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operations research an introduction
Operations Research An Introduction 8th Edition Hamdy A. Taha - Solutions
7. Population dynamics is impacted by the continual movement of people who are seeking better quality of life or better employment. The city of Mobile has an inner city population, a suburban population, and a surrounding rural population. The census taken in la-year intervals shows that 10% of the
(b) If the greater Mobile area currently includes 20,000 rural residents, 100,000 suburbanites, and 30,000 inner city inhabitants, what will the population distribution be in 10 years? In 20 years?
(c) Determine the long-run population picture of Mobile.
S. A car rental agency has offices in Phoenix, Denver, Chicago, and Atlanta. The agency allows one- and two-way rentals so that cars rented in one location may end up in another.Statistics show that at the end of each week 70% of all rentals are two-way. As for the one-way rentals: From Phoenix,
(b) If the agency starts the week with 100 cars in each location, what will the distribution be like in two weeks?
(c) If each location is designed to handle a maximum of 110 cars, would there be a longrun space availability problem in any of the locations?
(d) Determine the average number of weeks that elapse before a car is returned to its originating location.
9. A bookstore keeps daily track of the inventory level of a popular book to restock it to a level of 100 copies at the start of each day. The data for the last 30 days provide the following end-of-day inventory position: 1,2,0,3,2, 1,0,0,3,0, 1,1, 3,2,3,3,2,1,0,2,0,1,3, 0,0,3,2, 1,2,2.(a)
10. In Problem 9, suppose that the daily demand can exceed supply, which gives rise to shortage(negative inventory). The end-of-day inventory level for the past 30 days is given as:1,2,0, -2,2,2, -1, -1,3,0,0,1, -1, -2,3,3, -2, -1,0,2,0, -1, 3,0,0,3, -1, 1,2, -2.(a) Express the situation as a
11. A store starts a week with at least 3 PCS. The demand per week is estimated at 0 with probability .15,1 with probability .2,2 with probability .35,3 with probability .25, and 4 with probability .05. Unfilled demand is backlogged. The store's policy is to place an order for delivery at the start
(b) Suppose that the week starts with 4 PCS. Determine the probability that an order will be placed at the end of two weeks.
(c) Determine the long-run probability that no order will be placed in any week.
(d) If the fixed cost of placing an order is $200, the holding cost per PC per week is $5, and the penalty cost per shortage PC per week is $20, determine the expected inventory cost per week. .
12. Solve Problem 11 assuming that the order size, when placed, is exactly 5 pieces.
13. In Problem 12, suppose that the demand for the PCS is 0, 1,2,3,4, or 5 with equal probabilities.Further assume that the unfilled demand is not backlogged, but that the penalty cost is still incurred.(a) Express the situation as a Markov chain.
(b) Determine the long-run probability that a shortage will take place.
(c) If the fixed cost of placing an order is $200, the holding cost per PC per week is $5, and the penalty cost per shortage PC per week is $20, determine the expected ordering and inventory cost per week.
*14. The federal government tries to boost small business activities by awarding annual grants for projects. AIL bids are competitive, but the chance of receiving a grant is highest if the owner has not received any during the last three years and lowest jf awards were given in each of the last
(b) Determine the expected number of awards per owner per year.
15. Jim Bob has a history of receiving many fines for driving violations. Unfortunately for Jim Bob, modern technology can keep track of his previous fines. As soon as he has accumulated 4 tickets, his driving license is revoked until he completes a new driver education class, in which case he
(b) What is the average number of times Jim Bob is stopped by police before his license is revoked again?
(c) What is the probability that Jim Bob will lose his license?
(d) If each fine costs $100, how much, on the average, does Jim Bob pay between successive suspensions of his license?
*1. A mouse maze consists of the paths shown in Figure 17.3. Intersection 1 is the maze entrance and intersection 5 is the exit. At any intersection, the mouse has equal probabilities of selecting any of the available paths. When the mouse reaches intersection S, it will be allowed to recirculate
(b) Determine the probability that,starting at intersection 1, the mouse will reach the exit after three trials.
