Preview

Linear Programming

Good Essays
Open Document
Open Document
11494 Words
Grammar
Grammar
Plagiarism
Plagiarism
Writing
Writing
Score
Score
Linear Programming
LP (2003) 1

OPERATIONS RESEARCH: 343

1. LINEAR PROGRAMMING 2. INTEGER PROGRAMMING 3. GAMES

Books: Ð3Ñ IntroÞ to OR ÐF.Hillier & J. LiebermanÑ; Ð33Ñ OR ÐH. TahaÑ; Ð333Ñ IntroÞ to Mathematical Prog ÐF.Hillier & J. LiebermanÑ; Ð3@Ñ IntroÞ to OR ÐJ.Eckert & M. KupferschmidÑÞ

LP (2003) 2

LINEAR PROGRAMMING (LP)
LP is an optimal decision making tool in which the objective is a linear function and the constraints on the decision problem are linear equalities and inequalities. It is a very popular decision support tool: in a survey of Fortune 500 firms, 85% of the responding firms said that they had used LP. Example 1: Manufacturer Produces: Ingredients used in the production of A & C: Each ton of A requires: Each ton of C requires: Supply of X is limited to: Supply of Y is limited to: 1 ton of A sells for: 1 ton of C sells for: A (acid) and C (caustic) X and Y 2lb of X; 1lb of Y 1lb of X ; 3lb of Y 11lb/week 18lb/week £1000 £1000

Manufacturer wishes to maximize weekly value of sales of A & C. Market research indicates no more than 4 tons of acid can be sold each week. How much A & C to produce to solve this problem. The answer is a pair of numbers: x" Ðweekly production of AÑ, x# Ðweekly p.of CÑ There are many pairs of numbers Ðx" , x# Ñ: Ð0,0Ñ, Ð1,1Ñ, Ð3,5Ñ.... Not all pairs Ðx" , x# Ñ are possible weekly productions Ðex. x" œ 27, x# œ 2 are not possibleÑ Ð Ð27, 2Ñ is not a feasible set of production figuresÑ. The constraints on x" , x# are such that Ðx" , x# Ñ represent a possible set of production figures: The amount each product is produced is non-negative: The amount of ingredient X required to produce x" tons of A & x# tons of C is 2x"  x# . As X is limited to 11lb/week: The amount of ingredient Y required combined with the supply restriction: We cannot sell more than 4 tons of A/week:

x" 0 x # 0

2x"  x# Ÿ 11 x"  3x# Ÿ 18 x" Ÿ 4 Conversely any Ðx" , x# Ñ satisfying these

A possible set of production figures satisfies these

You May Also Find These Documents Helpful

  • Good Essays

    We've analyzed our potential suppliers for each gauge, taking into account their capacity, cost of production for each plant, and transportation costs (The model and our objective function can be found in the appendix.) Our goal was to allocate spinning production (at Filatoi Riuniti and six local mills) in a manner that would minimize overall costs, while meeting the demand and operating within the capacity constraint of each plant. Given the output of our optimization model, we should be outsourcing the spinning of our yarn in this way:…

    • 898 Words
    • 4 Pages
    Good Essays
  • Better Essays

    Acme

    • 1450 Words
    • 8 Pages

    Linear programming is the “several related mathematical techniques used to allocate limited resources among competing demands in an optimal way” (Jacobs & Chase, 2013, appendix…

    • 1450 Words
    • 8 Pages
    Better Essays
  • Good Essays

    Linear Programming

    • 737 Words
    • 3 Pages

    The EI Paso plant was expanded when the company began marketing its meters in Arizona, California, Nevada, and Utah. With the growth of the West Coast business, the Darby…

    • 737 Words
    • 3 Pages
    Good Essays
  • Good Essays

    Cheat sheet OMIS 355

    • 727 Words
    • 3 Pages

    The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. 
X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch 
MAX: 150 X1 + 250 X2
Subject to: 2 X1 + 5 X2 ≤ 200 − resource 1 
3 X1 + 7 X2 ≤ 175 − resource 2 X1, X2 ≥ 0. How many units of resource 1 are consumed by each unit of product 1 produced? 2…

    • 727 Words
    • 3 Pages
    Good Essays
  • Good Essays

    Quain Case Analysis

    • 1013 Words
    • 6 Pages

    The objective of this analysis is to determine what blend of the four chemicals will allow Quain to minimize the cost of a 50-lb bag of the fertilizer. To do this we have used Linear Programming (LP) – a technique specifically designed to help managers make decisions relative to the allocation of resources. In this case, C-30 = , C-92 = , D-21 = , and E-11 = . The constraints for this case were translated into linear equations (i.e. inequalities) to mathematically express their meaning. The first constraint that C-11 must constitute for at least 15% of the blend can be expressed as: . The second constraint that C-92 and C-30 must together constitute at least 45% of the blend can be expressed as: . The third constraint that D-21 and C-92 can together constitute no more than 30% of the blend can be expressed as: . Lastly, the fourth constraint is that Quain-Grow is packaged and sold in 50-lb bags can be expressed as: . These equations were obtained and entered into a POM LP as a minimizing function. The objective function of this case was calculated and expressed as . The following table is the input…

    • 1013 Words
    • 6 Pages
    Good Essays
  • Satisfactory Essays

    Mid Term Exams

    • 1230 Words
    • 5 Pages

    | The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.40, and for a bag of Vinegar chips $0.50. Which of the following is not a feasible production combination?Answer…

    • 1230 Words
    • 5 Pages
    Satisfactory Essays
  • Powerful Essays

    In linear programming, the unit profit or unit contribution associated with one decision variable can be affected by the quantity made of that variable or of any other variable in the problem.…

    • 1510 Words
    • 7 Pages
    Powerful Essays
  • Good Essays

    The Outdoor Furniture Corporation manufactures two products: benches and picnic tables for use in yards and parks. The firm has two main resources: its carpenters (labor) and a supply of redwood for use in the furniture. During the next production period, 1200 hours of manpower are available under a union agreement. The firm also has a stock of 5000 pounds of quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 pounds of redwood; each picnic table takes 7 labor hours and 35 pounds of redwood. Completed benches yield a profit of $9 each, and tables a profit of $20 each. We formulated the following linear program to solve this problem: Decision Variables:…

    • 766 Words
    • 4 Pages
    Good Essays
  • Good Essays

    4. The owner of Chips etc. produces 2 kinds of chips: Lime(L) and Vinegar(V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of saly, 6 ounces of flour, and 1 ounce of herbs to produce: while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.40, and for a bag of Vinegar Chips $0.50. Which of the following is not a feasible production combination?…

    • 894 Words
    • 6 Pages
    Good Essays
  • Satisfactory Essays

    Shelby Shelving is a manufacturing firm who produces shelves for grocery stores. They have two types of shelves named Model S and Model LX. These two models have three main steps in production which are: Forming, Stamping and Assembly. Forming and Stamping departments can produce both models on same line whereas Assembly lines are different for each model. Firm’s problem is to determine how much to produce monthly Model S and Model LX in order to maximize the profit. The decision depends on the optimal quantity that must be produced for each model. The key issue is the marketing department of the company and the controller of the firm have conflicts about Model S. The marketing department think that they should cut back Model S production because its’ cost is higher than the selling price which make them having losses but Controller disagreed and he argues that they will be worse-off with lower production because Model S has a contribution to make overhead. The solution is the optimal quantity to produce each model at which the total profit will be the maximum.…

    • 885 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Linear programming, the ability to find the smallest or biggest possible value under some constraints, is used in a typical optimization problem. In business this method helps to optimize solutions for politics, transportation scheduling, hospital stuffing, purchasing and many other problems. In biotechnology the same equation is used to optimize cell-factories, enabling the production of drugs, plastics, or any desired products.…

    • 501 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    EXCEL SOLVER TUTORIAL

    • 460 Words
    • 3 Pages

    Many firms face the problem of how to best use multiple scarce resources. Linear programming is designed to help find the product mix that maximizes profits in the short run when multiple constraints exist. While linear programming can be solved as a mathematical problem using pencil and paper, it is much more efficient to use Excel Solver. The key to using Excel Solver is to make certain you have modeled the problem correctly and then interpreted the results appropriately. In this problem we will practice the use of Solver.…

    • 460 Words
    • 3 Pages
    Satisfactory Essays
  • Good Essays

    Mr. Ramesh Chandra is a Software Development Project Manager in one of the renowned Indian Software Services Company, namely “ABC Technologies”. The company’s business division has recently won a project from a major…

    • 1401 Words
    • 6 Pages
    Good Essays
  • Good Essays

    The longer contract periods are more expensive because Workforce Unlimited experiences greater difficulty finding temporary workers who are willing to commit to longer work assignments.…

    • 743 Words
    • 3 Pages
    Good Essays
  • Powerful Essays

    Mathematical Programming

    • 18473 Words
    • 55 Pages

    Management science is characterized by a scientific approach to managerial decision making. It attempts to apply mathematical methods and the capabilities of modern computers to the difficult and unstructured problems confronting modern managers. It is a young and novel discipline. Although its roots can be traced back to problems posed by early civilizations, it was not until World War II that it became identified as a respectable and well defined body of knowledge. Since then, it has grown at an impressive pace, unprecedented for most scientific accomplishments; it is changing our attitudes toward decision-making, and infiltrating every conceivable area of application, covering a wide variety of business, industrial, military, and public-sector problems. Management science has been known by a variety of other names. In the United States, operations research has served as a synonym and it is used widely today, while in Britain operational research seems to be the more accepted name. Some people tend to identify the scientific approach to managerial problemsolving under such other names as systems analysis, cost–benefit analysis, and cost-effectiveness analysis. We will adhere to management science throughout this book. Mathematical programming, and especially linear programming, is one of the best developed and most used branches of management science. It concerns the optimum allocation of limited resources among competing activities, under a set of constraints imposed by the nature of the problem being studied. These constraints could reflect financial, technological, marketing, organizational, or many other considerations. In broad terms, mathematical programming can be defined as a mathematical representation aimed at programming or planning the best possible allocation of scarce resources. When the mathematical representation uses linear functions exclusively, we have a linear-programming model. In 1947, George B. Dantzig, then…

    • 18473 Words
    • 55 Pages
    Powerful Essays