Toll Booth HW10 UIUC

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O F F I C I A L

TO: The Bridge Commission

FROM: Group 29

DATE: April 28, 2012

SUBJECT: Tollway Design

The purpose of this document is to present the Bridge Commission with a design of a toll

queuing system. The system was designed with the objective of minimizing the total cost of the

system while ensuring an efficient traffic flow-rate during peak and non peak hours of operation.

To assist us in our design, we used a queue system simulator which allowed us to input data

recorded for different hours of operation. The hours of operation were split into 6 different

shifts. These shift hours were:

Shift 1: 0200 - 0600

Shift 2: 0600 - 1000

Shift 3: 1000 - 1400

Shift 4: 1400 - 1800

Shift 5: 1800 - 2200

Shift 6: 2200 - 0200

The simulator then allowed us to input how many toll booths would be operational during each

particular shift and simulated the flow of traffic. Upon conclusion of the simulation, a report wasgenerated which aided in the optimizing process. That report is attached to the end of this

document.

Since optimization was used to generate a solution to this queuing problem, we followed a 5 step

algorithm. The first of these steps was to clearly define a problem, which has already been stated

above.

Step 2 was to define our decision variables. They are labeled with subscripts corresponding to

shift number.

J = Annual operating cost per booth per year

K = Manning cost per booth per day

Gn = Number of operating booths


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The variable J, from the simulation, was found to be a constant value of $17,618.94 per year per

booth. For generalization purposes, this variable, although constant in value, will remain a

variable. K, however, is found based on the following criteria:

1. Each worker works a shift no longer than 8 and no less than 4 hours per shift.

2. Each worker earns a $10 per hour wage.

3. There will be no more than one worker on duty per operational booth.

The value K, defined by the following function, is equal to the number of workers on shift, times

the length of the shift, times the $10 per hour wage. K =

The rest of the decision variables are:

Bn = Total annual number of cars per shift

L = Total annual cost of queue system

Dn = Maximum Length of line in queue

Fn = Maximum Wait Time in queue

We quickly saw that the amount charged per car for the toll varied directly with total cost of the

system. Therefore, in minimizing cost of the system, we also minimize the amount each citizenpays for the toll.

Using these variables, the next step was to state an objective function. That function is:

L = { [(365)(K)] + [(J)(Gi, max)] }

(10)(4)(Gi) i = 1, 2, 3, 4, 5, 6


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The final value for the above function is the value L, which is the total annual cost of the queue

system. The above function produces the lowest cost when the variable Gi, max is a minimum,

since K will vary directly with G. That G value, however, is the highest number of working

booths in a full 24 hour day.

To avoid a value of negative infinity, the next step of our algorithm was formulated. This step

sets constraints for the function.

1. Bn, Dn, Fn, Gn 0

2. Dn 22 cars

3. Fn 20 minutes

The first constraint ensures that no values will be negative values. The constraints on Dn and Fn

are values picked based on a 5-day simulation. After 5 days, the report generated showed amaximum line length of 18 cars and a maximum wait time of 14 minutes. We realize that we can

not take into account excess traffic for political/recreational events. To try to accommodate for

such events, we allotted extra line length and time in queue. These constraints are non-binding,

and will not effect the cost of the system in any way. The limits on the number of toll booths

allowed to be open per shift were found using traffic data and simulation. We used our

knowledge of the mean inter-arrival times combined shift data to find a basis of how many

booths should be operating per shift, then used the simulation to adjust these numbers to produce

the lowest cost while still providing sufficient traffic flow. The final solution for the number of

booths (Gn-values) is as follows:

Shift 1: 4 Booths

Shift 2: 11 Booths

Shift 3: 5 Booths

Shift 4: 10 Booths

Shift 5: 7 Booths

Shift 6: 2 Booths

After finding this optimal solution, we used these numbers in our objective function and came up

with an annual system operating cost of $211,427.32 which is the cost of both manning and

operation of 11 booths, as well as an extra $17,618.94 for the purchase of one spare booth. The

simulation found an annual cost of $193,808.38 due to the exclusion of a spare booth. This spare

booth will remain idle and unmanned unless a situation should arise that one of the operating

booths must be shut down. Annual system manning costs $569,936.20 and combining this with

the price of operation we arrive at an annual total of $781,363.51. Our simulation report

calculated a total system cost of $763,744.58, again, a lower figure due to the exclusion of the

spare booth. The purchase of this booth raises the price of the toll per customer less than one


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cent, however due to decimal values the toll rose from $1.05 to $1.06. We found this to be an

insignificant increase, and went ahead with the purchase of the spare.