(c) Determine the long-run probability that the mouse will locate the exit intersection.(d) Determine the average number of trials needed to reach the exit point from intersection 1. 2 1 3 5 FIGURE 17.3 Mouse maze for Problem 1, Set 17.5a 4
2. In Problem 1, intuitively, if more options (routes) are added to the maze, will the average number of trials needed to reach the exit point increase or decrease? Demonstrate the answer by adding a route between intersections 3 and 4.
8. Second Time Around sells popular used items on consignment. Its operation can be viewed as an inventory problem in which the stock is replenished and depleted randomly according to Poisson distributions with rates Aand J.L items per day. Every time unit the item is out of stock, Second Time
7. Suppose in Problem 6 that the investor can choose any desired restaurant capacity based on a specific marginal cost for each additional capacity unit requested. Derive the associated general cost model, and define all its components and terms.
3. Jim and Joe start a game with five tokens, three for Jim and two for Joe. A coin is tossed and if the outcome is heads, Jim gives Joe a token, else Jim gets a token from Joe. The game ends when Jim or Joe has all the tokens. At this point, there is 30% chance that Jim and Joe will continue to
6. Pizza Unlimited sells two franchised restaurant models. Model A has a capacity of 20 groups of customers, and model B can seat 30 groups. The monthly cost of operating model A is $12,000 and that of model B is $16,000. An investor wants to set up a buffetstyle pizza restaurant and estimates that
5. Jobs arrive at a machine shop according to a Poisson distribution at the rate of 80 jobs per week. An automatic machine represents the bottleneck in the shop. It is estimated that a unit increase in the production rate of the machine will cost $250 per week. Delayed jobs normally result in lost
4. H&I Industry produces a special machine with different production rates (pieces per hour) to meet customer specifications. A shop owner is considering buying one of these machines and wants to decide on the most economical speed (in pieces per hour) to be ordered. From past experience, the owner
3. B&K Groceries is opening a new store boasting "state-of-the-art" check-out scanners.Mr. Bih, one of the owners of B&K, has limited the choices to two scanners: scanner A can process 10 items a minute, and the better-quality scanner B can scan 15 items a minute. The daily (10 hours) cost of
*2. Metaleo is in the process of hiring a repairperson for a 10-machine shop. Two candidates are under consideration. The first candidate can carry out repairs at the rate of 5 machines per hour and earns $15 an hour. The second candidate, being more skilled, receives$20 an hour and can repair 8
1. In Example 15.9-1, do the following:(a) Verify the values of JL2, JL3' and JL4 given in the example.(b) Suppose that the penalty of $80 per job per day is levied only on jobs that are not "in progress" at the end of the day. Which copier yields the lowest total cost per day?
*4. Optica, Ltd., makes prescription glasses according to orders received from customers.Each worker is specialized in certain types of glasses.The company has been experiencing unusual delays in the processing of bifocal and trifocal prescriptions. The worker in charge receives 30 orders per
3. Layson Roofing Inc. installs shingle roofs on new and old residences in Arkansas. Prospective customers request the service randomly at the rate of nine jobs per 30-day month and are placed on a waiting list to be processed on a FCFS basis. Homes sizes vary, but it is fairly reasonable to assume
2. Solve Example 15.7-1 assuming that the service-time distribution is given as follows:*(a) Uniform between 8 and 20 minutes.(b) Normal with J.1- := 12 minutes and cr = 3 minutes.(c) Discrete with values equal to 4,8, and 15 minutes and probabilities .2, .6, and .2, respectively.
1. In Example 15.7-1, compute the percentage of time the facility is idle.
8. Show that the rate of breakdown in the shop can be computed from the formula Acfr = J.tR where Ris the average number of busy repairpersons.
(b) Determine the probability that Joe will win in three coin tosses. That Jim will win in three coin tosses.
7. Verify the expression for Pn for the (MIMIR):(GDIKlK) model.
*6. After a long wait, the Newborns were rewarded with quintuplets, two boys and three girls, thanks to the wonders of new medical advances. During the first 5 months, the babies' life consisted of two states: awake (and mostly crying) and asleep. According to the Newborns, the babies
(c) Determine the probability that a game will end in Jim's favor. Joe's favor.
