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The document defines linear programming and describes its key components: an objective function to maximize or minimize, constraints in the form of linear inequalities, and non-negativity restrictions on variables. It provides examples of solving linear programming problems (LPPs) graphically and algebraically, including formulating an LPP for a production problem and a hotel planning problem. The document concludes that linear programming is an important mathematical technique that helps managers make optimal decisions.

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linear programming

The document provides an overview of linear programming, including its applications, assumptions, and mathematical formulation. Some key points:
- Linear programming is a tool for maximizing or minimizing quantities like profit or cost, subject to constraints. 50-90% of business decisions and computations involve linear programming.
- Applications in business include production, personnel, inventory, marketing, financial, and blending problems. The objective is to optimize variables like costs, profits, or resources while meeting constraints.
- Assumptions of linear programming include certainty, linearity/proportionality, additivity, divisibility, non-negativity, finiteness, and optimality at corner points.
- A linear programming problem is modeled mathemat

Integer Linear Programming

The document discusses integer programming and various methods to solve integer linear programming problems. It provides:
1) An overview of integer programming, defining it as an optimization problem where some or all variables must take integer values.
2) Three main types of integer programming problems - pure, mixed, and 0-1 integer problems.
3) Four methods for solving integer linear programming problems: rounding, cutting-plane, branch-and-bound, and additive algorithms.
4) A detailed example applying the cutting-plane and branch-and-bound methods to solve a sample integer programming problem.

Questions of basic probability for aptitude test

This document provides a collection of probability questions that are often asked in aptitude tests and competitive exams. It includes 14 questions with explanations and calculations of the probability for each. The purpose is to help students prepare for exams by understanding basic probability concepts and practicing sample questions. Links are provided at the end for additional free study materials on topics like reasoning, English, mathematics and general knowledge.

Goal Programming

The document discusses goal programming and its application to a case study of a medical manufacturing company. It introduces goal programming and describes how it can address multiple goals through deviations from target values. The case study establishes strategic, intermediate, and tactical goals for two medical products related to engineering cost, quality cost, production cost, setup time, delivery reliability and operations cost. A goal programming model is constructed to minimize deviations from these goals. The model is solved in steps to find optimal values for the decision variables to meet the goals.

Sequencing

The document describes three problems involving determining the optimal sequence of jobs through multiple machines to minimize the total elapsed time.
For the first problem involving two machines, the optimal sequence is job 2, 1, 6, 5, 4, 3 with a total elapsed time of 85 hours.
The second problem involving three machines is converted to two virtual machines, and the optimal sequence is job 3, 4, 2, 1, 5 with a total elapsed time of 51 hours.
The third problem involving four machines is also converted to two virtual machines, and the optimal sequence is job C, A, B, D with a total elapsed time of 82 hours.

Assignment Problem

This document discusses assignment problems and how to solve them using the Hungarian method. Assignment problems involve efficiently allocating people to tasks when each person has varying abilities. The Hungarian method is an algorithm that can find the optimal solution to an assignment problem in polynomial time. It involves constructing a cost matrix and then subtracting elements in rows and columns to create zeros, which indicate assignments. The method is iterated until all tasks are assigned with the minimum total cost. While typically used for minimization, the method can also solve maximization problems by converting the cost matrix.

Simplex Method

The Simplex Method is an algorithm for solving linear programming problems. It involves setting up the problem in standard form, constructing an initial simplex tableau, and then iteratively selecting pivot columns and performing row operations until an optimal solution is found. The method terminates when all indicators in the tableau are positive or zero, at which point the basic and non-basic variables can be identified to read the optimal solution.

Profit maximization

This document discusses profit maximization, which refers to determining the price and output level that generates the highest profit for a business. It defines profit as total revenue minus total costs. The document outlines two main methods for profit maximization: the marginal cost-marginal revenue method and the total cost-total revenue method. It explains that to maximize economic profits, a firm should produce the quantity where marginal revenue equals marginal costs. The document also notes that while profit maximization is good for businesses, it can be bad for consumers if companies cut costs or raise prices excessively.

Ch3

The document discusses modeling consumer preferences and indifference curves. It defines preference relations like strict preference, weak preference, and indifference. Indifference curves represent bundles that are equally preferred. They have specific properties like not intersecting and negatively or positively sloped depending on if goods are normal or inferior. The marginal rate of substitution is the slope of the indifference curve and represents the rate at which a consumer is willing to trade one good for another. Well-behaved preferences exhibit properties like monotonicity and convexity.

Linear Programming

This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.

Duality

- Duality theory states that every linear programming (LP) problem has a corresponding dual problem, and the optimal solutions of the primal and dual problems are related.
- The dual problem is obtained by converting the constraints of the primal to variables and vice versa.
- The dual simplex method starts with an infeasible but optimal solution and moves toward feasibility while maintaining optimality, unlike the regular simplex method which moves from a feasible to optimal solution.

Game theory and its applications

Game theory is a branch of applied mathematics that analyzes strategic interactions between rational decision-makers. It was developed by John von Neumann and Oskar Morgenstern in the 1940s. Game theory has applications in economics, military strategy, politics, and other domains involving conflict and cooperation between intelligent decision-makers. The document defines key concepts in game theory like Nash equilibrium, zero-sum games, prisoner's dilemma, and mixed strategies. It also discusses assumptions of game theory and provides examples of classic game theory models.

Transportation problem ppt

This document discusses transportation problems and three methods to solve them: the North West Corner Method, Least Cost Method, and Vogel Approximation Method. The objective of transportation problems is to minimize the cost of distributing products from sources to destinations while satisfying supply and demand constraints. The document provides examples to illustrate how each method works step-by-step to arrive at a basic feasible solution.

Assignment Problem

This is a special type of LPP in which the objective function is to find the optimum allocation of a number of tasks (jobs) to an equal number of facilities (persons). Here we make the assumption that each person can perform each job but with varying degree of efficiency. For example, a departmental head may have 4 persons available for assignment and 4 jobs to fill. Then his interest is to find the best assignment which will be in the best interest of the department.

Operations research-an-introduction

Operations research (OR) began during World War II when scientists applied analytical methods to solve complex military problems. Since then, OR has expanded to help organizations with strategic decision-making. OR uses interdisciplinary teams and quantitative techniques like linear programming to build mathematical models of systems. These models help optimize resource allocation and identify optimal solutions. OR aims to improve systems through objective, data-driven analysis and continues providing value as new problems emerge over time.

Simplex method concept,

The document discusses the Simplex method for solving linear programming problems involving profit maximization and cost minimization. It provides an overview of the concept and steps of the Simplex method, and gives an example of formulating and solving a farm linear programming model to maximize profits from two products. The document also discusses some complications that can arise in applying the Simplex method.

Chapter 19 decision-making under risk

This document provides an overview of key concepts for decision making under risk and uncertainty, including random variables, probability distributions, sampling, and Monte Carlo simulation. It introduces the concepts and outlines the steps for modeling problems that involve uncertain parameters through simulation. The goal is to simulate potential outcomes and evaluate alternatives while accounting for variation in inputs.

Ppt on decision theory

This document provides an overview of decision theory and various decision-making environments. It discusses the six steps in decision theory as applying to a case study about a lumber company expanding its product line. The types of decision-making environments covered are decisions under certainty, risk, and uncertainty. Decision-making under uncertainty further explores criteria for decisions like maximax, maximin, weighted average, equally likely, and minimax regret. Game theory and its applications to strategic decision-making between competitors are also briefly introduced.

Unit.2. linear programming

This document discusses linear programming and its concepts, formulation, and methods of solving linear programming problems. It provides the following key points:
1) Linear programming involves optimizing a linear objective function subject to linear constraints. It aims to find the best allocation of limited resources to achieve objectives.
2) Formulating a linear programming problem involves identifying decision variables, the objective function, and constraints. Problems can be solved graphically or algebraically using the simplex method.
3) The graphic method can be used for problems with two variables, involving plotting the constraints on a graph to find the optimal solution at a corner point of the feasible region.

Simplex Method

The document provides an overview of the simplex method for solving linear programming problems with more than two decision variables. It describes key concepts like slack variables, surplus variables, basic feasible solutions, degenerate and non-degenerate solutions, and using tableau steps to arrive at an optimal solution. Examples are provided to illustrate setting up and solving problems using the simplex method.

linear programming

linear programming

Integer Linear Programming

Integer Linear Programming

Questions of basic probability for aptitude test

Questions of basic probability for aptitude test

Goal Programming

Goal Programming

Sequencing

Sequencing

Assignment Problem

Assignment Problem

Simplex Method

Simplex Method

Profit maximization

Profit maximization

Ch3

Ch3

Linear Programming

Linear Programming

Duality

Duality

Game theory and its applications

Game theory and its applications

Transportation problem ppt

Transportation problem ppt

Assignment Problem

Assignment Problem

Operations research-an-introduction

Operations research-an-introduction

Simplex method concept,

Simplex method concept,

Chapter 19 decision-making under risk

Chapter 19 decision-making under risk

Ppt on decision theory

Ppt on decision theory

Unit.2. linear programming

Unit.2. linear programming

Simplex Method

Simplex Method

Lecture - Linear Programming.pdf

This document provides an introduction to linear programming. It defines linear programming as an optimization problem that involves maximizing or minimizing a linear objective function subject to linear constraints. Various terminology used in linear programming like decision variables, objective function, and constraints are explained. Several examples of linear programming problems from areas like production planning, scheduling, and resource allocation are presented and formulated mathematically. Graphical and algebraic solution methods for linear programming problems are discussed. The document also notes that integer programming problems cannot be solved using the same techniques as linear programs due to the discrete nature of the variables. Additional linear programming examples and problems from an operations research textbook are listed for further practice.

Lenier Equation

The document describes how to formulate and solve linear programming problems by defining the components of a linear programming problem, describing how to model real-world problems as linear programs, and outlining two methods, graphical and simplex, to solve linear programming problems. It then provides examples of solving linear programming problems using these steps and methods.

Decision making

The document provides an overview of a presentation on decision making and linear programming. It discusses farm management decisions that fall under organizational, administrative, and marketing categories. It then introduces quantitative analysis approaches and linear programming. Linear programming is defined as a technique to optimize performance under resource constraints. The assumptions and terminology of linear programming are explained. Finally, examples are provided to demonstrate how to formulate a linear programming model and solve it graphically.

introduction to Operation Research

The document discusses linear programming and provides examples to illustrate the process. It explains that linear programming involves optimizing a linear objective function subject to linear constraints. There are three basic components: decision variables, an objective to optimize, and constraints. Examples show how to formulate the objective function and constraints as linear equations or inequalities. The optimal solution is found by analyzing the feasible region defined by the constraints and determining which corner point gives the best value for the objective function.

Linear Programming.ppt

Here is the standard LP formulation of the problem:
Maximize: 3000X + 2000Y
Subject to:
X + 2Y ≤ 6 (Constraint on raw material A)
2X + Y ≤ 8 (Constraint on raw material B)
Y ≤ X + 1 (Demand for interior cannot exceed exterior by 1 ton)
Y ≤ 2 (Maximum demand for interior is 2 tons)
X, Y ≥ 0
Where:
X = Quantity of exterior paint produced (decision variable 1)
Y = Quantity of interior paint produced (decision variable 2)
The objective is to maximize total profit by choosing the optimal values of X and Y.

