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Introduction to Operations Research 10th edition Frederick S. Hillier, Gerald J. Lieberman - Solutions

You are given the following payoff table (in units of thousands of dollars) for a decision analysis problem:(a) According to Bayes€™ decision rule, which alternative should be chosen? (b) Find EVPI. (c) You are given the opportunity to spend $1,000 to obtain more information about which
Betsy Pitzer makes decisions according to Bayes€™ decision rule. For her current problem, Betsy has constructed the following payoff table (in units of dollars):(a) Which alternative should Betsy choose? (b) Find EVPI. (c) What is the most that Betsy should consider paying to obtain more
Using Bayes€™ decision rule, consider the decision analysis problem having the following payoff table (in units of thousands of dollars):(a) Which alternative should be chosen? What is the resulting expected payoff? (b) You are offered the opportunity to obtain information which will tell
Silicon Dynamics has developed a new computer chip that will enable it to begin producing and marketing a personal computer if it so desires. Alternatively, it can sell the rights to the computer chip for $15 million. If the company chooses to build computers, the profitability of the venture
Reconsider the Goferbroke Co. prototype example, including its analysis in Sec. 16.3. With the help of a consulting geologist, some historical data have been obtained that provide more precise information on the likelihood of obtaining favorable seismic soundings on similar tracts of land.
You are given the following payoff table (in units of dollars):You have the option of paying $100 to have research done to better predict which state of nature will occur. When the true state of nature is S1, the research will accurately predict S1 60 percent of the time (but will inaccurately
Reconsider Prob. 16.2-8. Suppose now that the Air Force knows that a similar type of engine was produced for an earlier version of the type of airplane currently under consideration. The order size for this earlier version was the same as for the current type. Furthermore, the probability
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 16.4. Briefly describe how decision analysis was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Reconsider Prob.16.3-2. The management of Silicon Dynamics now wants to see a decision tree displaying the entire problem. Construct and solve this decision tree by hand.
You are given the decision tree below, where the numbers in parentheses are probabilities and the numbers on the far right are payoffs at these terminal points. Analyze this decision tree to obtain the optimal policy.
The Athletic Department of Leland University is considering whether to hold an extensive campaign next year to raise funds for a new athletic field. The response to the campaign depends heavily upon the success of the football team this fall. In the past, the football team has had winning seasons
The comptroller of the Macrosoft Corporation has $100 million of excess funds to invest. She has been instructed to invest the entire amount for one year in either stocks or bonds (but not both) and then to reinvest the entire fund in either stocks or bonds (but not both) for one year more. The
On Monday, a certain stock closed at $10 per share. On Tuesday, you expect the stock to close at $9, $10, or $11 per share, with respective probabilities 0.3, 0.3, and 0.4. On Wednesday, you expect the stock to close 10 percent lower, unchanged, or 10 percent higher than Tuesday€™s close,
Use the scenario given in Prob.16.3-9. (a) Draw and properly label the decision tree. Include all the payoffs but not the probabilities. (b) Find the probabilities for the branches emanating from the event nodes. (c) Apply the backward induction procedure, and identify the resulting optimal policy.
Jean Clark is the manager of the Midtown Saveway Grocery Store. She now needs to replenish her supply of strawberries. Her regular supplier can provide as many cases as she wants. However, because these strawberries already are very ripe, she will need to sell them tomorrow and then discard any
Use the scenario given in Prob.16.3.-11. (a) Draw and properly label the decision tree. Include all the payoffs but not the probabilities. (b) Find the probabilities for the branches emanating from the event nodes. (c) Apply the backward induction procedure, and identify the resulting optimal
Use the scenario given in Prob.16.3-12. (a) Draw and properly label the decision tree. Include all the payoffs but not the probabilities. (b) Find the probabilities for the branches emanating from the event nodes. (c) Apply the backward induction procedure, and identify the resulting optimal policy.
Use the scenario given in Prob.16.3-13. (a) Draw and properly label the decision tree. Include all the payoffs but not the probabilities. (b) Find the probabilities for the branches emanating from the event nodes. (c) Apply the backward induction procedure, and identify the resulting optimal policy.
