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operations research an introduction

Operations Research An Introduction 11th Edition Hamdy A. Taha - Solutions

In linear regression, prove that the sum of differences between the predicted and estimated values over all the data points always equals zero—that is, n Σ(; – y ) = 0 - ni) 0 i=1
For the single-period model, show that for the discrete demand the optimal order quantity is determined from Р p+h P{D ≤ y" − 1} ≤ _P - ≤ P{D ≤ y*}
Two identical balls made of a tough alloy are tested for hardness. The test fails if a ball dropped from a floor in a 100-story building is dented upon impact. A ball can be reused in fresh drops only if undented. Using only these two identical balls, what is the smallest number of ball drops that
One of N machines must be selected for manufacturing Q units of a specific product. The minimum and maximum demands for the product are Q* and Q**, respectively. The total production cost for Q items on machine i involves a fixed cost Ki and a variable cost per unit ci, and it is given as TCi = Ki
A town is served by taxi services: Green Company and Blue Company. Statistics shows that Green controls 90% of the market. A hit-and-run night accident was reported by a witness who claimed it involved a Blue taxi. The police tested the witness claim under similar visibility conditions and
You have the chance to play the following game in a gambling casino. A fair die is rolled twice, leading to four outcomes: (1) Both rolls show the same even number, (2) Both rolls show the same odd number, (3) The two rolls show either even followed by odd or odd followed by even,
A test is available for detecting a certain genetic defect that is known to exist in 2% of the population. The test is false positive 10% of the time but detects true positives with probability .96. What is the probability that a person has the defective gene given that the test results are
Sunray Electric Coop uses a fleet of 20 trucks to service its electric network. The company wants to develop a preventive maintenance schedule for the fleet. The probability of a breakdown in year 1 is zero. For year 2, the breakdown probability is .03, increasing annually by .01 for years 3
Classify the states of the following Markov chains. If a state is periodic, determine its period:(a)(b)(c)(d) 01 10 0 0 1 100
Consider Problem 16-3. Suppose that Bank1 currently has $500,000 worth of outstanding loans. Of these, $100,000 have just been paid, $50,000 are 1 quarter late, $150,000 are 2 quarters late, $100,000 are 3 quarters late, and the rest are over 4 quarters late. What would the picture of these loans
Consider Problem 16-4.(a) For a patient who is currently on dialysis, what is the probability of receiving a transplant in 2 years?(b) For a patient who is currently a more-than-1-year survivor, what is the probability of surviving 4 more years?Problem 16-4Patients suffering from kidney failure can
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, move
A game involves four balls and two urns. A ball in either urn has probability 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.
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 three rows and two 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
The following table provides weather data in Fayetteville, Arkansas:Based on these records, use a Markov chain to determine the probability that a typical day in Fayetteville will be cloudy, sunny, rainy, or windy. Successive 90-day weather in Fayetteville, ARª CCSWR, RWSSC, CCRCS, SWRCR, RRRCW,
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%
A company reviews the state of one of its important products annually and decides whether it is successful (state 1) or unsuccessful (state 2). The company must decide whether to advertise the product to further promote sales. The following matrices, P1 and P2, provide the transition
A company can advertise through radio, TV, or newspaper. The weekly costs of advertising on the three media are estimated at $200, $900, and $300, respectively. The company can classify its sales volume during each week as (1) Fair, (2) Good, or (3) Excellent. A summary of the
In Problem 16-24, 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
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 3 years and lowest if awards were given in each of the last 3 years.
In Example 16.6-1, 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
In the gardener model, identify the matrices P and R associated with the stationary policy that calls for using fertilizer whenever the soil condition is fair or poor.
Identify all the stationary policies for the gardener model.
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 $18 and $25, 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
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
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
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 in hope of reducing waiting time.
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?
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
Prove that the mean and standard deviation of the exponential distribution are equal.
Prove that the mean and variance of the Poisson distribution during an interval t equal λt, where λ is the arrival rate.
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
A first-year 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
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 1/μ time units.
Derive the truncated Poisson distribution from the difference-differential equations of the pure death model using induction.
The time barber Joe takes to give a haircut is exponential with a mean of 12 minutes. Because of his popularity, customers arrive randomly (Poisson) at a rate much higher than Joe can handle: six per hour. Joe will feel comfortable if the arrival rate is effectively reduced to about four customers
McBurger fast-food restaurant has three 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 of 5 minutes. The waiting room inside the
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 post
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
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
Identify the discrete events needed to simulate the following situation: Two types of jobs arrive from two different sources. Both types are processed on a single machine, with priority given to jobs from the first source.
