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business
operations research an introduction
Operations Research An Introduction 8th Edition Hamdy A. Taha - Solutions
*(b) Suppose that the manufacturer ships batches to two customers, A and B. The contracts specify that the defectives for A and B should not exceed 5% and 8%, respectively.A penalty of $100 is incurred per percentage point above the maximum limit.Supplying better-quality batches than specified by
7. Historical data at Acme Manufacturing estimate a 5% chance that a manufactured batch of widgets will be unacceptable (bad). A bad batch has 15% defective items, and a good batch includes only 4% defective items. Letting a = lh (= 92) indicate that the batch is good (bad), the associated prior
(b) Determine the best course of action for AFe.
(b) Suppose that you contract a literary agent to conduct a survey concerning the potential success of the novel. From past experience, the agent advises you that when a novel is successful, the survey will predict the wrong outcome 20% of the time.When the novel is not successful, the survey will
*4. You are the author of what promises to be a successful novel. You have the option to either publish the novel yourself or through a publisher. 'The publisher is offering you $20,000 for signing the contract. If the novel is successful, it will sell 200,000 copies. If it isn't, it will sell only
3. In Example 13.2-2, suppose that you have the additional option of investing the original$10,000 in a safe certificate of deposit that yields 8% interest. Your friend's advice applies to investing in the stock market only.(a) Develop the associated decision tree.(b) What is the optimal decision
*2. Elektra receives 75% of its electronic components from vendor A and the remaining 25%from vendor B.lhe percentage of defectives from vendors A and Bare 1% and 2%, respectively.When a random sample of size 5 from a received lot is inspected, only one defective unit is found. Detennine the
1. Data in a community college show that 75% of new students who took calculus in high school do well, compared with 50% of those who did not take calculus. Admissions for the current academic year show that only 30% of the new students have completed a course in calculus. What is the probability
*19. Aspiration Level Criterion. Acme Manufacturing uses an industrial chemical in one of its processes. The shelf life of the chemical is 1 month, following which any amount left is destroyed.The use of the chemical by Acme (in gallons) occurs randomly according to the following distribution:{20~,
(b) The manufacturer can invest $1000 to obtain additional information about whether or not the price will increase. This information says that there is a 58% chance that the probability of price increase will be .9 and a 42% chance that the probability of price increase will be .3. Would you
18. (Rappaport, 1967) A manufacturer has used linear programming to determine the optimum production mix of the variousTV models it produces. Recent information received by the manufacturer indicates that there is a 40% chance that the supplier of a component used in one of the models may raise the
(b) Study the sensitivity of the solution to changes in the probability of good weather.
17. (Cohan and Associates, 1984) Modern forest management uses controlled fires to reduce fire hazards and to stimulate new forest growth. Management has the option to postpone or plan a burning. In a specific forest tract, if burning is postponed, a general administrative cost of $300 is incurred.
16. The outer diameter,d, of a cylinder is processed on an automatic machine with upper and lower tolerance limits of d + tv and d - tL . The production process follows a normal distribution with mean J.L and standard deviation CT. An oversized cylinder is reworked at the cost of c\ dollars. An
(b) Determine the value of a that maximizes the expected profit.
*15. An automatic machine produces a (thousands of) units of a product per day. As a increases, the proportion of defectives,p, goes up according to the following probability density function Each defective item incurs a loss of $50. A good item yields $5 profit.(a) Develop a decision tree for this
14. In Problem 13, suppose that the store wishes to extend the decision problem to a 2-day horizon. The alternatives for the second day depend on the demand in the first day. If demand on day 1 equals the amount stocked, the store will continue to order the same quantity for day 2; if it exceeds
*12. Sunray Electric Coop uses a fleet of 20 trucks to service its electric network. The company wants to develop a schedule of periodic preventive maintenance for the fleet. The probability of a breakdown in year 1 is zero. For year 2, the breakdown probability is .03, and it increases annually by
10. Rework Problem 9, assuming that the annual interest rate is 10% and that the decisions are made considering the time value of money. [Note: You need compound interest tables to solve this problem. You can use Excel function NPV(i, R) to compute the present value of cash flows stored in range R
8. Acme Manufacturing produces widget batches with .8%,1%,1.2%, and 1.4% defectives according to the respective probabilities .4, .3, .25, and .05. Three customers, A, B, and C, are contracted to receive batches with no more than .8%, 1.2%, and 1.4% defectives, respectively.If the defectives are
.8 if customers are receptive and $200,000 with probability .2 if they are not.(a) Draw the associated decision tree.(b) What course of action should AFC follow in launching the new product?A fair coin is flipped three successive times. You receive $1.00 for each head (ll) that turns up and an
5. APC is about to launch its new Wings 'N Things fast food nationally.The research department is convinced that Wings 'N Things will be a great success and wants to introduce it immediately in all AFC outlets without advertising. The marketing department sees"things" differently and wants to
4. You have the chance to invest your money in either a 7.5% bond that sells at face value or an aggressive growth stock that pays only 1% dividend. If inflation is feared, the interest rate will go up to 8%, in which case the principal value of the bond will go down by 10%, and the stock value
3. You have the chance to invest in three mutual funds: utility, aggressive growth, and global.The value of your investment will change depending on the market conditions. There is a 10% chance the market will go down, 50% chance it will remain moderate, and 40%chance it will perform well. The
*2. Farmer McCoy can plant either corn or soybeans. The probabilities that the next harvest prices of these commodities will go up, stay the same, or go down are .25, .30, and .45, respectively.If the prices go up, the corn crop will net $30,000 and the soybeans will net$10,000. If the prices
1. You have been invited to play the Fortune Wheel game on television. The wheel operates electronically with two buttons that produce hard (ll) or soft (5) spin of the wheel. The wheel itself is divided into white (W) and red (R) half-circle regions. You have been told that the wheel is designed
Determine the winning candidate and assess the consistency of the decision.