The executive search being conducted for Western Bank by Headhunters Inc. may finally be bearing fruit. The position to be filled is a key one—Vice President for Information Processing—because this person will have responsibility for developing a state-of-the-art management information system
Reconsider the original version of the Silicon Dynamics problem described in Prob.16.2-2. (a) Assuming the prior probabilities of the two levels of sales are both 0.5, use ASPE to construct and solve the decision tree for this problem. According to this analysis, which decision alternative should
Now reconsider the expanded version of the Silicon Dynamics problem described in Probs.16.3-2 and 16.4-2. (a) Use ASPE to construct and solve the decision tree for this problem. (b) Perform sensitivity analysis systematically by generating a data table that shows the optimal policy and the expected
Reconsider the decision tree given in Prob. 16.4-3. Use ASPE to construct and solve this decision tree.
Reconsider Prob.16.4-5. Use ASPE to construct and solve the decision tree for this problem.
Reconsider Prob.16.4-6. Use ASPE to construct and solve the decision tree for this problem.
Jose Morales manages a large outdoor fruit stand in one of the less affluent neighborhoods of San Jose, California. To replenish his supply, Jose buys boxes of fruit early each morning from a grower south of San Jose. About 90 percent of the boxes of fruit turn out to be of satisfactory quality,
Warren Buffy is an enormously wealthy investor who has built his fortune through his legendary investing acumen. He currently has been offered three major investments and he would like to choose one. The first one is a conservative investment that would perform very well in an improving economy and
The Morton Ward Company is considering the introduction of a new product that is believed to have a 50-50 chance of being successful. One option is to try out the product in a test market, at a cost of $5 million, before making the introduction decision. Past experience shows that ultimately
Chelsea Bush is an emerging candidate for her party’s nomination for President of the United States. She now is considering whether to run in the high-stakes Super Tuesday primaries. If she enters the Super Tuesday (S.T.) primaries, she and her advisers believe that she will either do well
Reconsider the Goferbroke Co. prototype example, including the application of utilities in Sec. 16.6. The owner now has decided that, given the company’s precarious financial situation, he needs to take a much more risk-averse approach to the problem. Therefore, he has revised the utilities given
You live in an area that has a possibility of incurring a massive earthquake, so you are considering buying earthquake insurance on your home at an annual cost of $180. The probability of an earthquake damaging your home during one year is 0.001. If this happens, you estimate that the cost of the
For your graduation present from college, your parents are offering you your choice of two alternatives. The first alternative is to give you a money gift of $19,000. The second alternative is to make an investment in your name. This investment will quickly have the following two possible
Reconsider Prob.16.6-3. You now are uncertain about what your true utility function for receiving money is, so you are in the process of constructing this utility function. So far, you have found that U(19) = 16.7 and U(30) = 20 are the utility of receiving $19,000 and $30,000, respectively. You
You wish to construct your personal utility function U(M) for receiving M thousand dollars. After setting U(0) = 0, you next set U(1) = 1 as your utility for receiving $1,000. You next want to find U(10) and then U(5). (a) You offer yourself the following two hypothetical alternatives: A1: Obtain
You are given the following payoff table:(a) Assume that your utility function for the payoffs is U(x) = ˆšx. Plot the expected utility of each alternative versus the value of p on the same graph. For each alternative, find the range of values of p over which this alternative maximizes
Dr. Switzer has a seriously ill patient but has had trouble diagnosing the specific cause of the illness. The doctor now has narrowed the cause down to two alternatives: disease A or disease B. Based on the evidence so far, she feels that the two alternatives are equally likely.Beyond the testing
You want to choose between decision alternatives A1 and A2 in the following decision tree, but you are uncertain about the value of the probability p, so you need to perform sensitivity analysis of p as well.Your utility function for money (the payoff received) is (a) For p = 0.25, determine which
Reconsider Prob. 16.2-4. Warren Buffy decides that Bayes’ decision rule is his most reliable decision criterion. He believes that 0.1 is just about right as the prior probability of an improving economy, but is quite uncertain about how to split the remaining probabilities between a stable
While El NiÃ±o is pouring its rain on northern California, Charlotte Rothstein, CEO, major shareholder and founder of Cerebrosoft, sits in her office, contemplating the decision she faces regarding her company€™s newest proposed product, Brainet. This has been a particularly
The CEO of Bay Area Automobile Gadgets is contemplating whether to add a road scanning device to the company’s driver support system. A series of decisions need to be made. Should basic research into the road scanning device be undertaken? If the research is successful, should the company develop
You are a contestant on “Who Wants to be a Millionaire?” and have just answered the $250,000 question correctly. If you decide to go on to the $500,000 question and then to the $1,000,000 question, you still have the option available of using the “phone a friend” lifeline on one of the
University Toys has developed a series of Engineering Professor Action Figures for the local engineering school and management needs to decide how to market the dolls in the face of uncertainty about the demand. One option is to immediately ramp up for full production, advertising, and sales.