The lead time for receiving an order can be 1 or 2 days, with equal probabilities. The demand per day assumes the values 0, 1, and 2 with the respective probabilities of .2, .7, and .1. Use the random numbers in Table 19.1 (starting with column 1) to estimate the joint distribution of the demand
The following table represents the variation in the number of waiting customers in a queue as a function of the simulation time.Compute the following measures of performance:(a) The average length of the queue.(b) The average waiting time in the queue for those who must wait. Simulation time, T
Jobs arrive at a constant rate at a carousel conveyor system. Three service stations are spaced equally around the carousel. If the server is idle when a job arrives at the station, the job is removed from the conveyor for processing. Otherwise, the job continues to rotate on the carousel until a
Two players, Juanita and Rafael, take turns in tossing a fair coin. If the outcome is heads, Rafael gets $10 from Juanita. Otherwise, Juanita gets $10 from Rafael.(a) How is the game simulated as a Monte Carlo experiment?(b) Run the experiment for 5 replications of 10 tosses each. Use the first
Cars arrive at a two-lane, drive-in bank, where each lane can house a maximum of four cars. If the two lanes are full, arriving cars seek service elsewhere. If at any time one lane is at least two cars longer than the other, the last car in the longer lane will jockey to the last position in the
The cafeteria at Elmdale Elementary provides a single-tray, fixed-menu lunch to all its pupils. Kids arrive at the dispensing window every 30 seconds. It takes 18 seconds to receive the lunch tray. Map the arrival–departure events on the time scale for the first five pupils.
Classify the following variables as either observation based or time based:(a) Time-to-failure of an electronic component.(b) Inventory level of an item.(c) Order quantity of an inventory item.(d) Number of defective items in a lot.(e) Time needed to grade test papers.(f) Number of cars in the
hypothetical triathlon competition includes three sequential activities: running, cycling, and swimming. Each participant attempts to break the fastest time achieved in previous competitions, currently set at T = 2 hr. The times for completing the three activities for this specific participant are
The demand for an expensive spare part of a passenger jet is 0, 1, 2, or 3 units per month with probabilities .2, .3, .4, and .1, respectively. The airline maintenance shop starts operation with a stock of 5 units and will bring the stock level back to 5 units immediately after it drops below 2
Simulate five wins or losses of the following game of craps: The player rolls two fair dice. If the outcome sum is 7 or 11, the player wins $10. Otherwise, the player records the resulting sum (called point) and keeps on rolling the dice until the outcome sum matches the recorded point, in which
Psychology professor Yataha is conducting a learning experiment in which mice are trained to find their way around a maze. The base of the maze is square. A mouse enters the maze at one of the four corners and must find its way through the maze to exit at the same corner it entered. The design of
Adapt file excelPr19-23.xlxs assuming that the triathlon times (in minutes) are normally distributed with the following means, E(ti), and standard deviations, Std(ti):The current record is T = 3 hours. Use the Box-Muller method to generate the sample times. i E(t₂) Std(ti) 1 2 25 100 3 10 3 55 6
Jobs arrive at Metalco Jobshop according to a Poisson distribution, with a mean of six jobs per day. Received jobs are assigned to five machines on a strict rotational basis; meaning, machine 1 will handle jobs 1 and 6. Determine one sample of the interval between the arrival of jobs at machine 1.
Develop a spreadsheet model for evaluating following integrals using a Monte Carlo experiment with 5 replications, 20 observations each:(a)(b)(c)(d) f(x)=√1 + √2x, 2 ≤ < x ≤ 5 <
In a simulation model, the subinterval method is used to compute batch averages. The transient period is estimated to be 100, and each batch has a time base of 100 time units as well. Using the following data, which provide the waiting times for customers as a function of the simulation time,
Solve the following simultaneous equations by converting the system to a nonlinear objective function with no constraints. x₂ - x² = 0 X2 X1 x₁ = 2
Find the maximum of each of the following functions by dichotomous search. Assume that Δ = .05.(a)(b) f(x) = x cos x, 0 ≤ x ≤ π(c) f(x) = x sin πx, 1.5 ≤ x ≤ 2.5(d) f(x) = −(x − 3)2, 2 ≤ x ≤ 4(e) f(x) = 1 |(x-3)³1' , 2 ≤ x ≤ 4
Determine the extreme points of the following functions.(a)(b) f(x) = x² + x² - 3x1x2
Determine the extreme points of the following functions.(a) f (x) = x3 + x(b) f (x) = x4 + x2(c) f (x) = 4x4 − x2 + 5(d) f (x) = (3x − 2)2 (2x − 3)2(e) f (x) = 6x5 − 4x3 + 10
Convert the following stochastic problem into an equivalent deterministic model.subject toAssume that a1 and a3 are independent and normally distributed random variables with means E{a1} = 2 and E{a3} = 5 and variances var{a1} = 9 and var{a3} = 16 and that b2 is normally distributed with mean 15
Prepare the input file RM3x.dat for the Reddy Mikks model (file RM3.txt), assuming the that the read statement is given as read (j in paint) (i in resource} ( rhs[i], (j in paint) aij [i, j] )
Consider the following stochastic programming model:subject toThe parameters a2 and a3 are independent and normally distributed random variables with means 5 and 2, and variance 16 and 25, respectively. Convert the problem into a (deterministic) separable programming form. Maximize z = x₁ +
Solve the following problem by the linear combinations method.subject to Minimize f(x) = x² + x² - 4x₁x ₂
