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In this problem, a train tunnel is required to be constructed underneath in a straight having a length of 7.5 Km (7500 m). There are three possible designs provided in this problem and calculations have to be done for the three design for the purpose of analysis. Below are the solutions for each of the 3 designs.

Design 1

Design 2:

Design 3:

A. The required cost to construct each tunnel with considering that the cost of 1 m3 of concrete is £120.

During the designing process of the tunnel, different design parameters must be taken into consideration which include: The geological aspects of the installation location, existing preliminary design, defining various tunnel configurations and constructions, estimating the construction cost (Peurifoy et al, 1975).

From the above analysis of the three designs, it is evident that design 2 attracted the highest construction cost because more volume of concrete than design 1 and 3 hence it will be eliminated. Despite design 1 requiring less volume of concrete and less construction cost than design 3, we will go for design 3 because this design has two tunnels which will reduce noise and warning out of tyre. In UK, the highest train is approximately 300mm and 150mm thick which makes design 2 a perfect choice. The design also can allow the passage of passengers through the side walk and the tunnel exit. This makes it easier for passenger evacuation in case of any emergency.

Reference

Norén, A. and Winér, J., 2003. Modelling Crowd Evacuation from Road and Train Tunnels-Data and design for faster evacuations. LUTVDG/TVBB--5127--SE.

Peurifoy, R.L., 1975. Estimating construction costs (No. Book).

Problem 2:

A. The table below shows timetable for customers at London station.

Table 1: Timetable for London station.

The second table below shows the customers at Birmingham station.

Table 2: Timetable for Birmingham station

The figures below show the timetables for London and Birmingham stations.

B. The two timetables are for a single day with 5 trains in both stations. The decisions that led to the development of the timetables are:

- Work in London station begins at 7:01 after train conditions are checked and tickets checked.

-The working time for the second train is taken as the departure time of the first train. Taking 20 minutes to complete all the tasks before the train leaves, the first train leaves at 7:01 and the next leaves at 7:21. 20 minutes is considered in this case because we are able to operate 3 trains in an hour as per the British government requirements.

- The arrival time for each train equals to the departure time plus 49 minutes.

- When a train arrives in the second station, it requires 6 minutes to empty it, 12 minutes to clean and prepare and 19 minutes to check passengers’ tickets adding up to 37 minutes

- Using Birmingham starting working time as 7:00 and 7:01 for London, there is a difference of 1 minute between departure and arrival time between the two stations.

- An extra train is reserved at each station to ensure that there is no delay in case of breaking down of the operating trains.

Reference

Aust, H., Oerder, M., Seide, F. and Steinbiss, V., 1995. The Philips automatic train timetable information system. Speech Communication, 17(3-4), pp.249-262.

Pena Alcaraz, M., Webster, M. and Ramos, A., 2011. An approximate dynamic programming approach for designing train timetables.

Problem 3:

A. The best length of train that can offer higher profit can be evaluated as follows;

The best train length will be; the passenger carriage to be all 9 first class passengers, but since the train has to have 9 carriages involving standard class passenger carriages and the appropriate selection will be 3 first class passenger and 6 standard class passengers.

Having the first-class seat profit as £8.20 and the standard class seat profit as £5, then a sample calculations of the total profit can be determined as follows;

The figure below shows the calculated profit for the proposed train length.

The assumptions made for the proposed length of train using the excel sheet are as follows;

Profits from the two engine carriages is constant which is equal to (147.6 + 240 = 387.6 pounds).

Number of seats of first class are 47.

Number of seats of standard class are 76.

Gross profit = Profit from first class + profit from standard class + profit from two engine carriages

The table below shows the calculated values for total profit

number of First class

number of standard class

number of seat of first class

number of seats of standard class

profit from first class

profit from standard class

profit from the two engine carriages

total profit

0

9

0

684

0

3420

387.6

3807.6

1

8

47

608

385.4

3040

387.6

3813

2

7

94

532

770.8

2660

387.6

3818.4

3

6

141

456

1156.2

2280

387.6

3823.8

4

5

188

380

1541.6

1900

387.6

3829.2

5

4

235

304

1927

1520

387.6

3834.6

6

3

282

228

2312.4

1140

387.6

3840

7

2

329

152

2697.8

760

387.6

3845.4

8

1

376

76

3083.2

380

387.6

3850.8

9

0

423

0

3468.6

0

387.6

3856.2

The data in the table above was then illustrated using a bar graph as shown below. It was found that the more profit could be obtained if the number of first class seats is 9 which means that there are no standard class carriages.

B. Since the journey is approximately 49 minutes, most people prefer standard class carriage for short trips as shown in the figure below. Therefore I would recommend 3-first class and 6-standard class so that all the seats are booked (London travel Watch 2010).

Reference

Fioole, P.J., Kroon, L., Maróti, G. and Schrijver, A., 2006. A rolling stock circulation model for combining and splitting of passenger trains. European Journal of Operational Research, 174(2), pp.1281-1297.

Schrijver, A., 1993. Minimum circulation of railway stock. Cwi Quarterly, 6(3), pp.205-217.

Problem 4:

This problem associated with the safety and health consideration in train in term of the size of the train wheel tyre. For this issue, it’s required to find at which point the tyre will wear down and become unsafe.

The carried-out calculations by the train operator includes the following data;

- The tyre size is 844 mm.

- It scrapped when its size become 838 mm.

- It looses 1 mm every 3,120,000 km.

To find the graphical relation between the distance covered and the wheel size, an excel sheet was used. After calculating the data in excel, the calculated results are shown in the table below.

Equation: D = 3120000*106 (844 - New size)

loss

new size

Distance Km

0

844

0

1

843

3120000

2

842

6240000

3

841

9360000

4

840

12480000

5

839

15600000

6

838

18720000

The calculated results are represented graphically as shown in the graph below showing the relation between distance travelled against the wheel size. From the graph, it shows that as the distance covered increases, the size of the tyre reduces. The tyre became a scrape after covering 18720000 Km.

From the linear equation of line:

M = = -

Y = - X + 844

With the proposed distance from London to Birmingham known to be 225 km, the number of trips that every train can travel before tyre scraping are calculated as shown below: The tyre scrapped after covering a distance of 18720000 Km, therefore the total number of journeys are calculated as:

s

From the calculation of trips, the train can travel 83,150 journeys before the wheel becomes a scrape. I recommend that the train wheel is changed after 83,100 journeys so that passengers’ lives are not put at risk.

Reference

Kane, M.E., Shockley, J.F. and Hickenlooper, H.T., Quantum Engineering Inc, 2004. Method and system for compensating for wheel wear on a train. U.S. Patent 6,701,228.

September 25, 2023

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