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operations research an introduction

Questions and Answers of Operations Research An Introduction

9 Winter’s method (with a 0.2, b 0.1, and g 0.5)is being used to forecast the number of customers served each day by Last National Bank. The bank is open Monday through Friday. At present,
9 The U.S. GNP during the years 1975–1984 is given in Table 32 in billions of dollars (file GNP.xls).a Plot x years after 1974 against GNP, and use the plot to describe how to fit a curve that
10 Suppose the true relationship between Y and time t is given by Y b0eb1t where b1 0. If we try to fit our usual linear model Y b0 b1t to the data, are we likely to encounter
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
3 We are using Winter’s method and monthly data to forecast the GDP. (All numbers are in billions of dollars.)At the end of January 2005, Lt 600 and Tt 5. We are given the following
2 We are predicting quarterly sales for soda at Gordon’s Liquor Store using Winter’s method. We are given the following information:Seasonality factors: fall 0.8 spring 1.2 winter 0.7
1 Simple exponential smoothing (with a 0.2) is being used to forecast monthly beer sales at Gordon’s Liquor Store. After observing April’s demand, the predicted demand for May is 4,000 cans of
8 Since September 11, 2001, each Pacer ticketholder’s clothing and handbags are searched. Assume that this takes exactly 10 seconds and occurs right after the ticketholder passes through the
7 At Indiana Pacer games, 10,000 fans must enter through 10 checkpoints in the hour before each game (interarrival times are exponential). It takes exactly 3 seconds to have a ticket processed. How
6 Solve Problem 5 if the service times followed a normal distribution with mean 3 minutes and standard deviation .5 minute. Use the syntax N(3,.5) to generate service times.
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
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
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.
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
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
A walk-in hospital clinic has four doctors. An average of 12 patients per hour arrive at the clinic (interarrival times are assumed to be exponential). A doctor can see an average of 4 patients per
An open queuing network consists of two servers: server 1 and server 2. An average of 8 customers per hour arrive from outside at server 1. An average of 17 customers per hour arrive from outside at
4 We are using the Holt method to predict monthly VCR sales at Highland Appliance. At the end of October, 2005, Lt 200 and Tt 10. During November, 2005, 230 VCRs are sold. At the end of November,
5 We are using simple exponential smoothing to predict monthly electric shaver sales at Hook’s Drug Store. At the end of October 2006, our forecast for December 2006 sales was 40. In November 50
6 We are using simple exponential smoothing to predict monthly auto sales at Bloomington Ford. The company believes that sales do not exhibit trend or seasonality, so simple exponential smoothing has
1 Table 25 gives data concerning pork sales (file Pork.xls).Price is in dollars per hundred lb sold, quantity sold is in billions of pounds, per-capita income is in dollars, U.S.population is in
7 The University Credit Union is open Monday through Saturday. Winter’s method is being used (with a b g 0.5) to predict the number of customers entering the bank each day. After incorporating
3 Table 27 gives the following data for monthly U.S.housing sales (in thousands of houses) for 1965–1972.a Use the years 1965–1966 to initialize the parameters for Winter’s method. Then find
17 Smalltown has two ambulances. Ambulance 1 is based at the local college, and ambulance 2 is based downtown. If a request for an ambulance comes from the college, the collegebased ambulance is sent
2 The polls are open in Gotham City from 11 A.M. to 6 P.M. The city has 3 voting machines. It takes an average of 1.5 minutes (exponentially distributed) for a voter to complete voting. The arrival
1 A single machine is used between 8 A.M. and 4 P.M. to perform EKGs (electrocardiograms). There are 3 waiting spaces, and any arrival finding no available waiting space is lost to the system. The
A small fast-food restaurant is trying to model the lunchtime rush. The restaurant opens at 11 A.M., and all customers wait in one line to have their orders filled. The arrival rate per hour at
1 English professor Jacob Bright has one typist, who types for 8 hours per day. He submits three types of jobs to the typist: tests, research papers, and class handouts. The information in Table 10
A copying facility gives shorter jobs priority over long jobs. Interarrival times for each type of job are exponential, and an average of 12 short jobs and 6 long jobs arrive each hour. Let type 1
8 A data communication network consists of three nodes, A, B, and C. Each packet transmitted contains 500 bits of information. The number of packets per second to be transmitted between each pair of
5 Suppose that interarrival times to a single-server system are exponential, but when n customers are present, there is a probability n n 1that an arrival will balk and leave the system before
5 Consider an M/G/1/GD/∞/∞ queuing system in which interarrival times are exponentially distributed with parameter l and service times have a probability density function s(t). Let Xi be the