Linear programming graphical method

1) The document discusses linear programming and its graphical solution method. It provides examples of forming linear programming models and using graphs to find the feasible region and optimal solution.
2) A toy manufacturing example is presented and modeled using linear programming with the objective of maximizing weekly profit. The feasible region is graphed and the optimal solution is identified.
3) Another example involving a wood products company is modeled and solved graphically to determine the optimal production mix to maximize profits. Corner points of the feasible region are identified and evaluated to find the optimal solution.

181_Sample-Chapter.pdf

The document introduces linear programming and provides examples to illustrate its basic concepts and formulation. It defines linear programming as a technique to optimally allocate limited resources according to a given objective function and set of linear constraints. It then provides definitions for key linear programming components - decision variables, objective function, and constraints. Examples are provided to demonstrate how to formulate linear programming problems from descriptions of resource allocation scenarios and how to represent them mathematically.

Introduction to Operations Research/ Management Science

Here are the steps to solve this problem:
Let x = number of inches of orange beads
Let y = number of inches of black beads
Constraints:
x >= 0
y >= 0
x + y <= 24 (total length must be <= 24 inches)
y >= 2x (black beads must be >= 2x the length of orange beads)
y >= 5 (black beads must be >= 5 inches)
Objective: Maximize x + y (total length of necklace)
To sketch the problem:
Plot the lines y = 2x, x + y = 24, y = 5 on a xy-plane.
The shaded region satisfying all constraints is the feasible

LP special cases and Duality.pptx

This document provides an overview of operations research and linear programming techniques. It begins with an introduction to the graphical method for solving linear programming problems with two variables by plotting the feasible region defined by the constraints. It then defines key terms like feasible solutions and optimal solutions. The document provides examples of using the graphical method to find the optimal solution for both maximization and minimization problems. It also discusses special cases that can occur with linear programs, such as alternative optimal solutions, unbounded solutions, infeasible solutions, and degenerate solutions. Finally, it provides an introduction to the concept of duality in linear programming.

Integer Programming PPt.ernxzamnbmbmspdf

The document discusses integer programming problems and various solution techniques. It begins by defining integer programming and noting that it allows for logical constraints using binary variables. Several examples of integer programming formulations are provided, including the knapsack problem, facility location problem, and mixed integer programs. The key solution techniques discussed are enumeration, branch and bound, and cutting planes. Branch and bound is explained in detail as a method that systematically enumerates a subset of feasible solutions to find the optimal solution.

LINEAR PROGRAMING

The document defines linear programming as a branch of mathematics used to find the optimal solution to problems with constraints. It provides examples of using linear programming to maximize profit or minimize costs in organizations. It also introduces drawing linear inequalities and solving simultaneous inequalities. The steps to formulate a linear programming problem are identified as defining variables and objectives, translating constraints, finding feasible solutions, and evaluating objectives to find optimal solutions.

LPP, Duality and Game Theory

The document provides information about linear programming problems (LPP), including:
- LPPs involve optimization of a linear objective function subject to linear constraints.
- Graphical and algebraic methods can be used to find the optimal solution, which must occur at a corner point of the feasible region.
- The simplex method is an algorithm that moves from one corner point to another to optimize the objective function.
- Examples are provided to illustrate LPP formulation, graphical solution, and use of the simplex method to iteratively find an optimal solution.

Management Science

The document introduces optimization problems and linear programming (LP). It defines the key components of an optimization problem as decisions variables, constraints, and an objective to maximize or minimize. LP problems involve linear (straight-line) relationships between variables and constraints. The document provides the general formulation of an LP problem and examples to illustrate how to model an optimization problem as an LP. It also describes graphical and algebraic methods for solving LP problems and identifying optimal solutions.

P

The document summarizes four problems involving linear programming models. Each problem provides data on production quantities, costs, profits, time constraints and defines the objective function and constraints. The problems involve maximizing profit from production of multiple products, minimizing production costs across two plants, and meeting required production quantities with limited resources. A linear programming model is formulated for each problem to optimize the given objective subject to the provided constraints.

