1 Introduction to Operations Research Prof. Fernando Augusto Silva Marins

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 recommend the course of action that is expected to yield the best outcome with what is available.

3 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) What Is Management Science?

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

5 The Management Science Process Management Science is a discipline that adopts the scientific method to provide management with key information needed in making 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 King to find how to best blend cuts of meat to minimize costs. Integer Linear Programming model was used by American Air Lines to determine an optimal flight schedule. The Shortest Route Algorithm was implemented by the Sony Corporation to developed an onboard car navigation system.

10 Examples of Management Science Applications Project Scheduling Techniques were 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 INFORMS 2007 Is Operations Research really important?

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

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

14 FINALISTAS EDELMAN

15 FINALISTAS EDELMAN

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 “  ”, “  ”, “=” 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 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. 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

19 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 The Galaxy Industries Production Problem 8(450) + 5(100)

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 : Decisions variables: The Galaxy Linear Programming Model – X 1 = Weekly production level of Space Rays – X 2 = 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 Objective Function: The Galaxy Linear Programming Model Max 8X 1 + 5X 2 – Weekly profit, to be maximized : Decisions variables: X 1 = Weekly production of Space Rays, X 2 = Weekly production of Zappers Space Ray- $8/dozen Zappers $5/dozen

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 Writing Constraints 2X 1 + 1X 2  1000 (Plastic) 3X 1 + 4X 2  2400 (Prod Time - Min) X 1 + X 2  700 (Total production) X 1 - X 2  350 (Mix) Decisions variables X 1 = Space Rays, X 2 = Zappers 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, Space Rays uses 2 of plastic and 3 min of labor Zappers uses 1 of plastic and 4 min of labor

28 Additional constraints Non negativity constraint - X  0 Lower bound constraint - X  L Upper bound constraint - X  U Integer constraint - X = integer Binary constraint - X = 0 or 1 Writing Constraints

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

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 The non-negativity constraints X2X2 X1X1 Graphical Analysis – the Feasible Region

Feasible X2X2 Infeasible Production Time 3X 1 +4X 2  2400 Total production constraint: X 1 +X 2  700 (redundant) The Plastic constraint 2X 1 +X 2  1000 X1X1 700 Graphical Analysis – the Feasible Region

Feasible X2X2 Infeasible Production Time 3X 1 +4X2  2400 Total production constraint: X 1 +X 2  700 (redundant) Production mix constraint: X 1 -X2  350 The Plastic constraint 2X 1 +X 2  1000 X1X1 700 Graphical Analysis – the Feasible Region There are three types of feasible points Interior points. Boundary points. Extreme points (5 Vertices).

35 The search for an optimal solution Start at some arbitrary profit, say profit = $2, Then increase the profit, if possible......and continue until it becomes infeasible Optimal Profit =$4,360 and optimal solution: X2X2 X1X1 8X 1 + 5X 2 = 2,000 Space Rays = 320 dozen Zappers = 360 dozen Current solution: Space Rays = 450, Zapper = 100 and Profit = $4,100 Max 8X 1 + 5X 2 8X 1 + 5X 2 = 3,

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37 Simulation

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.

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

40 – Approaches to developing a simulation model Using add-ins to Excel such 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.

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.

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

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. 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

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 Random number mapping – The Probability function Approach Demand

Random number mapping – The Cumulative Distribution Approach Daily demand X is determined by the random number Y between 0 and 1, such that X is the smallest value for which F(X)  Y. Y = F(1) =.25 <.34 F(2) =.45 > F(X) 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

47 Simulation of the JVC Problem Since we have two digit probabilities, we use the first two digits of each random number An illustration of generating a daily random demand.

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

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 H 0 : m = 16 Alternative hypothesis H A : m > 16