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Unit IIntroduction to Operations Research and Linear Programming Q1) Define Operations Research.A1) This is an analytical method that helps management make decisions. The name of this method is that research (O.R.) is comparatively new, but the tactic used for it's not new. Operations research involves the application of scientific principles and methods to strategic issues.The subject of operations research was born in the United Kingdom during World War II and was used in military strategy. During World War II, a group of scientists with representatives from mathematics, statistics, physics, and social sciences was commissioned to study various military operations. The team was very successful and contributed significantly to the meticulous response to the overall operation and related operational issues.As military strategies and their decisions were so important and costly that it became necessary to allocate such research to operations, the best scientists were grouped with the support of military institutions to develop scientific methods and methods. Adopted and provided quantitative information Decision.After World War II, its application began in industry, trade, agriculture, planning and various other economic areas.Operations research can be defined as:Definition:(I) Application of scientific methods, techniques, and tools to problems related to operating the system in order to provide system controllers with the best solution to the problem.(II) Research may be a tool for creating decisions that seek optimal results that are like the general purpose and constraints of the organization.(III) O.R. Is a scientific method that provides the executive department with a quantitative basis for making decisions about the operations under its control?(IV) A scientific approach to problem solving for management.(V) It helps executives make decisions by providing the necessary quantitative information based on scientific analysis.(VI) It is the requisition of the latest methods of mathematical science to complicated problems involving the management of huge systems of labor, machinery, materials, and money in industry, business, government, and defence. A distinctive approach is to develop a scientific model of the system that comes with measurement of things like chance and risk to predict and compare the result of alternative decisions, strategies, or controls.(VII) It is the application of scientists and subject experts to the study of specific manipulations of the scientific method. Its purpose is to provide management, which is the basis for quantitatively predicting the most effective outcomes of operations under a given set of variable conditions, thereby providing a sound basis for "decision-making". That is.In fact, operations research uses research and scientific methods for analysis, as well as for investigating current and future problems. Therefore, operations research provides management with an alternative plan for the problem to make a decision.It is very clear that Operations Research does not make management decisions, but instead, this method presents management with a careful scientific and quantitative analysis of the problem, allowing management to make healthier decisions. You will be during a better position to try to. Q2) Explain the phases in OR decision making process.A2) Since then, the main purpose of operations research has been to provide better quantitative information for decision making. Now our goal is to learn how to make better decisions.The decision-making process in OR research typically involves the following phases:Judgment stage: Operation decision. Determining the purpose. Judgment of the effectiveness of measures. Identify the type of problem, its cause, and the cause. B. Survey phase:Observation and data collection to better understand the problem Formulation of relevant hypotheses and models. Analysis of available information and verification of hypotheses. Generate and generate results, and consider alternatives. C. Action phase:Recommendations for corrective action for those who first raised the issue. This includes assumptions made, scope and limits, alternative course of action and its effects. Make the solution work: Implementation. In the absence of an OR, these phases are often performed completely, otherwise important steps are skipped. Judgment and subjective decision making are not enough. Therefore, the industry is looking for more objective ways to make decisions from operations research. It turns out that the method used also needs to take into account emotional and subjective factors.For example, skills and a creative workforce are key elements of our business, and if a manager wants to have a new place, he considers the employee's personal feelings about the place he chooses is needed. Q3) What are the some of the issues that operations research can analyse? A3) Here are some of the issues that operations research can analyse:1. Finance, budgeting, investment:Cash flow analysis, long-term capital requirements, investment portfolio, dividend policy, Credit policy. 2. Marketing:Product selection, competitive behaviour Number of salesmen, frequency of calling, Advertising strategy on cost and time. 3. Purchase:Purchasing policy, various prices, Determining the quantity and timing of purchases, Bid policy, Exchange policy, and Utilization of new material resources. 4. Production control:Logistics: Warehouse, distribution center, retail store location and size, delivery policy. Facility planning: Number and location of factories, warehouses, etc. Loading and unloading of facilities. Manufacturing: Production, employment, furlough, and stable production scheduling and ordering of optimal product configurations. Maintenance policy, crew size. Project schedule and resource allocation. 5. Personnel management:Combination of age and skill, Recruitment policy and Job assignment. 