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90-760: Decision & Risk Modeling Final Exam, Spring 2017Do not turn the page until you are told to do so. Note how much each problem is worthand budget your time. Total points possible: 200Name: ______________________________(Write answers in the space provided. If you need more space, you may use additionalsheet(s), but be sure to put your name on them and label with the problem number.)Q1: _________ out of 12 possible pointsQ2: _________ out of 12 possible pointsQ3: _________ out of 16 possible pointsQ4: _________ out of 15 possible pointsQ5: _________ out of 15 possible pointsQ6: _________ out of 20 possible pointsQ7: _________ out of 24 possible pointsQ8: _________ out of 18 possible pointsQ9: _________ out of 12 possible pointsQ10: _________ out of 10 possible pointsQ11: _________ out of 10 possible pointsQ12: _________ out of 24 possible pointsQ13: _________ out of 12 possible pointsQuestion #1: (12 points)What is the total waiting time if the following four jobs are serviced in the indicated service disciplines?Job #1: 5 minutesJob #2: 12 minutesJob #3: 1 minutesJob #4: 8 minutesa) FCFS (i.e., FIFO)b) SPTc) LCFS (i.e., LIFO)Question #2: (12 points)Customers have been arriving to an M/M/3 queueing system at an average rate of 10 per hour since dinosaursruled the world. Service times average 15 minutes. a) Fill in the blanked out cells E3:E4.b) If you arrived at a random time, how many customers would you see waiting for service?c) On average, how long do customers spend in the system before they complete service?d) Over an eight hour day, how long would a server expect to be idle?Question #3: (16 points)The Borough of Upper Scranton operates 100 buses that each break down completely at random an average ofonce a month. When a mechanic, who is paid $200 per day, works on a bus, the repair time is exponentiallydistributed with a mean of one day. When a bus is not in service, the Borough pays $500 a day to rent areplacement. I evaluated queueing performance characteristics with the Q.xls workbook for various numbers ofmechanics, obtaining the results below. a) If the Borough employs 3 mechanics, on average how long is a bus out of service when it breaks down?b) What is the expected total daily cost to the Borough if 4 mechanics are employed?c) What number of mechanics minimizes the Borough’s average daily cost?d) If the Q.xls spreadsheet didn’t give you the utilization, explain how you could compute it from othernumbers that it does give. (You’ll need to think this through from first principles, not apply any specificformula we used in class.)Information for questions #4 - #6. Suppose that three management science students (unwisely) divvy up theeight problems on a homework as indicated blow, work on their problems independently in order, and whenthey are all done staple the results together and submit the combined work. They also (unwisely) wait until sixhours before the homework is due to begin, and they are interested in whether they will finish the assignment ontime. Since each individual homework problem involves a very large number of separate tasks, each taking a variableamount of time, completion time for each problem can reasonably be modeled as being a normally distributedrandom variable with the following parameters.Question #4: (15 points)Suppose these unwise students forget all about uncertainty and just assume that every homework problem’scompletion time will be exactly equal to its expected value. E.g., the first problem will take exactly 2.5 hours,and so on.a) Draw the CPM project network for this problem augmented with dummy nodes as appropriate.b) Execute the forward and backward passes to find the earliest and latest starting times for each homeworkproblem.c) What is the critical path?Question #5: (15 points)If the students suddenly remember there is uncertainty but can’t remember how to run a simulation, and so takethe PERT approach to problem management:a) What do they think is the mean, variance, and standard deviation of the project completion time?b) What type of random variable do they think describes the project completion time and why?c) What do they judge to be the probability that they will finish the homework on time and how do youknow? (The explanation can be very brief but must include the key insight.) Question #6: (20 points)Suppose, much to the relief of their professor, the students suddenly remember the limitations of PERT and howto run a simulation. They build and run the simulation with 10,000 trials producing the output that is pastedbelow (with a few cells blanked out). Note: the formula in cell K5 is “=COUNTIF(J$24:J$10023,J5)/10000”,and similarly for cells K6 and K7.a) Now approximately what do they think is the probability that they will complete the homework on time?b) What is a balanced 90% confidence interval for how long it will take them to finish the homework?c) If they want to be 90% sure they will submit the homework on time, how long in advance of the duedate should they start?d) What is the Excel formula in cell I23?e) What is the Excel formula in cell I7?Question #7: (24 points)Sketch the typical form of the following graphs, labeling both axes and adding a smiley face in the ideal corner.a) Tradeoff curve when considering crashing a project in project management.b) A risk-return frontier for profit that shows the width of each alternative’s 90% confidence interval.c) Queue length as a function of utilization.d) Cumulative risk profile for two ways of saving lives with Alternative B exhibiting 1st order stochasticdominance over Alternative A.Question #8: (18 points)For each pair, circle the best answer.a) Which is more vulnerable to overfitting: (i) 8-period MA or (ii) 5-period WMA?b) Which is better for DA with three groups: (i) =TREND() with cut-offs or (ii) Centroid based method?c) What bias affected MIT MBA students’ bidding: (i) Availability or (ii) Anchoring?d) What kind of variation has a fixed period: (i) Seasonal or (ii) Secular trend?e) Which time series displays seasonality: (i) Daily movie theater ticket sales or (ii) Annual temperatures?f) Is an ROC curve more likely to pass through the point: (i) (0.2, 0.25) or (ii) (0.25, 0.2)?g) For which measure of risk are small numbers good: (i) Coefficient of variation or (ii) Return to risk?h) Which is the discrete random

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