A town has three gasoline stations , Petroco, National, and Gascorp. The residents purchase gasoline on a monthly basis. The following transition matrix contains the probabilities of the customers' purchasing a given brand of gasoline next month:
Using a decision tree, determine the probabilities of a customer's purchasing each brand of gasoline in month 3, given that the customer purchases National in the present month. Summarize the resulting probabilities in a table.
Discuss the properties that must exist for the transition matrix in Problem 1 to be considered a Markov process.
The only grocery store in a community stocks milk from two dairiesCreamwood and Cheesedale. The following transition matrix shows the probabilities of a customer's purchasing each brand of milk next week, given that he or she purchased a particular brand this week:
Given that a customer purchases Creamwood milk this week, use a decision tree to determine the probability that he or she will purchase Cheesedale milk in week 4.
Given the transition matrix in Problem 1, use matrix multiplication methods to determine the state probabilities for month 3, given that a customer initially purchases Petroco gasoline.
Determine the state probabilities in Problem 3 by using matrix multiplication methods.
A manufacturing firm has developed a transition matrix containing the probabilities that a particular machine will operate or break down in the following week, given its operating condition in the present week:
A city is served by two newspapersthe Tribune and the Daily News . Each Sunday readers purchase one of the newspapers at a stand. The following transition matrix contains the probabilities of a customer's buying a particular newspaper in a week, given the newspaper purchased the previous Sunday:
Determine the steady-state probabilities for this transition matrix algebraically and explain what they mean.
The Hergeshiemer Department Store wants to analyze the payment behavior of customers who have outstanding accounts. The store's credit department has determined the following bill payment pattern for credit customers from historical records:
A rural community has two television stations, and each Wednesday night the local viewers watch either the Wednesday Movie or a show called Western Times . The following transition matrix contains the probabilities of a viewer's watching one of the shows in a week, given that he or she watched a particular show the preceding week:
In Problem 3, assume that 600 gallons of milk are sold weekly, regardless of the brand purchased.
The manufacturing company in Problem 6 is considering a preventive maintenance program that would change the operating probabilities as follows:
The machine earns the company $1,000 in profit each week it operates. The preventive maintenance program would cost $8,000 per year. Should the company institute the preventive maintenance program?
In Problem 7, assume that 20,000 newspapers are sold each Sunday, regardless of the publisher.
The promotional campaign will cost $150 per week. Each newspaper sold earns the Daily News $0.05 in profit. Should the paper adopt the promotional campaign?
The following transition matrix describes the accounts receivable process for the Ewing-Barnes Department Store:
The states p and b represent an account that is paid and a bad account, respectively. The numbers 1 and 2 represent the fact that an account is either 1 or 2 months overdue, respectively. After an account has been overdue for 2 months, it becomes a bad account and is transferred to the store's overdue accounts section for collection. The company has sales of $210,000 each month. Determine how much the company will be paid and how many of the debts will become bad debts in a 2-month period.
The department store in Problem 13 will never be able to collect 20% of the bad accounts, and it costs the store an additional $0.10 per dollar collected to collect the remaining bad accounts. The store management is contemplating a new, more restrictive credit plan that would reduce sales to an estimated $195,000 per month. However, the tougher credit plan would result in the following transition matrix for accounts receivable:
Determine whether the store should adopt the more restrictive credit plan or keep the existing one.
In Westvale, a small, rural town in Maine, virtually all shopping and business are done in the town. The town has one farm and garden center that sells fertilizer to the local farmers and gardeners. The center carries three brands of fertilizer Plant Plus, Crop Extra, and Gro-fast and every person in the town who uses fertilizer uses one of the three brands. The garden center has 9,000 customers for fertilizer each spring. An extensive market research study has determined that customers switch brands of fertilizer according to the following probability transition matrix:
The number of customers presently using each brand of fertilizer is shown in the following table:
Westvale, a small community in Maine, has 7,000 bank patrons that do their banking business at one of three banks in town, the American National Bank, the Bank of Westvale and the Commerce Bank. The following transition matrix shows the probability that a bank customer will trade with the same bank next month or move to one of the other banks.
Determine the steady-state probabilities and the number of customers expected to trade at each bank in the long run.
Students switch among the various colleges of a university according to the following probability transition matrix:
Assume that the number of students in each college of the university at the beginning of the fall quarter is as follows:
A rental firm in the Southeast serves three statesVirginia, North Carolina, and Maryland. The firm has 700 trucks that are rented on a weekly basis and can be rented in any of the three states. The transition matrix for the movement of rental trucks from state to state is as follows:
Determine the steady-state probabilities and the number of trucks in each state in the long run.
