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operations research an introduction
Operations Research: An Introduction 10th Global Edition Hamdy A Taha - Solutions
The induction proof for deriving the general solution of the generalized model is applied as follows. ConsiderWe substitute for pn-1 and pn-2 in the general difference equation involving pn, pn-1, and pn-2 to derive the desired expression for pn. Verify this procedure. = k-1 Pk II o, k = 0, 1, 2,...
Consider the single-queue model where only one customer is allowed in the system.Customers who arrive and find the facility busy never return. Assume that the arrivals distribution is Poisson with mean l per unit time and that the service time is exponential with mean 1m time units.(a) Set up the
Consider a one-server queuing situation in which the arrival and service rates are given by ln = 10 - n, n = 0, 1, 2, 3 mn =n 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
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
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
First Bank of Springdale operates a one-lane drive-in ATM machine. Cars arrive according to a Poisson distribution at the rate of 10 cars per hour. The time per car needed to complete the ATM transaction is exponential with mean 5 minutes. The lane can accommodate a total of 10 cars. Once the lane
In the B&K model of Example 18.5-1, 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
In the B&K model of Example 18.5-1, suppose that the interarrival time at the checkout area is exponential with mean 8 minutes and that the checkout time per customer is also exponential with mean 12 minutes. Suppose further that B&K will add a fourth counter. Counters 1, 2, and 3 will open based
In Example 18.5-1, determine the following:(a) The probability distribution of the number of open counters.(b) The average number of busy counters.
Derive the truncated Poisson distribution from the difference-differential equations of the pure death model using induction. [Note: See the hint in Problem 18-28.]
Prove that the distribution of the time between departures corresponding to the truncated Poisson in the pure death model is an exponential distribution with mean 1m time units.
Demand for an item occurs according to a Poisson distribution with mean 3 per day.The maximum stock level is 25 items, which occurs on each Monday immediately after a new order is received. The order size depends on the number of units left at the end of the week on Saturday (business is closed on
A machine shop has just stocked 10 spare parts for the repair of a machine. Stock replenishment that brings the stock level back to 10 pieces occurs every 7 days. The time between breakdowns is exponential with mean 1 day. Determine the probability that the machine will remain broken for 2 days
Inventory is withdrawn from a stock of 80 items according to a Poisson distribution at the rate of 5 items per day. Determine the following:(a) The probability that 10 items are withdrawn during the first 2 days.(b) The probability that no items are left at the end of 4 days.(c) The average number
A freshman student receives a bank deposit of $100 a month from home to cover incidentals. Withdrawal checks of $20 each occur randomly during the month and are spaced according to an exponential distribution with a mean value of 1 week.Determine the probability that the student will run out of
Each morning, the refrigerator in a small machine shop is stocked with two cases (24 cans per case) of soft drinks for use by the shop’s 12 employees. The employees can quench their thirst at any time during the 8-hour work day (8:00 a.m. to 4:00 p.m.), and each employee is known to consume
The Springdale High School band is performing a benefit concert in its new 400-seat auditorium. Local businesses buy the tickets in blocks of 5 and donate them to youth organizations. Tickets go on sale to business entities for 5 hours only the day before the concert. The process of placing orders
Consider Example 18.4-2. In each of the following cases, first write the answer algebraically, and then use excelPoissonQ.xls or TORA to provide numerical answers.*(a) The probability that the stock is depleted after 3 days.(b) The average number of dozen roses left at the end of the second
In Example 18.4-2, use excelPoissonQ.xls or TORA to compute pn172, n = 1, 2,c, 18, and then verify manually that these probabilities yield E5n 0 t = 7 0 6 = .664 dozen.
Derive the Poisson distribution from the difference-differential equations of the pure birth model. Hint: The solution of the general differential equation is y' + a(t)y = b(t) y = e=Jar) { [b(1)/(0) "{b(t)e/n) dt + constant
Prove that the mean and variance of the Poisson distribution during an interval t equal lt, where l is the arrival rate.
The U of A runs two bus lines on campus: red and green. The red line serves north campus and the green line serves south campus with a transfer station linking the two lines. Green buses arrive randomly (according to a Poisson distribution) at the transfer station every 10 minutes. Red buses also
The Springdale Public Library receives new books according to a Poisson distribution with mean 25 books per day. Each shelf in the stacks holds 100 books. Determine the following:(a) The average number of shelves that will be stacked with new books each (30-day)month.(b) The probability that more
The time between arrivals at L&J restaurant is exponential with mean 5 minutes. The restaurant opens for business at 11:00 a.m. Determine the following:*(a) The probability of having 10 arrivals in the restaurant by 11:12 a.m., given that 4 customers arrived by 11:05 a.m.(b) The probability that a
In a bank operation, the arrival rate is 3 customers per minute. Determine the following:(a) The average number of arrivals during 10 minutes.(b) The probability that no arrivals will occur during the next minute.(c) The probability that at least one arrival will occur during the next minute.(d)
An art collector travels to art auctions once a month on the average. Each trip is guaranteed to produce one purchase. The time between trips is exponentially distributed. Determine the following:(a) The probability that no purchase is made in a 2-month period.(b) The probability that no more than
In Example 18.4-1, suppose that the clerk who enters the information from birth certificates into the computer normally waits until at least 6 certificates have accumulated. Find the probability that the clerk will be entering a new batch every hour.
