Questions and Answers of Principles Of Operations

For an automobile that you own or would like to own, estimate the correct values of the replacement cost, the rate of depreciation, the initial maintenance cost, and the rate at which the maintenance
A piece of industrial machinery costs $48,000 to replace and has essentially no salvage value. Over the first five years of operation, maintenance costs amounted to $8,000. If the maintenance cost
A collection of 30 Christmas tree lights are arranged in a pure series circuit; that is, if one of the lights burns out, then the entire string goes out. Suppose that each light fails completely at
A local cab company maintains a fleet of 10 cabs. Each time a cab breaks down, it is repaired the same day. Assume that breakdowns of individual cabs occur completely at random at a rate of two per
Applicational Materials sells several pieces of equipment used in the manufacture of silicon-based microprocessors. In 2003 the company filled 130 orders for model a55212. Suppose that the machines
Suppose that a particular light bulb is advertised as having an average lifetime of 2,000 hours and is known to satisfy an exponential failure law. Suppose for simplicity that the bulb is used
Notice that if k is an integer, this results in (k) (k 1)!. If k is not an integer, one must use the recursive definition for (k) coupled with the following table. For values of 1 k2, (k)is
The mean value of a Weibull random variable is given by the formula a1 (1 1), where represents the gamma function. The gamma function has the property that(k) (k 1)(k 1) for any
Assume that the bulbs in Problem 43 are not replaced as they fail.a. What fraction of the 6,000 bulbs are expected to fail in the first year?b. What fraction of the bulbs surviving the first year are
Continuing with the example of Problem 43, determine the following:a. The proportion of bulbs lasting more than two years.b. The probability that a bulb chosen at random fails in the first three
The owner of the hotel mentioned in Problem 43 falls on hard times and dispenses with replacement of the bulbs. She notices that more than half of the bulbs have burned out before the advertised
The owner of the hotel mentioned in Problem 43 has decided that in order to decrease the number of burned-out bulbs, she will replace all 6,000 bulbs at the start of each year in addition to
Six thousand light bulbs light a large hotel and casino marquee. Each bulb fails completely at random, and each has an average lifetime of 3,280 hours. Assuming that the marquee stays lit
A large national producer of appliances has traced customer experience with a popular toaster oven. A survey of 5,000 customers who purchased the oven early in 2000 has revealed the following:Number
Consider the case in which the failure mechanism for the product does not obey the exponential law. In that case, the cost under the free replacement warranty that is indifferent to the cost of
Habard’s, a chain of hardware stores, sells a variety of tools and home repair items.One of their best wrenches sells for $5.50. Habard’s will include a three-year free replacement warranty for
Zemansky’s sells tires with a pro rata warranty. The tires are warranted to deliver 50,000 miles with the rebate based on the remaining tread on the tire. The tires fail on the average after 35,000
For Problem 37, what length of period of the warranty equates the replacement costs of the calculator with and without the warranty?
A producer of pocket calculators estimates that the calculators fail at a rate of one every five years. The calculators are sold for $25 each with a one-year free replacement warranty but can be
For Example 13.13, what value of the warranty period for the full replacement warranty equates the full replacement and the pro rata warranties? (For what value of W1 is the consumer indifferent to
For Example 13.13, what value of the warranty period equates the full replacement warranty with no warranty? (That is, for what value of W1 is the consumer indifferent to purchasing from store A or
If replaced as a block, the parts cost $20 each, but cost $400 each when replaced after failure.a. Find the optimal block replacement strategy.b. Compare this to the cost of replacing the items as
A firm has purchased 30 Mactronics robots, described in Problem
Mactronics produces industrial robots. Each robot contains a part with a lifetime of five years at most, and at least one year. Lifetimes between one and five years are equally likely. If the part
A local newsletter is printed on a printer with a cartridge that may break or run out of ink during operation. The cartridges can be replaced for $7.50, but if they fail when the newsletter is being
The Navy uses a certain type of vacuum tube in a sonar scanning device. Based on past experience, the vacuum tube exhibits the following failure pattern:Number of Months of Operation Probability of
Tires that fail in service result in significantly higher replacement costs than those replaced before failure. For an 18-wheeler (that is, a truck with 18 tires), failure on the road costs $300,
A large electronic pipe organ contains 100 fuses. Because of the power demands of the organ, the fuses burn out at a fairly regular rate, but newer fuses last longer than older ones. The probability
An expensive piece of equipment is used in the masking operation for semiconductor manufacture.A capacitor in the equipment fails randomly. The capacitor costs $7.50, but if it burns out while the
Repeat the calculations for the Harley Brown trucking company using the following parameter values: 0.0156, 1.8, c1 1,000, c2 600.
Consider Example 13.11 of Harley Brown, Inc.Without performing the calculations, discuss what the effect on the optimal replacement policy would likely be if the parameter values were c1 $800, c2
Try to determine the optimal replacement agewhen C(u) au and S(u) K bu.What difficulty do you encounter?