4 Show that if p is not equal to 1 1+p+p++ p = 1-p 1-P (Hint: Recall how we evaluated 1+p+ p + ....)
9 Each machine on Widgetco’s assembly line gets out of whack an average of once a minute. Laborers are assigned to reset a machine that gets out of whack. The company pays each laborer cs dollars
2 The time between buses follows the mass function shown in Table 2. What is the average length of time one must wait for a bus? TABLE 2 Time Between Buses 30 minutes 1 hour Probability 2 hours 2
11 At the beginning of each year, an aircraft engine is in good, fair, or poor condition. It costs $500,000 to run a good engine for a year, $1 million to run a fair engine for a year, and $2 million
9 At any time, the size of a tree is 0, 1, 2, or 3. We must decide when to harvest the tree. Each year, it costs $1 to maintain the tree. It costs $5 to harvest a tree. The sales price for a tree of
Robert Blue is trying to find a parking place near his favorite restaurant. He is approaching the restaurant from the west, and his goal is to park as nearby as possible. The available parking places
For a price of $1/gallon, the Safeco Supermarket chain has purchased 6 gallons of milk from a local dairy. Each gallon of milk is sold in the chain’s three stores for $2/gallon.The dairy must buy
19 (Requires use of a spreadsheet) An average of 140 people per year apply for public housing in Boston. An average of 20 housing units per year become available.During a given year, there is a 10%
18 An average of 10 people per hour arrive (interarrival times are exponential) intending to swim laps at the local YMCA. Each intends to swim an average of 30 minutes.The YMCA has three lanes open
16 Podunk University Operations Research Department has two phone lines. An average of 30 people per hour try to call the OR Department, and the average length of a phone call is 1 minute. If a
15 Each hour, an average of 3 type 1 and 3 type 2 customers arrive at a single-server station. Interarrival times for each customer type are exponential and independent.The average service time for a
14 The mail order firm of L. L. Pea receives an average of 200 calls per hour (times between calls are exponentially distributed). It takes an L. L. Pea operator an average of 3 minutes to handle a
13 A printing shop receives an average of 1 order per day.The average length of time required to complete an order is .5 day. At any time, the print shop can work on at most one job.a On the average,
12 An average of 40 jobs per day arrive at a factory. The time between arrivals of jobs is exponentially distributed.The factory can process an average of 42 jobs per day, and the time to process a
11 Ships arrive at a port facility at an average rate of 2 ships every 3 days. On the average, it takes a single crew 1 day to unload a ship. Assume that interarrival and service times are
10 The owner of an exclusive restaurant has two tables but only one waiter. If the second table is occupied, the owner waits on that table himself. Service times are exponentially distributed with
9 At an exclusive restaurant, there is only one table and waiting space for only one other group; others that arrive when the waiting space is filled are turned away. The arrival rate follows an
8 The Newcoat Painting Company has for some time been experiencing high demand for its automobile repainting service. Since it has had to turn away business, management is concerned that the limited
7 An automated car wash will wash a car in 10 minutes.Arrivals occur an average of 15 minutes apart (exponentially distributed).a On the average, how many cars are waiting in line for a wash?b If the
6 The manager of a large group of employees must decide if she needs another photocopying machine. The cost of a machine is $40 per 8-hour day whether or not the machine is in use. An average of 4
5 There are 5 students and one keg of beer at a wild and crazy campus party. The time to draw a glass of beer follows an exponential distribution, with an average time of 2 minutes. The time to drink
4 Each year, an average of 500 people pass the New York state bar exam and enter the legal profession. On the average, a lawyer practices law in New York State for 35 years.Twenty years from now, how
3 At the Smalltown post office, patrons wait in a single line for the first open window. An average of 100 patrons per hour enter the post office, and each window can serve an average of 45 patrons
2 Registration at State University proceeds as follows:Upon entering the registration hall, the students first wait in line to register for classes. A single clerk handles registration for classes,
1 Buses arrive at the downtown bus stop and leave for the mall stop. Past experience indicates that 20% of the time, the interval between buses is 20 minutes; 40% of the time, the interval is 40
4 Consider a computer system to which two types of computer jobs are submitted. The mean time to run each type of job is m1 . The interarrival times for each type of job are exponential, with an
3 Four doctors work in a hospital emergency room that handles three types of patients. The time a doctor spends with each type of patient is exponentially distributed, with a mean of 15 minutes.