Prove that the mean and standard deviation of the exponential distribution are equal.
The U of A runs two bus lines on campus: red and green. The red line serves north campus, and the green line serves south campus with a transfer station linking the two lines. Green buses arrive randomly (exponential interarrival time) at the transfer station every 10 minutes. Red buses also arrive
The time between failures of a Kencore refrigerator is known to be exponential with mean value 9000 hrs (about 1 year of operation), and the company issues a 1-year warranty on the refrigerator. What are the chances that a breakdown repair will be covered by the warranty?
A customer arriving at a McBurger fast-food restaurant within 4 minutes of the immediately preceding customer will receive a 10% discount. If the interarrival time is between 4 and 5 minutes, the discount is 6%. If the interarrival time is longer than 5 minutes, the customer gets 2% discount. The
In Problem 18-14, suppose that Ann pays Jim 2 cents if the next arrival occurs within 1 minute and 3 cents if the interarrival time is between 1 and 1.5 minutes. Ann receives from Jim 5 cents if the interarrival time is between 1.5 and 2 minutes and 6 cents if it is larger than 2 minutes. Determine
Suppose that in Problem 18-14 the rules of the game are such that Jim pays Ann 2 cents if the next customer arrives after 1.5 minutes, and Ann pays Jim an equal amount if the next arrival is within 1 minute. For arrivals within the range 1 to 1.5 minutes, the game is a draw. Determine Jim’s
Ann and Jim, two employees in a fast-food restaurant, play the following game while waiting for customers to arrive: Jim pays Ann 2 cents if the next customer does not arrive within 1 minute; otherwise, Ann pays Jim 2 cents. Determine Jim’s average payoff in an 8-hr period. The interarrival time
The manager of a new fast-food restaurant wants to quantify the arrival process of customers by estimating the fraction of interarrival time intervals that will be (a) less than 1 minutes, (b) between 1 and 2 minutes, and (c) more than 2 minutes. Arrivals in similar restaurants occur at the rate of
The time between arrivals at the game room in the student union is exponential, with mean 10 minutes.(a) What is the arrival rate per hour?(b) What is the probability that no students will arrive at the game room during the next 15 minutes?(c) What is the probability that at least one student will
Suppose that the time between breakdowns for a machine is exponential with mean 5 hours. If the machine has worked without failure during the last 4 hours, what is the probability that it will continue without failure during the next 2 hours? That it will break down during the next hour?
The time between arrivals at the State Revenue Office is exponential with mean value.04 hour. The office opens at 8:00 a.m.*(a) Write the exponential distribution that describes the interarrival time.*(b) Find the probability that no customers will arrive at the office by 8:15 a.m.(c) It is now
In Example 18.3-1, determine the following:(a) The average number of failures per day, assuming the service is offered 24 hours a day, 7 days a week.(b) The probability of at least one failure in a 3-hour period.(c) The probability that the next failure will not occur within 4 hours.(d) If no
(a) Explain your understanding of the relationship between the arrival rate l and the average interarrival time. What are the units describing each parameter?(b) In each of the following cases, determine the average arrival rate per hour, l, and the average interarrival time in hours.*(i) One
In each of the situations in Problem 18-3, discuss the possibility of the customers jockeying, balking, and reneging.
True or False?(a) An impatient waiting customer may not elect to renege.(b) If a long waiting time is anticipated, an arriving customer may not elect to balk.(c) Jockeying from one queue to another is exercised in hope of reducing waiting time.
