The British/Europeans refer to "operational research", the Americans to Therefore, to give a formal definition of the term Operations Research is a difficult task. Introduction to. Operations Research. Deterministic Models. JURAJ STACHO. Department of Industrial Engineering and Operations Research. PDF Drive is your search engine for PDF files. As of today we have 78,, eBooks for you to download for free. No annoying ads, no download limits, enjoy .

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𝗣𝗗𝗙 | This is an introductory text for Operations Research with focus on methods used to solve Linear Programming Problems (LPP). Albright, Winston & Zappe, Data Analysis and Decision Making. Albright, VBA for Modelers: Developing Decision Support Systems with Microsoft Excel. help the students with a book on Operations research. OPERATIONS RESEARCH, with other chapters to students, with a hope that it will.

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About Welcome to EasyEngineering, One of the trusted educational blog. Get New Updates Email Alerts Enter your email address to subscribe to this blog and receive notifications of new posts by email. Search Your Files. There are some situations in business that require whole numbers rather than fractions.

Integer programming models are often classified as being either mixed integer programming models or pure integer programming models. If all the variables must take only integer values, it is pure integer programming. If only some specified variables have to be integers and others may be non-integers, we have a mixed integer programming problem.

Integer programming problems can be solved using Branch and Brand method or Gomory Algorithm Ekoko, Goal Programming This is a type of linear mathematical programming that can be used to analyze decision situations involving multiple goals, which sometimes are complementary or conflicting.

Linear programming model has only one objective function, as such, it is inappropriate for many decision making situations in which a manager is concerned with several potentially conflicting objectives. Goal programming provides a method for extending linear programming to accommodate models in which there is more than one objective.

Operations Research Perspectives

For example, there are times when management may want to maximize profits and increase wages paid to employees or to upgrade product quality and reduce product cost. The pervasiveness of multi-objective decisions make goal programming an important tool for managers. It requires management to set some estimated targets for each goal and assign priorities to them, that is, to rank them in order of importance.

With this information, goal programming tries to minimize the deviations from the targets that were set. It can be solved using the Modified Simplex Method Anderson et al, Decision Analysis Decision analysis is a systematic approach to decision making in situations where there are a number of alternative courses of action and some uncertainty as to the precise outcomes of the various possible options Agbadudu, One of the features of problems which exist in the real-world is the existence of uncertainty, this is, at the time the decision is made, the decision maker is uncertain as to which state of nature will occur in the future and thus has no control over them.

In dealing with such problems, it is essential to obtain a measure of the uncertainty. Most business decision situations can be categorized into two classes, decision-making situations where probabilities cannot be assigned to future occurrences and decision situations where probabilities can be assigned.

Within this context, several decision-making criteria are available. In the case where probabilities could not be assigned to the occurrence of outcomes, the maximax, maximin, minimax regret, equal likelihood and Hurwicz decision criteria can be used.

In the case where probabilities could be assigned to the states of nature of a decision situation, the expected value criteria and decision trees are used Agbadudu, In the business world, many decisions are made in competitive situations where the decision of a competitor affects the decision of a firm.

According to Camerer all situations in which at least one agent can only act to maximize his utility through anticipating either consciously or just implicitly in his behaviour the responses to his actions by one or more other agents is called a game. The term player is used to denote each firm, which takes part in games related to decision- making.

Each player in a game faces a choice among two or more possible strategies. A strategy is a predetermined programme of play that tells a firm what actions to take in response to every possible strategy its competitors might use.

The best strategy for each player is known as the optimal strategy. When each player in the game adopts a single strategy as an optimal strategy, then pure strategy game exists. On the otherhand, when the player adopts a mixture of strategies, then it is a mixed strategy game.

A pure strategy game can be solved according to the minimax decision criteria while a mixed strategy game can be solved using expected gain and loss method or linear programming Ekoko, Markov Analysis Markov analysis is a technique for analyzing the current behavior of some variables in an effort to predict the future behavior of that same variable. According to Taylor and Bernard a markov chain is a collection of random variables xt where the index t runs through 0, 1, … having the property that given the present, the future is conditionally independent of the past.

