1. Review for Exam II Tues. July 29, 2003 (50% multiple choice, 50% problems)

2. Bring b an orange scantron sheet b pencil b calculator b nothing else (exam is closed book)

3. Exam will cover b Chapter 6 b Chapter 9, M/M/1, M/M/K, M/G/1

4. Chapter 6 b DMUU—Decision Making Under Uncertainty b DMUR—Decision Making Under Risk b UT—Utility Theory b GT—Game Theory

5. DMUU & DMUR Components b A set of Future States of Nature, S Mutually exclusiveMutually exclusive collective exhaustivecollective exhaustive b A set of alternatives, A Mutually exclusiveMutually exclusive b A set of payoffs defined on A x S payoffs for each alternative/state pair payoffs for each alternative/state pair b A Decision Criterion

6. DMUU Criteria b Pessimist what is the payoff for minimizing losses or costs???what is the payoff for minimizing losses or costs??? b Optimist What is the payoff for minimizing losses or costs??What is the payoff for minimizing losses or costs?? b Inbetweenist b Regrettist b Insufficient Reason

7. DMUR Criteria b EREV -- expected value b Expected Regret b ERPI -- expected return with perfect information b EVPI -- expected value of perfect information b ERSI -- expected return with sample information b EVSI -- expected value of sample information

8. Relationships between b EVSI, ERSI, EVPI, ERPI b EREV & Expected Regret b Minimal Expected Regret and EVPI b Expected Regret + EREV, for any alternative =

9. EVPI = Expected Value of Perfect Information b = ERPI – EREV for the optimal alternative b = minimum expected regret

10. EVPI and EVSI b EVPI = ERPI - EREV for the optimal alternative b EVSI = ERSI - EREV for the optimal alternative

11. Bayesian Revision b When is it needed? b Starts with what? b Ends with what what? b Combines prior and sample information as inputs b Outputs posterior (revised) information

12. Two ways to do Bayesian Revision b Tables b Probability trees

13. Probability Trees b Construct backward-looking tree Find joint probabilities at the end nodes by taking the product of all probabilities leading out to the end nodeFind joint probabilities at the end nodes by taking the product of all probabilities leading out to the end node b Construct forward-looking tree Move joint probabilities to their appropriate end nodesMove joint probabilities to their appropriate end nodes Calculate marginal probabilities of indicator statesCalculate marginal probabilities of indicator states Calculate posterior conditional probabilitiesCalculate posterior conditional probabilities

14. The revised probabilities b Are posterior probabilities b Replace the prior probabilities when A particular predictive state is known (like success or failure)A particular predictive state is known (like success or failure)

15. Decision Trees b For multi-stage decisions b Two node types, basically--decision nodes and chance nodes b Know how to solve the decision tree

16. Utility Theory b What is it for? b What kinds of decision makers are there?

17. Procedure b The highest payoff is assigned a value of ___ b The lowest payoff is assigned a value of b Intermediate payoffs are assigned value by asking the DM an indifference question

18. Most DM’s b Are risk-taking for small dollar amounts and risk-averse when the dollar amounts are large, as compared to their net worth

19. Most large corporations b Are risk-neutral for small-to-medium-sized risks, relative to the market capitalization of the firm b In such situations payoffs work well in place of utiles

20. Games b Where do such constructs make sense in business? b What is a fair game? b For what kinds of games can pure strategies be employed? b How are mixed strategies obtained?

21. Examples b Airlines competing on a particular route b Retail grocers competing in a particular market, like Gold Beach in the book b Television networks competing for market share in prime time

22. History and Rationale b First formalized and explained by von Neumann and Morgenstern in 1947 who also made important contributions to utility theory b Resembles a decision theory problem in which the states of nature are managed by a malevolent opponent who actively chooses his states or strategies so as to minimize the decision maker’s expected payoff or utility.

23. b Classification of Games Number of PlayersNumber of Players –Two players - Chess –Multi-player - More than two competitors (Poker) Total returnTotal return –Zero Sum - The amount won and amount lost by all competitors are equal (Poker among friends) –Nonzero Sum -The amount won and the amount lost by all competitors are not equal (Poker In A Casino) Sequence of MovesSequence of Moves –Sequential - Each player gets a play in a given sequence. –Simultaneous - All players play simultaneously.

