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Operations Research is an interdisciplinary field that uses scientific methods to make optimal decisions. It began developing in the 1930s and 1940s to help with logistics problems in World War II. Some common application areas of operations research include forecasting, production scheduling, inventory control, capital budgeting, and transportation. The document provides examples of transportation problems and assignment problems, which are types of optimization problems addressed by operations research techniques. It also discusses solution methods for transportation problems like the Northwest Corner method and MODI method.
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Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
help.mbaassignments@gmail.com
or
call us at : 08263069601
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The document discusses different types of graphs including Eulerian graphs, Hamiltonian graphs, planar graphs, and matchings. It defines Eulerian graphs as graphs containing an Eulerian cycle, and Hamiltonian graphs as graphs containing a Hamiltonian cycle that visits every vertex. It describes planar graphs as graphs that can be drawn on a plane without intersecting edges. The document provides theorems for determining when a graph is Eulerian based on vertex degrees and decomposing the edge set into cycles. It also defines matchings as independent line sets and provides Hall's Marriage Theorem conditions for a matching saturating vertices in a bipartite graph.
Graph Theory
This document discusses concepts in graph theory including subgraphs, walks, paths, cycles, connectivity, and k-connectivity. It defines key terms such as:
- Subgraphs, spanning subgraphs, trivial subgraphs, and induced subgraphs.
- Walks, paths, and cycles as sequences of edges in a graph.
- Connected graphs and their components as well as articulation vertices and bridges.
- Distance in connected graphs and k-connectivity as relating to the minimum number of vertices or edges that must be removed to disconnect a graph.
Or graphical method, simplex method
The document describes the linear programming model and the simplex method for solving linear programming problems.
The linear programming model involves defining decision variables, an objective function to maximize, and constraints on the variables. The simplex method is then used to iteratively find the optimal solution. It involves setting up an initial tableau, identifying entering and leaving variables, performing row operations to create new tableaus, and checking for optimality until reached.
As an example, the document formulates a linear programming model to maximize profits for a company producing two products subject to resource constraints. It then applies the simplex method through multiple iterations to arrive at the optimal solution of 270 units of product 1 and 75 units of product 2 for maximum profits
Operations Research
The document summarizes transportation problems and assignment problems. It discusses how transportation problems seek to minimize shipping costs by determining optimal routes from sources to destinations. Assignment problems similarly aim to minimize costs by optimally assigning workers to jobs. The document outlines the linear programming formulations and network representations of these problems. It also describes specialized algorithms like the Hungarian method for solving assignment problems and the MODI method for obtaining reduced costs in transportation problems.
Game theory
Two-person zero-sum games involve two players where the total gains and losses sum to zero. Such games can be represented using a payoff matrix which shows the payoffs for each combination of strategies. The value of the game is the expected payoff when both players use optimal strategies. If a saddle point exists in the matrix, where the maximin and minimax values are equal, it represents the optimal strategies and value of the game.
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- 1. Operations Research BY Mrs.K.Eswari M.Sc.,M.Phil, ASSISTANT PROFESSOR, DEPARTMENT OF MATHEMATICS(SELF), V.V.VANNIAPERUMAL COLLEGE FOR WOMEN, VIRUDHUNAGAR.
- 2. Introduction • Operations Research is an Art and Science • It had its early roots in World War II and is flourishing in business and industry with the aid of computer • Primary application areas of Operations Research include forecasting, production scheduling, inventory control, capital budgeting, and transportation. 2
- 3. What is Operations Research? 3 Operations The activities carried out in an organization. Research The process of observation and testing characterized by the scientific method.Situation, problem statement, model construction,validation,experimentation,candidate solutions. Operations Research is a quantitative approach to decision making based on the scientific method of problem solving.
- 4. History of OR • OR is a relatively new discipline. • 70 years ago, it would have been possible to study mathematics, physics or engineering at university but it would not have been possible to study OR. • It was really only in the late 1930's that operations research began in a systematic way. 4
- 5. Transportation Problem (TP) Distributing any commodity from any group of supply centers, called sources, to any group of receiving centers, called destina tions, in such a way to minimize the cost(shipping cost). If the total supply is equal to the total demand, then the TP is said to be balanced. Otherwise it is said to be unbalanced.
- 6. Example :
- 7. There are three methods to find an initial basic solution for a transportation problem. North West Corner method Least Cost method Vogel’s Approximation method To find an optimum solution for a transportation problem, MODI method is used.
- 8. The Assignment Problem (AP) - a special case of TP with number of sources is equal to the number of destinations i.e m = n & si=dj for all i and j. The Hungarian Algorithm => method used to solve the assignment problem
- 9. Example:
- 14. THANK YOU