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operations research an introduction
Operations Research An Introduction 8th Edition Hamdy A. Taha - Solutions
*5. 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
4. 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 10 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
3. The Springdale High School band is performing a benefit jazz concert in its new 4OO-seat auditorium. Local businesses buy the tickets in blocks of 10 and donate them to youth organizations.Tickets go on sale to business entities for 4 hours only the day before the concert. The process of placing
2. Consider Example 15.4-2. In each of the following cases, first write the answer algebraically, and then use excelPoissonQ.xls orTORA 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
1. In Example 15.4-2, use excelPoissonQ.xls orTORA to compute Pn(7), n = 1,2, ... , 18, and then verify manually that these probabilities yield E{nlt = 7} = .664 dozen.
7. Prove that the mean and variance of the Poisson distribution during an interval t equal At, where A is the arrival rate.8. Derive the Poisson distribution from the difference-differential equations of the pure birth model. Hint: The solution of the general differential equation
(b) Determine the average number of weeks I keep a book before returning it to the library.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
6. 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
5. 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
4. 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 8 customers arrived by 11:05 A.M.(b) The probability that
3. In a bank operation, the arrival rate is 2 customers per minute. Determine the following:(a) The average number of arrivals during 5 minutes.(b) The probability that no arrivals will occur during the next .5 minute.(c) The probability that at least one arrival will occur during the next .5
2. 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 3-month period.(b) The probability that no more
*1. In Example 15.4-1, suppose that the clerk who enters the information from birth certificates into the computer normally waits until at least 5 certificates have accumulated. Find the probability that the clerk will be entering a new batch every hour.
13. Prove that the mean and standard deviation of the exponential distribution are equal.
(b) Determine the average number of bets until the game ends.
U. The U ofA 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
(c) Determine the probability of ending the game with $6. Of losing all $3.
4. 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 his pursuing the degree full
11. The time between failures of a Kencore refrigerator is known to be exponential with mean value 9000 hours (about 1 year of operation) and the company issues a I-year warranty on the refrigerator. What are the chances that a breakdown repair will be covered by the warranty?
*10. 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 1 between 4 and 5 minutes, the discount is 6%. If the interarcival time is longer than i 5 minutes, the customer gets 2%
9. In Problem 7, 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
8. Suppose that in Problem 7 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 expected
*7. 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-hour period. The interarrival
6. 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 2 minutes, (b) between 2 and 3 minutes, and (c) more than 3 minutes. Arrivals in similar restaurants occur at the rate
(c) What is the probability that at least one student will visit the game room during the next 20 minutes?
(b) What is the probability that no students will arrive at the game room during the next 15 minutes?
5. 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?
4. Suppose that the time between breakdowns for a machine is exponential with mean 6 hours. If the machine has worked without failure during the last 3 hours, what is the probability that it will continue without failure during the next hour? That it will break down during the next .5 hour?
3. The time between arrivals at the State Revenue Office is exponential with mean value.05 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
2. In Example 15.3-1,determine the following:(a) The average number of failures in 1 week, assuming the service is offered 24 hours a day, 7 days a week.(b) The probability of at least one failure in a 2-hour period.(c) The probability that the next failure will not occur within 3 hours.(d) If no
(b) Determine the expected number of academic years before Jim's student life comes to an end.
(c) In each of the following cases, determine the average service rate per hour, IL, and the average service time in hours.*(i) One service is completed every 12 minutes.(ii) Two departures occur every 15 minutes.(iii) Number of customers served in a 30-minute period is 5.(iv) The average service
L (a) Explain your understanding of the relationship between the arrival rate Aand the average interarrival time. What are the units describing each variable?(b) In each of the following cases, detennine the average arrival rate per hour, A, and the average interarrival time in hours.*(i) One
(c) Determine the probability that Jim will end his academic journey with only a master's degree.
S. In each of the situations in Problem 1, discuss the possibility of the customers jockeying, balking, and reneging.
(d) If Jim's fellowship pays an annual stipend of $15,000 (but only when he attends school), how much will he be paid before ending up with a degree?s. 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
4. True or False?(a) An impatient waiting customer may elect to renege.(b) If a long waiting time is anticipated, an arriving customer may elect to balk.(c) Jockeying from one queue to another is exercised to reduce waiting time.
3. 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
2. For each of the situations in Problem 1, 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 ~ime, (f)
(b) What is the probability that the employee stay with the company until planned retirement at age 62?(c) At age 57, what is the probability the employee will call it quits'?(d) At age 58, what is the expected number of years before the employee is off the payroll?
1. 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.(I)
6. In Problem 3, Set 17.1a,(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 six months old, determine the number of quarters until its status
7. 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 anyone set.(a) Express the match as a Markov chain.(b) On the average, how
*8. Students at U of A have expressed dissatisfaction with the fast pace at which the math department is teaching the one-semester Call. To cope with this problem, the math department is now offering Call in 4 modules. Students will set their individual pace for each module and, when ready, will
(b) On the average, would a student starting with module 1 at the beginning of the current semester be able to take Cal II the next semester (Call is a prerequisite for Cal II)?(c) Would a student who has completed only one module last semester be able to finish Call by the end of the current
(d) Would you recommend that the use of the module idea be extended to other basic math classes? Explain.
