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operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
An average of 20 ambulance calls per hour are received by Gotham City Hospital. An ambulance requires an average of 20 minutes to pick up a patient and take the patient to the hospital. The ambulance is then available to pick up another patient. How many ambulances should the hospital have to
12 Suppose we eliminate disk 2. What will happen to the system response time? In this problem, you may assume that all requests that leave the CPU go to disk 1.
11 Suppose we install a cache for disk 2. This will increase the mean time taken for a CPU visit by 30% and the mean time for a visit to disk 2 by 10%. On the other hand, the cache for disk 2 ensures that half the time the job was going to go to disk 2 the job will actually stay at the CPU and be
10 Suppose the file server in Problem 9 now has 8 clients.Answer the questions in Problem 9.
9 Jobs arrive to a file server consisting of a CPU and two disks (disk 1 and disk 2). Currently there are six clients, and an average of three jobs per second arrive. Each visit to the CPU takes an average of .01 second, each visit to disk 1 takes an average of .02 second, and each visit to disk 2
7 Before completing production, a product must pass through three stages of production. On the average, a new product begins at stage 1 every 6 minutes. The average time it takes to process the product at each stage is as follows:stage 1, 3 minutes; stage 2, 2 minutes; stage 3, 1 minute.After
6 Consider a queuing system consisting of three stations in series. Each station consists of a single server, who can process an average of 20 jobs per hour (processing times at each station are exponential). An average of 10 jobs per hour arrive (interarrival times are exponential) at station
5 An average of 10 jobs per hour arrive at a job shop.Interarrival times of jobs are exponentially distributed. It takes an average of 1 30 minutes (exponentially distributed) to complete a job. Unfortunately, 1 3of all completed jobs need to be reworked. Thus, with probability 1 3, a completed job
4 An average of 120 students arrive each hour (interarrival times are exponential) at State College’s Registrar’s Office to change their course registrations. To complete this process, a person must pass through three stations. Each station consists of a single server. Service times at each
3 Consider the following queuing systems:System 1 An average of 40 customers arrive each hour;interarrival times are exponential. Customers must complete two types of service before leaving the system. The first server takes an average of 30 seconds (exponentially distributed) to perform type 1
2 Consider an automobile assembly line in which each car undergoes two types of service: painting, then engine installation. Each hour, an average of 22.4 unpainted chassis arrive at the assembly line. It takes an average of 2.4 minutes to paint a car and an average of 3.75 minutes to install an
1 A Social Security Administration branch is considering the following two options for processing applications for social security cards:Option 1 Three clerks process applications in parallel from a single queue. Each clerk fills out the form for the application in the presence of the applicant.
Consider two servers. An average of 8 customers per hour arrive from outside at server 1, and an average of 17 customers per hour arrive from outside at server 2. Interarrival times are exponential. Server 1 can serve at an exponential rate of 20 customers per hour, and server 2 can serve at an
The last two things that are done to a car before its manufacture is complete are installing the engine and putting on the tires. An average of 54 cars per hour arrive requiring these two tasks. One worker is available to install the engine and can service an average of 60 cars per hour. After the
10 Bectol, Inc. is building a dam. A total of 10 million cu ft of dirt is needed to construct the dam. A bulldozer is used to collect dirt for the dam. Then the dirt is moved via dumpers to the dam site. Only one bulldozer is available, and it rents for $100 per hour. Bectol can rent, at $40 per
9 An army has 200 tanks. Tanks need maintenance 10 times per year, and maintenance takes an average of 2 days.The army would like to have an average of at least 180 tanks working. How many repairmen are needed? Assume exponential interarrival and service times. (Hint: Use a oneway data table.)
8 Allbest airlines has 100 planes. Planes break down an average of twice a year and take one week to fix. Assuming the times between breakdowns and repairs are exponential, how many repairmen are needed to ensure that there is at least a 95% chance that 90 or more planes are available?(Hint: Use a
7 (Requires use of a spreadsheet or LINGO) The machine repair model may often be used to approximate the behavior of a computer’s CPU (central processing unit). Suppose that 20 terminals (assumed to always be busy) feed the CPU.After the CPU responds to a user, he or she takes an average of 80
