# Introduction to Management Science Study Set 3

## Quiz 14 :Simulation

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The U.S. Army Corps of Engineers has historically constructed dams on various rivers in the southeastern United States. Its primary instrument for evaluating and selecting among many projects under consideration is benefit-cost analysis. The Corps estimates both the annual benefits deriving from a project in several different categories and the annual costs and then divides the total benefits by the total costs to develop a benefit-cost ratio. This ratio is then used by the Corps and Congress to compare numerous projects under consideration and select those for funding. A benefit-cost ratio greater than 1.0 indicates that the benefits are greater than the costs; and the higher a project's benefit-cost ratio, the more likely it is to be selected over projects with lower ratios. The Corps is evaluating a project to construct a dam over the Spradlin Bluff River in southwest Georgia. The Corps has identified six traditional areas in which benefits will accrue: flood control, hydroelectric power, improved navigation, recreation, fish and wildlife, and area commercial redevelopment. The Corps has made three estimates (in dollars) for each benefit-a minimum possible value, a most likely value, and a maximum benefit value. These benefit estimates are as follows: There are two categories of costs associated with a construction project of this type-the total capital cost, annualized over 100 years (at a rate of interest specified by the government), and annual operation and maintenance costs. The cost estimates for this project are as follows: Using Crystal Ball, determine a simulated mean benefit-cost ratio and standard deviation. What is the probability that this project will have a benefit-cost ratio greater than 1.0
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Calculate the benefit cost ratio of the project, as shown below:
Step1: Estimate the total of benefits accruing from the projects as shown under:
The average or the most likely event is taken as the mean of the benefit for each criterion.
Sum of the benefits is the total of the benefit from each 6 criteria's.
Step2: Estimate the total costs in undertaking the project, as shown below:
The total cost is the sum of the cost accruing from each operational and maintenance source.
Step3: Calculate the Benefit cost ratio by using the formula, as shown below:
B/C ratio = 25,915,800/19,040,500
B/C ratio = 1.361
Step 5: Estimate the probability of the Benefit cost ratio greater than 1, is shown as below:
P (x = 1)
=-2.48
Calculate the probability for the given z value table. The probability for z 2.48 is 0.4934.
Thus, the probability of benefit cost ratio greater than 1 = 0.493+0.5 = 0.993

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First American Bank is trying to determine whether it should install one or two drive-through teller windows. The following probability distributions for arrival intervals and service times have been developed from historical data: Assume that in the two-server system, an arriving car will join the shorter queue. When the queues are of equal length, there is a 50-50 chance the driver will enter the queue for either window. a. Simulate both the one- and two-teller systems. Compute the average queue length, waiting time, and percentage utilization for each system. b. Discuss your results in (a) and suggest the degree to which they could be used to make a decision about which system to employ.
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(a)The bank with a single drive through line can be simulated as shown below:
The average queue length is.444 cars. The average waiting time is 1.27 minutes. The utilization is 1.05 x 100 = 105%.
The average waiting times for the double drive through windows at the bank are shown below:
The average queue length is.06 cars. The average waiting time is 1 minutes. The utilization is.52 x 100 = 52%.
(b) The average waiting time for the single teller system is not unreasonable at this time, however to manage future customer growth, the second teller is advisable since the utilization is over 100%.

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Sound Warehouse in Georgetown sells CD players (with speakers), which it orders from Fuji Electronics in Japan. Because of shipping and handling costs, each order must be for five CD players. Because of the time it takes to receive an order, the warehouse outlet places an order every time the present stock drops to five CD players. It costs $100 to place an order. It costs the warehouse$400 in lost sales when a customer asks for a CD player and the warehouse is out of stock. It costs $40 to keep each CD player stored in the warehouse. If a customer cannot purchase a CD player when it is requested, the customer will not wait until one comes in but will go to a competitor. The following probability distribution for demand for CD players has been determined: The time required to receive an order once it is placed has the following probability distribution: The warehouse has five CD players in stock. Orders are always received at the beginning of the week. Simulate Sound Warehouse's ordering and sales policy for 20 months, using the first column of random numbers in Table. Compute the average monthly cost. Table Free Essay Answer: Answer: Use the Excel spreadsheet below to simulate the demand for CD players each month as well as the time to receive an order. Random numbers have been inputted from the table with two decimal places due to the probability used in the problem. This is to ensure the lookup table works correctly. The average monthly cost is$594 a month.