Study the following system and identify the associated queuing situations. For each situation, define the customers, the server(s), the queue discipline, the service time, the maximum queue length, and the calling source. Orders for jobs are received at a workshop for processing. On receipt, the
For each of the situations in Problem 18-3, identify the following: (a) nature of the calling source (finite or infinite), (b) nature of arriving customers (individually or in bulk), (c) type of the interarrival time (probabilistic or deterministic), (d) definition and type of service time, (f)
In each of the following situations, identify the customer and the server:*(a) Planes arriving at an airport.*(b) Taxi stand serving waiting passengers.(c) Tools checked out from a crib in a machining shop.(d) Letters processed in a post office.(e) Registration for classes in a university.(f) Legal
Acme Metal Jobshop is in the process of purchasing a multipurpose drill press. Two models, A and B, are available with hourly operating costs of $20 and $35, respectively.Model A is slower than model B. Queuing analysis of similar machines shows that when A is used, the average number of jobs in
Suppose that further analysis of the McBurger restaurant (Example 18.1-1) reveals the following additional results:Number of cashiers 1 2 3 4 5 6 7 Idleness (%) 0 8 12 18 29 36 42(a) What is the productivity of the operation (expressed as the percentage of time the employees are busy) when the
Consider Problem 17-4, dealing with patients suffering from kidney failure. Determine the following measures:(a) The expected number of years a patient stays on dialysis.(b) The longevity of a patient who starts on dialysis.(c) The life expectancy of a patient who survives 1 year or longer after a
An NC machine is designed to operate properly with power voltage setting between 108 and 112 volts. If the voltage falls outside this range, the machine will stop. The power regulator for the machine can detect variations in increments of one volt.Experience shows that change in voltage takes place
Pfifer and Carraway (2000). A company targets its customers through direct mail advertising. During the first year, the probability that the customer will make a purchase is .5, which decreases to .4 in year 2, .3 in year 3, and .2 in years 4. If no purchases are made in four consecutive years, the
At U of A, promotion from assistant to associate professor requires the equivalent of five points (years) of acceptable performance. Performance reviews are conducted once a year, and the candidate is given an average rating, a good rating, or an excellent rating. An average rating is the same as
Students at U of A have expressed dissatisfaction with the fast pace at which the math department is teaching the one-semester Cal I. To cope with this problem, the math department is now offering Cal I in 4 modules. Students will set their individual pace for each module and, when ready, will take
In a men’s singles tennis tournament, Andre and John are playing a match for the championship. The match is won when either player wins three out of five sets. Statistics show that there is 60% chance that Andre will win any one set.(a) Express the match as a Markov chain.(b) On the average, how
In Problem 17-3,(a) Determine the expected number of quarters until a debt is either repaid or lost as bad debt.(b) Determine the probability that a new loan will be written off as bad debt. Repaid in full.(c) If a loan is 6 months old, determine the number of quarters until its status is settled.
An employee who is now 55 years old plans to retire at the age of 62, but does not rule out the possibility of quitting earlier. At the end of each year, he weighs his options(and state of mind regarding work). The probability of quitting after one year is only .1 but seems to increase by
Jim must make five years worth of progress to complete his doctorate degree at ABC University. However, he enjoys the life of a student and is in no hurry to finish his degree. In any academic year there is a 50% chance he may take the year off and a 50%chance of pursuing the degree full time.
In Casino del Rio, a gambler can bet in whole dollars. Each bet will either gain $1 with probability .4 or lose $1 with probability .6. Starting with three dollars, the gambler will quit if all money is lost or the accumulation is doubled.(a) Express the problem as a Markov chain.(b) Determine the
When I borrow a book from the city library, I try to return it after one week. Depending on the length of the book and my free time, there is a 30% chance that I keep it for another week. If I have had the book for two weeks, there is a 10% chance that I’ll keep it for an additional week. Under
In Example 17.6-1, suppose that the labor cost for machines I and II is $25 per hour and that for inspection is only $15 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
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
An amateur gardener with training in botany is tinkering with cross-pollinating pink irises with red, orange, and white irises. 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% orange, 50% pink, and 25%
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 play
In Problem 17-29, 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.
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 5, the experiment is repeated by
The daily weather in Fayettville, Arkansas, can be cloudy (C), sunny (S), rainy (R), or windy (W). Records over the past 90 days are CCSWRRWSSCCCRCSSWRCRRRR CWSSWRWWRCRRRRCWSSWRWCCSWRRWSSCCCRCSSWSSWRWWRCR RRRCWSSWRWCCSWRRWSSS. Based on these records, use a Markov chain to determine the probability
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 starts
The federal government tries to boost small business activities by awarding annual grants for projects. All 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 if awards were given in each of the last three
In Problem 17-24, 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
Solve Problem 17-23, assuming that the order size, when placed, is exactly 5 pieces.
