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business
operations research an introduction
Operations Research Applications And Algorithms 4th Edition Wayne L. Winston - Solutions
4 Using x as the independent variable and y as the dependent variable, find the least squares line for the following three data points:x y 1 2 4 5 7 2
5 We are trying to predict the number of uses of automatic bank teller machines as a function of time. The data are given in Table 29. Here, x number of years after 1980 and y number of monthly uses of ATMs (in millions) during the given year. The estimated regression equation is yˆ 102.3
6 Carboco puts metal coatings on jet propeller blades. The harder the coating, the higher the quality of the coating. The coating is shot onto the blade using pressurized gas contained in an F-gun. Carboco can control the temperature and gas pressure in the F-gun and can also control the room
7 We have been assigned to determine how the total weekly production cost for Widgetco depends on the number of widgets produced during the week. The following model has been proposed:Y b0 b1X b2X2 b3X3 where X number of widgets produced during the week and Y total production cost
8 Let Yt sales during month t (in thousands of dollars)for a photography studio (SALES in Table 31) and Pt price charged for portraits during month t (PRICE). Use a computer to fit the following model to the data in Table 31(file Portrait.xls):Yt b0 b1Yt1 b2Pt t Thus, last month’s
3 Quarterly sales for a department store over a six-year period are given in Table 22 (file Sales.xls).a Use multiple regression to develop a model that canbe used to predict future quarterly sales. (Hint: Use dummy variables and an independent variable for the number of the quarter (quarter 1,
2 We are trying to determine the number of labor hours required to produce a unit of a product. We are given the information in Table 21 (file Learn.xls). For example, the 2nd unit produced required 517 labor hours, and the 600th unit produced required 34 labor hours.a Try to determine a
8 The Holt method (exponential smoothing with trend and without seasonality) is being used to forecast weekly car sales at TOD Ford. Currently, the base is estimated to be 50 cars per week, and the trend is estimated to be 6 cars per week. During the current week, 30 cars are sold. After observing
We want to predict maintenance expense ( y) for a truck during the current year, from the independent variables x1 = miles driven (in thousands) during the current year and x2 =age of the truck (in years) at the beginning of the current year. We are given the information in Table 15. TABLE 15 Truck
3 Suppose that the number of customers entering the bank is growing at around 20% per year. How could we incorporate this fact into the forecasting procedure described in this section?
2 Suppose again that the bank is a college credit union, but now the staff gets paid every other Friday. Again, bank traffic is much higher than usual on staff paydays. How could we incorporate this fact into the forecasting procedure described in this section?
1 Suppose the bank is a college credit union and that on days when the college’s professors get paid, bank traffic is much higher than usual. Assuming that college professors are paid on the first weekday of each month, how could we incorporate this fact into the forecasting procedure described
16 A version of simple exponential smoothing can be used to predict the outcome of sporting events. To illustrate, consider pro football. We first assume that all games are played on a neutral field. Before each day of play, we assume that each team has a rating. For example, if the Bears’ rating
15 Holt’s method assumes an additive trend. For example, a trend of 5 means that the base will increase by 5 units per period. Suppose there is actually a multiplicative trend.Thus, if the current estimate of the base is 50 and the current estimate of the trend is 1.2, we would predict demand to
14 Winter’s method assumes a multiplicative seasonality but an additive trend. For example, a trend of 5 means that the base will increase by 5 units per period. Suppose there is actually a multiplicative trend. Then (ignoring seasonality)if the current estimate of the base is 50 and the current
13 In our discussion of Winter’s method, a monthly seasonality of (say) 0.80 for January means that during January, air conditioner sales are expected to be 80% of the sales during an average month. An alternative approach to modeling seasonality is to let the seasonality factor for each month
12 Simple exponential smoothing with a 0.3 is being used to predict sales of radios at Lowland Appliance.Predictions are made on a monthly basis. After observing August radio sales, the forecast for September is 100 radios.a During September, 120 radios are sold. After observing September sales,
11 Winter’s method is being used to forecast quarterly U.S. retail sales (in billions of dollars). At the end of the first quarter, Lt 300, Tt 30, and the seasonal indexes are as follows: quarter 1, 0.90; quarter 2, 0.95; quarter 3, 0.95;quarter 4, 1.20. During the second quarter, retail
10 We have been assigned to forecast the number of aircraft engines ordered each month by Engine Company.At the end of February, the forecast is that 100 engines will be ordered during April. During March, 120 engines are ordered.a Using a 0.3, determine (at the end of March) a forecast for the
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, the following seasonalities have been estimated: Monday, 0.80; Tuesday, 0.90; Wednesday, 0.95;
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 could be used to predict GNP during future years.b When the regression on transformed data is done, we
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 autocorrelation? Heteroscedasticity? Multicollinearity?
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
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 seasonalities: January, 0.80; February, 0.85; December, 1.2. During February 2005, the GDP is at a level of
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 summer 1.3 Current base estimate 400 cases per quarter Current trend estimate 40 cases per quarter
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 beer.a At the beginning of May, what is the prediction for July’s beer sales?b Actual demand
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 checkpoint. Four people are available at each checkpoint to do the searching. How long does the average
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 long does an average ticketholder spend from arrival to passing through the checkpoint?
