Review for Exam II

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

18. Most DM’s • 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 • Are risk-neutral for small-to-medium-sized risks, relative to the market capitalization of the firm • In such situations payoffs work well in place of utiles

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

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

22. History and Rationale • First formalized and explained by von Neumann and Morgenstern in 1947 who also made important contributions to utility theory • 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. Classification of Games • Number of Players • Two players - Chess • Multi-player - More than two competitors (Poker) • Total 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 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 • Games which possess a saddle point are said to have pure strategies • Example: • This game has a Saddle Point--because there is convergence upon the same cell of the table

25. Mixed Strategies • Most games do not possess saddle points and hence cannot use pure strategies • For such situations, it becomes necessary for both players to use mixed (probabilistic) strategies • 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 • A queuing system consists of three basic components: • 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. • Service: Customers receive service and leave the system.

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

29. The Arrival Process • Under three conditions the arrivals can be modeled as a Poisson process • 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. • Independence : the arrival of one customer has no influence on the arrival of another.

30. The Poisson Arrival Process (lt)ke- lt k! P(X = k) = Where l = 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).

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

32. 0 HANK’s HARDWARE – An illustration of the Poisson distribution. • Input to the Poisson distribution l = 6 customers per hour.t = 0.5 hour.lt = (6)(0.5) = 3. 1 2 3 4 5 6 7 8 k 1 0 2 3 (lt) e- lt k ! 1 0 = P(X = k )= 2 0.224042 0.224042 0.149361 0.049787 3 1! 0! 2! 3!

33. HANK’s HARDWARE – Using Excel for the Poisson probabilities • Solution • We can use the POISSON function in Excel to determine Poisson probabilities. • Point probability: P(X = k) = ? • Use Poisson(k, lt, FALSE) • Example: P(X = 0; lt = 3) = POISSON(0, 1.5, FALSE) • Cumulative probability: P(X£k) = ? • Example: P(X£3; lt = 3) = Poisson(3, 1.5, TRUE)

34. HANK’s HARDWARE – Excel Poisson

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

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

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

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

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

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

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

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

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

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

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

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

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

48. The Exponential Distribution -Characteristics • 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. • If customer arrivals follow a Poisson distribution with mean rate l, their interarrival times are exponentially distributed with mean time 1/l.

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

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