1. Consider the data of Problem 1, Set 13.la. Copy the weights in a logical order into the solution summary section of the spreadsheet excelAHExls, then develop the formula for evaluating the first alternative, VA, and copy it to evaluate the remaining two alternatives.
1. In Example 16.2-1, estimate the area of the circle using the first two columns of the (0, 1)random numbers in Table 16.1. (For convenience, go down each column, selecting R1 first and then R2.) How does this estimate compare with the ones given in Figure 16.2?
2. Suppose that the equation of a circle is(x - 3f + (y + 2)2 = 16(a) Define the corresponding distributionsf(x) andf(y), and then show how a sample point (x, y) is determined using the (0, 1) random pair (RJ, R2 ).
(b) Use excelCircle.xls to estimate the area and the associated 95% confidence interval given n = 100,000 and N = 10.
3. Use Monte Carlo sampling to estimate the area of the lake shown in Figure 16.3. Base the estimate on the first two columns of (0, 1) random numbers in Table 16.1.
4. Consider the game in which two players, Jan and Jim, take turns in tossing a fair coin. If the outcome is heads, Jim gets $10 from Jim. Otherwise, Jan gets $10 from Jan.*(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 five columns of the (0,1) random numbers in Table 16.1, with each column corresponding to one replication.(c) Establish a 95% confidence interval on Jan's winnings.
(d) Compare the confidence interval in (c) with Jan's expected theoretical winnings.
5. Consider the following definite integral:(a) Develop the Monte Carlo experiment to estimate the integral. x dx
(b) Use the first four columns in Table 16.1 to evaluate the integral based on 4 replications of size 5 each. Compute a 95% confidence interval, and compare it with the exact value of the integral.
*7. 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 16.1 (starting with column 1) to estimate the joint distribution of the demand
8. Consider the Buffon needle experiment. A horizontal plane is ruled with parallel lines spaced D cm apart. A needle of length d em (d < D) is dropped randomly on the plane.The objective of the experiment is to determine the probability that either end of the needle touches or crosses one of the
(b) Design the Monte Carlo experiment, and provide an estimate of the desired probability.(c) Use Excel to obtain 4 replications of size 10 each of the desired probability. Determine a 95% confidence interval for the estimate. Assume D = 20 em and d = 10 em.(d) Prove that the theoretical
1. Categorize the following situations as either discrete or continuous (or a combination of both). In each case, specify the objective of developing the simulation model.*(a) Orders for an item arrive randomly at a warehouse. An order that cannot be filled immediately from available stock must
(b) World population is affected by the availability of natural resources, food production, environmental conditions, educational level, health care, and capital investments.
(c) Goods arrive on pallets at a receiving bay of an automated warehouse. The pallets are loaded on a lower conveyor belt and lifted through an up-elevator to an upper conveyor that moves the pallets to corridors. The corridors are served by cranes that pick up the pallets from the conveyor and
2. Explain why you would agree or disagree with the following statement: "Most discrete event simulation models can be viewed in some form or another as queuing systems consisting of sources from which customers are generated, queues where customers may wait, and facilities where customers are
*1. In Example 16.3-2, suppose that the first customer arrives at time 0. Use the first three random numbers in column 1 of Table 16.1 to generate the arrival times of the next 3 customers and graph the resulting events on the time scale.
*2. Uniform Distribution. Suppose that the time needed to manufacture a part on a machine is described by the foHowing uniform distribution:1 f(t) = --, a $. t $. b b-a Determine an expression for the sample t given the random number R.
