Introduction to Operations Research Prof. Fernando Augusto Silva Marins feg.unesp.br/~fmarins

1. Introduction to Operations Research Prof. Fernando Augusto Silva Marins www.feg.unesp.br/~fmarins fmarins@feg.unesp.br

2. What Is Management Science (Operations Research, Operational Research ou ainda Pesquisa Operacional)? • Management Science is the discipline that adapts the scientific approach for problem solving to help managers make informed decisions. • The goal of management science is to recommendthe course of action that is expected to yield the best outcome with what is available.

3. What Is Management Science? • The basic steps in the management science problem solving process involves • Analyzing business situations (problem identification) • Building mathematical models to describe them • Solving the mathematical models • Communicating/implementing recommendations based on the models and their solutions (reports)

4. The Management Science Process Problem definition • The four-step management science process Mathematical modeling Solution of the model Communication/implementationof results

5. The Management Science Process • Management Science is a discipline that adopts the scientific method to provide management with key information needed inmaking informed decisions. • The team concept calls for the formation of (consulting) teams consisting of members who come from various areas of expertise.

6. The Management Science Approach • Logic and common sense are basic components in supporting the decision making process. • The use of techniques such as: • Statistical inference • Mathematical programming • Probabilistic models • Network and computer science • Simulation

7. Using Spreadsheets in Management Science Models • Spreadsheets have become a powerful tool in management science modeling. • Several reasons for the popularity of spreadsheets: • Data are submitted to the modeler in spreadsheets • Data can be analyzed easily using statistical (Data Analysis Statistical Package) and mathematical tools (Solver Optimization Package) readily available in the spreadsheet. • Data and information can easily be displayed using graphical tools.

8. Classification of Mathematical Models • Classification by the model purpose • Optimization models • Prediction models • Classification by the degree of certainty of the data in the model • Deterministic models (Mathematical Programming) • Probabilistic (stochastic) models (Simulation)

9. Examples of Management Science Applications • Linear Programming was used by Burger Kingto find how to best blend cuts of meat to minimize costs. • Integer Linear Programming model was used byAmerican Air Linesto determine an optimal flight schedule. • The Shortest Route Algorithm was implemented by the Sony Corporationtodeveloped an onboard car navigation system.

10. Examples of Management Science Applications • Project Scheduling Techniqueswere used by a contractor to rebuild Interstate 10 damaged in the 1994 earthquake in the Los Angeles area. • Decision Analysis approach was the basis for the development of a comprehensive framework for planning environmental policy in Finland. • Queuing models are incorporated into the overall design plans for Disneyland and Disney World, which lead to the development of ‘waiting line entertainment’ in order to improve customer satisfaction.

11. Is Operations Research really important? INFORMS 2007

12. Sucessos da Pesquisa Operacional em Logística 61 trabalhos = 42%

13. Edelman: métodos empregados Todos finalistas Somente logística Simulação estocástica discreta é popular na indústria...

14. FINALISTAS EDELMAN 1984-2007

15. FINALISTAS EDELMAN 1984-2007

16. Optimization Models • Many managerial decision situations lend themselves to quantitative analyses. • A Mathematical Model consists of • Objective function with one or more Control /Decision Variables to be optimised. • Constraints (Functional constraints “£”, “³”, “=” restrictions that involve expressions with one or more Control /Decision Variables)

17. The Galaxy Industries Production Problem • Galaxy manufactures two toy doll models: • Space Ray. • Zapper. • Resources are limited to • 1000 pounds of special plastic. • 40 hours of production time per week.

18. Technological input • Space Rays uses 2 of plastic and 3 min of labor • Zappers uses 1 of plastic and 4 min of labor Galaxy Industries Production Problem Marketing requirement • Total production cannot exceed 700 dozens. • Number of dozens of Space Rays cannot exceed number of dozens of Zappers by more than 350.