2 Suppose a supermarket uses a system in which all customers wait in a single line for the first available cashier.Assume that the service time for a customer who purchases k items is exponentially
On the Podunk U computer system, faculty jobs (type 1) always preempt student jobs(type 2). The length of each type of job follows an exponential distribution, with mean 30 seconds. Each hour, an
Gotham Township has 5 police cars. The police department receives two types of calls:emergency (type 1) and nonemergency (type 2) calls. Interarrival times for each type of call are exponentially
5 Southbest Airlines has an average of 230 customers per hour arriving at a ticket counter where 8 agents are working.Each agent can serve an average of 30 customers per hour.The squared coefficient
4 Suppose a teller costs $30 per hour. Suppose the bank values a customer’s time at NBD at $c per hour. Show how variations in c affect the number of tellers that NBD should use.
3 Using a two-way data table, determine how changes in the squared coefficient of variation for interarrival and service times affect the average number of customers in the NBD branch.
2 Show how the average time a customer must wait for a teller depends on the number of servers.
1 NBD believes the congestion level is satisfactory if the average number of customers in line equals the number of servers. For the information given in the example, what is the maximum arrival rate
The NBD Bank branch in Bloomington, Indiana has 6 tellers. At peak times, an average of 4.8 customers per minute arrive at the bank. It takes a teller an average of 1 minute to serve a customer. The
2 A manufacturing process always has 8 parts in process.A part must successfully complete two steps (step 1 and step 2) to be completed. A single machine performs step 1 and can process an average of
1 Jobs arrive to a file server consisting of a CPU and two disks (disk 1 and disk 2). With probability 13/20, a job goes from CPU to disk 1, and with probability 6/20, a job goes from CPU to disk 2.
Consider a flexible manufacturing system in which 10 parts are always in process. Each part requires two operations. Each part begins by having operation 1 done at machine 1.Then, with probability
1 A travel agency wants to determine if the length of customers’ phone calls can be adequately modeled by an exponential distribution. Last week, the agency recorded the length of all phone calls
The following interarrival times (in minutes) have been observed: 0.01, 0.07, 0.03, 0.08, 0.04, 0.10, 0.05, 0.10, 0.11, 1.17, 1.50, 0.93, 0.54, 0.19, 0.22, 0.36, 0.27, 0.46, 0.51, 0.11, 0.56, 0.72,
8 A company’s warehouse can store up to 4 units of a good. Each month, an average of 10 orders for the good are received. The times between the receipt of successive orders are exponentially
7 On the average, 26 patrons per year come to the I.U.library to borrow the I Ching (assume that interarrival times are exponential). Borrowers who find the book unavailable leave and never return. A
6 (Requires the use of a spreadsheet or LINGO) US Airlines receives an average of 500 calls per hour from customers who want to make a reservation (time between calls follows an exponential
5 (Requires the use of a spreadsheet or LINGO) At the peak usage time, an average of 200 people per hour attempt to log on the Jade Vax. The average length of time somebody spends on the Vax is 20
4 A phone system is said to receive 1 Erlang of usage per hour if callers keep lines busy for an average of 3,600 seconds per hour. Suppose a phone system receives 2 Erlangs of usage per hour. If you
3 In Example 15, suppose the hospital had 10 ambulances.On the average, how many ambulances would be en route or returning from a call?
2 A telephone order sales company must determine how many telephone operators are needed to staff the phones during the 9-to-5 shift. It is estimated that an average of 480 calls are received during
1 Suppose that a fire department receives an average of 24 requests for fire engines each hour. Each request causes a fire engine to be unavailable for an average of 20 minutes.To have at most a 1%
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
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
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
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
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
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.
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
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
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
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
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.
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
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
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
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
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
6 For the machine repair model, show that W K/l- (1/l).

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