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Each semester, the students in the college of business at State University must have their course schedules approved by the college adviser. The students line up in the hallway outside the adviser's office. The students arrive at the office according to the following probability distribution: The time required by the adviser to examine and approve a schedule corresponds to the following probability distribution: Simulate this course approval system for 90 minutes. Compute the average queue length and the average time a student must wait. Discuss these results.
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A robbery has just been committed at the Corner Market in the downtown area of the city. The market owner was able to activate the alarm, and the robber fled on foot. Police officers arrived a few minutes later and asked the owner, "How long ago did the robber leave" "He left only a few minutes ago," the store owner responded. "He's probably 10 blocks away by now," one of the officers said to the other. "Not likely," said the store owner. "He was so stoned on drugs that I bet even if he has run 10 blocks, he's still only within a few blocks of here! He's probably just running in circles!" Perform a simulation experiment that will test the store owner's hypothesis. Assume that at each corner of a city block there is an equal chance that the robber will go in any one of the four possible directions: north, south, east, or west. Simulate for five trials and then indicate in how many of the trials the robber is within 2 blocks of the store.
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The Paymore Rental Car Agency rents cars in a small town. It wants to determine how many rental cars it should maintain. Based on market projections and historical data, the manager has determined probability distributions for the number of rentals per day and rental duration (in days only) as shown in the following tables: Design a simulation experiment for the car agency and simulate using a fleet of four rental cars for 10 days. Compute the probability that the agency will not have a car available upon demand. Should the agency expand its fleet Explain how a simulation experiment could be designed to determine the optimal fleet size for the Paymore Agency.
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Solve Problem at the end of Chapter by using simulation. Problem The financial success of the Downhill Ski Resort in the Blue Ridge Mountains is dependent on the amount of snowfall during the winter months. If the snowfall averages more than 40 inches, the resort will be successful; if the snowfall is between 20 and 40 inches, the resort will receive a moderate financial return; and if snowfall averages less than 20 inches, the resort will suffer a financial loss. The financial return and probability, given each level of snowfall, follow: A large hotel chain has offered to lease the resort for the winter for $40,000. Compute the expected value to determine whether the resort should operate or lease. Explain your answer. Essay Answer: Tags Choose question tag The Dynaco Manufacturing Company produces a product in a process consisting of operations of five machines. The probability distribution of the number of machines that will break down in a week follows: a. Simulate the machine breakdowns per week for 20 weeks. b. Compute the average number of machines that will break down per week. Essay Answer: Tags Choose question tag The time between arrivals of oil tankers at a loading dock at Prudhoe Bay is given by the following probability distribution: The time required to fill a tanker with oil and prepare it for sea is given by the following probability distribution: a. Simulate the movement of tankers to and from the single loading dock for the first 20 arrivals. Compute the average time between arrivals, average waiting time to load, and average number of tankers waiting to be loaded. b. Discuss any hesitation you might have about using your results for decision making. Essay Answer: Tags Choose question tag The Saki automobile dealer in the Minneapolis-St. Paul area orders the Saki sport compact, which gets 50 miles per gallon of gasoline, from the manufacturer in Japan. However, the dealer never knows for sure how many months it will take to receive an order once it is placed. It can take 1, 2, or 3 months, with the following probabilities: The dealer orders when the number of cars on the lot gets down to a certain level. To determine the appropriate level of cars to use as an indicator of when to order, the dealer needs to know how many cars will be demanded during the time required to receive an order. Simulate the demand for 30 orders and compute the average number of cars demanded during the time required to receive an order. At what level of cars in stock should the dealer place an order Essay Answer: Tags Choose question tag A CPM/PERT project network has probabilistic activity times (x) as shown on each branch of the network; for example, activity 1-3 has a.40 probability that it will be completed in 6 weeks and a.60 probability it will be completed in 10 weeks: Simulate the project network 10 times and determine the critical path each time. Compute the average critical path time and the frequency at which each path is critical. How does this simulation analysis of the critical path method compare with regular CPM/PERT analysis Essay Answer: Tags Choose question tag State University is playing Tech in their annual football game on Saturday. A sportswriter has scouted each team all season and accumulated the following data: State runs four basic plays- a sweep, a pass, a draw, and an off tackle; Tech uses three basic defenses-a wide tackle, an Oklahoma, and a blitz. The number of yards State will gain for each play against each defense is shown in the following table: The probability that State will run each of its four plays is shown in the following table: The probability that Tech will