9. At U of A, promotion from assistant to associate professor requires the equivalent of five years of seniority. Performance reviews are conducted once a year and the candidate is given either an average rating, a good rating, or an excellent rating. An average rating is the same as probation and
*10. (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 reduces to .4 in year 2, .3 in year 3, and .2 in year 4. If no purchases are made in four consecutive years,
(b) Determine the expected number of years a new customer will be on the mailing list.
(c) If a customer has not made a purchase in two years, determine the expected number of years on the mailing list.
11. 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 take
(b) Determine the probability that the machine will stop because the voltage is low. High.
(c) What should be the ideal voltage setting that will render the longest working duration for the machine?
12. Consider Problem 4, Set 17.1a, dealing with patients suffering from kidney failure. Determine the following measures:(a) TIle 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 one year or
1. Consider Example 18.2-1.(a) Compute acf by the two methods presented in the example, using aX2 = .001 instead ofax2 = .01. Does the effect of linear approximation become more negligible with the decrease in the value ofaxl?
*(b) Specify a relationship among the elements of ax = (ax}. aX2, aX3) at the feasible point X o = (1,2,3,) that will keep the point Xo + ax feasible.
(c) lty = (Xl, X3) and Z = Xl> what is the value of ax! that will produce the same value of ocf given in the example?
1. Suppose that Example 18.2-2 is solved in the following manner. First, use the constraints to express Xl and X2 in terms of X3; then use the resulting equations to express the objective function in terms of X3 only. By taking the derivative of the new objective function with respect to X3, we can
2. Apply the Jacobian method to Example 18.2-1 by selecting Y = (X2' X3) and Z = (xd.
*3. Solve by the Jacobian method:We can draw some general conclusions from the application of the Jacobian method to the linear programming problem. From the numerical example, the necessary conditions require the independent variables to equal zero. Also, the sufficiency conditions indicate that
4. Solve by the Jacobian method:Minimize f(X) = 5xf + x~ + 2XtX2 subject to(a) Find the change in the optimal value off(X) if the constraint is replaced by XlX2 - 9.99 = O.(b) Find the change in value off (X) if the neighborhood of the feasible point (2,5)given that XIX2 = 9.99 and (lxl == .01.
5. Consider the problem:Maximize f(X) = xi + 2x~ + lOxj + 5X1X2 subject to gl (X) = Xl + x~ + 3X2X3 - 5 = 0 g2(X) = XI + 5XIX2 + x~ - 7 = 0 Apply the Jacobian method to find af(X) in the neighborhood of the feasible point (1, 1; 1).Assume that this neighborhood is specified by ogl = -.01, ag2 =
6. Consider the problem Minimize f(X) = xI + x~ + xj + x~subject to gl(X) = Xl + 2X2 + 3X3 + 5X4 - 10 = 0 g2(X) = Xl + 2x2 + 5X3 + 6X4 - 15 = 0(a) Show that by selecting X3 and X4 as independent variables, the Jacobian method fails to provide a solution and state the reason.
TIle inside diameter of a cylinder is normally distributed with a mean of 1 cm and a standard deviation of .01 em. A solid rod is assembled inside eaeh cylinder. The diameter of the rod is also normally distributed with a mean of .99 cm and a standard deviation of.01 cm. Determine the percentage of
The weights of individuals who seek a helicopter ride in an amusement park have a mean of 180 lb and a standard deviation of 15 lb. The helicopter can carry five persons but has a maximum weight capacity of 1000 lb. What is the probability that the helicopter will not take off with five persons
*(b) Solve the problem using Xl and x3 as independent variables and apply the sufficiency condition to determine the type of the resulting stationary point.
1. The college of engineering at U of A requires a minimum ACf score of 26. The test score among high school seniors in a given school district is normally distributed with mean 22 and standard deviation 2.(a) Determine the percentage of high school seniors who are potential en!9.neering
2. Prove the formulas for the mean and variance of the exponential distribution.
:~1. Walmark Store gets its customers from within town and the surrounding rural areas.Town customers arrive at the rate of 5 per minute, and rural customers arrive at the rate of 7 per minute. Arrivals are totally random. Determine the probability that the interarrival time for all customers is
2. The Poisson distribution with parameter A approximates the binomial distribution with parameters (n, p) when n ~ 00, p ~ 0, and np ~ A. Demonstrate this result for the situation where a manufactured lot is known to contain 1% defective items. If a sample of 10 items is taken from the lot,
*1. Customers arrive at a service facility according to a Poisson distribution at the rate of four per minute. What is the probability that at least one customer will arrive in any given 30-second interval?
6. Prove the formulas for the mean and variance of the binomial distribution.Repair jobs arrive at a small-engine repair shop in a totally random fashion at the rate of 10 per day.a. What is the average number of jobs that are received daily at the shop?b. What is the probability that no jobs will
S. Suppose that you play the following game: You throw 2 fair dice. If there is no match, you pay 10 cents. If there is a match, you get 50 cents. What is the expected payoff for the game?