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
In Problem 17-21, 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
A bookstore restocks a popular book 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) Represent the daily inventory as a
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, 20%
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 10-year intervals shows that 10% of the
Warehouzer owns a renewable forest land for growing pine trees. Trees can fall into one of four categories depending on their age: baby (0–5 years), young (5–10 years), mature(11–15 years), and old (more than 15 years). Ten percent of baby and young trees die before reaching the next age
There are three categories of income tax filers in the United States: those who never evade taxes, those who sometimes do it, and those who always do it. An examination of audited tax returns from 1 year to the next shows that of those who did not evade taxes last year, 95% continue to be in the
A store sells a special item whose daily demand can be described by the following pdf:Daily demand, D 0 1 2 3 P{D} .1 .3 .4 .2 The store, using daily review, is comparing two ordering policies: (1) Order up to 3 units if the stock level is less than 2; else do not order. (2) Order 3 units if the
Some ex-cons spend the rest of their lives either free, on trial, in jail, or on probation.At the start of each year, statistics show that there is 50% chance that a free ex-con will commit a new crime and go on trial. The judge may send the ex-con to jail with probability.6 or grant probation with
Joe loves to eat out in area restaurants. His favorite foods are Mexican, Italian, Chinese, and Thai. On the average, Joe pays $12.00 for a Mexican meal, $17.00 for an Italian meal, $11.00 for a Chinese meal, and $13.00 for a Thai meal. Joe’s eating habits are predictable: There is a 70% chance
On a sunny day, MiniGolf can gross $2000 in revenues. If the day is cloudy, revenues drop by 20%. A rainy day will reduce revenues by 80%. If today’s weather is sunny, there is an 80% chance it will remain sunny tomorrow with no chance of rain. If it is cloudy, there is a 20% chance that tomorrow
A museum has six rooms of equal sizes arranged in the form of a grid with two rows and three columns. Each interior wall has a door that connects adjacent rooms. Museum guards move about the rooms through the interior doors. Represent the movements of each guard in the museum as a Markov chain, and
A game involves four balls and two urns. A ball in either urn has 50:50 chance of being transferred to the other urn. Represent the game as a Markov chain, and show that its states are periodic with period t = 2.
Classify the states of the following Markov chains. If a state is periodic, determine its period: 0 1 *(a) 0 0 1 1 0 0 *(b) 100 O 120130 4 1 OETIT 0 1 3 0 0 1 .2 (c) (d) 0 0 0 0 .1 122 .7 0.5 .5 0 .7 100100 053000 157000 030 00 0 0 0 0 0 0 .4.6 .2 8 .9 0 .7 .1
A die-rolling game uses a 4-square grid. The squares are designated clockwise as A, B, C, and D with monetary rewards of $4, – $2, – $6, and $9, respectively. Starting at square A, roll the die to determine the next square to move to in a clockwise direction. For example, if the die shows 2, we
Consider Problem 17-4.(a) For a patient who is currently on dialysis, what is the probability of receiving a transplant in two years?(b) For a patient who is currently a more-than-one-year survivor, what is the probability of surviving four more years?
Suppose that Bank1 currently has $1,000,000 worth of outstanding loans. Of these, $300,000 have just been paid, $150,000 are one quarter late, $250,000 are two quarters late, $200,000 are three quarters late, and the rest are over four quarters late. What would the picture of these loans be like
Consider Problem
If the police car is currently at a call scene, determine the probability that an apprehension will take place in two patrols.
Consider Problem
Determine the probability that the professor will purchase the current model in 4 years.
Consider Problem
Pliskin and Tell (1981). Patients suffering from kidney failure can either get a transplant or undergo periodic dialysis. During any one year, 30% undergo cadaveric transplants, and 10% receive living-donor kidneys. In the year following a transplant, 30% of those who receive the cadaveric
Cyert and Associates (1963). Bank1 offers loans which are either paid when due or are delayed. If the payment on a loan is delayed by more than 4 quarters (1 year), Bank1 considers the loan a bad debt and writes it off. The following table provides a sample of Bank1’s past experience with
A police car is on patrol in a neighborhood known for its gang activities. During a patrol, there is a 60% chance of responding in time to the location where help is needed; else regular patrol will continue. Upon receiving a call, there is a 10%chance for cancellation (in which case normal patrol
An engineering professor acquires a new computer once every two years. The professor can choose from three models: M1, M2, and M3. If the present model is M1, the next computer can be M2 with probability .25 or M3 with probability .1. If the present model is M2, the probabilities of switching to M1
Consider the infinite-horizon inventory situation with zero delivery lag and backlogged demand. Develop the optimal inventory policy based on the minimization of cost given that Holding cost for z units = hz2 Penalty cost for z units = px2 Show that for the special case where h = p, the optimal
The pdf of the demand per period in an infinite-horizon inventory model is given as f1D2 = .08D, 0 … D … 5 The unit cost parameters are Unit selling price = $10 Unit purchase price = $8 Unit holding cost per month = $1 Unit penalty cost per month = $10 Discount factor = .9 Determine the optimal
Consider a two-period probabilistic inventory model in which the demand is backlogged, and orders are received with zero delivery lag. The demand pdf per period is uniform between 0 and 10, and the cost parameters are given as Unit selling price = $2 Unit purchase price = $1 Unit holding cost per
Work Problem 16-12, assuming that there is a fixed cost of $10 associated with the delivery of donuts.
In the single-period model in Section 16.2.1, suppose that the model maximizes profit and that a setup cost K is incurred. Given that r is the unit selling price and using the information in Section 16.2.1, develop an expression for the expected profit, and determine the optimal order quantity.
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