Integer Programming, Goal Programming, and Nonlinear Programming

Integer Programming, Goal Programming, and Nonlinear Programming
Salah Skaik: Marketing & Business Development Consultant

Linear progarmming part 1

This document provides an overview of linear programming (LP). It begins with definitions, including that LP is a technique to optimize an objective function subject to constraints. It provides an example LP problem about production allocation. It then discusses key LP concepts like decision variables, objective functions, constraints, feasible and optimal solutions. Matrix notation for representing LP problems is introduced. Finally, it formulates three sample LP problems: production allocation, material blending, and diet formulation.

Linear Programming Problems : Dr. Purnima Pandit

Linear programming problems involve optimizing an objective function subject to constraints on variables. They can be modeled and solved using techniques like the simplex method. The simplex method works by moving from one basic feasible solution to an adjacent extreme point through an exchange of variables in and out of the basis. It begins with an initial basic feasible solution and proceeds iteratively until an optimal solution is reached.

Linear Programming

The document provides an example of using linear programming to solve an optimization problem for a machine shop owner. It outlines the constraints on machine time, raw materials, and labor and calculates the contribution per unit for two products. It then formulates the problem as a linear programming model to determine the number of each product to maximize total contribution. The summary formulates the optimization problem to determine how many of each type of coffee table a furniture company should produce to maximize contribution subject to capacity constraints on cutting, assembly and finishing hours and minimum demand for small tables.

Linear Programming

The document provides an example of using linear programming to solve an optimization problem for a machine shop owner. It outlines maximizing contributions by determining the optimal number of units to produce of two products (A and B) given constraints on machine time, raw materials, and labor available. It presents the linear programming formulation and solution showing the feasible region and optimal point that maximizes total contribution. The document also provides another example of using the simplex method to solve a multi-variable optimization problem for a furniture company to maximize total contribution under constraints.

Linear Programming

The document provides an example of using linear programming to solve an optimization problem for a machine shop owner. It outlines maximizing contributions by determining the optimal number of units to produce of two products (A and B) given constraints on machine time, raw materials, and labor available. It presents the linear programming formulation and solution showing the feasible region and optimal point that maximizes total contribution. The document also provides another example of using the simplex method to solve a multi-variable optimization problem for a furniture company to maximize total contribution under constraints.

Lecture - Linear Programming.pdf

Lecture - Linear Programming.pdf

Lenier Equation

Lenier Equation

Decision making

Decision making

introduction to Operation Research

introduction to Operation Research

Linear Programming.ppt

Linear Programming.ppt

Linear programming graphical method

Linear programming graphical method

181_Sample-Chapter.pdf

181_Sample-Chapter.pdf

Introduction to Operations Research/ Management Science

Introduction to Operations Research/ Management Science

LP special cases and Duality.pptx

LP special cases and Duality.pptx

Integer Programming PPt.ernxzamnbmbmspdf

Integer Programming PPt.ernxzamnbmbmspdf

LINEAR PROGRAMING

LINEAR PROGRAMING

LPP, Duality and Game Theory

LPP, Duality and Game Theory

Management Science

Management Science

P

P

Integer Programming, Goal Programming, and Nonlinear Programming

Integer Programming, Goal Programming, and Nonlinear Programming

Linear progarmming part 1

Linear progarmming part 1

Linear Programming Problems : Dr. Purnima Pandit

Linear Programming Problems : Dr. Purnima Pandit

Linear Programming

Linear Programming

Linear Programming

Linear Programming

Linear Programming

Linear Programming

Chauvinism

This document discusses the concepts of chauvinism and racism. It defines chauvinism as an exaggerated form of patriotism and support for one's own group, often accompanied by a prejudiced or hostile attitude towards other groups. The document then provides historical context for the term from a French soldier named Nicolas Chauvin. It examines perspectives on chauvinism from Indian society and includes an excerpt from an interview with a self-described misogynist that illustrates extreme chauvinist views. The document concludes by discussing laws and acts passed in India to prevent domestic violence, sexual harassment, discrimination, and protect women's rights in an effort to reduce chauvinism in society.