You are given the following payoff table (in units of thousands of dollars) for a decision analysis problem:(a) Which alternative should be chosen under the maximin payoff criterion? (b) Which alternative should be chosen under the maximum likelihood criterion? (c) Which alternative should be
Dwight Moody is the manager of a large farm with 1,000 acres of arable land. For greater efficiency, Dwight always devotes the farm to growing one crop at a time. He now needs to make a decision on which one of four crops to grow during the upcoming growing season. For each of these crops, Dwight
A new type of airplane is to be purchased by the Air Force, and the number of spare engines to be ordered must be determined. The Air Force must order these spare engines in batches of five, and it can choose among only 15, 20, or 25 spares. The supplier of these engines has two plants, and the Air
Vincent Cuomo is the credit manager for the Fine Fabrics Mill. He is currently faced with the question of whether to extend $100,000 credit to a potential new customer, a dress manufacturer. Vincent has three categories for the creditworthiness of a company: poor risk, average risk, and good risk,
Verify the following relationships for an M/M/1 queueing system:
It is necessary to determine how much in-process storage space to allocate to a particular work center in a new factory. Jobs arrive at this work center according to a Poisson process with a mean rate of 3 per hour, and the time required to perform the necessary work has an exponential distribution
For each of the following statements about the queue in a queueing system, label the statement as true or false and then justify your answer by referring to a specific statement in the chapter. (a) The queue is where customers wait in the queueing system until their service is completed. (b)
Consider the following statements about an M/M/1 queueing system and its utilization factor p. Label each of the statements as true or false, and then justify your answer. (a) The probability that a customer has to wait before service begins is proportional to p. (b) The expected number of
Customers arrive at a single-server queueing system in accordance with a Poisson process with an expected interarrival time of 25 minutes. Service times have an exponential distribution with a mean of 30 minutes. Label each of the following statements about this system as true or false, and then
For each of the following statements about an M/M/1 queueing system, label the statement as true or false and then justify your answer by referring to specific statements in the chapter. (a) The waiting time in the system has an exponential distribution. (b) The waiting time in the queue has an
The Friendly Neighbor Grocery Store has a single checkout stand with a full-time cashier. Customers arrive randomly at the stand at a mean rate of 30 per hour. The service-time distribution is exponential, with a mean of 1.5 minutes. This situation has resulted in occasional long lines and
The Centerville International Airport has two runways, one used exclusively for takeoffs and the other exclusively for landings. Airplanes arrive in the Centerville air space to request landing instructions according to a Poisson process at a mean rate of 10 per hour. The time required for an
The Security & Trust Bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson process at a mean rate of 2 per minute. However, business is growing and management projects that the mean arrival rate will be 3 per minute a year from now. The transaction time between the
Consider the M/M/s model. (a) Suppose there is one server and the expected service time is exactly 1 minute. Compare L for the cases where the mean arrival rate is 0.5, 0.9, and 0.99 customers per minute, respectively. Do the same for Lq, W, Wq, and P{W > 5}. What conclusions do you draw about the
Consider the M/M/s model with a mean arrival rate of 10 customers per hour and an expected service time of 5 minutes. Use the Excel template for this model to obtain and print out the various measures of performance (with t = 10 and t = 0, respectively, for the two waiting time probabilities) when
A gas station with only one gas pump employs the following policy: If a customer has to wait, the price is $3.50 per gallon; if she does not have to wait, the price is $4.00 per gallon. Customers arrive according to a Poisson process with a mean rate of 20 per hour. Service times at the pump have
You are given an M/M/1 queueing system with mean arrival rate λ and mean service rate μ. An arriving customer receives n dollars if n customers are already in the system. Determine the expected cost in dollars per customer.
Midtown Bank always has two tellers on duty. Customers arrive to receive service from a teller at a mean rate of 40 per hour. A teller requires an average of 2 minutes to serve a customer. When both tellers are busy, an arriving customer joins a single line to wait for service. Experience has shown
Section 17.6 gives the following equations for the M/M/1 model:Show that Eq. (1) reduces algebraically to Eq. (2).
Derive Wq directly for the following cases by developing and reducing an expression analogous to Eq. (1) in Prob. 17.6-17. (a) The M/M/1 model (b) The M/M/s model
Consider an M/M/2 queueing system with λ = 4 and μ = 3. Determine the mean rate at which service completions occur during the periods when no customers are waiting in the queue.
