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This paper begins by looking at the relationships between mathematics, Operational Research and management. It then examines a number of difficulties in the application of mathematics to management-difficulties of determining an optimal solution, and of implementation. Some innovative mathematical methods for management are discussed in this paper. Finally, an attempt is made to distinguish areas of application of mathematics in management by reference to the scale of resources at stake, the complexity of the problem, and the adequacy of the predictive data.
Optimization techniques, statistical methods and some mathematical models.
Mathematics is an essential subject and knowledge to enhances a person's reasoning, problem-solving skills and in general, ability to think logically. Hence it enables an easy grasp of most subjects, whether science and technology, medicine, the economy or business and finance. Mathematics is one of the popular managerial science tools used by profit and non-profit organizations. As the global environment becomes fiercely competitive, Mathematics has gained significance in applications like world-class Manufacturing systems, Lean production, Benchmarking, Just-in-time inventory techniques Mathematical tools and techniques such as the Theory of Chaos are used for mapping and forecasting market trends. Statistics and probability, which are very important branches of mathematics, are used in everyday business and economics. Another important tool in this field is Management science, it is concerned with developing and applying models and concepts that may prove useful in helping to elucidate management issues and solve managerial problems, as well as designing and developing new and better models of organisational excellence. The application of these models within the corporate sector became known as management science.
Financial mathematics and business mathematics are considered two important branches of mathematics in today's world and these are examples of the direct application of mathematics to business and economics. Examples of applied maths such as probability theory and management science, queuing theory, time-series analysis, linear programming all are vital for business.
Table shows mathematical area with their application
Computational fluid dynamics Aircraft and automobile design
Differential equations Aerodynamics, porous media, finance
Discrete mathematics Communication and information security
Formal systems and logic Computer security, verification
Geometry Computer-aided engineering and design
Nonlinear control Operation of mechanical and electrical systems
Numerical analysis Essentially all applications
Optimization Asset allocation, shape and system design
Parallel algorithms Weather modelling and prediction, crash simulation
Statistics Design of experiments, analysis of large data sets
Stochastic processes Signal analysis
Some Important innovative tools of mathematics in management
The term Operations Research (OR) describes the discipline that is focused on the application of information technology for informed decision-making. In other words, OR represents the study of optimal resource allocation. The goal of OR is to provide rational bases for decision making by seeking to understand and structure complex situations, and to utilize this understanding to predict system behavior and improve system performance. Much of the actual work is conducted by using analytical and numerical techniques to develop and manipulate mathematical models of organizational systems that are composed of people, machines, and procedures. This article introduces some of the methods and application that are affiliated with OR, and elaborates on some of the benefits that may be gained by incorporating OR into theactual business framework
The growth of global markets and the resulting increase in competition have highlighted the need for Operation Research. In order to be competitive, businesses must meet the challenges present in a global market by offering products and services that offer good value to their customers. Good value is a combination of low cost, high quality, rapid availability and real time information on these.
In order to enhance the role of operational research and speed up the process and methodologies of different stakeholders, they should work closely and complement each others effort. In this process, the academicians should take the lead in the design, development and demonstration of sustainable operational research models. Industry should support this initiative and accelerate the transmission of this methodology
TECHNIQUES USED IN OPERATION RESEARCH
Decision Analysis: Decision analysis refers to a set of quantitative methods for analyzing decisions that use expected utility as the criterion for identifying the preferred alternative.
Decision analysis provides tools for quantitatively analyzing decisions with uncertainty and/or multiple conflicting objectives, and these tools can be especially useful when there is limited directly relevant data so that expert judgment plays a significant role in the decision making process. It provides a systematic quantitative approach to making better decisions, rather than a description of how unaided decisions are made.
A general decision making process can be divided into the following 8 steps:
1. Define the problem
2. Determine the requirements
3. Establish Goals
4. Identify alternatives
5. Define criteria
6. Select a decision making tool
7. Evaluate alternatives against criteria
8. Validate solutions against problem statement
The above steps are illustrated through a Flow Diagram as given below:
Linear programming arose as a mathematical model developed during Second World War to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. In Operation Research optimization means to find out the maximum profit and minimum loss in any deal which we can done in Quantitative Techniques, in this we can narrowing our choices to the very best when there are virtually immeasurable feasible options. This is a constrained optimization technique, which optimize some criterion within some constraints. In Linear programming the objective function (profit, loss or return on investment) and constraints are linear.
