Group A 2 A news vendor sells newspapers and tries to maximize profits. The number of papers sold each day is a random variable. However, analysis of the past month’s data shows the distribution of daily demand in Table 16. A paper costs the vendor 20¢. The vendor sells the paper for 30¢. Any unsold papers are returned to the publisher for a credit of 10¢. Any unsatisfied demand is estimated to cost 10¢ in goodwill and lost profit. If the policy is to order a quantity equal to the preceding day’s demand, determine the average daily profit of the news vendor by simulating this system. Assume that the demand for day 0 is equal to 32.
3 An airport hotel has 100 rooms. On any given night, it takes up to 105 reservations, because of the possibility of no-shows. Past records indicate that the number of daily reservations is uniformly distributed over the integer range [96, 105]. That is, each integer number in this range has an equal probability, .1, of showing up. The no-shows are represented by the distribution in Table 17. Develop a simulation model to find the following measures of performance of this booking system: the expected number of rooms used per night and the percentage of nights when more than 100 rooms are claimed. 4 The university library has one copying machine for the students to use. Students arrive at the machine with the distribution of interarrival times shown in Table 18. The time to make a copy is uniformly distributed over the range [16, 25] seconds. Analysis of past data has shown that the number of copies a student makes during a visit has the distribution in Table 19. The librarian feels that under the present system, the lines in front of the copying machine are too long and that the time a student spends in the system (waiting time + service time) is excessive. Develop a simulation model to estimate the average length of the waiting line and the expected waiting time in the system. 5 A salesperson in a large bicycle shop is paid a bonus if he sells more than 4 bicycles a day. The probability of selling more than 4 bicycles a day is only .40. If the number of bicycles sold is greater than 4, the distribution of sales is as shown in Table 20. The shop has four different models of bicycles. The amount of the bonus paid out varies by type. The bonus for model A is $10; 40% of the bicycles sold are of this type. Model B accounts for 35% of the sales and pays a bonus of $15. Model C has a bonus rating of $20 and makes up 20% of the sales. Finally, model D pays a bonus of $25 for each sale but accounts for only 5% of the sales. Develop a simulation model to calculate the bonus a salesperson can expect in a day. 6 A heart specialist schedules 16 patients each day, 1 every 30 minutes, starting at 9 A.M. Patients are expected to arrive for their appointments at the scheduled times. However, past experience shows that 10% of all patients arrive 15 minutes early, 25% arrive 5 minutes early,
50% arrive exactly on time, 10% arrive 10 minutes late, and 5% arrive 15 minutes late. The time the specialist spends with a patient varies, depending on the type of problem. Analysis of past data shows that the length of an appointment has the distribution in Table 21. Develop a simulation model to calculate the average length of the doctor’s day.
Group B 7 Suppose we are considering the selection of the reorder point, R, of a (Q, R) inventory policy. With this policy, we order up to Q when the inventory level falls to R or less. The probability distribution of daily demand is given in Table 22. The lead time is also a random variable and has the distribution in Table 23. We assume that the “order up to” quantity for each order stays the same at 100. Our interest here is to determine the value of the reorder point, R, that minimizes the total variable inventory cost. This variable cost is the sum of the expected inventory carrying cost, the expected ordering cost, and the expected stockout cost. All stockouts are backlogged. That is, a customer waits until an item is available. Inventory carrying cost is estimated to be 20¢/unit/day and is charged on the units in inventory at the end of a day. A stockout costs $1 for every unit short. The cost of ordering is $10 per order. Orders arrive at the beginning of a day. Develop a simulation model to simulate this inventory system to find the best value of R. 8 A large car dealership in Bloomington, Indiana, employs five salespeople. All salespeople work on commission; they are paid a percentage of the profits from the cars they sell. The dealership has three types of cars: luxury, midsize, and subcompact. Data from the past few years show that the car sales per week per salesperson have the distribution in Table 24. If the car sold is a subcompact, a salesperson is given a commission of $250. For a midsize car, the commission is either $400 or $500, depending on the model sold. On the midsize cars, a commission of $400 is paid out 40% of the time, and $500 is paid out the other 60% of the time. For a luxury car, commission is paid out according to three separate rates: $1,000 with a probability of 35%, $1,500 with a probability of 40%, and $2,000 with a probability of 25%. If the distribution of type of cars sold is as shown in Table 25, what is the average commission for a salesperson in a week?