3. Jobs are received randomly at a one-machine shop. The time between arrivals is exponential with mean 2 hours. The time needed to manufacture a job is uniform between 1.1 and 2 hours. Assuming that the first job arrives at time 0, determine the arrival and departure time for the first five jobs
4. 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
5. In a simulation situation,TV units are inspected for possible defects. TIlere is an 80%chance that a unit will pass inspection, in which case it is sent to packaging. Otherwise, the unit is repaired. We can represent the situation symbolically in one of two ways.goto REPAIRl.2, PACKAGEI.S goto
6. A player tosses a fair coin repeatedly until a head occurs. The associated payoff is 2/1, where n is the number of tosses until a head comes up.(a) Devise the sampling procedure of the game.(b) Use the random numbers in column 1 of Table 16.1 to determine the cumulative payoff after two heads
7. Triangular Distribution. In simulation, the lack of data may make it impossible to determine the probability distribution associated with a simulation activity. In most of these situations, it may be easy to describe the desired variable by estimating its smallest, most likely, and largest
8. Consider a probability distribution that consists of a rectangle flanked on the left and right sides by two symmetrical right triangles. The respective ranges for the triangle on the left, the rectangle, and the triangle on the right are [a, b], [b, c], and [c, d], a < b < c
*9. Geometric distribution. Show how a random sample can be obtained from the following geometric distribution:f(x) = p(l - p)X, x = 0,1,2, ...The parameter x is the number of (Bernoulli) failures until a success occurs, and p is the probability of a success, 0 < p < 1. Generate five samples for p
10. Weibull distribution. Show how a random sample can be obtained from the Weibull distribution with the following probability density function:where lX > 0 is the shape parameter, and f3 > 0 is the scale parameter.
*1. In Example 16.3-3, compute an Erlang sample, given m = 4 and A= 5 events per hour.
2. In Example 16.3-4, generate three Poisson samples during a 2-hour period, given that the mean of the Poisson is 5 events per hour.
3. In Example 16.4-5, generate two samples from N(8, 1) by using both the convolution method and the Box-Muller method.
4. Jobs arrive at Metaleo jobshop according to a Poisson distribution, with a mean of six jobs per day. Received jobs are assigned to the five machining centers of the shop on a strict rotational basis. Determine one sample of the interval between the arrival of jobs at the first machine center.
5. The ACT scores for the 1994 senior class at Springdale High are normal, with a mean of 27 points and a standard deviation of 3 points. Suppose that we draw a random sample of six seniors from that class. Use the Box-Muller method to determine the mean and standard deviation of the sample.
*6. 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 point where it entered. The
1. In Example 16.3-5, continue the steps of the procedure until a valid sample is obtained.Use the (0,1) random numbers in Table 16.1 in the same order in which they are used in the example.
2. Consider the beta pdf of Example 16.3-6. Determine a two-step pyramid majorizing function g(x) with two equal jumps of height liS = .75 each. Obtain one beta sample based on the new majorizing function using the same (0, 1) random sequence in Table 16.1 that was employed in Example 16.3-6.The
3. Detennine the functions g(x) and hex) for applying the acceptance-rejection method to the following function:Use the (0, 1) random numbers from column 1 in Table 16.1 to generate two samples fromf(x). [Hint: For convenience, use a rectangular g(x) over the defined range off(x).] f(x) sin(x) +
4. The interarrival time of customers at HairKare is described by the following distribution: f(4) = 12 20
*1. Use excelRN.xls with the following sets of parameters and compare the results with those in Example 16.4-1:b = 17, C = 111, m = 103, seed = 7
1. Suppose that the barbershop of Section 16.5.1 is operated by two barbers, and customers are served on a FCFS basis. Suppose further that the time to get a haircut is uniformly distributed between 15 and 30 minutes. TIle interarrival time of customers is exponential, with a mean of 10 minutes.
4. Suppose that the barbershop of Example 16.5-1 is operated by three barbers. Assume further that the utilization of the servers (barbers) is summarized as given in the following table: Simulation time, 7 (hr) No. of busy servers 10 0 0
1. Using the input data in Section 16.5.1, run the Excel simulator for 10 arrivals and graph the changes in facility utilization and queue length as a function of the simulation time.Verify that the areas under the curves equal the sum of the service times and the sum of the waiting times,
2. Simulate the MIMll model for 500 arrivals given the arrival rate A = 4 customers per hour and the service rate J.l. = 6 departures per hour. Run 5 replications (by refreshing the spreadsheet-pressing F9) and determine a 95% confidence interval for all the measures of performance of the model.
3. Television units arrive on a conveyor belt every 15 minutes for inspection at a singleoperator station. Detailed data for the inspection station are not available. However, the operator estimates that it takes 10 minutes "on the average" to inspect a unit. Under the worst conditions, the
(b) Based on 5 replications, estimate the average number of units awaiting inspection and the average utilization of the inspection station.
1. In Example 16.6-1, use the subinterval method to compute the average waiting time in the queue for those who must wait.