19. 8(450) + 5(100) The Galaxy Industries Production Problem • The current production plan calls for: • Producing as much as possible of the more profitable product, Space Ray ($8 profit per dozen). • Use resources left over to produce Zappers ($5 profit per dozen), while remaining within the marketing guidelines. • The current production plan consists of: • Space Rays = 450 dozen • Zapper = 100 dozen • Profit = $4,100 per week

20. Management is seeking a production schedule that will increase the company’s profit.

21. A Linear Programming model can provide an insight and an intelligent solution to this problem.

22. Defining Control/Decision Variables • Ask, “Does the decision maker have the authority to decide the numerical value (amount) of the item?” • If the answer “yes” it is a control/decision variable. • By very precise in the units (and if appropriate, the time frame) of each decision variable.

23. The Galaxy Linear Programming Model • Decisions variables: • X1 = Weekly production level of Space Rays • X2 = Weekly production level of Zappers (in dozens)

24. Objective Function • The objective of all optimization models, is to figure out how to do the best you can with what you’ve got. • “The best you can” implies maximizing something (profit, efficiency...) or minimizing something (cost, time...).

25. The Galaxy Linear Programming Model Decisions variables: X1 = Weekly production of Space Rays, X2 = Weekly production of Zappers • Objective Function: Space Ray- $8/dozen Zappers $5/dozen • Weekly profit, to be maximized Max 8X1 + 5X2

26. Writing Constraints • Create a limiting condition in words in the following manner:(The amount of a resource required) (Has some relation to) (The availability of the resource) • Make sure the units on the left side of the relation are the same as those on the right side. • Translate the words into mathematical notation using known or estimated values for the parameters and the previously defined symbols for the decision variables.

27. Decisions variables X1 = Space Rays, X2 = Zappers Writing Constraints Space Rays uses 2 of plastic and 3 min of labor Zappers uses 1 of plastic and 4 min of labor 3X1 + 4X2£ 2400 (Prod Time - Min) 2X1 + 1X2£ 1000 (Plastic) X1 + X2£ 700 (Total production) There is 1000 of special plastic and 40 hours (2,400 min) of production time/week. Total production  700, Number Space Rays cannot exceed number of dozens of Zappers by more than 350, X1 - X2£ 350 (Mix)

28. Writing Constraints Nonnegativity constraint - X ³ 0 Lower bound constraint - X ³ L Upper bound constraint - X £ U Integer constraint - X = integer Binary constraint - X = 0 or 1 • Additional constraints

29. The Galaxy Linear Programming Model Nonnegativity constraint Lower bound constraint - Upper bound constraint - Integer constraint Binary constraint Max 8X1 + 5X2 (Weekly profit) subject to (the constraints) 2X1 + 1X2£ 1000 (Plastic) 3X1 + 4X2£ 2400 (Production Time - Min) X1 + X2£ 700 (Total production) X1 - X2£ 350 (Mix) Is there Additional Constraints? Xj  0, j = 1,2 (Nonnegativity) Integers??

30. The Graphical Analysis of Linear Programming The set of all points that satisfy all the constraints of the model is called a FEASIBLE REGION

31. Using a graphical presentation we can represent: • All the constraints • The objective function • The three types of feasible points.

32. Graphical Analysis – the Feasible Region The non-negativity constraints X2 X1

33. The Plastic constraint 2X1+X2 £ 1000 Graphical Analysis – the Feasible Region X2 1000 700 Total production constraint: X1+X2 £700 (redundant) 500 Infeasible Feasible Production Time 3X1+4X2 £2400 X1 500 700

34. Graphical Analysis – the Feasible Region X2 The Plastic constraint 2X1+X2 £ 1000 1000 700 Total production constraint: X1+X2 £ 700 (redundant) 500 Infeasible Production mix constraint: X1-X2 £350 Feasible Production Time 3X1+4X2£ 2400 X1 500 700 • Boundary points. Extreme points (5 Vertices). • Interior points. • There are three types of feasible points

35. Max 8X1 + 5X2 The search for an optimal solution Start at some arbitrary profit, say profit = $2,000... Then increase the profit, if possible... X2 ...and continue until it becomes infeasible 1000 Optimal Profit =$4,360 and optimal solution: 700 • Space Rays = 320 dozen • Zappers = 360 dozen 600 8X1 + 5X2 = 3,000 Current solution: Space Rays = 450, Zapper = 100 and Profit = $4,100 8X1 + 5X2 = 2,000 400 X1 250 500