6. R & D:Focused field of research and development. Reliability and alternative decisions. Determine time-cost trade-offs and manage development projects. Q4) What are the features of OR?A4) Features of operations research (features):The main features of operations research (O.R.) are as follows. Interdisciplinary team approach: This requires an interdisciplinary team that includes individuals with skills in mathematics, statistics, economics, engineering, materials science, computers, and more. 2. A holistic approach to the system:When assessing decisions, look at key interactions and their impact on the organization as a whole on the functions they were originally involved in. 3. Methodological approach:Use scientific methods to solve O.R. problems 4. Objective approach:We will try to find the best or best solution to the problem under consideration, taking into account the goals of the O.R. organization. Operations Research Methodology:Operations research is a scientific approach to decision making, so you need to follow these steps: 1. Problem formulation:First you need to clearly define the problem. It is common to start O.R. as study with a tentative formulation of the problem. This is reformulated many times during the study. Economic aspects should also be considered in this study. Developing O.R. in a study, analysts need to analyse the following key components: (I) Environment:The environment includes physical, social, and economic factors that can affect the issue under consideration. The O.R. team or analyst should investigate the content of the organization, including men, materials, machinery, suppliers, consumers, competitors, governments, and the general public. (II) Decision maker:Operations analysts need to investigate the relationship between decision makers and the problem at hand. (III) Purpose:You need to define the purpose with the whole problem in mind. (IV) Alternative:O.R. studies determine which alternative behavioural policies are most effective in achieving the desired objectives. You should also consider the expected reaction of your competitors to the alternatives. 2. Solution derivation:The model is used to determine the solution, either by simulation or mathematical analysis. Mathematical analysis to derive the optimal solution involves analytical or numerical procedures and uses different disciplines of mathematics. 3. Model and solution testing:A well-formulated and properly manipulated model can help predict the impact of control variable changes on system-wide effectiveness. The validity of the solution is checked by comparing the results with the results obtained without the use of a model. 4. Establishing control over the solution:The solution derived from the model remains valid as long as the uncontrolled variables hold their values and relationships. When the value of one or more variables changes or the relationships between variables change, the solution goes out of control. In this situation, you need to change the model to take the changes into account. 5. Solution implementation:The solution thus obtained should be translated into operating procedures so that stakeholders can easily understand and apply it. After applying the solution to the system, O.R the group should investigate the system's response to the changes made. Q5) Explain Operations research modelA5) Operations Research model:An operations research model is an ideal representation of a real-life situation, representing one or more aspects of reality. The purpose of the model is to provide a means to analyse system behaviour and improve performance. Model classification:Models can be classified on the following elements: 1. Depends on the degree of abstraction:Mathematical model. Language model. 2. by function:Descriptive model. Predictive model. A normative model of repetitive problems. 3. by structure:Physical model. Analog (graphical) model. Symbolic or mathematical model. 4. Depends on the nature of the environment:Deterministic model. Probabilistic model. 5. by time horizon:Static model. Dynamic model. Q6) Write short note on Model building.A6) A mathematical model is a set of equations that describes a system or problem. The equations represent the objective functions and constraints. The objective function is the formula for the purpose (cost or benefit of the operation), and the constraint is the formula for the limit on the achievement of the purpose.These expressions consist of controllable and uncontrollable variables. The general form of a mathematical model is: O = f (xi, yi) Where O = objective function xi = controllable variablesyi = variables that cannot be controlled Relationship between f = O and xi, yi.Not all variables are included because the model is only an approximation of the actual situation. Simplification of operations research model:When building a model, you should try to simplify the model, but only to the extent that the accuracy is not significantly reduced.Some of the common simplifications are:Omit specific variables. Aggregation (or grouping) of variables. Change the nature of the variable. For example, consider a variable as a constant or continuous. Change the relationship between variables. That is, consider the variable as linear or straight. Change the constraint. Q7) What are the methods of OR?A7) The important methods of operations research are explained below.(I) Inventory management model:Operations research balances inventory costs with one or more of the following costs:Shortage cost. Order cost. Storage cost. Interest expense. This study will help you make decisions about:Amount to buy When to order Make or buy, that is, make decisions and buy. The most well-known usage is the form of economic order quantity equations for finding economic lot sizes. (II) Standby line model:These models are used with their associated costs to minimize latency and idle time.There are two types of standby line models:(A) Queuing theory. This can be applied to determine the number of service facilities and / or the timing of arrival for the service.(B) An ordering theory that can be applied to determine the order of services. (III) Replacement model:These models are used to determine when items should be replaced or maintained.(I) Abolished or(II) Usage efficiency deteriorates(III) It becomes uneconomical to repair or maintain.