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

Transcription

1 Overview of OR Modeling Approach & Introduction to Linear Programming Métodos Cuantitativos 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 2

3 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 3

4 Introduction to Linear Programming History of OR M. En C. Eduardo Bustos Farías Métodos Cuantitativos M. En C. Eduardo Bustos Farías 4

5 Jordan, Minkowsky y Farkas worked about linnear models Métodos Cuantitativos M. En C. Eduardo Bustos Farías 5

6 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 6

7 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 7

8 Origins of OR Management of WW II by England A team of scientists Worked on strategic and tactical problems Land and air defenses Métodos Cuantitativos M. En C. Eduardo Bustos Farías 8

9 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 9

10 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 10

11 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 11

12 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 12

13 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 13

14 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 14

15 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 15

16 Decisions Most effective use of limited military resources Deployment of radar defenses Maximize bomber effectiveness Métodos Cuantitativos M. En C. Eduardo Bustos Farías 16

17 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 17

18 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 18

19 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 19

20 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 20

21 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 21

22 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 22

23 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 23

24 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 24

25 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 25

26 The Problem Solving Process Identify Problem Formulate & Implement Model Analyze Model Test Results Implement Solution unsatisfactory results Métodos Cuantitativos M. En C. Eduardo Bustos Farías 26

27 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 27

28 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 28

29 Model Formulation 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 29

30 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 30

31 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 31

32 Model Formulation General Model Classifications Iconic Analog Symbolic, or mathematical Métodos Cuantitativos M. En C. Eduardo Bustos Farías 32

33 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 33

34 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 34

35 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. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 35

36 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 36

37 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 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 37

38 Structure Math models have three basic sets of elements: Decision variables and parameters Constraints or restrictions Objective function(s) Métodos Cuantitativos M. En C. Eduardo Bustos Farías 38

39 Decision variables and parameters Decision variables are the unknowns which are to be determined from the solution of the model Parameters are the known decision variable coefficients and/or resource availability Métodos Cuantitativos M. En C. Eduardo Bustos Farías 39

40 Constraints or restrictions To account for the physical limitations of the system, the model must include constraints which limit the decision variables to their feasible (or permissible) values Métodos Cuantitativos M. En C. Eduardo Bustos Farías 40

41 Objective function This defines the measure of effectiveness of the system as a mathematical function of its decision variables The optimum solution to the model has been obtained if the corresponding values of the decision variables yield the best value of the objective function while satisfying all the constraints Métodos Cuantitativos M. En C. Eduardo Bustos Farías 41

42 Find the values of the decision variables x j, j =1,2,..., n which will optimize: z = f subject to : g i ( x 1 ( x, x 1 2,..., and x, x 2,..., x n j x n 0 for i = 1,2,...,m for j = 1,2,...,n Métodos Cuantitativos M. En C. Eduardo Bustos Farías 42 ) ) b i

43 The objective function and constraints in math model may take on many forms, depending upon the system being modeled Functions may be linear or non-linear, the decision variables may be continuous or discrete, and parameters may be deterministic or stochastic. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 43

44 Solution of the Model MATHEMATICAL PROGRAMMING Linear Programming Nonlinear programming Integer Programming Stochastic Programming Dynamic Programming Métodos Cuantitativos M. En C. Eduardo Bustos Farías 44

45 Solution of the Model The Optimization Process The optimal solution to a mathematical model cannot generally be obtained in one step Rather it requires: An initial solution A set of computational rules (an algorithm) Iteration of the algorithm to reach an optimal, feasible solution (provided one exists) Métodos Cuantitativos M. En C. Eduardo Bustos Farías 45

46 Validation of the Model Testing the Model Solution Thoroughly check the model structure, assumptions and process Compare model solutions with historical data from the existing system being modeled Compare model solutions with forecasts for planned new systems Correct any errors in the formulation and re-solve the model Métodos Cuantitativos M. En C. Eduardo Bustos Farías 46

47 Validation of the Model Model Solution Reliability The reliability of the solution obtained from the model depends on the validity of this model in representing the real system, i.e., how well the resulting solution actually applies to the assumed real system represented by the model (accuracy) Real-life system Solution Model Métodos Cuantitativos M. En C. Eduardo Bustos Farías 47

48 Application of the Model Preparing to Apply the Model Document the system for applying the model for use in decision making Solution procedure Operating procedures Database and MIS interfaces Interaction with other models for decision support Métodos Cuantitativos M. En C. Eduardo Bustos Farías 48

49 Implementation of the Model Implementation Garner support of top management, operating management, and the OR team Explain the new system and its relationship to operating realities Indoctrinate user personnel Monitor initial experiences and correct problems discovered in usage Maintain currency and accuracy of documentation Métodos Cuantitativos M. En C. Eduardo Bustos Farías 49