Suppose that 5 components (one unit per product unit) are used in the production of 10 products according to the following schedule:The unit assembly cost of each product is a function of the component used: $9, $4, $6, $5, and $8 for components 1 through 5, respectively. The maximum demand for any
For the Reddy Mikks model, explain why the following read statement is cumbersome: read (i in resource} ( rhs[i], (j in paint) (unitprofit[j], aij [i,j]))
Suppose that the contents of file RM4rhs.tab read asMake the necessary changes in RM4.txt and execute the model. ampl. tab 1 1 constrName Availability m1 m2 24 6 demand 1 market 2
Givenfind(a) A + 7B(b) 2A − 3B(c) (A + 2B)T A = 1 9 2 5-8, B = 37 2 4 7 9 3 -1 4 6 2 8 10
Modify RM2.txt so that rhs[“m1”] will assume the values 20 to 35 tons in steps of 5 tons. All solve commands must be executed from within the command file cmd.txt in the following manner:The command file cmd.txt is developed using the three different versions to construct the loop:(a) for{}.(b)
A fair 6-faced die is rolled 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 and less than 6, and F is odd and greater than
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 have
Two dice are rolled independently, and the two numbers that turn up are recorded. Determine the following:(a) The probability that the two numbers are even.(b) The probability that the sum of the two numbers is 10.(c) The probability that the two numbers differ by at least 3.
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 Nancy 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. Nancy’s past winning record against John is one out of three. Determine the following:(a) The probability that Jim will win the tournament.(b)
Bayes’ theorem. Given the two events A and B, show that P{A|B} = P{B\A}P{A}, P{B} > 0 P{B}
The owner of a newspaper stand receives 50 copies of Al Ahram newspaper every morning. The number of copies sold, x, varies randomly according to the following probability distribution:(a) Determine the probability that the owner will sell out completely.(b) Determine the expected number of unsold
The stock of WalMark Stores, Inc., trades on the New York Stock Exchange under the symbol WMS. Historically, the price of WMS goes up(down) with the Dow 60% (25%) of the time. There is also a 5% chance that WMS will go up when the Dow goes down and 10% that it will go down when the Dow goes up.(a)
Graduating high school seniors with an ACT score of at least 26 can seek admission in two universities, A and B. 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 accepted in B
Show that the mean and variance of a uniform random variable x, a ≤ x ≤ b, are E{x} var {x} = b + a 2 (b - a)2 12
Prove that if the probability P{A | B} = P{A}, then A and B must be independent.
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) If
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) Determine the pdf and CDF of x, and graph the resulting functions.(b) Find the probability that x is an even
For the pdf f(x), prove that var {x} = E{x²} - (E{x})²
Given the pdf f(x) and y = cx + d, where c and d are constants, prove that E{y} = cE{x} + d var{y} = c² var {x}
The daily demand for unleaded gasoline is uniformly distributed between 750 and 1250 gallons. The 1100-gallon gasoline tank is refilled daily at midnight. What is the probability that the tank will be empty just before a refill?
A fair die is rolled 10 times. What is the probability that the rolled die will not show an even number?
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?
A fortune-teller claims to predict whether 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 millionaire and one
In a gambling casino, you play the game of selecting 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. Determine your long-run expected
Suppose that you throw 2 fair dice simultaneously. If there is a match, you receive 50 cents. Otherwise, you pay 10 cents. Determine the expected payoff of the game.
Prove the formulas for the mean and variance of the binomial distribution.
Customers shopping at Walmark Store are both urban and suburban. Urban customers arrive at the rate of 5 per minute, and suburban customers arrive at the rate of 7 per minute. Arrivals are totally random. Determine the probability that the interarrival time for all customers is less than 5 seconds.
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?
The inside diameter of a cylinder is normally distributed with a mean of 1 cm and a standard deviation of .01 cm. A solid rod is assembled inside each 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
An automatic device is used to count the volume of traffic at a busy intersection. The arrival time is recorded and translated into an absolute time starting from zero. The following table provides the arrival times (in minutes) for the first 60 cars. Use Excel to construct a suitable histogram.
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 1b. What is the probability that the helicopter will not take off with five persons
The following data represent the period (in seconds) needed to transmit a message.Use Excel to construct a suitable histogram. Test the hypothesis that these data are drawn from a uniform distribution at a 95% confidence level, given the following additional information about the theoretical
Consider Problem 16-2. If the police car is currently at a call scene, determine the probability that an apprehension will take place in two patrols.Problem 16-2A police car is normally on regular patrol circulating a neighborhood. During the patrol, there is a 60% chance of responding in time to
Consider Problem 16-1. Determine the probability that the professor will purchase the current model in 4 years.Problem 16-1An engineering professor acquires a new computer every 2 years. The professor can choose from three models: M1, M2, and M3. If the present model is M1, the next computer can be

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