Problems and exercises in Operations Research All the solutions, however, are by the author .. 11 Easy modelling problems: solutions. Request PDF on ResearchGate | Operations research problems. Statements and solutions | The objective of this book is to provide a valuable. weekly value of sales of A & C. Market research indicates no more maximize This is called a.: A problem of optimizing (maximizing or minimizing) a linear .. Basic variables in a basic solution are not necessarily nonzero: Definition: If one or.
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Operations Research - Problems and Solutions - JK Sharma - Ebook download as PDF File .pdf) or read book online. Operations Research - Problems and. help the students with a book on Operations research. . problems and their solutions have consequences in several dimensions, such as. Therefore, to give a formal definition of the term Operations Research is a O.R. is an interdisciplinary discipline which provided solutions to problems of military.
Who jointly use the OR tools to obtain a optimal solution of the problem. The tries to analyze the cause and effect relationship between various parameters of the problem and evaluates the outcome of various alternative strategies. System Approach. The main aim of the system approach is to trace for each proposal all significant and indirect effects on all sub-system on a system and to evaluate each action in terms of effects for the system as a whole. Use of Computers. The models of OR need lot of computation and therefore, the use of computers becomes necessary.
With the use of computers it is possible to handle complex problems requiring large amount of calculations. The objective of the operations research models is to attempt and to locate best or optimal solution under the specified conditions.
For the above purpose, it is necessary that a measure of effectiveness has to be defined which must be based on the goals of the organization. These measures can be used to compare the alternative courses of action taken during the analysis.
Formulate the Problem 9 OR analyst first defines the organization's problem. Defining the problem includes specifying the organization's objectives and the parts of the organization or system that must be studied before the problem can be solved.
Step 2. Observe the System Next, the analyst collects data to estimate the values of parameters that affect the organization's problem. These estimates are used to develop in Step 3 and evaluate in Step 4 a mathematical model of the organization's problem. Step 3. Formulate a Mathematical Model of the Problem The analyst, then, develops a mathematical model in other words an idealized representation of the problem. In this class, we describe many mathematical techniques that can be used to model systems.
Step 4. Verify the Model and Use the Model for Prediction The analyst now tries to determine if the mathematical model developed in Step 3 is an accurate representation of reality. To determine how well the model fits reality, one determines how valid the model is for the current situation.
Step 5. Select a Suitable Alternative Given a model and a set of alternatives, the analyst chooses the alternative if there is one that best meets the organization's objectives.
Sometimes the set of alternatives is subject to certain restrictions and constraints. In many situations, the best alternative may be impossible or too costly to determine. Step 6. Present the Results and Conclusions of the Study In this step, the analyst presents the model and the recommendations from Step 5 to the decision making individual or group.
This may result from incorrect definition of the problem on hand or from failure to involve decision maker s from the start of the project. In this case, the analyst should return to Step 1, 2, or 3. Step 7. Implement and Evaluate Recommendation If the decision maker s has accepted the study, the analyst aids in implementing the recommendations. The summary of steps have been depicted in Fig1. R as a concept 11 "OR is the representation of real-world systems by mathematical models together with the use of quantitative methods algorithms for solving such models, with a view to optimizing.
It can be used to find the best solution to any problem be it simple or complex. It is useful in every field of human activities, where optimization of resources is required in the best way.
Thus, it attempts to resolve the conflicts of interest among the components of organization in a way that is best for the organization as a whole. The main fields where OR is extensively used are given below, however, this list is not exhaustive but only illustrative. Industrial Establishment and Private Sector Units OR can be effectively used in plant location and setting finance planning, product and process planning, facility planning and construction, production planning and control, purchasing, maintenance management and personnel management etc.
Business Management and Competition OR can help in taking business decisions under risk and uncertainty, capital investment and returns, business strategy formation, optimum advertisement outlay, optimum sales force and their distribution, market survey and analysis and market research techniques etc. Transportation Transportation models of OR can be applied to real life problems to forecast public transport requirements, optimum routing, forecasting of income and expenses, project management for railways, railway network distribution, etc.
In the same way it can be useful in the field of communication.
Home Management and Budgeting OR can be effectively used for control of expenses to maximize savings, time management, work study methods for all related works. Linear Programming LP is a mathematical technique. It is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. It takes all kinds of factors into consideration to determine the best combination of a purchasing or manufacturing process, to either maximize profit, minimize cost or some other goal.
Therefore, LP is a very important part of any business. The transportation problem is a special type of linear programming problem, where the objective is to minimize the cost of distributing a product from a number of sources to a number of destinations.
In a few words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem. Assignment deals with the question how to assign n object to m other object in an injective fashion in the best possible way.
The queuing problem is identified by the presence of a group of customers who arrive randomly to receive some service. Queuing theory deals with problems which involve queuing or waiting.
This theory helps in calculating the expected number of people in the queue, expected waiting time in the queue, expected idle time for the server, etc. It is used for decision making under conflicting situations where there are one or more opponents i. In the game theory, we consider two or more persons with different objectives, each of whose actions influence the outcomes of the game.