Standard form of describing a linear programming problem consists of the following three parts:
A linear function to be maximized
Problem constraints of the following form
Non-negative variables ,
Statistics is the science of good decision making in the face of uncertainty and is used in many disciplines such as financial analysis, econometrics, auditing, production and operations including services improvement, and marketing research.
Importance of Statistics in Different Fields:-
Statistics plays a vital role in every fields of human activity. Statistics has important role in determining the existing position of per capita income, unemployment, population growth rate, housing, schooling medical facilities etcâ€¦in a country. Now statistics holds a central position in almost every field like Industry, Commerce, Trade, Economics, Mathematics, Astronomy etcâ€¦, so application of statistics is very wide.
Important fields in which statistics is commonly applied. Â
(1) Business:Â Statistics play an important role in business. A successful businessman must be very quick and accurate in decision making. He knows that what his customers wants, he should therefore, know what to produce and sell and in what quantities. Statistics helps businessman to plan production according to the taste of the costumers, the quality of the products can also be checked more efficiently by using statistical methods. So all the activities of the businessman based on statistical information. He can make correct decision about the location of business, marketing of the products, financial resources etcâ€¦
(2) In Economics: Statistics play an important role in economics. Economics largely depends upon statistics. National income accounts are multipurpose indicators for the economists and administrators. Statistical methods are used for preparation of these accounts. In economics research statistical methods are used for collecting and analysis the data and testing hypothesis. The relationship between supply and demands is studies by statistical methods, the imports and exports, the inflation rate, the per capita income are the problems which require good knowledge of statistics.
(4) In Banking:Â Statistics play an important role in banking. The banks make use of statistics for a number of purposes. The banks work on the principle that all the people who deposit their money with the banks do not withdraw it at the same time. The bank earns profits out of these deposits by lending to others on interest.
(5) In State Management (Administration):Statistics is essential for a country. Different policies of the government are based on statistics. Statistical data are now widely used in taking all administrative decisions. Suppose if the government wants to revise the pay scales of employees in view of an increase in the living cost, statistical methods will be used to determine the rise in the cost of living. Preparation of federal and provincial government budgets mainly depends upon statistics because it helps in estimating the expected expenditures and revenue from different sources. So statistics are the eyes of administration of the state. Â Â Â
(6) In Accounting and Auditing: Accounting is impossible without exactness. But for decision making purpose, so much precision is not essential the decision may be taken on the basis of approximation, know as statistics. The correction of the values of current asserts is made on the basis of the purchasing power of money or the current value of it. In auditing sampling techniques are commonly used. An auditor determines the sample size of the book to be audited on the basis of error. Â Â Â
Some real time examples for this case can be customers waiting in the queue in banks or to buy groceries in departmental stores. The contribution of the computer here is to maintain the queue according to the arrival time of the event, in this case the customers, and process each event one after the other according to their arrival time. Simulation represents the full extent of the models covering all perceivable systems which incorporate characteristics of a queue. We identify the unit demanding service, whether it is human or otherwise, as customer. The unit providing service is known as the server. This terminology of customers and servers is used in a generic sense regardless of the nature of the physical context. Some examples are given below.
In communication systems, voice or data traffic queue up for lines for transmission. A simple example is the telephone exchange.
In a manufacturing system with several work stations, units completing work in one station wait for access to the next.
Vehicles requiring service wait for their turn in a garage.
Patients arrive and wait at a doctor's clinic for treatment.
Queuing System is used in situations where the queue is formed (for example customers waiting for service, aircrafts waiting for landing, jobs waiting for processing in the computer system, etc). The objective here is minimizing the cost of waiting without increasing the cost of servicing. The term "classical" queuing theory refer to descriptive models of queuing systems, usually based on Markovian assumptions, in which the goal is to derive an explicit expression for the queue-length or waiting-time distribution (or its transform), usually in steady state