(IV) Assignment model: (IV) Competitive strategy:This type of strategy is adopted when the efficiency of one institution's decisions depends on the decisions of another institution. Examples of such strategies are card and chespel games, price fixing in competitive markets where these strategies are called "theory". (V) Linear programming:These techniques are used to solve the problem of operations with many variables that are subject to certain restrictions. For such issues, the objectives are profit, cost, production quantity, etc., but the limits are as follows: Government policy, plant capacity, product demand, raw material availability, water or electricity, storage capacity, etc. (VI) Sequence model:These involve choosing the right sequence to run a set of jobs running on a service facility or machine in order to optimize the efficiency measurement of system performance. (VII) Simulation model:Simulation is an exploratory way to study behaviour over time. (VIII) Network model:This is an approach for planning, scheduling, and controlling complex projects. Q8) What techniques are to apply for wide selection of problems?A8) These techniques apply to a really wide selection of problems.(I) Distribution or transportation issues:For such issues, we are given different centers in demand and we also know different warehouses with inventory locations. By using linear programming, you can find the most economical distribution of products from different warehouses to different centers. (II) Product composition:You can apply these techniques to determine the best product and available resource combination for your plant to get the maximum profit or the lowest production cost. (III) Production plan:These techniques can also be applied to assign different jobs to different machines to maximize profits, maximize production, and minimize total production time. (IV) Personnel allocation:Similarly, this technique can be applied to assign different people with different aptitudes to different jobs in order to complete a task in a minimum amount of time. (V) Agricultural production:You can also apply this technique to maximize the grower's interests. This involves growing a large number of items with different returns and harvest times on different types of land with different fertility. (VI) Financial application:Many financial decision-making problems are often solved by using applied mathematics.Some of them are:(I) select the optimal portfolio to maximize the return on investment from alternative investment opportunities such as bonds and stocks. Such problems are commonly faced by managers of investment trusts, banks and insurance companies.(II) Determining a financial mix strategy, including the choice of means such as funding companies, projects and inventories. Q9) What’s linear programming?A9) So, what's linear programming? Applied mathematics may be a simple technique that uses linear functions to represent complex relationships and find the simplest points. The important words within the preamble are drawn. The particular relationship could also be far more complicated, but it is often simplified to a linear relationship.Linear programming applications are everywhere around you. Use applied mathematics on a private and professional side. If you're driving from home to figure and need to use the shortest route, you're using applied mathematics. Or, when delivering a project, develop a technique to figure efficiently in order that your team can deliver on time. Example of applied mathematics problemSuppose you've got six packages delivered daily by a FedEx courier. The warehouse is at point A. The six destinations are indicated by U, V, W, X, Y, Z. The numbers on the road indicate the space between cities. To save lots of fuel and time, delivery personnel want to use the shortest route.Therefore, the courier calculates different routes to succeed in all six destinations and comes up with the shortest route. This method of selecting the littlest way is named applied mathematics.In this case, the deliveryman's purpose is to deliver the parcel to all or any six destinations on time. The method of selecting the simplest route is named research. Research may be a decision-making approach that has a group of methods for operating a system. Within the example above, my system was a delivery model.Linear programming is employed to seek out the optimal solution to a given constrained problem. Applied mathematics formulates a true problem into a mathematical model. This includes objective functions and linear inequalities that are subject to constraints.Does the 6-point linear representation above represent the important world? Yes, No. the particular route isn't a line, so it's oversimplified. Multiple turns, U-turns, traffic lights, and traffic jams can occur. However, with simple assumptions, we are creating an answer that greatly reduces the complexity of the matter and works in most scenarios. Q10) Mention the uses of Linear Programming. A10) Linear programming and optimization are used in a variety of industries. Manufacturing and service industries regularly use linear programming.Manufacturing uses linear programming to analyse supply chain operations. Their motivation is to maximize efficiency with minimal operating costs. Following the recommendations of the linear programming model, manufacturers can reconfigure their storage layouts and adjust their employees to reduce bottlenecks. This is a case study of Cequent's small warehouse based in the United States. Watch this video for a clearer understanding. Linear programming is also used in retail stores organized to optimize shelf space. With the number of products on the market growing exponentially, it's important to understand what your customers want. Optimization is actively used in stores such as Wal-Mart, Hyper city, Reliance, and Big Bazaar. The products in the store are strategically placed with the customer's shopping patterns in mind. The goal is to make it easier for customers to find and select the right product. This has limitations such as limited shelf space and various products. Optimization is also used to optimize delivery routes. The service industry uses optimization to find the best route for multiple salespeople travelling to multiple cities. With the help of clustering and greedy algorithms, delivery routes are determined by companies such as FedEx and Amazon. The goal is to minimize operational costs and time. Optimization is also used