5. Kleen All is a service company that performs a variety of odd jobs, such as yard work, tree pruning, and house painting. The company's four employees leave the office with the first assignment of the day. After completing an assignment, the employee calls the office requesting instruction for
*4. An operator attends five automatic machines. After each machine completes a batch run, the operator must reset it before a new batch is started. The time to complete a batch run is exponential with mean 45 minutes. The setup time is also exponential with mean 8 minutes.(a) Determine the average
(d) Determine the average number of coin tosses needed before Jim wins. Joe wins.
3. In the computations in Figure 15.9, it may appear confusing that the average rate of machine breakdown in the shop, Aerr, increases with the increase in R. Explain why the increase in Aeff should be expected.
4. An amateur gardener with training in botany is experimenting with scientific crosspollination of pink irises with red, orange, and white irises. His annual experiments show that pink can produce 60% pink and 40% white, red can produce 40% red, 50%pink, and 10% orange, orange can produce 25%
2. In Example 15.6-8, define and compute the productivity of the repairpersons for R := 1, 2,3, and 4. Use this information in conjunction with the measure of machine productivity to decide on the number of repairpersons Tooleo should hire.
4. Repeat Problem 3 for large p = 9 and show that the same conclusion holds except that the value of c must be higher (at least 14). From the results of Problems 3 and 4, what 1. In Example 15.6-8, do the following:(a) Verify the values of Aeff given in Figure 15.9.*(b) Compute the expected number
3. Show (by using excelPoissonQ.xls orTORA) that for small p = .1, the values of Ls, Lq, Ws, Wq , and Pn for the (MIMjc):(GDloo/oo) model can be estimated reliably using the less cumbersome formulas of the (M1M100 ): (GD1(0 / 00 ) model for c as small as 4 servers.
2. New drivers are required to pass written tests before they are given a road driving test.These tests are usually administered by the city police department. Records at the City of Springdale show that the average number of written tests is 100 per 8-hour day.The average time needed to complete
1. In Example 15.6-7, compute the following:(a) The probability that the investor will sell out completely.(b) The probability that the investor will own at least 10 securities.(e) The probability that the investor will own between 30 and 40 securities, inclusive.(d) The investor's net annual
*(b) What is the probability that an arriving car will wait in the lanes?(c) What is the probability that an arriving car will occupy the only remaining parking space on the lot?
4. At U of A, newly enrolled freshmen students are notorious for wanting to drive their cars to class (even though most of them are required to live on campus and can conveniently make use of the university free transit system). During the first couple of weeks of the fall semester, traffic havoc
3. A small engine repair shop is run by three mechanics. Early in March of each year, people bring in their tillers and lawn mowers for service and maintenance. The shop is willing to accept all the tillers and mowers that customers bring in. However, when new customers see the floor of the shop
2. Eat & Gas convenience store operates a two-pump gas station. The lane leading to the pumps can house at most 3 cars, excluding those being serviced. Arriving cars go elsewhere if the lane is full. The distribution of arriving cars is Poisson with mean 20 per hour. The time to fill up and pay for
1. In Example 15.6-6, determine the following:(a) The expected number of idle cabs.(b) The probability that a calling customer will be the last on the list.(c) The limit on the waiting list if it is desired to keep the waiting time in the queue to below 3 minutes..~:.:.
