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Chapter 1 Introduction to Model Building

Chapter 1 Introduction to Model Building. What is Operations Research? What is Management Science?. World War II : British military leaders asked scientists and engineers to analyze several military problems Deployment of radar

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Chapter 1 Introduction to Model Building

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  1. Chapter 1Introduction to Model Building

  2. What is Operations Research? What is Management Science? • World War II : British military leaders asked scientists and engineers to analyze several military problems • Deployment of radar • Management of convoy, bombing, antisubmarine, and mining operations. • The result was called Military Operations Research, later Operations Research • MIT was one of the birthplaces of OR • Professor Morse at MIT was a pioneer in the US • Founded MIT OR Center and helped found ORSA

  3. What is Management Science (Operations Research)? • Today: Operations Research and Management Science mean • “the use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information, or in seeking further information if current knowledge is insufficient to reach a proper decision.” • Decision science, systems analysis, operational research, systems dynamics, operational analysis, engineering systems, systems engineering, and more.

  4. Voices from the past • Waste neither time nor money, but make the best use of both. • Benjamin Franklin • Obviously, the highest type of efficiency is that which can utilize existing material to the best advantage. • Jawaharlal Nehru • It is more probable that the average man could, with no injury to his health, increase his efficiency fifty percent. • Walter Scott

  5. Operations Research Over the Years • 1947 • Project Scoop (Scientific Computation of Optimum Programs) with George Dantzig and others. Developed the simplex method for linear programs. • 1950's • Lots of excitement, mathematical developments, queuing theory, mathematical programming.A.I. in the 1960's • 1960's • More excitement, more development and grand plans. A.I. in the 1980's.

  6. Operations Research Over the Years • 1970's • Disappointment, and a settling down. NP-completeness. More realistic expectations. • 1980's • Widespread availability of personal computers. Increasingly easy access to data. Widespread willingness of managers to use models. • 1990's • Improved use of O.R. systems.Further inroads of O.R. technology, e.g., optimization and simulation add-ins to spreadsheets, modeling languages, large scale optimization. More intermixing of A.I. and O.R.

  7. Operations Research in the 00’s • LOTS of opportunities for OR as a field • Data, data, data • E-business data (click stream, purchases, other transactional data, E-mail and more) • Need for more automated decision making • Need for increased coordination for efficient use of resources (Supply chain management)

  8. Optimization is everywhere • Models, Models, Models • The goal of models is “insight” not numbers • paraphrase of Richard Hamming • Algorithms, Algorithms, Algorithms

  9. Optimization is Everywhere • It is embedded in language, and part of the way we think. • firms want to maximize value to shareholders • people want to make the best choices • We want the highest quality at the lowest price • When playing games, we want the best strategy • When we have too much to do, we want to optimize the use of our time • etc.

  10. Mathematical Optimization is nearly everywhere • Finance • Marketing • E-business • Telecommunications • Games • Operations Management • Production Planning • Transportation Planning • System Design • Look for it! You will see opportunities for its use.

  11. 1.1 - An Introduction to Modeling • Operations Research (management science) is a scientific approach to decision making that seeks to best design and operate a system, usually under conditions requiring the allocation of scarce resources. • A system is an organization of interdependent components that work together to accomplish the goal of the system.

  12. 1.1 - An Introduction to Modeling A Modeling Example Eli Daisy produces the drug Wozac in huge batches by heating a chemical mixture in a pressurized container. Each time a batch is produced, a different amount of Wozac is produced. The amount produced is the process yield (measured in pounds). Daisy is interested in understanding the factors that influence the yield of Wozac production process. The solution on subsequent slides describes a model building process for this situation.

  13. 1.1 - An Introduction to Modeling • Daisy is interested in determining the factors that influence the process yield. This would be referred to as a descriptive model since it describes the behavior of the actual yield as a function of various factors. • Daisy might determine the following factors influence yield: • Container volume in liters (V) • Container pressure in milliliters (P) • Container temperature in degrees centigrade (T) • Chemical composition of the processed mixture

  14. 1.1 - An Introduction to Modeling Letting A, B, and C be the percentage of the mixture made up of chemical A, B, and C, then Daisy might find , for example, that: Yield = 300 + 0.8V +0.01P + 0.06T + 0.001T*P - 0.01T2 – 0.001P2 + 11.7A + 9.4B + 16.4C + 19A*B + 11.4A*C – 9.6B*C

  15. 1.1 - An Introduction to Modeling To determine this relationship, the yield of the process would have to be measured for many different combinations of the previously listed factors. Knowledge of this equation would enable Daisy to describe the yield of the production process once volume, pressure, temperature, and chemical composition were known.