The army is attempting to determine the optimal replacement age for a piece of field equipment. The equipment costs $280,000 to replace. The manufacturer will supply a rebate toward the next purchase
Suppose for the simple age replacement model that the maintenance cost is the same every year [that is, C(u) a for all u 0]. What is the optimal replacement age? Why is this so?
For the basic age replacement model, derive the optimal replacement age when the maintenance cost rate C(u) has the form C(u) au for some constant a 0.
For the basic age replacement model, consider a piece of equipment that costs$18,000 to replace. The total maintenance costs for five years of operation are estimated to be $2,400. Assuming a linear
An aircraft engine fails with probability p. Assume that for an aircraft to successfully complete a flight, at least half the engines must operate. Show that for 0 p , a four-engine plane is
Consider the system of six identical components pictured in Figure 13–8. If each component has constant failure rate , derive the distribution function of the time until failure of the system.
A design engineer is considering the number of levels of redundancy to build into a particular circuit. The circuit will be part of a sensitive piece of equipment with cost estimated at $500 per
Consider the following three systems: (a) a single component with failure rate one per year, (b) two components in series, each of which has failure rate one every two years, and (c) two components
Show that the time until failure of a series system of n identical components, each of which has the Weibull lifetime distribution, also has a distribution of the Weibull type.
A subassembly in an industrial robot consists of 12 components in series, each of which fails completely at random at a rate of once every 50 years.a. What is the mean time between failures for this
For Problem 13 determine the following:a. The probability that a single carrier sent on a six-month mission will not require replacement of the module during that time.b. The probability that the
An electronic module used by the Navy in a sonar device requires replacement on the average once every 16 months and fails according to a Poisson process. Suppose that the Navy places these sonar
Herman’s Hardware uses a neon light in its store window that is left burning continuously.The light has an average lifetime of 1,250 hours and fails completely at random. Lights that burn out are
Automobiles arrive at a tollbooth on a highway completely at random according to a Poisson process with rate 4 cars per hour. Determine the following:a. The probability that the time between any
A sample of high-capacity resistors is tested until failure and the results fitted to a Weibull probability model. Based on these tests, the reliability function of a resistor is estimated to be R(t)
The Air Force maintains enormous amounts of data on engine failure times. A particular engine has experienced a failure pattern whose failure rate function is closely approximated by r (t)
For the equipment mentioned in Problem 7,a. Derive the failure rate function r (t), and draw a graph of the function.b. Without using the failure rate function, determine the probability that a piece
A piece of equipment has a lifetime T (measured in years) that is a continuous random variable with cumulative distribution function F(t) 1 et10 (t10) et10 for all t 0.a. What is the
For the pressure transducer mentioned in Problem 5, use the failure rate function to estimate the likelihood that a transducer that has been operating for six years fails in the seventh year. How
A pressure transducer regulates a climate control system in a factory. The transducer fails according to an exponential distribution with rate one failure every five years on average.a. What is the
A microprocessor that controls the tuner in color TVs fails completely at random(that is, according to the exponential distribution). Suppose that the likelihood that a microprocessor that has
The items have a failure rate function given by r (t) 1.105 0.30t, where t is measured in years of operation.a. Derive R(t) and F(t).b. If 300 items are still operating at time t 1 year,
A large number of identical items are placed into service at time
Suppose that the cumulative distribution function of the lifetime of a piece of operating equipment is given by F(t) 1 e0.6t 0.6te0.6t, where t is measured in years of continuous operation.a.
Three hundred identical cathode ray tubes (CRTs) placed into service simultaneously on January 1, 1976, experienced the following numbers of failures through December 31, 1988:Year 1983 1984 1985
a. Develop a spreadsheet from which one may obtain a graph such as Figure 12–17 for sequential sampling. Store the values of p0, p1, , and in cell locations so that these can be altered at will.
Samples of size 50 are drawn from lots of 1,000 items. The lot is rejected if there are more than two defectives in the sample. Using a binomial approximation, graph the OC curve as a function of p,
A p chart is used to monitor the number of riding lawn mowers produced. The numbers that are sent back for rework because they did not pass inspection are Day Number Produced Number Rejected 1 400 23
Twenty sets of four measurements of the diameters in inches of Hot Shot golf balls were Sample 1 2.13 2.18 2.05 1.96 2 2.08 2.10 2.02 2.20 3 1.93 1.98 2.03 2.06 4 2.01 1.94 1.91 1.99 5 2.00 1.90 2.14
For the single sampling plan derived in part (a) of Problem 67, suppose lots that are not passed are returned to Hammerhead.a. Estimate the graph of the AOQ curve by computing AOQ(p) for various
Hammerhead produces heavy-duty nails, which are purchased by Modulo, a maker of prefabricated housing. Modulo buys the nails in lots of 10,000 and subjects a sample to destructive testing to
A double sampling plan is constructed as follows. From a lot of 200 items, a sample of 10 items is drawn. If there are zero defectives, the lot is accepted. If there are two or more defectives, the
A single sampling plan is used to determine the acceptability of shipments of a bearing assembly used in the manufacture of skateboards. For lots of 500 bearings, samples of n 20 are taken. The lot
A p chart is used to monitor the fraction defective of an integrated circuit to be used in a commercial pacemaker. A sample of 15 circuits is taken from each day’s production for 30 consecutive
X control charts are used to maintain control of the manufacture of the cases used to house a generic brand of personal computer. Separate charts are maintained for length, width, and height. The
Graph the OC curves for the sampling plan described in Problem 58 for both manufacturers A and B.
Assuming AQL 5 percent and LTPD 10 percent, determine the values of and for the plan described in Problem 58 for manufacturers A and B, and for the plan described in Problem 60 for
Would a fairer plan be to reject the lot if more than 10 percent of the chips in a sample are defective? Which of the two manufacturers mentioned would be at an advantage if this plan were adopted?
Consider the sampling plan discussed in Problem
Estimate the values of the AOQL in each case.
Graph the AOQ curves for manufacturers A and B mentioned in Problem
A maker of personal computers, Noname, purchases 64K DRAM chips from two different manufacturers, A and B. Noname uses the following sampling plan: A sample of 10 percent of the chips is drawn and
A manufacturer of large appliances maintains a p chart for the production of washing machines. The machines may be rejected because of cosmetic or functional defectives.Based on sampling 30 machines
The construction of the p chart described in this chapter requires specification of both UCL and LCL levels. Is the LCL meaningful in this context? In particular, what does it mean when an observed
Discuss the advantages and disadvantages of the following strategies in control chart design. In particular, what are the economic trade-offs attendant to each strategy?a. Choosing a very small value
Consider the data presented in Table 12–1 for the tracking arm example. Suppose that an R chart is constructed based on sample numbers 1 to 15 only.a. What is the estimate of obtained from these
A process that is in statistical control has an estimated mean value of 180 and an estimated standard deviation of 26.a. Based on subgroups of size 4, what are the control limits for the X and R
The target value is 0.37 inch, and the upper and lower limits are 0.35 and 0.39 inch, respectively(assume that these are based on three-sigma limits). Wheeler, which purchases the bearings from XYZ
XYZ produces bearings for bicycle wheels and monitors the process with an X chart for the diameter of the bearings. The X chart is based on subgroups of size
X and R charts are maintained on a single quality dimension. A sudden shift in the process occurs, causing the process mean to increase by 2, where is the true process standard deviation. No
Figure 12–1 presents a conceptual picture of the trade-off between process cost and the costs of losses due to poor quality. What are the costs of poor quality and what difficulties might arise
In what ways could each of the factors listed contribute to poor quality?a. Managementb. Laborc. Equipment maintenanced. Equipment designe. Control and monitoringf. Product design
On average, how many of the microprocessors would have to be tested ifa. p .001?b. p .005?c. p .01?
Estimate the ASN curve for the sampling plan derived in Problem
Devise a sequential sampling plan for Sayle that results in .05 and .20. What are the advantages and disadvantages of this plan over the Sayle sampling plan derived in Problem 32?
Consider the example of Hemispherical Conductor and the Sayle Company discussed in Problem
According to your curve, what is the expected number of microprocessors that must be tested when the true proportion of defectives in the lot isa. 0.1 percent?b. 1.0 percent?c. 10 percent?
Estimate the ASN curve for the plan derived in Problem
Suppose that a lot of 1,000 microprocessors is inspected. Suppose that the 31st, 89th, 121st, and 122nd chips tested are found defective. Assuming the sequential sampling plan derived in Problem 41,
Consider the sequential sampling plan described in Problem
A manufacturer of aircraft engines uses a sequential sampling plan to accept or reject incoming lots of microprocessors used in the engines. Assume an AQL of 1 percent and an LTPD of 5 percent.
For the double sampling plan described in Problem 39, determine the following:a. The probability that the lot is rejected based on the first sample.b. The probability that the lot is rejected based
If there are four or more defectives in the sample, reject the lot. If there is one or fewer defective, accept the lot. If there are two or three defectives, sample an additional five items and
Consider the following double sampling plan. First select a sample of 5 from a lot of
By trial and error devise a double sampling plan for Spire CDs that achieves .10 and .10.
Graph the OC curve for the double sampling plan with n1 20, n2 10, c1 3, and c2 c3 5, as described in this section. If you are doing this by hand, evaluate the curve at p 0, .2, .4, .6,
Consider the double sampling plan for Spire CDs described in this section. Over a period of one year, 3,860 boxes of records are subject to inspection using this plan.If 60 percent of these batches
For the double sampling plan for Spire CDs presented in this section, what is the probability that a lot is rejected on the first sample? Perform the computation for both p p0 and p p1.

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