50 Introduction to Mathematical Programming We all face decision about how to use limited resources such as: Oil in the earth Land for dumps Time Money Workers Métodos Cuantitativos M. En C. Eduardo Bustos Farías 50

51 Mathematical Programming... MP is a field of management science that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual of a business. a.k.a. Optimization Métodos Cuantitativos M. En C. Eduardo Bustos Farías 51

52 Characteristics of Optimization Problems Decisions Constraints Objectives Métodos Cuantitativos M. En C. Eduardo Bustos Farías 52

53 Solution of the Model MATHEMATICAL PROGRAMMING Linear Programming Nonlinear programming Integer Programming Stochastic Programming Dynamic Programming Métodos Cuantitativos M. En C. Eduardo Bustos Farías 53

54 Linear Programming Linear means that all the mathematical functions in the model are required to be linear functions Programming is essentially a synonym for planning Thus, linear programming means planning using a mathematical model containing only linear functions Métodos Cuantitativos M. En C. Eduardo Bustos Farías 54

55 Concern of LP LP deals with the problem of allocating limited resources among competing activities in an optimal manner Métodos Cuantitativos M. En C. Eduardo Bustos Farías 55

56 General Approach Formulating and solving an LP model requires: Optimizing (maximizing or minimizing) a linear function of variables called the objective function Subject to a set of linear equalities and/or inequalities called constraints Métodos Cuantitativos M. En C. Eduardo Bustos Farías 56

57 Mathematical Expression of an Max (or Min) Subject to: LP Model Z = c x + c x c x a x + a x a x (, =, ) b n n 1 a x + a x a x (, =, ) b n n 2 a x + a x a x (, =, ) b m1 1 m2 2 mn n m and x 0, j = 1, 2,..., n j Métodos Cuantitativos M. En C. Eduardo Bustos Farías 57 n n Objecive Function Constraints

58 Parameters and Variables In the preceding formulation, c j,b i, and a ij, for (i=1,2,...,m; j=1,2,...,n) are constants which are determined depending on the technology of the problem the x j s are the decision variables only one of the signs (<, =, >) holds for each constraint Métodos Cuantitativos M. En C. Eduardo Bustos Farías 58

59 Canonical Form of an LP Model Max (or Min) Z = S. T. n j= 1 ax x j ij j = b i j= 1, i = 1,2,...,m 0, j = 1,2,...,n n c x j j Métodos Cuantitativos M. En C. Eduardo Bustos Farías 59

60 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 60 Matrix Form of LP Model Max Z cx st Ax b x =.. 0 c c c c n =[,,..., ] 1 2 x x x x n = 1 2 b b b b m = 1 2 A a a a a a a a a a n n m m mn = =

61 Observations c j is the increase or decrease in the overall measure of effectiveness (Z) that results from each unit increase or decrease in x j The number of relevant scarce resources is m, so that each of the first m linear inequalities corresponds to a constraint on the availability of one of these resources Métodos Cuantitativos M. En C. Eduardo Bustos Farías 61

62 Observations b i is the amount of resource i available to the n activities a ij is the amount of resource i consumed by each unit of activity j The left side of the constraint inequalities is the total usage of the respective resource The non-negativity constraints rule out the possibility of negative activity levels Métodos Cuantitativos M. En C. Eduardo Bustos Farías 62

63 Formulation The key to successful application of linear programming is the ability to recognize when a problem can be solved by linear programming, and to formulate the corresponding (and appropriate) model Métodos Cuantitativos M. En C. Eduardo Bustos Farías 63

64 An Example LP Problem Blue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes. Aqua-Spa Hydro-Lux Pumps 1 1 Labor 9 hours 6 hours Tubing 12 feet 16 feet Unit Profit $350 $300 There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 64

65 5 Steps In Formulating LP Models: 1. Understand the problem. 2. Identify the decision variables. X 1 =number of Aqua-Spas to produce X 2 =number of Hydro-Luxes to produce 3. State the objective function as a linear combination of the decision variables. MAX: 350X X 2 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 65

66 5 Steps In Formulating LP Models (continued) 4. State the constraints as linear combinations of the decision variables. 1X 1 + 1X 2 <= 200 } pumps 9X 1 + 6X 2 <= X X 2 <= 2880 } labor } tubing 5. Identify any upper or lower bounds on the decision variables. X 1 >= 0 X 2 >= 0 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 66

67 LP Model for Blue Ridge Hot Tubs MAX: 350X X 2 S.T.: 1X 1 + 1X 2 <= 200 9X 1 + 6X 2 <= X X 2 <= 2880 X 1 >= 0 X 2 >= 0 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 67

68 LP Formulation (Blending Problem) The manager of an oil refinery must decide on the optimal mix of two possible blending processes, of which the inputs and outputs per production run of each process are as follows: Input Output Process Crude A Crude B Gasoline X Gasoline Y Métodos Cuantitativos M. En C. Eduardo Bustos Farías 68