This operating and manning cost is not the only cost the toll will pay for. The Bridge

Commission has estimated a cost of $1,417,000 for annual cost of bridge maintenance as well asa $1,320,000 cost for interest and sold bonds. Altogether, the total cost of the bridge, and

tollway is $3,518,363.51.

In order to acquire the necessary cash inflow, a toll is collected from each vehicle passing. We

are aware that these people are not just dots on a graph, so in the design of this system, many

factors were considered that had no mathematical value, and therefore no decision variables. We

have a reasonable wait time at all hours, ensuring the good morale of most customers, and most

importantly, we are aware that the toll should be as low as possible.

To ensure this is the case, we simply divided the total cost, L, by the annual number of cars, B,which was a number generated by the simulator. This generated number used for our calculation

was based on a 5-day simulation and turned out to be 3,344,760 vehicles. Putting it all together,

we derived a minimum toll of $1.06 per customer.

Finally, following is a list of assumptions made for our calculations. Note that some assumptions

appearing earlier in this report are repeated here.

1. The total annual number of vehicles found in the 5 day simulation remained the same for

further simulations.2. The extra line length and queue time proposed is sufficient time to allow for small periods of

increased traffic flow.

3. Each toll operator works no longer than an 8 hour shift, as well as no less than 4. (Note: The

amount of money needed to pay 1 operator for an 8-hour shift or 2 workers for 4-hour each

shifts is the same.).

4. Toll operators receive no extra pay for holiday/overtime.

5. Toll operators receive no raises or incentives.

6. No more than one worker per operating booth is on duty at one time.

7. In the event of an emergency or maintenance, no more than 1 booth will be forfeited.

In conclusion, the members of Group 29 feel that the aforementioned solution is the most

efficient solution for the toll way problem. It allows for adequate traffic flow at every hour of

the day as well as room for emergencies and low idle time percentages. This means that the

Bridge commission will not endure excess costs for unessential toll booths. This solution has a

total of 31.5% idle time. This number is averaged from the very high idle times during off-peak

hours and 0% idle time during peak hours.


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Furthermore, the operation of more toll booths would directly effect the amount each citizen pays

for access to the bridge. Minimizing that number is as important as any other aspect of the

design. The price of the toll has to fund the building, operation and maintenance costs of both

the bridge and the tollway while still being low enough to prevent travelers from seeking

alternate routes. Price, however, is not the only customer satisfaction related factor considered.

Wait time was also a main factor. Running different numbers of toll booths during different

shifts changed the maximum wait time of our system. Wait times were longest during peak

traffic flow hours, and were limited to a maximum of 20 minutes, although the simulation report

generated a maximum waiting time of 14 minutes and an average time in line of 2.2 minutes with

an average line length of 2 vehicles.

Our final result called for the construction of 11 toll booths as well as 1 emergency booth to be

operated as specified for each of the 6 shifts per 24-hour period. Each vehicle will be charged atoll of $1.06, which will, due to the nature of the calculations, provide a slight excess of revenue

which will be dispersed throughout the Bridge Commission as seen fit.


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Simulation Data For Selected Solution

Run using data file:

C:\Users\Bilal\Desktop\Systems Engineering and Economics\Finished Assignments\Assignment10\FULLDAY.DAT

Simulated 5 run(s) with 24 hours per run.

Total Vehicles = 45,819 vehiclesTotal Vehicles Annually = 3,344,760 vehicles / yrAverage Line Length = 2 vehiclesMaximum Line Length = 18 vehiclesAverage Total Time in Line = 2.2 minutesMaximum Waiting Time in Line = 14 minutes

Average Number of Booths Open = 6.51 booth (s)Number of Booths Needed = 11 booth (s)Percent Idle Time in Open Booths = 31.5%Annual Cost of Booths = $193,808.38 / yrAnnual Labor Cost = $569,936.20 / yrTotal Annual Cost = $763,744.57 / yrCollection Cost per Vehicle = $ .23 / vehicle

Shift Time No. Open Booths02:00 406:00 1110:00 514:00 1018:00 722:00 2

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Toll Booth HW10 UIUC