use each of its defenses follows: The sportswriter estimates that State will run 40 plays during the game. The sportswriter believes that if State gains 300 or more yards, it will win; however, if Tech holds State to fewer than 300 yards, it will win. Use simulation to determine which team the sportswriter will predict to win the game. Essay Answer: Tags Choose question tag Concerned about recent weather-related disasters, fires, and other calamities at universities around the country, university administrators at Tech have initiated several planning projects to determine how effectively local emergency facilities can handle such situations. One of these projects has focused on the transport of disaster victims from campus to the five major hospitals in the area: Montgomery Regional, Raeford Memorial, County General, Lewis Galt, and HGA Healthcare. The project team would like to determine how many victims each hospital might expect in a disaster and how long it would take to transport victims to the hospitals. However, one of the problems the project team faces is the lack of data on disasters, since they occur so infrequently. The project team has looked at disasters at other schools and has estimated that the minimum number of victims that would qualify an event as a disaster for the purpose of initiating a disaster plan is 10. The team has further estimated that the largest number of victims in any disaster would be 200, and based on limited data from other schools, they believe the most likely number of disaster victims is approximately 50. Because of the lack of data, it is assumed that these parameters best define a triangular distribution. The emergency facilities and capabilities at the five area hospitals vary. It has been estimated that in the event of a disaster situation, the victims should be dispersed to the hospitals on a percentage basis based on the hospitals' relative emergency capabilities, as follows: 25% should be sent to Montgomery Regional, 30% to Raeford Memorial, 15% to County General, 10% to Lewis Galt, and 20% to HGA Healthcare. The proximity of the hospitals to Tech also varies. It is estimated that transport times to each of the hospitals is exponentially distributed with an average time of 5 minutes to Montgomery Regional, 10 minutes to Raeford Memorial, 20 minutes to County General, 20 minutes to Lewis Galt, and 15 minutes to HGA Healthcare. (It is assumed that each hospital has two emergency vehicles, so that one leaves Tech when the other leaves the hospital, and consequently, one arrives at Tech when the other arrives at the hospital. Thus, the total transport time will be the sum of transporting each victim to a specific hospital.) a. Perform a simulation analysis using Crystal Ball to determine the average number of victims that can be expected at each hospital and the average total time required to transport victims to each hospital. b. Suppose that the project team believes they cannot confidently assume that the number of victims will follow a triangular distribution using the parameters they have estimated. Instead, they believe that the number of victims is best estimated using a normal distribution with the following parameters for each hospital: a mean of 6 minutes and a standard deviation of 4 minutes for Montgomery Regional; a mean of 11 minutes and a standard deviation of 4 minutes for Raeford Memorial; a mean of 22 minutes and a standard deviation of 8 minutes for County General; a mean of 22 minutes and a standard deviation of 9 minutes for Lewis Galt; and a mean of 15 minutes and a standard deviation of 5 minutes for HGA Healthcare. Perform a simulation analysis using this revised information. c. Discuss how this information might be used for planning purposes. How might the simulation be altered or changed to provide additional useful information Not Answered There is no answer for this question Tags Choose question tag Loebuck Grocery orders milk from a dairy on a weekly basis. The manager of the store has developed the following probability distribution for demand per week (in cases): The milk costs the grocery$10 per case and sells for $16 per case. The carrying cost is$0.50 per case per week, and the shortage cost is $1 per case per week. Simulate the ordering system for Loebuck Grocery for 20 weeks. Use a weekly order size of 16 cases of milk and compute the average weekly profit for this order size. Explain how the complete simulation for determining order size would be developed for this problem. Essay Answer: Tags Choose question tag Every time a machine breaks down at the Dynaco Manufacturing Company (Problem), 1, 2, or 3 hours are required to fix it, according to the following probability distribution: a. Simulate the repair time for 20 weeks and then compute the average weekly repair time. b. If the random numbers that are used to simulate breakdowns per week are also used to simulate repair time per breakdown, will the results be affected in any way Explain. c. If it costs$50 per hour to repair a machine when it breaks down (including lost productivity), determine the average weekly breakdown cost. d. The Dynaco Company is considering a preventive maintenance program that would alter the probabilities of machine breakdowns per week as shown in the following table: The weekly cost of the preventive maintenance program is $150. Using simulation, determine whether the company should institute the preventive maintenance program. Problem The Dynaco Manufacturing Company produces a product in a process consisting of operations of five machines. The probability distribution of the number of machines that will break down in a week follows: a. Simulate the machine breakdowns per week for 20 weeks. b. Compute the average number of machines that will break down per week. Essay Answer: Tags Choose question tag The time between arrivals of cars at the Petroco Service Station is defined by the following probability distribution: a. Simulate the arrival of cars at the service station for 20 arrivals