69 LP Formulation (Blending Problem-continued) The maximum amounts available of crudes A and B are 200 units and 150 units, respectively. Market requirements show that at least 100 units of gasoline X and 80 units of gasoline Y must be produced. The profits per production run from process 1 and process 2 are 3 and 4, respectively Métodos Cuantitativos M. En C. Eduardo Bustos Farías 69

70 Solution Métodos Cuantitativos M. En C. Eduardo Bustos Farías 70

71 Identify the decision variables x1 = number of production runs of process 1 to be made x2 = number of production runs of process 2 to be made Métodos Cuantitativos M. En C. Eduardo Bustos Farías 71

72 State the objective function as a linear combination of the decision Max Z = 3x + 4x 1 2 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 72

73 State the constraints as linear combinations of the decision variables 5x + 4x x + 5x x + 4x x + 4x crude A crude B gasoline X gasoline Y Métodos Cuantitativos M. En C. Eduardo Bustos Farías 73

74 Identify any upper or lower bounds on the decision variables x 0, x Métodos Cuantitativos M. En C. Eduardo Bustos Farías 74

75 Blending Problem Formulation Max s.t. Z= 3x + 4x 5x + 4x 200 3x + 5x 150 5x + 4x 100 8x + 4x 80 x , x Métodos Cuantitativos M. En C. Eduardo Bustos Farías 75

76 LP Formulation (Diet Problem) An individual wants to decide on the constituents of a diet which will satisfy his daily needs of proteins, fats, and carbohydrates at the minimum cost. Choices from 5 different types of foods can be made. The yields per unit of these foods are given in the following table: Métodos Cuantitativos M. En C. Eduardo Bustos Farías 76

77 Yields Per Unit Food Type Proteins Fats Carbohydrates Cost/Unit 1 p 1 f 1 c 1 d 1 2 p 2 f 2 c 2 d 2 3 p 3 f 3 c 3 d 3 4 p 4 f 4 c 4 d 4 5 p 5 f 5 c 5 d 5 Min. Daily Req mt. P F C Métodos Cuantitativos M. En C. Eduardo Bustos Farías 77

78 Solution Métodos Cuantitativos M. En C. Eduardo Bustos Farías 78

79 Identify the decision variables The decision variables x1, x2, x3, x4, x5 represents the number of units used of the first, second, third, fourth and fifth type of food, respectively Métodos Cuantitativos M. En C. Eduardo Bustos Farías 79

80 State the objective function as a linear combination of the decision Min Z d x d x d x d x d x = Métodos Cuantitativos M. En C. Eduardo Bustos Farías 80

81 State the constraints as linear combinations of the decision variables px+ px + px + px + px P fx+ fx+ fx+ fx+ fx F cx + cx + cx + cx + cx C Métodos Cuantitativos M. En C. Eduardo Bustos Farías 81

82 Identify any upper or lower bounds on the decision variables x, x, x, x, x Métodos Cuantitativos M. En C. Eduardo Bustos Farías 82

83 Diet Problem Formulation Min Z = d1x1+ d2x2 + d3x3+ d4x4 + d5x5 s.t. px+ px + px + px + px P fx+ fx+ fx+ fx+ fx F cx+ cx + cx + cx + cx C x1, x2, x3, x4, x5 0 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 83

84 Limitations or suppositions of LP Métodos Cuantitativos M. En C. Eduardo Bustos Farías 84

85 Limitations of LP Proportionality or Linearity The objective function and the constraints are linear expressions of the decision variables. This means that the contribution of each activity is directly proportional to the level of the activity Métodos Cuantitativos M. En C. Eduardo Bustos Farías 85

86 Limitations of LP Additivity The total measure of effectiveness (objective function) and each total resource usage (constraint) resulting from the joint performance of the activities must equal the respective sums of these quantities resulting from each activity being conducted individually Métodos Cuantitativos M. En C. Eduardo Bustos Farías 86

87 Limitations of LP Divisibility Fractional levels of the decision variables must be permitted Métodos Cuantitativos M. En C. Eduardo Bustos Farías 87

88 Limitations of LP Deterministic All of the coefficients in the LP model (,, )are assumed to be known constants a b c ij i j Métodos Cuantitativos M. En C. Eduardo Bustos Farías 88

89 Economic Significance of Linearity The simplifying assumption of linearity causes some problems: 1. In profit-maximizing production problem linearity of the objective function implies constant profit rate per unit as output increases; this means a) that selling price is constant; and Métodos Cuantitativos M. En C. Eduardo Bustos Farías 89

90 b) that average variable cost is constant; the law of diminishing returns does not influence the production process, in other words it implies constant input prices perfect competition in output and input markets 2. Linear resource constraints imply constant combination of inputs; this means constant returns to scale. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 90