10. For the (M/Mlc):(GDjoojoo) model, Morse (1958,p.l03) shows that as ~-1, L =_Pq c - P
9. In the United States, the use of single-line, multiple-server queues is common in post offices and in passenger check-in counters at airports. However, both grocery stores and banks (especially in smaller communities) tend to favor single-line, single-server setups, despite the fact that
8. Drake Airport services rural, suburban, and transit passengers. The arrival distribution for each of the three groups is Poisson with mean rates of 15, 10, and 20 passengers per hour, respectively. The time to check in a passenger is exponential with mean 6 minutes.Determine the number of
7. U of A computer center is equipped with four identical mainframe computers. The number of users at any time is 25. Each user is capable of submitting a job from a terminal every 15 minutes, on the average, but the actual time between submissions is exponential.Arriving jobs will automatically go
6. A small post office has two open windows. Customers arrive according to a Poisson distribution at the rate of 1 every 3 minutes. However, only 80% of them seek service at the windows. The service time per customer is exponential, with a mean of 5 minutes. All arriving customers form one line and
*5. McBurger fast food restaurant has 3 cashiers. Customers arrive according to a Poisson distribution every 3 minutes and form one line to be served by the first available cashier.The time to fill an order is exponentially distributed with a mean of5 minutes. The waiting room inside the restaurant
4. Customers arrive at Thrift Bank according to a Poisson distribution, with a mean of 45 customers per hour. Transactions per customer last about 5 minutes and are exponentially distributed. The bank wants to use a single-line multiple-teller operation, similar to the ones used in airports and
3. Determine the minimum number of parallel servers needed in each of the following(Poisson arrivaUdeparture) situations to guarantee that the operation of the queuing situation will be stable (i.e., the queue length will not grow indefinitely):(a) Customers arrive every 5 minutes and are served at
*2. In the cab company example, suppose that the average time per ride is actually about 14.5 minutes, so that the utilization (= :c) for the 2- and 4-cab operations increases to more than 96%. Is it still worthwhile to consolidate the two companies into one? Use the average waiting time for a ride
1. Consider Example 15.6-5.(a) Show that the remarkable reduction in waiting time by more than 50% for the consolidated case is coupled with an increase in the percentage of time the servers remain busy.(b) Determine the number of cabs that the consolidated company should have to limit the average
*5. Customers tend to exhibit loyalty to product brands but may be persuaded through clever marketing and advertising to switch brands. Consider the case of three brands: A, B, and C. Customer "unyielding" loyalty to a given brand is estimated at 75%, giving the competitors only a 25% margin to
6. Patients arrive at a I-doctor clinic according to a Poisson distribution at the rate of 20 patients per hour. The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential, with a mean of 8 minutes.(a) What is the probability that an arriving patient
5. A cafeteria can seat a maximum of 50 persons. Customers arrive in a Poisson stream at the rate of 10 per hour and are served (one at a time) at the rate of 12 per hour.(a) What is the probability that an arriving customer will not eat in the cafeteria because it is full?(b) Suppose that three
*4. The final assembly of electric generators at Electro is produced at the Poisson rate of 10 generators per hour. The generators are then conveyed on a belt to the inspection department for final testing. The belt can hold a maximum of 7 generators. An electronic sensor will automatically stop
3. The time barber Joe takes to give a haircut is exponential witb a mean of 12 minutes.Because of his popularity, customers usually arrive (according to a Poisson distribution)at a rate much higher than Joe can handle: six customers per hour. Joe really will feel comfortable if the arrival rate is
2. Consider the car wash facility of Example 15.6-4. Determine the Dumber of parking spaces such that the percentage of cars that cannot find a space does not exceed 1%.
*1. In Example 15.6-4, determine the following:(a) Probability that an arriving car will go into the wash bay immediately on arrival.(b) Expected waiting time until a service starts.(c) Expected number of empty parking spaces(d) Probability that all parking spaces are occupied.(e) Percent reduction
*5. Consider Problem 5, Set l5.6b. To attract more business, the owner of the restaurant will give free soft drinks to any customer who waits more than 5 minutes. Given that a drink costs 50 cents, how much will it cost daily to offer free drinks? Assume that the restaurant is open for 12 hours a
3. In Example 15.6-3, determine the service rate JL that satisfies the condition Ws < 10 minutes.4. In Example 15.6-3, determine the service rate IL that will satisfy the condition P{T> 10 minutes}
2. In Example 15.6-3, compute the following:(a) The standard deviation of the waiting time T in the system.(b) The probability that the waiting time in the system will vary by half a standard deviation around the mean value.
*1. In Problem 3, Set 15.6b, determine the probability that detective Columbo will take more than 1 week to solve a crime case.