  16. 1.1 - An Introduction to Modeling • Prescriptive or Optimization Models • Prescriptive models “prescribes” behavior for an organization that will enable it to best meet its goals. Components of this model include: • objective function(s) • decision variables • constraints • An optimization model seeks to find values of the decision variables that optimize (maximize or minimize) an objective function among the set of all values for the decision variables that satisfy the given constraints.

  17. 1.1 - An Introduction to Modeling The Objective Function The Daisy example seeks to maximize the yield for the production process. In most models, there will be a function we wish to maximize or minimize. This function is called the model’s objective function. To maximize the process yield we need to find the values of V, P, T, A, B, and C that make the yield equation (below) as large as possible. Yield = 300 + 0.8V +0.01P + 0.06T + 0.001T*P - 0.01T2 – 0.001P2 + 11.7A + 9.4B + 16.4C + 19A*B + 11.4A*C – 9.6B*C

  18. 1.1 - An Introduction to Modeling • In many situations, an organization may have more than one objective. • For example, in assigning students to the two high schools in Bloomington, Indiana, the Monroe County School Board stated that the assignment of students involve the following objectives: • equalize the number of students at the two high schools • minimize the average distance students travel to school • have a diverse student body at both high schools

  19. 1.1 - An Introduction to Modeling The Decision Variables Variables whose values are under our control and influence system performance are called decision variables. In the Daisy example, V, P, T, A, B, and C are decision variables. Constraints In most situations, only certain values of the decision variables are possible. For example, certain volume, pressure, and temperature conditions might be unsafe. Also, A, B, and C must be nonnegative numbers that sum to one. These restrictions on the decision variable values are called constraints.

  20. 1.1 - An Introduction to Modeling • Suppose the Daisy example has the following constraints: • Volume must be between 1 and 5 liters • Pressure must be between 200 and 400 milliliters • Temperature must be between 100 and 200 degrees centigrade • Mixture must be made up entirely of A, B, and C • For the drug to perform properly, only half the mixture at most can be product A.

  21. 1.1 - An Introduction to Modeling • Mathematically, these constraints can be expressed: V ≤ 5 V ≥ 1 P ≤ 400 P ≥ 200 T ≤ 200 T ≥ 100 A ≥ 0 B ≥ 0 C ≥ 0 A + B + C = 1.0 A ≤ 0.5

  22. 1.1 - An Introduction to Modeling The Complete Daisy Optimization Model Letting z represent the value of the objection function (the yield), the entire optimization model may be written as: maximize z = 300 + 0.8V +0.01P + 0.06T + 0.001T*P - 0.01T2 – 0.001P2 + 11.7A + 9.4B + 16.4C + 19A*B + 11.4A*C – 9.6B*C Subject to (s.t.) V ≤ 5 V ≥ 1 P ≤ 400 P ≥ 200 T ≤ 200 T ≥ 100 A + B + C = 1.0 A ≤ 0.5 A ≥ 0 B ≥ 0 C ≥ 0

  23. 1.1 - An Introduction to Modeling Any specification of the decision variables that satisfies all the model’s constraints is said to be in the feasible region. For example, V = 2, P = 300, T = 150, A = 0.4, B = 0.3 and C = 0.3 is in the feasible region. An optimal solution to an optimization model any point in the feasible region that optimizes (in this case maximizes) the objective function. Using LINGO, it can be determined that the optimal solution to its model is V = 5, P = 200, T = 100, A = 0.294, B = 0, C = 0.706, and z = 209.384 as shown on the next slides.

  24. 1.1 - An Introduction to Modeling Daisy problem formulation in LINGO 7.0

  25. 1.1 - An Introduction to Modeling Solution extract of Daisy example (shown without slack, surplus, or dual prices) using LINGO 7.0:

  26. 1.1 - An Introduction to Modeling Static and Dynamic Models A static model is one in which the decision variables do not involve sequences of decisions over multiple periods. A dynamic model is a model in which the decision variables do involve sequences of decisions over multiple periods. In a static model, we solve a one shot problem whose solutions are optimal values of the decision variables at all points in time. The Daisy problem is an example of a static model.