in machine learning. Supervised learning addresses the basics of linear programming. The system is trained to fit a mathematical model of a function from labelled input data that can predict values from unknown test data. The application of linear programming does not end here. There are many other real-world linear programming applications, such as those applied by shareholders, sports, stock markets, and more. Please continue to explore further. Q11) What are the components of LP?A11) Let's use the instance above to define some terms utilized in applied mathematics.Determinants: Determinants are variables that determine the output. They represent my ultimate solution. To unravel the matter, you initially got to identify the coefficient of determination. Within the above example, the entire number of units A and B, represented by X and Y, respectively, is my coefficient of determination.Objective function: Defined because the objective of deciding. Within the example above, the corporate wants to extend the gross profit margin, represented by Z. Therefore, profit is my objective function.Constraint: A constraint may be a limit or limit on a choice variable. These usually limit the worth of the choice variable. Within the above example, limiting the supply of Milk and Chocó resources is my constraint.Non-negative limit: altogether applied mathematics, the choice variable should take a non-negative value. This suggests that the worth of the choice variable must be greater than or adequate to 0.The process of formulating an applied mathematics problemLet's take a glance at the steps to generally define an applied mathematics problem.Identify the coefficient of determination Write an objective function Mention constraints Explicitly state non-negative limits For the matter to be an applied mathematics problem, the choice variables, objective functions, and constraints must all be linear functions.If all three conditions are met, it's called an applied mathematics problem.In order for the corporate to form the foremost profit, the above inequality must be met. When converted into a mathematical model is called formulating a real-world problem The other two factors are resource availability and technical factors. These can be better explained using the example below. The executable solution of a linear programming problem must meet the constraints and non-negative constraints. The executable solution of LPP with the maximization problem is the optimal solution when the objective function value is maximum (maximum). Similarly, an LPP executable solution with a minimization problem is optimal when the objective function value is the minimum (minimum). Q12) Explain the assumptions of LP.A12) There are some assumptions that linear programming works, these are: Proportional: The basic assumption underlying linear programming is that as the constraint inequality changes, so does the objective function. That is, if the product provides Rs 20 for profit, the total contribution is equal to 20x1. Where x1 is the number of units in the product. For example, if the product has 5 units, the contribution will be Rs 100, and if it has 10 units, it will be Rs 200. Therefore, if the output (sales) is doubled, the profit is also doubled.2. Additivity: The additive assumption argues that the total profit of the objective function is determined by the sum of the profits provided individually by each product. Similarly, the total amount of resources used is determined by the total amount of resources each product uses individually. This means that there is no interaction between the decision variables. 3. Continuity: Another assumption of linear programming is that the coefficient of determination is continuous. This means that you can use a combination of outputs with integer values as well as decimal values.For example, if 52/3 units of product A and 10 1/3 units of product B are produced in a week. In this case, a fraction of production is considered work in progress and the remaining production portion is acquired the following week. Therefore, producing 17 units of product A and 31 units of product B in 3 weeks means 52/3 units of product A and 10 1/3 units of product B per week4. Certainty: Another basic assumption of linear programming is certainty. That is, the coefficients of the objective function coefficient parameters and the constraint inequality are definitely known. The profit per unit of a product, the availability of materials and workforce per unit, the requirements for materials and workforce per unit, etc. are known and given in a linear programming problem. 5. Finite choice: This assumption means that the decision maker has a particular choice and the decision variable assumes a non-negative value. The non-negative assumption is true in a sense, and the output of a production problem cannot be negative. Therefore, this assumption is considered feasible.Therefore, when solving a linear programming problem, you need to keep these assumptions in mind so that the best alternative is selected. Q13) What terminology is used in applied mathematics problem?A13) Terminology utilized in applied mathematics problems 1. LP Problem Components: All LPPs are made from the coefficient of determination, b. Objective function, c. Constraints. 2. Optimization: applied mathematics attempts to maximise or minimize the worth of the target function. 3. Cost Factor Benefit: The coefficient of the target function variable represents the speed at which the worth of the target function increases or decreases by including one unit of every decision variable within the solution.Coefficient of determinationObjective functionConstraint 4. Constraints: Maximization (or minimization) is performed consistent with a group of constraints. Therefore, LP is often defined as a constrained optimization problem. They reflectResource limits.5. Input / output coefficient: The coefficient of the constraint variable is named the input / output coefficient. These indicate the speed at which a specific resource is unitized or depleted. They are showed the left of the constraint. 6. Capacity: The capacity or availability of varied resources is shown to the proper of the constraint.
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