World trade organisation

The document provides an introduction to the World Trade Organization (WTO). It discusses that the WTO is an international organization that oversees and liberalizes global trade. The WTO aims to ensure trade flows freely, predictably, and openly for all members. It operates a system of trade rules and provides a forum for negotiating trade agreements and settling disputes between member countries. The document then discusses the history, structure, functions, objectives, achievements, advantages, and disadvantages of the WTO.

Primary research on baby care product by using quantitative research method

The document discusses primary research conducted by Johnson & Johnson India Ltd. on baby care products using quantitative methods. It provides details on the survey conducted, including questions asked about brand and product preferences, new product ideas, and customer satisfaction. The results showed most customers prefer J&J baby products and are satisfied with the quality.

Amul Inventory Techniques & Other Details

This document outlines different categories of items such as vital, essential, and desirable items. It also discusses types of items like fast moving, slow moving and non-moving items. Finally, it mentions some innovative ideas and food items like parathas, chocolate and pickles.

Cost Sheet At Corporate Level

The document discusses the components of a cost sheet, including direct costs, indirect costs, and calculations. It provides a sample cost sheet for a company producing 1 lakh units of bread. Direct costs include raw materials, direct labor, and direct expenses. Indirect costs include factory overheads, office administration overheads, and sales/distribution overheads. The cost sheet calculates costs per unit and totals for raw materials, labor, overheads, cost of production, sales, and profit.

Export procedure

This document outlines the export procedure and documentation for Adinath Exports, an Indian exporting company. It begins with an introduction to exports and exporters. It then discusses the importance of export documents and procedures, providing examples of common documents. It introduces Adinath Exports, including what they export. The main body describes the step-by-step export procedure, including obtaining necessary licenses and memberships, creating invoices and packing lists, shipping goods, and collecting payment. In conclusion, following the correct export procedure and documentation is essential for smooth and successful international trade.

Sbi mutual gold fund

This document discusses SBI Gold Fund, a mutual fund scheme offered by SBI Mutual Fund. It provides an overview of the fund, including its objectives to track returns of SBI Gold Exchange Traded Fund, asset allocation of 95-100% in SBI GETS and 0-5% in money market instruments, and features like no requirement for a demat account, liquidity, and availability of systematic investment plans starting from Rs. 100 per month. The document also covers details like net asset value calculations, historical NAV data, and commissions disclosed to distributors for selling various SBI MF schemes.

Retailing & Promotion Mix

The document discusses retailing and promotion mix. It defines retailing as the sale of goods or services to the ultimate consumer. It notes several types of retailers like supermarkets, specialty stores, franchising. Promotion mix is defined as the combination of tools a company uses to communicate with audiences, which includes advertising, personal selling, sales promotion, publicity and public relations. Factors influencing the promotion mix are also outlined.

Dabur

This document summarizes Dabur India Limited, a leading FMCG company in India. It discusses Dabur's origins in 1884 as a health care products manufacturer. Over time, Dabur expanded its Ayurvedic products and research laboratories. By 2000, Dabur achieved a turnover of Rs. 1000 crores and established market leadership in India. The document also outlines Dabur's major product categories including health care, oral care, foods, hair care and skin care. It positions Dabur as one of the leading FMCG companies in India with a presence in over 60 overseas countries.

Can mumbai be shanghai

The document compares and contrasts the cities of Mumbai and Shanghai. It provides an overview of the population, economy, culture, and transportation infrastructure of both cities. It also discusses the results of a survey on what changes would be needed for Mumbai to become more like Shanghai. The document concludes by considering whether Mumbai can become Shanghai.