91 Summary of the Economic Implications of the Linearity Assumption Linear objective function Constant gross profit per unit (GP = P - AVC = constant) Price (P) is constant Constant returns to variable inputs Constant input prices Firm is a price taker in the output market Métodos Cuantitativos M. En C. Eduardo Bustos Farías 91

92 LP Formulation (Product Mix Problem) Suppose we must decide on the number of units to be manufactured of two different products. The profits per unit of product 1 and product 2 are 2 and 5, respectively. Each unit of product 1 requires 3 machine hours and 9 units of raw material. Each unit of product 2 requires 4 machine hours and 7 units of raw material. The maximum available machine hours and raw material units are 200 and 300, respectively. A minimum of 20 units is required of product 1. Métodos Cuantitativos M. En C. Eduardo Bustos Farías 92

93 Solution Métodos Cuantitativos M. En C. Eduardo Bustos Farías 93

94 Identify the decision variables x1 = number of units of product 1 to be produced x2 = number of units of product 2 to be produced Métodos Cuantitativos M. En C. Eduardo Bustos Farías 94

95 State the objective function as a linear combination of the decision variables Max Z = 2x + 5x 1 2 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 95

96 State the objective function as a linear combination of the decision variables Max Z = 2x + 5x 1 2 Proportionality This means that the contribution of each activity is directly proportional to the level of the activity Métodos Cuantitativos M. En C. Eduardo Bustos Farías 96

97 State the objective function as a linear combination of the decision variables Max Z = 2x + 5x 1 2 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 97

98 State the objective function as a linear combination of the decision variables Max Z = 2x + 5x 1 2 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 98

99 State the constraints as linear combinations of the decision variables 3x + 4x 1 2 9x + 7x machine hours raw material x 1 20 A minimum of 20 units is required of product 1 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 99

100 State the constraints as linear combinations of the decision variables 3x + 4x 1 2 9x + 7x x 1 20 Proportionality. This means that the contribution of each activity is directly proportional to the level of the activity Métodos Cuantitativos M. En C. Eduardo Bustos Farías 100

101 State the constraints as linear combinations of the decision variables 3x + 4x 1 2 9x + 7x x 1 20 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 101

102 State the constraints as linear combinations of the decision variables 3x + 4x 1 2 9x + 7x x 1 20 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 102

103 Identify any upper or lower bounds on the decision variables x1> 0 x 0 2 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 103

104 LP Model Product Mix Formulation Max Z= 2x + 5x S.T. 3x + 4x 200 9x + 7x x x1> 0, x 2 0 Métodos Cuantitativos M. En C. Eduardo Bustos Farías 104

105 Problems Métodos Cuantitativos M. En C. Eduardo Bustos Farías 105

106 EXAMPLE: An Investment Problem: Retirement Planning Services, Inc. A client wishes to invest $750,000 in the following bonds. Years to Company Return Maturity Rating Acme Chemical 8.65% 11 1-Excellent DynaStar 9.50% 10 3-Good Eagle Vision 10.00% 6 4-Fair Micro Modeling 8.75% 10 1-Excellent OptiPro 9.25% 7 3-Good Sabre Systems 9.00% 13 2-Very Good Métodos Cuantitativos M. En C. Eduardo Bustos Farías 106

107 Investment Restrictions No more than 25% can be invested in any single company. At least 50% should be invested in longterm bonds (maturing in 10+ years). No more than 35% can be invested in DynaStar, Eagle Vision, and OptiPro. FORMULATE THE LP MODEL Métodos Cuantitativos M. En C. Eduardo Bustos Farías 107

108 EXAMPLE:A Blending Problem:The Agri-Pro Company Agri-Pro has received an order for 8,000 pounds of chicken feed to be mixed from the following feeds. Percent of Nutrient in Nutrient Feed 1 Feed 2 Feed 3 Feed 4 Corn 30% 5% 20% 10% Grain 10% 3% 15% 10% Minerals 20% 20% 20% 30% Cost per pound $0.25 $0.30 $0.32 $0.15 The order must contain at least 20% corn, 15% grain, and 15% minerals. FORMULATE THE LP MODEL Métodos Cuantitativos M. En C. Eduardo Bustos Farías 108

109 LP Formulation -Blending Problem The manager of an oil refinery must decide on the optimal mix of two possible blending processes, of which the inputs and outputs per production run of each process are as follows: Input Output Process Crude A Crude B Gasoline X Gasoline Y The maximum amounts available of crudes A and B are 200 units and 150 units, respectively. Market requirements show that at least 100 units of gasoline X and 80 units of gasoline Y must be produced. The profits per production run from process 1 and process 2 are 3 and 4, respectively Métodos Cuantitativos M. En C. Eduardo Bustos Farías 109 FORMULATE THE LP MODEL