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 take 10 minutes to fill.a What percentage of the time is the pharmacist busy?b On average, how long
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 at the Credit Union (interarrival times are exponential).a How long do customers have to wait for a
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. Then each patient is processed by one of two registration clerks. This takes an average of 7
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 exponential interarrival times). Each X-ray machine can handle an average of 150 passengers per hour
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 available to perform stage 1.After completing stage 1, the job immediately passes to stage 2. Stage 2
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 hour, with the standard deviation of service times being 8.66 minutes.Simulate the operation of this
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 server 2. Interarrival times are exponential. Server 1 can serve at an exponential rate of 20
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, MAD 25, and we are 95% sure that VCR sales for December, 2005 will be between _____ and _____.
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 shavers were sold, and during December 45 shavers were sold. Suppose a 0.50. At the end of December,
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 yielded satisfactory forecasts for the most part. Each March, however, Bloomington Ford 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 millions, and GNP is in billions of dollars.a Use this data to develop a regression equation that could
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 the arrivals of October 16, Lt 200 customers, Tt 1 customer, and the seasonalities are as
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 values ofa, b, and g that yield a MAD (for 1967–1972) of less than 3.5. (Hint: It may be necessary
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 if it is available. Otherwise, the downtown-based ambulance is sent (if available). If no ambulance
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 rate of voters throughout the day is as shown in Table 12. What is the probability that all voting
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 arrival rate per hour at time t (t 0 is 8 A.M., and t 8 is 4 P.M.) is given byAssume that service
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 different times is as shown in Table 11. Arrivals follow a nonhomogeneousPoisson process. The restaurant
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 is available. Professor Bright has told the typist that tests have priority over research papers, and
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 job = short job and type 2 job = long job. Then we are given that Determine the average length of time
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 nodes is as follows:a Compute the expected delay for a packet.b If a total of 75,000 bits/second 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 entering service. Also assume exponential service times.a Find the probability distribution of the number
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 number of customers present an instant after the ith customer completes service.a Explain why X1, X2, .
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 per hour and estimates that each hour of idle machine time costs the company cm dollars in lost
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 to run a poor engine for a year. A fair engine can be overhauled for $2 million, and it immediately
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 each size is as follows:The transition probability matrix for the size of the tree is as follows:0 1
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 are pictured in Figure 2. Robert is nearsighted and cannot see ahead;he can only see whether the
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 back for 50¢/gallon any milk that is left at the end of the day. Unfortunately for Safeco, demand
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% chance that a family on the waiting list will find private housing and remove themselves from the
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 for lap swimming. If one swimmer is in a lane, he or she swims up and down the right side of the
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 person attempts to call when both lines are busy, he or she hangs up and is lost to the system. Assume
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 type 1 customer is 6 minutes, and the average service time for a type 2 customer is 3 minutes (all
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 call. If a caller gets a busy signal, L. L.Pea assumes that he or she will call Seas Beginning (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, how many jobs are present in the print shop?b On the average, how long will a person who places an
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 job is exponentially distributed.a What is the probability that exactly 180 jobs arrive at the
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 exponential. The shipping company owns the port facility as well as the ships using that facility. It is
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 mean 1 hour, and the time between arrivals is exponentially distributed with mean 1.5 hours. When the
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 exponential distribution with a rate of one group per hour. It takes the average group 1
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 space available to store cars awaiting painting has cost lost revenue. A small vacant lot next to
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 car wash could be speeded up, what wash time would reduce the average waiting time to 5 minutes?
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 people per hour need to use the copying machine. Each person uses the copier for an average of 10
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 a beer also follows an exponential distribution, with a mean of 18 minutes. After finishing a beer,
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 many lawyers would you expect there to be in New York State?
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 per hour. The post office estimates a cost of 10¢ for each minute a patron waits in line and
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, and it takes the clerk an average of 2 minutes to handle a student’s registration. Next, the
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 minutes; and 40% of the time, the interval is 2 hours. If I have just arrived at the downtown bus stop,
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 average of li type i jobs arriving each hour. Consider the following three situations.a Type 1 jobs have
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. Interarrival times for each customer type are exponential, with the average number of arrivals per hour
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 distributed, with mean k seconds.Also, a customer who purchases k items feels that the cost of
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 average of 10 faculty and 50 student jobs are submitted. What is the average length of time between the
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 distributed, with an average of 10 emergency and 20 nonemergency calls being received each hour. Each
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 of variation for the interarrival times is 1.5 and 2 for the service times.a On average, how many
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 that can be satisfactorily handled with 6 servers?
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 squared coefficient of variation for both interarrival times and service times is .5. Estimate 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 8 parts per minute. A single machine performs step 2 and can process 11 parts per minute.
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. With probability 1/20, a job is finished after its CPU operation and is immediately replaced by
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 .75 the part has operation 2 processed on machine 2, and with probability .25 the part has operation 2
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 and obtained the following results(in seconds): 4, 6, 5, 8, 9, 10, 12, 8, 16, 20, 24, 27, 33, 37, 43,
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, 0.29, 0.04, 0.73. Does it seem reasonable to conclude that these observations come from an
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 distributed. When an item is used to fill an order, a replacement item is immediately ordered, and it
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 borrower keeps a copy of the I Ching for an average of 4 weeks.a If the library has only one copy,
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 distribution). It takes an average of 3 minutes to handle each call. Each customer who buys a ticket
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 minutes. If the Indiana University Computing Service wants to ensure that during peak usage only 1% of
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 want only 1% of all calls blocked, how many phone lines do you need?†
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 this period and that the average call lasts for 6 minutes. If the company wants to have at most 1
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% chance of being unable to respond to a request, how many fire engines should the fire department have?
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