  27. 1.1 - An Introduction to Modeling Static and Dynamic Models For a dynamic model, consider a company (SailCo) that must determine how to minimize the cost of meeting (on-time) the demand for sail boats it produces during the next year. SailCo must determine the number of sail boats to produce during each of the next four quarters. SailCo’s decisions must be made over multiple periods and thus posses a dynamic model.

  28. 1.1 - An Introduction to Modeling Linear and Nonlinear models Suppose that when ever decision variables appear in the objective function and in the constraints of an optimization model the decision variables are always multiplied by constants and then added together. Such a model is a linear model. The Daisy example is a nonlinear model. While the decision variables in the constraints are linear, the objective function is nonlinear since the objective function terms: 0.001T*P, - 0.01T2, – 0.001P2, 19A*B, 11.4A*C, and – 9.6B*C are nonlinear. In general, nonlinear models are much harder to solve.

  29. 1.1 - An Introduction to Modeling Integer and Noninteger Models If one or more of the decision variables must be integer, then we say that an optimization model is an integer model. If all the decision variables are free to assume fractional values, then an optimization model is a noninteger model. The Daisy example is a noninteger example since volume, pressure, temperature, and percentage composition are all decision variables which may assume fractional values. If decision variables in a model represent the number of workers starting during each shift, then clearly we have a integer model. Integer models are much harder to solve then noninteger models.

  30. 1.1 - An Introduction to Modeling Deterministic and Stochastic Models Suppose that for any value of the decision variables the value of the objective function and whether or not the constraints are satisfied is known with certainty. We then have a deterministic model. If this is not the case, then we have a stochastic model. If we view the Daisy example as a deterministic model, then we are making the assumption that for given values of V, P, T, A, B, and C the process yield will always be the same. Since this is unlikely, the objective function can be viewed as the average yield of the process for given decision variable values.

  31. 1.2 – The Seven-Step Model-Building Process Operations research used to solve an organization’s problem follows a seven-step model building procedure: • 1. Formulate the Problem • Define the problem. • Specify objectives. • Determine parts of the organization to be studied. • 2. Observe the System • Determines parameters affecting the problem. • Collect data to estimate values of the parameters.

  32. 1.2 – The Seven-Step Model-Building Process • 3. Formulate a Mathematical Model of the Problem • 4. Verify the Model and Use the Model for Prediction • Does the model yield results for values of decision variables not used to develop the model? • What eventualities might cause the model to become invalid? • 5. Select a Suitable Alternative • Given a model and a set of alternative solutions, determine which solution best meets the organizations objectives.

  33. 1.2 – The Seven-Step Model-Building Process • 6. Present the Results and Conclusion(s) of the Study to the Organization • Present the results to the decision maker(s) • If necessary, prepare several alternative solutions and permit the organization to chose the one that best meets their needs. • Any non-approval of the study’s recommendations may have stemmed from an incorrect problem definition or failure to involve the decision maker(s) from the start of the project. In such a case, return to step 1, 2, or 3.

  34. 1.2 – The Seven-Step Model-Building Process • 7. Implement and Evaluate Recommendations • Upon acceptance of the study by the organization, the analyst: • Assists in implementing the recommendations. • Monitors and dynamically updates the system as the environment and parameters change to ensure that recommendations enable the organization to meet its goals.

  35. Linear Programming (our first tool, and probably the most important one.) • minimize or maximize a linear objective • subject to linear equalities and inequalities • maximize 3x + 4y • subject to 5x + 8y24 • x, y  0 A feasible solution satisfies all of the constraints. x = 1, y = 1 is feasible; x = 1, y = 3 is infeasible. An optimal solution is the best feasible solution. The optimal solution is x = 4.8, y = 0.

