This formulation is used to derive an integer linear programming model that is relaxed in a Lagrangian way. In this paper, we propose a graph theoretic formulation for the problem using a directed multigraph in which nodes correspond to departures/arrivals at a certain station at a given time instant. Trains can overtake each other only in correspondence of an intermediate station, and a minimum time interval between two consecutive departures and arrivals of trains in each station is specified. Each train connects two given stations along the track (possibly different from the two major stations) and may have to stop for a minimum time in some of the intermediate stations.
![operational research modelling operational research modelling](https://slideplayer.com/slide/5036762/16/images/2/Learning+Objectives+Students+will+be+able+to%3A.jpg)
In particular, we concentrate on the problem of a single, one-way track linking two major stations, with a number of intermediate stations in between.
![operational research modelling operational research modelling](https://slidetodoc.com/presentation_image_h/94edebfa479d852378943d330028632e/image-22.jpg)
![operational research modelling operational research modelling](https://s2.studylib.net/store/data/010120652_1-8b8bd08469241b076e24c1907fe36bba.png)
Evidently the main objective of Operations research is to provide a scientific basis to the decision-makers for solving the problems involving the interaction of various components of organization, by employing a team of scientists from different disciplines, all working together for finding a solution which is the best in the interest of the organization as a whole. The application of Operations research methods helps in making decisions in such complicated situations. Operations research does mathematical scoring of consequences of a decision with the aim of optimizing the use of time, efforts and resources and avoiding blunders. Doing preparative calculations is the purpose of Operations research. Preparative calculations may avoid long and costly research.
#Operational research modelling trial
The search of a decision may also be done by trial and error but such a search may be cumbersome and costly.
![operational research modelling operational research modelling](http://acronymsandslang.com/acronym_image/304/4f85155fa3edc0af55867bee70728cba.jpg)
Certainly in such situations also decision may be arrived at intuitively from experience and common sense, yet they are more judicious if backed up by mathematical reasoning. Examples of such decision are finding the appropriate product mix when there are large numbers of products with different profit contributions and production requirement or planning public transportation network in a town having its own layout of factories, apartments, blocks etc. But here the decisions we are concerned with are rather complex and heavily loaded with responsibility. In simple situations decisions are taken simply by common sense, sound judgment and expertise without using any mathematics. Decision-making is one of the main activity of a manager or executive. We make decisions in our every day life without even noticing them. The ultimate goal of all such decisions is to either maximize the desired benefit or minimize the effort required. In various practical problems we may have to take many technical or managerial decisions at several stages. Optimization is the act of obtaining the best result under any given circumstance. Operations Research approach of problem solving