Markov chain is a sequence of events or experiments in which the probability of occurrence for an event depends on the immediately preceding event. It is also referred to as markov process. The controlling factor in a markov chain is the transition probability. It is a conditional probability for the system to go to a particular new state, given the current state of the system.

Thus, markov analysis is specifically applicable to systems that exhibit probabilistic movement from one state or condition of the system to another overtime. It has been successfully applied to a wide variety of decision situations such as examining and predicting the behavior of consumers in terms of their brand loyalty and their switching from one brand to another, assessing the behavior of prices, manpower planning, estimating bad debts or credit management and maintenance planning.

Queuing Model Queuing models are techniques for analyzing problems concerned with providing service to customers in a line.

All queuing situations involve the arrival of customers at a service facility where some time is spent waiting for and receiving the desired service. The customers could be persons or objects like unfinished items proceeding to the next stage of production Agbadudu, Given a model of a queuing system, some of the descriptions of the system that can be obtained from the model are as follows: The description above can be invaluable in business decision making.

Queuing can be used for solving problems such as the determination of the optimum number of servers, repairs and maintenance of equipment subject to breakdown and outsourcing decisions. A cost is usually associated with waiting because it involves consumption of time. Long queues may cause customer annoyance and perhaps a switch to another firm.

However, if there are too many service points, they may often be idle and also involve high cost to the organization. The objective of queuing models is to strike an optimal balance in minimizing waiting costs to both the customer and the service facility. Simulation Simulation is a quantitative technique for evaluating alternative courses of action through experimentation performed via a mathematical model in order to represent actual decision making under conditions of uncertainty Agbadudu, It involves the operation of a model or simulator, which is a representation of the system.

The model is amenable to manipulation, which would be impossible, too expensive or impractical to perform on the entity it portrays. The operation of the model can be studied and the properties concerning the behaviour of the actual system or its sub-system can be inferred. Also, in situations where certain assumptions will make the suggested optimal solution not to be accurate, simulation would be the appropriate method for tackling such problems.

It can be applied to decisions regarding manufacturing of alternative kinds of products, choosing location of a plant, selecting suitable processes of production or arranging for the sales of the finished product. In all these situations, management must examine and weigh alternatives. When alternatives are few in number and management has experience, decisions may be simple. However, when he has little experience, lacks information or has a problem not amenable to simple numerical analysis or trial and error methods, actual experimentation may be too expensive and time consuming, simulation procedures may then be extremely helpful.

Forecasting A forecast is a prediction of what will occur in the future Barry and Stair, Managers are constantly trying to predict the future regarding a number of factors in order to make decisions in the present that will assure the continued success of their firms. Often, a manager will use judgement, opinion or past experience to forecast what will occur in the future. However, a number of mathematical methods are also available to aid the manager in making such decisions.

Forecasting is the process of developing assumptions or premises about the future that managers can use in planning or decision-making. There are a variety of forecasting methods, the applicability of which are dependent on the time frame of the forecast, the existence of patterns in the forecast and the number of variables the forecast is related to.

Basically, these methods include moving averages, exponential smoothing, the decomposition of seasonal data and regression analysis Levin et al, Inventory Models The term inventory is used to denote the quantity of goods or materials kept on hand for future use in production or sales Kalavathy, By carrying large inventories, one can minimize the chances of shortage or make more profit in case of a price rise or enjoy economics of scale due to downloading in bulk.

However, large inventories require more carrying costs in terms of insurance charges, taxes, storage, obsolescence and deterioration. Inventory models help to strike a balance between having too many quantities of an item on hand and running out of stock. It deals with the following decisions: It helps in identifying the order quantity that can minimize the total cost of carrying inventory.

Operations Research Books

It is given as: In a situation where the firm produces the items, the model becomes: Some of these activities can be performed simultaneously while others are done one after the other. Network models enable the manager to consider the nature of the activities that are required for the completion of a project, to define the critical path and then to make decisions consistent with his resources and requirements.

