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# Math138 lectures 3rd edition scoolbook

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### Math138 lectures 3rd edition scoolbook

1. 1. De La Salle University- Dasmariñas College of Science Mathematics Department EDWIN S. BUNAG Asst. Professor 5 esbunag@dlsud.edu.ph 1997-present Master of Arts in Mathematics (2001) Bachelor of Science in Mathematics (1996) College Algebra, Trigonometry, Math of Investment, Statistics, Business Calculus, Quantitative Techniques, Math of Accounting and Finance, Theory of Interest, Plane and Solid Mensuration
2. 2. Course Description  This course deals with analysis of decision-making situations in business environment using probabilities, inventory, forecasting, linear programming, and structured linear programming as applied in business processes. This course will equip the students with the necessary skills and knowledge of the different Management Science/Operation Research techniques, which will develop their decision-making capabilities. Completion and mastery of the course is a great tool in decision making for future business executives.
3. 3. Course Outline  Introduction  Probability and Probability Distribution  Decision Analysis  Utility and Game Theory  Forecasting  Linear Programming  Transportation and Assignment Problems  Inventory Models  Waiting Line models  Simulation
4. 4. Over View (Need for Good Decision) Managers Success A Manager Should Be a Good Decision Maker Increasing competition Changing markets Changing customers requirements More complex business environment Complex information needs and system Increased uncertainty Larger error costs
5. 5. Problem solving can be defined as the process of identifying a difference between the actual and the desired state of affairs and then taking action to resolve the difference. Problem Solving and Decision Making
6. 6. Over view (Decision Making Process) SCIENTIFIC METHOD OF SOLVING PROBLEM Observation Define the Problem Formulation of Hypothesis Experimentation Verification
7. 7. Over view (Decision Making Process) Decision Making Structuring the Problem Analyzing the Problem Define the problem Identify the alternatives Determine the criteria Evaluate the alternatives Choose an alternative Evaluate the results Implement the decision Qualitative Analysis Quantitative Analysis
8. 8. MANAGEMENT SCIENCE / OPERATIONS RESEARCH/DECISION SCIENCE/QUANTITATIVE TECHNIQUES  the discipline of using mathematics, and other analytical methods or quantitative methods, to help make better business decisions.
9. 9. WHEN DO MANAGERS USE QUANTITATIVE TECHNIQUES 1. The problem is complex. 2. The problem is especially important (e.g., a great deal of money is involved). 3. The problem is new. 4. The problem is repetitive.
10. 10. History and Practical Application of Operations Research / Quantitative Methods
11. 11. Chapter 2 Introduction to Probability  Experiments and the Sample Space  Assigning Probabilities to Experimental Outcomes  Events and Their Probability  Some Basic Relationships of Probability  Bayes’ Theorem
12. 12. Probability as a Numerical Measure of the Likelihood of Occurrence 0 1.5 Increasing Likelihood of Occurrence Probability: The event is very unlikely to occur. The occurrence of the event is just as likely as it is unlikely. The event is almost certain to occur.
13. 13. REVIEW OF THE BASIC PROBABILITY CONCEPTS Experiment -any process that generates outcome. Sample Space - the set of all possible outcomes of a given experiment. Event - one or more possible outcomes of an experiment. Mutually Exclusive Events - two events that can not occur at the same time. Otherwise not mutually exclusive.
14. 14. THREE TYPES OF PROBABILITY N nEP )( Where: P(E) refers to the probability of an event will occur. n refers to the number of elements in the event. N refers to the number of elements in the sample space. The Classical Approach •classical probability defines the probability that an event will occur as
15. 15. THREE TYPES OF PROBABILITY The Relative Frequency Approach this method of defining probability uses the relative frequencies of past occurrences as probabilities. The Subjective Approach subjective probabilities are based on the personal belief or feelings of the person who makes the probability estimate.
16. 16. PROBABILITY RULES P(A or B) = P(A) + P(B) If two events A and B are mutually exclusive, the Special Rule of Addition states that the Probability of A or B occurring equals the sum of their respective probabilities.
17. 17. Addition Rule for Mutually Exclusive Events Arrival Frequency Early 100 On Time 800 Late 75 Canceled 25 Total 1000 New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York: What is the probability that a flight is either early or late? What is the probability that a flight is either late or cancelled?
18. 18. PROBABILITY RULES Person Sex Age 1 Male 31 2 Male 33 3 Female 46 4 Female 29 5 Male 41 •The Addition Rule for Not Mutually Exclusive Events P(A or B) = P(A) + P(B) – P(A and B) Example: The data below refers to the number of persons in the city council. The members of the council decided to elect a chairperson by random draw. What is the probability that the chairperson will be either female or over 35? What is the probability that the chairperson will be either Male or over 32?
19. 19. PROBABILITY RULES •Multiplication Rule with Independent Events •Multiplication Rule with Dependent Events P(A and B) = P(A) P(B) Example: If two coins are flipped once, what is the probability that both coins will turn up heads? A nationwide survey found that 72% of people in the United States like pizza. If 3 people are selected at random, what is the probability that all three like pizza?
20. 20. PROBABILITY RULES Example: A bag of fruits contains six mangoes, four atis and five guavas. If you are sampling without replacement, what is the probability of getting a mango and an atis in that order?
21. 21. PROBABILITY RULES The probability of event A occurring given that the event B has occurred is written P(A|B) A Conditional Probability is the probability of a particular event occurring, given that another event has occurred. P(A|B) = P(A and B)/P(B) P(B|A) = P(A and B)/P(A)
22. 22. PROBABILITY RULES Major Male Female Total Accounting 170 110 280 Finance 120 100 220 Marketing 160 70 230 Management 150 120 270 Total 600 400 1000 The Dean of the School of Business at Owens University collected the following information about undergraduate students in her college: Given that the student is a female, what is the probability that she is an accounting major? Given that the student is a male, what is the probability that he is a Marketing major?
23. 23. Bayes’ Theorem New Information Application of Bayes’ Theorem Posterior Probabilities Prior Probabilities  Often we begin probability analysis with initial or prior probabilities.  Then, from a sample, special report, or a product test we obtain some additional information.  Given this information, we calculate revised or posterior probabilities.  Bayes’ theorem provides the means for revising the prior probabilities.
24. 24. A manufacturing firm receives shipments of parts from different suppliers. There is a 65% chance that a part is from supplier 1 and 35% that a part is from supplier 2. Additional information is given on the conditional probability of receiving good and bad parts from two suppliers: Good Parts Bad Parts Supplier 1 0.98 0.02 Supplier2 0.95 0.05 Find the probability that a bad part is from supplier 1? Find the probability that a bad part is from supplier 2? Example (page 43)
25. 25. Bayes’ Theorem 1 1 2 2 ( ) ( | ) ( | ) ( ) ( | ) ( ) ( | ) ... ( ) ( | ) i i i n n P A P B A P A B P A P B A P A P B A P A P B A  To find the posterior probability that event Ai will occur given that event B has occurred, we apply Bayes’ theorem.  Bayes’ theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space.
26. 26. Tabular Approach  Step 1 Prepare the following three columns: Column 1 The mutually exclusive events for which posterior probabilities are desired. Column 2 The prior probabilities for the events. Column 3 The conditional probabilities of the new information given each event.
27. 27. Tabular Approach  Step 2 Column 4 Compute the joint probabilities for each event and the new information B by using the multiplication law. Multiply the prior probabilities in column 2 by the corresponding conditional probabilities in column 3. That is, P(Ai IB) = P(Ai) P(B|Ai).
28. 28. Tabular Approach  Step 3 Column 4 Sum the joint probabilities. The sum is the probability of the new information, P(B).
29. 29.  Step 4 Column 5 Compute the posterior probabilities using the basic relationship of conditional probability. The joint probabilities P(Ai IB) are in column 4 and the probability P(B) is the sum of column 4. Tabular Approach )( )( )|( BP BAP BAP i i
30. 30. Assignment 1. Give 2 example for each probability rules. 2. Examples should be business related problems. 3. Provide complete solution. 4. Discuss how it can be used in decision making.
31. 31. Random Variable Binomial Probability Distribution Poisson Probability Distribution Exponential Probability Distribution Normal Probability Distribution
32. 32. Random Variable Is a function whose value is a real number determined by each element in the sample space. Example A coin is tossed three times. List down the elements of the sample space. List down the possible values of the following random variables: X: the number of heads that fall Y: the number of tails that fall W: the number of heads minus the number of tails A random variable is a numerical description of the outcome of an experiment.
33. 33. Random Variable Sample space HHH HHT HTH THH HTT THT TTH TTT Random Variable X: no. of heads 3 2 2 2 1 1 1 0 Random Variable Y: no. of tails 0 1 1 1 2 2 2 3 Random Variable W: X – Y 3 1 1 1 -1 -1 -1 -3
34. 34. Probability Distribution Probability distribution is a list of all possible outcome of a random variable with their corresponding probabilities. Example Probability distribution of the following random variable: Random Variable X P(X=0) = 1/8 P(X=1) = 3/8 P(X=2) = 3/8 P(X=3) = 1/8 Random Variable Y P(Y=0) = 1/8 P(Y=1) = 3/8 P(Y=2) = 3/8 P(Y=3) = 1/8 Random Variable W P(W=-3) = 1/8 P(W=-1) = 3/8 P(W=1) = 3/8 P(W=3) = 1/8 Note: Discrete Random Variable : defined over a discrete sample space. Continuous Random Variable: defined over a continuous sample space.
35. 35. Expected Value and Variance The expected value, or mean, of a random variable is a measure of its central location. The variance summarizes the variability in the values of a random variable. The standard deviation, , is defined as the positive square root of the variance. Var(x) = 2 = (x - )2f(x) E(x) = = xf(x)
36. 36.  a tabular representation of the probability distribution for TV sales was developed.  Using past data on TV sales, … Example: JSL Appliances Number Units Sold of Days 0 80 1 50 2 40 3 10 4 20 200 x f(x) 0 .40 1 .25 2 .20 3 .05 4 .10 1.00
37. 37. Types of Probability Distribution Discrete Probability Distribution 1. Binomial Probability Distribution 2. Poisson Probability Distribution Continuous Probability Distribution 3. Exponential Probability Distribution 4. Normal Probability Distribution
38. 38. The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable. The required conditions for a discrete probability function are: Discrete Probability Distributions f(x) > 0 f(x) = 1
39. 39. The Binomial Probability Distribution It describes discrete data resulting from an experiment called a Bernoulli process. Properties of the Binomial Distribution 1. The sample consists of a fixed number of observations, n. 2. Each trial has only two possible outcomes. 3. The probability of a success and failure on any trial remains fixed over time. 4. The trials are statistically independent.
40. 40. The Binomial Probability Distribution Where: P(r) is the probability of r successes in n trials. n is the total number of trials. r represents a certain number of successes. p represents the probability of success. q represents the probability of failure. rnr rn qpCP(r) Binomial Formula ( )! ( ) (1 ) !( )! x n xn f x p p x n x
41. 41. Binomial Probability Distribution (1 )np p  Expected Value  Variance  Standard Deviation E(x) = = np Var(x) = 2 = np(1 p)
42. 42. Example: Evans Electronics  Binomial Probability Distribution Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year. Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year?
43. 43. The Binomial Probability Distribution Sample Problem 2 5 employees are required to operate a chemical process; the process cannot be started until all 5 workstations are manned. Employee records indicate there is a 0.4 chance of any one employee being late, and we know that they all come to work independently of each other. Management is interested in knowing the probabilities of 0, 1, 2, 3, 4, or 5 employees being late, so that a decision concerning the number of backup personnel can be made.
44. 44. The Poisson Probability Distribution Named after the mathematician and physicist Siméon Poisson (1781-1840). It describes the distribution of arrivals per unit time at a service facility. A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space. It is used to describe a number managerial situations including the demand (arrivals) of patience at a health clinic, the distribution of telephone calls going through a central switching system, the arrivals of vehicles at a toll booth, the number of accidents at an intersection, and the number of looms waiting for service in a textile mill.
45. 45. The Poisson Probability Distribution THE POISSON FORMULA x e – P(x) = x Where: P(x) the probability of exactly x occurrences. x Lambda (the average number of occurrences per interval of time) raised to the x power. e– e (2.71828…) raised to the negative lambda power. x x factorial ! )( x e xf x
46. 46. The Poisson Probability Distribution Sample Problem 1 The manager of DWEIN BANK records the arrival of customers and on the average; three costumers arrive per minute at the bank during the noon to 1 P.M. hour. What is the probability that in a given minute exactly two customers will arrive? What is the probability that more than two customers will arrive in a given minute?
47. 47. Example: Mercy Hospital Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? MERCY
48. 48. The Exponential Probability Distribution It describes a continuous random variable of the interarrival time. It describes the distribution of service time at a service facility. Some Applications Used in waiting line theory to model the length of time between arrivals in processes such as customers at a bank’s ATM, clients in a fast-food restaurant and patients entering a hospital emergency room.
