Term Paper on "Consumption Problem Introduction to the Amtrak Trains"

Term Paper 8 pages (2491 words) Sources: 1+

[EXCERPT] . . . .

Consumption Problem

Introduction to the Amtrak Trains Fuel Consumption Problem

Formulation of the Problem

Objective of this project is to use the mathematical solution to solve the problem facing Amtrak Train. One of the problems facing Amtrak Trains is the blocking problem making the company to face constant increase in fuel and the issue makes the company to face challenges in allocation of scarce resources to cut costs and increase the profitability. The paper uses the integer programming to solve the blocking problem facing the company. Typically, blocking determines the train schedule, which also determines the major resource costs such as car costs, locomotive costs, crew costs and yard operating costs. Solving the blocking problem facing Amtrak Train to near optimality is very critical to the efficiency of Amtrak Train operations. The integer programming below is used to solve the blocking problem facing Amtrak Train

Formulation of Integer Programming

Minimize "k" K ?(i, j)?Acij + xkij + ?i-N ?(i, j) ?O (i) hiyij" subject to:

"?(i, j) ?O (i) xkij ?(i, j) ?I (i) xkij = { vk if I = o (k), -vk }"

= "{0 if I ?o (k) or d (k) }" for all k ? K

={ if I = d (k)}

k? K. xkij ? uijyij for all (i, j) ? a

(i, j) ?O (i) yij ? bi for all I ? N

k? K ? (i, j) ?I (i) xkij ? di for all I ? N

"yij = 0 or 1 and xkij" = 0 or vk

The railroad blocking model is based on the following constraints and objective functions.

Constraints

Maximum number of blocks that could be built at a node is limited.

Maximum volume of shipments that could pass through a node is limited.

Objective Function:

Shipments distances traveled

Shipments Intermediate handlings

US railroad blocking problem is as follows:

Routing problem and multi-commodity flow network design.

3,000 nodes

50,000 commodities

Over a million 0-1 network design variables (yij)

Variables that consist of Hundreds of billions of integer flow ( xkij )

Substantial amount of costs involved

Cost of flow: $1,000 - $2,000 million

Cost of handling: $500 - $1,000 million

Network Analysis and Linear Programming Algorithm

The paper also uses network analysis and linear programming algorithm to solve Amtrak Train problem, the paper presents the variables and constraints as follows:

Decision Variables

The decision variables are as follows:

Train origins, routes and destinations,

Train times and train days operations

Train block-to-train assignment during the day of the week

Trip plans for all train cars

Locomotive assignment

Crew assignment

Other constraints are as follows:

Yard capacity constraints

Line capacity constraints

Train capacity constraints

Business rules

Fig 1: Train Schedule Design Problem

Blocks Trains

Shipments Block-to-Train

Shipment Block

Trip Plan

Crew Balance Crew

Locomotive Balance Locomotive

The railcar, crew, and locomotive are the resources to maintain three time-space networks.

Weekly size problem:

Number of railcars= 100,000 -- 200,000

Number of locomotives= 2,000 -- 4,000

Number of crew districts= 300-400

Number of crews= 4,000-6,000

The paper also uses the integer programming to achieve optimization approach to provide the excellent approach to track the problem associated with Amtrak Train. Using integer programming model, Amtrak Train will be able to reduce the cost of fuel used to run its traditional train. By declining the costs of fuel expenses yearly, the company will be able to generate profitability. Integer programming is a mathematical technique that is concerned with the allocation of scarce resources to the best advantages of organizations. Linear programming is a procedure that assists an organization to optimize the value by declining the costs and maximize the profits. Thus, the linear programming will be used to allocate Amtrak Train resources to the best advantages of the company. Allocation problems are concerned with the utilization of scarce resources to the best advantages of organizations. Within contemporary business environment, major preoccupation of management is resources allocation decision in order to cut costs to enhance organization profitability.

One of the major problems facing Amtrak is inability to allocate its track to achieve the highest optimal advantages to decline the cost of fuel. The company is also facing the management problem because the company management is facing the daunting task to address the optimization problems, which make the company to face challenges in achieving costs reduction due to the constant increase in the fuel costs. Amtrak Rail Company is also facing the problem of rescheduling. The real time problem of train schedules has caused constant increase in overall company expenses which is higher than the total revenue making the company to run at a loss annually. Mathematical model and optimization techniques are effective to assist Amtrak to achieve costs reduction as well as enhancing service improvements.

Mathematical Model

The paper develops the integer programming based on the:

Microscopic model

The paper defines equation governing microscopic model as follows:

G = (V; E).

G standards for trains

R stands for the set of all given routes in G

The full microscopic equation is as follows:

an undirected infrastructure graph is denoted as G = (V; E),

a set of directed train's routes R, is { e1; e2…. enr } with ei ?, E,

a set of train types C, a mapping ? from the routes R. To train types C, positive running time on der on edges e ? E. For all routes r ? R. measured in ?

orientation edges induced by traversing routes in one or both directions, stop possibilities for some nodes vi ? V induced by traversing routes.