The Koher Company manufactures precision machine tools. It has a quality management program for maintaining product quality that relies heavily on statistical process control techniques. A machine center operator takes a sample at the end of every hour to see if the process is within statistical control limits. If the process is not within the preestablished limits, it is out of control; if it is within the limits, the process is in control. If the process remains out of control for 2 hours, it is shut down by the operator. The following transition matrix provides probabilities that a machine center will be in control (C), will be out of control (O), or will shut down (S) in the next hour, given the process status in the current hour :
In Problem 19 the Koher Company is considering a new operator training program that will alter the probability that a machine center process will be out of control, as follows:
Determine the annual savings in the number of hours a process is out of control or shut down with this new program.
Whitesville, where State University is located, has three bookstores that sell textbooksthe State Bookstore, the Eagle Bookstore, and Books n' Things. The State Bookstore is operated by the university. Don Williams, the manager of the State Bookstore, is in the process of placing book orders with his book distributors for the next semester. There are 17,000 undergraduate students at State, and they all purchase their textbooks at one of the three stores. Students often change from which bookstore they buy from one semester to the next. Don has sampled a group of students and developed the following transition matrix for student movements between the stores:
In this semester, if 9,000 students bought their textbooks at the State Bookstore, 5,000 bought their books at the Eagle Bookstore, and 3,000 students bought their books at Books n' Things, how many are likely to buy their books at these stores next semester? How many students will purchase their books from each store in the long-run future?
Don Williams, the manager of the State Bookstore from Problem 21, would like to increase his volume of textbook business. He believes that if he reduces textbook prices by 10%, he could increase his sales volume. Individual student textbook purchases currently average $175. He is not sure the increase in volume would make up the loss of revenue from the price cut; however, he does think that the additional customers would increase sales for other items, such as clothing, supplies , and computer software. The other stores cannot as easily implement a price cut because they do not carry all the other items the State Bookstore does. Don estimates that a price reduction would alter the transition matrix for the movement of students between stores as follows:
However, the other bookstores frequently complain that the State Bookstore has an unfair competitive advantage because it is located on campus and does not charge state sales tax. The university is sensitive to these local business complaints and has indicated to Don that he should keep his total market share in the long run to about 50% or less of the total student body market share.
At 4:00 P.M. each weekday, the three local television stations in Salem have no network obligations and can schedule whatever shows they choose. WALS runs the Ofrah Williams talk show, WBDJ runs the Josie Donald talk show, and WCXI runs episodes of the Barney Fife show. The transition matrix of probabilities that a regular viewer will watch the same show or change shows from one day to the next is as follows:
The Josie Donald show in Problem 23 is contemplating an advertising campaign to increase its viewers, at a cost of $25,000. Each viewer generates $0.12 in commercial revenue per day. The station manager knows that the effects of any ad campaign will last only 4 months. The revised transition matrix resulting from the ad campaign is as follows:
Should the station undertake the ad campaign?
Klecko's Copy Center uses several copy machines that deteriorate rather rapidly in terms of the quality of copies produced as the volume of copies increases . Each machine is examined at the end of each day to determine the quality of the copies being produced, and the results of that inspection are classified as follows:
The costs associated with each classification are for maintenance and repair and redoing unacceptable copies. When a machine reaches classification 4 and copies are unacceptable, major maintenance is required (resulting in downtime), after which the machine resumes making excellent copies. The transition matrix showing the probabilities of a machine's being in a particular classification state after inspection is as follows:
Determine the expected daily cost of machine maintenance.
Determine the steady-state probabilities for the transition matrix in Problem 1.
When freshmen at Tech attend orientation, the university's president tells each freshman that one of the two students next to him or her will not graduate. The freshmen interpret this as meaning that two thirds of the entering freshmen will graduate. The following transition probabilities have been developed from data gathered by Tech's registrar. They show the probability of a student's moving from one class to the next during an academic year and eventually graduating (G) or dropping out (D):
Libby Jackson is a carpenter who works for a large construction company that has a number of housing developments under way in the metropolitan Washington, DC, area. Each day Libby is assigned to one of the company's developments, either hanging and finishing drywall, doing trim work, framing, or roofing. The following transition matrix describes the probability that she will move from a job one day to the same job or another the next:
Frank Beamish, the head football coach at Tech, has had his staff scout State University for most of its games this season to get ready for the annual season -ending game. The Tech coaching staff has developed the following transition matrix of the probabilities that State will change its defense from one play to the next: