Math138 lectures 3rd edition scoolbook

Quantitative Techniques

Qualitative analysis is based primarily on the managers judgment and experience; it includes the managers intuitive feel for the problem and is more an art than a science.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 ProcessModel DevelopmentData PreparationModel SolutionReport GenerationTypes of criteria:Single-criterion decision problems are those in which the objective is to find the best solution with respect to one criteria.Multicriteria decision problems are those that involve more than one criterion.Course DescriptionThis 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. Course OutlineIntroductionProbability and Probability DistributionDecision AnalysisUtility and Game TheoryForecastingLinear ProgrammingTransportation and Assignment ProblemsInventory ModelsWaiting Line modelsSimulationOver View (Need for Good Decision)

ManagersSuccess

A Manager Should Be a GoodDecision MakerIncreasing competitionChanging marketsChanging customers requirementsMore complex business environmentComplex information needs and systemIncreased uncertaintyLarger error costs

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 MakingOver view (Decision Making Process)SCIENTIFIC METHOD OF SOLVING PROBLEM

ObservationDefine the ProblemFormulation of HypothesisExperimentationVerification

Over view (Decision Making Process)Decision Making

Structuring the Problem

Analyzing the Problem

Define the problemIdentify the alternativesDetermine the criteriaEvaluate the alternativesChoose an alternativeEvaluate the resultsImplement the decisionQualitative AnalysisQuantitative AnalysisMANAGEMENT SCIENCE / OPERATIONS RESEARCH/DECISION SCIENCE/QUANTITATIVE TECHNIQUESthe discipline of using mathematics, and other analytical methods or quantitative methods, to help make better business decisions.WHEN DO MANAGERS USE QUANTITATIVE TECHNIQUESThe problem is complex.The problem is especially important (e.g., a great deal of money is involved).The problem is new.The problem is repetitive.History and Practical Application of Operations Research / Quantitative MethodsChapter 2 Introduction to ProbabilityExperiments and the Sample SpaceAssigning Probabilities to Experimental OutcomesEvents and Their ProbabilitySome Basic Relationships of ProbabilityBayes TheoremProbability as a Numerical Measureof the Likelihood of Occurrence01.5Increasing Likelihood of OccurrenceProbability:The eventis veryunlikelyto occur.The occurrenceof the event is just as likely asit is unlikely.The eventis almostcertainto occur.REVIEW OF THE BASIC PROBABILITY CONCEPTSExperiment -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.THREE TYPES OF PROBABILITY

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 Approachclassical probability defines the probability that an event will occur as

THREE TYPES OF PROBABILITYThe Relative Frequency Approachthis method of defining probability uses the relative frequencies of past occurrences as probabilities. The Subjective Approachsubjective probabilities are based on the personal belief or feelings of the person who makes the probability estimate. PROBABILITY RULESP(A or B) = P(A) + P(B)

If two events A and B are mutually exclusive, theSpecial Rule of Addition states that the Probability of A or B occurring equals the sum of their respective probabilities. Addition Rule for Mutually Exclusive Events

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?PROBABILITY RULESPersonSexAge 1Male 31 2Male 33 3Female 46 4Female 29 5Male 41

The Addition Rule for Not Mutually Exclusive EventsP(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?PROBABILITY RULESMultiplication Rule with Independent EventsMultiplication Rule with Dependent EventsP(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?

21PROBABILITY RULESExample: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?PROBABILITY RULESThe 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)PROBABILITY RULES

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? Bayes TheoremNewInformationApplicationof BayesTheoremPosteriorProbabilitiesPriorProbabilities 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.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 PartsBad PartsSupplier 10.980.02Supplier20.950.05

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)Bayes Theorem

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.Tabular ApproachStep 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.Tabular ApproachStep 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). Tabular ApproachStep 3 Column 4 Sum the joint probabilities. The sum is theprobability of the new information, P(B).Step 4 Column 5 Compute the posterior probabilities using the basic relationship of conditional probability.

The joint probabilities P(Ai I B) are in column 4 and the probability P(B) is the sum of column 4.Tabular Approach

Assignment1. Give 2 example for each probability rules.Examples should be business related problems.Provide complete solution.Discuss how it can be used in decision making.PROBABILITY DISTRIBUTIONRandom VariableBinomial Probability DistributionPoisson Probability DistributionExponential Probability DistributionNormal Probability Distribution

Random VariableIs a function whose value is a real number determined by each element in the sample space.ExampleA coin is tossed three times. List down the elements of thesample space. List down the possible values of the followingrandom variables:X: the number of heads that fallY: the number of tails that fallW: the number of heads minus the number of tails A random variable is a numerical description of the outcome of an experiment.Random VariableSample space

HHHHHTHTHTHHHTTTHTTTHTTTRandom VariableX: no. of heads

32221110Random VariableY: no. of tails

01112223Random VariableW: X Y

3111-1-1-1-3Probability DistributionProbability distribution is a list of all possible outcome of a random variable with their corresponding probabilities.ExampleProbability distribution of the following random variable:Random Variable XP(X=0) = 1/8P(X=1) = 3/8P(X=2) = 3/8P(X=3) = 1/8Random Variable YP(Y=0) = 1/8P(Y=1) = 3/8P(Y=2) = 3/8P(Y=3) = 1/8Random Variable WP(W=-3) = 1/8P(W=-1) = 3/8P(W=1) = 3/8P(W=3) = 1/8Note:Discrete Random Variable : defined over a discrete sample space.Continuous Random Variable: defined over a continuous sample space.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) 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 Days0 80 1 50 2 40 3 10 4 20 200 x f(x) 0 .40 1 .25 2 .20 3 .05 4 .10 1.00Types of Probability DistributionDiscrete Probability Distribution

1. Binomial Probability DistributionPoisson Probability Distribution

Continuous Probability Distribution

3. Exponential Probability Distribution4. Normal Probability Distribution 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 Distributionsf(x) > 0f(x) = 1The Binomial Probability DistributionIt describes discrete data resulting from an experiment called a Bernoulli process.

Properties of the Binomial Distribution1. The sample consists of a fixed number of observations, n.2. Each trial has only two possible outcomes.The probability of a success and failure on any trial remains fixed over time.4. The trials are statistically independent.The Binomial Probability DistributionWhere: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.prepresents the probability of success.qrepresents the probability of failure.

Binomial Formula

Binomial Probability Distribution

Expected Value

Variance

Standard DeviationE(x) = = npVar(x) = 2 = np(1 - p)Example: Evans ElectronicsBinomial 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?The Binomial Probability DistributionSample 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.The Poisson Probability DistributionNamed after the mathematician and physicist Simon 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.

The Poisson Probability Distribution

The Poisson Probability DistributionSample Problem 1The 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? Example: Mercy HospitalPatients arrive at the emergency room of MercyHospital at the averagerate of 6 per hour onweekend evenings.

What is theprobability of 4 arrivals in30 minutes on a weekend evening?MERCYThe Exponential Probability DistributionIt describes a continuous random variable of the interarrival time.It describes the distribution of service time at a service facility. Some ApplicationsUsed in waiting line theory to model the length of time between arrivals in processes such as customers at a banks ATM, clients in a fast-food restaurant and patients entering a hospital emergency room. The Exponential Probability Distribution

The Exponential Probability Distribution

The Exponential Probability Distribution

The Normal Probability DistributionThe normal probability distribution is frequently referred to as the Gaussian distribution (named after the mathematician-astronomer Karl Gauss 1777-1855).

Standard ScoreIt 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 1The Normal Probability DistributionApplications: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

The Normal Probability Distribution2. 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?The Normal Probability Distributionc. 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?

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.What proportion of the shafts manufactured by this process meet the specifications?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.Types of models:Iconic models are physical replicas of real objects.Ex.

Model Development

Types of models: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

Types of models:Mathematical models are representations of a problem by a system of symbols and mathematical relationships or expressions.

Model DevelopmentModels Used in EconomicsCost 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).Breakeven Analysis

Linear Programming ModelObjective Function: Maximize Profit: Z = 8x + 6y

Subject to:(Assembly constraint) 4x + 2y 60(Finishing constraint)2x + 4y 48(Implicit constraint) x, y 0ArrivalFrequency

Early100

On Time800

Late75

Canceled25

Total1000

MajorMaleFemaleTotal

Accounting170110280

Finance120100220

Marketing16070230

Management150120270

Total6004001000

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?