Production planning & control (ppc)

This document discusses production planning and control (PPC) in a manufacturing plant. It outlines the importance of PPC for steady production flow, maintaining delivery schedules, cost control, and quality improvement. The key steps of PPC include production planning, routing, scheduling, dispatching, and follow-up to efficiently manage processes, materials, and the shop floor for maximizing profits.

HOW TO CONTROL YOUR ANGER

The document summarizes a book on how to control anger. It is written by M.K. Gupta and discusses the causes, manifestations, and effects of anger on the body and mind. It provides short and long-term measures to control anger through self-introspection and avoiding provoking situations. It emphasizes that forgiveness can help quench the fire of anger by releasing negative feelings without seeking revenge. The conclusion states that the book gives readers strategies to overcome their deadly enemy of anger.

Skill of successful entrepreneur

The document introduces entrepreneurship and discusses the skills and qualities needed for success. It defines an entrepreneur as an individual who runs a small business and assumes the risk. Successful entrepreneurs have both soft skills like managing perceptions and emotional intelligence, as well as hard skills like leadership, marketing, and risk-taking. Qualities like patience, self-discipline, passion, and determination are also important. The document also outlines some advantages like excitement and independence but also disadvantages like irregular income and workload of being an entrepreneur.

Marginal costing

This document provides an introduction to the concept of marginal costing. It defines marginal costing as accounting that distinguishes between fixed and variable costs, charging variable costs to cost units and writing off fixed costs against the total contribution. The document outlines the key features, advantages, and disadvantages of marginal costing. It also provides an example of calculating the break-even point and profit-volume ratio of a company called NHC Foods Ltd. The document concludes that marginal costing supports managerial decision making.

Chauvinism

Chauvinism

World trade organisation

World trade organisation

Primary research on baby care product by using quantitative research method

Primary research on baby care product by using quantitative research method

Amul Inventory Techniques & Other Details

Amul Inventory Techniques & Other Details

Cost Sheet At Corporate Level

Cost Sheet At Corporate Level

Export procedure

Export procedure

Sbi mutual gold fund

Sbi mutual gold fund

Retailing & Promotion Mix

Retailing & Promotion Mix

Dabur

Dabur

Can mumbai be shanghai

Can mumbai be shanghai

Production planning & control (ppc)

Production planning & control (ppc)

