 15 is the retired community, so they do not make any trips to work or school. Zone 49 is the university, where there is a population of students who live on campus. Because their occupation is solely being a student, they do not make trips to work and do not make trips to school because they are intrazonal.

The table below summarizes the trip production and attraction for Other-other trips, which include a different set of zones than the home-based trips. The percentage represents the proportion of total Other-other trips in that particular zone. There are 21 total “other” zones, which include retail, restaurants, recreational areas, golf courses, sports stadium, airport, and amusement park.

Trip Distribution:

In order to generate trip arrays, we implemented a gravity model that assumes total trip production at the origin and total attractions at the destination is proportional to the trips produced at the origin and attracted to a destination.

According to the Gravity Model formula, the number of trips T between each origin zone i to destination zone j is equal to:

Where:

Tij = trips produced at I and attracted at j

Pi = total trip production at I

Aj = total trip attraction at j

Fij = a calibration term for interchange ij, (friction factor) or travel time factor (Fij =C/ tijn)

C= calibration factor for the friction factor

Kij = a socioeconomic adjustment factor for interchange ij

i = origin zone

j = destination zone

n = total number of zones

For our calculations, all Kij = 1. This means there is no socioeconomic adjustment factor for interchange ij.

Calculating the trip array is a multi-step process that requires many matrices to be utilized and calculated. The matrices necessary for each trip type are P, Ainput, D, F, I, Sum, P/Sum, [P/Sum]transpose, [P][A]transpose, [P/Sum][A]transpose, and [A]transpose. Also mandatory to step through the iterations is C = Aoutput, A desired, C previous, and A new.

Three matrices are the same for all trip types except for a slight change for Other-other trips:

D: This is the distance matrix. It calculates the difference between each zone by using a grid system, similar to Manhattan distance. We take the difference in the distance between the x-coordinates of each zone's centroid and add it to the difference between their y-coordinates. For intra-zonal distances, the square root of the land area was used for most trip arrays and this was multiplied by ten for the Other-other trip array generation.

F: This is the friction factor matrix, which is the aversion or reluctance factor that each person experiences of traveling to a particular zone. For all trip types, this was kept constant at 1/D2.

I: This is the identity matrix.

Using excel, we calculated the Ainput necessary to reach our Adesired as an output. To do this we use our trip Production vector (P) and our Adesired as our initial Ainput. Looping through the following process, we update our Ainput after each iteration. First, we calculate the Sum vector by matrix multiplying F by Ainput. Following this, we calculate all the values for Sum, P/Sum, [P/Sum]transpose, [P][A]transpose, [P/Sum][A] transpose, and [A]transpose, using MMULT and TRANSPOSE in excel. Then we find each cell of a potential trip array Tij by multiplying ([P/Sum][A]transpose)ij by F ij. Summing up the columns of this trip array will give the [A]output vector. If this output is within 1% error of the desired A for each zone then the process is over and this is the trip array we use. However, if it is not, then the [A]input must be updated to [A]new. This [A]new will be equal to the Adesired multiplied by Ainput and divided by Aoutput. This process repeats until we have calculated a trip array that produces an Aoutput that converges close to our Adesired within a 1% error.

Trip Demand for the city varies for each trip purposes. Therefore, the PersonTrips and PersonTripMiles were determined for each trip purpose. The PersonTrips is the number of trips taken by a community member as predicted by the Gravity Model. Therefore, this is equivalent to the Trip Array matrix calculated, with the assumption that each trip is taken by a single person. Below is a table that includes the PersonTrips for each mile range.

According to the distribution of trips taken for each purpose, it appears that a large majority (68%) of the trips taken overall are less than five miles long. This is expected because people typically prefer destinations that are closer to their current location. Because the number of zones for HB-School is particularly small, the trip distances deviates slightly from the others. The concentration of residential areas forces some students to travel to schools that are farther away from their home. However, those that are willing to drive the further distance are rewarded by receiving a stellar education to prepare for entrance into the prestigious university located in the northeast part of Paradise City.

The PersonTripMiles depicts the average trip length traveled per person. This can be calculated by dividing the total trip length by the total number of trips taken for each trip purpose. To find the total trip length, we took the sum of all trips in each zone for a particular trip type. For the non-redundant six trip purposes (HB-work, HB-school, HB-other, Work-other, School-other, Other-other), the total trip length is equivalent to the matrix multiplication of the distance array transposed by the number of trips. For school-home and work-home, the total trip length is proportional to the HB-school and HB-work.

Finally for Other-home, the total trip length could not be computed directly using a linear combination of other total trip lengths because of the possibility of taking an Other-other trip. Therefore, a matrix was created by multiplying each value from the Other-home trip array by the corresponding value in the D array to obtain a matrix of total tripDistance from zone to zone. Then, this matrix was summed to get the total Other-home miles traveled. Though this method could have been used for every trip type, it was not necessary because the other methods were easier to implement.

Below is a summary of the total trip distance, total number of trips, and average trip length for each trip purpose. The average trip length is equivalent for HB-work/Work-Home and HB-school/School-Home because the distance between trips to these zones remains the same. The HB-other/Other-home do not produce the same average trip length because people can make an Other-other trip before returning home. It is evident that the average trip length for Other-other is the shortest at 3.58 miles, and the HB-work trips are the longest at 4.55 miles. The total average trip length per Paradise City member is 4.29 miles.

Looking at the distribution of the PersonTrips for each trip purpose, the cumulative distribution was determined. The following graph displays the cdf for each of the six trip purposes.

All cumulative trip distributions, regardless of trip purpose, follow a similar trend. Initially, the percentage of trips taken increases quickly because of people’s preference to take trips that are shorter in length. As the trip length increases, the number of people taking those trips declines, resulting in the decreasing slope of the graph. Overall, the analysis of trip generation and distribution of Paradise City represents a simplified approach to solving transportation models faced in the real world.

*Further details on our analysis can be found in the Appendix and corresponding Excel spreadsheet.

Appendix

Trip Production and Attraction Vectors

Trip Arrays:

HB-Work

HB-School

HB-Other

Work-Home

Work-Other

School-Home

School-Other

Other-Home