Thus, this model is concerned with the acquisition, storage, handling of inventories so as to ensure the availability of material whenever needed and minimize wastage and losses. It is a powerful tool to tackle multiple and incompatible goals of an enterprise.
Goal programming models are very similar to linear programming models but whereas linear programs have one objective goal programs can have several objectives. It is a technique that involves setting up a model of real situation and then performing experiments. Simulation is used where it is very risky, cumbersome, or time consuming to conduct real experiment to know more about a situation. These methods may be used when either the objective function or some of the constraints are not linear in nature.
Non-Linearity may be introduced by factors such as discount on price of purchase of large quantities. These methods may be used when one or more of the variables can take only integral values. The Integer Programming problem IP is that of deciding whether there exists an integer solution to a given set of m rational inequalities on n variables.
Dynamic programming is a methodology useful for solving problems that involve taking decisions over several stages in a sequence. It is related to Waiting Line Theory. It is applicable when the facilities are fixed, but the order of servicing may be controlled. The scheduling of service or sequencing of jobs is done to minimize the relevant costs.
These models are concerned with the problem of replacement of machines, individuals, capital assets, etc. Network scheduling is a technique used for planning, scheduling and monitoring large projects. Such large projects are very common in the field of construction, maintenance, computer system installation, research and development design, etc. It is an analytical process transferred from the electrical communications field to operations research.
It seeks to evaluate the effectiveness of information flow within a given system and helps in improving the communication flow.
In most of the cases a large number of iterations are required to reach optimal solution. Manually this task becomes time consuming and single mistake at any point can generate erroneous results. With the development of computers and P. The computational time requirements are also less and no paper work is required.
The reliability of solutions is also high. For the large size problems, where simulation was to be used, it was not possible to carry it out manually, which is now possible with the use of computers. To handle linear programming problem with multiple variables use to be cumbersome and time taking; can be done at wink of moment without any manual efforts.
It enables organizations to consider more alternative actions and scenarios, and determine the best allocation of resources and the best plans for accomplishing goals. It is a software package developed by Chang and Sullivan for problem solving algorithms for OR as well as modules of statistics, non —linearity programming and financial analysis.
It is an interactive linear, quadratic, and integer programming system useful to a wide range of users. CPLEX solves linear and convex quadratic programs by simplex or interior-point methods, and linear and convex quadratic integer programs by a branch-and-bound procedure GUROBI: Gurobi solves linear programs by simplex and interior-point methods, and linear mixed- integer programs by a branch-and-bound procedure.
Support for convex quadratic programs, both continuous and mixed-integer, is planned for version 4. The application reality is a little more complex and a little less dramatic. The model of OR as an activity conducted for executives by internal OR groups with a good deal of choice as to which issue to tackle. A new model of highly specific investigations and developments conducted by external specialist firms and management consultancies is alive and flourishing.
This is not an unequivocally happy outcome. There was, and still is, great merit in the sort of grounded, detailed investigation that is the hallmark of good traditional OR. The technically efficient, goal oriented, socially aware OR consultants of tomorrow may get things done and satisfy their clients but they may miss the potential implications or the valuable insight as they rush to the next challenge. However, the limitations are related to the problem of model building and the time and money factors involved in application rather than its practical utility.
The models of OR strive to find out optimal solutions taking into account all the factors. These factors may be huge and state them in quantity and establish relationships among the factors require ample calculations which can be effectively solved by computers. OR provides solution only when all elements related to a problem can be quantified.
All relevant variables do not lend themselves to quantification. Factors which cannot be quantified find no place in OR study. Similarly, a manager fails to understand the complex working of OR. Thus there is a gap between the User and Analyst. OR models are a costly proposition as they require time and money for scientific application. The computational time increases depending upon the size of the problem and accuracy of results desired.
Implementation of any decision is a delicate task. It must take into account the complexities of human relations and behavior.
Sometimes, resistance is offered due to psychological factors which may not have any bearing on the problem as well as its solution. The process of testing and improving a model to increase its validity is commonly referred as Model validation. The OR group doing this review should preferably include at least one individual who did not participate in the formulation of model to reveal mistakes. A systematic approach to test the model is to use Retrospective test.
This test uses historical data to reconstruct the past and then determine the model and the resulting solution. Comparing the effectiveness of this hypothetical performance with what actually happened, indicates whether the model tends to yield a significant improvement over current practice. Preparing to apply the model After the completion of testing phase, the next step is to install a well-documented system for applying the model.
This system will include the model, solution procedure and operating procedures for implementation. The system usually is computer-based. Databases and Management Information System may provide up-to-date input for the model. An interactive computer based system called Decision Support System is installed to help the manager to use data and models to support their decision making as needed.
A managerial report interprets output of the model and its implications for applications. Implementation The last phase of an OR study is to implement the system as prescribed by the management. The success of this phase depends on the support of both top management and operating management.
The implementation phase involves several steps 1. OR team provides a detailed explanation to the operating management 2. If the solution is satisfied, then operating management will provide the explanation to the personnel, the new course of action.
The OR team monitors the functioning of the new system 4. Feedback is obtained.