You are given an M/M/2 queueing system with λ = 4 per hour and μ = 6 per hour. Determine the probability that an arriving customer will wait more than 30 minutes in the queue, given that at least 2 customers are already in the system.
In the Blue Chip Life Insurance Company, the deposit and withdrawal functions associated with a certain investment product are separated between two clerks, Clara and Clarence. Deposit slips arrive randomly (a Poisson process) at Clara’s desk at a mean rate of 16 per hour. Withdrawal slips arrive
People’s Software Company has just set up a call center to provide technical assistance on its new software package. Two technical representatives are taking the calls, where the time required by either representative to answer a customer’s questions has an exponential distribution with a mean
Consider a generalization of the M/M/1 model where the server needs to “warm up” at the beginning of a busy period, and so serves the first customer of a busy period at a slower rate than other customers. In particular, if an arriving customer finds the server idle, the customer experiences a
For each of the following models, write the balance equations and show that they are satisfied by the solution given in Sec. 17.6 for the steady-state distribution of the number of customers in the system. (a) The M/M/1 model. (b) The finite queue variation of the M/M/1 model, with K = 2. (c) The
Consider a telephone system with three lines. Calls arrive according to a Poisson process at a mean rate of 6 per hour. The duration of each call has an exponential distribution with a mean of 15 minutes. If all lines are busy, calls will be put on hold until a line becomes available. (a) Print out
Janet is planning to open a small car-wash operation, and she must decide how much space to provide for waiting cars. Janet estimates that customers would arrive randomly (i.e., a Poisson input process) with a mean rate of 1 every 4 minutes, unless the waiting area is full, in which case the
Explain why the utilization factor p for the server in a single-server queueing system must equal 1 = P0, where P0 is the probability of having 0 customers in the system.
Consider the finite queue variation of the M/M/s model. Derive the expression for Lq given in Sec. 17.6 for this model.
For the finite queue variation of the M/M/1 model, develop an expression analogous to Eq. (1) in Prob. 17.6-17 for the following probabilities: (a) P {W > t}. (b) P {Wq > t}.
George is planning to open a drive-through photodeveloping booth with a single service window that will be open approximately 200 hours per month in a busy commercial area. Space for a drive-through lane is available for a rental of $200 per month per car length. George needs to decide how many car
At the Forrester Manufacturing Company, one repair technician has been assigned the responsibility of maintaining three machines. For each machine, the probability distribution of the running time before a breakdown is exponential, with a mean of 9 hours. The repair time also has an exponential
Reconsider the specific birth-and-death process described in Prob. 17.5-1. (a) Identify a queueing model (and its parameter values) in Sec. 17.6 that fits this process. (b) Use the corresponding Excel template to obtain the answersfor parts (b) and (c) of Prob. 17.5-1.
Consider the M/G/1 model. (a) Compare the expected waiting time in the queue if the servicetime distribution is (i) exponential, (ii) constant, (iii) Erlang with the amount of variation (i.e., the standard deviation) halfway between the constant and exponential cases. (b) What is the effect on the
Consider the M/G/1 model with λ = 0.2 and μ = 0.25. T (a) Use the Excel template for this model (or hand calculations) to find the main measures of performance—L, Lq, W, Wq—for each of the following values of σ: 4, 3, 2, 1, 0. (b) What is the ratio of Lq with σ = 4 to Lq with σ = 0? What
Consider the following statements about an M/G/1 queueing system, where 2 is the variance of service times. Label each statement as true or false, and then justify your answer. (a) Increasing σ2 (with fixed λ and μ) will increase Lq and L, but will not change Wq and W. (b) When choosing between
Marsha operates an expresso stand. Customers arrive according to a Poisson process at a mean rate of 30 per hour. The time needed by Marsha to serve a customer has an exponential distribution with a mean of 75 seconds. (a) Use the M/G/1 model to find L, Lq, W, and Wq. (b) Suppose Marsha is replaced
Antonio runs a shoe repair store by himself. Customers arrive to bring a pair of shoes to be repaired according to a Poisson process at a mean rate of 1 per hour. The time Antonio requires to repair each individual shoe has an exponential distribution with a mean of 15 minutes. (a) Consider the
Identify the customers and the servers in the queueing system in each of the following situations: (a) The checkout stand in a grocery store. (b) A fire station. (c) The tollbooth for a bridge. (d) A bicycle repair shop. (e) A shipping dock. (f) A group of semiautomatic machines assigned to one
The maintenance base for Friendly Skies Airline has facilities for overhauling only one airplane engine at a time. Therefore, to return the airplanes to use as soon as possible, the policy has been to stagger the overhauling of the four engines of each airplane. In other words, only one engine is