6 For the machine repair model, show that W K/l- (1/l).
4 This problem illustrates balking. The Oryo Cookie Ice Cream Shop in Dunkirk Square has three competitors. Since people don’t like to wait in long lines for ice cream, the arrival rate to the Oryo Cookie Ice Cream Shop depends on the number of people in the shop. More specifically, while j 4
3 Gotham City has 10,000 streetlights. City investigators have determined that at any given time, an average of 1,000 lights are burned out. A streetlight burns out after an average of 100 days of use. The city has hired Mafia, Inc., to replace burned-out lamps. Mafia, Inc.’s contract states that
2 My dog just had 3 frisky puppies who jump in and out of their whelping box. A puppy spends an average of 10 minutes (exponentially distributed) in the whelping box before jumping out. Once out of the box, a puppy spends an average of 15 minutes (exponentially distributed) before jumping back into
1 A laundromat has 5 washing machines. A typical machine breaks down once every 5 days. A repairer can repair a machine in an average of 2.5 days. Currently, three repairers are on duty. The owner of the laundromat has the option of replacing them with a superworker, who can repair a machine in an
The Gotham Township Police Department has 5 patrol cars. A patrol car breaks down and requires service once every 30 days. The police department has two repair workers, each of whom takes an average of 3 days to repair a car. Breakdown times and repair times are exponential.1 Determine the average
4 Consider an M/G/1/GD/∞/∞ queuing system in which an average of 10 arrivals occur each hour. Suppose that each customer’s service time follows an Erlang distribution, with rate parameter 1 customer per minute and shape parameter 4.a Find the expected number of customers waiting in line.b
3 An average of 40 cars per hour arrive to be painted at a single-server GM painting facility. 95% of the cars require 1 minute to paint; 5% must be painted twice and require 2.5 minutes to paint. Assume that interarrival times are exponential.a On the average, how long does a car wait before being
2 Using the fact that Ls m l, demonstrate that for an M/G/1/GD/∞/∞ queuing system, the probability that the server is busy is r m l.
1 An average of 20 cars per hour arrive at the drive-in window of a fast-food restaurant. If each car’s service time is 2 minutes, how many cars (on the average) will be waiting in line? Assume exponential interarrival times.
3 There are at present 40 solar energy construction firms in the state of Indiana. An average of 20 solar energy construction firms open each year in the state. The average firm stays in business for 10 years. If present trends continue, what is the expected number of solar energy construction
2 The State U doctoral program in business admits an average of 25 doctoral students each year. If a doctoral student spends an average of 4 years in residence at State U, how many doctoral students would one expect to find there?
1 Each week, the Columbus Record Club attracts 100 new members. Members remain members for an average of one year (1 year 52 weeks). On the average, how many members will the record club have?
During each year, an average of 3 ice cream shops open up in Smalltown. The average time that an ice cream shop stays in business is 10 years. On January 1, 2525, what is the average number of ice cream shops that you would find in Smalltown? If the time between the opening of ice cream shops is
15 (Requires the use of a spreadsheet or LINGO) The Carco plant in Bedford produces windshield wipers for Fords. In a given day, each machine in the plant can produce 1,000 wipers. The plant operates 250 days per year, and Ford will need 3 million wipers per year. It costs $50,000 per year to
14 Consider the following two queuing systems:System 1 An M/M/1 system with arrival rate l and service rate 3m.System 2 An M/M/3 system with arrival rate l and each server working at rate m.Without doing extensive calculations, which system will have the smaller Wand L? (Hint: Write down the
13 A muffler shop has three mechanics. Each mechanic takes an average of 45 minutes to install a new muffler.Suppose an average of 1 customer per hour arrives. What is the expected number of mechanics that are busy at any given time? Answer this question without assuming that service times and
12 An average of 100 customers per hour arrive at Gotham City Bank. It takes a teller an average of 2 minutes to serve a customer. Interarrival and service times are exponential.The bank currently has four tellers working. The bank manager wants to compare the following two systems with regard to
11 A Northwest Airlines ticket counter forecasts that 200 people per hour will need to check in. It takes an average of two minutes to service a customer. Assume that interarrival times and service times are exponential and that all customers wait in a single line for the first available agent a If
10 A data storage system consists of 3 disk drives sharing a common queue. An average of 50 storage requests arrive per second. The average time required to service a request is .03 second. Assuming that interarrival times and service times are exponential, determine:a The probability that a given
9 An average of 40 students per hour arrive at the MBA computing lab. The average student uses a computer for 20 minutes. Assume exponential interarrival and service times.a If we want the average time a student waits for a PC to be at most 10 minutes, how many computers should the lab have?b If we
8 An average of 300 customers per hour arrive at a huge branch of bank 2. It takes an average of 2 minutes to serve each customer. It costs $10 per hour to keep open a teller window, and the bank estimates that it will lose $50 in future profits for each hour that a customer waits in line.How many
7 An average of 50 customers per hour arrive at a small post office. Interarrival times are exponentially distributed.Each window can serve an average of 25 customers per hour. Service times are exponentially distributed. It costs$25 per hour to open a window, and the post office values the time a
6 An average of 90 patrons per hour arrive at a hotel lobby(interarrival times are exponential), waiting to check in. At present, there are 5 clerks, and patrons are waiting in a single line for the first available clerk. The average time for a clerk to service a patron is 3 minutes (exponentially
5 An average of 100 customers arrive each hour at the Gotham City Bank. The average service time for each customer is 1 minute. Service times and interarrival times are exponential. The manager wants to ensure that no more than 1% of all customers will have to wait in line for more than 5 minutes.
4 MacBurger’s is attempting to determine how many servers (or lines) should be available during the breakfast shift. During each hour, an average of 100 customers arrive at the restaurant. Each line or server can handle an average of 50 customers per hour. A server costs $5 per hour, and the cost
3 In this problem, all interarrival and service times are exponential.a At present, the finance department and the marketing department each have one typist. Each typist can type 25 letters per day. Finance requires that an average of 20 letters per day be typed, and marketing requires that an
2 A small bank is trying to determine how many tellers to employ. The total cost of employing a teller is $100 per day, and a teller can serve an average of 60 customers per day.An average of 50 customers per day arrive at the bank, and both service times and interarrival times are exponential. If
1 A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 18 customers arrive each hour, and the average checkout time for a customer is 4 minutes. Interarrival times and service times are exponential, and the system may be modeled as an M/M/s/GD/∞/∞
The manager of a bank must determine how many tellers should work on Fridays. For every minute a customer stands in line, the manager believes that a delay cost of 5¢ is incurred.An average of 2 customers per minute arrive at the bank. On the average, it takes a teller 2 minutes to complete a
Consider a bank with two tellers. An average of 80 customers per hour arrive at the bank and wait in a single line for an idle teller. The average time it takes to serve a customer is 1.2 minutes. Assume that interarrival times and service times are exponential. Determine 1 The expected number of
7 A small mail order firm Seas Beginnings has one phone line. An average of 60 people per hour call in orders, and it takes an average of 1 minute to handle a call. Time between calls and time to handle calls are exponentially distributed.If the phone line is busy, Seas Beginnings can put up to c
6 Two one-man barber shops sit side by side in Dunkirk Square. Each can hold a maximum of 4 people, and any potential customer who finds a shop full will not wait for a haircut. Barber 1 charges $11 per haircut and takes an average of 12 minutes to complete a haircut. Barber 2 charges $5 per
5 Use the answer to Problem 3 to derive the steady-state probabilities for the M/M/1/GD/c/∞ system given in Equation (34).
3 An average of 125 packets of information per minute arrive at an internet router. It takes an average of .002 second to process a packet of information. The router is designed to have a limited buffer to store waiting messages.Any message that arrives when the buffer is full is lost to the
2 An average of 40 cars per hour (interarrival times are exponentially distributed) are tempted to use the drive-in window at the Hot Dog King Restaurant. If a total of more than 4 cars are in line (including the car at the window) a car will not enter the line. It takes an average of 4
1 A service facility consists of one server who can serve an average of 2 customers per hour (service times are exponential). An average of 3 customers per hour arrive at the facility (interarrival times are assumed exponential).The system capacity is 3 customers.a On the average, how many
A one-man barber shop has a total of 10 seats. Interarrival times are exponentially distributed, and an average of 20 prospective customers arrive each hour at the shop. Those customers who find the shop full do not enter. The barber takes an average of 12 minutes to cut each customer’s hair.
15 From the time a request for data is submitted until the request is fulfilled, a database takes an average of 3 seconds to respond to a request for data. We find that the database is idle around 20% of the time. Answer the following questions, assuming that the database can be modeled as an M/M/1
14 For the M/M/1/GD/∞/∞ queuing model, show that the following results hold:a W (L 1)Ws.b Wq LWs.c Interpret the results in (a) and (b).
13 A worker at the State Unemployment Office is responsible for processing a company’s forms when it opens for business. The worker can process an average of 4 forms per week. In 2002, an average of 1.8 companies per week submitted forms for processing, and the worker had a backlog of .45 week.
12 A tire plant must produce an average of 100 tires per day. The plant produces tires in a batch of size x. The plant manager must determine the batch size x that minimizes the time a batch spends in the plant. From the time a batch of tires arrives, it takes an average of 2 1 0 of a day to set up
11 A bank is trying to determine which of two machines should be rented to process checks. Machine 1 rents for$10,000 per year and processes 1,000 checks per hour.Machine 2 rents for $15,000 per year and processes 1,600 checks per hour. Assume that the machines work 8 hours a day, 5 days a week, 50
10 Consider an airport where taxis and customers arrive(exponential interarrival times) with respective rates of 1 and 2 per minute. No matter how many other taxis are present, a taxi will wait. If an arriving customer does not find a taxi, the customer immediately leaves.a Model this system as a
8 Referring to Problem 1, suppose the airline wants to determine how many checkpoints to operate to minimize operating costs and delay costs over a ten-year period.Assume that the cost of delaying a passenger for 1 hour is$10 and that the airport is open every day for 16 hours per day. It costs $1
7 Assume that an average of 125 packets per second of information arrive to a router and that it takes an average of.002 second to process each packet. Assuming exponential interarrival and service times, answer the following questions.a What is the average number of packets waiting for entry into
6 Our local maternity ward delivers 1,500 babies per year.On the average, 5 beds in the maternity ward are filled. How long does the average mother stay in the maternity ward?
5 On a typical Saturday, Red Lobster serves 1,000 customers. The restaurant is open for 12 hours. On average, 150 customers are present. How long does an average customer spend in the restaurant?
4 A fast-food restaurant has one drive-through window.An average of 40 customers per hour arrive at the window.It takes an average of 1 minute to serve a customer. Assume that interarrival and service times are exponential.a On the average, how many customers are waiting in line?b On the average,
3 For an M/M/1/GD/∞/∞ queuing system, suppose that both l and m are doubled.a How is L changed?b How is W changed?c How is the steady-state probability distribution changed?
2 The Decision Sciences Department is trying to determine whether to rent a slow or a fast copier. The department believes that an employee’s time is worth $15 per hour. The slow copier rents for $4 per hour and it takes an employee an average of 10 minutes to complete copying(exponentially
1 Each airline passenger and his or her luggage must be checked to determine whether he or she is carrying weapons onto the airplane. Suppose that at Gotham City Airport, an average of 10 passengers per minute arrive (interarrival times are exponential). To check passengers for weapons, the airport
A local computer store sells $300,000 worth of computers per year. On average accounts receivable are $45,000. On average, how long does it take from the time a customer is billed until the store receives payment?
Our local MacDonald’s uses an average of 10,000 pounds of potatoes per week. The average number of pounds of potatoes on hand is 5,000. On the average, how long do potatoes stay in the restaurant before being used?
Machinists who work at a tool-and-die plant must check out tools from a tool center.† An average of ten machinists per hour arrive seeking tools. At present, the tool center is staffed by a clerk who is paid $6 per hour and who takes an average of 5 minutes to handle each request for tools. Since
Suppose that all car owners fill up when their tanks are exactly half full.† At the present time, an average of 7.5 customers per hour arrive at a single-pump gas station. It takes an average of 4 minutes to service a car. Assume that interarrival times and service times are both exponential.1
An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the average service time for each customer is 4 minutes, and both interarrival times and service times are exponential. Answer the following questions:1 What is the probability that the teller is idle?2 What is
3 You are doing an industry analysis of the Bloomington pizza industry. The rate (per year) at which pizza restaurants enter the industry is given by p, where p price of a pizza in dollars. The price of a pizza is assumed to be max(0, 16 .5F), where F number of pizza restaurants in
2 My home uses two light bulbs. On average, a light bulb lasts for 22 days (exponentially distributed). When a light bulb burns out, it takes an average of 2 days (exponentially distributed) before I replace the bulb.a Formulate a three-state birth–death model of this situation.b Determine the
1 Show that the values of the pj’s given in (16) do indeed satisfy the flow balance equations (14) and (14 ).
Indiana Bell customer service representatives receive an average of 1,700 calls per hour.The time between calls follows an exponential distribution. A customer service representative can handle an average of 30 calls per hour. The time required to handle a call is also exponentially distributed.
7 An average of 12 jobs per hour arrive at our departmental printer.a Use two different computations (one involving the Poisson and another the exponential random variable) to determine the probability that no job will arrive during the next 15 minutes.b What is the probability that 5 or fewer jobs
6 Suppose it is equally likely that a plane flight is 50%, 60%, 70%, 80%, or 90% full.a What fraction of seats on a typical flight are full?This is known as the flight load factor.b We are always complaining that there are never empty seats on our plane flights. Given the previous information, what
5 During the year 2000, there was an average of .022 car accident per person in the United States. Using your knowledge of the Poisson random variable, explain the truth in the statement, “Most drivers are better than average.”
4 The time between arrivals of buses follows an exponential distribution, with a mean of 60 minutes.a What is the probability that exactly four buses will arrive during the next 2 hours?b That at least two buses will arrive during the next 2 hours?c That no buses will arrive during the next 2
3 There are four sections of the third grade at Jefferson Elementary School. The number in each section is as follows: section 1, 20 students; section 2, 25 students;section 3, 35 students; section 4, 40 students. What is the average size of a third-grade section? Suppose the board of education
1 Suppose I arrive at an M/M/7/FCFS/8/∞ queuing system when all servers are busy. What is the probability that I will complete service before at least one of the seven customers in service?
3 Find the probability that the time between two consecutive orders is between 1 and 3 minutes.
Find the mean and standard deviation of the number of beers ordered between 9 P.M.and 1 A.M.
1 Find the probability that exactly 60 beers are ordered between 10 P.M. and 12 midnight.
The number of glasses of beer ordered per hour at Dick’s Pub follows a Poisson distribution, with an average of 30 beers per hour being ordered.
13 You have invented a new product: the HAL DVD player.Each of 1,000 potential customers places a different value on this product. A consumer’s valuation is equally likely to be any number between $0 and $1,000. It costs $100 to produce the HAL player. During a year in which we set a price p for
12 A syndicate of college students spends weekends gambling in Las Vegas. They begin week 1 with W dollars.At the beginning of each week, they may wager any amount of their money at the gambling tables. If they wager d dollars, then with probability p, their wealth increases by d dollars, and with
10 For $50, we can enter a raffle. We draw a certificate containing a number 100, 200, 300, . . . , 1,000. Each number is equally likely. At any time, we can redeem the highest-numbered certificate we have obtained so far for the face value of the certificate. We may enter the raffle as many times
8 The Dallas Mavericks trail by two points and have the ball with 10 seconds remaining. They must decide whether to take a two- or a three-point shot. Assume that once the Mavericks take their shot, time expires. The probability that a two-point shot is successful is TWO, and the probability that a
7 A country now has 10 units of capital. Each year, it may consume any amount of the available capital and invest the rest. Invested capital has a 50% chance of doubling and a 50% chance of losing half its value. For example, if the country invests 6 units of capital, there is a 50% chance that the
6 A machine in excellent condition earns $100 profit per week, a machine in good condition earns $70 per week, and a machine in bad condition earns $20 per week. At the beginning of any week, a machine may be sent out for repairs at a cost of $90. A machine that is sent out for repairs returns in
5 I am a contestant on the popular TV show “Tired of Fortune.” During the bonus round, I will be asked up to four questions. For each question that is correctly answered, I win a certain amount of money. One incorrect answer, however, means that I lose all the money I have previously won, and
4 At the beginning of 2004, the United States has B barrels of oil. If x barrels of oil are consumed during a year, then consumers earn a benefit (measured in dollars) of u(x). The United States may spend money on oil exploration. If d dollars are spent during a year on oil exploration, then there
3 At Hot Dog Queen Restaurant, the following sequence of events occurs during each minute:a With probability p, a customer arrives and waits in line.b Hot Dog Queen determines the rate s at which customers are served. If any customers are in the restaurant, then with probability s, one of the
2 At the beginning of each period, a company must determine how many units to produce. A setup cost of $5 is incurred during each period in which production takes place.The production of each unit also incurs a $2 variable cost.All demand must be met on time, and there is a $1 per-unit holding cost
1 A company has five sales representatives available for assignment to three sales districts. The sales in each district during the current year depend on the number of sales representatives assigned to the district and on whether the national economy has a bad or good year (see Table 12). In the
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