Overview of OR Modeling Approach & Introduction to Linear Programming

Transcription

1 Overview of OR Modeling Approach & Introduction to Linear Programming M. En C. Eduardo Bustos Farías 1

2 What is Operations Research? A field of study that uses computers, statistics, and mathematics to solve business problems. Also known as: Management Science Decision Science M. En C. Eduardo Bustos Farías 2

3 The Importance of Management Science Management science The discipline of applying advanced analytical methods to help make better decisions. Devoted to solving managerial-type problems using quantitative models M. En C. Eduardo Bustos Farías 3

4 OPERATIONS RESEARCH First applied to research on (military) problems Use of scientific knowledge through interdisciplinary team effort for the purpose of deciding the best utilization of limited resources M. En C. Eduardo Bustos Farías 4

5 Introduction to Linear Programming History of OR M. En C. Eduardo Bustos Farías M. En C. Eduardo Bustos Farías 5

6 Jordan, Minkowsky y Farkas worked about linnear models M. En C. Eduardo Bustos Farías 6

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8 Linear programming was developed as a discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. M. En C. Eduardo Bustos Farías 8

9 Origins of OR Management of WW II by England A team of scientists Worked on strategic and tactical problems Land and air defenses M. En C. Eduardo Bustos Farías 9

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11 The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. M. En C. Eduardo Bustos Farías 11

12 SCOOP US Air Force wanted to investigate the feasibility of applying mathematical techniques to military budgeting and planning. George Dantzig had proposed that interrelations between activities of a large organization can be viewed as a LP model and that the optimal program (solution) can be obtained by minimizing a (single) linear objective function. Air Force initiated project SCOOP (Scientific Computing of Optimum Programs) NOTE: SCOOP began in June 1947 and at the end of the same summer, Dantzig and associates had developed: 1) An initial mathematical model of the general linear programming problem. 2) A general method of solution called the simplex method. M. En C. Eduardo Bustos Farías 12

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15 Nobel prize in econonmics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and the economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role. M. En C. Eduardo Bustos Farías 15

16 Simplex Today A large variety of Simplex-based algorithms exist to solve LP problems. Other (polynomial time) algorithms have been developed for solving LP problems: Khachian algorithm (1979) Kamarkar algorithm (AT&T Bell Labs, mid 80s) See Section 4.10 BUT, none of these algorithms have been able to beat Simplex in actual practical applications. HENCE, Simplex (in its various forms) is and will most likely remain the most dominant LP algorithm for at least the near future M. En C. Eduardo Bustos Farías 16

17 Decisions Most effective use of limited military resources Deployment of radar defenses Maximize bomber effectiveness M. En C. Eduardo Bustos Farías 17

18 OR in the USA The US adopted OR methods in WW II based on Britain s successes Following the war, industry adopted these OR methods and approaches The US assumed leadership in the development of OR as an academic discipline M. En C. Eduardo Bustos Farías 18

19 Progress in OR Applications The development and application of the tools of OR are due in large part to the parallel development of digital computers to handle large scale computational problems M. En C. Eduardo Bustos Farías 19

20 Example OR Application Areas Military planning Industrial planning Healthcare analysis Financial institutional and investment planning Governmental planning at all levels Transportation systems planning Logistics system design and operation Emergency system planning M. En C. Eduardo Bustos Farías 20

21 Some Applications of LPs Production Planning: Given several products with varying production requirements and cost structures, determine how much of each product to produce in order to maximize profits. Scheduling: Given a staff of people, determine an optimal work schedule that maximizes worker preferences while adhering to scheduling rules. Portfolio Management: Determine bond portfolios that maximize expected return subject to constraints on risk levels and diversification. And an incredible number more. M. En C. Eduardo Bustos Farías 21

22 Typical Applications of Linear Programming 1. A manufacturer wants to develop a production schedule and inventory policy that will satisfy sales demand in future periods and same time minimize the total production and inventory cost. 2. A financial analyst must select an investment portfolio from a variety of stock and bond investment alternatives. He would like to establish the portfolio that maximizes the return on investment. M. En C. Eduardo Bustos Farías 22

23 Typical Applications of Linear Programming continued 3. A marketing manager wants to determine how best to allocate a fixed advertising budget among alternative advertising media such as radio, TV, newspaper, and magazines. The goal is to maximize advertising effectiveness. 4. A company has warehouses in a number of locations throughout the country. For a set of customer demands for its products, the company would like to determine how much each warehouse should ship to each customer so that the total transportation costs are minimized. M. En C. Eduardo Bustos Farías 23

24 Successful Applications of OR Merril Lynch 5 million customers 16,000 financial advisors Developed a model to design product features and pricing options to better reflect customer value Benefits: $80 million increase in annual revenue $22 billion increase in net assets M. En C. Eduardo Bustos Farías 24

25 Successful Applications of OR Jan de Wit Co. Brazil s largest lily farmer Annually plants 3.5 million bulbs and produces 420,000 pots & 220,000 bundles of lilies in 50 varieties. Developed model to determine what to plant, when to plant it, and how to sell it. Benefits: 26% increase in revenue 32% increase in contribution margin M. En C. Eduardo Bustos Farías 25