Reconsider Prob. 17.7-6. Management has adopted the proposal but now wants further analysis conducted of this new queueing system. (a) How should the state of the system be defined in order to formulate the queueing model in terms of transitions that only involve exponential distributions (b)
The McAllister Company factory currently has two tool cribs, each with a single clerk, in its manufacturing area. One tool crib handles only the tools for the heavy machinery; the second one handles all other tools. However, for each crib the mechanics arrive to obtain tools at a mean rate of 24
Consider a single-server queueing system with a Poisson input, Erlang service times, and a finite queue. In particular, suppose that k = 2, the mean arrival rate is 2 customers per hour, the expected service time is 0.25 hour, and the maximum permissible number of customers in the system is 2. This
Consider the E2/M/1 model with λ = 4 and μ = 5. This model can be formulated in terms of transitions that only involve exponential distributions by dividing each interarrival time into two consecutive phases, each having an exponential distribution with a mean of 1/(2λ) = 0.125, and then
A company has one repair technician to keep a large group of machines in running order. Treating this group as an infinite calling population, individual breakdowns occur according to a Poisson process at a mean rate of 1 per hour. For each breakdown, the probability is 0.9 that only a minor repair
Consider the finite queue variation of the M/G/1 model, where K is the maximum number of customers allowed in the system. For n = 1, 2, . . . , let the random variable Xn be the number of customers in the system at the moment tn when the nth customer has just finished being served. (Do not count
Southeast Airlines is a small commuter airline serving primarily the state of Florida. Their ticket counter at a certain airport is staffed by a single ticket agent. There are two separate lines—one for first-class passengers and one for coach-class passengers. When the ticket agent is ready for
Consider the model with nonpreemptive priorities presented in Sec. 17.8. Suppose there are two priority classes, with λ1 = 2 and λ2 = 3. In designing this queueing system, you are offered the choice between the following alternatives: (1) one fast server (μ = 6) and (2) two slow servers (μ =
Consider the single-server variation of the nonpreemptive priorities model presented in Sec. 17.8. Suppose there are three priority classes, with λ1 = 1, λ2 = 1, and λ3 = 1. The expected service times for priority classes 1, 2, and 3 are 0.4, 0.3, and 0.2, respectively, so μ1 = 2.5, μ2 = 3
Consider a single-server queueing system with any servicetime distribution and any distribution of interarrival times (the GI/G/1 model). Use only basic definitions and the relationships given in Sec. 17.2 to verify the following general relationships: (a) L = Lq + (1 – P0). (b) L = Lq + p. (c)
A particular work center in a job shop can be represented as a single-server queueing system, where jobs arrive according to a Poisson process, with a mean rate of 8 per day. Although the arriving jobs are of three distinct types, the time required to perform any of these jobs has the same
Reconsider the County Hospital emergency room problem as analyzed in Sec. 17.8. Suppose that the definitions of the three categories of patients are tightened somewhat in order to move marginal cases into a lower category. Consequently, only 5 percent of the patients will qualify as critical cases,
Reconsider the queueing system described in Prob. 17.4-6. Suppose now that type 1 customers are more important than type 2 customers. If the queue discipline were changed from first-come-first-served to a priority system with type 1 customers being given nonpreemptive priority over type 2
Consider the queueing model with a preemptive priority queue discipline presented in Sec. 17.8. Suppose that s = 1, N = 2, and (λ1 + λ2) < μ and let Pij be the steady-state probability that there are i members of the higher-priority class and j members of the lower-priority class in the queueing
Read the referenced article that fully describes the OR study summarized in the application vignette presented in Sec. 17.9. Briefly describe how queueing theory was applied in this study. Then list the various financial and nonfinancial benefits that resulted from this study.
Consider a queueing system with two servers, where the customers arrive from two different sources. From source 1, the customers always arrive 2 at a time, where the time between consecutive arrivals of pairs of customers has an exponential distribution with a mean of 20 minutes. Source 2 is itself
Consider a system of two infinite queues in series, where each of the two service facilities has a single server. All service times are independent and have an exponential distribution, with a mean of 3 minutes at facility 1 and 4 minutes at facility 2. Facility 1 has a Poisson input process with a
Under the assumptions specified in Sec. 17.9 for a system of infinite queues in series, this kind of queueing network actually is a special case of a Jackson network. Demonstrate that this is true by describing this system as a Jackson network, including specifying the values of the aj and the pij,

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