4. In a gambling casino game you are required to select a number from 1 to 6 before the operator rolls three fair dice simultaneously. The casino pays you as many dollars as the number of dice that match your selection. If there is no match, you pay the casino only$1. What is your long-run expected
*3. A fortune teller claims to predict whether or not people will amass financial wealth in their lifetime by examining their handwriting. To verify this claim, 10 millionaires and 10 university professors were asked to provide samples of their handwriting.The samples are then paired, one
2. Suppose that five fair coins are tossed independently. What is the probability that exactly one of the coins will be different from the remaining four?
*1. A fair die is rolled 10 times. What is the probability that the rolled die will not show an even number?
4. Given the pdf f(x), a ~ x sb, prove that var{x} = E{x2} - (E{X})2 5. Given the pdf f(x), a s x sb, and y = ex +d, where c and d are constants. Prove that E{y} = eE{x} + d var{y} = e2 var{x}
3. Show that the mean and variance of a uniform random variable x, a ~ x ~ h, are E{x} = b + a 2(b - a)2 var{x} = 12
2. Compute the mean and variance of the random variable in Problem 2, Set 12.2a.
1. Compute the mean and variance of the random variable defined in Problem 1, Set 12.2a.
*3. The owner of a newspaper stand receives 50 copies of Ai Ahram newspaper every morning.ll1e number of copies sold daily, X, varies randomly according to the following probability distribution:{-ls, x = 35, 36, , 49 p(x) = f'x: 50,51, ,59 33'X - 60,61, ,70(a) Determine the probability that the
2. The results of Example 12.3-1 and of Problem 1 show positive averages for both the surplus and shortage of stamps. Are these results inconsistent? Explain.
1. In Example 12.3-1, compute the average number of extra stamps needed to meet your maximum possible demand.
*3. The daily demand for unleaded gasoline is uniformly distributed between 750 and 1250 gallons. The gasoline tank, with a capacity of 1100 gallons, is refilled daily at midnight.What is the probability that the tank will be empty just before a refill?
1. The number of units, x, needed of an item is discrete from 1 to 5. The probability, p(x), is directly proportional to the number of units needed. The constant of proportionality is K.(a) Oetermine the pdf and COF of x, and graph the resulting functions.
*7. Statistics show that 70% of all men have some form of prostate cancer. The PSA test will show positive 90% of the time for afflicted men and 10% of the time for healthy men.What is the probability that a man who tested positive does have prostate cancer?
6. A retailer receives 75% of its batteries from Factory A and 25% from Factory B. The percentages of defectives produced by A and B are known to be 1% and 2%, respectively.A customer has just bought a battery randomly from the retailer.(a) What is the probability that the battery is defective?(b)
4. Prove that if the probability p{A1B} = P{A}, then A and B must be independent.
*3. Graduating high school seniors with an ACf score of at least 26 can apply to two universities, A and B, for admission. The probability of being accepted in A is .4 and in B .25.The chance of being accepted in both universities is only 15%.(a) Determine the probability that the student is
2. The stock of WalMark Stores, Inc., trades on the New York Stock Exchange under the symbol WMS. Historically, the price ofWMS goes up with the increase in the Dow average 60% of the time and goes down with the Dow 25% of the time. There is also a 5%chance that WMS will go up when the Dow goes
1. In Example 12.1-2, suppose that you are told that the outcome is less than 6.(a) Determine the probability of getting an even number.(b) Determine the probability of getting an odd number larger than one.
You can toss a fair coin up to 7 times. You will win $100 if three tails appear before a head is encountered. What are your chances of winning?Ann, Jim, John, and Liz are scheduled to compete in a racquetball tournament. Ann is twice as likely to beat Jim, and Jim is at the same level as John.
Suppose that you roll two dice independently and record the number that turns up for each die. Determine the following:(a) The probability that both numbers are even.(b) The probability that the sum of the two numbers is 10.(c) TIle probability that the two numbers differ by at least 3.
1. A fair 6-faced die is tossed twice. Letting E and F represent the outcomes of the two tosses, compute the following probabilities:(a) The sum of E and F is 11.(b) The sum of E and F is even.(c) The sum of E and F is odd and greater than 3.(d) E is even less than 6 and Fis odd greater than 1.(e)
*3. Answer Problem 2 assuming that two or more persons share your birthday.
*2. Consider a random gathering of n persons. Determine the smallest n such that it is more likely than not that two persons or more have the same birthday. (Hint: Assume no leap years and that all days of the year are equally likely to be a person's birthday.)
(c) Determine the sensitivity coefficients given the solution in (b).
*1. In a survey conducted in the State of Arkansas high schools to study the correlation between senior year scores in mathematics and enrollment in engineering colleges, 400 out of 1000 surveyed seniors have studied mathematics. Engineering enrollment shows that, of the 1000 seniors, 150 students
2. In the cost model in Section 15.9.1, it is generally difficult to estimate the cost parameter C2 (cost of waiting). As a result, it may be helpful to compute the cost C2 implied by the aspiration levels. Using the aspiration level model to determine c*, we can then estimate the implied C2 by
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