It also enables management to plan ahead, to take stock of the situation at every stage of planning and may even alert him in time to obviate future sources of trouble. Thus management may be in a position to plan the best possible use of resources in a way that the milestone is reached within the appointed time and given costs. CPM is used for scheduling and controlling projects where the time needed to complete each of the scheduled activities is known.

It assumes that the duration of each activity is known with certainty but PERT assumes that each activity occurs with probability Hira, Hence, it is capable of dealing with in-built uncertainties. This is achieved by using three time estimates, optimistic, most probable or normal and pessimistic.

The average of these three time estimates is usually used for analysis. The main objectives of these techniques are to minimize total completion time, costs involved and idle resources. These techniques have been applied to many management and allocation problems such as planning, scheduling and evaluation of projects.

Dynamic Programming Dynamic programming is a technique which is useful when a decision maker is faced with a multistage decision. It is often concerned with the allocation of scarce resources. The states at a stage are often the different resource levels available at that stage.

Given each state, a number of decisions are possible, each of which results in a return. For each state, the best decision is determined as the one that results in the greatest return.

These states and decisions are then related to the next stage in the solution process with a transition function.

Schaum's Outline of Operations Research

This means that the solution approach of dynamic programming is to breakdown a problem into smaller sub problems called stages and then solves these stages sequentially. The outcome of a decision at one stage will affect the decision made at the next stage in the sequence. For example, when a manager specifies a production level for his assembly line at the start of each month, the choice of a production level for the nth month can be influenced by and in turn can influence production levels selected for prior and subsequent months.

The principle behind dynamic programming is known as the principle of optimality. Dynamic programming is a unique approach to problem solving. It uses other techniques within its overall solution approach. As such, it is applicable to a wide variety of problems. It has been successfully applied to such areas as equipment replacement, production scheduling, inventory management, and for planning advertising expenditures. As a result of its wide application, there is no single algorithm that can be used to solve all problems and so a separate algorithm is normally developed for each problem.

Non-linear Programming Non-linear programming is a mathematical technique for analyzing problems in which the objective function and constraints are not linear. The solution approach for problems with non-linear relationships requires the application of substitution method which is used for models that contain only equality constraints and calculus. A number of studies have examined the extent of use of these techniques in business organizations in several countries.

Turban examined Operations Research activities at the corporate level in business organizations. The results indicated how widely the techniques had been applied to current projects in the United States of America and that simulation and linear programming were the most widely used techniques. Cook and Russel reported that in , they did a study to examine the Operation Research techniques that are used by companies in the United States of America.

Ledbetter and Cox studied companies in Britain and got similar results that simulation and linear programming were the most frequently used techniques. Schweigman studied Tanzanian companies and Wright studied companies in Ghana and got similar results. Lane et al investigated the techniques that practitioners and educators have found to be most important and most useful. The study was based on questionnaires and they found that the techniques that are used in order of importance are statistical methods, linear programming, simulation, network models, decision analysis, waiting line models, inventory models and dynamic programming.

Morgan studied 12 companies and 3 practitioners. The results gave further support to linear programming, simulation and network www. In Nigeria, several researchers have recommended one Operations Research technique or the other to companies for efficiency and effective decision making.

Unyimadu did a study on how games theory can be applied to the functional management strategies of firms in the banking industry. He recommended that games theory should be adopted for monitoring competition in the banking industry in Nigeria. Alimhinaga applied markov chain on data from Guinness Nigeria Plc in order to determine the brand loyalty of consumers to its products.

The study revealed that there was a dwindling trend in the brand loyalty of consumers to Guinness. He therefore recommended that firms in the brewery sector should apply markov chain model in analyzing brand switching pattern and market shares of brands at regular intervals for better marketing decisions and policy formulation to obtain repeat download of their products. In addition, Ighomereho conducted a study to determine the extent to which Nigerian quoted companies actually use these techniques and it was found that managers in Nigerian business organizations use the techniques but the extent of use is small.