24. Classification on the basis of pure vs. mixed strategies b Games which possess a saddle point are said to have pure strategies b Example: b This game has a Saddle Point--because there is convergence upon the same cell of the table

25. Mixed Strategies b Most games do not possess saddle points and hence cannot use pure strategies b For such situations, it becomes necessary for both players to use mixed (probabilistic) strategies b The probabilities with which the players execute each of their strategies can be determined as the solution to two linear programming problems-- one for each player

26. Queuing Theory

27. 9.2 Elements of the Queuing Process b A queuing system consists of three basic components: Arrivals: Customers arrive according to some arrival pattern. Arrivals: Customers arrive according to some arrival pattern. Waiting in a queue: Arriving customers may have to wait in one or more queues for service.Waiting in a queue: Arriving customers may have to wait in one or more queues for service. Service: Customers receive service and leave the system.Service: Customers receive service and leave the system.

28. The Arrival Process b There are two possible types of arrival processes Deterministic arrival process.Deterministic arrival process. Random arrival process.Random arrival process. b The random process is more common in businesses.

29. b Under three conditions the arrivals can be modeled as a Poisson process Orderliness : one customer, at most, will arrive during any time interval.Orderliness : one customer, at most, will arrive during any time interval. Stationarity : for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length.Stationarity : for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length. Independence : the arrival of one customer has no influence on the arrival of another.Independence : the arrival of one customer has no influence on the arrival of another. The Arrival Process

30. P(X = k) = Where = mean arrival rate per time unit. t = the length of the interval. e = 2.7182818 (the base of the natural logarithm). k! = k (k -1) (k -2) (k -3) … (3) (2) (1).  t  k e - t k! The Poisson Arrival Process

31. HANK’s HARDWARE – Arrival Process b Customers arrive at Hank’s Hardware according to a Poisson distribution. b Between 8:00 and 9:00 A.M. an average of 6 customers arrive at the store. b What is the probability that k customers will arrive between 8:00 and 8:30 in the morning (k = 0, 1, 2,…)?

32. kk b Input to the Poisson distribution  = 6 customers per hour. t = 0.5 hour. t = (6)(0.5) = 3.  t  e - t k !  0  0.049787 0 1! 1 0.149361 2 2! 0.224042 3 3! 0.224042 1234567 P(X = k )= 8  1 2 3 HANK’s HARDWARE – An illustration of the Poisson distribution.

33. HANK’s HARDWARE – Using Excel for the Poisson probabilities b Solution We can use the POISSON function in Excel to determine Poisson probabilities.We can use the POISSON function in Excel to determine Poisson probabilities. Point probability: P(X = k) = ?Point probability: P(X = k) = ? –Use Poisson(k, t, FALSE) –Example: P(X = 0; t = 3) = POISSON(0, 1.5, FALSE) Cumulative probability: P(X  k) = ?Cumulative probability: P(X  k) = ? –Example: P(X  3; t = 3) = Poisson(3, 1.5, TRUE)

34. HANK’s HARDWARE – Excel Poisson

35. b Factors that influence the modeling of queues –Line configuration –Jockeying –Balking The Waiting Line Characteristics – Priority – Tandem Queues – Homogeneity

36. b A single service queue. b Multiple service queue with single waiting line. b Multiple service queue with multiple waiting lines. b Tandem queue (multistage service system). Line Configuration

37. b Jockeying occurs when customers switch lines once they perceived that another line is moving faster. b Balking occurs if customers avoid joining the line when they perceive the line to be too long. Jockeying and Balking

38. b These rules select the next customer for service. b There are several commonly used rules: First come first served (FCFS).First come first served (FCFS). Last come first served (LCFS).Last come first served (LCFS). Estimated service time.Estimated service time. Random selection of customers for service.Random selection of customers for service. Priority Rules

39. Tandem Queues b These are multi-server systems. b A customer needs to visit several service stations (usually in a distinct order) to complete the service process. b Examples Patients in an emergency room.Patients in an emergency room. Passengers prepare for the next flight.Passengers prepare for the next flight.

40. b A homogeneous customer population is one in which customers require essentially the same type of service. b A non-homogeneous customer population is one in which customers can be categorized according to: Different arrival patternsDifferent arrival patterns Different service treatments.Different service treatments. Homogeneity

41. b In most business situations, service time varies widely among customers. b When service time varies, it is treated as a random variable. b The exponential probability distribution is used sometimes to model customer service time. The Service Process

42. f(t) =  e -  t  = the average number of customers who can be served per time period. Therefore, 1/  = the mean service time. The probability that the service time X is less than some “t.” P(X  t) = 1 - e -  t The Exponential Service Time Distribution

43. Schematic illustration of the exponential distribution The probability that service is completed within t time units P(X  t) = 1 - e -  t X = t

44. HANK’s HARDWARE – Service time  Hank’s estimates the average service time to be 1/  = 4 minutes per customer. b Service time follows an exponential distribution. b What is the probability that it will take less than 3 minutes to serve the next customer?