9 Consider a bank with 4 tellers. Customers arrive at an exponential rate of 60 per hour. A customer goes directly into service if a teller is idle. Otherwise, the arrival joins a waiting line. There is only one waiting line for all the tellers. If an arrival finds the line too long, he or she may decide to leave immediately (reneging). The probability of a customer reneging is shown in Table 26. If a customer joins the waiting line, we assume that he or she will stay in the system until served. Each teller serves at the same service rate. Service times are uniformly distributed over the range [3, 5]. Develop a simulation model to find the following measures of performance for
this system: (1) the expected time a customer spends in the system, (2) the percentage of customers who renege, and (3) the percentage of idle time for each teller. 10 Jobs arrive at a workshop, which has two work centers (A and B) in series, at an exponential rate of 5 per hour. Each job requires processing at both these work centers, first on A and then on B. Jobs waiting to be processed at each center can wait in line; the line in front of work center A has unlimited space, and the line in front of center B has space for only 4 jobs at a time. If this space reaches its capacity, jobs cannot leave center A. In other words, center A stops processing until space becomes available in front of B. The processing time for a job at center A is uniformly distributed over the range [6, 10]. The processing time for a job at center B is represented by the following triangular distribution:
Develop a simulation model of this system to determine the following measures of performance: (1) the expected number of jobs in the workshop at any given time, (2) the percentage of time center A is shut down because of shortage of queuing space in front of center B, and (3) the expected completion time of a job. 11 Tankers arrive at an oil port with the distribution of interarrival times shown in Table 12. The port has two terminals, A and B. Terminal B is newer and therefore more efficient than terminal A. The time it takes to unload a tanker depends on the tankerâ€™s size. A supertanker takes 4 days to unload at terminal A and 3 days at terminal B. A midsize tanker takes 3 days at terminal A and 2 days at terminal B. The small tankers take 2 days at terminal A and 1 day at terminal B. Arriving tankers form a single waiting line in the port area until a terminal becomes available for service. Service is given on an FCFS basis. The type of tankers and the frequency with which they visit this port is given by the distribution in Table 13. Develop a simulation model for this port. Compute such statistics as the average number of tankers in port, the average number of days in port for a tanker, and the percentage of idle time for each of the terminals.
Group C 1 At a manufacturing assembly line, 30 jobs arrive per hour. Each job must pass through two production stages: stage 1 and stage 2. Stage 1 takes an average of 1 minute to complete, and 1 worker is available to perform stage 1. After completing stage 1, the job immediately passes to stage 2. Stage 2 takes an average of 2 minutes to complete, and 2 workers are available to work on stage 2. After completing stage 2, each job is inspected. Inspection takes an average of 3 minutes, and 3 workers are available to perform inspection. After inspection, 10% of the jobs must be returned to stage 1, and they then repeat both stages 1 and 2. After inspection, 20% of all
jobs return to stage 2 and repeat stage 2. Assume that interarrival times and service times are exponential. a What is the average time a job spends in the system from arrival to completion? b What percentage of the time is each worker busy?
2 The United Airlines security station for Terminal C in Indianapolis has 3 X-ray machines. During the busy early morning hours, an average of 400 passengers per hour arrive at Terminal C (with exponential interarrival times). Each X-ray machine can handle an average of 150 passengers per hour (with exponential service times for X-ray machines). After going through security, 90% of the customers are free to go to their flight, but 10% must be â€œwanded.â€? Three people are available to do the wanding. Wanding requires a mean of 4 minutes, with a standard deviation of 2 minutes. a How long does it take the average passenger to pass through security? b If there were no wanding, how long would it take the average passenger to pass through security? c Which would improve the situation more: adding an X-ray machine or adding an additional person to perform wanding?
3 Consider an emergency room. An average of 10 patients arrive per hour (interarrival times are exponential). Upon entering, the patient fills out a form. Assume that this always takes 5 minutes. Then each patient is processed by one of two registration clerks. This takes an average of 7 minutes (exponentially distributed). Then each patient walks 2 minutes to a waiting room and waits for one of 4 doctors. The time a doctor takes to see a patient averages 20 minutes, with a standard deviation of 10 minutes. a On the average, how long does a patient spend in the emergency room b On the average, how much of this time is spent waiting for a doctor? c What percentage of the time is each doctor busy?
4 The Indiana University Credit Union has 4 tellers working. It takes an average of 3 minutes (exponentially distributed) to serve a customer. Assume that an average of 60 customers per hour arrive at the Credit Union (interarrival times are exponential). a How long do customers have to wait for a teller? b What percentage of the time is a teller busy?
5 A pharmacist has to fill an average of 15 orders per hour (interarrival times are exponentially distributed). 80% of the orders are relatively simple and take 2 minutes to fill. 20% of the orders take 10 minutes to fill. a What percentage of the time is the pharmacist busy? b On average, how long does it take to get a prescription filled?