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TALLER 2 APLICACIONES DE EXCEL 1.

Use the linear congruential generator to obtain a sequence of 10, 100 and 1000 random numbers, given that a=17, c=43, m=100 and x0=31. What happened y each sequence? 2.

Consider a continuos random variable with the following pdf:

Develop a process generator for these breakdown times using the inverse transformation method and the acceptance-rejection method 3.

A job shop manager wants to develop a simulation model to help schedule jobs through the shop. He has evaluated the completion times for all the different types of jobs. For one particular job, the times to completion can be represented by the following triangular distribution:

Develop a process generator for this distribution using the inverse transformation method. 4.

In August 2007, a car dealer is trying to determine how many 2008 cars should be ordered. Each car ordered in August 2007 cost $10000. The demand for dealer’s 2008 models has the probability distribution shown in Table 1. Each car sells for $15000. If demand for 2008 car exceeds the number of cars ordered in August, the dealer must reorder at a cost of $12000 per car. Excess cars may be disposed of at $9000 per car. Use simulation to determine how many cars should be ordered in August. For your optimal order quantity find a 95% confidence interval for your expected profit. Table 1 Nº Demand Probability 20 .30 25 .15 30 .15 35 .20 40 .20

5.

Six months before its annual convention the American Medical Association must determine how many rooms to reserve. At this time the AMA can reserve rooms at a cost of $50 per room. The AMA believes the number of doctors attending the convention will be normality distributed with a mean of 50000 and a standard deviation of 1000. If the number of people attending the convention exceeds the number of rooms reserved, extra rooms must be reserved at a cost of $80 per room. Use simulation to determine the number of rooms that should be reserved to minimize the expected cost the AMA.

6.

A doctor’s office schedules patients at 15-minutre intervals beginning at 9:00 and ending at 4:00. Patients are equally likely to arrive at any time within 5 minutes of their appointment. The number of minutes the doctor spends with a patient is governed by the distribution in Table 2: Estimate the probability that the doctor will be able to leave by 5:00 On the average, how many patients are presented in the office? Table 2 Time in minutes Probability 10 .6 20 .2 30 .2

7.

You currently have $100. Each week you can invest any amount of money you currently have in a risky investment. With the probability 0.4 the amount you invest is tripled (if you invest $100, you receive $300), and with probability 0.6 the amount you invest is lost. Consider the following investment strategies: a) each week invest 10% of you money b) each week invest 30% of you money c) Each week invest 50%of you money. Simulate 100 weeks of each strategy 50 times. Which strategy appears to be best? Compare with the Kelly criteria.

8.

A news vendor sells newspaper 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 en Table 3. A paper cost the vendor 0.20. The vendor sells the paper for 0.30. Any unsatisfied demand is estimated to cost 0.10 in goodwill and lost profit. If the policy is 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. Table 3 Demand per day Probability 30 .05 31 .15 32 .22 33 .38 34 .14 35 .06

9.

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 reservation is uniformly distributed over the integer range has [96, 105]. That is, each integer number in this range has an equal probability. The no-shows are represented by the distribution in Table 4. 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. Table 4 Number of Probability No-shows 0 .10 1 .20 2 .25 3 .30 4 .10 5 .05

10. 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 0.40. If the number of bicycles sold is greater than 4, the distribution of sales is as shown in Table 5. 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. Table 5 Nº of Probability Bicycles sold 5 .35 6 .45 7 .15 8 .05 11. 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 schedules times. However, past experience shows that 10% of all patients arrive 15 minutes early, 25% arrive 5 minutes early, 50% arrives 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 appointment has the Table 6. Develop a simulation model to calculate the average length of the doctor’s day. Table 6 Length of Probability appointment (minutes)

24 27 30 33 36 39

.10 .20 .40 .15 .10 .05

12. 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 7. The lead time is also a random variable and has the distribution in Table 8. 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 cost $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. Table 7 Daily Probability demand (units) 12 .05 13 .15 14 .25 15 .35 16 .15 17 .05 Table 8 Lead Probability time (days) 1 .20 2 .30 3 .35 4 .15 13. A large car dealership in Bloomington, Indiana employs five salespeople. All salespeople work on commission; they are paid a percentage of the profits from de cars they sell. The dealership has three types of cars: luxury, midsize, and subcompact. Data from the past few years show that he car sales per week per salesperson have the distribution in Table 9. 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 the three separates rates: $1000 with a probability of 35%, $1500 with a probability of 40% and $2000 with a probability of 25%. If the distribution of type of cars sold is shown in Table 10, what is the average commission for a salesperson in a week? Table 9 Nº Probability of cars sold 0 .10 1 .15 2 .20 3 .25 4 .20 5 .10

Table 10 Type of car sold Subcompact Midsize Luxury

Probability .40 .35 .25

14. 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 11. If a customer joins the waiting line, we assume that he or she 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 the 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. Table 11 Length of Probability of Queue (q) reneging 6≤q≤8 .20 9 ≤ q ≤ 10 .40 11 ≤ q ≤ 14 .60 q > 14 .80 15. Jobs arrive at 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 time. If this space reaches its capacity, jobs cannot leave center A. In other words, center A stops processing until spaces become 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 job at center B is represented by the following triangular distribution: