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operations research an introduction

Questions and Answers of Operations Research An Introduction

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
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
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
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 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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
5 Ethan Sherwood owns two printing presses, on which he prints two types of jobs. At the beginning of each day, there is a.5 probability that a type 1 job will arrive, a .1 probability that a type 2
4 During any day, I may own either 0 or 1 share of a stock.The price of the stock is governed by the Markov chain shown in Table 11. At the beginning of a day in which I own a share of stock, I may
3 Suppose you are using the policy iteration method to determine an optimal policy for an MDP. How might you use LINDO to solve the value determination equations?
2 Priceler Auto Corporation must determine whether or not to give consumers 8% or 11% financing on new cars. If Priceler gives 8% financing during the current month, the probability distribution of
1 A warehouse has an end-of-period capacity of 3 units.During a period in which production takes place, a setup cost of $4 is incurred. A $1 holding cost is assessed against each unit of a period’s
At the beginning of each week, a machine is in one of four conditions (states): excellent(E), good (G), average (A), or bad (B). The weekly revenue earned by a machine in each type of condition is as
15 In Example 9, assume T 10 and pt |t|/10. Using a spreadsheet, solve for Robert’s optimal strategy.
14 In Example 10, suppose pt t/60 and dt t. Using a spreadsheet, solve for Dirk’s optimal strategy. (Hint: The possible states are 50, 51, . . . , 1,880 (thousands).)
13 At 7 A.M., eight people leave their cars for repair at Harry’s Auto Repair Shop. If person i’s car is ready by time t (7 A.M. time 0, and so on), he will pay Harry ri(t) dollars.For example,
12 Bailey Bliss is the campaign manager for Walter Glenn’s presidential campaign. He has D dollars to allocate to T winner-take-all primaries. If xt dollars are allocated to primary t, then with
11 Georgia Stein is the new owner of the New York Yankees. Each season, Georgia must decide how much money to spend on the free agent draft. During each season, Georgia can spend any amount of money
10 An advertising firm has D dollars to spend on reaching customers in T separate markets. Market t consists of kt people. If x dollars are spent on advertising in market t, the probability that a
9 Martha and Ken Allen want to sell their house. At the beginning of each day, they receive an offer. We assume that from day to day, the sizes of the offers are independent random variables and that
8 Consider the following probabilistic inventory model:a At the beginning of each period, a firm observes its inventory position.b Then the firm decides how many units to produce during the current
7 The National Cat Foundling Home encourages people to adopt its cats, but (because of limited funds) it allows each prospective owner to inspect only four cats before choosing one of them to take
6 I own a single share of Wivco stock. I must sell my share at the beginning of one of the next 30 days. Each day, the price of the stock changes. With probability q(x), the price tomorrow will
5 At the beginning of each week, a machine is either running or broken down. If the machine runs throughout the week, it earns revenues of $100. If the machine breaks down during a week, it earns no
4 In the time interval between t and t 1 seconds before the departure of Braneast Airlines Flight 313, there is a probability pt that the airline will receive a reservation for the flight and a
3 Consider a machine that may be in any one of the states 0, 1, 2, . . . . At the beginning of each month, the state of the machine is observed, and it is decided whether to replace or keep the

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