37. Simulation 37

38. Overview of Simulation When do we prefer to develop simulation model over an analytic model? When not all the underlying assumptions set for analytic model are valid. When mathematical complexity makes it hard to provide useful results. When “good” solutions (not necessarily optimal) are satisfactory (In general it is the interest of the Enterprises). - A simulation develops a model to numerically evaluate a system over some time period. • - By estimating characteristics of the system, the best alternative from a set of alternatives under consideration (sceneries) can be selected. 38

39. Continuous simulation systems monitor the system each time a change in its state takes place. Overview of Simulation - Discrete simulation systems monitor changes in a state of a system at discrete points in time. • Simulation of most practical problems requires the use of a computer program. 39

40. Approaches to developing a simulation model Using add-ins to Excel such as @Risk or Crystal Ball Using general purpose programming languages such as: FORTRAN, PL/1, Pascal, Basic. Using simulation languages such as GPSS, SIMAN, SLAM. Using a simulator software program (ARENA, SIMUL8, PROMODEL). Overview of Simulation - Modeling and programming skills, as well as knowledge of statistics are required when implementing the simulation approach. 40

41. Monte Carlo Simulation Monte Carlo simulation generates random events. Random events in a simulation model are needed when the input data includes random variables. To reflect the relative frequencies of the random variables, the random number mapping method is used. 41

42. Jewel Vending Company (JVC) installs and stocks vending machines. Bill, the owner of JVC, considers the installation of a certain product (“Super Sucker” jaw breaker) in a vending machine located at a new supermarket. JEWEL VENDING COMPANY – an example for the random mapping technique 42

43. Data The vending machine holds 80 units of the product. The machine should be filled when it becomes half empty. Bill would like to estimate the expected number of days it takes for a filled machine to become half empty. JEWEL VENDING COMPANY • Daily demand distribution is estimated from similar vending machine placements. • P(Daily demand = 0 jaw breakers) = 0.10 • P(Daily demand = 1 jaw breakers) = 0.15 • P(Daily demand = 2 jaw breakers) = 0.20 • P(Daily demand = 3 jaw breakers) = 0.30 • P(Daily demand = 4 jaw breakers) = 0.20 • P(Daily demand = 5 jaw breakers) = 0.05 43

44. Random number mapping – The Probability function Approach Random number mapping uses the probability function to generate random demand. A number between 00 and 99 is selected randomly. The daily demand is determined by the mapping demonstrated below. 34 34 0.30 34 34 34 0.20 0.20 34 34 0.15 34 34 0.10 0.05 45-74 26-44 10-25 75-94 00-09 Demand 95-99 0 1 2 2 3 4 5 44 26-44

45. Random number mapping – The Cumulative Distribution Approach F(X) Daily demand X is determined by the random number Y between0 and 1, such that X is the smallest value for which F(X) ³ Y. 1.00 1.00 0.95 0.75 34 0.45 Y = 0.34 F(1) = .25 < .34 F(2) = .45 > .34 0.34 0.25 0.10 0.00 45 2 0 1 2 3 4 5 X

46. A random demand can be generated by hand (for small problems) from a table of pseudo random numbers. Using Excel a random number can be generated by The RAND() function The random number generation option (Tools>Data Analysis) Simulation of the JVC Problem 46

47. Simulation of the JVC Problem 45-74 26-44 10-25 75-94 00-09 95-99 3 • An illustration of generating a daily random demand. • Since we have two digit probabilities, we use the first two digits of each random number. 0 1 2 3 4 5 47

48. Simulation of the JVC Problem Simulation is repeated and stops once total demand reaches 40 or more. The number of “simulated” days required for the total demand to reach 40 or more is recorded. 48

49. The purpose of performing the simulation runs is to find the average number of days required to sell 40 jaw breakers. Each simulation run ends up with (possibly) a different number of days. Simulation Results and Hypothesis Tests • Hypothesis test is conducted to test • whether or not m = 16. Null hypothesis H0 : m = 16 Alternative hypothesis HA : m > 16 49