45. b We can use the EXPDIST function in Excel to determine exponential probabilities. b Probability density: f(t) = ? Use EXPONDIST(t, , FALSE)Use EXPONDIST(t, , FALSE)  Cumulative probability: P(X  k) = ? Use EXPONDIST(t, , TRUE)Use EXPONDIST(t, , TRUE) Using Excel for the Exponential Probabilities

46. b The mean number of customers served per minute is ¼ = ¼(60) = 15 customers per hour. b P(X <.05 hours) = 1 – e -(15)(.05) = ? b From Excel we have: EXPONDIST(.05,15,TRUE) =.5276EXPONDIST(.05,15,TRUE) =.5276 HANK’s HARDWARE – Using Excel for the Exponential Probabilities 3 minutes =.05 hours

47. HANK’s HARDWARE – Using Excel for the Exponential Probabilities =EXPONDIST(B4,B3,TRUE) =EXPONDIST(A10,$B$3,FALSE) Drag to B11:B26

48. The memoryless property.The memoryless property. –No additional information about the time left for the completion of a service, is gained by recording the time elapsed since the service started. –For Hank’s, the probability of completing a service within the next 3 minutes is (0.52763) independent of how long the customer has been served already. The Exponential and the Poisson distributions are related to one another.The Exponential and the Poisson distributions are related to one another. –If customer arrivals follow a Poisson distribution with mean rate, their interarrival times are exponentially distributed with mean time 1 /  The Exponential Distribution - Characteristics

49. 9.3 Performance Measures of Queuing System b Performance can be measured by focusing on: Customers in queue.Customers in queue. Customers in the system.Customers in the system. b Performance is measured for a system in steady state.

50. Roughly, this is a transient period… n Time 9.3 Performance Measures of Queuing System b The transient period occurs at the initial time of operation. b Initial transient behavior is not indicative of long run performance.

51. This is a steady state period……….. n Time 9.3 Performance Measures of Queuing System The steady state period follows the transient period. Meaningful long run performance measures can be calculated for the system when in steady state. Roughly, this is a transient period…

52.  k  Each with service rate of   k  Each with service rate of       …   For k servers with service rates        …   For k servers with service rates    For one server  For one server In order to achieve steady state, the effective arrival rate must be less than the sum of the effective service rates. 9.3 Performance Measures of Queuing System k servers

53. P 0 = Probability that there are no customers in the system. P n = Probability that there are “n” customers in the system. L = Average number of customers in the system. L q = Average number of customers in the queue. W = Average time a customer spends in the system. W q = Average time a customer spends in the queue. P w = Probability that an arriving customer must wait for service. P w = Probability that an arriving customer must wait for service.  = Utilization rate for each server (the percentage of time that each server is busy).  = Utilization rate for each server (the percentage of time that each server is busy). Steady State Performance Measures

54. b Little’s Formulas represent important relationships between L, L q, W, and W q. b These formulas apply to systems that meet the following conditions: Single queue systems, Single queue systems, Customers arrive at a finite arrival rate  and Customers arrive at a finite arrival rate  and The system operates under a steady state condition. The system operates under a steady state condition. L =  W L q = W q L = L q + L =  W L q = W q L = L q + Little’s Formulas For the case of an infinite population

55. b Queuing system can be classified by: Arrival process.Arrival process. Service process.Service process. Number of servers.Number of servers. System size (infinite/finite waiting line).System size (infinite/finite waiting line). Population size.Population size. b Notation M (Markovian) = Poisson arrivals or exponential service time.M (Markovian) = Poisson arrivals or exponential service time. D (Deterministic) = Constant arrival rate or service time.D (Deterministic) = Constant arrival rate or service time. G (General) = General probability for arrivals or service time.G (General) = General probability for arrivals or service time. Example: M / M / 6 / 10 / 20 Example: M / M / 6 / 10 / 20 Classification of Queues

56. 9.4 M  M  1 Queuing System - Assumptions Poisson arrival process.Poisson arrival process. Exponential service time distribution.Exponential service time distribution. A single server.A single server. Potentially infinite queue.Potentially infinite queue. An infinite population.An infinite population.

57. The probability that a customer waits in the system more than “t” is P(X>t) = e - (  - )t The probability that a customer waits in the system more than “t” is P(X>t) = e - (  - )t P 0 = 1 – ( ) P n = [1 – ( )]( ) n L =  (  – ) L q = 2  [  (  – )] W = 1  (  – ) W q =  [  (  – )] P w =   =  M / M /1 Queue - Performance Measures