This finding is not surprising as Nigeria is a developing country that is still growing in terms of technology and computer application.

Development in computer technology has made it possible for Operations Research techniques to be easy to use. Computations in many Operations Research techniques are usually too large to carry out manually because they require large amount of data and many decision variables. Currently, there are softwares that make the computation to be very easy and also allow the decision maker to combine the results with executive judgement and experience to make better decisions.

The low level of use can also be attributed to the few experts in Operations Research that we have in Nigeria. With respect to the most frequently used techniques, it was found that managers in Nigeria use forecasting techniques, network models, inventory models, simulation, linear programming, goal programming, games theory, decision analysis, markov analysis, queuing models and transportation model.

The techniques that they do not use include assignment model, integer programming, dynamic programming and nonlinear programming. This finding is similar to that of Lane et al except for dynamic programming.

It however contradicted the findings of Schweigman and Wright that the most commonly used technique in developing countries is linear programming. This study revealed that the most commonly used technique is forecasting techniques. This may be as a result of increased uncertainty and complexity of the Nigerian business environment. Areas of Application of Operations Research in Business Organizations Operations Research as a problem solving and decision-making science can be applied to a variety of issues in business organizations.

Levin et al and Gupta and Hira classified the application of Operations Research in organizations based on the functions of management. Based on this foundation, this paper outlines the various areas of activities where Operations Research can be applied in Nigerian business organizations. Conclusion and Recommendations From the above areas of application, it can be concluded that the techniques of Operations Research are useful tools for decision making in business.

Although, the techniques are tools for decision making and not the complete decision making process, its role and usefulness in decision making cannot be overemphasized. It is no longer possible to depend solely on subjective decision making. Therefore, Operations Research techniques should be taken as an aid to decision making and a supplement to judgement, experience and intuition. It is therefore recommended that: Real life problems should be solved in the classroom so that students can begin to see it as a problem solving tool rather than a compulsory course that must be passed.

It is not abstract in nature because the issues it addresses are things we experience on a daily basis.

Recent Operations Research Perspectives Articles

So when teaching Operations Research, it should be demystified. This is very important because the extent of use cannot really increase without the use of computer. All the techniques discussed in the literature review can be solved using computer.

With the software, the manager need not bother to solve the mathematics involved. All that is required is to input the data and the computer will bring out the result. This could be achieved through regular workshops, training, seminars and conferences.

They should also market these programmes and encourage managers of business organizations to attend. They should be encouraged to be members of the institute.

References [1] Agbadudu, A.

Elementary Operations Research, Benin City: Operations Research, Mathematics and Social Sciences: Centre for Management Development. R, Sweeney, D. West Publishing Company.

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Quantitative Analysis for Management, New York: Prentice Hall, Inc. Princeton University Press. Introduction to Operations Research, New York: Introduction to Management Science, New Jersey: Prentice — Hall.

Managerial Economics, Benin City: March Publishers. United City Press. Management, Boston: Houghton Mifflin Company. Operations Research, New Delhi; S. Operational Research: Operations Research, New Delhi: Operations Research Techniques, Interfaces, Vol 23, Industrial Engineering, February: S Quantitative Approaches to Management, New York: McGraw- Hill, International.

Quantitative Techniques. Thompson Learning. Understanding Business,New York: McGraw-Hill [26] Saaty, T. Elements of Queuing Theory, New York: McGraw-Hill Book Company. Administrative Behavior: The Free Press. Operations Research: An Introduction. Pearson Education Inc. Introduction to Management Science. New Jersey: Prentice Hall. Thesis, University of Benin, Benin City.Chapter-9 deals with Waiting line model and its application with certain useful problems and their solutions.

Applications and Algorithms, Belmont California: Magee reviewed the phases through which it has developed. This involves choosing a course of action that is satisfactory or good enough under the circumstances.

Competitive games, rectangular game, saddle point, minimax maximin method of optimal strategies, value of the game.