49. 49. The Exponential Probability Distribution THE EXPONENTIAL FORMULA P(T t) = 1 – e– t Where: P(T t) the probability that the service time T will be less than or equal to t. the mean service rate e 2.71828…
50. 50. The Exponential Probability Distribution Sample Problem 1 Suppose the distribution in the figure represents the time it takes for a microcomputer repair facility to repair 1 unit, and suppose the mean repair time has been found to be 3 hours. What is the probability that the service on a faulty microcomputer will be completed in 2 or fewer hours? 0 p(t) t
51. 51. The Exponential Probability Distribution Sample Problem 2 Suppose that customers arrive at a bank’s ATM at the rate of 20 per hour. If a customer has just arrived, what is the probability that the next customer will arrive within 6 minutes?
52. 52. The Normal Probability Distribution The normal probability distribution is frequently referred to as the Gaussian distribution (named after the mathematician-astronomer Karl Gauss 1777-1855). x z Standard Score It is characterized by a normal curve. * it is bell shaped * it has a single highest peak * it is symmetrical about the center * the curve is asymptotic to the horizontal line * the area under the curve is 100% or 1
53. 53. The Normal Probability Distribution Applications: 1. Lifetimes of batteries in a certain application are normally distributed with mean 50 hours and standard deviation 5 hours. a. Find the probability that a randomly selected battery lasts between 42 and 52 hours. b. Find the 40th percentile of battery lifetimes
54. 54. The Normal Probability Distribution 2. A process manufactures ball bearings whose diameters are normally distributed with mean 2.505 cm and standard deviation 0.008 cm. Specifications call for the diameter to be in the interval 2.5 0.01 cm. a. What proportion of the ball bearings will meet the specification? b. The process can be recalibrated so that the mean will be equal to 2.5 cm, the center of the specification interval. The standard deviation of the process remains 0.008 cm. What proportion of the diameters will meet the specification?
55. 55. The Normal Probability Distribution c. The process has been recalibrated so that the mean diameter is now 2.5 cm. To what value must the standard deviation be lowered so that 95% of the diameters will meet the specification?
56. 56. The Normal Probability Distribution Shaft manufactured for use in optical storage devices have diameters that are normally distributed with mean 0.652 cm and standard deviation 0.003 cm. The specification for the shaft diameter is 0.650 0.005 cm. a. What proportion of the shafts manufactured by this process meet the specifications? b. The process mean can be adjusted through calibration. If the mean is set to 0.650 cm, what proportion of the shafts will meet the specifications? c. From part b, how many shaft will be rejected if there are 100,000 shaft produced.
57. 57. Qualitative analysis is based primarily on the manager’s judgment and experience; it includes the manager’s intuitive “feel” for the problem and is more an art than a science.
58. 58. Quantitative analysis will concentrate on the quantitative facts or data associated with the problem and develop models or mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problem. Quantitative Analysis Process  Model Development  Data Preparation  Model Solution  Report Generation
59. 59. Types of models: 1. Iconic models are physical replicas of real objects. Ex. Model Development
60. 60. Types of models: 2. Analog models are physical in form but do not have the same physical appearance as the object being modeled. Ex. The position of the needle on the dial of a speedometer represents the speed of the automobile. Model Development
61. 61. Types of models: 3. Mathematical models are representations of a problem by a system of symbols and mathematical relationships or expressions. Model Development
62. 62. Models Used in Economics Cost function C(x) is the cost of producing x units of the commodity. Revenue function R(x) is the revenue obtained from selling x units of the commodity R(x)=xp(x). Profit function P(x) is the profit obtain from selling x units of the commodity P(x)=R(x) – C(x).
63. 63. Breakeven Analysis Market research indicates that consumers will buy x thousand units of a particular kind of coffee maker when the unit price is dollars. The cost of producing the x thousand units is thousand dollars. a. What are the revenue and profit functions for this production process? b. For what values of x is production of coffee makers profitable?
64. 64. Linear Programming Model Objective Function: Maximize Profit: Z = 8x + 6y Subject to: (Assembly constraint) x + 2y 60 (Finishing constraint) x + 4y (Implicit constraint) x, y
65. 65. Types of criteria: 1. Single-criterion decision problems are those in which the objective is to find the best solution with respect to one criteria. 2. Multicriteria decision problems are those that involve more than one criterion.