Consumption Problem Introduction to the

¶ … 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 objective function for the implementation of the model is to minimize the accumulated delays between the original timetable (before happening of the incident) and the new timetable (solution of the calculation) over all the nodes and the trains: "f = ? delay (c, n), where delay (c, n) = a (c, n) - Ao (c, n)" (5) Size of the Problems The size of the problems provides the amount of constraints.

To provide the number of constraints and variables, the paper provides the study of Amtrak rail system between Delaware, and New Jersey. The entire datasets held with the 99 trains are 42 tracks and 43 nodes. The datasets gives a problem of 220,000 variables and 380,000 constraints. With available constraints and variables, the paper uses all possibilities to solve the problems using the algorithms. To arrive at the solution to the problem, the study uses the new traffic management concepts and other potential solving methods.

3-Part: Structure of the Model; The paper uses the Ilog Cplex tool to solve the problem. The linear programming as follows: The algorithms check all constraints and simplify the problem as much as possible using the mathematical point-of-view. Using the system, the study attempts to find the first solution and refine it in order to find the best solution as being revealed in Fig 1.

Fig 1: Linear Programming Algorithms Based on the linear programming algorithms, the study uses the pre-solve phase to reduce problem from the 220,000 variables and 380,000 constraints to approximately 64.000 variables and 300.000 constraints. The Ilog Cyplex uses the branch and cut algorithm to derive the optimal solution. The calculation time is improved to meet real time processing and large scale dataset.

The algorithms rescheduling module allows this study to perform traffic management with real world data and the experiment is carried out using SISYFE train simulator where the software stimulates the train running and the details reveals how the system would evolves in real life situation. The operation is carried out using the algorithms which take into account the train's dynamic performances, the tracks layout, and distances to be covered.

The rescheduling module is a large traffic control stimulator named LIPARI which contains three different modules: The paper first seeks to detect the abnormal situation and compare the original timetable produced by the train simulator with the real world life situation. When the study detects the incident, the study sends the data to the re-scheduling module. The solution provided is to minimize delays by providing new speed and a new routing, which assists in enhancing traffic management.

This study uses linear programming to arrive at the optimal solution to the problems. Using network flows & linear programming operations research techniques, the paper uses a two-stage decomposition process: Train schedule without time Train routes Block-train assignment Locomotive assignment Crew assignment Train schedule with time Train routes Block-train assignment Locomotive assignment Crew assignment Using the train route optimization, the paper determines the.