  36. Terminology • Decision variables: e.g., x and y. • In general, there are quantities you can control to improve your objective which should completely describe the set of decisions to be made. • Constraints:e.g.,5x + 8y24 , x  0 , y  0 • Limitations on the values of the decision variables. • Objective Function. e.g., 3x + 4y • Value measure used to rank alternatives • Seek to maximize or minimize this objective • examples: maximize NPV, minimize cost

  37. Introduction to Linear Programming • Diet Problem: A decision maker wants to minimize the cost of meeting a set of requirements • Available foods for consumption: brownies, chocolate ice cream, cola, and pineapple cheese cake • Costs: 50 cents/brownie, 20 cents/scoop of ice cream, 30 cents/bottle of cola, 80 cents/piece of pineapple cheesecake • Daily Requirements: at least 500 calories, 6 oz of chocolate, 10 oz of sugar, and 8 oz of fat

  38. Introduction to Linear Programming • Diet Problem(Continued) • Nutritional Values • Type of calories chocolate sugar Fat • Food (unit) (oz) (oz) (oz) • Brownie 400 3 2 2 • Chocolate 200 2 2 4 • ice cream • Cola 150 0 4 1 • Pineapple 500 0 4 5 • Cheese cake

  39. Introduction to Linear Programming • Diet Problem (Continued) • Decision Variables: • x1 = number of brownies eaten daily • x2 = number of scoops of ice cream eaten daily • x3 = bottles of cola drunk daily • x4 = pieces of pineapple cheese cake eaten daily • Total cost of diet = 50x1 + 20x2 + 30x3 + 80x4

  40. Introduction to Linear Programming • Diet Problem (Continued) • Constraint 1: Daily calories intake must be at least 500 calories • 400x1 + 200x2 + 150x3 + 500x4  500 • Constraint 2: Daily chocolate intake must be at least 6 oz • 3x1 + 2x2  6 • Constraint 3: Daily sugar intake must be at least 10 oz • 2x1 + 2x2 + 4 x3 + 4x4  10 • Constraint 4: Daily fat intake must be at least 8 oz • 2x1 + 4x2 + x3 + 5x4  8 • Non-negativity constraints • xi  0 for i=1, 2, 3, 4

  41. Some Success Stories • Optimal crew scheduling saves American Airlines $20 million/yr. • Improved shipment routing saves Yellow Freight over $17.3 million/yr. • Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 million/yr. • GTE local capacity expansion saves $30 million/yr.

  42. Other Success Stories (cont.) • Optimizing global supply chains saves Digital Equipment over $300 million. • Restructuring North America Operations, Proctor and Gamble reduces plants by 20%, saving $200 million/yr. • Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hours/yr. • Better scheduling of hydro and thermal generating units saves southern company $140 million.

  43. Success Stories (cont.) • Improved production planning at Sadia (Brazil) saves $50 million over three years. • Production Optimization at Harris Corporation improves on-time deliveries from 75% to 90%. • Tata Steel (India) optimizes response to power shortage contributing $73 million. • Optimizing police patrol officer scheduling saves police department $11 million/yr. • Gasoline blending at Texaco results in saving of over $30 million/yr.

  44. Scheduling Postal Workers • Each postal worker works for 5 consecutive days, followed by 2 days off, repeated weekly. Minimize the number of postal workers (for the time being, we will permit fractional workers on each day.)

  45. Formulating as an LP • Select the decision variables • Let x1 be the number of workers who start working on Monday, and work till Friday • Let x2 be the number of workers who start on Tuesday … • Let x3, x4, …, x7 be defined similarly.

  46. The linear program Minimize z = x1 + x2 + x3 + x4 + x5 + x6 + x7 x1 + x4 + x5 + x6 + x7  17 subject to x1 + x2 + x5 + x6 + x7  13 x1 + x2 + x3 + x6 + x7  15 x1 + x2 + x3 + x4 + x7  19 x1 + x2 + x3 + x4 + x5 14 x2 + x3 + x4 + x5 + x6 16 x3 + x4 + x5 + x6 + x7  11 xj 0 for j = 1 to 7

  47. On the selection of decision variables • Would it be possible to have yj be the number of workers on day j? • Workers on day j is at least dj. • Each worker works 5 days on followed by 2 days off. • Conclusion: sometimes the decision variables incorporate constraints of the problem. • Hard to do this well, but worth keeping in mind • We will see more of this in integer programming.

  48. Some Enhancements of the Model • Suppose that there was a pay differential. The cost of workers who start work on day j is cj per worker. Minimize z = c1 x1 + c2 x2 + c3 x3 + … + c7 x7

  49. Overview • 5 in 7 scheduling problem • The model • Practical enhancements or modifications

  50. Some Enhancements of the Model • Suppose that one can hire part time workers (one day at a time), and that the cost of a part time worker on day j is PTj. • Let yj = number of part time workers on day j

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