From the mathematical point-of-view, managing a network regulation poses a significant hard problem for management. Thus, it is not possible to rely on computer improvement to solve such problem. The use of operation research is the most relevant technique to solve this problem. The first step to be used in solving the problem is modeling. The model is good in assisting in train reschedule problem which is built on the SNCF, and is used to make decision support to make capacity study for railway. (Semet, Schoenauer ).

Variables

The project uses the mixed integer programming model that contains both numerical and binary variables to discuss the railway network operations that include the rescheduling decisions:

First the paper considers the times of arrival and departure of each train using the second as schedule time unit and the variables are put in numerical values. D will represent the departure time of the train and a represents the arrival time.

Second, the paper considers regulation decision which is represented by pure 0-1 values. The paper considers three types of actions that include: track choice, re-ordering, and extra stop:

track choice variable expresses whether a given train uses the track or not, re-ordering variable expresses whether the train passes before another node of the network revealing a crossing example.

extra stop variable expresses whether a given train stops in the new schedule while the train should not be in the original one. The technique is specifically used to allow fast trains to pass over the slower trains and stopped at sidings or loops.

Constraints

The paper provides constraints that represent the traffic management process. The following stipulated constraints are many which, obviously not exhaustive, however the paper represents the major ones necessary to the construction of a new schedule:

The following constraints are associated with each node (n), train (c), of the network.

The following constraints are associated with each train (c) at each node (n) of the network.

1. Original schedule:

A train cannot depart from the station earlier than what is previously defined in the original schedule:

D (c, n) ?Do (c, n) (1)

Due to operating and commercial purpose for example maintenance, the stopping times must be bounded:

"Min_stop ? D (c, n) - a (c, n) ?, Max_stop" (2)

2. Headways:

To prevent conflicts, it is compulsory to space the trains. By considering each type of potential conflict between each pair of trains, the paper imposes a specific separation time between departures and arrivals of the two trains.

"Min_spacing ?a (c1,n) - a (c2,n) and Min_spacing ? D (c1,n) - D (c2,n)" (3)

(Gely, Dessagne and Lerin 2 ).

3. Running times:

Based on the rolling stock and infrastructure characteristics, there are maximum speeds allotted to each train on each track. Since the paper is not allowing trains to slow less than a minimal speed, it is critical to consider a minimal and a maximal running time that is needed to reach one node from another:

"Min_time ?a (c, n2) - D (c, n1) ?Max_time" (4)

(Gely, Dessagne and Lerin 2 ).

Apart from constraints stipulated above, other specific constraints that must be treated include: shuttles, connections between two trains, & #8230;

Finally, due to decision variables, there is a need to refine most of the constraints stated above. For example, each spacing constraint must be taken into account which is the order between the choice between many tracks and the trains.

Objective function

The… READ MORE

How to Reference "Consumption Problem Introduction to the Amtrak Trains" Term Paper in a Bibliography

Related Term Papers:

Amtrak Trains System Fuel Consumption Problem Term Paper

Amtrak Trains System Fuel

Consumption

Problem of Amtrak Trains System Fuel Consumption

Amtrak operates and manages for-profit intercity rail passenger transportation in the United States. The company manages the network… read more

Term Paper 5 pages (1598 words) Sources: 1+ Topic: Business / Corporations / E-commerce

---

Consumption Problem Selection of a Planning Horizon Term Paper

Consumption Problem

Selection of a planning horizon: time period

Selection of a geographical focus: location

Motivation

Reasons others are interested in the project

Formulation of the Problem

Discussion of the… read more

Term Paper 5 pages (1579 words) Sources: 1+ Topic: Transportation / Mass Transit

---

Company Problem Introduction Have Been Working Term Paper

Company Problem

Introduction have been working for a Public Relations Firm. And it would be an understatement to say that the arrival of the Internet and its additional mechanisms might… read more

Term Paper 4 pages (1152 words) Sources: 0 Topic: Journalism / Media / PR / News

---

Trains for a Legally Valid Contract Term Paper

Trains

For a legally valid contract to exist, there must be consideration (i.e., a good or service of value must be exchanged for money or something else of value), there… read more

Term Paper 1 pages (448 words) Sources: 1+ Topic: Law / Legal / Jurisprudence

---

Problem Solving Process Research Proposal

Problem Solving Process - Why a Standard Work Schedule Does Not Work for Every Federal Employee in the National Capitol Commuting Area

The modern day employers are faced with rapidly… read more

Research Proposal 7 pages (2136 words) Sources: 8 Style: APA Topic: Career / Labor / Human Resources

---

View 100+ more related papers  >>