and compute the average time between arrivals. b. Simulate the arrival of cars at the service station for 1 hour, using a different stream of random numbers from those used in (a) and compute the average time between arrivals. c. Compare the results obtained in (a) and (b). Essay Answer: Tags Choose question tag A city is served by two newspapers-the Tribune and the Daily News. Each Sunday readers purchase one of the newspapers at a stand. The following matrix contains the probabilities of a customer's buying a particular newspaper in a week, given the newspaper purchased the previous Sunday: Simulate a customer's purchase of newspapers for 20 weeks to determine the steady-state probabilities that a customer will buy each newspaper in the long run. Essay Answer: Tags Choose question tag Simulate the decision situation described in Problem 1(a) at the end of Chapter for 20 weeks, and recommend the best decision. Problem 1(a) A concessions manager at the Tech versus A M football game must decide whether to have the vendors sell sun visors or umbrellas. There is a 30% chance of rain, a 15% chance of overcast skies, and a 55% chance of sunshine, according to the weather forecast in College Junction, where the game is to be held. The manager estimates that the following profits will result from each decision, given each set of weather conditions: a. Compute the expected value for each decision and select the best one. Essay Answer: Tags Choose question tag James Banks was standing in line next to Robin Cole at Klecko's Copy Center, waiting to use one of the copy machines. "Gee, Robin, I hate this," he said. "We have to drive all the way over here from Southgate and then wait in line to use these copy machines. I hate wasting time like this." "I know what you mean," said Robin. "And look who's here. A lot of these students are from Southgate Apartments or one of the other apartments near us. It seems as though it would be more logical if Klecko's would move its operation over to us, instead of all of us coming over here." James looked around and noticed what Robin was talking about. Robin and he were students at State University, and most of the customers at Klecko's were also students. As Robin suggested, a lot of the people waiting were State students who lived at Southgate Apartments, where James also lived with Ernie Moore. This gave James an idea, which he shared with Ernie and their friend Terri Jones when he got home later that evening. "Look, you guys, I've got an idea to make some money," James started. "Let's open a copy business! All we have to do is buy a copier, put it in Terri's duplex next door, and sell copies. I know we can get customers because I've just seen them all at Klecko's. If we provide a copy service right here in the Southgate complex, we'll make a killing." Terri and Ernie liked the idea, so the three decided to go into the copying business. They would call it JET Copies, named for James, Ernie, and Terri. Their first step was to purchase a copier. They bought one like the one used in the college of business office at State for$18,000. (Terri's parents provided a loan.) The company that sold them the copier touted the copier's reliability, but after they bought it, Ernie talked with someone in the dean's office at State, who told him that the University's copier broke down frequently and when it did, it often took between 1 and 4 days to get it repaired. When Ernie told this to Terri and James, they became worried. If the copier broke down frequently and was not in use for long periods while they waited for a repair person to come fix it, they could lose a lot of revenue. As a result, James, Ernie, and Terri thought they might need to purchase a smaller backup copier for $8,000 to use when the main copier broke down. However, before they approached Terri's parents for another loan, they wanted to have an estimate of just how much money they might lose if they did not have a backup copier. To get this estimate, they decided to develop a simulation model because they were studying simulation in one of their classes at State. To develop a simulation model, they first needed to know how frequently the copier might break down- specifically, the time between breakdowns. No one could provide them with an exact probability distribution, but from talking to staff members in the college of business, James estimated that the time between breakdowns was probably between 0 and 6 weeks, with the probability increasing the longer the copier went without breaking down. Thus, the probability distribution of breakdowns generally looked like the following: Next, they needed to know how long it would take to get the copier repaired when it broke down. They had a service contract with the dealer that "guaranteed" prompt repair service. However, Terri gathered some data from the college of business from which she developed the following probability distribution of repair times: Finally, they needed to estimate how much business they would lose while the copier was waiting for repair. The three of them had only a vague idea of how much business they would do but finally estimated that they would sell between 2,000 and 8,000 copies per day at$0.10 per copy. However, they had no idea about what kind of probability distribution to use for this range of values. Therefore, they decided to use a uniform probability distribution between 2,000 and 8,000 copies to estimate the number of copies they would sell per day. James, Ernie, and Terri decided that if their loss of revenue due to machine downtime during 1 year was \$12,000 or more, they should purchase a backup copier. Thus, they needed to simulate the breakdown and repair process for a number of years to obtain an average annual loss of revenue. However, before programming the simulation model, they decided to conduct a manual simulation of this process for 1 year to see if the model was working correctly. Perform this manual simulation for JET Copies and determine the loss of revenue for 1 year.