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Transportation Systems and
How would your typical day be changed without availability of your principal mode of transportation? Consider both personal transportation as well as goods and services that you rely on.
A typical day in my life would be significantly different without the airplane. Although I do not use this mode daily, goods and services that I do purchase are transported via this mode. Other modes, such as trucking, trains, and the automobile, could serve as replacements to the airplane; however, the airplane significantly lowers the transit time for shipping goods. For example, I mail a letter to California. Typical transit time for this letter using the airplane is three days. By using another mode other than the airplane, the transit time for the same letter would probably exceed seven days. As for my personal transportation, long distance travel is accomplished by using the airplane. For example, I take a vacation to Europe. If I travel using a cruise ship, it would take me in excess of seven days to reach Europe. However, if I fly, I can arrive in Europe within nine hours. Having the ability to transport people and goods quickly allows the international trade market to prosper, which in turn provides me with goods in a timely and efficient manner.
What are the most central problems in your state concerning one of the following: (a) air transportation, (b) railroads, (c) water transportation, (d) highways, or (e) public transportation. (To answer this question, obtain a copy of the governor’s plan for transportation in your state or contact a key official in the transportation department.)
- A problem in Virginia concerning air transportation is the high cost associated with short haul flights from airports such as Richmond and Norfolk to connection hubs for major airlines. Another problem is that our air transportation system is aging while the demand continues to increase; our air transportation system is approaching capacity and requiring substantial capital investment to provide modern terminals, increase the number of gates and available parking.
- Virginia is experiencing a new dilemma with its railroads. For the first time in nearly 30 years, freight railroads are expanding their operations and growing to serve their market segment. To continue to compete with other
railroads in neighboring states, Virginia must investigate the possibility of providing rail clearances to facilitate double stacking of containers into the Port of Hampton Roads. Another problem associated with the increase in freight rail transportation is the conflict encountered with passenger trains running on freight company-owned tracks. As the demand for passenger rail service increases and the freight market share increases, more conflicts will likely occur and the passenger services may require parallel or additional track kilometers to meet demand.
- The most central problem concerning water transportation in Virginia is the increased build-up of silt in our channels. In order for Virginia to remain competitive, it will have to continue to dredge our navigable waterways. Another problem is the increased volume of pleasure crafts and cargo vessels. The increased interaction between these types of vessels will likely result in more serious accidents. To mitigate this, more boater safety classes should be provided to ensure all boat operators are responsible on the water.
- Virginia’s highways are experiencing increased volumes and delays while the overall infrastructure is continuing to age. The volume of trucks on Virginia’s highways are significantly increasing annually. As a result, Virginia is experiencing an accelerated deterioration of our highways as well as more serious accidents.
- The major problem concerning public transportation is that modern systems such as the ones in Atlanta and San Francisco are not present in Virginia. Only Northern Virginia and the suburbs of Washington, D.C. have rapid rail transit in form of the Metro system that is now facing major renovations. Virginia does not have a sophisticated rural public transportation system that provides all individuals with a means of transportation.
A bridge has been constructed between the mainland and an island. The total cost (excluding tolls) to travel across the bridge is expressed as C = 50 + 0.5V, where V is the number of veh/hr and C is the cost/vehicle in cents. The demand for travel across the bridge is V = 2500 −10C.
- Determine the volume of traffic across the bridge.
- If a toll of 25 cents is added, what is the volume across the bridge? What volume would be expected with a 50-cent increase?
- A tollbooth is to be added, thus reducing the travel time to cross
the bridge. The new cost function is C = 50 + 0.2V. Determine the volume of traffic that would cross the bridge.
- Determine the toll to yield the highest revenue for demand and supply function in part (a), and the associated demand and revenue.
- Substitute the total cost function into the demand function and solve for V.
V = 2500 − 10(50 + 0.5V)
V = 2500 − 500 − 5V
6V = 2000
V = 333.33 vehicles/ hour
Therefore, the number of vehicles wanting to cross this bridge is 334 vehicles/hour.
- Add 25 cents to the original cost function.
C = 50 + 0.5V + 25
C = 75 + 0.5V
Substitute the above cost function into the demand function and solve for V.
V = 2500 − 10(75 + 0.5V)
V = 2500 − 750 − 5V
6V = 1750
V = 291.667
Therefore, the new volume crossing the bridge will now be 292 vehicles / hour with a 25 cents toll.
- Add 50 cents to the original cost function.
C = 50 + 0.5V + 50
C = 100 + 0.5V
Substitute the above cost function into the demand function and solve for V.
V = 2500 − 10(100 + 0.5V)
V = 2500 − 1000 − 5V
6V = 1500 V = 250
With no toll, the volume would be 334 vehicles/hour; with 50 cents toll, the volume would be 250 vehicles/hour. That means that an increase of toll by 50 cents reduces traffic by 334 – 250 = 84 vehicles/hour.
- Substitute the new cost function into the demand function and solve for V.
V = 2500 −10 (50 + 0.2V)
V = 2500 − 500 − 2V
3V = 2000
V = 666.67 vehicles/ hour
Therefore, the new number of vehicles wanting to cross this bridge is 667 vehicle/hour.
- Assume toll rate at T. The new cost function will be C = 50 + 0.5V + T. Since the revenue generated is the toll rate, T, time the volume, V, first solve for V with the new cost function.
V = 2500 − 10(50 + 0.5V + T)
V = 2500 − 500 − 5V − 10T
V = (2000 − 10T) / 6
Since the revenue generated is R = T × V, substitute the above expression into the revenue formula and differentiate with respect to T.
R = T × ((2000 − 10T) / 6) R = (2000T − 10T2) / 6 dR/dT (2000T − 10T2) / 6 = 0
(2000 − 20T) / 6 = 0
Therefore, the toll which would yield the maximum revenue is T = 100, or T = ($) 1.00.
R = T × V
R = (2000T − 10T2) / 6
R = (2000(100) − 10(100)2 / 6
R = 16,666.67
Therefore, a toll of ($) 1.00 will yield a revenue of 166.67 per hour.
A toll bridge carries 6,000 veh/day. The current toll is ($) 3.50/vehicle. Studies have shown that for each increase in toll of 50 cents, the traffic volume will decrease by 500 veh/day. It is desired to increase the toll to a point where revenue will be maximized.
- Write the expression for travel demand on the bridge, related to toll increase and current volume.
- Determine toll charge to maximize revenues.
- Determine traffic in veh/day after toll increase.
- Determine total revenue increase with new toll.
- V = 6000 − 500(x / 50)
- Since the original toll was 350 cents per vehicle, the new toll charge will be
T = 350 + x
The revenue (R) is generated by the equation R = V × T. Substitute the above expressions into the revenue function and differentiate with respect to x, setting the derivative equal to zero.
R = (6000 − 500(x / 50)) × (350 + x)
R = (6000 − 10x) × (350 + x) R = 2100000 + 6000x − 3500x − 10x2 dR/dx (2100000 + 2500x − 10x2) = 0
2500 − 20x = 0 x = 125
Therefore, an increase in toll of 125 cents will maximize revenues.
- Now, substitute the new toll, x, into the demand function developed in part a.
V = 6000 − 500(x / 50)
V = 6000 − 500(125/50)
V = 6000 − 1250
V = 4750 vehicles per day
The new demand for the bridge will be 4,750 vehicles per day.
- R = V × T
R = 4750 × (350 + 125)
R = 4750 × 475
R = $2,256,250
The total revenue to be generated with the new toll will be ($) 2,256,250 per day.