HOW TO CONTROL YOUR ANGER

HOW TO CONTROL YOUR ANGER

Skill of successful entrepreneur

Skill of successful entrepreneur

Marginal costing

Marginal costing

- 1. OPERATIONAL RESEARCH Topic: Linear Programming Problem Submitted To : Prof. Nilesh Coordinators : Zeel Mathkiya (19) Dharmik Mehta (20) Sejal Mehta (21) Hirni Mewada (22) Varun Modi (23) Siddhi Nalawade (24)
- 2. DEFINITION OF LINEAR PROGRAMMING The Mathematical Definition of LP: “It is the analysis of problem in which a linear function of a number of variables is to maximised (minimised), when those variables are subject to a number of restraints in the form of linear inequalities”.
- 3. TERMINOLOGY OF LINEAR PROGRAMMING A typical linear program has the following components An objective Function. Constraints or Restrictions. Non-negativity Restrictions.
- 4. TERMS USED TO DESCRIBE LINEAR PROGRAMMING PROBLEMS Decision variables. Objective function. Constraints. Linear relationship. Equation and inequalities. Non-negative restriction.
- 5. FORMATION OF LPP Objective function Constraints Non-Negativity restrictions Solution Feasible Solution Optimum Feasible Solution
- 6. SOLVED EXAMPLE -1 A Company manufactures 2 types of product H₁ & H₂. Both the product pass through 2 machines M₁,M₂.The time requires for processing each unit of product H₁,H₂.On each machine & the available capacity of each machine is given below: Product Machine M₁ M₂ H₁ 3 2 H₂ 2 7 Available Capacity(hrs) 1800 1400 The availability of materials is sufficient to product 350 unit of H₁ & 150 of H₂.Each unit of H₁ gives a profit of Rs.25,each unit of H₂ gives profit of Rs.20.Formulate the above problem as LPP.
- 7. SOLUTION From manufactures point of view we need to maximise the profit.The profit depend upon the number of unit of product H₁ &H₂ produced. Let x₁= no of unit of H₁ produce x₂=no of unit of H₂ produce x₁ ≥ 0 1 x₂ ≥ 0 2 3x₁ + 2x₂ ≤ 1800 3 2x₁ + 7x₂ ≤ 1400 4 Z= 25x₁ + 20x₂ LPP is formed as follows: Maximise Z= 25x₁ + 20x₂
- 8. CONTI….. Subject to: x₁ ≥ 0 x₂ ≥ 0 3x₁ + 2x₂ ≤ 1800 2x₁ + 7x₂ ≤ 1400
- 9. CONTI…... A Manager of hotel dreamland plans and extancison not more than 50 groups attleast 5 must be executive single rooms the number of executive double rooms should be atleast 3 times the number of executive single rooms. He charges Rs.3000 for executive double rooms and Rs.1800 executive single rooms per day.
- 10. CONTI….. Formulate the above problume for LPP SOLUTION → The LPP is formulated as follows ; Let X1 = Total No. of single executive rooms Let X2 = Total No. of Double executive rooms ... X1 + x2 < 50 X1 > 5 x2 > 3 X1 Maximise ; Z = 1800 X1 + 3000 x2
- 11. The LPP is formulated as follows Maximise ; Z = 1800 X1 + 3000 x Subject to ; X1 + x2 < 50 X1 > 5 x 2 > 3 X1
- 12. GRAPHICAL METHOD 1. Arrive at a graphical solution for the following LPP. Maximize Z = 40x1 + 35x2 Subject to : 2x1 + 3x2 < 60 4x1 + 3x2 < 96 x1 , x 2 > 0
- 13. Solution : Let us consider the equation 1) 2x1 + 3x2 = 60 Put x2 = 0: 2x1 = 60 x1 = 30 A = (30 , 0) Put x1 = 0 : 3x2 = 60 x2 = 20 B = (0 , 20)
- 14. 2) 4x1 + 3x2 < 96 Put x2 = 0 : 4x1 = 96 x1 = 24 C = (24 , 0) Put x1 = 0 : 3x2 = 96 x2 = 32 D = (0 , 32)
- 15. Y axis 40 Scale : Xaxis = 1 cm = 5 units 35 Yaxis = 1 cm = 5 units 30 D 25 20 B 15 10 p 5 C A X axis O 5 10 15 20 25 30 35 40
- 16. OBPC is the feasible region Points x1 x2 z O 0 0 z=0 B 0 20 z = 40(0) + 35 (20) = 700 P 18 8 z = 40(18) + 35(8) = 1000 C 24 0 z = 40(24) + 35(0) = 960 Thus, the optimal feasible solution is x1 = 18 , x2 = 8 and z = 1000
- 17. CONTI….. Find the feasible solution to following LPP Minimize Z = 6x + 5y Subject to = x + y > 7 x<3,y<4 x<0,y>0
- 18. Solution : Removing Inequality in given equation 1. x + y > 7 Put y = 0 : x = 7 Put x = 0 : y = 7 The two points are : A = (7 , 0) & B = (0 , 7) Further, X=3,y=4
- 19. Y axis 8 7 B Scale : X axis = 1cm = 1 unit Y axis = 1cm = 1 unit 6 5 P 4 3 2 1 A 1 2 3 4 5 6 7 8 X axis O
- 20. CONTI….. As all the 3 lines intersect each other at a common point P( 3 , 4) it is the feasible solution to LPP Z = 6(3) + 5(4) = 18 + 20 = 38
- 21. CONCLUSION Linear programming is very important mathematical technique which enables managers to arrive at proper decisions regarding his area of work. Thus it is very important part of operations research.