8. For the (MIM/I):(GDjoojoo), derive the expression for Lq using the basic definition 2::=2(n - I)Pno
7. In the (MIMI1):(GD/oo/oo), give a plausible argument as to why Ls does not equal Lq + 1, in general. Under what condition will the equality hold?
*(d) How many car spaces should be provided in front of the window (including the car being served) so that an arriving car can fmd a space there at least 90% of the time?
6. Customers arrive at a one-window drive-in bank according to a Poisson distribution, with a mean of 10 per hour.The service time per customer is exponential, with a mean of 5 minutes. There are three spaces in front of the window, including the car being served.Other arriving cars line up outside
*5. A fast-food restaurant has one drive-in window. Cars arrive according to a Poisson distribution at the rate of 2 cars every 5 minutes. The space in front of the window can accommodate at most 10 cars, including the one being served. Other cars can wait outside this space if necessary. The
4. Cars arrive at the Lincoln Tunnel toll gate according to a Poisson distribution, with a mean of 90 cars per hour. The time for passing the gate is exponential with mean 38 seconds.Drivers complain of the long waiting time, and authorities are willing to reduce the average passing time to 30
3. Over the years, Detective Columbo, of the Fayetteville Police Department, has had phenomenal success in solving every single crime case. It is only a matter of time before any case is solved. Columbo admits that the time per case is "totally random," but, on the average, each investigation will
*2. John Macko is a student at Ozark U. He does odd jobs to supplement his income. Job requests come every 5 days on the average, but the time between requests is exponential.The time for completing a job is also exponential with mean 4 days.(a) What is the probability that John will be out of
1. In Example 15.6-2, do the following.(a) Determine the percent utilization of the wash bay.(b) Determine the probability that an arriving car must wait in the parking lot prior to entering the wash bay.(c) If there are seven parking spaces, determine the probability that an arriving car will find
1. In Example 17.6-1, suppose that the labor cost for machines I and II is $20 per hour and that for inspection is only $18 per hour. Further assume that it takes 30 minutes and 20 minutes to process a piece on machines I and II, respectively. The inspection time at each of the two stations is 10
2. Solve Example 15.6-1 using the following data: number of parking spaces = 6, number of temporary spaces == 4, A == 10 cars per hour, and average parking time == 45 minutes.
1. In Example 15.6-1, do the following:*(a) Compute Lq directly using the formula 2:::c+l(n - c)Pn.(b) Compute Ws from Lq.*(c) Compute the average number of cars that will not be able to enter the parking lot during an 8-hour period.*(d) Show that c - (Ls - Lq ), the average number of empty spaces,
7. Consider a one-server queuing situation in which the arrival and service rates are given by An = 10 - n,n = 0,1,2,3 n/-Ln = 2" + 5, n = 1,2, 3,4 This situation is equivalent to reducing the arrival rate and increasing the service rate as the number in the system, n, increases.(a) Set up the
*6. A barbershop serves one customer at a time and provides three seats for waiting customers.If the place is full, customers go elsewhere. Arrivals occur according to a Poisson distribution with mean four per hour. The/time to get a haircut is exponential with mean 15 minutes. Determine the
5. Have you ever heard someone repeat the contradictory statement, "The place is so crowded no one goes there any more"? This statement can be interpreted as saying that the opportunity for balking increases with the increase in the number of customers seeking service. A possible platform for
4. First Bank of Springdale operates a one-lane drive-in ATM machine. Cars arrive according to a Poisson distribution at the rate of 12 cars per hour. The time per car needed to complete the ATM transaction is exponential with mean 6 minutes. The lane can accommodate a total of 10 cars. Once the
*3. In the B&K model of Example 15.5-1, suppose that all three counters are always open and that the operation is set up such that the customer will go to the first empty counter.Determine the following:(a) The probability that all three counters will be in use.(b) The probability that an arriving
2. In the B&K model of Example 15.5-1, suppose that the interarrival time at the checkout area is exponential with mean 5 minutes and that the checkout time per customer is also exponential with mean 10 minutes. Suppose further that B&K will add a fourth counter and that counters will open based on
1. In Example 15.5-1, determine the following:(a) The probability distribution of the number of open counters.(b) The average number of busy counters.
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