26 Successful Applications of OR Samsung Electronics Leading DRAM manufacturer Semiconductor facilities cost $2-$3 billion High equipment utilization is key Developed comprehensive planning and scheduling system to control WIP Benefits: Cut cycle times in half $1 billion increase in annual revenue M. En C. Eduardo Bustos Farías 26

27 Table Successful Applications of Management Science M. En C. Eduardo Bustos Farías 27

28 Table Successful Applications of Management Science (cont d) M. En C. Eduardo Bustos Farías 28

29 Problem Solving Approaches Managers tend to use a qualitative approach to problem solving when 1. The problem is fairly simple. 2. The problem is familiar. 3. The costs involved are not great. Managers tend to use a quantitative approach when 1. The problem is complex. 2. The problem is not familiar. 3. The costs involved are substantial. 4. Enough time is available to analyze the problem. M. En C. Eduardo Bustos Farías 29

30 Advantages of the Quantitative Approach Directs attention to the essence of an analysis: to solve a specific problem. Improves planning which helps prevent future problems Results in more objective decisions than purely qualitative analysis. Incorporates advances in computational technologies to managerial problemsolving. M. En C. Eduardo Bustos Farías 30

31 The Problem Solving Process Identify Problem Formulate & Implement Model Analyze Model Test Results Implement Solution unsatisfactory results M. En C. Eduardo Bustos Farías 31

32 The Management Science Approach M. En C. Eduardo Bustos Farías 32

33 Phases of an OR Study Define the problem, gather relevant data Formulate a mathematical model Develop a method for solving the model (usually computer-based) Test the model solution (validation), revise as necessary Prepare for ongoing applications of the model in decision making Implement M. En C. Eduardo Bustos Farías 33

34 Define the Problem Usually a very difficult process Vague, imprecise concepts of the problem Data does not exist, or in an inappropriate form Determine Objectives Constraints Interrelationships Alternatives Time constraints M. En C. Eduardo Bustos Farías 34

35 Models A Model An abstraction of reality. It is a simplified, and often idealized, representation of reality. Examples : an equation, an outline, a diagram, and a map By its very nature a model is incomplete. Provides an alternative to working with reality M. En C. Eduardo Bustos Farías 35

36 Modeling in OR A model in the sense used in OR is defined as an idealized representation of a real-life situation Real-life system Model M. En C. Eduardo Bustos Farías 36

37 Model Formulation Descriptive Model The objective of a descriptive model is to provide the means for analyzing the behavior of an existing system for the purpose of improving its performance M. En C. Eduardo Bustos Farías 37

38 Model Formulation Prescriptive Model The objective of a prescriptive model is to define the ideal structure of a future system, which includes functional relationships among its components M. En C. Eduardo Bustos Farías 38

39 Model Formulation General Model Classifications Iconic Analog Symbolic, or mathematical M. En C. Eduardo Bustos Farías 39

40 Model Formulation Iconic Models Iconic models represent the system by scaling it up or down, e.g., a toy airplane is an iconic model of a real one M. En C. Eduardo Bustos Farías 40

41 Model Formulation Analog Models Analog models require the substitution of one property for another for the ultimate purpose of achieving convenience in manipulating the model. After the problem is solved, the results are reinterpreted in terms of the original system. M. En C. Eduardo Bustos Farías 41

42 Model Formulation Symbolic, or Mathematical Models Mathematical models employ a set of mathematical symbols to represent the decision variables of the system. These variables are related by appropriate mathematical functions to describe the behavior of the system. The solution of the problem is then obtained by applying welldeveloped mathematical techniques. M. En C. Eduardo Bustos Farías 42

43 Models Symbolic models Use numbers and algebraic symbols Mathematical models Decision variables Uncontrollable variables M. En C. Eduardo Bustos Farías 43

44 Deterministic versus Probabilistic Models Deterministic models Used for problems in which information is known with a high degree of certainty. Used to determine an optimal solution to the problem. Probabilistic models Used when it cannot be determined precisely what values (requiring probabilities) will occur (usually in the future). M. En C. Eduardo Bustos Farías 44

45 Model Formulation Additional OR Model Types Simulation models digital representations which imitate the behavior of a system using a computer Heuristic models some intuitive rules or guidelines are applied to generate new strategies which yield improved solutions to the model M. En C. Eduardo Bustos Farías 45

46 Categories of Mathematical Models Model Independent OR/MS Category Form of f(. ) Variables Techniques Prescriptive known, known or under LP, Networks, IP, well-defined decision maker s CPM, EOQ, NLP, control GP, MOLP Predictive unknown, known or under Regression Analysis, ill-defined decision maker s Time Series Analysis, control Discriminant Analysis Descriptive known, unknown or Simulation, PERT, well-defined uncertain Queueing, Inventory Models M. En C. Eduardo Bustos Farías 46