3. In Example 16.6-2, suppose that the start point for each observation is the point in time where all the servers have just become idle. Thus, in Figure 16.13, these points correspond to t = 10, 17, 24, and 33. Compute the 95% confidence interval for the utilization of the servers based on the new
4. In a single-server queuing situation, the system is simulated for 100 hours. The results of the simulation show that the server was busy only during the following time intervals: (0, 10), (15, 20), (25,30), (35,60), (70, 80), and (90,95). The length of the transient period is estimated to be 10
1. Patrons arrive randomly at a three-clerk post office. TIle interarrival time is exponential with mean 5 minutes. The time a clerk spends with a patron is exponential with a mean of 10 minutes. All arriving patrons form one queue and wait for the first available free clerk.Run a simulation model
2. Television units arrive for inspection on a conveyor belt at the constant rate of 5 units per hour. The inspection time takes between 10 and 15 minutes, uniformly distributed. Past experience shows that 20% of inspected units must be adjusted and then sent back for reinspection. The adjustment
3. A mouse is trapped in a maze and desperately "wants out." After trying between 1 and 3 minutes, uniformly distributed, there is a 30% chance that it will find the right path. Otherwise, it will wander around aimlessly for between 2 and 3 minutes, uniformly distributed, and eventually end up
4. In the final stage of automobile manufacturing, a car moving on a transporter is situated between two parallel workstations to allow work to be done on both the left and right sides of the car simultaneously. TIle operation times for the left and right sides are uniform between 15 and 20 minutes
5. Cars arrive at a one-bay car wash facility where the interarrival time is exponential, with a mean of 10 minutes. Arriving cars line up in a single lane that can accommodate at most five waiting cars. If the lane is full, newly arriving cars will go elsewhere. It takes between 10 and 15 minutes,
1. An engineering professor purchases a new computer every two years with preferences for three models: MI, M2, and M3. If the present model is Ml, the next computer may be M2 with probability .2 or M3 with probability .15. If the present model is M2, the probabilities of switching to Ml and M3 are
*2. A police car is on patrol in a neighborhood known for its gang activities. During a patrol, there is a 60% chance that the location where help is needed can be responded to in time, else the car will continue regular patrol. Upon receiving a call, there is a 10% chance for cancellation (in
3. (eyert and Associates, 1963) Bank1 offers loans which are either paid when due Or are delayed. If the payment on a loan is delayed more than 4 quarters (1 year), Bank1 COnsiders the loan a bad debt and writes it off. TIle following table provides a sample of Bank! '8 past experience with
4. (Pliskin and Tell, 1981) Patients suffering from kidney failure can either get a transplant or undergo periodic dialysis. During anyone year, 30% undergo cadaveric transplants and 10% receive living-donor kidneys. In the year following a transplant, 30% of the cadaveric transplants and 15% of
1. Consider Problem 1, Set 17.1a. Determine the probability that the professor will purchase the current model in four years.
*2. Consider Problem 2, Set 17.1a.1f the police car is currently at a call scene, determine the probability that an apprehension will take place in two patrols.
3. Consider Problem 3, Set 17.1a. Suppose that Bankl currently has $500,000 worth of outstanding loans. Of these, $100,000 are new, $50,000 are one quarter late, $150,000 are two quarters late, $100,000 are three quarters late, and the rest are over four quarters late.What would the picture of
4. Consider Problem 4, Set 17.1a.(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?
*1. On a sunny Spring 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
(b) Determine the average number of days the weather will not be sunny.
2. Joe loves to eat out in area restaurants. His favorite foods are Mexican, Italian, Chinese, and Thai. On the average, Joe pays $10.00 for a Mexican meal, $15.00 for an Italian meal,$9.00 for a Chinese meal, and $11.00 for a Thai meal. Joe's eating habits are predictable:There is a 70% chance
(b) How often does Joe eat Mexican food?
3. Some ex-cons spend the rest of their lives in one fOUT of states: 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 Qr
4. A store sells a special item whose daily demand can described by the following pdf:The store is comparing two ordering policies: (1) Order up to 3 units every 3 days if the stock level is less than 2, else do not order. (2) Order 3 units every 3 days if the stock level is zero, else do not
*5. 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 one year to the next shows that of those who did not evade taxes last year, 95% continue in the
(b) In the long run, what would be the percentages of "never," "sometimes," and"always" tax categories?
(c) Statistics show that a taxpayer in the "sometimes" category evades taxes on about$5000 per return and in the "always" category on about $12,000. Assuming an average income tax rate of 12% and a filers population of 70 million, determine the annual reduction in collected taxes due to evasion.
6. 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 group.
(b) If the forest land can hold a total of 500,000 trees, determine the long-run composition of the forest.
(c) If a new tree is planted at the cost of $1 per tree and a harvested tree has a market value of $20, determine the average annual income from the forest operation.
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